diff --git a/.gitattributes b/.gitattributes
index 07c9b8d739fe1224abf07beac05ed612c01f9bef..cb1f0cfbbec5827b5cf3e09f9afb2e8bffe6b1ba 100644
--- a/.gitattributes
+++ b/.gitattributes
@@ -185,3 +185,9 @@ V9AzT4oBgHgl3EQfJ_vP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 x9FJT4oBgHgl3EQfhSy6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 XNAyT4oBgHgl3EQf9PpX/content/2301.00870v1.pdf filter=lfs diff=lfs merge=lfs -text
 o9FMT4oBgHgl3EQf7jFB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf filter=lfs diff=lfs merge=lfs -text
+c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf filter=lfs diff=lfs merge=lfs -text
+dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf filter=lfs diff=lfs merge=lfs -text
+MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf filter=lfs diff=lfs merge=lfs -text
+BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf filter=lfs diff=lfs merge=lfs -text
+2tFKT4oBgHgl3EQf7y43/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
diff --git a/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/2301.13234v1.pdf.txt b/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/2301.13234v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d7a76dc4e5e0f6c3e1a0c5522bcb4d1d3663af80
--- /dev/null
+++ b/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/2301.13234v1.pdf.txt
@@ -0,0 +1,10427 @@
+arXiv:2301.13234v1  [cond-mat.str-el]  30 Jan 2023
+Field Theoretic Aspects of Condensed Matter Physics: An
+Overview⋆
+Eduardo Fradkin
+Department of Physics and Institute for Condensed Matter Theory,
+University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana Illinois 61801-3080, U.S.A.
+Abstract
+In this chapter I discuss the impact of concepts of Quantum Field Theory in modern Condensed
+Physics. Although the interplay between these two areas is certainly not new, the impact and
+mutual cross-fertilization has certainly grown enormously with time, and quantum Field Theory
+has become a central conceptual tool in Condensed Matter Physics. In this chapter I cover how
+these ideas and tools have influenced our understanding of phase transitions, both classical and
+quantum, as well as topological phases of matter, and dualities.
+Keywords:
+Contents
+1
+Introduction
+3
+2
+Early Years: Feynman Diagrams and Correlation Functions
+3
+3
+Critical Phenomena
+4
+3.1
+Classical Critical Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3.2
+Landau-Ginzburg Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+3.3
+The Renormalization Group
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3.3.1
+Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3.3.2
+The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . .
+6
+3.3.3
+Fixed Points
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+3.3.4
+Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.3.5
+RG flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+8
+3.3.6
+Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+4
+Quantum Criticality
+11
+4.1
+Dynamic Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.2
+The Ising Model in a Transverse Field . . . . . . . . . . . . . . . . . . . . . . .
+12
+4.3
+Quantum Antiferromagnets and Non-Linear Sigma Models . . . . . . . . . . . .
+14
+4.3.1
+Spin coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+14
+4.3.2
+Path integral for a spin-S degree of freedom . . . . . . . . . . . . . . . .
+15
+4.3.3
+Quantum Ferromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+4.3.4
+Quantum Antiferromagnet . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+5
+Topological Excitations
+17
+5.1
+Topological Excitations: Vortices and Magnetic Monopoles . . . . . . . . . . . .
+18
+5.1.1
+Vortices in two dimensions . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+5.1.2
+Magnetic monopoles in compact electrodynamics . . . . . . . . . . . . .
+22
+5.2
+Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains . . . . .
+25
+5.3
+Topology and open integer-spin chains . . . . . . . . . . . . . . . . . . . . . . .
+26
+⋆Work was supported in part by the US National Science Foundation through grant No. DMR 1725401 at the Uni-
+versity of Illinois.
+Preprint submitted to Encyclopedia of Condensed Matter Physics 2e
+February 1, 2023
+
+6
+Duality in Ising Models
+27
+6.1
+Duality in the 2D Ising Model
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+6.2
+The 3D duality: Z2 gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . .
+28
+7
+Bosonization
+31
+7.1
+Dirac fermions in one space dimensions . . . . . . . . . . . . . . . . . . . . . .
+31
+7.2
+Chiral symmetry and chiral symmetry breaking . . . . . . . . . . . . . . . . . .
+34
+7.3
+The chiral anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+36
+7.4
+Bosonization, anomalies and duality . . . . . . . . . . . . . . . . . . . . . . . .
+38
+8
+Fractional Charge
+42
+8.1
+Solitons in one dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+8.2
+Polyacetylene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+8.3
+Fractionally charged solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+43
+9
+Fractional Statistics
+45
+9.1
+Basics of fractional statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+46
+9.2
+What is a topological field theory . . . . . . . . . . . . . . . . . . . . . . . . . .
+46
+9.3
+Chern-Simons Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+47
+9.4
+BF gauge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+48
+9.5
+Quantization of Abelian Chern-Simons Gauge Theory . . . . . . . . . . . . . . .
+48
+9.6
+Vacuum degeneracy a torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+50
+9.7
+Fractional Statistics and Braids . . . . . . . . . . . . . . . . . . . . . . . . . . .
+51
+10 Topological Phases of Matter
+54
+10.1 Topological Insulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+54
+10.1.1 Dirac fermions in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . .
+54
+10.1.2 Dirac Fermions and Topological Insulators
+. . . . . . . . . . . . . . . .
+55
+10.1.3 Chern invariants
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+56
+10.1.4 The quantum Hall effect on a lattice . . . . . . . . . . . . . . . . . . . .
+58
+10.1.5 The Anomalous Quantum Hall Effect . . . . . . . . . . . . . . . . . . .
+61
+10.1.6 The Parity Anomaly
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+62
+10.2 Three-dimensional Z2 topological insulators . . . . . . . . . . . . . . . . . . . .
+67
+10.2.1 Z2 Topological Invariants
+. . . . . . . . . . . . . . . . . . . . . . . . .
+67
+10.2.2 The Axial Anomaly and the Effective Action
+. . . . . . . . . . . . . . .
+68
+10.2.3 Theta terms, and Domain walls: Anomaly and the Callan-Harvey Effect .
+71
+10.3 Chern-Simons Gauge Theory and The Fractional Quantum Hall Effect . . . . . .
+73
+10.3.1 Landau levels and the Integer Hall effect . . . . . . . . . . . . . . . . . .
+73
+10.3.2 The Laughlin Wave Function . . . . . . . . . . . . . . . . . . . . . . . .
+77
+10.3.3 Quasiholes have fractional charge . . . . . . . . . . . . . . . . . . . . .
+78
+10.3.4 The Jain States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+79
+10.3.5 Quasiholes have fractional statistics . . . . . . . . . . . . . . . . . . . .
+80
+10.3.6 Hydrodynamic Effective Field Theory . . . . . . . . . . . . . . . . . . .
+80
+10.3.7 Composite Boson Field Theory
+. . . . . . . . . . . . . . . . . . . . . .
+82
+10.3.8 Composite Fermion Field Theory
+. . . . . . . . . . . . . . . . . . . . .
+85
+10.3.9 The Compressible States . . . . . . . . . . . . . . . . . . . . . . . . . .
+90
+10.3.10 Fractional Quantum Hall Wave Functions and Conformal Field Theory
+.
+91
+10.3.11 Edge States and Chiral Conformal Field Theory . . . . . . . . . . . . . .
+99
+11 Particle-Vortex Dualities in 2+1 dimensions
+105
+11.1 Electromagnetic Duality
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+11.2 Particle-Vortex Duality in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . 105
+11.2.1 The 3D XY Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
+11.2.2 Scalar QED in 3D
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
+11.2.3 The duality mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+11.3 Bosonization in 2+1 dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 109
+11.3.1 Bosonization of the Dirac theory in 2+1 dimensions
+. . . . . . . . . . . 110
+11.3.2 Bosonization of the Fermi Surface . . . . . . . . . . . . . . . . . . . . . 114
+12 Conclusions
+116
+2
+
+1. Introduction
+Many (if not most) puzzling problems in Condensed Matter Physics involve systems with
+a macroscopically large number of degrees of freedom often in regimes of large fluctuations,
+thermal and/or quantum mechanical. The description of the physics of systems of this type
+requires the framework provided by Quantum Field Theory. Although quantum field theory has
+its origins in high-energy physics, notably in the development of Quantum Electrodynamics, it
+has found a nurturing home in Condensed Matter Physics.
+There is a long history of of cross-fertilization between both fields. Since the 1950’s in many
+of the most significant developments in Condensed Matter Physics, Quantum Field Theory has
+played a key role if not in the original development but certainly in the eventual understanding the
+meaning and the further development of the discoveries. As a result, many of the discoveries and
+concepts developed in Condensed Matter have had a reciprocal impact in Quantum Field theory.
+One can already see this interplay in the development of the Bardeen-Cooper-Schrieffer theory
+of superconductivity [1] and its implications in the theory of dynamical symmetry breaking in
+particle physics by Nambu [2].
+The close and vibrant relationship between both fields has continued to these days, and it
+is even stronger today than before. Many textbooks have been devoted to teaching these ideas
+and concepts to new generations of condensed matter physicists (and field theorists as well).
+The earlier texts focused on Green functions which are computed in perturbation theory using
+Feynman diagrams [3] [4] [5], while the more modern ones have a broader scope, use path
+integrals and attack non-perturbative problems [6, 7] [8] [9, 10] [11]. In two recent books I have
+discussed many aspects of the interrelation between condensed matter physics and quantum field
+theory in more depth than I can do in this chapter [9, 10].
+2. Early Years: Feynman Diagrams and Correlation Functions
+Quantum field theory played a key role in the development of the Theory of the Fermi Liquid
+[12, 13]. The theory of the Fermi liquid was first formulated by Landau using the framework of
+hydrodynamics and the quantum Boltzmann equation. Landau’s ideas were later given a micro-
+scopic basis using Green functions and Feynman diagrams [3], including the effects of quantum
+fluctuations at finite temperature and non-equilibrium behavior [14, 15]. Linear response theory
+was developed which allowed the computation of response functions (such as electrical conduc-
+tivities and magnetic susceptibilities) from the computation of correlation functions for a given
+microscopic theory [16]. In turn, correlation functions can be computed in terms of a set of Feyn-
+man diagrams. These concepts and tools borrowed many concepts from field theory including
+the study of the analytic structure of the generalized susceptibilities and the associated spectral
+functions (together with the use of dispersion relations). These developments led to the deriva-
+tion of the fluctuation-dissipation theorem. These ideas were widely applied to metals [13] and
+superconductors [1], as well as to quantum magnets [17].
+The spectrum of an interacting system has low energy excitations characterized by a set of
+quantum numbers associated with the symmetries of the theory. These low energy excitations are
+known as quasiparticles. In the case of the Landau theory of the Fermi liquid the quasiparticle is
+a “dressed” electron: it is a low energy excitation with the same quantum numbers (charge and
+spin) and an electron but with a renormalized effective mass. There are many such quasiparticles
+in condensed matter physics. The correlation functions (the propagators) of a physical system
+has a specific analytic structure. In momentum (and frequency) space, the quasiparticle spectrum
+is given by the poles of the correlators.
+The role of symmetries and, in particular of gauge invariance, in the structure of correlation
+functions was investigated extensively. A direct consequence of symmetries is the existence of
+Ward identities which must be satisfied by all the correlation functions of the theory. Ward iden-
+tities are exact relations that relate different correlation functions. Such identities contain a host
+of important results. For example, in a theory with a globally conserved charge, the Hamiltonian
+(and the action) have a global U(1) symmetry associated with the transformation of the local field
+operator φ(x) (which can be fermionic or bosonic) to a new field φ′(x) = eiθφ(x) (where θ is a
+constant phase). Theories with a global continuous symmetry have a locally conserved current
+(and satisfy a continuity equation). The Ward identity requires the correlators of these currents
+(and densities) to be be transverse (i.e. they should have vanishing divergence). In the absence
+of so-called quantum anomalies (which we will discuss below) global symmetries can be made
+local and become gauge symmetries.
+3
+
+In many circumstances a global symmetry can be spontaneously broken. If the global sym-
+metry is continuous, then the Ward identities imply the existence of gapless excitations known
+as Goldstone bosons. For example in the case of a superfluid, which has a spontaneously broken
+U(1) symmetry the Goldstone boson is the gapless phase mode. Instead, a the N´eel phase of a
+quantum antiferromagnet has two gapless Goldstone bosons, the magnons of the spontaneously
+broken SO(3) global symmetry of this state of matter. Another example is the Ward identity of
+quantum electrodynamics, which relates the electron self-energy to the electron-photon vertex
+function, which also holds in non-relativistic electron fluids. In addition, these identities implied
+the existence of sum rules that the spectral functions must satisfy. All of these results became part
+of the standard toolkit of condensed matter experimentalists in analyzing their data and for the-
+orists to make predictions. Ward identities and sum rules also imply restrictions on the allowed
+approximations which are often needed to obtain predictions from a microscopic model.
+3. Critical Phenomena
+3.1. Classical Critical Phenomena
+The late 1960s and particularly 1970s brought about an intense back and forth between con-
+densed matter physics and field theory in the context of the problem of classical critical phenom-
+ena and phase transitions. This was going to become a profound revolution on the description of
+macroscopic physical systems with large-scale fluctuations. The problem of continuous (“sec-
+ond order”) phase transitions has a long history going back to the work of Landau [18, 19] who
+introduced the concept of an order parameter field. This turned out to be a powerful concept of
+broad applicability in many physical systems sometimes quite different from each other at the
+microscopic level.
+A simple example is that of a ferromagnet with uniaxial anisotropy in which the spins of the
+atoms in a crystal are strongly favored to be aligned (or anti-aligned) along certain directions of
+the crystal. The simplest microscopic model for this problem is the Ising model, a spin system in
+which the individual spins are allowed to take only two values, σ = ±1. The partition function
+of the Ising model (in any dimension) is
+Z =
+�
+[σ]
+exp
+− J
+T
+�
+⟨r,r′⟩
+σ(r)σ(r′)
+
+(1)
+where J is the exchange coupling constant and T is the temperature (measured in energy units);
+here [σ] denotes the sume over the 2N spin configurations (for a lattice with N sites), and ⟨r, r′⟩
+are nearest neighbor sites of the lattice. The order parameter of the Ising model is the local mag-
+netization ⟨σ(r)⟨ which, in the case of a ferromagnet, is uniform. The partition function of the
+Ising model can be computed trivially in one dimension. The solution of the two-dimensional
+Ising model by Onsager constituted a tour-de-force in theoretical physics [20]. Its actual meaning
+remained obscure for some time. The work of Schultz, Mattis and Lieb [21] evinced a deep con-
+nection between Onsager’s solution and the problem if the spectrum of one-dimensional quantum
+spin chains [22] (specifically the one-dimensional Ising model in a transverse field [23]). One
+important result was that the Ising model was in fact a theory of (free) fermions which, crudely
+speaking, represented the configurations of domain walls of the magnet. However, even this sim-
+ple model cannot be solved exactly in general dimension, and approximate mean field theories
+of various sorts were devised over time to understand its physics.
+3.2. Landau-Ginzburg Theory
+Landau’s approach assumed that close enough to a phase transition, the important spin con-
+figurations are those for which the local magnetization varies slowly on lattice scales. In this
+picture the local magnetization, on long enough length scales, becomes an order parameter field
+that takes values on the real numbers, and can be positive of negative. Thus, the order parameter
+field is effectively the average of local magnetizations on some scale large compared to the lattice
+scale, which we will denote by a real field φ(x). The thermodynamics properties of a system of
+this type in d dimensions can be described in terms of a free energy
+F[φ] =
+�
+ddx
+�κ
+2 (▽φ(x))2 + a(T − Tc)φ2(x) + uφ4(x) + . . .
+�
+(2)
+4
+
+which is known as the Ginzburg-Landaufree energy. Here κ is the stiffness of the order parameter
+field, Tc is the (mean-field) critical temperature; a and u are two (positive) constants. This
+expression make sense if the transition is continuous and hence that the order parameter is small
+near the transition. The energy of the Ising model is invariant under the global symmetry [σ] �→
+[−σ]. This is the symmetry of the group Z2. Likewise, the Ginzburg-Landau free energy has the
+global (discrete) symmetry [φ(x)] �→ [−φ(x)], and also has a Z2 global symmetry.
+In Landau’s approach, which was a mean field theory, the equilibrium state is the global
+minimum of this free energy. The nature of the equilibrium state depends on whether T > Tc
+or T < Tc: for T > Tc the global minimum is the trivial configuration, ¯φ(x) = 0 (this is the
+paramagnetic state), whereas for T < Tc the equilibrium state is two fold degenerate, ¯φ(x) =
+±(a(Tc − T)/2)β, with the two degenerate states being related by the Z2 symmetry (this is the
+ferromagnetic state). In the Landau theory the critical exponent of the magnetization is β = 1/2
+and the critical exponent of the correlation length is ν = 1/2. However, in the case of the 2D Ising
+model the order parameter exponent is β = 1/8 [24] and the correlation length exponent is ν = 1.
+These (and other) apparent discrepancies led many theorists for much of the 1960s believe that
+each model was different and that these behaviors reflected microscopic differences. In addition,
+Landau’s theory was regarded as phenomenological and believed to be of questionable validity.
+3.3. The Renormalization Group
+This situation was to change with the development of the Renormalization Group, due pri-
+marily to the work of Leo P. Kadanoff [25, 26, 27] and Kenneth G. Wilson [28, 29, 30, 31, 32, 33].
+The renormalization Group was going to have (and still has) a profound effect both in Condensed
+Matter Physics and in Quantum Field Theory (and beyond).
+3.3.1. Scaling
+Several phenomenological theories were proposed in the 1960s to describe the singular be-
+havior of physical observables near a continuous phase transition [34, 35, 36]. These early works
+argued that in order to explain the singular behavior of the observables the free energy density
+had to have a singular part which should be a homogeneous function of the temperature, mag-
+netic field, etc. A function f(x) is homogeneous if it satisfies the property that it transforms
+irreducibly under dilations, i.e. f(λx) = λk f(x), there λ is a real positive number (a scale) and
+k is called the degree. These heuristic ideas then implied that the critical exponents should obey
+several identities. In 1966 Kadanoff wrote and insightful paper in which he showed that the ho-
+mogeneity hypothesis implied that in that regime these systems should obey scaling. He showed
+that this can be justified by performing a sequence of block-spin transformations in which the
+configurations that vary rapidly at the lattice scale a become averaged at the scale of a larger sized
+block of length scale ba > a which resulted in a renormalization of the coupling constants from
+{K} at scale a to {K′} at the new scale ba [25, 27]. In other words, the block spin transformation
+amounts to a scale transformation and a renormalization of the couplings (and operators). From
+this condensed matter/statistical physics perspective the important physics is in the long distance
+(“infrared”) behavior.
+A significant consequence of these ideas was that close enough to a critical point, if the
+distance |x − y| between two local observables O(x) and O(y) is large compared to the lattice
+spacing a but small compared to the correlation length ξ, their correlation function takes the
+form of a power law
+⟨O(x)O(y)⟩ ∼
+const.
+|x − y|2∆O
+(3)
+where ∆O is a positive real number known as the scaling dimension of the operator O [34, 25, 37].
+These conjectures were known to be satisfied in the non-trivial case of the 2D Ising model [26],
+as well as in the Landau-Ginzburg theory once the effects of Gaussian fluctuations were included
+[37]. For example, in the 2D Ising model, the scaling dimensions of the local magnetization σ is
+∆σ = 1/8 and of the energy density ε is ∆ε = 1, which were sufficient to explain all the singular
+behaviors known at that time.
+The concept of renormalization actually originated earlier in quantum field theory as part
+of the development of Quantum Electrodynamics (QED). In QED the notion of renormaliza-
+tion was used to “hide” the short distance (“ultraviolet”) divergencies of the Feynman diagrams
+needed to compute physical processes involving electrons (and positrons) and photons, i.e. their
+strong, divergent, dependence of an artificially introduced short-distance cutoff or regulator. In
+particular the sum of the leading diagrams that enter in the electron-photon vertex amounted to
+5
+
+a redefinition (renormalization) of the coupling constant. It was observed by Murray Gell-Mann
+and Francis Low that this renormalization was equivalent to the solution of a first order differ-
+ential equation that governed the infinitesimal change of the coupling, the fine structure constant
+α = e2/4π, under an infinitesimal change of the UV cutoff Λ [38]
+Λ dα
+dΛ ≡ β(α) = 2
+3πα2 + O(α3)
+(4)
+where β(α) is the Gell-Mann-Low beta function. Except for the work by Nikolai Bogoliubov
+and coworkers [39], this reinterpretation by Gell-Mann and Low was not actively pursued, partly
+because it predicted that the renormalized coupling became very large at short distances, α → ∞,
+and, conversely, it vanished in the deep long distance regime, α → 0 (if the electron bare mass is
+zero). In other terms, QED is strongly coupled in the UV and trivial in the IR. The same behavior
+was found in the case of the theory of a scalar field φ(x) with an φ4 interaction which is relevant
+in the theory of phase transitions. In addition to these puzzles, the1960s saw the experimental
+development of the physics of hadrons which involve strong interactions. For these reasons, for
+much of that decade most high-energy theorists had largely abandoned the use of quantum field
+theory, and explored other, phenomenologically motivated, approaches (which led to an early
+version of string theory.) At any rate the notion that the physics may depend on the scale was
+present as was the notion that in some regimes field theories may exhibit scale-invariance at least
+in an approximate form.
+3.3.2. The Operator Product Expansion
+The next stage of the development of these ideas was the concept of the operator product
+expansion (OPE). If we denote by {O j(x)} the set of all possible local operators in a theory (a field
+theory or a statistical mechanical system near criticality), then the product of two observables on
+this list closer to each other than to any other observable (and to the correlation length ξ) obeys
+the expansion
+lim
+x→y O j(x)Ok(y) = lim
+x→y
+�
+l
+C jkl
+|x − y|∆j+∆k−∆l Ol
+� x + y
+2
+�
+(5)
+where this equation should be understood as a weak identity, valid inside an expectation value.
+Remarkably, this concept was derived independently and simultaneously by Leo Kadanoff [40]
+(who was working in critical phenomena), by Kenneth Wilson [41] (who was interested in the
+short distance singularities arising in Feynman diagrams), and by Alexander Polyakov [42, 43]
+(also working in critical phenomena). In Eq.(5) {∆j} are the scaling dimensions of the operators
+{O j}. The coefficients C jkl are (like the dimensions) universal numbers. In a follow up paper
+Polyakov showed that if the theory has conformal invariance, i.e. scale invariance augmented by
+conformal transformations which preserve angles, then, provided the operators O j are suitably
+normalized, the coefficients C jkl of the OPE are determined by a three point correlator
+⟨O j(x)Ok(y)Ol(z)⟩ =
+C jkl
+|x − y|∆jk|y − z|∆kl|z − x|∆l j
+(6)
+where ∆jk = ∆j + ∆k − ∆l. These results constitute the beginnings of Conformal Field Theory.
+In a nontrivial check, Kadanoff and Ceva showed that the OPE holds for the local observables of
+the 2D Ising model [44].
+3.3.3. Fixed Points
+The next and crucial step in the development of the renormalization Group was made by
+Kenneth Wilson. Wilson was a high-energy theorist who wanted to know how to properly define
+a quantum field theory and the physical meaning of renormalization.
+In a Lorentz invariant quantum field theory one is interested in the computation of the ex-
+pectation value of time-ordered operators. In the case of a self-interacting scalar field φ(x) in
+D-dimensional Euclidean space-time, obtained by analytic continuation from Minkowski space-
+time to imaginary time, the observables are computed from the functional (or path) integral by
+functional differentiation of the partition function
+Z =
+�
+Dφ exp
+�
+−S (φ, ∂µφ) +
+�
+dDx J(x)φ(x)
+�
+(7)
+6
+
+with respect to the local sources J(x). For a scalar field the Euclidean action is
+S =
+�
+dDx
+�1
+2(∂µφ(x))2 + m2
+2 φ2(x) + λ
+4!φ4(x)
+�
+(8)
+which has the same form as the free energy of the Landau-Ginzburg theory of phase transitions
+shown in Eq.(2). It is apparent that the Landau-Ginzburg theory is the classical limit of the theory
+of the scalar field whose partition function is a sum over all histories of the field. It is easy to
+see that en expansion of the partition function (or of a correlator) in powers of the the coupling
+constant λ can be cast in the form of a sum of Feynman diagrams. To lowest order in λ a typical
+Feynman diagram involves a one-loop integral in momentum space of the form
+I(p) =
+�
+dDq
+(2π)D
+1
+(q2 + m2)((q − p)2 + m2)
+(9)
+As noted by Wilson in his Nobel Lecture [33], this integral has large contributions from the IR
+region of small momenta q ∼ 0, but for any dimension D ≥ 4 has a much larger contribution
+form large the UV region of large momenta, which requires the introduction of a UV cutoff Λ
+in momentum space (or a lattice spacing a in real space by defining the theory on a hypercubic
+lattice). In quantum field theory one then has to require that somehow one takes the limit a → 0
+(or Λ → ∞). To take this limit in the field theory is very much analogous to the definition of
+a conventional integral in terms of a limit of a Riemann sum. The difference is that this is a
+functional integral. While for a function of bounded variation in a finite interval (a, b) the limit
+of a partition of the interval into N steps each of length ∆x, such that N∆x = b − a, exists and
+defines the integral of the function
+lim
+∆x→0 lim
+N→∞
+N
+�
+j=1
+f(x j)∆x j =
+� b
+a
+dx f(x),
+(10)
+the analogous statement does not obviously exists in general for a functional integral, i.e. an
+integral over a space of functions which is what is required. In fact, although thousands of
+integrals of a function are known to exist, there are extremely few examples for a functional
+integral. Moreover, in order to take the continuum limit the lattice spacing must approach zero,
+a → 0. This means that physical scales, such as the correlation length ξ, must diverge in lattice
+units so that they can be fixed in physical units. But to do that one has to be asymptotically
+close to a continuous phase transition! Hence, the problem of defining a quantum field theory is
+equivalent to the problem of critical phenomena at a continuous phase transition!
+Wilson gave a systematic formulation to the Renormalization Group by generalizing the ear-
+lier ideas introduced by Kadanoff and the earlier work by Gell-Mann and Low. Wilson’s key
+contribution was the introduction of the concept of a fixed point of the Renormalization Group
+transformation [28, 29, 30, 31]. As we saw, the block-spin transformation is a procedure for
+coarse graining the degrees of freedom of a physical system resulting in a renormalization of the
+coupling constants. Upon the repeated action of the RG transformation its effect can be pictured
+as a flow in the space of coupling constants. However, in addition of integrating-out short dis-
+tance degrees of freedom one needs to restore the units of length which have changed under that
+process. This requires a rescaling of lengths. Once this is done, Wilson showed that the resulting
+RG flows necessarily have fixed points, special values of the couplings which are invariant (fixed)
+under the action of the RG transformation. He then deduced that at a fixed point the theory has
+no scales, aside from the linear size L of the system and the microscopic UV cutoff (the lattice
+spacing a in a spin system).
+This analysis means that for length scales long compared to a → 0 but short compared to
+L → ∞ the theory acquired a new, emergent, symmetry: scale invariance. Therefore, at a fixed
+point the correlators of all local observables must be homogeneous functions (hence, must scale).
+3.3.4. Universality
+A crucial consequence of the concept if the fixed point is that phase transitions can be clas-
+sified into universality classes. Universality means that a large class of physical systems with
+different microscopic properties have fixed points with the same properties, i.e. the same scaling
+dimensions, operator product expansions and correlation functions at long distances independent
+on how they are defined microscopically. Although the renormalization group transformation is
+a transformation scheme that we define and, because of that the location in coupling constant
+7
+
+space of the fixed point itself does depends on the scheme we choose, its universal properties are
+the same. Thus, universality classes depend only on features such as the space (and spacetime)
+dimension and the global symmetries of the system. But the systems themselves may be quite
+different. This we speak if the Ising universality class in 2D, on the superfluid (or XY) transi-
+tion class in 3D, etc. This concept, which originated in the theory of phase transition, has been
+adopted and generalized in the development of conformal field theory.
+3.3.5. RG flows
+Combined with the condition that the correlators decay at long separations, homogeneity
+implies that the correlators must have the form of Eq.(3). In addition, this equation also implies
+that at a fixed point the operators (the local observables) have certain scaling dimensions. Let
+us consider a theory close to a fixed point whose action we will denote by S ∗. Let {O j(x)} be
+a complete set of local observables whose scaling dimensions are {∆j}. The total action of the
+theory close to the fixed point then can be expanded as a linear combination of the operators with
+dimensionless coupling constants {g j}
+S = S ∗ +
+�
+dDx
+�
+j
+g ja∆j−DO j(x)
+(11)
+Under a change of length scale x → x′ = bx, with b > 1, the operators (which must transform
+homogeneously)change as O j(bx) = b−∆jO j(x). Since the phase space changes as dDx′ = bDddx,
+we can keep the form of the action provided the coupling constants also change to compensate
+for these changes as g′
+j = bD−∆jg j. Let b = |x′|/|x| = 1+da/a, where da is an infinitesimal change
+of the UV cutoff a. Then, if we integrate-out the degrees of freedom in the range a < |x| < a+da,
+the rate of change of the coupling constants {g j} under this rescaling is
+adg j
+da ≡ β(g j) = (D − ∆j)g j + . . .
+(12)
+which we recognize as a Gell-Mann Low beta function for each coupling constant.
+This result says that if the scaling dimension ∆j < D, then the renormalized coupling will
+increase as we increase the length scale, g′
+j > g j, and along this direction in coupling constant
+space the RG flows away from the fixed point. Conversely, if ∆j > D the renormalized coupling
+flows to smaller values, g′
+j < g j and the RG flows into the fixed point. We then say that an
+operator is relevant if its scaling dimension satisfies ∆j < D, and that it is irrelevant if ∆j > D.
+If ∆j = D then we say that operator is marginal.
+To go beyond this simple dimensional analysis one has to include the effects of fluctuations.
+To lowest orders in the couplings one finds [45]
+adg j
+da = (D − ∆j)g j +
+�
+k,l
+C jkl gkgl + . . .
+(13)
+where {C jkl} are the coefficients of the OPE shown in Eq.(6). This expression is the general form
+of a perturbative renormalization group and it is valid close enough to a fixed point.
+Wilson and Fisher [30] used a similar approach to analyze how fluctuations alter the results
+of the Landau-Ginzburg theory. They considered the partition function of Eq.(7) with the action
+of Eq.(8). Instead of working in real space they considered the problem in momentum space and
+partitioned the field configurations into slow and fast modes
+φ(x) = φ<(x) + φ>(x)
+(14)
+where φ>(x) are configurations whose Fourier components have momenta in the range bΛ < |p| <
+Λ, where Λ is a UV momentum cutoff and b < 1. Hence, if we choose b → 1, the fast modes φ >
+(x) have components in a thin momentum shell near the UV cutoff Λ. Conversely, the slow modes
+φ<(x) have momenta in the range 0 ≤ |p| < bΛ. One can then use Feynman diagrams to integrate
+out the fast modes and derive an effective low-energy action with renormalized couplings. In the
+case of a free field theory (with λ = 0) the scaling dimension of the φ4 operator is ∆4 = 2(D − 2)
+whereas the φ2 operator has dimension ∆2 = D − 2. Upon defining a dimensionless mass and
+coupling constant by m2 = tΛ2 and λ = gΛ4−D, the beta functions are found to be [10]
+β(t) = − Λ dt
+dΛ = 2t + g
+2 − gt
+2 + . . .
+(15)
+β(g) = − Λ dg
+dΛ = (4 − D)g − 3
+2g2 + . . .
+(16)
+8
+
+(where we absorbed an uninteresting factor if the definition of the coupling constant).
+The RG flows of Eq.(16) show that the free-field (Gaussian) fixed point at g = 0 is stable for
+D > 4 and the asymptotic IR behavior is the same as predicted by the Landau-Ginzburg theory.
+However, for D < 4, the free-field fixed point becomes unstable and a new fixed point arises at
+g∗ = 2
+3ǫ + O(ǫ2), where we have set ǫ = 4 − D. This is the Wilson-Fisher fixed point. At this
+fixed point the correlation length diverges with an exponent ν = 1
+2 + ǫ
+12 + O(ǫ2), which deviates
+from the predictions of the Landau-Ginzburg theory. The small parameter of this expansion is ǫ,
+and this is known as the ǫ expansion.
+The Wilson-Fisher (WF) fixed point is an example of a non-trivial fixed point at which the
+correlation length is divergent. It has only one relevant operator, the mass term, which in the IR
+flows into the symmetric phase for t > 0 and flows to the broken symmetry for t < 0. Conversely,
+in the UV it flows into the WF fixed point. For these reasons condensed matter physicists say
+that this is an IR unstable (or critical) fixed point while high-energy physicists say that it is the
+UV fixed point. At this fixed point a non-trivial field theory can be defined with non-trivial
+interactions. UV fixed points also define examples of what in high-energy physics are called
+renormalizable field theories and can be used to define a continuum field theory.
+The D = 4-dimensional theory is special in that the φ4 operator is marginal. As can be seen
+in Eq.(16), at D = 4 the beta function for the dimensionless coupling constant g does not have
+a linear term and is quadratic in g. In this case the operator is marginally irrelevant, and its
+beta function has the same behavior as the beta function of Gell-Mann and Low for QED. Such
+theories are said to have a “triviality problem” since, up to logarithmic corrections to scaling,
+there are no interactions in the IR and, conversely, become large in the UV.
+There are also fixed points at which the correlation length ξ → 0. These fixed points are
+IR stable (and in a sense trivial). These stable fixed points are sinks of the IR RG flows. Such
+fixed points define stable phases of matter, e.g. the broken symmetry state, the symmetric (or
+unbroken state), etc. However in the UV they are unstable and in high-energy physics such fixed
+points correspond to non-renormalizable field theories.
+More sophisticated methods are needed to go beyond the lowest order beta functions of
+Eq.(13), and the computation of critical exponents beyond the leading non-trivial order. Pos-
+sibly the record high-precision calculations have been done for φ4 theory for which the beta
+function is know to O(ǫ5). This has been achieved using the method of dimensional regulariza-
+tion [46, 47, 48] (with minimal subtraction). Special resummation methods (Borel-Pad´e) have
+been used to do these calculations in D = 3 dimensions [49]. Remarkably, these results are
+so precise that in the case of the superfluid transition, which is well described by a φ4 theory
+with a complex field, the results could only be tested in the microgravity environment of the
+International Space Station!
+3.3.6. Asymptotic Freedom
+There are several physical systems systems of great interest whose beta function has the form
+β(g) = Ag2 + . . .
+(17)
+The coupling constant has a different interpretation in each theory and the constant A > 0,
+opposite to the sign of the beta function found in QED and φ4 theory in D = 4 dimensions.
+This beta function means that while the associated operator is marginal, with this sign is actually
+marginally relevant. This also means that the fixed point is unstable in the IR but the departure
+from the fixed point is logarithmically small. Conversely, the in the UV the RG flows into the
+fixed point and the effective constant is weak at short distances. This is the origin of the term
+asymptotic freedom [50]. The paradigmatic examples of theories with a beta function of this
+form are the Kondo problem, the 2D non-linear sigma model, and the D = 4 dimensional Yang-
+Mills non-abelian gauge theory.
+The Kondo problem is the theory of a localized spin-1/2 degree of freedom in a metal. the
+electrons of the metal couple to this quantum impurity through an exchange interaction of the
+impurity and the magnetic moment density of the mobile electrons in the metal with coupling
+constant J. This problem is actually one dimensional since only the s wave channel of the
+mobile (conduction band) electrons actually couple to the localized impurity. In 1970 Philip
+Anderson developed a theory of the Kondo problem in terms of the renormalization of the Kondo
+coupling constant g as a function of the energy scale [51]. Anderson used perturbation theory
+in J to progressively integrated out the modes of the conduction electrons close to an effective
+9
+
+bandwidth Ec and found that the beta function has the form of Eq.(17)
+Ec
+dJ
+dEc
+= −ρJ2 + . . .
+(18)
+where ρ is the density of states at the Fermi energy of the conduction electrons. This work
+implied that the free-impurity fixed point is IR unstable and that the effective coupling constant
+J increases as the energy cutoff Ec is lowered. He argued that at some energy scale, the Kondo
+scale, perturbation theory breaks down and that there is a crossover to a strong coupling regime
+which is not accessible in perturbation theory.
+Shortly thereafter, in 1973 Wilson developed a numerical renormalization group approach
+which showed that the Kondo problem is indeed a crossover from the free impurity fixed point
+to the “renormalized” Fermi liquid [32]. In addition, Wilson use the numerical renormalization
+group to examine the approach to the strong coupling fixed point and showed that it is charac-
+terized by a Wilson ratio, a universal number obtained from the low temperature specific heat
+and the impurity magnetic susceptibility (in suitable units). Wilson’s numerical RG predicted
+a number close to 2π for the this ratio. In 1980 N. Andrei and P. Wiegmann showed (indepen-
+dently) that the Kondo problem is an example of an integrable field theory that can be solved
+by the Bethe ansatz [52, 53]. Their exact result was consistent with Wilson’s RG, including the
+numerical value of the Wilson ratio.
+In 1972 Gerard ’t Hooft and Martinus Veltman showed that Yang-Mills gauge theory is renor-
+malizable [46]. This groundbreaking result opened the door to use quantum field theory to de-
+velop the theory of strong interactions in particle physics known as Quantum Chromodynamics
+(QCD). In 1973 David Gross and Frank Wilczek [50] and, independently, David Politzer [54]
+computed the renormalization group beta function of Yang-Mills theory with gauge group G and
+found it to be of the same form as Eq.(17),
+Λ dg
+dΛ = − g3
+16π2
+11
+3 C2(G) + . . .
+(19)
+where g is the Yang-Mills coupling constant and Λ is a UV momentum scale. Here C2(G) is the
+quadratic Casimir for a gauge group G. For S U(3), the case of physical interest, C2(S U(3)) = 3.
+This result implies that under the RG at large momenta (short distances) the Yang-Mills coupling
+constant flows to zero (up to logarithmic corrections). This result holds in the presence of quarks
+provided the number of quark flavors is less than a critical value. Hence at short distances the
+effective coupling is weak. Gross and Wilczek called this phenomenon asymptotic freedom.
+This behavior was consistent with the observation of weakly coupled quarks in deep inelastic
+scattering experiments. However, the flip side of asymptotic freedom is that at low energies
+(long distances) the coupling constant grows without limit, which implies that at low energies
+perturbation theory is not applicable. This strong infrared behavior suggested that in QCD quarks
+are permanently confined in color neutral bound states (hadrons). However, unlike the Kondo
+problem we just discussed, QCD is not an integrable theory (so far as we know) and to show that
+it confines has required the development of Lattice Gauge Theory [55, 56]. To this date the best
+evidence for quark confinement has been obtained using large-scale Monte Carlo simulations in
+Lattice Gauge Theory [57].
+We close this subsection with a discussion of an important case: the non-linear sigma models.
+The O(N) non-linear sigma model is the continuum limit of the classical Heisenberg model for a
+spin with N components. Historically, the non-linear sigma model is the effective field theory for
+pions in particle physics. We will discuss its role in the theory of quantum antiferromagnets in
+subsection 4.3 and especially in the case of quantum antiferromagnetic spin chains in subsection
+5.2.
+The simplest non-linear sigma model is a theory of an N-component scalar field n(x) which
+satisfies the unite length local constraint, n2(x) = 1. The Euclidean Lagrangian is
+L = 1
+2g(∂µn(x))2
+(20)
+where g is the coupling constant (the temperature in the classical Heisenberg model). At the
+classical level, i.e. in the broken symmetry phase, where ⟨n⟩ � 0, this model describes the N − 1
+massless modes (Goldstone bosons) of the spontaneously broken O(N) symmetry. Dimensional
+analysis shows that the coupling constant has units of ℓD−2. Hence, we expect to find marginal
+behavior at D = 2. In 1975 Alexander Polyakov used a momentum shell renormalization group
+10
+
+in D = 2 dimensions and showed that the beta function of this model is (here a is the short-
+distance cutoff) [58]
+β(g) = adg
+da = N − 2
+2π g2 + O(g3)
+(21)
+Hence, in D = 2 dimensions also this theory is asymptotically free. As in the other examples we
+just discussed, asymptotic freedom here also implies that the coupling constant g grows to large
+values in the low-energy (long-distance) regime. In close analogy with Yang-Mills theory in
+D = 4 dimensions, Polyakov conjectured that the O(N) non-linera sigma model also undergoes
+dynamical dimensional transmutation [50], that the global O(N) symmetry is restored and that
+for all values of the coupling constant g the theory is in a massive with a finite correlation length
+ξ ∼ exp((N − 2)/2πg). Extensive numerical simulations, used to construct a renormalization
+group using Monte Carlo simulations [59], showed that there is indeed a smooth crossover from
+the weak coupling (low temperature) regime to the high temperature regime where the correlation
+length is finite. The non-linear sigma model is a renormalizable field theory in D = 2 dimensions
+[60]. For D > 2 dimensions it can be studied using the 2 + ǫ expansion [60, 49], which predicts
+the existence of a nontrivial UV fixed point and a phase transition from a Goldstone phase to a
+symmetric phase.
+It turns out that there is a significant number of asymptotically free non-linear sigma models
+in D = 2 dimensions, many of physical interest [61], in particular non-linear sigma models
+whose target manifold is a coset space, a quotient of a group G and a subgroup H. The O(N) non
+linear sigma model is an example since the broken symmetry space leaves the O(N −1) subgroup
+unbroken (the manifold of the Goldstone bosons). In that case the quotient is O(N)/O(N − 1)
+which is isomorphic to the N −1 dimensional sphere S N−1. In later sections we will discuss other
+examples in which more general non-linear sigma models play an important role.
+Models on coset spaces arise in the theory of Anderson localization in D = 2 dimensions.
+Anderson localization is the problem of a fermion (an electron) in a disordered system in which
+the electron experiences a random electrostatic potential. In the limit of strong disorder Philip
+Anderson showed that all one-particle states are exponentially localized and the diffusion con-
+stant (and the conductivity) vanishes[62]. There was still the question of when it is possible
+for the electron to have a finite diffusion constant (and conductivity). In D = 2 dimensions
+the conductivity is a dimensionless number which suggests that this may be the critical dimen-
+sion for diffusion. Abrahams, Anderson, Licciardello and Ramakrishnan used a weak disorder
+calculation to construct a scaling theory that implied that in D = 2 dimensions the RG flow
+of the conductivity at long distances (large samples) flows to zero and all states are localized
+[63]. Shortly thereafter Wegner gave strong arguments that showed that the existence of diffu-
+sion implied that there are low-energy “diffusson” modes which behaved as Goldstone modes of
+a non-linear sigma model on the quotient manifold O(N+ + N−)/O(N+) × O(N−) in the “replica
+limit” N± → 0 [64]. A field theory approach to this non-linear sigma model was developed by
+McKane and Stone [65] and by Hikami [66].
+4. Quantum Criticality
+Quantum criticality is the theory of a phase transition of a system (e.g. a magnetic system)
+at zero temperature that occurs as a coupling constant (or parameter) is varied continuously.
+Although not necessarily under that name, this question has existed as a conceptual problem for
+a long time, In particular, already in 1973 Wilson considered the problem of the behavior of
+quantum filed theories blow four spacetime dimensions and their phase transitions [67].
+The modern interest in condensed matter physics stems from discoveries made since the late
+1980s. Since that time he behavior of condensed matter systems at a quantum critical point has
+emerged as a major focus in the field. There were several motivations for this problem. One
+was (and still is) to understand the behavior of quantum antiferromagnets in the presence of
+frustrating interactions. Frustrating interactions are interactions which favor incompatible types
+of antiferromagnetic orders. The result is the presence of intermediate non-magnetic “valence
+bond” phases that favor the formation of spin singlets between nearby spins. these phases typ-
+ically either break the point group symmetry of the lattice or are spin liquids (which will be
+discussed below). Another motivation is that doped quantum antiferromagnets typically harbor
+superconducting phases (among others) whose high-temperature behavior is a “strange” metal
+that violates the basic assumptions (and behaviors) of Fermi liquids. The most studied version
+of this problem is the case of the copper oxide high temperature superconductors. It was con-
+jectured that there is a quantum critical point inside the superconducting phase at which the
+11
+
+antiferromagnetic order (or other orders) disappears and which may be the reason for the strange
+metal behavior above the superconducting critical temperature. Many of these questions are
+discussed in depth by Sondhi and coworkers [68] and in the textbook by Sachdev [69].
+4.1. Dynamic Scaling
+We will consider a general quantum phase transition and assume that it is scale invariance.
+However, except for the case of relativistic quantum field theories, in condensed matter systems
+space and time do not need to scale in the same way. Let us assume that the system of interest has
+just one coupling constant g and that the system of interest has a quantum phase transition (at zero
+temperature) at some critical value gc between two phases, for instance one with a spontaneously
+broken symmetry and a symmetric phase. If the quantum phase transition is continuous then the
+correlation length ξ will diverge at gc and so will the correlation time ξt. However these two
+scales are in general different and do not necessarily diverge at the same rate. So, in general,
+if some physical quantity is measured at the quantum critical point at some momentum p and
+frequency ω, the length scale of the measurement is 2π/|p| and at a frequency is ω ∼ |p|z, where
+z is the dynamic critical exponent. Let us say that we measure the observable O at momentum p
+and frequency ω at gc. Scale invariance in both space and time means that at gc the observable
+O(p, ω) at momentum p and frequency ω must scale as
+O(p, ω) = |p|−∆O ˜O(|p|z/ω)
+(22)
+where ∆O is the scaling dimension of the observable O.
+The situation changes at finite temperature T. A quantum field theory at temperature T is
+described by a path integral on a manifold which along the imaginary time direction τ is finite
+of length 2π/T and periodic for a theory bosonic fields and anti-periodic for fermionic fields
+[10]. Since the imaginary time direction is finite, the behavior for correlation times ξτ < 2π/T
+and ξτ > 2π/T must be different. Indeed, in the first regime the behavior is essentially the same
+as at T = 0, while in the second it should be given by the classical theory in the same space
+dimension.At the quantum critical point gc there is only one time scale ξτ ∼ 2π/T and only one
+length scale ξ ∼ (2π/T)1/z.
+4.2. The Ising Model in a Transverse Field
+The prototype of the quantum phase transition is the Ising model in a transverse field. This
+model describes a system of spin-1/2 degrees of freedom with ferromagnetic interactions with
+uniaxial anisotropy in the presence of a transverse uniform magnetic field. The Hamiltonian is
+H = −J
+�
+⟨r,r′⟩
+σ3(r)σ3(r′) − h
+�
+r
+σ1(r)
+(23)
+where J and h are the exchange coupling constant and the strength of the transverse field, re-
+spectively. Here σ1 and σ3 are the two Pauli matrices defined on the sites {r} of a lattice with
+ferromagnetic interactions between spins on nearest neighboring sites. The Hilbert space is the
+tensor product of the states of the spins at each site of the lattice. At each site there are two nat-
+ural bases of states: the eigenstates of σ3, which we denote by | ↑⟩ and | ↓⟩ (whose eigenvalues
+are ±1), and the eigenstates of σ1, which we denote by |±⟩ (whose eigenvalues are also ±1).
+It is well known that the Ising Model in a Transverse Field on a hypercubic lattice in D
+dimensions is equivalent to a classical Ising model in D + 1 dimensions [70, 71]. These two
+models are related through the transfer matrix. Indeed, a classical Ising model can be regarded
+as a path integral representation of the quantum model in one dimension less. For simplicity
+we will see how this work for the 2D the classical ferromagnetic Ising model of Eq.(1), but the
+construction is general. We will regard the configuration of spins on a row of the 2D lattice as
+a state of a quantum system, and the set of states on all rows as the evolution of the state along
+the perpendicular direction that we will regard as a discretized imaginary time. The contribution
+from two adjacent rows to the partition function defines the matrix element of a matrix between
+two arbitrary configurations. In Statistical Physics this matrix is known as the Transfer Matrix ˆT
+and the full partition function (with periodic boundary conditions) is
+Z = tr ˆT Nτ
+(24)
+where Nτ is the number of rows. For the case of the ferromagnetic Ising model (actually, for any
+unfrustrated model) the transfer matrix can always be constructed to be hermitian. This property
+12
+
+holds in fact for any theory that satisfies a property known as reflection positivity which requires
+that all (suitably defined) correlation functions be positive. For theories of this type, and the Ising
+model is an example, the matrix elements of the transfer matrix can be identified with the matrix
+element of the evolution operator of a quantum theory for a small imaginary time step [70]. Also,
+the positivity of the correlators is equivalent to the condition of positivity of the norm of states in
+the quantum theory.
+For the classical models that satisfy these properties, all directions of the lattice are equiv-
+alent. Moreover, asymptotically close to the critical point, the behavior of all the correlators
+becomes isotropic, i.e. invariant under the symmetries of Euclidean space. This means that
+the arbitrary choice of the direction for the transfer matrix is irrelevant. Consequently, tin the
+quantum model its equal-time correlators behave the same way as its correlation functions in
+imaginary time. In other words space and time behave in the same way and the quantum the-
+ory is relativistically invariant. This implies at the quantum critical point the energy ε(p) of its
+massless excitations should behave as ε(p) = v |p|. In a relativistic theory the dynamical critical
+exponent must be z = 1 and the coefficient v is the speed of the excitations (the “speed of light”).
+We should note that this is not necessarily always the case. There are in fact classical systems,
+e.g. liquid crystals [72], which are spatially anisotropic and map onto quantum mechanical theo-
+ries in one less dimension for which the dynamical critical exponent z � 1. One such example are
+the Lifshitz transitions of nematic liquid crystals in three dimensions and the associated quantum
+Lifshitz model in D = 2 dimensions, for which the dynamical exponent is z = 2 [73].
+Just as in the classical counterpart in D + 1 dimensions, the quantum model in D ≥ 1 has
+two phases: a broken symmetry ferromagnetic phase for J ≫ h and a symmetric paramagnetic
+phase for h ≫ J. In the symmetric phase the ground state is unique (asymptotically is the
+eigenstate of σ1 with eigenvalue +1), while in the broken symmetry phase the ground state
+is doubly degenerate (and asymptotically is an eigenstate of σ3) and there is a non-vanishing
+expectation value of the local order parameter ⟨σ3(r)⟩ � 0. In the symmetric phase the correlation
+function of the local order parameter decays exponentially with distance with a correlation length
+ξ, as does the connected correlation function in the broken symmetry phase. The model has
+a continuous quantum phase transition at a critical value of the ratio h/J. For general space
+dimensions D > 1 this model is not exactly solvable and much of what we know about it is due
+to large-scale numerical simulations.
+This problem was solved exactly in one-dimension [23] using the Jordan-Wigner transfor-
+mation that maps a one dimensional quantum spin system to a theory of free fermions [22]. The
+fermion operators at site j are
+χ1(j) = K(j − 1)σ3(j),
+χ2(j) = i ˆK(j)σ3(j)
+(25)
+where K(j) is the kink creation operator (i.e. the operator that creates a domain wall between
+sites j and j + 1 [70], and is given by
+K(j) =
+�
+n≤ j
+σ1(n)
+(26)
+The operators χ1(j) and χ2(j) are hermitian, χ†
+j(n) = χ j(n), and obey the anticommutation algebra
+{χ j(n), χ j′(n′)} = 2δ j j′δnn′
+(27)
+Hence, they are fermionic operators are hermitian, anti-commute with each other and square to
+the identity. Operators of this type are called Majorana fermions.
+Alternatively, we can use the more conventional (Dirac) fermion operators c(n) and its adjoint
+c†(n) which are related to the Majorana fermions as
+c(n) = χ1(n) + iχ2(n),
+c†(n) = χ1(n) − iχ2(n)
+(28)
+which obey the standard anticommutation algebra
+{c(n), c(n′)} = {c†(n), c†)n′)} = 0,
+{c(n), c†(n′)} = δ − nn′
+(29)
+In this sense, a Majorana fermion is half of a Dirac fermion.
+In terms of the Majorana operators the Hamiltonian of Eq.(23) becomes
+H = i
+�
+j
+χ1(j)χ2(j) + ig
+�
+j
+χ2(j)χ1(j + 1)
+(30)
+13
+
+where we have rescaled the Hamiltonian by a factor of h and the coupling constant is g = J/h.
+Here we have not specified the boundary conditions (which depend on the fermion parity). Qual-
+itatively, the Majorana fermions can be identified with the domain walls of the classical models.
+In the Ising model the number of domain walls on each row is not conserved but their parity is.
+Likewise, the number of Majorana fermions NF is not conserved either but the fermion parity,
+(−1)NF, is conserved.
+It is an elementary excercise to show that the spectrum of this theory has a gap G(g) which
+vanishes at gc = 1 as G(g) ∼ |g − gc|ν, with an exponent ν = 1. Since the Hamiltonian of
+Eq.(30) is quadratic in the Majorana operators, these operators obey linear equations of motion.
+In the scaling regime we take the limit of the lattice spacing a → 0 and the coupling constant
+g → gc = 1, while keeping the quantity m = (g − gc)/a fixed. In this regime, the two-component
+hermitian spinor field χ = (χ1, χ1), and χ† = χ, satisfies a Dirac equation
+(i/∂ − m)χ = 0
+(31)
+where we set the speed � = 1, and where defined the 2 × 2 Dirac gamma-matrices γ0 = σ2,
+γ1 = iσ3, and γ5 = σ1. Upon defining ¯χ = χTγ0, we find that the Lagrangian of this field theory
+is
+L = ¯χi/∂χ − 1
+2m ¯χχ
+(32)
+which indeed becomes massless at the quantum phase transition of the Ising spin chain. For these
+considerations, we say that the phase transition of the Ising model (2D classical or 1D quantum)
+is in the universality class of massless Majorana fermions where m → 0. In Eq.(32) we have
+used the standard Feynman slash notation, /a = γµaµ, where aµ is a vector.
+4.3. Quantum Antiferromagnets and Non-Linear Sigma Models
+As we noted above, the discovery of high temperature superconductors in the copper ox-
+ide compounds prompted the study of the behavior of these strongly correlated materials at low
+temperatures and of possible quantum phase transitions which they may host. The prototypical
+cuprate material La2CuO4 is a quasi two-dimensional Mott insulator which exhibits long-range
+antiferromagnetic order below a critical temperature Tc. A simple microscopic model is a spin-S
+quantum Heisenberg antiferromagnet on the 2D square lattice of the Cu atoms, whose Hamilto-
+nian is
+H = 1
+2
+�
+r,r′
+J(|r − r′|) S(r) · S(r′)
+(33)
+where S are the spin-S angular momentum operators. We will consider the case where the ex-
+change interaction for nearest neighbors J is dominant and a weaker J′ ≪ J for next nearest
+neighbors. In this section we do not consider the regime J′ ≃ J in which the interactions com-
+pete for incompatible ground states due to frustration effects.
+4.3.1. Spin coherent states
+The simplest way to see the physics of this antiferromagnet is to construct a path-integral
+representation for a spin-S system using spin coherent states [74, 75, 76]. For details see Ref.[9]
+which we follow here. A coherent state of the (2S + 1-dimensional) spin-S representation of
+SU(2) is the state |n⟩, labeled by the spin polarization unit vector n
+|n⟩ = eiθ(n0×n)·S |S, S ⟩
+(34)
+where n2 = 1. The states of the spin-S representation are spanned by the eigenstates of S 3 and
+S2,
+S 3 |S, M⟩ = M|S, M⟩,
+S2|S, M⟩ = S (S + 1)|S, M⟩
+(35)
+and |S, S ⟩ is the highest weight state with eigenvalues S and S (S + 1). In Eq.(34) n0 is a unit
+vector along the axis of quantization (the direction e3), and θ is the colatitude, such that n · n0 =
+cos θ. Two spin coherent states, |n1⟩ and |n2⟩, are not orthonormal,
+⟨n1|n2⟩ = eiΦ(n1,n2,n0) S
+�1 + n1 · n2)
+2
+�S
+(36)
+where Φ(n1, n2, n0) is the area of the spherical triangle of the unit sphere spanned by the unit
+vectors n1, n2 and n0. However, there is an ambiguity in the definition of the area of the spherical
+14
+
+triangle since the sphere is a 2-manifold without boundaries: if the “inside” triangle has spherical
+area Φ, the complement (“outside”) triangle has area 4π − Φ. Thus, the ambiguity of the phase
+prefactor of Eq.(36) is
+ei4πS = 1
+(37)
+since S is an integer or a half-integer. So, the quantization of the representations of SU(2) makes
+the ambiguity unobservable. In addition, the spin coherent states |n⟩ satisfy the resolution of the
+identity
+I =
+�
+|n⟩⟨n|
+�2S + 1
+4π
+�
+δ(n2 − 1) d3n
+(38)
+and
+⟨n|S|n⟩ = S n
+(39)
+4.3.2. Path integral for a spin-S degree of freedom
+As an example consider problem of a spin-S degree of freedom coupled to an external mag-
+netic field B(t) that varies slowly in time. The (time-dependent) Hamiltonian is given by the
+Zeeman coupling
+H(t) = B(t) · S
+(40)
+As usual, the path-integral is obtained by inserting the (over-complete) set of coherent states at
+a large number of intermediate times. The resulting path integral is a sum of the histories of the
+spin polarization vector n(t)
+Z = tr exp
+�
+i
+� T
+0
+dt H(t)
+�
+=
+�
+Dn exp (iS[n])
+�
+t
+δ(n2(t) − 1)
+(41)
+where the action is
+S = S SWZ[n] − S
+� T
+0
+dt B(t) · n(t)
+(42)
+where SWZ[n] is the Wess-Zumino action
+SWZ[n] =
+� T
+0
+dt A[n] · ∂tn
+(43)
+=
+� 1
+0
+dτ
+� T
+0
+dt n(t, τ) · ∂tn(t, τ) × ∂τn(t, τ)
+(44)
+where A[n] is the vector potential of a Dirac magnetic monopole (of unit magnetic charge) at
+the center of the unit sphere. The vector potential A[n] has a singularity associated with the
+Dirac string of the monopole. We can write an equivalent expression which is singularity-free
+using Stokes Theorem. We did this in the second line of Eq.(44) which required to extend the
+circulation of A on the closed path described by n(t) to the flux of the vector potential through
+the submanifold Σ of the unit sphere S 2 whose boundary is the history n(t), i.e. the area of Σ.
+The smooth (and arbitrary) extension of configuration n(t) to the interior of Σ is done by defining
+n(t, τ) such that n(t, 1) = n0, n(t, 0) = n(t), and n(0, τ) = n(T, τ). Since SWZ is the area of the
+submanifold Σ of the unit sphere S 2, just as in Eq.(37), here too there is an ambiguity of 4π in
+the definition of the area. Here too, this ambiguity is invisible since the spin S is an integer or a
+half-integer.
+The path integral of Eq.(41) was derived first by Michael Berry [77] (and extended by Barry
+Simon [78]). The first term (which we called Wess-Zumino by analogy with its field theoretic
+versions) is called the Berry Phase. The role of this term, which is first order in time derivatives,
+is to govern the quantum dynamics of the spin which, in presence of a uniform magnetic field,
+executes a precessional motion of the (Bloch) sphere. It is also apparent from this expression that
+in the large-S limit, the path integral can be evaluated by means of a semiclassical approximation.
+The coherent-state construction shows that this problem is equivalent to the path integral
+of a formally massless non-relativistic particle of unit electric charge on the surface of the unit
+sphere with a magnetic monopole of magnetic charge S in its interior! This is not surprising
+since the Hilbert space of a non-relativistic particle moving on the surface of a sphere with and
+radial magnetic field (the field of a magnetic monopole) has a Landau level type spectrum with
+a degeneracy given by the flux. The condition of a massless particle means that only the lowest
+Landau level survives and all other levels have an infinite energy gap.
+The coherent state approach has been used to derive a path integral formulation for ferromag-
+nets and antiferromagnets. A detailed derivation can be found in Ref. [9].
+15
+
+4.3.3. Quantum Ferromagnet
+We will consider first the simpler case of a quantum ferromagnet and in Eq.(33) we will set
+J = −|J| < 0 for nearest neighbors and zero otherwise. The action for the path-integral for the
+spin-S quantum Heisenberg ferromagnet on a hypercubic lattice is
+S = S
+�
+r
+SWZ[n(r, t)] − |J|S 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt �n(r, t) − n(r′, t)�2
+(45)
+where we have subtracted the classical ground state energy. The oder parameter for this theory
+is the expectation value of the local magnetization, n = ⟨n(r)⟩, which is constant in space but
+points in an arbitrary direction in spin space.
+In the low energy regime the important configurations are slowly varying in space and we
+can simply approximate the action of Eq.(45) by its continuum version in d space dimensions
+S = S
+ad
+0
+�
+ddx SWZ[n(x, t)] − |J|S 2
+2ad
+0
+�
+ddx
+� T
+0
+dt (▽n(x, t))2
+(46)
+where a0 is the lattice spacing. As before, the path integral; is done for a field which satisfies ev-
+erywhere in space-time the constraint n2(x, t) = 1. This action can be regarded as non-relativistic
+non-linear sigma model.
+It is straightforward to show that the classical equations of motion for this theory are the
+Landau-Lifshitz equations
+∂tn = |J|S a2
+0 n × ▽2n
+(47)
+subject to the constraint n2 = 1. Due to the constraint, the Landau-Lifshitz equation is non-linear.
+We will a decomposition of the field into a longitudinal and two transverse components, σ and
+π, respectively
+n =
+�σ
+π
+�
+(48)
+subject to the constraint σ2 +π2 = 1. The linearized Landau-Lifshitz equations become (to linear
+order in π)
+∂tπ1 ≃ −|J|S a2
+0 ▽2 π2,
+∂tπ2 ≃ +|J|S a2
+0 ▽2 π1
+(49)
+The solution to these equations are ferromagnetic spin waves (magnons or Bloch waves) which
+satisfy the dispersion relation
+ω(p) ≃ |J|S a2
+0p2 + O(p4)
+(50)
+which shows that the dynamic exponent for a ferromagnet is z = 2. Notice that in this case the
+two transverse components are not independent (they are effectively a dynamical pair). These
+are the Goldstone bosons of a ferromagnet.
+4.3.4. Quantum Antiferromagnet
+Formally, the quantum antiferromagnet has a coherent state path integral whose action is
+S = S
+�
+r
+SWZ[n(r, t)] − JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt n(r, t) · n(r′, t)
+(51)
+with J > 0. For a bipartite lattice, e.g. the 1D chain, and the square and cubic lattices, the classi-
+cal ground state is an antiferromagnet with a N´eel order parameter, the staggered magnetization.
+Let m(r) be the expectation value of the local magnetization. A bipartite lattice is the union of
+two interpenetrating sublattices, and the local magnetization is staggered, i.e. it takes values with
+opposite signs (with equal values) on the two sublattices. Thus, we make the change of variables,
+n(r, t) → (−1)rn(r, t) in Eq.(51) and find
+S = S
+�
+r
+(−1)rSWZ[n(r, t)] − JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt (n(r, t) − n(r′, t))2
+(52)
+We want to obtain the low energy effective action for the field n(r, t). To this end, we decompose
+this field into a slowly varying part, that we will call m(r, t), and a small rapidly varying part
+l(r, t) (which represents ferromagnetic fluctuations)
+n(r, t) = m(r, t) + (−1)ra0l(r, t)
+(53)
+16
+
+Since n2(r, t) = 1, we will demand that the slowly varying part also obeys the constraint,
+m2(r, t) = 1, and require that the two components be orthogonal to each other, m · l = 0.
+Due to the behavior of the staggered Wess-Zumino terms of Eq.(52), the resulting continuum
+field theory turns out to have subtle but important differences between one dimension and higher
+dimensions. Here we will state the results for two and higher dimensions. We will discuss in
+detail the one-dimensional below when we discuss the role of topology.
+It turns out that if the dimension d > 1, the contribution of the staggered Wess-Zumino terms
+for smooth field configurations is [75, 79, 80]
+lim
+a0→0 S
+�
+r
+(−1)rSWZ[n(r, t)] = S
+�
+d3x l(x, t) · m(x, t) × ∂tm(x, t)
+(54)
+The continuum limit of the second term of Eq.(52) in the case of a two-dimensional system is
+lim
+a0→0
+JS 2
+2
+�
+⟨r,r′⟩
+� T
+0
+dt �n(r, t) − n(r′, t)�2 = a0
+JS 2
+2
+�
+d3x
+�
+(▽m(x, t))2 + 4l2(x, t)
+�
+(55)
+The massive field l[x, t] represents ferromagnetic fluctuations. Since this is a massive field it can
+be integrated-out leading to an effective field theory for the antiferromagnetic fluctuations m(x, t)
+whose Lagrangian is that of a non-linear sigma model
+L = 1
+2g
+� 1
+vs
+(∂tm(x, t) − vs(▽m(x, t))2
+�
+(56)
+where the coupling constant is g = 2/S and the spin-wave velocity is vs = 4a0JS . If we to allow
+for a weak next-nearest-neighbor interaction J′ > 0, the coupling constant g and the spin wave
+velocity vs become renormalized to g′ ≃ g/ √1 − 2J′/J and v′
+s ≃ vs
+√1 − 2J′/J.
+We conclude that that the quantum fluctuations about a N´eel state are well described by a non-
+linear sigma model. Provided the frustration effects of the next-nearest-neighbor interactions are
+weak enough, the long-range antiferromagnetic N´eel order should extend up to a critical value
+of the coupling constant gc where the RG beta function has a non-trivial zero, which signals a
+quantum phase transition to a strong coupling phase without long-range antiferromagnetic order.
+Motivated by the discovery of high temperature superconductivity in the strongly correlated
+quantum antiferromagnet La2CuO4 (at finite hole doping) in 1988 Chakravarty, Halperin and
+Nelson [81] utilized a quantum non-linear sigma model to analyze this system and its quantum
+phase transition. La2CuO4 is a quasi-two-dimensional material and so it exhibits strong quantum
+and thermal fluctuations. The upshot of this analysis is that while at T = 0 the non-linear sigma
+model has a quantum phase transition, at T > 0 the long range order is absent in a strictly 2D
+system but present in the actual material due to the weak-three-dimensional interaction. So, in
+the strict 2D case there is no phase transition but two different crossover regimes: a renormal-
+ized classical regime (without long range order), a quantum disordered regime and a quantum
+critical regime. La2CuO4 has long range N´eel (antiferromagnetic) order at T = 0 and is in the
+renormalized classical regime (with long range order due to the weak 3D interaction).
+The non-linear sigma model does not describe the nature of the ground state for g > gc
+beyond saying that there is no long range order. The problem is that, unlike the Ising model in a
+transverse field, the microscopic tuning parameter is the next nearest neighbor antiferromagnetic
+coupling J′, and to reach the regime g ≃ gc one has to make J′ ≃ J. This is the regime in which
+frustration effects become strong. In this regime the assumption that the important configurations
+are smooth and close to the classical N´eel state is incorrect. The nature of the ground state turns
+out to depend on the value of S .
+5. Topological Excitations
+Topology has come to play a crucial role both in Condensed Matter Physics and in Quan-
+tum Field Theory. Topological concepts have been used to classify topological excitations such
+as vortices and dislocations and to provide a mechanism for phase transitions, quantum num-
+ber fractionalization, tunneling processes in field theories, and nonperturbative construction of
+vacuum states. Here we will discuss a few representative cases of what has become a very vast
+subject.
+17
+
+5.1. Topological Excitations: Vortices and Magnetic Monopoles
+In Condensed Matter Physics topological excitations play a central role in the description of
+topological defects and on their role in phase transitions. Here topology integers in the classifi-
+cation of the configuration space into equivalence classes characterized by topological invariants
+[82]. The most studied example are vortices. Vortices play a key role in the mixed phase of type
+II superconductors in a uniform magnetic field [83]. Vortices also play a key role in the Statistical
+Mechanics of 2D superfluids and the the 2D classical XY model [84, 85] [86, 87] [88]. Disloca-
+tions and disclinations play an analogous role in the theory of classical melting [84, 89, 90], and
+2D and 3D classical liquid crystals [91, 92, 72].
+A similar problem occurs in Quantum Field Theory. Theories with global symmetries, such
+as the two-dimensional O(3) non-linear sigma model discussed above, when formulated in Eu-
+clidean space-time have instantons. Typically instantons are finite Euclidean action configu-
+rations, which are also classified into equivalence classes (associated with homotopy groups)
+labeled by topological invariants [93, 94]. Instantons play a central role in understanding the
+non-perturbative structure of gauge theories. Gauge theories with a compact gauge group cou-
+pled to matter fields have non-trivial vortex [95] and monopole [96, 97, 98] configurations, as
+do non-abelian Yang-Mills gauge theories [99]. Instantons have also played a central role in
+Condensed Matter Physics as well, notably in Haldane’s work on 1D quantum antiferromag-
+nets (discussed below), and in the problem of macroscopic quantum tunneling and coherence
+[100, 101].
+5.1.1. Vortices in two dimensions
+In this section I will focus on the the problem of the superfluid transition in 2D and the
+closely related problem of the phase transition of a magnet with an easy-plane anisotropy, the
+classical XY model. A superfluid is described by an order parameter that is a one-component
+complex field φ(x). If electromagnetic fluctuations are ignored, this description also applies
+to a superconductor. The complex field can be written in terms of an amplitude |φ(x)|, whose
+square represents the local superfluid density, and a phase θ(x) = arg(φ(x)).
+Deep in the
+superfluid phase the amplitude is essentially constant, that we will set to be a real positive
+number φ0, while the phase field θ(x) is periodic with period 2π and can fluctuate.
+Simi-
+larly, an easy-plane ferromagnet is described by a two-component real order parameter field
+M(x) = (M1(x), M2(x)) = |M(x)|(cosθ(x), sin θ(x)). Deep in the ferromagnetic phase the ampli-
+tude |M| is essentially constant but the phase field θ(x) can fluctuate.
+We will assume that we are in a regime where the local superfluid density |φ0|2 is well formed
+(or, equivalently that |M| is locally well formed) but that the phase field is fluctuating. In this
+regime the problem at hand is an O(2) ≃ U(1) non-linear sigma model, and its partition function
+takes the form
+Z =
+�
+Dθ exp
+�
+−
+�
+d2x 1
+2g
+�
+∂µθ(x)
+�2�
+(57)
+where we defined the coupling constant g = T/J|φ0|2, where T is the temperature, J is an in-
+teraction strength, and |φ0|2 is the magnitude (squared) of the amplitude of the order parameter,
+which we will take to be constant; κ = J|φ0|2 is the phase stiffness.
+Except for the requirement that the phase field be locally periodic, θ ≃ θ + 2π, superficially
+this seems to be a trivial free (Gaussian) field theory. We will see that the periodicity (or, com-
+pactification) condition makes this theory non-trivial. Indeed, configurations of the phase field
+that are weak enough that that do not see the periodicity condition, for all practical purposes, can
+regarded as being non-compact and ranging from −∞ to +∞. However there are many configu-
+rations for which the periodicity condition is essential. Such configurations are called vortices.
+Even in the absence of vortices, the periodic (compact) nature of the phase field is essential
+to the physics of this problem. In fact the only allowed observables must be invariant under
+local periodic shifts of the phase field. This implies that the phase field θ itself is not a physical
+observable but that exponentials of the phase of the form exp(inθ(x)) are physical. This operator
+is just the order parameter field of the XY model. In Conformal Field theory operators of this
+type are called vertex operators [102, 103]. We will see below that this theory has a dual field ϑ,
+associated with vortices, and that there are vertex operators of the dual field. In String Theory the
+model of a compactified scalar is known as the compactified boson and represents the coordinate
+of a string on a compactified space, in this case a circle S 1 [104].
+To picture a vortex consider a large closed curve C on the 2D plane. Hence, topologically
+a closed curve is isomorphic to a circle, C ≃ S 1. The phase field θ(x) is equivalent to a unit
+18
+
+circle S 1. Therefore the configuration space are maps of S 1 (the large circle) onto S 1 (the unit
+circle of the order parameter space). The configurations can be classified by the number of times
+the phase winds on the large circle C. The winding number is an integer called the topological
+charge of the configuration, the vorticity. Thus, a vortex is a configuration of the phase field θ(x)
+that winds by 2πm (where m is an integer):
+(∆θ)C
+2π
+= 1
+2π
+�
+C
+dx · ▽θ(x) ≡ i
+� 2π
+0
+dϕ
+2π eiθ(ϕ)∂ϕe−iθ(ϕ) = m
+(58)
+where ϕ ∈ (0, 2π] is the azimuthal angle for a vector at the center of the large circle C. Here
+n is the vorticity or winding number of the configuration; m > 0 is a vortex and m < 0 is an
+anti-vortex. The vorticity is a topological invariant of the field the configuration θ(x) which does
+not change under smooth changes.
+The winding number of a vortex is a topological invariant that classifies the configurations
+of the phase field as continuous maps of a large circle S 1 onto the unit circle S 1 defined by the
+phase field. In Topology such continuous maps are called homotopies. The winding number
+classifies these maps into a discrete set of equivalence classes, which form a homotopy group
+under the composition of two configurations. In this case the homotopy group is called Π1(S 1).
+Since the equivalence classes are classified by a topological invariant that takes integer values,
+the homotopy group Π1(S 1) is isomorphic to the group of integers, Z [82].
+The field jµ(x) = ∂µθ(x) is the superfluid current, and the vorticity ω(x) is the curl of the
+current, i.e.
+ω(x) = ǫµν∂µ jν(x) = ǫµν∂µ∂νθ(x)
+(59)
+which vanishes unless θ(x) has a branch-cut singularity across which the phase field jumps by
+2πn. Let ω(x) be the vorticity field with singularities at the locations {x j} of vortices with topo-
+logical charge m j
+ω(x) = 2π
+�
+j
+m jδ2(x − x j)
+(60)
+which is satisfied by the phase field configuration
+θ(x) =
+�
+j
+2πm jIm ln(z − zj)
+(61)
+where we have used the complex coordinates z = x1 + ix2. Away from the singularities {x j},
+this configuration obeys the Laplace equation. Hence, it has a Cauchy-Riemann dual field ϑ(x)
+which satisfies the Cauchy-Riemnann equation
+∂µϑ = ǫµν∂νθ
+(62)
+which satisfies the Poisson equation
+− ▽2ϑ(x) = ω(x)
+(63)
+whose solution is
+ϑ(x) =
+�
+d2y G(|x − y|) ω(y)
+(64)
+where G(|x − y|) is the Green function of the 2D Laplacian
+− ▽2G(|x − y|) = δ2(x − y)
+(65)
+In 2D this Green function is
+G(|x − y|) = 1
+2π ln
+�
+a
+|x − y|
+�
+(66)
+where a is a short distance cutoff (a lattice spacing). In what follows we will assume that the
+Green function of Eq.(66) has been cutoff so that G(|x − y|) = 0 for |x − y| ≤ a.
+The energy of a configuration of vortices {n j} with vanishing total vorticity, �
+j m j = 0, is
+E[θ] = Jφ2
+0
+2
+�
+d2x
+�
+∂µθ
+�2
+= Jφ2
+0
+2
+�
+d2x
+�
+d2y ω(x) G(|x − y|) ω(y) = 2πJφ2
+0
+�
+j>k
+m jmk ln
+�
+a
+|x j − xk
+�
+(67)
+19
+
+where we used that configurations with non-vanishing vorticity do not contribute to the partition
+function since they have infinite energy in the thermodynamic limit. We conclude that, up to an
+unimportant prefactor, that the partition function of Eq.(57) is the same as the partition function
+a gas of charges {m j} (the vortices) with total vanishing vorticity, �
+j m j = 0,
+Z2DCG =
+�
+[{mj}]
+exp
+−2π Jφ2
+0
+T
+�
+j<k
+m jmk ln
+�|x j − xk|
+a
+�
+(68)
+which is known as the two-dimensional (neutral) Coulomb gas [84].
+Kosterlitz and Thouless argued that the neutral two dimensional Coulomb gas has a phase
+transition a critical temperature TKT between a low temperature phase dielectric phase in which
+vortices and anti-vortices are bound into neutral pairs and a high temperature phase in which
+pairs unbind, the vortex charges are screened and the vortex gas is in a plasma phase. A simple
+estimate of the critical temperature is found by computing the free energy of a single vortex
+Fvortex = Evortex − TS vortex, where Evortex of a single vortex and S vortex is the vortex entropy. In
+2D the energy of a single vortex of unit topological charge, n = 1, is πJφ2
+0 ln(L/a), where L is the
+linear size of the system, and the vortex entropy is S vortex = ln(L/a)2, where (L/a)2 is the number
+of places where a vortex can be located, i.e. the area of the system in units of the spacing a.
+Vortices unbind (proliferate) when the entropy beats the energy. Hence, the critical point occurs
+at a temperature TKT where the vortex free energy vanishes, Fvortex[TKT] = 0. This yields the
+estimate
+TKT = π
+2 Jφ2
+0
+(69)
+which is the Kosterlitz-Thouless critical temperature.
+There is an alternative, equivalent way to see this physics. Let is consider the theory if
+Eq.(57) in the background of a U(1) gauge field Aµ(x)
+Z[A] =
+�
+Dθ exp
+�
+− 1
+2g
+�
+d2x
+�
+∂µθ − Aµ
+�2�
+(70)
+where the curl of the background gauge field Aµ is chosen to be equal to the local vorticity of
+Eq.(59),
+ǫµν∂µAν(x) = ω(x)
+(71)
+In other words, the gauge Aµ field forces the phase field to have vortices at the locations {x j}.
+Next we will use a Hubbard-Stratonovichtransformation (a Gaussian identity) to write Eq.(70)
+in the equivalent form
+Z[A] =
+�
+Dθ
+�
+Daµ exp
+�
+−g
+2
+�
+d2x a2
+µ + i
+�
+d2x aµ(x)
+�
+∂µθ(x) − Aµ(x)
+��
+(72)
+where we neglected an unimportant prefactor. In Eq.(72) the phase field can be integrated-out
+(after an integration by parts) to yield the constraint ∂µaµ = 0. This constraint is solved by
+introducing the dual field ϑ(x) such that
+aµ(x) = ǫµν∂νϑ(x)
+(73)
+Hence, the partition function Z[A] becomes
+Z[A] =
+�
+Dϑ exp
+�
+−g
+2
+�
+d2x (∂µϑ)2 + i
+�
+d2x ϑ(x) ω(x)
+�
+(74)
+where we have assumed that the total vorticity is zero,
+�
+d2x ω(x) = 2π �
+j m j = 0. This identity
+show that the quantization of the vortex topological charge requires that the dual field ϑ should
+also be periodic (compact) with period 1.
+It is elementary to see that after performing the gaussian integral over the field ϑ in Eq.(74)
+and summing over all vortex configurations (with total vanishing vorticity) we reproduce the
+partition function of the 2D neutral Coulomb gas [105, 106]. Thus, the theory of Eq.(70) defined
+in terms of the phase field θ and the theory of Eq.(74) are equivalent. This is an example of a
+duality transformation which amounts to the exchange of the fields
+θ ↔ ϑ,
+and
+g ↔ 1/g
+(75)
+20
+
+A more careful analysis done on the lattice shows that the dual field ϑ is defined on the dual of
+the square lattice (which is also a square lattice) [88, 105]. In addition, by differentiating both
+the partition function of the phase field of Eq.(70) and the partition function of the dual field
+ϑ of Eq.(74) with respect to the background field Aµ we obtain the duality identification of the
+currents
+1
+g∂µθ ←→ gǫµν∂νϑ
+(76)
+as an operator identity. This identity has the same form as the duality of forms in geometry.
+In this case, the dual of the current ∂µθ, which is a 1-form field, is another 1-form field, ∂µϑ.
+In both cases the field θ and its dual ϑ are associated with global symmetries. Later we will
+encounter similar duality identifications but their content will be different in different dimensions.
+Moreover, running these identities backwards it is easy to see that the insertion of the order
+parameter operator exp(ipθ(x)) (with p and integer), which acts as a source on the phase field θ,
+in the partition function is is equivalent to minimally couple the dual field ϑ to a singular gauge
+field Bµ(x) whose curl is p at the location x. In other words, charges are dual to vortices.
+We will now use the identity we derived in Eq.(74) to compute the full partition function
+which is a sum over all vortex configurations. Now we will assume that the vortices have a large
+core energy (i.e. the energy associated with the short distance behavior of the Green function of
+Eq.(66)). For a vortex with unit topological charge we will call this energy u, and for a vortex
+with charge m the core energy will be um2. The effects of the core energy can be accounted for
+in terms of a vortex fugacity z = exp(−un2/T). The total partition function now is
+Z =
+�
+{mj}
+�
+Dϑ exp
+−g
+2
+�
+d2x (∂µϑ)2 + i
+�
+j
+2πm jϑ(x j) − u
+T
+�
+j
+m2
+j
+
+(77)
+where, once again, we assumed global charge neutrality.
+In the limit in which z = exp(−u/T) ≪ 1, only the vortices with the lowest topological
+charges m = ±1 contribute and, furthermore, they are dilute. In this limit we can perform over
+the configurations with the lowest topological charge and derive the effective field theory of the
+dual field ϑ:
+ZSG =
+�
+Dϑ exp
+�
+−
+�
+d2x
+�g
+2(∂µϑ)2 − � cos(2πϑ)
+��
+(78)
+which is known as the sine-Gordon field theory. The coupling constant of the sine-Gordon theory
+is � = 2 exp(−u/T)/a2, where a is the core size (the lattice spacing). Hence, the cosine operator
+of the sine-Gordon theory represents the effects of the vortices.
+We can now analyze the role of vortices by looking at the renormalization group of the sine-
+Gordon theory. The first question is what is the scaling dimension of the cosine operator. This
+can be computed by looking at the vortex operator correlator in the theory with � = 0:
+⟨exp(i2πϑ(x)) exp(−i2πϑ(y))⟩ =
+const.
+|x − y|2π/g
+(79)
+which implies that the vortex scaling dimension is
+∆vortex = π
+g
+(80)
+Eq.(12) implies that in D = 2 dimensions the operator is irrelevant if ∆ > 2, relevant for ∆ < 2
+and marginal for ∆ = 2. Hence, vortices are marginal if ∆vortex = 2 which happens if g = π/2.
+This is the same as to say that the system is at the Kosterlitz-Thouless critical temperature T =
+TKT. Hence, vortices are irrelevant if T < TKT and relevant for T > TKT.
+This RG analysis tells us that T > TKT, when vortices proliferate, the coupling constant ν
+flows to strong coupling to a regime where ϑ is pinned to an integer value and the theory is in a
+massive phase. In this phase the connected vortex correlator decays exponentially with distance
+which is the same as to say that the vortex charge is screened. This is why in this phase the
+vortices proliferate. It can be shown that in this phase the correlator of the spins of the XY
+model, i.e. the correlator of the vertex operator exp(iθ(x)), decays exponentially with distance
+and the systems is in its disordered phase.
+There is still the question of the nature of the phase with T < TKT. The Mermin-Wagner The-
+orem [107] (and its generalizations by Hohenberg [108] and Coleman [109]) states that classical
+21
+
+statistical mechanical systems with a global continuous symmetry group cannot undergo spon-
+taneous symmetry breaking in space dimensions D ≤ 2 (and quantum systems with space-time
+dimensions D ≤ 2). Does this theory violate this the Mermin-Wagner Theorem? The answer is
+no. It is easy to see that in the phase in which the vortices are irrelevant, i.e. for T < TKT, the
+correlator of the order parameter operator is always a power law of the distance |x − y|,
+⟨exp(iθ(x)) exp(−iθ(y))⟩ =
+const.
+|x − y|g/2π
+(81)
+with an exponent that depends on temperature and satisfies
+g
+2π =
+T
+2πJφ2
+0
+≤ 1
+4
+(82)
+In other words, the entire low temperature phase is not an ordered phase of matter since the cor-
+relator is not constant at long distance. On the other hand, the correlators of the order parameter
+exp(iθ) and of the vortices exp(iϑ) have a power la behavior for T < TKT, we conclude that in
+this temperature range the system is scale invariant and that it has a line of critical points.
+In summary, we succeeded in expressing the partition function as a sum over configurations
+of the topological excitations, the vortices. We succeeded in doing that because vortices are
+labeled by their coordinates on the plane. In addition, we found a non-trivial phase transition
+since the entropy and the energy both scale logarithmically with the linear size of the system
+or, equivalently, that vortices became marginally relevant at a critical temperature. This is the
+mechanism behind the Kosterlitz-Thouless transition.
+One may ask if this construction is generic and the answer is no. As an example consider the
+Abelian Higgs model in D = 2. This model has a complex scalar field minimally coupled to a
+Maxwell gauge field and, hence, its gauge group is U(1). In the classical spontaneously broken
+phase, the gauge field becomes massive. In this phase the long range coherence of the phase field
+of the vortices of the scalar field is screened at the scale of the penetration depth. As a result
+the Euclidean action of the vortices is now finite. Furthermore, on longer scales the interaction
+energy between vortices becomes short ranged. Hence, instead of a 2D Coulomb gas now one
+has a gas of particles (vortices and anti-vortices) with short range interactions. In this case the
+entropy always dominates and the vortices proliferate [110]. This behavior is quite analogous to
+the restoration of symmetry by proliferation of domain walls in one-dimensional classical spin
+chains [19] and to the analogous problem of tunneling in the path integral formulation of quantum
+mechanical double-well potentials [93]. As a caveat, we should note that, in spite of the obvious
+similarities, the 2D Abelian Higgs model does not describe correctly a 2D superconductor (i.e. a
+superconducting film) since the electromagnetic field is not confined to the film and this renders
+the electromagnetic action nonlocal.
+5.1.2. Magnetic monopoles in compact electrodynamics
+Instantons are of great interest in Quantum Field Theory since they provide for a mechanism
+to understand the non-perturbative structure of these theories. For this reasons they have been
+used to understand the mechanisms of quark confinement and the role of quantum anomalies
+in non-Abelian gauge theories [99, 97, 98, 96, 111]. Instantons in non-Abelian gauge theories
+are magnetic monopoles and the condensation of monopoles have long been argued to be the
+mechanism behind quark confinement.
+The simplest non-Abelian gauge theory that has magnetic monopoles is the Georgi-Glashow
+[112]. This model has a three-component real field φ and an SU(2) Yang-Mills gauge field Aµ. In
+its Higgs phase the scalar field φ acquires an expectation value which breaks the gauge symmetry
+group SU(2) down to its diagonal U(1) subgroup. Since U(1) ⊂ SU(2), this Abelian gauge group
+is compact, meaning that its magnetic fluxes are quantized. Polyakov [113] and ’t Hooft [111]
+showed that in 2+1 Euclidean dimensions have non-singular instanton solutions which at long
+distances resemble the magnetic monopole originally proposed by Dirac in 1931 [114]
+Bi(x) = q
+2
+xi
+|x|2 − 2πqδi,3δ(x1)δ(x2)θ(−x3)
+(83)
+The first term in Eq.(83) is the magnetic radial field of a monopole of magnetic charge q. The
+second term represents an infinitely long infinitesimally thin solenoid ending at the location of
+the monopole, x = 0, that supplies the quantized magnetic flux 2πq. This singular term is
+known as the Dirac string. The string itself (and its orientation) is physically unobservable to any
+22
+
+electrically charged particle that obeys the Dirac quantization condition, qe = 2π (in units where
+ℏ = c = 1).
+In the language of a lattice gauge theory [32], a theory with a compact (i.e. periodic) U(1)
+gauge fields on a D = 3 cubic lattice, describing a compact gauge field in 2 + 1 dimensions.
+This theory should have instantons that resemble magnetic monopoles much in the same way as
+a theory with a compact global U(1) symmetry has vortices. The simplest example is Polyakov’s
+compact electrodynamics [97] whose partition function is
+Z =
+�
+x,µ
+� 2π
+0
+dAµ(x)
+2π
+exp
+
+1
+4e2
+�
+x,µ,ν
+cos(Fµν(x))
+
+(84)
+where Fµν(x) = ∆µAν(x)−∆νAµ(x) ≡ �
+µ Aµ is the magnetic flux through the elementary plaquette
+labeled by a site x and a pair of directions, µ and ν, with µ = 1, 2, 3. This theory is invariant under
+local gauge transformations Aµ(x) → Aµ(x) + ∆µΦ(x) and it is also invariant under local periodic
+shifts of the gauge fields Aµ(x) → Aµ(x) + 2πℓµ(x), where ℓµ(x) ∈ Z. The plaquette flux operator
+satisfies the lattice version of the Bianchi identity that the product of exponentials of the flux on
+the faces of every elementary cube of the lattice is
+�
+cubefaces
+eiFµν(x) = 1
+(85)
+which says that the theory can have magnetic monopoles of integer magnetic charge.
+We will analyze this theory following an approach analogous to what we used for vortices in
+section 5.1.1.To this end we will consider the partition function
+Z[Bµν] =
+�
+DAµ exp
+�
+− 1
+4e2
+�
+d3x
+�
+Fµν(x) − Bµν(x)
+�2�
+(86)
+where Fµν(x) = ∂µAν − ∂νAµ is the field strength of the abelian U(1) Maxwell gauge field Aµ.
+Here Bµν(x) is an (anti-symmetric) two-form background gauge field. The coupling constant of
+this theory is e2. Since Aµ is a connection it has units of length−1, and F2
+µν is a dimension 4 field.
+Then, in D = 3 dimensions, e2 has units of length−1.
+The theory is invariant under two local transformations, namely the usual invariance under
+gauge transformations
+Aµ(x) → Aµ(x) + ∂µΦ(x),
+Bµν(x) → Bµν(x)
+(87)
+where Φ(x) is an arbitrary smooth function of x. The presence of the background two-form field
+Bµν now requires invariance under one-form gauge transformations
+Aµ(x) → Aµ(x) + aµ(x),
+Bµν(x) → Bµν(x) + ∂µaν − ∂νaµ
+(88)
+The two-form gauge field Bµν essentially represents the magnetic monopoles. Let {m j} be a
+configuration of monopoles of charges m j with coordinates {x j}, with total vanishing monopole
+charge, �
+j m j = 0. Let M(x) be the magnetic monopole density at x,
+M(x) = 2π
+�
+j
+m j δ3(x − x j)
+(89)
+which can be expressed as the curl of the two-form gauge field Bµν,
+M(x) = 1
+2ǫµνλ∂µBνλ(x)
+(90)
+We will proceed next much in the same way as in Eq.(72) and rewrite the partition function
+of Eq.(86) in terms of a two-form Hubbard-Stratonovich field bµν(x) such that
+Z[B] =
+�
+DAµ
+�
+Dbµν exp
+�
+−e2
+4
+�
+d3x b2
+µν(x) + i
+�
+d3x 1
+2bµν(x)
+�
+Fµν(x) − Bµν(x)
+��
+=
+�
+DAµ
+�
+Dbµν exp
+�
+−e2
+4
+�
+d3x b2
+µν(x) + i
+�
+d3x
+�
+Aµ(x)∂νbµν(x) − 1
+2bµν(x)Bµν(x)
+��
+(91)
+23
+
+Thus, the gauge field Aµ plays the role of a Lagrange multiplier field the enforces the constraint
+∂νbµν(x) = 0
+(92)
+which is solved in terms of a compact scalar field ϑ(x)
+bµν(x) = ǫµνλ∂λϑ(x)
+(93)
+Using this identity and the definition of the monopole density M(x) we find that the partition
+function Z[Bµν] of Eq.(86) becomes
+Z[B] =
+�
+Dϑ exp
+�
+−
+�
+d3x
+�e2
+2 (∂µϑ(x))2 + iM(x)ϑ(x)
+��
+=
+�
+Dϑ exp
+−e2
+2
+�
+d3x(∂µϑ(x))2 + 2πi
+�
+j
+m jϑ(x j)
+
+(94)
+which requires that the field ϑ obeys the compactification condition ϑ → ϑ + n, where n is an
+arbitrary integer. Eq.(94) says that the magnetic monopole instantons of the compact U(1) gauge
+theory are dual to charges of the dual phase field ϑ, which has a compact U(1) global symmetry.
+The full partition function is obtained by summing over all monopole configurations satis-
+fying the total neutrality condition, �
+j m j = 0. As in section section 5.1.1, we will weigh the
+configurations with a coupling u and find
+Z =
+�
+{mj}
+Z{m j}
+�
+Dϑ exp
+−e2
+2
+�
+d3x (∂µϑ)2 +
+�
+j
+2πim jϑ(x j) − u
+�
+j
+m2
+j
+
+(95)
+which is the same theory we found in Eq.(77) except that now we are in 3D. Moreover, summing
+only over dilute configurations of monopoles and anti-monopoles we find, once again the sine-
+Gordon theory but now in D = 3 dimensions:
+Z =
+�
+Dϑ exp
+�
+−
+�
+d3x
+�e2
+2 (∂µϑ)2 − � cos(2πϑ)
+��
+(96)
+with � = 2 exp(−u)/a3.
+In spite of the similarities between Eq.(96) and the sine-Gordon theory in 2D, Eq.(78), the
+physics is very different. It is straightforward to see that, in the limit v = 0, the monopole operator
+correlator is
+⟨exp �2πiϑ(x)) exp(−2πiϑ(y)� = exp
+�4π2
+e2 [G(|x − y|) − G(0)]
+�
+≃ exp
+� π
+2e2
+� 1
+R − 1
+a
+��
+(97)
+where a is the short-distance cutoff. Unlike the behavior of the correlator of the vortex operators
+in 2D found in Eq.(79), Eq.(97) does not show a power-law behavior. The reason is that at � = 0
+the compactified field ϑ should be regarded as the Goldstone boson of a spontaneously broken
+U(1) symmetry. However, the cosine operator is now always relevant and the field ϑ is pinned
+and its fluctuations are actually massive.
+Looking back at the partition function of Eq.(95), we could integrate out the field ϑ and
+obtain an expression with the same form as the Coulomb gas of Eq.(68) except that now this is
+the three-dimensional neutral Coulomb gas [97]
+Z3DCG =
+�
+[{mj}]
+exp
+− π
+2e2
+�
+j<k
+m jmk
+�
+1
+|x j − xk| − 1
+a
+�
+(98)
+Now the energy of an isolated monopole is finite (in the infrared) while the entropy is still log-
+arithmically divergent. The conclusion is that the monopoles proliferate and the 3D neutral
+Coulomb gas is always in a plasma phase withe screened interactions. These results also imply
+that the expectation value of the Wilson loop operator in the U(1) gauge theory decays exponen-
+tially with the area of the loop. The conclusion is that in 2+1 dimensions compact electrody-
+namics is in a confining phase for all values of the coupling constant e2 [97, 98].
+24
+
+5.2. Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains
+We will now discuss the one-dimensional case in which topology plays a crucial role. In one
+space dimension, the sites of the chain are labelled by an integer j = 1, . . ., N (with N even).
+After some simple manipulations, the continuum limit of the staggered Wess-Zumino terms of
+Eq.(52) is
+lim
+a0→0 S
+N
+�
+j=1
+(−1) jSWZ[n(j, t)] =S
+2
+�
+d2x m(x, t) · ∂tm(x, t) × ∂xm(x, t)
++ S
+�
+d2x l(x, t) · m(x, t) × ∂tm(x, t)
+(99)
+The continuum limit of the second term of Eq.(52) is essentially the same as in Eq.(55).
+By putting together the results of Eq.(99) and Eq.(55), we find that in the case of a 1D chain
+the continuum field theory has the Lagrangian density
+L[m, l] = −2a0JS 2l2 + S l · m × ∂tm − a0JS 2
+2
+(∂xm)2 + S
+2 m · ∂tm × ∂xm
+(100)
+Finally, we integrate-out the ferromagnetic fluctuations represented by the massive field l(x, t) to
+find that the effective low-energy Lagrangian has the form
+L = 1
+2g
+� 1
+vs
+(∂tm)2 − vs (∂xm)2
+�
++ θ
+4π m · ∂xm × ∂tm
+(101)
+which is a non-linear sigma model with N = 3 components (and hence has a global O(3) sym-
+metry) with a topological term whose coupling constant is θ. This result was first derived by
+Haldane [115].
+In Eq.(101) the effective coupling constant g, the spin-wave velocity vs, and the θ-angle are
+given by g = 2
+S , vs = 2a0JS , and θ = 2π. As in the 2D case, since g = 2/S , large S implies that
+the non-linear sigma model is in the weak coupling regime. Provided that vs > 0, one can always
+rescale the time and space coordinates without affecting the form of the Lagrangian, including
+the coupling constant or, as we will see, the value of the θ angle. In what follows we will assume
+that we have done the rescaling in such a way that we set vs = 1 and, that time and space scale as
+lengths. Thus we will use a relativistic notation and, after an analytic continuation to imaginary
+time, we label the time coordinate by x2 and the space direction by x1. The partition function
+now takes the form
+Z =
+�
+Dmexp
+�
+− 1
+2g
+�
+Ω
+d2x
+�
+∂µn
+�2 + iθ Q[m]
+�
+(102)
+where Ω is the spacetime manifold. In this notation the first term of the exponent is called the
+Euclidean action of the non-linear sigma model. In Eq.(102) we denoted by Q[m] the quantity
+Q[m] = 1
+8π
+�
+Ω
+d2x ǫµνm · ∂µm × ∂νm
+(103)
+Here we will consider the case in which the spacetime manifold Ω is closed. In particular
+we will assume that it is a two-sphere S 2. The quantity Q[m] is the integral of a total derivative
+which counts the number of times the field configuration m(x1, x2) wraps around the sphere S 2.
+In other words, it yields a non-vanishing result only for “large” configurations which wind (or
+wrap) around the sphere S 2. Since Q[n] is an integer, it has the same for all smooth the field
+configuration m that can be smoothly deformed into each other and are homotopically equiva-
+lent. If we demand that the field configurations m have finite Euclidean action, which requires
+that at infinity the configurations take the same (but arbitrary) value of m, we have effectively
+compactified the x1 − x2 plane into a two sphere S 2. On the other hand the field m is restricted
+by the constraint m2 = 1 to take values on a two-sphere S 2. Therefore, the field configurations
+m(x1, x2) are smooth maps of the S 2 of the coordinate space to the S 2 of the target space of the
+field m. Hence, the integer Q[m] classifies the smooth field configurations into a set of equiva-
+lence classes each labeled by the integer Q. Under composition homotopies form groups, and the
+equivalence classes themselves also form a group which, in this case, is isomorphic to the group
+of integers, Z. In Topology, these statements are summarized by the notation Π2(S 2) ≃ Z. The
+25
+
+configurations with non-zero values of Q are called instantons which, in the quantum problem,
+represent tunneling processes of the non-linear sigma model.
+Another consequence of Q being an integer is that the contribution of its term to the weight of
+the path integral of Eq.(103) is a periodic function of θ angle. On the other hand, since the only
+allowed values of the θ are θ = 0 (mod 2π) for S integer, or θ = π (mod 2π) for S a half-integer,
+the contribution of the topological invariant Q to the weight of the path integral is
+exp(iθ Q[m]) = (−1)2S Q
+(104)
+Therefore, for spin chains with S integer, the weight is 1 and the topological invariant does not
+contribute to the path integral. But, if S is a half-integer, the weight is (−1)Q[m], and it does
+contribute. Moreover, its contribution is the same for all half-integer values of S .
+These results have important consequences for the physics of spin chains which led Haldane
+to some startling conclusions [115]. In the weak coupling regime, g ≪ 1 (equivalently, for
+large S ), we can use the perturbative renormalization group and derive the beta function for all
+these O(3) non-linear sigma models (with or without topological terms) and find that their beta
+functions are the same as in Eq.(21) with N = 3. Hence, for all S , the effective coupling flows
+to large values. We can make this inference since the topological term yields no contribution for
+all configurations which are related by smooth deformations. Thus, we infer that all spin chains
+with S integer are in a massive phase with an exponentially small energy gap ∼ exp(−2πS ). This
+result is nowadays known as the Haldane gap.
+On the other hand, these results also imply that all spin chain with half-integer spin S are
+also the same and, in particular, th same as spin-1/2 chains. However, the Hamiltonian of the
+quantum spin chain with spin-1/2 degrees of freedom is an example of an integrable system and
+its spectrum is known to be gapless from its Bethe Ansatz solution [116, 117]. However, the
+spin-1/2 chain is not only gapless but its low energy states are gapless solitons with a relativistic
+spectrum. The low energy description of a theory of this type must be described by a conformal
+field theory. Haldane concluded that the RG flow for spin-1/2 chains must have an IR stable fixed
+point at some finite (and large) value of the coupling constant. This conjecture was confirmed by
+Affleck and Haldane [118] who showed that the spin-1/2 Heisenberg chains are in the universality
+class of the SU(2)1 Wess-Zumino-Witten model [119] whose CFT was solved by Knizhnik and
+Zamolodchikov [120].
+5.3. Topology and open integer-spin chains
+In the preceding section we showed that integer spin chains have a Haldane gap. We did that
+by showing that in that case the topological term is absent. However, the derivation is correct
+provided the spacetime manifold is closed, e.g. a sphere, a torus, etc. What happens if the system
+has a boundary? Let us denote the boundary of Ω by Γ = ∂Ω. For example we will take Γ to be
+along the imaginary time direction and hence that it is a circle of circumference 1/T, where T
+is the temperature. The topological term has the same form as the Berry phase term of the path
+integral for spin, the “Wess-Zumino” term of Eq.(44), but its prefactor is 1/2 as big. Thus, for
+a system with an open boundary the topological term yields a net contribution equal to a Berry
+phase with a net prefactor of S/2.
+In other words, the boundary of the integer spin chain behaves as a localized degree of free-
+dom whose spin is 1/2 (mod an integer). Form the periodicity requirement we also see that if
+the spin chain is made of odd-integer degrees of freedom, there should be a spin 1/2 degree of
+freedom localized at the open boundary!. Conversely, if the chain is made of even-integer spins,
+there is no boundary degree of freedom! This line of argument implies that an antiferromagnetic
+chain with odd-integer spins must have a spin-1/2 degree of freedom at the boundary whereas
+a chain of even-integer spins should not. Notice that the “bulk” behavior is the same for both
+odd and even integer spin chains. The difference is whether or not they have a non-trivial “zero-
+mode” state at the boundary. In particular, the existence of this state is robust, i,.e. it cannot
+be removed by making smooth changes to the quantum Hamiltonian or, what is the same, the
+boundary state is topologically protected.
+A simple system that displays a protected spin-1/2 zero mode at the boundary is a generalized
+S = 1 spin chain with Hamiltonian
+H = α
+N
+�
+j=1
+S(j) · S(j + 1) + β
+N
+�
+j=1
+(S(j) · S(j + 1))2
+(105)
+26
+
+where S(j) = (S x(j), S y(j), S z(j) are the spin 1 matrices at each lattice site j. This problem was
+examined in great detail by Affleck, Kennedy, Lieb and Tasaki [121] who showed that at the
+special value of the parameters α = 1/2 and β = 1/6 this Hamiltonian takes the form of a sum of
+projection operators
+H =
+�
+j
+P2(S(j) + S(j + 1))
+(106)
+where P2 is an operator that projects out the spin 2 states. These authors constructed the exact
+ground state, known as the AKLT state, of this Hamiltonian by writing each spin 1 degree of
+freedom of two spin-1/2 degrees of freedom at each site. They showed that the ground state is a
+projected product state in which the “constituent’ spin-1/2 degrees of freedom (each labeled by
++ and − respectively) on nearby sites j and j + 1 are in a valence bond singlet state of the form
+1√
+2(| ↑j,+, ↓j+1,−⟩− ↓j,+, ↑j+1,−⟩ (and symmetrizing at each site to project onto a spin 1 state).
+This state of the spin-1 chain is translation invariant and gapped and hence agrees with Haldane’s
+result. Moreover, it has a spin-1/2 degree of freedom at each open boundary [122].
+The arguments discussed above show that the AKLT state is a gapped topological state. The
+gapless spin-1/2 boundary states are an example of spin fractionalization. The spin-1/2 boundary
+degrees of freedom are an example of an edge state which are present in many, thought not all,
+topological phases of matter. One may ask what symmetry protects the gaplessness of these edge
+degrees of freedom. The only perturbation that would give a finite energy gap to the spin-1/2
+edge states is an external magnetic field. However, this perturbation would break the global
+SU(2) symmetry of the Hamiltonian as well as time reversal invariance.
+6. Duality in Ising Models
+Duality plays a significant role of our understanding of statistical physics and of quantum
+field theory. Many seemingly unrelated correspondences between different theories have come
+to be called dualities.
+6.1. Duality in the 2D Ising Model
+One of the earliest versions of duality transformations was used to relate the high-temperature
+expansion of the 2D classical Ising model and its low-temperature expansion [123]. In all dimen-
+sions, for concreteness we will think of a hypercubic lattice, the high temperature expansion is a
+representation of the partition function as a sum of contributions of closed loops on the lattice. At
+temperature T, a configuration of loops γ contributes with a weight which in the Ising model has
+the form C(γ) tanhL(γ)(β), where β = 1/T and L(γ) is the length of the loop γ, i.e. the number
+of links on the loop, and C(γ) is an entropic factor that counts the number of allowed loops with
+fixed perimeter L(γ). However not all loop configurations are allowed as in the Ising model they
+satisfy constraints such as being non-overlapping, etc.
+In section 4.2 we noted that the partition function of the classical Ising model (in any dimen-
+sion) can be interpreted as the path-integral of a quantum spin model in one dimension less on a
+lattice with a discretized imaginary time. In this picture, the loops γ can be regarded as processes
+in which pairs of particles are created at some initial (imaginary) time, evolve and eventually are
+annihilated at a later (imaginary) time. In other words, the high temperature phase is a theory of
+a massive scalar field. The restrictions on the allowed loop configurations represents interactions
+among these particles. In the temperature range in which the expansion in loops is convergent
+the particles are massive as the loops are small. As the radius of convergence of the expansion is
+approached, longer and increasingly fractal-like loops begin to dominate the partition function,
+and concomitantly the mass of the particles decreases. This process is the signal of the approach
+to a continuous phase transition where the particles become massless. In fact, right at the critical
+point the particle interpretation is lost as the associated fields acquire anomalous dimensions.
+Returning to the 2D classical Ising model, Kramers and Wannier also considered the low
+temperature expansion. This is an expansion around one of the broken symmetry state, e.g.
+the state with all spins up. In this low temperature regime the partition function is a sum of
+configurations of flipped spins. A typical configuration is a set of clusters of flipped spins. In the
+absence of a uniform field, a configuration of flipped spins has a energy cost only on links of the
+lattice with oppositely aligned spins (“broken bonds”). Thus a cluster of flipped spins costs an
+energy equal to 2 (I assumed that I set J = 1) for each broken bond at the boundary of the cluster.
+This boundary is domain wall which is a closed loop on the dual lattice. In 2D the dual of the
+square lattice is the square lattice of the dual sites (the centers of the elementary plaquettes of the
+27
+
+2D lattice). In 2D the links of the direct lattice pierce the links of the dual lattice. We can see
+that this analogous to the the geometric duality of forms: sites (“0-forms”) are dual to plaquettes
+(“2-forms”) and links (“1-forms”) are dual to links (also “1-forms”).
+Thus, in 2D the low temperature expansion is an expansion in the loops γ∗ of the dual lattice
+that represent the domain walls. We can also regard the domain walls (the dual loops) as the
+histories of of pairs of particles on he dual lattice. However, the weight of each dual loop is
+exp(−2β) per link of γ∗. Except for that, the counting and restrictions on the dual loops γ∗
+are the same as those of γ. This means that there is a one-to-one correspondence between the
+two expansions with the replacement tanh β ↔ exp(−2β∗). This mapping means that the dual
+of the 2D classical Ising model (with global Z2 symmetry) at inverse temperature β is a dual
+Ising model (also with global Z2 symmetry) on the dual lattice at inverse temperature β∗. This
+correspondence is in close analogy with what we discussed in section 5.1.1.
+In particular if one assumes that there is a transition at βc = 1/Tc then there should also be a
+transition at β∗
+c = −1/2 lntanh βc. Moreover, if one further assumes (as Kramers and Wannier did
+[123]) that there is a unique transition (correct in the Ising model but not in other cases), then the
+critical point must be such that exp(−2βc) = tanh βc, which yields the value Tc = 2/ ln(
+√
+2 + 1),
+which agrees with the Onsager result [20].
+In section 4.2 we showed that the classical 2D Ising model is equivalent to the one-dimensional
+Ising model in a transverse field whose Hamiltonian is given in Eq.(23). The 1D Ising model on
+a transverse field has spin degrees of freedom defined on the sites of a one-dimensional chain
+labeled by an integer-valued variable n. The 1D Hamiltonian is expressed in terms of local op-
+erators, the Pauli matrices σ3(n) and σ1(n). The 1D chain has a dual lattice whose sites are the
+midpoints of the chain. Thus in 1D sites (“0-forms”) are dual to links (“1-forms”) and viceversa.
+We will now see that there is a Hamiltonian version of the Kramers-Wannier duality [70].
+In Eq.(26) we introduced the kink creation operator which flips are σ3 operators to the left and
+including site j. For clarity we will denote the kink creation operator operator as τ3(˜n), where
+˜n is the site of the dual lattice between the sites n and n + 1 of the original lattice. We will now
+define an operator τ1(˜n) = σ3(n)σ3(n+1). The operators τ1(˜n) and τ3(˜n) satisfy the same algebra
+os the Pauli operators σ1(n) and σ3(n). Furthermore, we readily find the Hamiltonian of the dual
+theory is the same (up to boundary conditions) as the original Hamiltonian of Eq.(23) except
+that the dual coupling constant is λ∗ = 1/λ. So, once again, if we assume that there is a single
+(quantum) phase transition we require λ∗
+c = λc which is only satisfied by λc = 1. In this language
+the kink creation operator plays the role of a disorder operator [70].
+6.2. The 3D duality: Z2 gauge theory
+We will now discuss the role of duality in the 3D Ising model on a cubic lattice. This case,
+and its generalizations to higher dimensions, was considered first by Franz Wegner [124].
+In section 6.1 we discussed the loop representation of the high temperature expansion and
+showed that it has the form form in all dimensions. Hence, in 3D the high temperature expansion
+is an expansion in closed loops γ with weight of tanh β per unit link of loop. Hence, in 3D as
+well, the high temperature phase can be regarded as field theory of a massive scalar field. As in
+the 2D case, the weight of a loop configuration is tanh β for each link of the loop γ.
+However, the low temperature expansion has a radically different form and physical inter-
+pretation. Much as in 2D, the low temperature expansion is an expansion in clusters of flipped
+spins. Here too, in the absence of an external field, the only energy cost resides at the boundary
+of the clusters of overturned spins, the domain walls. But in 3D the clusters occupy volumes
+whose boundaries are closed surfaces Σ∗. In 3D a link (bond) is dual to a plaquette (expressing
+the fact that in 3D a 1-form is dual to a 2-form). Hence the closed domain walls of overturned
+spins is dual to a closed surface Σ∗ on the dual lattice. The weight of each configuration of closed
+surfaces is exp(−2β) for each plaquette of the surface Σ∗.
+These facts mean that the dual of the 3D Ising model is a theory on the dual lattice with
+coupling constant β∗, with exp(−2β∗) = tanh β, such that its expansion for small β∗ is a sum over
+configurations of closed surfaces σ∗, and for large β∗ is a sum of configurations of closed loops
+γ∗ on the dual lattice. The dual theory is the naturally defined on (dual) plaquettes, not on links.
+To this end, let us define a set of Ising-like degrees of freedom σµ(x) = ±1 located on the links
+(x, µ) of the dual lattice. These degrees of freedom are coupled on each plaquette of the lattice.
+Since each plaquette has four links, the coupling involves the degrees of freedom on all four links
+28
+
+of each plaquette. The partition function of the dual theory is
+Z =
+�
+{σµ(x)}
+exp
+β∗
+�
+plaquettes
+σµ(x)σν(x + eµ)σµ(x + eµ + eν)σν(x)
+
+(107)
+where the sum in th exponent (the negative of the “action”) runs on all the plaquettes of the dual
+lattice.
+The action of the theory of Eq.(107) is invariant under the reversal of all six Ising degrees of
+freedom on links sharing a given site x. This is a local symmetry. Unlike the 3D Ising model,
+which has a global Z2 symmetry of flipping all spins simultaneously, this theory has local (or
+gauge) Z2 symmetry. This is the simplest example of a lattice gauge theory in which the degrees
+of freedom are gauge fields that take values on the Z2 gauge group [124, 32, 71].
+We saw that in the spin model the high temperature expansion (i.e. the expansion in powers of
+tanh β) is a sum over loop configurations which can be interpreted in terms of processes in which
+pairs of particles are created, evolve (in imaginary time) and then are destroyed (also in pairs).
+The analogous interpretation of the expansion of the Z2 gauge theory in powers of tanh β∗ =
+exp(−2β) as a sum over the configurations of closed surfaces is not in terms of the histories of
+particles but in terms of the histories of closed strings which, as they evolve, sweep the closed
+surfaces. The physics is, however, more complex as the sum over surfaces runs over all surfaces
+of arbitrary topology with arbitrary number of handles (or genus). Thus, over (imaginary) time a
+closed strong is created, evolves, splits into two closed strings, etc.
+Therefore, the 3D Ising model, which has a Z2 global symmetry is dual to a Z2 gauge theory
+which has a Z2 local symmetry. The duality maps the high temperature (disordered) phase of
+the Ising model to the strong coupling (small β∗) phase of the gauge theory. In the gauge theory
+language this is the confining phase. This can be seen by computing the expectation value of the
+Wilson loop operator on the closed loop Γ, which here reads ⟨�
+(x,µ)∈Γ σµ(x)⟩. In the small β∗
+phase this expectation value decays exponentially with the size of the minimal surface bounded
+by the loop Γ. This behavior of the area law of the Wilson loop. Under duality the insertion of
+this operator is equivalent to an Ising model with a domain wall terminating on the loop Γ, and
+the area law of the Wilson loop is the consequence of the fact that if the Ising model has long
+range order, the free energy cost of the domain wall scales with its area. Moreover, in this phase
+of the gauge theory the closed strings are small meaning that the string tension (the energy per
+unit length of string) is finite. In the Ising model language, the string tension becomes surface
+tension of the domain wall.
+In the preceding subsection we discussed a Hamiltonian version of the duality. We will briefly
+do the same in the case of the 2+1 dimensional Ising model. The hamiltonian of the Ising model
+in a transverse field on a square lattice is
+H2D−TFIM = −
+�
+r
+σ1(r) − λ
+�
+r, j=1,2
+σ3(r)σ3(r + e j)
+(108)
+where r labels the sites of the square lattice and e j (with j = 1, 2) are the two (orthonormal)
+primitive unit vectors of the lattice. Here too, at each site we have a two-level system (the spins),
+σ1 and σ3 are Pauli matrices acting on these states at each site and λ is the coupling constant.
+Just as in the 1D case, this system has two phases: an ordered phase for λ > λc and a disordered
+phase for λ < λc, where λc is a critical coupling. This Hamiltonian is invariant under the global
+Z2 symmetry generated by the global spin flip operator Q = �
+r σ1(r) which commutes with the
+Hamiltonian, [Q, H2D−TFIM] = 0.
+Let is consider now a Z2 gauge theory on the dual of the square lattice. We will now define
+a two-dimensional Hilbert space on each link, denoted by (˜r, j) (with j = 1, 2) of the dual lattice
+with sites labelled by ˜r. We will further define at each link (˜r, j) the Pauli operators σ1
+j(˜r) and
+σ3
+j(˜r). The Hamiltonian of the Z2 gauge theory is
+HZ2gauge = −
+�
+˜r, j
+σ1
+j(˜r) − g
+�
+˜r
+σ3
+1(˜r)σ3
+2(˜r + ˜e1)σ3
+1(˜r + ˜e1 + ˜e2)σ3
+2(˜r)
+(109)
+where ˜e j = ǫjkek (with j, k = 1, 2).
+Unlike the Hamiltonian of Eq.(108), the Hamiltonian of Eq.(109) has a local symmetry. Let
+˜Q(˜r) be the operator that flips the Z2 gauge degrees of freedom on the four links that share the
+site ˜r,
+˜Q(˜r) =
+�
+j=1,2
+σ1
+j(˜r)σ1
+j(˜r − ˜e j)
+(110)
+29
+
+These operators commute with each other, [ ˜Q(˜r), ˜Q(˜r′)] = 0, and commute with the Hamiltonian,
+[HZ2gauge, ˜Q(˜r)] = 0. The Hilbert space of gauge-invariant states are eigenstates of ˜Q(r) with unit
+eigenvalue,
+˜Q(r)|Phys⟩ = |Phys⟩
+(111)
+(for all r). This constraint is the the Z2 Gauss Law. The Hamiltonian HZ2gauge has two phases: a
+confining phase for g < gc, and a deconfined phase for g > gc (where gc is a critical coupling).
+The duality transformation is defined so that
+σ1(r) =
+�
+plaquette(r)
+σ3
+j(˜r)
+(112)
+is the product of the σ3
+j operators of the gauge theory on a dual plaquette centered at the site r,
+and
+σ1
+1(˜r) = σ3(r)σ3(r + e2),
+σ1
+2(˜r) = σ3(r)σ3(r + e1)
+(113)
+These identities imply that the gauge theory constraint of the Z2 Gauss Law, Eq.(111), is satisfied.
+This also means that duality is a mapping of the gauge-invariant sector of the Z2 gauge theory
+to the Hilbert space of the Ising model in a transverse field.
+Eq.(112) and Eq.(113) imply that the Hamiltonians H2D−TFIM and HZ2gauge are equal to each
+other with the identification of the coupling constants g = 1/λ. Hence, the ordered phase of the
+Ising model, λ > λc, maps onto the confining phase of the Z2 gauge theory and, conversely, the
+disordered phase of the Ising model maps onto the deconfined phase of the gauge theory.
+Eq.(113) also implies that the operator σ3(r) of the Ising model can be identified with the
+operator
+σ3(r) =
+�
+γ(r)
+σ1
+j(˜r)
+(114)
+where the product is on the links of the dual lattice pierced by the path γ(r) on the direct lattice
+ending at the site r.
+While σ3(r) is of course just the order parameter of the Ising model, the dual operator defined
+by Eq.(114) anti-commutes with the plaquette term of the Hamiltonian of Eq.(109) and creates an
+Z2 flux excitation at the plaquette. With some abuse of language, this operator can be regarded as
+creating a Z2 “magnetic monopole”. Since in the confining phase this operator has an expectation
+value, we can regard this phase as a magnetic condensate.
+We will now discuss the role of boundary conditions which is different in both theories. Let
+us examine the behavior of the Z2 gauge theory with periodic boundary conditions. Periodic
+boundary conditions means that the 2D space is topologically a 2-torus. It is straightforward to
+show that the ground state in the confining phase is unique and insensitive to boundary condi-
+tions. The physics of the deconfined phase is more subtle. We will now see that on a 2-torus
+it has a four-fold degenerate ground state and that the degeneracy is not due to the spontaneous
+breaking of a global symmetry.
+Let γ1 and γ2 be two non-contractible loops on the torus along the directions 1 and 2 re-
+spectively. Let us consider the (“electric”) Wilson loop operators along γ1 and γ2, W[γ1] =
+�
+(r, j)∈γ1 σ3
+j(r) and similarly for γ2. Similarly let us consider the (“magnetic”) ’t Hooft operators
+˜W[˜γ1] and ˜W[γ2] defined on the non-contractible closed paths of the dual lattice ˜γ1 and ˜γ2, such
+that ˜W[˜γ1] = �
+˜γ1 σ1
+2(˜r) and ˜W[˜γ2] = �
+˜γ2 σ1
+1(˜r). It is easy to see that the Wilson and’t Hooft
+operators satisfy that W[γi]2 = ˜W[˜γ j]2 = I, and that
+[W[γi], W[γ j] = [ ˜W[˜γi], ˜W[˜γ j]] = 0,
+{W[γ1], ˜W[˜γ2]} = {W[γ2], ˜W[˜γ1]} = 0
+(115)
+Let us consider the special limit of g → ∞. In this limit the Wilson loop operators W[γi] on
+the two non-contractibleloops of the 2-torus commute with the Hamiltonian HZ2gauge of Eq.(109).
+Therefore the eigenstates of the Hamiltonian can be chosen to be the eigenstates of these two Wil-
+son loops. Since the Wilson loop operators are hermitian and obey W[γi]2 = I, the spectrum of
+each loop is two-dimensional | ± 1⟩i (i = 1, 2) with eigenvalues ±1, respectively. At g → ∞ these
+states are also eigenstates of HZ2gauge. Thus, the ground state of HZ2gauge is four dimensional. The
+’t Hooft loops ˜W[˜γi] act as ladder operators in this restricted Hilbert space.
+The conclusion is that, on a 2-torus, at least in the limit g → ∞, the ground state of HZ2gauge
+is degenerate. However, this degeneracy is not due to the spontaneous breaking of a global
+symmetry. Rather, it reflects the topological character of the theory. To see this, one can extend
+this analysis to a theory to be on a more general two-dimensional and instead of a 2-torus we can
+30
+
+consider a surface with g handles (not to be confused with the coupling constant!), e.g. g = 0 for
+a sphere (or disk), g = 1 for a 2-torus, g = 2 for a pretzel, etc. For each of these two-dimensional
+surfaces the number different number of non-contractible loops is g, and the Wilson loops defined
+on them commute with each other (and with HZ2gauge). Hence, in the limit g → ∞ HZ2gauge has
+a ground state degeneracy of 2g which, clearly, depends only on the topology of the surface. We
+conclude that, at least as g → ∞, the Hamiltonian HZ2gauge is in a topological phase.
+However, is the exact degeneracy found in the limit g → ∞ a property of the entire deconfined
+phase? For a finite system the expansion in powers of 1/g is convergent. On the other hand, in
+the thermodynamic limit, L → ∞, the expansion has a finite radius of convergence with gc being
+an upper bound. Let us consider the ground state at g = ∞ with eigenvalue +1 for the Wilson
+loop W[γ1]. The degenerate state with eigenvalue −1 is created by the ’t Hooft loop ˜W[˜γ2], i.e.
+| − 1⟩1 = ˜W[˜γ2] | + 1⟩1. The ’t Hooft operator is a product of σ1 operators on links along the
+direction 1 crossed by the path ˜γ2 of the dual lattice, which involves (at least) L links. Then, it
+takes L orders in the expansion in powers of 1/g to mix the state | + 1⟩1 with the state | − 1⟩1,
+and this amplitude is of order 1/gL, which is exponentially small. The same argument applies to
+the mixing between all four states. For a finite but large system of linear size L, the degeneracy
+is lifted but the energy splitting is exponentially small in the system size. Hence, the topological
+protection is a feature of the deconfined phase in the thermodynamic limit L → ∞.
+We conclude that the deconfined phase of the Z2 lattice gauge theory is a topological phase.
+This result extends to the case of a gauge theory with discrete gauge group Z, the cyclic group
+of k elements. In general spacetime dimension D > 2 the Zk gauge theory also has a confined
+and a deconfined phase and if k ≳ 4 it also has an intermediate “Coulomb phase”[125, 126]. In
+section 9.4 we will see that the low-energy (IR) regime of the deconfined phase is described by
+a topological quantum field theory known as the level k BF theory.
+7. Bosonization
+The behavior of one-dimensional electronic systems is of great interest in Condensed Matter
+Physics for many reasons. One is that, even for infinitesimally weak interactions, these one-
+dimensional metals violate the basic principles of the Landau Theory of the Fermi Liquid [13].
+A central assumption of Femi Liquid theory is that at low energies the excitation energy of the
+fermion (electron) quasiparticle is always much larger than its width. Hence, at asymptotically
+low energies the quasiparticle excitations become increasingly sharp.
+A manifestation of this feature is that the quasiparticle propagator (the “Green function”) has
+a pole on the real frequency axis with a finite residue Z. In one-dimensional metals this assump-
+tion always fails since the residue Z vanishes, the Fermi field acquires a non-trivial anomalous
+dimension, and the pole is replaced by a branch cut. To this date this is the best example of what
+is called a ”non-Fermi liquid”. In one-dimensional metals this non-Fermi liquid is often called a
+“Luttinger Liquid” [127]. A detailed analysis can be found in chapter 6 of Ref. [9], and in other
+books.
+Bosonization provides for a powerful tool to understand the physics of these non-trivial sys-
+tems. Bosonization is a duality between a massless Dirac field in 1+1 dimensions and a (also
+massless) relativistic Bose (scalar) field [128, 129, 130, 131]. In this context, bosonization is a set
+of operator identities relating observables between two different (dual) continuum field theories.
+These identities have a close resemblance to the Jordan-Wigner transformation which relates op-
+erators of a theory of spinless (Dirac) fermions on a one-dimensional lattice to a theory of bosons
+with hard cores (i.e. spins) on the same lattice [22].
+7.1. Dirac fermions in one space dimensions
+To understand how these operator identities come about and what these fields mean in the
+Condensed Matter context we will consider the simple problem of a system of non-interacting
+spinless fermions c(n) on a one-dimensional chain of length L (with L → ∞), for simplicity with
+periodic boundary conditions. The Hamiltonian is
+H0 = −t
+L
+�
+n=1
+c(n)†c(n + 1) + h.c.
+(116)
+In momentum space, and in the thermodynamic limit L → ∞, the Hamiltonian becomes
+H0 =
+� π
+−π
+dp
+2π (ε0(p) − µ) c†(p)c(p)
+(117)
+31
+
+where −π ≤ p ≤ π, µ is the chemical potential (which fixes the fermion number) and ε0(p) is the
+(free) quasiparticle energy which, in this case, is
+ε0(p) = −2t cos p
+(118)
+This simple system has a global internal symmetry
+c′(n) = eiαc(n),
+c′(n)† = e−iαc(c)
+(119)
+where α is a constant phase (with period 2π). This global symmetry reflects the conservation of
+fermion number
+NF =
+�
+n
+c†(n)c(n)
+(120)
+The free fermion Hamiltonian of Eq.(116) is also invariant under lattice translations.
+The ground state of this system is obtained by occupying all single particle states with energy
+below the chemical potential, E ≤ µ ≡ EF, which defines the Fermi energy EF. In what follows
+we will redefine the zero of the energy at the Fermi energy. In the thermodynamic limit the
+one-particle states are labeled by momenta defined in the first Brillouin zone [−π, π). The ground
+state |G⟩ of this system is obtained by filling up all single-particle states with momentum p in
+the range [−pF, pF), where pF is the Fermi momentum. Hence, the occupied states have energy
+ε0(p) ≤ EF. The ground state is
+|G⟩ =
+�
+|p|≤pF
+c†(p)|0⟩
+(121)
+and is called the filled Fermi sea. We will further assume that the fermionic system is dense in
+the sense that the occupied states are a finite fraction of the available states, i.e. we assume that
+|EF| is a finite fraction of the bandwidth W = 4t.
+The single-particle excitations have low energy if |ε0(p) − EF| ≪ |EF|. This range can only
+be accessed by quasiparticles with momenta p close to ±pF. For states wit p close to pF we can
+a linearized approximation and write
+ε0(p) ≃ �F(p − pF) + . . .
+(122)
+and similarly for the states near −pF. Here �F = ∂ǫ0
+∂p
+���pF = 2t sin pF is the Fermi velocity. Let q
+being the momentum measured from pF (or −pF in the other case), This approximation is correct
+in a range of momenta −Λ ≤ q ≤ Λ, where 2π
+L ≪ Λ ≪ π. In this regime we can approximate
+the lattice fields c(n) with two continuum right moving ψR(x) and left moving ψL(x) Fermi fields,
+such that (with x = na0)
+c(n) ∼ eipF xψR(x) + e−ipF xψL(x)
+(123)
+in terms of which the Hamiltonian takes the continuum form
+Hcontinuum =
+�
+dx
+�
+ψ†
+R(x)(−i�F)∂xψR(x) − ψ†
+L(x)(−i�F)∂xψL(x)
+�
+=
+� dq
+2πq�F
+�
+ψ†
+R(q)ψR(q) + ψ†
+R(q)ψR(q)
+�
+(124)
+In what follows I will rescale time and space in such a way as to set �F = 1.
+Eq.(124) is the Hamiltonian of a massless Dirac field in 1+1 dimensions. It describes the
+effective low energy behavior of the excitations of the fermionic system defined on a lattice by
+the Hamiltonian of Eq.(116). Here low energy means asymptotically close to the Fermi energy
+(which we have set to zero) and momenta close to ±pF. We will see that this effective relativistic
+field theory gives a complete description of the universal low energy physics encoded in the
+microscopic model of Eq.(116) up to some subtle issues associated with what are called quantum
+anomalies.
+Let us denote by ψ(x) the bi-spinor field ψ(x) = (ψR(x), ψL(x)) and the 2 × 2 Dirac gamma-
+matrices in terms of the three Pauli matrices are
+γ0 = σ1,
+γ1 = −iσ2,
+γ5 = σ3
+(125)
+which obey the Dirac algebra
+{γµ, γν} = 2gµν
+(126)
+32
+
+where gµν is the metric tensor in 1+1-dimensional Minkowski spacetime
+gµν =
+�1
+0
+0
+−1
+�
+(127)
+Using the standard notation ¯ψ(x) = ψ†(x)γ0 (where I left the spinor index α = 1, 2 implicit)
+the Lagrangian density of the free massless Dirac fermion is
+L = ¯ψi/∂ψ
+(128)
+where we have used the Feynman slash notation which denotes the contraction of a vector field,
+say Aµ with the Dirac gamma matrices with a slash, Aµγµ ≡ /A.
+Formally, the Dirac lagrangian of Eq.(128) (and the Dirac Hamiltonian of Eq.(124)) are in-
+variant under two separate global transformations. It has a global U(1) (gauge) symmetry trans-
+formation
+ψ′(x) = eiθψ(x)
+(129)
+(where θ is constant) under which the two components of the bi-spinor ψ transform in the same
+way
+ψ′
+R(x) = eiθψR(x),
+ψ′
+L(x) = eiθψL(x)
+(130)
+The massless Dirac theory is also invariant under a global gauge U(1) chiral transformation
+ψ′(x) = eiθγ5ψ(x)
+(131)
+which in components it reads
+ψ′
+R(x) = eiθψR(x),
+ψ′
+L(x) = e−iθψL(x)
+(132)
+In general, if a system has a global continuous symmetry it should have a locally conserved
+current and a globally conserved charge. This is the content of Noether’s Theorem. Since the
+massless Dirac theory has these two global symmetries one would expect that it should have two
+separately conserved currents.
+The global U(1) symmetry has an associated current jµ = (j0, j1) given by
+jµ = ¯ψγµψ
+(133)
+which is invariant under the global U(1) symmetry. In terms of the right and left moving Dirac
+fields, ψR and ψL, the components of the current are
+j0 = ψ†
+RψR + ψ†
+LψL,
+j1 = ψ†
+RψR − ψ†
+LψL
+(134)
+This current is locally conserved and satisfies the continuity equation
+∂µ jµ = 0
+(135)
+It has an associated conserved global charge, which we will call fermion number
+Q =
+� ∞
+−∞
+dxj0(x) =
+� ∞
+−∞
+dx
+�
+ψ†
+RψR + ψ†
+LψL
+�
+(136)
+This is the continuum version of the conservation of fermion number NF of the free fermion
+lattice model discussed above.
+Since this current is conserved it can be coupled to an electromagnetic field through a term
+in the Lagrangian
+Lint = −ejµAµ
+(137)
+where Aµ is the electromagnetic vector potential, which in 1+1 dimensions has only two com-
+ponents, Aµ = (A0, A1). Here e is a coupling constant which is interpreted as the electric charge.
+This coupling amounts to making the U(1) symmetry a local gauge symmetry under which the
+fields transform as follows
+ψ′(x) = eiθ(x)ψ(x),
+A′
+µ(x) = Aµ(x) + 1
+e∂µθ(x)
+(138)
+33
+
+Now the conservation of fermion number becomes the conservation of the total electric charge
+Qe = −eQ
+(139)
+In a continuum non-relativistic model the Fermi field is ψ(x) (ignoring spin), e.g. an electron
+gas in one dimension such as a quantum wire, the analog of decomposition shown in Eq.(123) of
+the electron field ψ(x) becomes
+ψ(x) = eipF xψR(x) + e−ipF xψL(x)
+(140)
+The electron field ψ(x) transforms as ψ′(x) = eiαψ(x) under the global U(1) gauge symmetry of
+Eq.(130). In particular, the local density operator ρ(x) = ψ†(x)ψ(x) is invariant under this sym-
+metry. Under both decompositions of Eqs. (123) and (140), the local density operator becomes
+ρ(x) = ψ†
+R(x)ψR(x) + ψ†
+L(x)ψL(x) + ei2pF xψ†
+R(x)ψL(x) + e−i2pF xψ†
+L(x)ψR(x)
+(141)
+Clearly the non-relativistic density operator ρ(x) is invariant under the global U(1) gauge sym-
+metry. The same considerations applies to the lattice version of the density operator (the local
+occupation number).
+Thus, the density operator ρ(x) can be decomposed into a slowly varying part (the first two
+terms in Eq.(141)) and the last two terms which oscillate with wave vectors Q = ±2pF. These
+observations imply that we can express ρ(x) in the form
+ρ(x) = ¯ρ + j0(x) + eiQxψQ(x) + e−iQxψ−Q(x)
+(142)
+This decomposition can be interpreted as a Fourier expansion of the density operator in term of
+newly defined slowly-varying fields. In Eq.(142) Q = 2pF and ¯ρ is the average density, where we
+assumed that the Dirac density operator j0(x) has vanishing expectation value (i.e. it is normal-
+ordered). In Eq.(142) we defined the (bosonic) operators
+ψQ(x) = ψ†
+R(x)ψL(x),
+ψ−Q(x) = ψ†
+L(x)ψR(x)
+(143)
+which characterize the oscillatory component of the density. The operators ψ±Q(x) are the order
+parameters of a charge density wave state in one dimension and Q is the ordering wavevector.
+Repulsive interactions between the electrons can cause scattering process between the right
+and left moving components to become relevant (in the Renormalization Group sense) leading
+to the spontaneous breaking of translation invariance. The resulting state is known as a charge-
+density-wave (CDW). In this state the operators ψ±Q(x) (or a linear combination of them) acquire
+a non-vanishing expectation value, and the expectation value of the density operator ρ(x) has a
+(static) modulated component, and the Dirac Hamiltonian density becomes
+H = ψ†
+R(x)(−i)∂xψR(x) − ψ†
+L(x)(−i)∂xψL(x) + m
+�
+ψ†
+R(x)ψL(x) + ψ†
+L(x)ψR(x)
+�
+(144)
+where m is the Dirac mass. The operator in the second term of Eq.(144) mixes the right and left
+moving components of the Dirac spinor. This hermitian operator is known as the Dirac mass
+term. In relativistic notation this operator is written
+¯ψψ = ψ†
+R(x)ψL(x) + ψ†
+L(x)ψR(x)
+(145)
+When m � 0, i.e. when ⟨ ¯ψψ⟩ � 0, the fermionic spectrum has a mass gap, and the electronic
+states with momenta ±pF have an energy gap. Alternatively we could have considered a state
+with the in which the (hermitian) operator that has an expectation value is
+i ¯ψγ5ψ = i
+�
+ψ†
+R(x)ψL(x) − ψ†
+L(x)ψR(x)
+�
+(146)
+which is known as the γ5 mass term. In a more general CDW state both mass terms can be
+present.
+7.2. Chiral symmetry and chiral symmetry breaking
+The CDW states we introduced have different symmetries. To see this let us observe that the
+operators ψ±Q(x) transform non-trivially under a global U(1) chiral transformation
+ψ′
+±Q(x) = e∓2iθψ±Q(x)
+(147)
+34
+
+Upon substituting this transformation in the expansion of the density ρ(x) of Eq.(142), we see
+that a chiral transformation is equivalent to a uniform displacement of the density operator by
+θ/pF,
+ρ(x) → ρ (x + θ/pF)
+(148)
+Under a chiral transformation with arbitrary angle θ, the Dirac and γ5 mass term operators trans-
+form as an orthogonal transformation of a two-component vector. In particular for a chiral trans-
+formation with θ = π/4,
+¯ψψ �→ i ¯ψγ5ψ,
+i ¯ψγ5ψ �→ − ¯ψψ
+(149)
+which is equivalent to a displacement of the density by 1/4 of the period of the CDW. In this
+sense these two states are equivalent, as is the state with a more general linear combination.
+The microscopic lattice model (and the continuum non-relativistic model) are invariant under
+the spatial inversion symmetry x ↔ −x. This also implies a symmetry under the exchange of
+right and left moving components of the Dirac spinor, ψR ↔ ψL. In the massless Dirac theory
+this operation is equivalent to the multiplication of the spinor by the Pauli matrix σ1. In the
+massless theory the multiplication by σ2 (followed by a chiral transformation with θ = π/2) has
+the same effect.
+These symmetries have an important effect on the fermionic spectrum. To understand what
+they do let us consider the one-particle Dirac Hamiltonian with a Dirac mass m (jn momentum
+space)
+h =
+�p
+m
+m
+−p
+�
+(150)
+This operator anti-commutes with the Pauli matrix σ2, {h, σ2} = 0. Let |E⟩ be an eigenstate of
+the Hamiltonian h with energy eigenvalue E =
+�
+p2 + m2. Let us consider the state σ2|E⟩. It is
+also an eigenstate of h but with energy −E, i.e.
+hσ2|E⟩ = −σ2h|E⟩ = −E|E⟩,
+⇒ σ2|E⟩ = | − E⟩
+(151)
+Hence if |E⟩ is an eigenstate of energy E, then σ2|E⟩ is an eigenstate of energy −E. This means
+that the spectrum is invariant under charge conjugation symmetry. Notice that under this opera-
+tion the spinor transforms as
+σ2
+�ψR
+ψL
+�
+=
+�−iψL
+iψR
+�
+= e−iγ5π/2
+�ψL
+ψR
+�
+(152)
+In other words, this theory is invariant under charge conjugation C and parity P which, combined,
+it implies that it is invariant under time-reversal T .
+The same consideration applies in the case of a γ5 mass term in which case the one-particle
+Dirac Hamiltonian now is
+h =
+� p
+−im5
+im5
+−p
+�
+(153)
+This Hamiltonian now anti-commutes with the Pauli matrix σ1 which also implies that the same
+charge conjugation symmetry C, |E⟩ ↔ |−E⟩, is present in the spectrum of the case of a γ5 mass.
+We can also repeat the argument on parity invariance P, which is now multiplication by σ1, Thus
+the theory is invariant under time-reversal T .
+However, if the theory has both a Dirac mass m and a γ5 mass m5 these symmetries are
+broken. Indeed, the one-particle Dirac Hamiltonian now is
+h =
+�
+p
+m − im5
+m + im5
+−p
+�
+= pσ3 + mσ1 + m5σ2
+(154)
+which no longer has a spectral symmetry. In this case CP is broken and, hence, so it T since
+CPT remains unbroken (as it should).
+In a lattice system this is a symmetry transformation only if θ = πn/pF is a lattice displace-
+ment, which restricts the allowed values of the chiral angle to be discrete. Although this is true
+interactions play a significant role in the actual behavior. In fact, there are physical situations
+in which an effective continuous symmetry actually emerges in this he infrared (long-distance)
+limit. This is what happens when the CDW is incommensurate and, as we will see in the next
+subsection, it slides under the action of an electric field.
+In the case of a half-filled system (with only nearest-neighbor hopping matrix elements) the
+Fermi wave vectors are ±π/2. In this case the allowed discrete chiral transformation has a chiral
+35
+
+angle π/2 under which ¯ψψ �→ − ¯ψψ (and similarly for i¯γ5ψ) corresponding by a translation by one
+lattice spacing. The allowed four fermion operator is ( ¯ψψ)2. If the lattice fermions are spinless
+this operator reduces to
+( ¯ψψ)2 = −2 jR(x)jL(x) + lim
+y→x ψ†
+R(x)ψ†
+L(x)ψL(y)ψR(y) + h.c.
+(155)
+where introduced the right and left moving (chiral) components of the current operator
+jR(x) = 1
+2(j0(x) + j1(x)) = ψ†
+R(x)ψR(x),
+jL(x) = 1
+2(j0(x) − j1(x)) = ψ†
+L(x)ψL(x)
+(156)
+The first term in Eq.(155) is known as the backscattering interaction and has scaling dimension
+2. Hence, it is a marginal operator. The theory with only the first term is known in Condensed
+Matter Physics as the Luttinger model and in High-Energy Physics at the (massless) Thirring
+model. The second operator in Eq.(155) formally violates momentum conservation as its total
+momentum is 4pF = 2π which is a reciprocal lattice vector and, as such, it is equivalent to
+zero (mod 2π). Such an operator is not formally allowed in a (naive) continuum theory. This
+operator is due to a lattice Umklapp process and breaks the formal continuous chiral symmetry
+to a discrete Z2 subgroup. Although the naive scaling dimension of this operator is 2 (and hence
+it is formally marginal). If the fermions are spinless, the leading operator actually vanishes and
+the leading non-vanishing operator actually has dimension 4, which is irrelevant. However, as
+shown above, backscattering processes of the form jR(x) jL(x) are part of this operator and are
+exactly marginal.
+If the interaction is strong enough the backscattering interaction can make the Umklapp op-
+erator relevant. When this happens the fermionic system has a quantum phase transition to an
+insulating state with a period 2 (commensurate) CDW state. On the other hand, for spin 1/2
+fermions the operator ( ¯ψψ)2 is allowed and is marginally relevant. If the interactions are re-
+pulsive the resulting state is an antiferromagnetic N´eel state at quantum criticality, while for
+attractive interactions it is a period 2 CDW. This is what happens in the 1D Hubbard model (see,
+e.g. Ref. [9] for a detailed analysis).
+There are two theories in relativistic systems which are closely related to this problem. One
+if the Gross-Neveu model [132] which is a theory of N species of massless Dirac spinors with
+Lagrangian density
+LGN = ¯ψai/∂ψa + g( ¯ψaψa)2
+(157)
+with a = 1, . . ., N (summation of repeated induces is implies). The spin-1/2 Hubbard model
+corresponds to the case N = 2. This Lagrangian is invariant only under the discrete chiral
+symmetry ¯ψaψa �→ − ¯ψaψa. This is a discrete, Z2, symmetry and as such it can be spontaneously
+broken in 1+1 dimensions. For N ≥ 2, the resulting state has a (dynamically) broken Z2 chiral
+symmetry and that there is a chiral condensate ⟨ ¯ψaψa⟩ � 0 corresponding to a period 2 CDW.
+The other theory is known as the chiral Gross-Neveu model whose Lagrangian density is
+LcGN = ¯ψai/∂ψa + g
+�
+( ¯ψaψa)2 − ( ¯ψaγ5ψa)2�
+(158)
+which has the full continuous chiral symmetry. For N = 1 this theory is equivalent to the Lut-
+tinger model (and to the Gross-Neveu model if we ignore the umklapp term). For N ≥ 2 the chi-
+ral symmetry is formally broken. If this were true this theory would violate the Mermin-Wagner
+theorem. However a detailed study (most easily done using bosonization methods) shows that
+instead of long range order the correlator of both mass terms decay as a power law as a function
+of distance, consistent with the requirements by this theorem.
+We close this subsection by noting that if the lattice model is not half filled but its density
+is either incommensurate or has a higher degree of commensurability, say p/q, the chiral sym-
+metry is actually continuous (in the incommensurate case) or effectively continuous since the
+requisite umklapp terms are strongly irrelevant. However, if the lattice model is not at half filling
+charge conjugation symmetry C is broken at the lattice scale (in the UV), where it is equivalent
+to particle-hole symmetry, but it is recovered in the low-energy, IR, regime (up to irrelevant op-
+erators). In this sense, both the continuous chiral symmetry and charge conjugation symmetry
+can be regarded as emergent IR symmetries
+7.3. The chiral anomaly
+In section 7.2 we showed that the theory of massless Dirac fermions, in addition to a global
+U(1) gauge gauge symmetry, has a second conservation law which we called a global U(1) chiral
+36
+
+symmetry, shown in Eq.(131). This symmetry implies that there is a locally associated chiral
+current j5
+µ, given by
+j5
+µ(x) = ¯ψ(x)γµγ5ψ(x)
+(159)
+which also satisfies a continuity equation
+∂µ j5
+µ = 0
+(160)
+and there is a globally conserved chiral charge Q5
+Q5 =
+� ∞
+−∞
+dx j5
+0(x) =
+� ∞
+−∞
+dx
+�
+ψ†
+RψR − ψ†
+LψL
+�
+(161)
+It is easy to check that the Dirac current jµ and the chiral current j5
+µ are related by
+j5
+µ = ǫµν jµ
+(162)
+where ǫµν is the second rank Levi-Civita tensor.
+The simultaneous conservation of both currents jµ and j5
+µ in the massless Dirac theory implies
+that the right and left moving densities jR and jL, defined in Eq.(156), should be separately
+conserved. In fact, if the Dirac theory has a mass term
+L = ¯ψi/∂ψ − m ¯ψψ
+(163)
+it is straightforward to show that
+∂µ j5
+µ = 2mi ¯ψγ5ψ
+(164)
+which means that in the massive theory the axial current is not conserved. This is easy to under-
+stand since the mass term mixes the right and left moving components of the Dirac spinor and,
+hence, the right and left moving densities are not conserved.
+What happens in the massless limit, m → 0, is more subtle. This problem was investigated
+in the late 1960’s in 3+1 dimensions by S. Adler [133] and by J. S. Bell and R. Jackiw [134]
+who were interested in the anomalous decay of a neutral pion into two photons, π0 → 2γ.
+This process appears at third order of perturbation theory and it involves the computation of a
+triangle diagram (a fermion loop). In 3+1 dimensions this process has a UV divergence which
+needed to be regulated. These authors showed that it is not possible to find a regularization
+in which both the Dirac (gauge) current jµ = ¯ψγµψ and the axial current j5
+µ = ¯ψγµγ5ψ are
+conserved. In other words, if gauge invariance is preserved then the axial current is not and has
+an anomaly and results in a non-conservation of the axial current, ∂µ j5
+µ � 0. On the other hand,
+at least in the case of the physical gauge-invariant regularization, the obtained expression for
+the anomaly in the axial current is universal, independent of the value of the UV regulator (the
+cutoff). Sometime later G. ’t Hooft showed that in non-abelian gauge theories instant processes
+also lead to anomalies and, furthermore, the result was also universal [96, 111]. Since the result
+is universal and, hence, independent of the UV scale, this led to the concept of anomaly matching
+conditions.
+We will examine this problem in 1+1 dimensions (although it plays a key role in the theory of
+topological insulators in three space dimensions [135]). As we noted above, the lattice model is
+gauge-invariant and has only one conserved current. The conserved axial current appeared only
+in the low-energy regime in which the lattice model is described by a theory of massless Dirac
+fermions. To understand this problem we will consider the theory of fermions in 1+1 dimen-
+sions coupled to a U(1) gauge field field. In the presence of a background (i.e. not quantized)
+electromagnetic field in the A0 = 0 gauge the free fermion Hamiltonian of Eq.(116) becomes
+H[A] = −t
+L
+�
+n=1
+c†(n)eieA(n,t)/ℏc(n) + h.c.
+(165)
+An uniform and constant electric field is E is represented by a vector potential A = −cEt (with
+c being the speed of light). The net effect of this gauge field is to shift of the momentum of the
+fermion quasiparticles p → p + ecEt/ℏ or, what is the same, to displace the fermion dispersion
+relation in momentum by the ecEt/ℏ. This means that the Fermi points are also shifted by that
+amount, pF → pF + ecEt/ℏ and −pF → −pF + ecEt/ℏ. This means that the single particle states
+between pF and pF +ecEt/ℏ that were empty for E = 0 are now occupied and the states between
+37
+
+−pF and −pF + ecEt/ℏ that were occupied for E = 0 are now empty. This means that number
+of right-moving fermions is increasing at a rate of ecE/ℏ and that the number of left-moving
+fermions decreases at the same rate. This results in a net current. Throughout this process the
+total number of fermions is not changed, gauge invariance is satisfied, but the number of right
+and left moving fermions are not separately concerned. Notice that this is an effect that involved
+the entire Femi sea but the net effect is at low energies.
+Let us see now how this plays out in the effective Dirac theory. The massless Dirac La-
+grangian density in the background of an unquantized electromagnetic field Aµ is
+L = ¯ψ(i/∂ − e /A)ψ
+(166)
+Since there is no mass term the Dirac equation still decouples into two equations, for the right
+and left moving components of the Dirac spinor. In the A0 = 0 gauge they are
+i∂0ψR = (−i∂1 − A1)ψR,
+i∂0ψL = (i∂1 − A1)ψL
+(167)
+In the temporal gauge, A0 = 0, a uniform electric field E = ∂0A1, and A1 increases linearly with
+time. As A1 increases, the Fermi momentum pF (which is equal to the Fermi energy EF) also
+increases at the rate eE. The density of states of a system of length L is L/(2π). So, the rate of
+change of the number of right-moving fermions is
+dNR
+dx0
+= e
+2πE
+(168)
+where we defined NR =
+� L
+0 dx jR and similarly for NL. If the UV regulator of the theory is
+compatible with gauge invariance, then the total fermion number must be conserved and the total
+vacuum charge must remain equal to zero,
+Q =
+� L
+0
+dx j0(x) = NR + NL = 0
+(169)
+Thus, if NR increases, then NL must decrease by the same amount. Or, equivalently, the electric
+field E creates an equal number of particles NR and of antiparticles ¯NL = −NL.
+On the other hand, the chiral charge Q5 = NR − NL must increase at the rate
+dQ5
+dx0
+= dNR
+dx0
++ d ¯NL
+dx0
+= e
+π E
+(170)
+Again, the details of the UV regularization do not matter, only that it is gauge-invariant. We can
+also interpret Eq.(170) as the rate of particle-antiparticle pair creation by an electric field.
+In a relativistic notation these results are expressed as
+∂µ j5
+µ = e
+2πǫµνFµν
+(171)
+Hence, the formally conserved current j5
+µ has an anomaly and is not conserved due to quantum
+effects. Since it is not conserved, we cannot gauge the chiral symmetry. In the next subsection
+we will see that the the anomaly is closely related with bosonization.
+7.4. Bosonization, anomalies and duality
+We will reexamine the problem at hand from the point of view of the fermionic currents of
+the Dirac theory jµ as operators. Since the currents obey the continuity equation, ∂µ jµ = 0 one
+expects that this would imply that it may be possible to write them as a curl of a scalar field, i.e.
+jµ(x) = ǫµν∂νφ(x) where φ(x) should be a scalar field. Since this should be an operator identity
+we will need to understand how the currents act on the physical Hilbert space.
+The Fermi-Bose mapping in 1D systems is closely related to the chiral anomaly we just
+discussed. Bosonization of a system of 1+1 dimensional massless Dirac fermions is a set of
+operator identities understood as matrix elements of the observables in the physical Hilbert space.
+These identities were first derived by Daniel Mattis and Elliott Lieb [128], based on earlier work
+by Julian Schwinger [136]. These identities were rediscovered (and their scope greatly expanded)
+by Alan Luther and Victor Emery [129], by Sidney Coleman [131], by Stanley Mandelstam
+[130], and by Edward Witten [137]. A non-abelian version of bosonization was subsequently
+derived by Witten [119].
+38
+
+The physical Hilbert space is defined as follows. Let |FS⟩ ≡ |0⟩ denote the filled Fermi
+sea. In what follows we will assume that the physical system is macroscopically large abd that
+local operators create the physical states by acting finitely on the filled Fermi sea. Physical
+observables, such as the right and left moving densities jR(x) and jL(x), need to be normal-
+ordered with respect to the physical vacuum state, the filled Fermi (Dirac) sea |0⟩. The normal
+ordered densities are : jR(x) : ≡ jR(x) − ⟨0|jR(x)|0⟩ and : jL(x) : ≡ jL(x) − ⟨0|jR(x)|0⟩. Since
+the densities are products of fermion operators they need to be defined as a limit in which the
+operators are separated by a short distance η. Crucial to this construction is that the computation
+the expectation values be regularized in such a way that the charge (gauge) current jµ(x) is locally
+conserved and satisfies the continuity equation ∂µ jµ = 0 as an operator identity. In what follows
+all expectation values will refer to the filled Fermi sea state |0⟩.
+The propagators of the right and left moving Fermi fields are given by
+⟨ψ†
+R(x0, x1)ψR(0, 0)⟩ =
+−i
+2π(x0 − x1 + iǫ),
+⟨ψ†
+L(x0, x1)ψL(0, 0)⟩ =
+i
+2π(x0 + x1 + iǫ)
+(172)
+The expectation value of the currents at a space location x1 are
+⟨jR(x1)⟩ = lim
+η→0⟨ψ†
+R(x1 + η)ψR(x1 − η)⟩ =
+i
+4πη,
+⟨jL(x1)⟩ = lim
+η→0⟨ψ†
+L(x1 + η)ψL(x1 − η)⟩ = −i
+4πη
+(173)
+which are divergent at short distances. It follows that the normal-ordered right and left moving
+current densities satisfy the equal-time commutation relations
+[jR(x1), jR(x′
+1)] = − i
+2π∂1δ(x1 − x′
+1),
+[jL(x1), jL(x′
+1)] = + i
+2π∂1δ(x1 − x′
+1)
+(174)
+These identities imply that the normal-ordered space-time components of the current jµ = (j0, j1)
+satisfy the equal-time commutation relations
+[j0(x1), j1(x′
+1)] = − i
+π∂1δ(x1 − x′
+1),
+[j0(x1), j0(x′
+1)] = [ji(x1), j1(x′
+1)] = 0
+(175)
+The non-vanishing right-hand sides of these commutators are known as Schwinger terms. These
+identities define the the U(1) (Kac-Moody) current algebra.
+We should note that in a theory of non-relativistic Fermi fields ψ(x, t) in all dimensions, the
+charge density ρ(x) and the current operators j(x) =
+1
+2i
+�
+ψ†(x)▽ψ(x) − ▽ψ†(x)▽ψ(x)
+�
+satisfy a
+similar expression (also at equal times)
+[ρ(x), jk(x′)] = −i e2
+mc2 ∂k
+�δ(x − x′)ρ(x)� ,
+[ρ(x), ρ(x′)] = 0,
+[jk(x), jl(x′)] = 0
+(176)
+In one dimension, and in the regime in which the fermions have a macroscopic density so that
+ρ(x) ≃ ⟨ρ(x)⟩, after a multiplicative rescaling of the operators, the non-relativistic identities of
+Eq.(176) are equivalent to the U(1) current algebra of Eq.(175).
+The U(1) current algebra of Eq,(175) is reminiscent of the equal-time canonical commutation
+relations of a scalar field. Indeed, if φ(x) is a scalar field and Π(x) = ∂0φ(x) is its canonically
+conjugate momentum, then they obey the equal-time canonical commutation relations
+[φ(x1), Π(x′
+1)] = iδ(x1 − x′
+1)
+(177)
+We can then identify the charge density j0 and the current density j1 with the scalar field operators
+j0(x) =
+1√π∂1φ(x),
+j1(x) = − 1√πΠ(x) = − 1√π∂0φ(x)
+(178)
+which obey the U(1) current algebra of Eq.(175) as a consequence of the canonical commutation
+relations, Eq. (177). Furthermore, we can rewrite Eq.(177) in the Lorentz covariant form
+jµ(x) =
+1√πǫµν∂νφ(x)
+(179)
+which is clearly consistent with the local conservation of the current jµ,
+∂µ jµ(x) = 0
+(180)
+39
+
+Let us examine now the question of the conservation of the chiral current j5
+µ. In Eq.(162)
+we showed that the gauge current and the chiral current are related by j5
+µ = ǫµν jν. Therefore the
+divergence of the chiral current is
+∂µ j5
+µ = ǫµν∂µ jν =
+1√πǫµνǫνλ∂µ∂λφ = − 1√π∂2φ
+(181)
+where we used the identification of the gauge current in terms of the scalar field φ, Eq.(179).
+Therefore we conclude that
+∂µ j5
+µ = 0 ⇔ ∂2φ = 0
+(182)
+This equation states that the chiral current as an operator identity is conserved if and only if the
+field φ is a free massless scalar field, whose Lagrangian density is
+LB = 1
+2(∂µφ)2
+(183)
+In Eq.(166) we considered the free massless Dirac Lagrangian coupled to a background (not
+quantized) gauge field Aµ through the usual minimal coupling which here is Lint = −ejµAµ. using
+the bosonization identity for the gauge current, Eq.(179) we see that the Lagrangian density of
+the bosonized theory now becomes
+LB[A] =1
+2(∂µφ)2 −
+e√πǫµν∂νφ(x) Aµ(x)
+≡1
+2(∂µφ)2 + J(x)φ(x)
+(184)
+where the source J(x) is
+J(x) =
+e√πǫµν∂νAµ(x) =
+e
+√
+4π
+F∗(x)
+(185)
+where F∗(x) = ǫµνFνµ(x) is the (Hodge) dual of the field strength Fµν. Hence, F∗ is (essentially)
+the source for the scalar field φ(x). This implies that the equation of motion of the scalar field
+must be
+− ∂2φ(x) = J(x) =
+e√πǫµν∂νAµ
+(186)
+Retracing our steps we find that the chiral current j5
+µ obeys
+∂µ j5
+µ = − 1√π∂2φ = e
+2πǫµνFµν
+(187)
+which reproduces the the chiral anomaly given in Eq.(170).
+These results suggest that the theory of a free massless Dirac spinor must be equivalent to
+the theory of the free massless scalar field. This statement is known as bosonization. However
+for this identification to be correct there must be an identification of the Hilbert spaces and of all
+the operators of each theory. We will not do this detailed analysis here but we will highlight the
+most significant statements.
+Let us begin with the fermion number of the Dirac theory. Consider a system of fermions
+of total length L. Using the bosonization identity of Eq.(179) we find that the fermion number
+NF ≡ Q is given by
+NF =
+� L
+0
+dx1 j0(x0, x1) =
+1√π
+� L
+0
+dx1 ∂1φ(x0, x1) =
+1√π(φ(x0, L) − φ(x0, 0))
+(188)
+Thus, the vacuum sector of the Dirac theory, with NF = 0, corresponds to the theory of the scalar
+field with periodic boundary conditions, φ(x1 = 0) = φ(x1 = L). Furthermore, since the fermion
+number is quantized, NF ∈ Z, changing the fermion number is the same as twisting the boundary
+conditions of the scalar so that
+φ(x1 + L) = φ(x1) + √πNF
+(189)
+In String Theory [104] the scalar field is interpreted as the coordinate of a string. Compactifying
+the space where the string lives to be a circle of radius R means that the string coordinate is
+defined modulo 2πRn, where n is an integer. We see that the condition imposed by Eq.(189)
+40
+
+is equivalent to say that the scalar field is compactified and that the compactification radius is
+R = 1/
+√
+4π. This identification also imposes the restriction that the allowed operators of the
+scalar field must obey the identification
+φ(x) ∼ φ(x) + 2πRn
+(190)
+as an equivalency condition.
+The simplest bosonic operators that obey the compactification condition are the vertex oper-
+ators Vα(x),
+Vα(x) = exp(iαφ(x))
+(191)
+The compactification condition then requires that the allowed vertex operators should have α =
+n/R =
+√
+4π n, where n is an integer. Since the propagator of the scalar field in 1+1-dimensional
+(Euclidean) spacetime is
+G(x − x′) = − 1
+2π ln
+�|x − x′|
+a
+�
+(192)
+where a is a short-distance cutoff, we find that the scaling dimension of the vertex operator is
+∆α = α2/(4π) = n2. We will see shortly that the vertex operator with α =
+√
+4π is essentially the
+Dirac mass operator (which has scaling dimension 1).
+The free massless scalar field can be decomposed into right and left moving components, φR
+and φL respectively,
+φ = φR + φL,
+ϑ = −φR + φL
+(193)
+where
+ϑ(x0, x1) =
+� x1
+−∞
+dx′
+1Π(x0, x′
+1)
+(194)
+is the Cauchy-Riemann dual of the field φ(x) since they satisfy the Cauchy-Riemann equation
+∂µφ = ǫµν∂νϑ
+(195)
+The right and left moving component of the Dirac spinor are found to have the bosonized expres-
+sion [130]
+ψR(x) =
+1
+√
+2πa
+: exp(i
+√
+4πφR(x)) :,
+ψL(x) =
+1
+√
+2πa
+: exp(−i
+√
+4πφL(x)) :
+(196)
+It is easy to check that the propagators of these operators agree with the expressions given in
+Eq.(172), and that they have scaling dimension 1/2 and spin 1/2.
+How does a chiral transformation act on the scalar field? A chiral transformation by an angle
+θ, c.f. Eq.(132), acts on the right and moving fermions as ψ′
+R = exp(iθ)ψR and ψ′
+L = exp(−iθ)ψL.
+From Eq.(196) we see that the right and left moving components of the scalar field transform as
+φ′
+R = φR + θ/
+√
+4π and φ′
+L = φL + θ/
+√
+4π. This means that a chiral transformation by an angle
+θ of the Dirac fermion by an angle θ is equivalent to a translation (a shift) of the scalar field
+φ′ = φ + 2θ/
+√
+4π.
+We can use the Operator Product Expansion discussed in section 3.3.2 to show that the
+fermion mass terms ¯ψψ and i ¯ψγ5ψ are given by the following identifications
+¯ψψ =
+1
+2πa : cos(
+√
+4πφ) :,
+i ¯ψγ5ψ =: sin(
+√
+4πφ) :
+(197)
+These operators have scaling dimension 1 and transform properly under chiral transformations.
+These identifications imply that a theory of free massive Dirac fermions
+LD = ¯ψi/∂ ψ − m ¯ψψ
+(198)
+is equivalent to the sine-Gordon field theory [131] whose Lagrangian is
+LSG = 1
+2(∂µφ)2 − g : cos(
+√
+4πφ) :
+(199)
+where g = m/(2πa).
+Given the central role played by the current algebra identities of Eqs. (175) one may wonder
+if a similar approach might apply in higher dimensions. Schwinger terms in current algebra play
+an important role in relativistic field theories. However in higher dimensions their structure is
+41
+
+more complex and does not lead to identities of the type we have discussed. The reason at the
+root of this problem is largely kinematical. The Bose (scalar) field φ is qualitatively a bound
+state, a collective mode in the language of Condensed Matter Physics. In 1+1 dimensions this
+collective mode exhausts the spectrum at low energies due to the strong kinematical restriction
+on one spatial dimension.
+The equivalency between the theory of free massive Dirac fermions and the sine-Gordon
+theory is an example of the power of bosonization. On the Dirac side the mapping the theory is
+free and its spectrum is well understood. But on the sine-Gordon side the theory is non-linear.
+In fact in the sine-Gordon theory the fermions are essentially solitons, domain walls of the scalar
+field. For these and many other reasons that we do not have space here bosonization plays a
+huge role in understanding the non-perturbative behavior of systems both in Condensed Matter
+Physics and in Quantum Field Theory in 1+1 dimensions. We will see in section 11.3.1 that to
+an extent some of these ideas can and have been extended to relativistic systems and classical
+statistical mechanical systems in 2+1 dimensions.
+8. Fractional Charge
+8.1. Solitons in one dimensions
+We begin by returning to the equivalency between the theory of free massive Dirac fermions
+and sine-Gordon theory, in 1+1 dimensions. In Eq.(188) we showed that the boundary conditions
+of the compactified scalar field φ(x) are determined by the fermion number NF of the dual Dirac
+theory, and that the vacuum sector of the Dirac theory maps onto the sine-Gordon theory with
+periodic boundary conditions. We will now examine the sector with one fermion, NF = 1. This
+sector of the Dirac theory maps onto the sine-Gordon theory with twisted boundary conditions,
+φ(L) − φ(0) = √π
+(200)
+The Hamiltonian of the sine-Gordon theory is
+HSG =
+� ∞
+−∞
+dx
+�1
+2Π2(x) + 1
+2 (∂xφ(x)) + g cos
+� √
+4πφ(x)
+��
+(201)
+In the sector with periodic boundary conditions the classical ground states are static and uniform
+configurations that minimize the potential energy. Since the potential energy is a periodic func-
+tional of φ(x) the classical minima are at φn(x) = (n + 1/2) √π, where n is an arbitrary integer.
+The classical energy of these ground states is extensive and is given by Egnd = −gL where L is
+the linear size of the system.
+The classical ground state in the twisted sector is a domain wall (or soliton) which interpo-
+lates between the static and uniform ground states φ(x) = ± √π/2. The classical ground state in
+this sector is the static solution of the Euler-Lagrange equation
+d2φ
+dx2 = −2g √π sin
+�
+2 √πφ(x)
+�
+(202)
+such that asymptotically satisfies limx→±∞ = ± √π/2. The solution is the classical soliton con-
+figuration
+φ(x) =
+2√π tan−1 �
+exp(2 √πg(x − x0))
+�
+−
+√π
+2
+(203)
+The soliton solution represents a domain wall between two symmetry-related classical ground
+states with φ = ± √π/2.
+The energy of the soliton (measured from the energy of the ground state in the trivial sector)
+is finite and is given by
+Esoliton = 4
+�
+g
+π
+(204)
+where x0 is a zero mode of the soliton solution and represents its coordinate. By coupling the
+bosonized theory to a weak electromagnetic field Aµ, as given in Eq.(184), it is easy to check that
+it has electric charge −e and, in this sense represents the electron. Then the identities of Eq.(196)
+can be used that as a quantum state it is indeed a fermion.
+42
+
+8.2. Polyacetylene
+In section 7.4 we saw that solitons of a scalar field can be regarded as being equivalent to
+electrons, fermions with charge −e. We will now see that in a theory of fermions coupled to
+a domain wall of a scalar field, the soliton carries fractional charge. This problem has been
+extensively studied in one-dimensional conductors such as polyacetylene, in particular by the
+work of Wu-Pei Su, J. Robert Schrieffer and Alan Heeger [138] and by Roman Jackiw and J.
+Robert Schrieffer [139]. In quantum field theory this problem was first discussed by Roman
+Jackiw and Claudio Rebbi [140] and by Jeffrey Goldstone and Frank Wilczek [141].
+In section 7.1 we showed that the physics of lattice fermions in one dimension at low energies
+is well described by a theory of massless Dirac fermions. In a one-dimensional conductor, such as
+polyacetylene, the fermions couple to the lattice vibrations (phonons). Su, Schrieffer and Heeger
+(SSH) [138] proposed a simple model in which the electrons couple to the lattice vibrations
+through a modulation of the hopping amplitude between two consecutive sites n and n + 1,
+instead of being a constant t, becomes tn,n+1 = t − g(un+1 − un), where un is the displacement of
+the ion (a CH group in polyacetylene) at site n from its classical equilibrium position and g is
+the electron-phonon coupling constant. In polyacetylene there number of electrons (which are
+spin-1/2 fermions) is equal to the number of sites of the lettuce and the electronic band is half-
+filled. At half filling this simple band structure is invariant under a particle-hole transformation.
+If the coupling to the lattice vibrations is included this symmetry remains respected provided the
+displacements change sign un → −un for all lattice sites. In polyacetylene the lattice dimerizes
+(a process known as a Peierls distortion) and the discrete translation symmetry by one lattice
+spacing is spontaneously broken: the system becomes a period 2 CDW on the bonds of the
+lattice. The broken symmetry state is still invariant under a particle-hole transformation.
+The effective field theory of this system is a theory of two Dirac spinors ψα,σ(x), where
+α = 1, 2 denotes right and left-moving fermions, and σ =↑, ↓ are the two spin polarizations. The
+Lagrangian density of this system is
+L = ¯ψσi/∂ψσ − gφ(x) ¯ψσ(x)ψσ(x) − 1
+2φ(x)2
+(205)
+The real scalar field φ(x) represents the distortion field of the polyacetylene chain. Here we will
+assume that the chains has spontaneously distorted and we will regard the scalar field as static
+and classical. This is a good approximation since the masses of the CH complexes is much
+bigger than the electron mass. This continuum model is due to Takayama, Lin-Liu and Maki
+[142] and further developed by Campbell and Bishop [143, 144]. Many of the results fund in
+this (adiabatic) approximation remain qualitatively correct upon taking into account the quantum
+dynamics of the chain, even in the limit in which the ions are treated as being “light” (provided
+the spin of the fermions is taken into account) [145, 146].
+In the field theory the discrete symmetry of displacements by one lattice spacing becomes
+the Z2 symmetry φ → −φ. This is a symmetry of the electron phonon system once combined
+with the discrete chiral transformation ψ → γ5ψ under which ¯ψψ → − ¯ψψ. The ground state is
+two fold degenerate ±φ0 with
+φ0 = 2Λ�F
+g
+exp
+�
+−π�F
+g2
+�
+(206)
+and the Dirac fermion (the electron) has a exponentially small mass, m = gφ0.
+8.3. Fractionally charged solitons
+Jackiw and Rebbi showed that the 1+1-dimensional classical φ4 theory has the following
+soliton solution which interpolates between the two classically ordered states at ±φ0 [140]
+φ(x) = φ0 tanh
+� x − x0
+ξ
+�
+(207)
+where ξ is the correlation length of φ4 theory, and x0 is the (arbitrary) location of the soliton.
+They further showed that, when coupled to a theory of massless relativistic fermions through a
+Yukawa coupling, as in Eq.(205), this soliton carries fractional charge. The argument goes as
+follows. The one-particle Dirac Hamiltonian for a Dirac fermion with a position-dependent mass
+m(x) is
+H = −iσ1∂x + m(x)σ3 =
+�m(x)
+−i∂x
+−i∂x
+−m(x)
+�
+(208)
+43
+
+where m(x) = gφ(x), with φ(x) being the soliton solution of Eq.(207). This Hamiltonian is her-
+mitian and real. Furthermore, this Hamiltonian anti-commutes with the Pauli matrix σ2. This
+implies that for every positive-energy state |E⟩ with energy +E there is a negative-energy eigen-
+state with energy −E given by σ2|E⟩. Hence, the spectrum is particle-hole symmetric (or, what
+is the same, charge-conjugation invariant). In addition, and consistent with charge-conjugation
+symmetry, the Hamiltonian of Eq.(208) has state with E = 0, a zero-mode, with a normalizable
+spinor wave function
+ψ0(x) =
+1√
+2
+�−i
+1
+�
+exp
+�
+−sgn(m)
+� x
+0
+dx′ m(x′)
+�
+(209)
+which exists for an arbitrary function m(x) which changes sign once at some location (which
+we took to be x0 = 0). Jackiw and Rebbi further showed that the soliton (anti-soliton) carries
+fractional charge
+Q = ∓e
+2
+(210)
+This result follows from the spectral asymmetry identity of the density of states ρS (E) in the
+presence of the soliton
+Q = −e
+2
+� ∞
+0
+(ρS (E) − ρS (−E)) = −e
+2η
+(211)
+where η is the spectral asymmetry of the Dirac operator in the soliton background, and it is known
+as the APS η-invariant of Atiyah-Patodi-Singer [147]. Given the one-to-one correspondence
+that exists between positive and negative energy states in the spectrum, the spectral asymmetry
+follows from the condition that the zero mode be half-filled, which is required by normal-ordering
+or, what is the same, by charge neutrality. Another way to understand this result is that adding (or
+removing) a fermion of charge −e results in the creation of a soliton-antisoliton pair, with each
+topological excitation carrying half of the charge of the electron. In other words, in the dimerized
+phase the electron is fractionalized. In this analysis we ignored the spin of the electron. If we
+take it into account the spin degree of freedom the soliton is instead a boson with charge ∓e.
+There is an alternative, complementary, way to think about the charge of the soliton. Gold-
+stone and Wilczek [141] considered a theory in which the (massless) Dirac fermion is coupled to
+two real scalar fields, ϕ1 and ϕ2, with Lagrangian
+L = ¯ψi/∂ψ − gϕ1 ¯ψψ − igϕ2 ¯ψγ5ψ ≡ ¯ψi/∂ψ − g|ϕ| ¯ψ exp(iθγ5)ψ
+(212)
+where |ϕ|2 = ϕ2
+1 + ϕ2
+2 and θ = tan−1(ϕ2/ϕ1). They considered a soliton in which gϕ1 = m is the
+constant (in space) Dirac mass and ϕ2 winds slowly between two values ±ϕ0 for x → ±∞. In
+this theory the one-particle Dirac Hamiltonian is
+H = −iσ1∂x + gϕ1σ3 + gϕ2σ2
+(213)
+which is hermitian and complex and, hence, it violates CP invariance.
+A perturbative calculation of the induced (gauge-invariant) current jµ, which is given by the
+triangle diagram of a fermion loop with two gauge field insertions and a coupling of the scalar
+fields, yields the result (with a = 1, 2)
+⟨jµ(x)⟩ = 1
+2πǫµνǫab
+ϕa∂νϕb
+|ϕ|2
+= 1
+2πǫµν∂νθ
+(214)
+which is locally conserved. Notice, however, that the induced axial current j5
+µ is not conserved,
+∂µ⟨j5
+µ⟩ = − 1
+2π∂2θ � 0 and, hence, this current is anomalous. We can now compute the total
+charge accumulated as the soliton is created adiabatically to be given by the Goldstone-Wilczek
+formula
+Q = −e∆θ
+2π
+(215)
+where ∆θ = θ(+∞) − θ(−∞). Since limx→±∞ ϕ2(x) = ±ϕ0, we obtain the result
+Q = − e
+π tan−1 �gϕ0
+m
+�
+(216)
+In the limit m = gϕ → 0, where CP (or T) invariance is recovered, we get
+lim
+m→0 Q = −e
+2
+(217)
+44
+
+which is the Jackiw-Rebbi result for the fractional charge of a soliton of Eq.(210).
+The results for the fractional charge of the soliton can also be derived using the bosonization
+identities of section 7.4. Indeed, the bosonized expression for the Lagrangian of Eq.(212) is
+L = 1
+2
+�
+∂µφ
+�2 − g|ϕ|
+2πa cos
+� √
+4πφ − θ
+�
+(218)
+Deep in the phase in which the Bose field φ is massive, the non-linear term in Eq.(218) locks this
+field to the chiral angle θ, i.e.
+φ =
+1
+√
+4π
+θ
+(219)
+However, the bosonization identities also tell us that the gauge current jµ is given by the curl of
+the scalar field φ. Therefore, in this state the current jµ is
+jµ =
+1√πǫµν∂νφ = 1
+2πǫµν∂νθ
+(220)
+which is the same as the Goldstone-Wilczek result of Eq.(214). That these two seemingly differ-
+ent approaches yield the same result is not accidental as they both follow from the axial anomaly.
+9. Fractional Statistics
+A fundamental axiom of Quantum Mechanics is that identical particles are indistinguish-
+able [148, 149]. In non-relativistic Quantum Mechanics this leads to the requirement that the
+quantum states of a system of identical particles must be eigenstates of the pairwise particle ex-
+change operator. Since two exchanges are equivalent to the identity operation this implies that the
+states must be even or odd under pairwise exchanges. This result, in turn, implies that particles
+can be classified as either being bosons (whose states are invariant under pairwise exchanges)
+or fermions (whose states change sign under pairwise particle exchanges). A consequence is
+that bosons obey the the Bose-Einstein (and can condense into a single particle state) whereas
+fermions obey the Fermi-Dirac distribution and must obey the Pauli exclusion principle. It is
+an implicit assumption of this line of reasoning that all relevant states of a system of identical
+particles can be efficiently represented by a (suitably symmetrized or antisymmetrized) product
+state.
+This classification is present at an even deeper level in (relativistic) Quantum Field Theory
+where locality, unitarity and Lorentz invariance require that the fields be classified as represen-
+tations of the Lorentz group and obey the Spin-Statistics Theorem [150]. The Spin-Statistics
+Theorem is actually an axiom of local relativistic Quantum Field theory which requires that
+fields that transform with an integer spin representation of the Poincar´e group (i.e. scalars, gauge
+fields, gravitational fields, etc) must be bosons while fields that transform with a half-integer
+spin representation (i.e. Dirac spinors, etc) must be fermions. This spin-statistics connection is
+intrinsic to the construction of String Theory [104].
+Given these considerations there was a general consensus that fermions and bosons were
+the only possible types of statistics. Nevertheless several exceptions to this rule were known
+to exist. One is the construction of the magnetic monopole in 3+1 dimensional gauge theory
+by Tai-Tsun Wu and Chen-Ning Yang who showed that a scalar coupled to a Dirac magnetic
+monopole behaves as a Dirac spinor [151]. This was an early example of statistical transmutation
+by coupling a matter field to a non-trivial configuration of a gauge field. As we will note below,
+the construction of anyons (particles with fractional statistics) in 2+1 dimensions has a close
+parentage to the Wu-Yang example. Examples of statistical transmutation were known to exist
+in 1+1 dimensional theories where a system of hard-core bosons was shown to be equivalent to a
+theory of free fermions using the Jordan-Wigner transformation [22] which represents a fermion
+as a composite operator of a hard-core boson and an operator that creates a kink (or soliton).
+This construction also underlies the fermion-boson mapping in 1+1 dimensional field theories
+[128, 129, 131, 130] that we discussed in section 7.4. Finally, in the late 1970s it was found that
+1+1 dimensional ZN spin systems harbor operators known as parafermions which obey the same
+algebra shortly afterwards found to be obeyed by anyons in 2+1 dimensions [152].
+45
+
+9.1. Basics of fractional statistics
+Jon Magne Leinaas and Jan Myrheim wrote an insightful paper in 1977 in which they exam-
+ined the structure of the configuration space of the histories of a system of N identical particles
+[153]. Using the Feynamn path-integral approach, they showed that if the worldlines of the iden-
+tical particles are not allowed to cross, then the configuration space is topologically non-trivial.
+Through a detailed analysis they showed that the three and higher dimensions under a pairwise
+particle exchange the states must be either even or odd and hence the particles are either bosons
+or fermions. Leinaas and Myrheim also showed that in one and two space dimensions the wave
+functions can change by a phase, nowadays known as the statistical angle. In retrospect this re-
+sult could have been anticipated (but was not) in an earlier paper by Michael G. G. Laidlaw and
+C´ecile Morette DeWitt [154] who did a similar analysis of the configuration space of identical
+particles in the Feynman path integral.
+Frank Wilczek generalized the Aharonov-Bohm effect [155] to describe the quantum me-
+chanics of composite objects made of electric charge and magnetic flux in two space dimensions
+[156]. Wilczek showed that composite objects made of a non-relativistic particle of charge q
+bound to a magnetic flux of (magnetic) charge Φ behaves as an object with fractional angu-
+lar momentum qΦ/2π. Here Φ is measured in units of the flux quantum 2π (in units in which
+ℏ = c = e = 1). Furthermore, in a subsequent paper Wilczek [157] showed that, upon an
+adiabatic process in which the two composites exchange positions without their worldlines co-
+inciding, the wave function of two identical flux-charge composites changes by a phase factor
+exp(±iqΦ). For instance if Φ = π (i.e. a half-flux quantum) the acquired phase is equal to
+exp(iπ) = −1. thus if the particle was a boson, the composite becomes a fermion and viceversa.
+Wilczek coined the term anyon to describe the behavior of an arbitrary charge-flux composite.
+Furthermore, this construction also implies that not only fractional statistics but also fractional
+spin, consistent with a generalization of the Spin-Statistics Theorem. In other terms, “flux-
+attachment” implies fractional statistics (and fractional spin). Clearly Wilczek’s construction
+gave an explicit physical grounding to the general arguments of the 1977 paper by Leinaas and
+Myrheim. It is worth note the close analogy between this construction in 2+1 dimensions and the
+Wu-Yang construction in 3+1 dimensions (whose consistency with the Spin-Statistics Theorem
+had been shown earlier on by Alfred Goldhaber [158]).
+In this description the statistics of the composites (the anyons) enters in the form of complex
+weights (phases) given in terms of the linking numbers of the worldlines. Hence, the concept of
+fractional statistics is intimately related to the theory of knots and of the representations of the
+braid group. These concepts were originally introduced in physics to describe statistical trans-
+mutation in the theory of solitons [159] in the context of the Skyrme model [160]. Yong-Shi Wu
+[161] developed an explicit connection between the work of Leinaas and Myrheim and Wilczek’s
+work on anyons in terms of operations acting on the worldlines of the anyons and described by
+the Braid Group. Wu’s work and the somewhat early paper by Wilczek and Zee on the statistics
+of solitons [162] marked the definite entry of the theory of knots (and of the braid group) in
+physics in general and in condensed matter physics in particular. The classification of anyons
+in terms of representations of the braid group labeled by the fractional statistics (determined by
+linking numbers) as well as of fractional (or topological) spin (determined by the writhing num-
+ber of the worldlines) leads to a rich set of physical consequences. As it will turn out, Wilczek’s
+anyons are described one-dimensional representations of the braid group. As we will see below,
+additional and intriguing (non-abelian) representations will also play a role.
+9.2. What is a topological field theory
+We will now consider a special class of gauge theories known as topological field theories.
+These theories often (but not always) arise as the low energy limit of more complex gauge theo-
+ries. In general, one expects that at low energies the phase of a gauge theory be either confining
+or deconfined. While confining phases have (from really good reasons!) attracted much atten-
+tion, deconfined phases are often regarded as trivial, in the sense that the general expectation
+is that their vacuum states be unique and the spectrum of low lying states is either massive or
+massless.
+Let us consider a gauge theory whose action on a manifold M with metric tensor tensor
+gµν(x) is
+S =
+�
+M
+dDx √g L(g, Aµ)
+(221)
+46
+
+At the classical level, the the energy-momentum tensor T µν(x) is the linear response of the action
+to an infinitesimal change of the local metric,
+T µν(x) ≡
+δS
+δgµν(x)
+(222)
+That a theory is topological means that depends only on the topology of the space in which is
+defined and, consequently, it is independent of the local properties that depend on the metric, e.g.
+distances, angles, etc. Therefore, at least at the classical level, the energy-momentum tensor of a
+topological field theory must vanish identically,
+T µν = 0
+(223)
+In particular, if the theory is topological, the energy (or Hamiltonian) is also zero. Furthermore, if
+the theory is independent of the metric, it is invariant under arbitrary coordinate transformations.
+Thus, if the theory is a gauge theory, the expectation values of Wilson loops will be independent
+of the size and shape of the loops. Whether or not a theory of this type can be consistently defined
+at the quantum level is a subtle problem which we will briefly touch on below.
+It turns out that, due to the non-local nature of the observables of a gauge theory, the low
+energy regime of a theory in its deconfined phase can have non-trivial global properties. In what
+follows, we will say that a gauge theory is topological if all local excitations are massive (and in
+fact we will send their mass gaps to infinity). The remaining Hilbert space of states is determined
+by global properties of the theory, including the topology of the manifold of their space-time. In
+several cases, the effective action of a topological field theory does not depend on the metric of
+the space-time, at least at the classical level. In all cases, the observables are non-local objects,
+Wilson loops and their generalization.
+9.3. Chern-Simons Gauge Theory
+Gauge theories play a key role in physics. In 2+1 dimensions it is possible to define a special
+gauge theory which is odd under time reversal invariance and parity: Chern-Simons gauge theory.
+Originally introduced in Quantum Field Theory in1982 by Stanley Deser, Roman Jackiw and
+Stephen Templeton [163, 164], Chern-Simons gauge theory can be defined for any compact Lie
+group, as well as an extension of Einstein’s gravity in 2+1 dimensions. In 1989 Edward Witten
+[165] showed that Chern-Simons gauge theory computes the expectation values of configurations
+of Wilson loops, regarded as the worldlines of heavy particles, in terms of a set of topological
+invariants known as the Jones polynomial that classify knots in three dimensions.
+We will consider the simplest case, the U(1) Chern-Simons gauge theory. The Chern-Simons
+action for a U(1) gauge field Aµ in 2+1 dimension is
+S [A] = k
+4π
+�
+Ω
+d3x ǫµνλAµ∂νAλ +
+�
+Ω
+d3x JµAµ
+(224)
+where Jµ is a set of conserved currents (representing the worldlines of a set of heavy particles).
+On a closed 3-manifold Ω (e.g. a sphere, a torus, etc) the Chern-Simons action is gauge invariant
+provided the parameter k (known as the level) is an integer. If the manifold Ω has a boundary,
+the action is not gauge invariant at the boundary. Gauge invariance is restored by additional
+boundary degrees of freedom. This structure is general, and not just a feature of the U(1) theory.
+Since the Chern-Simons action is first order in derivatives it is not invariant under time re-
+versal and under parity (which in 2+1 dimensions is a reflection). These symmetries make this
+theory relevant to the description of the fractional quantum Hall effect. In the absence of exter-
+nal sources, Jµ = 0, at the classical level the Chern-Simons action is invariant under arbitrary
+changes of coordinates. This means that the theory is, at least classically, a topological field
+theory.
+For a general non-abelian gauge group G the Chern-Simons action becomes
+S = k
+4π
+�
+M
+d3x tr
+�
+AdA + 2
+3A ∧ A ∧ A
+�
+(225)
+Here, the cubic term is shorthand for
+tr
+�
+A ∧ A ∧ A
+�
+≡ tr
+�
+ǫµνλAµAνAλ�
+(226)
+for a gauge field Aµ that takes values on the algebra of the gauge group G.
+47
+
+9.4. BF gauge theory
+A closely related (abelian) gauge theory is the so-called BF theory [166] which, in a general
+spacetime dimension D (even or odd), is a theory of a vector field Aµ (a one-form) and an an-
+tisymmetric tensor field B with D − 2 Lorentz indices (a D2 form), known as a Kalb-Ramond
+field. In 2+1 dimensions the action of the BF theory is
+S = k
+2π
+�
+M
+d3x ǫµνλBλ∂µAν
+(227)
+where, once again, k is an integer.
+The BF gauge theory has the same content as the topological sector of a discrete Zk gauge
+theory. To see this we will consider the theory of compact electrodynamics which is a U(1)
+gauge theory minimally coupled to charged scalar field φ. The will assume that the gauge theory
+is defined for a compact U(1) gauge group (meaning that the gauge flux is quantized) and that
+the complex scalar field has charge integer k ∈ Z. As usual minimal coupling is implemented by
+introducing the covariant derivative Dµ − ∂µ + ikAµ, where Aµ is the U(1) gauge field. Deep in
+the phase in which the global U(1) symmetry is spontaneously broken, usually called the Higgs
+regime, the amplitude of the scalar field is frozen at its (real) vacuum expectation value φ0 but its
+phase ω, representing the Goldstone mode, is unconstrained. In this limit the Lagrangian of this
+theory becomes
+L = |φ0|2 �
+∂µω − kAµ
+�2 − 1
+4e2 F2
+µν
+(228)
+where e is the coupling constant of the gauge field (the electric charge) and Fµν is the field
+strength of the gauge field Aµ. In general spacetime dimension D the charge e has units of
+length(D−4)/2. In this limit the gauge field becomes massive (this is the Higgs mechanism). This
+theory has fluxes quantized in units of 2π/k and has only k distinct fluxes. This is the Zk gauge
+theory.
+Here we will consider this theory in 2+1 dimensions. We will use a gaussian (Hubbard-
+Stratonovich) decoupling of the first term of Eq.(228) in terms of a gauge field Cµ to write the
+Lagrangian in the equivalent form
+L = −
+1
+4|φ0|2C2
+µ + Cµ(∂µω − kAµ) − 1
+4e2 F2
+µν
+(229)
+Up to an integration by parts, we see that the phase field ω plays the role of a Lagrange multiplier
+field which forces the vector field Bµ to obey the constraint ∂µCµ = 0. This constraint is solved
+by writing Cµ as
+Cµ = 1
+2πǫµνλ∂νBλ
+(230)
+of a 1-form gauge field Bµ. Upon solving the constraint the Lagrangian of Eq.(229) becomes
+L = k
+2πǫµνλAµ∂νBλ − 1
+4e2 Fµν(A)2 −
+1
+32π2|φ0|2 Fµν(B)2
+(231)
+For spacetime dimensions D < 4 the IR the Maxwell terms for the fields Aµ and Bµ are irrelevant
+in the IR and, in this limit, this theory reduces to the BF theory at level k of Eq.(227). Therefore,
+Zk gauge theory is equivalent to the BF theory at level k. In fact, this result is essentially valid
+in all dimensions with the main difference being that the field Bµ is, in general, a rank D − 2
+Kalb-Ramond antisymmetric field.
+9.5. Quantization of Abelian Chern-Simons Gauge Theory
+By expanding the action of Eq.(224) the Lagrangian density becomes
+L = k
+4πǫi jAi∂0A j + A0
+� k
+2π B − J0
+�
+− JiAi
+(232)
+where B = ǫi j∂iA j is the local flux, J0 is a local classical density and Ji a local classical current.
+Then, the first term of Eq.(232) implies that the spatial components of the gauge field obey
+equal-time canonical commutation relations
+[A1(x), A2(x′)] = i2π
+k δ(x − x′)
+(233)
+48
+
+The second term of the Lagrangian enforces the Gauss Law which for this theory simply
+implies that the states in the physical Hilbert space obey the constraint
+B(x) = 2π
+k J0(x)
+(234)
+Thus the Gauss Law requires that a charge density necessarily has a magnetic flux attached to it.
+In other terms, the physical states are charge-flux composites as postulated in Wilczek’s theory
+[157]. This is the theoretical basis to the concept of flux attachment which, as we will see in
+section 10.3, is widely used in the theory of the fractional quantum Hall effect.
+The third term in Eq.(232) simply states that the Hamiltonian density is just
+H = JiAi
+(235)
+Hence, in the absence of sources the Hamiltonian vanishes, H = 0.
+The Chern-Simons action is locally gauge-invariant, up to boundary terms. To see this let us
+perform a gauge transformation, Aµ → Aµ + ∂µΦ, where Φ(x) is a smooth, twice differentiable
+function. Then,
+S [Aµ + ∂µΦ] =
+�
+M
+(Aµ + ∂µΦ)ǫµνλ∂ν(Aλ + ∂λΦ)
+=
+�
+M
+d3x ǫµνλAµ∂νAλ +
+�
+M
+d3x ǫµνλ∂µΦ∂νAλ
+(236)
+Therefore, the change is
+S [Aµ + ∂µΦ] − S [Aµ] =
+�
+M
+d3x ∂µΦF∗
+µ =
+�
+M
+d3x ∂µ(ΦF∗
+µ) −
+�
+M
+d3x Φ∂µF∗
+µ
+(237)
+where F∗
+µ = ǫµνλ∂νAλ, is the dual field strength. However, in the absence of magnetic monopoles,
+this field satisfies the Bianchi identity, ∂µF∗µ = ∂µ(ǫµνλ∂νAλ) = 0. Therefore, using the Gauss
+Theorem, we find that the change of the action is a total derivative and integrates to the boundary
+δS =
+�
+M
+d3x ∂µ(ΦF∗
+µ) =
+�
+Σ
+dS µΦF∗
+µ
+(238)
+where Σ = ∂M is the boundary of M. In particular, if Φ is a non-zero constant function on M,
+then the change of the action under such a gauge transformation is
+δS = Φ × flux(Σ)
+(239)
+Hence, the action is not invariant if the manifold has a boundary, and the theory must be supplied
+with additional degrees of freedom at the boundary.
+Indeed, the flat connections, i.e. the solution of the equations of motion, Fµν = 0, are pure
+gauge transformations, Aµ = ∂µφ, and have an action that integrates to the boundary. Let the
+M = D × R where D is a disk in space and R is time. The boundary manifold is Σ = S 1 × R,
+where S 1 is a circle. Thus, in this case, the boundary manifold Σ is isomorphic to a cylinder. The
+action of the flat configurations reduces to
+S =
+�
+S 1×R
+d2x k
+4π∂0φ∂1φ
+(240)
+This implies that the dynamics on the boundary is that of a scalar field on a circle S 1, and obeys
+periodic boundary conditions.
+Although classically the theory does not depend on the metric, it is invariant under arbitrary
+transformations of the coordinates. However, any gauge fixing condition will automatically break
+this large symmetry. For instance, we can specify a gauge condition at the boundary in the form
+of a boundary term of the form Lgauge fixing = A2
+1. In this case, the boundary action of the field ϕ
+becomes
+S [ϕ] =
+�
+S 1×R
+d2x k
+4π
+�
+∂0φ∂1φ − (∂1φ2)
+�
+(241)
+The solutions of the equations of motion of this compactified scalar field have the form φ(x1∓x0),
+and are right (left) moving chiral fields depending of the sign of k. This boundary theory is not
+topological but is is conformally invariant.
+49
+
+A similar result is found in non-abelian Chern-Simons gauge theory. In the case of the
+S U(N)k Chern-Simons theory on a manifold D × R, where D is a disk whose boundary is Γ, and
+R is time, the action is
+S CS[A] =
+�
+D×R
+d3x
+� k
+8πtr
+�
+ǫµνλAµ∂νAλ + 2
+3ǫµνλAµAνAλ
+��
+(242)
+This theory integrates to the boundary, Γ×R where it becomes the chiral (right-moving) S U(N)k
+Wess-Zumino-Witten model (at level k) at its IR fixed point, λ2
+c = 4π/k
+S WZW[g] =
+1
+4λ2c
+�
+Γ×R
+d2x tr
+�
+∂µg∂µg−1�
++
+k
+12π
+�
+B
+ǫµνλtr
+�
+g−1∂µg g−1∂νg g−1∂λg
+�
+(243)
+Here, g ∈ S U(N) parametrizes the flat configurations of the Chern-Simons gauge theory. The
+boundary theory is a non-trivial CFT, the chiral Wess-Zumino-Witten CFT [165].
+9.6. Vacuum degeneracy a torus
+We will now construct the quantum version of the U(1) Chern-Simons gauge theory on a
+manifold M = T 2 × R, where T 2 is a spatial torus, of linear size L1 and L2. Since this manifold
+does not have boundaries, the flat connections, ǫi j∂iA j = 0 do not reduce to local gauge transfor-
+mations of the form Ai = ∂iΦ. Indeed, the holonomies of the torus T 2, i.e. the Wilson loops on
+the two non-contractible cycles of the torus Γ1 and Γ2 are gauge-invariant observables:
+� L1
+0
+dx1A1 ≡ ¯a1,
+� L2
+0
+dx1A2 ≡ ¯a2
+(244)
+where ¯a1 and ¯a2 are time-dependent. Thus, the flat connections now are
+A1 = ∂1Φ + ¯a1
+L1
+,
+A2 = ∂2Φ + ¯a2
+L2
+(245)
+whose action is
+S = k
+4π
+�
+dx0ǫi j¯ai∂0¯a j
+(246)
+Therefore, the global degrees of freedom ¯a1 and ¯a2 at the quantum level become operators that
+satisfy the commutation relations
+[¯a1, ¯a2] = i2π
+k
+(247)
+We find that the flat connections are described by the quantum mechanics of ¯a1 and ¯a2. A
+representation of this algebra is
+¯a2 ≡ −i2π
+k
+∂
+∂¯a1
+(248)
+Furthermore, the Wilson loops on the two cycles become
+W[Γ1] = exp
+�
+i
+� L1
+0
+A1
+�
+≡ ei¯a1,
+W[Γ2] = exp
+�
+i
+� L2
+0
+A2
+�
+≡ ei¯a2
+(249)
+and satisfy the algebra
+W[Γ1]W[Γ2] = exp(−i2π/k)W[Γ2]W[Γ1]
+(250)
+Under large gauge transformations
+¯a1 → ¯a1 + 2π,
+¯a2 → ¯a2 + 2π
+(251)
+Therefore, invariance under large gauge transformations on the torus implies that ¯a1 and ¯a2 define
+a two-torus target space.
+Let us define the unitary operators
+U1 = exp(ik¯a2),
+U2 = exp(−ik¯a1)
+(252)
+which satisfy the algebra
+U1U2 = exp(i2πk)U2U1
+(253)
+50
+
+The unitary transformations U1 and U2 act as shift operators on ¯a1 and ¯a2 by 2π, and hence
+generate the large gauge transformations. Moreover, the unitary operators U1 and U2 leave the
+Wilson loop operators on non-contractible cycles invariant,
+U−1
+1 W[Γ1]U1 = W[Γ1],
+U−1
+2 W[Γ2]U2 = W[Γ2]
+(254)
+Let |0⟩ be the eigenstate of W[Γ1] with eigenvalue 1, i.e. W[Γ1]∥0⟩ = |0⟩. The state W[Γ2]|0⟩ is
+also an eigenstate of W[Γ1] with eigenvalue exp(−i2π/k), since
+W[Γ1]W[Γ2]|0⟩ = ei2π/kW[Γ2]W[Γ1]|0⟩ = e−i2π/kW[γ2|0⟩
+(255)
+More generally, since
+W[Γ1]W p[Γ2]∥0⟩ = e−i2πp/kW p[Γ2]|0⟩
+(256)
+we find that, provided k ∈ Z, there are k linearly independent vacuum states |p⟩ = W p[Γ2]|0⟩, for
+the U(1) Chern-Simons gauge theory at level k. It is denoted as the U(1)k Chern-Simons theory.
+Therefore the finite-dimensional topological space on a two-torus is k-dimensional. It is trivial
+to show that, on a surface of genus g, the degeneracy is kg.
+We see that in the abelian U(1)k Chern-Simons theory the Wilson loops must carry k possible
+values of the unit charge. This property generalizes to the non-abelian theories, which are tech-
+nically more subtle. We will only state some important results. For example, if the gauge group
+is SU(2) we expect that the Wilson loops will carry the representation labels of the group SU(2),
+i.e. they will be labelled by (j, m), where j = 0, 1
+2, 1, . . . and the 2 j+1 values of m satisfy |m| ≤ j.
+However, it turns out SU(2)k Chern-Simons theory has fewer states, and that the values of j are
+restricted to the range j = 0, 1
+2, . . . , k
+2.
+9.7. Fractional Statistics and Braids
+Another aspect of the topological nature of Chern-Simons theory is the behavior of expec-
+tation values of products of Wilson loop operators. Let us compute the expectation value of
+a product of two Wilson loop operators on two positively oriented closed contours γ1 and γ2.
+We will do this computation in the abelian Chern-Simons theory U(1)k in 2+1-dimensional Eu-
+clidean space. Note that the Euclidean Chern-Simons action is pure imaginary since the action
+is first-order in derivatives. The expectation value to be computed is
+W[γ1 ∪ γ2] =
+�
+exp
+�
+i
+�
+γ1∪γ2
+dxµAµ
+��
+CS
+(257)
+The result changes depending on whether the loops γ1 and γ2 are linked or unlinked. In this
+section we will compute the contribution to this expression for a pair of contours γ1 and γ2. Here
+we will not include the contribution to this expectation value for each contours. We will return
+to this problem in section 11.3.1 where we discuss the problem of fractional spin.
+The expectation value of a Wilson loop on the union of two contours, as in the present case,
+γ can be written as
+�
+exp
+�
+i
+�
+γ
+dxµAµ
+��
+CS =
+�
+exp
+�
+i
+�
+d3xJµAµ
+��
+CS
+(258)
+where the current Jµ is
+Jµ(x) = δ(xµ − zµ(t))dzµ
+dt
+(259)
+Here zµ(t) is a parametrization of the contour γ. Therefore, the expectation value of the Wilson
+loop is [165]
+�
+exp
+�
+i
+�
+γ
+dxµAµ
+��
+CS ≡ exp(iI[γ]CS) = exp
+�
+− i
+2
+�
+d3x
+�
+d3y Jµ(x)Gµν(x − y)Jν(y)
+�
+(260)
+where Gµν(x − y) = ⟨Aµ(x)Aν(y)⟩CS is the propagator of the Chern-Simons gauge field. Since the
+loops are closed, the current Jµ is conserved, i.e. ∂µJµ = 0, and the effective action I[γ]CS of the
+loop γ is gauge-invariant.
+The Euclidean propagator of Chern-Simons gauge theory (in the Feynman gauge) is
+Gµν(x − y) = 2π
+k G0(x − y)ǫµνλ∂λδ(x − y)
+(261)
+51
+
+where G0(x − y) is the propagator of the massless Euclidean scalar field, which satisfies
+− ∂2G0(x − y) = δ3(x − y)
+(262)
+Using these results, we find the following expression for the effective action
+I[γ]CS =π
+k
+�
+d3x
+�
+d3y Jµ(x)Jν(y)G0(x − y)ǫµνλ∂λδ(x − y)
+=π
+k
+�
+γ
+dxµ
+�
+γ
+dyνǫµνλ∂λG0(x − y)
+(263)
+Since the current Jµ is conserved, it can be written as the curl of a vector field, Bµ, as
+Jµ = ǫµνλ∂νBλ
+(264)
+In the Lorentz gauge, ∂µBµ = 0, we can write
+Bµ = ǫµνλ∂νφλ
+(265)
+Hence,
+Jµ = −∂2φµ
+(266)
+where
+φµ(x) =
+�
+d3y G0(x − y)Jµ(y)
+(267)
+Upon substituting this result into the expression for Bµ, we find
+Bµ =
+�
+d3yǫµνλ∂νG0(x − y)Jλ(y) =
+�
+γ
+ǫµνλ∂νG0(x − y)dyλ
+(268)
+Therefore, the effective action I[γ]CS becomes
+I[γ]CS = π
+k
+�
+γ
+dxµ
+�
+γ
+dyνǫµνλ∂λG0(x − y) = π
+k
+�
+γ
+dxµBµ(x)
+(269)
+Let Σ be an oriented open surface of the Euclidean three-dimensional space whose boundary is
+the oriented loop (or union of loops) γ, i.e. ∂Σ = γ. Then, using Stokes Theorem we write in the
+last line of Eq.(269) as
+I[γ]CS = π
+k
+�
+Σ
+dS µǫµνλ∂νBλ = π
+k
+�
+Σ
+dS µJµ
+(270)
+The integral in the last line of this equation is the flux of the current Jµ through the surface Σ.
+Therefore, this integral counts the number of times nγ the Wilson loop on γ pierces the surface Σ
+(whose boundary is γ), and therefore it is an integer, nγ ∈ Z. We will call this integer the linking
+number (or Gauss invariant) of the configuration of loops. In other words, the expectation value
+of the Wilson loop operator is
+W[γ]CS = eiπnγ/k
+(271)
+The linking number is a topological invariant since, being an integer, its value cannot be changed
+by smooth deformations of the loops, provided they are not allowed to cross.
+We will now see that this property of Wilson loops in Chern-Simons gauge theory leads to the
+concept of fractional statistics. Let us consider a scalar matter field that is massive and charged
+under the Chern-Simons gauge field. The excitations of this matter field are particles that couple
+minimally to the gauge field. Here we will be interested in the case in which these particles are
+very heavy. In that limit, we can focus on states that have a few of this particles which will be in
+their non-relativistic regime.
+Consider, for example, a state with two particles which in the remote past, at time t = −T →
+−∞, are located at two points A and B. This initial state will evolve to a final state at time t = T →
+∞, in which the particles either go back to their initial locations (the direct process), or to another
+one in which they exchange places, A ↔ B. At intermediate times, the particles follow smooth
+worldlines. These two processes, direct and exchange. There we see that the direct process is
+equivalent to a history with two unlinked loops (the worldlines of the particles), whereas in the
+exchange process the two loops form a link. It follows from the preceding discussion that the
+52
+
+two amplitudes differ by the result of the computation of the Wilson loop expectation value for
+the loops γ1 and γ2. Let us call the first amplitude Wdirect and the second Wexchange. The result is
+Wexchange = Wdirecte±iπ/k
+(272)
+where the sign depends on how the two worldlines wind around each other.
+An equivalent interpretation of this result is that if Ψ[A, B] is the wave function with the two
+particles at locations A and B, the wavefunction where their locations are exchanged is
+Ψ[B, A] = e±iπ/kΨ[A, B]
+(273)
+where the sign depends on whether the exchange is done counterclockwise or clockwise. Clearly,
+for k = 1 the wave function is antisymmetric and the particles are fermions, while for k → ∞
+they are bosons. At other values of k the particles obey fractional statistics and are called anyons
+[156, 153]. The phase factor φ = ±π/k is called the statistical phase.
+Notice that, while for fermions and bosons the statistical phase ϕ = 0, π is uniquely defined
+(mod 2π), for other values of k the statistical angle is specified up to a sign that specifies how
+the worldlines wind around each other. Indeed, mathematically the exchange process is known
+as a braid. Processes in which the worldlines wind clock and counterclockwise are braids that
+are inverse of each other. Braids can also be stack sequentially yielding multiples of the phase
+ϕ. In addition to stacking braids, Wilson loops can be fused: seen from some distance, a pair of
+particles will behave as a new particle with a well defined behavior under braiding. This process
+of fusion is closely related to the concept of fusion of primary fields in Conformal Field Theory.
+Furthermore, up to regularization subtleties [167], the self-linking terms (those with a = b)
+yield a topological spin 1/2k, consistent with the spin-statistics connection [168]. For k = 1 this
+means that the flux-charge composites have spin 1/2.
+What we have just described is a mathematical structure called the Braid group. The example
+that we worked out using abelian Chern-Simons theory yields one-dimensional representations
+of the Braid group with the phase ϕ being the label of the representations. For U(1)k there are k
+types of particles (anyons). That these representations are abelian means that, in the general case
+of U(1)k, acting on a one-dimensional representation p (defined mod k) with a one-dimensional
+representation q (also defined mod k) yields the representation one-dimensional p+q (mod k). We
+will denote the operation of fusing these representations (particles!) as [q]mod k × [p]mod k = [q +
+p]mod k. These representations are in one-to-one correspondence with the inequivalent charges of
+the Wilson loops, and with the vacuum degeneracy of the U(1)k Chern-Simons theory on a torus.
+A richer structure arises in the case of the non-abelian Chern-Simons theory at level k [165],
+such as SU(2)k. For example, for SU(2)1 the theory has only two representations, both are one-
+dimensional, and have statistical angles ϕ = 0, π/2.
+However, for SU(2)k, the content is more complex. In the case of SU(2)2 the theory has
+a) a trivial representation [0] (the identity, (j, m) = (0, 0)), b) a (spinor) representation [1/2]
+((j, m) = (1/2, ±1/2)), and c) a the representation [1] ((j, m) = (1, m), with m = 0, ±1). These
+states will fuse obeying the following rules: [0] × [0] = [0], [0] × [1/2] = [1/2], [0] × [1] = [1],
+[1/2]×[1/2] = [0]+[1], [1/2]×[1] = [1/2], and [1]×[1] = [0] (note the truncation of the fusion
+process!).
+Of particular interest is the case [1/2] × [1/2] = [0] + [1]. In this case we have two fusion
+channels, labeled by [0] and [1]. The braiding operations now will act on a two-dimensional
+Hilbert space and are represented by 2 × 2 matrices. This is an example of a non-abelian repre-
+sentation of the braid group. These rather abstract concepts have found a physical manifestation
+in the physics of the fractional quantum Hall fluids, whose excitations are vortices that carry
+fractional charge and anyon (braid) fractional statistics.
+Why this is interesting can be seen by considering a Chern-Simons gauge theory with four
+quasi-static Wilson loops. For instance in the case of the SU(2)2 Chern-Simons theory the Wilson
+loops carry the spinor representation, [1/2]. If we call the four particles A, B, C and D, we
+would expect that their quantum state would be completely determined by the coordinates of
+the particles. This, however, is not the case since, if we fuse A with B, the result is either a
+state [0] or a state [1]. Thus, if the particles were prepared originally in some state, braiding
+(and fusion) will lead to a linear superposition of the two states. This braiding process defines
+a unitary matrix, a representation of the Braid Group. The same is true with the other particles.
+However, it turns out that for four particles there are only two linearly independent states. This
+two-fold degenerate Hilbert space of topological origin is called a topological qubit.
+53
+
+Moreover, if we consider a system with N (even) number of such particles, the dimension of
+the topologically protected Hilbert space is 2
+N
+2 −1. Hence, for large N, the entropy per particle
+grows as 1
+2 ln 2 = ln
+√
+2. Therefore the qubit is not an “internal” degree of freedom of the
+particles but a collective state of topological origin. Interestingly, there are physical systems,
+known as non-abelian fractional quantum Hall fluids that embody this physics and are accessible
+to experiments! For these reasons, the non-abelian case has been proposed as a realization of a
+topological qubit [169, 170].
+10. Topological Phases of Matter
+10.1. Topological Insulators
+We will now give a brief discussion of the physics of Topological Insulators from a field-
+theoretic perspective. Topological insulators are systems whose electronic states (band struc-
+tures) have special topological properties which manifest in the existence of symmetry-protected
+edge states. For this reason these systems are known as symmetry-protected topological states
+(or SPTs). The simplest example is found in one space dimension where it is related to the fas-
+cinating problem of fractionally charged solitons and electron fractionalization. In several ways
+many of the concepts involved in these 1+1-dimensional systems can and have been extended to
+higher dimensions.
+10.1.1. Dirac fermions in 2+1 dimensions
+We will now see that, in spite of the formal similarities withe the 1+1 dimensional case, this
+theory has different symmetries, particularly concerning parity and time reversal invariance. In
+addition, in 2+1 dimensions it is not possible to define a γ5 Dirac matrix which implies that there
+is no chiral symmetry and no chiral anomaly. We will consider first a theory of a Dirac field
+which in 2+1 dimensions is a bi-spinor (as is in 1+1 dimensions). The Lagrangian of the Dirac
+theory coupled to a background gauge field Aµ is
+L = ¯ψiγµ∂µψ − m ¯ψψ − eAµ ¯ψγµψ ≡ ¯ψiγµDµ(A)ψ − m ¯ψψ
+(274)
+where ¯ψ = ψ†γ0, ¯ψγµψ ≡ jµ is the gauge-invariant (and conserved) Dirac current, and Dµ(A) =
+∂µ + ieAµ is the covariant derivative.
+In Eq.(274), γµ are the three 2 × 2 Dirac matrices which obey the algebra, {γµ, γν} = 2gµν,
+where gµν = diag(1, −1, −1) is the metric of 2+1 dimensional Minkowski spacetime. The Dirac
+matrices γµ can be written in terms of the three 2 × 2 Pauli matrices. For instance we can choose
+γ0 = σ3, γ1 = iσ2 and γ2 = −iσ1 which satisfy the Dirac algebra. Since the three gamma
+matrices involve all tree Pauli matrices, it is not possible to define a γ5 matrix which would
+anticommute with the gamma matrices. In this sense, it is not possible to define a chirality for
+bi-spinors in 2+1 dimensions
+We could have also chosen a different set of gamma matrices. For instance, we could have
+chosen γ0 = σ3, γ1 = iσ2 and γ2 = +iσ1 which also satisfy the Dirac algebra. These two choices
+are equivalent to the 2D parity transformation x1 → x1 and x2 → −x2. In other words, we can
+choose the three gamma matrices to be defined in terms of a right handed or a left handed frame
+(or triad). Thus, the choice of handedness of the frame used to define the gamma matrices can
+be regarded as a chirality.
+The parity transformation is also equivalent to a unitary transformation (a change of basis)
+U = γ1 = iσ1 which flips the sign of both γ2 and γ0. Thus, a parity transformation is equivalent
+to a change of the sign of γ0 or, what is the same, as changing the sign of the Dirac mass
+m → −m. Since the Dirac theory is charge conjugation invariant, the parity transformation is
+equivalent to time reversal. It is easy to see that in 2+1 dimensions the massive Dirac theory
+is not invariant under time reversal since this the single particle Dirac Hamiltonian H(p) for
+momentum p involves all three Pauli matrices,
+H(p) = α · p + βm
+(275)
+where, as usual, α = γ0γ and β = γ0. In fact, time reversal is equivalent to the change of the
+sign of the Dirac mass. These considerations also apply to the massless Dirac theory coupled to
+a background gauge field.
+54
+
+10.1.2. Dirac Fermions and Topological Insulators
+In addition systems in one space dimension, discussed in section 7.1, in which Dirac fermions
+naturally describe the low energy electronic degrees of freedom, Dirac fermions also play a role
+in many other system in Condensed Matter Physics. Such systems range from the nodal Bogoli-
+ubov fermionic excitations of d-wave superconductors in 2D and p-wave superfluids in 3D, to
+Chern and Z2 topological insulators in 2D and Z2 topological insulators and Weyl semimetals in
+3D. The also play a role in spin liquid phases of frustrated spin-1/2 quantum antiferromagnets,
+such as Dirac and chiral spin liquids. Dirac fermions play also a significant role in our under-
+standing of the compressible limit of 2D electron gases (2DEG) in large magnetic fields (see
+section 10.3). We will not cover here all of these examples. Instead we will focus of the 2D
+Chern topological insulators (which exhibit the anomalous quantum Hell effect) and on the 3D
+Z2 topological insulators.
+In Condensed matter Physics the important low energy electronic degrees of freedom belong
+to the electronic states close to the Fermi energy. In such systems the lattice periodic potential
+determines the properties of their band structures. The only significant exception to this general
+rule is the case of the 2DEGs in GaAs-AlAs heterostructures (the most commonly used platform
+for the study of quantum hall effects) whose electronic densities are low enough so that the lattice
+periodic potential can for practical purposes almost always be ignored.
+In general, Dirac (and Weyl) fermions arise when the conduction and valence bands cross
+at isolated points of the Brillouin zone. In such situations, the near the crossing points the low
+energy electronic states locally (in momentum space) look like cones and, ignoring possible
+anisotropies, look like the states of a Dirac-Weyl theory. For these reasons, Dirac fermions play
+a central role in the theory of graphene type materials [171] and, more significantly, in the theory
+of topological insulators [172].
+Dirac fermions play a central role in Quantum Electrodynamics and in the Standard Model of
+Particle Physics. The problem of quark confinement requires an understanding of these theories
+in a regime inaccessible to Feynman-diagrammatic perturbation theory and an intrinsically non-
+perturbative formulation, known as Lattice Gauge Theory [55, 56, 173], needed to be developed.
+For this reason in the 1970s fermionic Hamiltonians with crossings at points became of great
+interest in High Energy Physics as a way to describing the short-distance dynamics of quarks in
+Lattice Gauge Theory.
+This program run into difficulties when extended to the theory of Weak Interactions which
+have an odd number of species of chiral (Weyl) Dirac fermions. All local discretizations of the
+Dirac equation yield an even number of species. This fact came to be known as the fermion
+doubling problem. These results are special cases of a general theorem, due to Holger Nielsen
+and Masao Ninomiya [174, 175, 176] (and extended by Daniel Friedan [177]), which proves that
+for systems whose kinetic energy is local in space (as it must be) there must always be an even
+number of crossings. Therefore, it is impossible to write a local theory with an odd number of
+chiral fermionic species.
+In the context of Condensed Matter Physics, the simplest example of Dirac fermions is found
+in the electronic structure of graphene (a single layer of graphite) discussed by A. H. Castro Neto
+and coworkers [171]. Graphene is an allotrope of crystalline carbon in which the carbon atoms
+are arranged in a 2D hexagonal lattice, which is a lattice with two inequivalent sites in its unit
+cell, labeled by A and B. Let rA and rB be the sites of the two sublattices. Each site rA has three
+nearest neighbors on the B sublattice located at ri
+B = rA + di, where i = 1, 2, 3 and
+d1 =
+� 1
+2
+√
+3
+, 1
+2
+�
+,
+d2 =
+� 1
+2
+√
+3
+, −1
+2
+�
+,
+d3 =
+�
+− 1√
+3
+, 0
+�
+(276)
+(in units in which the lattice spacing is a = 1). The nearest neighbor A sites are related by the
+vectors
+a1 =
+
+√
+3
+2 , −1
+2
+ ,
+a2 = (0, 1),
+a3 =
+−
+√
+3
+2 , −1
+2
+
+(277)
+where a1 and a2 are the primitive lattice vector of the hexagonal lattice. The nearest neighbor
+sites of the B sublattice are also related by these same three vectors.
+The low energy electronic degrees of freedom of graphene can be described by a single
+fermionic state (ignoring spin) at each carbon atom . Let c†(rA) and c†(rB) be the fermion oper-
+ators that creates an electron at the site rA and at the site nearest neighbor sites rB, respectively.
+The Hamiltonian is
+H = t1
+�
+rA,i=1,2,3
+c†(rA)c(rA + di) + h.c.
+(278)
+55
+
+where t1 is the hopping matrix element between nearest neighbor A and B sites.
+In Fourier space
+c(rA) =
+�
+BZ
+d2k
+(2π)2ψA(k) exp(ik · rA),
+c(rB) =
+�
+BZ
+d2k
+(2π)2 ψB(k) exp(ik · rB)
+(279)
+where BZ is the first Brillouin zone of the hexagonal lattice, a hexagonal region of reciprocal
+space with vertices at ±2π/
+√
+3(1, 1/
+√
+3) (which are denoted by K and K′, respectively) and their
+two images under 2π/3 rotations. In momentum space the Hamiltonian is [178]
+H = t1
+�
+BZ
+d2k
+(2π)2
+�
+ψA(k)
+ψB(k)
+� �
+0
+�
+j=1,2,3 exp(ik · d j)
+�
+j=1,2,3 exp(−ik · d j)
+0
+� �ψA(k)
+ψB(k)
+�
+(280)
+The one-particle states of this Hamiltonian have eigenvalues E(k) = ±
+����
+�
+j=1,2,3 exp(ik · d j
+����.
+Clearly E(k) vanishes at the K and K′ points, the corners of the Brillouin zone. Hence, there
+is a crossing of the two bands at the K and K′ points near which the energy eigenvalues are a two
+cones at K and K′, respectively. Thus, the low energy states of graphene consists of two massless
+(gapless) Dirac bi-spinors, with opposite chirality/parity.
+Other examples with similar fermion content are a theory of fermions on a 2D square lattice
+with a π magnetic flux (1/2 of the flux quantum) per plaquette which arises, for instance, in the
+theory of the integer quantum Hall effect on lattices by David Thouless, Mahito Kohmoto, Peter
+Nightingale and Marcel den Nijs [179] (albeit for more general flux per plaquette), and in the
+theory of flux phases of 2D spin liquids of Ian Affleck and J. Brad Marston [180]. It also arises
+in the theory of the chiral spin liquid of Xiao-Gang Wen, Frank Wilczek and Anthony Zee [181].
+We will discuss these theories in section 10.
+10.1.3. Chern invariants
+The single-particle quantum states of free fermions (electrons) in the periodic potential of a
+crystal are Bloch wave functions labeled by the lattice momentum k and band indices m. The
+Bloch states are periodic functions of the lattice momenta whose periods are the Brillouin zones.
+Topologically each Brillouin zone is a torus: in 1D is a 1-torus, in 2D a 2-torus, etc. The sym-
+metries (and shapes) of the Brillouin zone are dictated by the symmetries of the host crystal.
+In this section we will consider the quantum states of systems of fermions on a 2D lattice with
+M bands with eigenvalues {Em(k)} with m = 1, . . ., M. The eigenstates are Bloch states {|um(k)⟩}
+such that the wave functions ar ψm(x) = um(k) exp(ik · x), where k is a (quasi) momentum in the
+first Brillouin zone [182]. We will assume that the band spectrum is such that all the bands are
+separated from each other by a finite gap for all momenta in the first Brillouin zone. Hence, the
+eigenvalues obey the inequality |Em(k) − En(k)| > 0. We will assume that the system of interest
+has N < M filled bands and, consequently, that there is a gap separating the top-most occupied
+band N (the “valence band”) and the lowest unoccupied band N + 1 (the “conduction band”) that
+does not close everywhere in the first Brillouin zone.
+Let us consider specifically a 2D system and a Bloch state |um(k)⟩ at a momentum k, and let
+∂k|um(k)⟩ be an infinitesimally close Bloch state with momentum k′ = k+dk. The inner product
+of these two infinitesimally close Bloch states is the vector field (defined on the first Brillouin
+zone for each band m)
+A(m)
+j (k) = i⟨um(k)|∂ j|um(k)⟩
+(281)
+where j = 1, 2 are the orthogonal direction in momentum space and we have used the notation
+∂i = ∂/∂ki. This vector field is known as the Berry Connection of the electronic states in the mth
+band.
+The theory of the quantum states of non-interacting electrons in periodic potentials as devel-
+oped by Felix Bloch [182] is the foundation of much of what we know about the physics of many
+semiconductors and simple metals. This theory is the core subject of many textbooks [183, 184].
+This theory is based on the classification of the electronic states as representations of the group
+of lattice translations and point (and spatial) group symmetry transformations. A key unstated
+assumption in much of this body of work is that the Bloch states are globally well-defined func-
+tions on the Brillouin zone of a given crystal. It is rather remarkable that this assumption was not
+stated explicitly since the original work by Bloch in 1929 until the work by Thouless, Kohmoto,
+Nightingale and den Nijs on the quantum states of electrons on 2D lattices in the presence of an
+uniform magnetic field [179]. The realization that there is a topological obstruction to define the
+56
+
+electronic states globally in the Brillouin zone is the key to the development of the the theory of
+topological insulators and, more generally, of the modern theory of band structures [185, 186].
+In quantum mechanics states are defined up to a phase. This means that Bloch states that
+differ by a phase describe the same quantum state. In other words each quantum state |um(k)⟩ is
+a member of a ray (or fiber) of states each labeled by a phase,
+|um)(k)⟩ �→ exp(ifm(k)) |um(k)⟩
+(282)
+Changing the states by phase factors defines a U(1)M unitary transformation of physically equiva-
+lent states. Since we have a fibre at each point k, this amount at defining a fibre bundle. However,
+under this transformation the Berry connection A(m)
+j (k) changes by a gradient of the phases
+A(m)
+j (k) �→ A(m)
+j (k) + ∂ j fm(k)
+(283)
+where we assumed that the phases fm(k) are continuous and differentiable functions on the entire
+first Brillouin zone. Since the physical states cannot change by redefinitions of the phases of
+the basis states we are led to the condition that only the data that is invariant under the gauge
+transformations defined by Eqs. (282) and (283) is physically meaningful, which is encoded in
+the gauge-invariant pseudo-scalar quantity F (m)(k)
+F (m)(k) = ǫi j∂iA(m)
+j (k)
+(284)
+which is known as the Berry curvature.
+In what follows we will be interested in the quantity
+C(m)
+1
+= 1
+2π
+�
+Γ
+dk jA(m)
+j (k) = 1
+2π
+�
+BZ
+d2k F (m)(k)
+(285)
+where Γ is the boundary of the 1st Brillouin zone (which we denote by BZ). We will now show
+that the quantity C(m)
+1 , which measure the flux of the Berry connection A(m)
+j (k) through the first
+Brillouin zone (in units of 2π), is an integer independent of the particular connection A(m)
+j (k) that
+we have chosen. This integer-valued quantity is a topological invariant known as the first Chern
+number.
+However, since the first Brillouin zone is a 2-torus ,which is a closed manifold, a Berry
+connection with a net flux cannot obey periodic boundary conditions. Instead, we must allow
+for generalized (large) gauge transformations that wrap around the 2-torus of the first Brillouin
+zone. This problem is similar (and closely related to) the problem of the wave functions of a
+charged particle moving in the presence of a magnetic monopole [151]. We will consider a 2D
+system whose first Brillouin zone is spanned by two reciprocal lattice vectors b1 and b2, related
+by the two primitive lattice vectors a1 and a2 by the relation bi · a j = 2πδi j (with i, j = 1, 2) The
+generalized gauge transformations now are
+|um(k + b1)⟩ = exp(if (1)
+m (k)) |um(k)⟩,
+|um(k + b2)⟩ = exp(if (2)
+m (k)) |um(k)⟩
+A(m)
+j (k + b1) =A(m)
+j (k) + ∂ j f (1)
+m (k),
+A(m)
+j (k + b2) = A(m)
+j (k) + ∂ j f (2)
+m (k)
+(286)
+where f (1)
+m (k) and f (2)
+m (k) are smooth functions of k.
+We will now show that the Bloch states cannot be defined globally over the Brillouin zone if
+the flux of the Berry connection does not vanish. More specifically let us assume that at some
+point k0 ∈ T we know the Bloch state |um(k0)⟩ which satisfies the generalized periodic boundary
+conditions of Eq.(286). The question is, can we determine the Bloch state at some other also
+arbitrary point k′
+0? We will now that there is a topological obstruction that does not allow it if
+the flux of the Berry phase is not zero. The reason is that the phase of the Bloch state cannot be
+defined since, in general, the Bloch state will vanish at some point of the Brillouin zone where
+the phase is undefined.
+We will prove that this si true by following an elegant construction due to Mahito Kohmoto
+which goes as follows [187]. The Brillouin zone is a torus, that we will denote by T, can be
+regarded as the tensor product of two circles, T ≡ S 1 × S 1. Let us split the torus T into two
+disjoint subsets (or patches) HI and HII such that T = HI
+� HII. We will assume that in region
+HI. Let us assume that the Bloch state vanishes at some point k0 ∈ TI and that the Bloch state
+does not vanish for all points k ∈ HII. This means that we can choose the Bloch state |um(k)⟩
+to be real for all k ∈ HII. On the other hand we can always assign some arbitrary phase to the
+57
+
+Bloch state at k0 ∈ TI. Once we have done that we can extend the definition of the phase to a
+neighborhood of k0 wholly included in HI. If we the assume that there is only one zero, then
+the phase can be defined over all of HI. The result is that we now have two different definitions
+of the phase of the Bloch state on HI and on HII. Let |um(k)⟩I and |um(k)⟩II be the two resulting
+definitions of the Bloch state. Let the closed curve γ be the common boundary of regions HI
+and HII, γ = ∂HI = ∂HII. However on the common boundary γ these the two definitions of the
+Bloch state must be a gauge transformation,
+|um(k)⟩I = exp(ifm(k)) |um(k)⟩II
+(287)
+on all points k ∈ γ. The gauge transformation fm(k), known as the transition function, is a
+smooth periodic function of the points k on the closed curve γ. Likewise, the Berry connection
+A(m)
+j (k) has two definitions on the regions HI and HII which differ by a gauge transformation on
+all the points k of the common boundary γ,
+A(m)
+j (k)
+����I − A(m)
+j (k)
+����II = ∂ j fm(k)
+(288)
+The transition function defines a mapping of the closed curve γ, which is topologically equivalent
+to a circle S 1, to the phase of the Bloch state which is defined mod 2π and hence is also a circle
+S 1. Hence the transition functions fm(k) are homotopies which can be classified by the homotopy
+group Π1(S 1) ≃ Z. We recognize that the classification of the the transition functions is the same
+that we used for the vortices of section 5.1.1. Hence, the change of the transition function on a
+full revolution of the closed curve γ must be an integer multiple of 2π.
+We will now compute the quantity C(m)
+1 , given in Eq.(285), which measures the flux of the
+Berry connection through the 2-torus T that defines the first Brillouin zone. Using the partition
+of the torus T = TI
+� TII we can write
+C(m)
+1
+= 1
+2π
+�
+T
+d2k F (m)(k) = 1
+2π
+��
+HI
+d2k F (m)(k) +
+�
+HII
+d2k F (m)(k)
+�
+(289)
+Using Stokes Theorem on the two regions HI and HII we get
+C(m)
+1
+= 1
+2π
+�
+γ
+dk j
+�
+A j(k)
+����I − A j(k)
+����II
+�
+(290)
+Since the two definitions of the Berry connection differ by a gauge transformation, Eq.(288), we
+can express C(m)
+1
+in terms of the transition function fm(k) on the curve γ
+C(m)
+1
+= 1
+2π
+�
+γ
+dk · ∂fm(k) = 1
+2π (∆fm(k))γ = n
+(291)
+where we used that the transition functions are classified by the integer-valued quantity n ∈ Z.
+We conclude that the flux of the Berry curvature C(m)
+1
+takes only integer values. It is known of
+the first Chern number C(m)
+1
+which is a topological invariant since it cannot change by a smooth
+redefinition of the curve γ. Since C(m)
+1
+� 0 implies that the Bloch states must vanish at least
+at some point (or points) k ∈ HI, then the integer-valued Chern number can only change if a
+redefinition of the curve γ crosses at least one point k at which the Bloch state vanishes.
+The conclusion of this analysis is that whenever the flux of the Berry curvature through the
+Brillouin zone does not vanish the Bloch states cannot be defined globally on the Brillouin zone.
+A direct consequence is that in this case the band is characterized by a topological invariant
+called the first Chern number. This number is a property of a given band and, in general, it is
+different for each band. This is a particular case of a more general topological classification of
+the states of free fermions on a lattice which depends on on the dimensionality of the system
+[188, 189, 190].
+10.1.4. The quantum Hall effect on a lattice
+We will now show that 2D free fermion systems with Chern bands, i.e. bands characterized
+by a non-vanishing Chern number, are insulators that have a quantized Hall conductivity. We will
+do this for the theory of the integer quantum Hall effect on a 2D lattice of Thouless, Kohmoto,
+Nightingale and den Nijs (TKNN) [179, 187] (see, also, Ref. [191]). Although this problem
+played a key role in the development of the theory of topological phases of matter, for many
+58
+
+years it was viewed as an academic problem since it would require gigantic magnetic fields to
+be in the regime of interest for typical solids. The situation changed with the development of
+twisted bilayer graphene and similar systems which have unit cells large enough for this physics
+to be observable. These new materials has allowed to study this problem experimentally.
+TKNN considered a system of free fermions on a planar (actually square) lattice in a uniform
+magnetic field B with flux 2πp/q per plaquette (in units in which the flux quantum φ0 = eℏ/c = 1)
+with p and q two co-prime integers. The tight-binding Hamiltonian for this simple problem is
+H =
+�
+r, j=1,2
+t j c†(r) eiA j(r) c(r + e j) + h.c.
+(292)
+where c(r) and c†(r) are fermion creation and annihilation operators on the lattices sites {r =
+(m, n)}, e j are lattice unit vectors, and t j, with j = x, y, is a hopping matrix element between
+nearest neighbor sites of the square lattice along the x and y directions. On each link of the
+lattice we defined a vector potential A j(r), where the oriented sum of the lattice vector potentials
+on each plaquette 2πφ, where φ = p
+q, is the flux on each plaquette of the square lattice. In the
+axial (Landau) gauge A1(m, n) = 0, A2(m, n) = 2πmφ is a periodic function of m with period q.
+In this gauge the Hamiltonian becomes
+H =t
+�
+m,n
+�
+c†(m + 1, n)c(m, n) + c†(m, n)c(m + 1, n)
+�
++t
+�
+m,n
+�
+c†(m, n + 1) ei2πφm c(m, n) + c†(m, n) e−i2πφm c(m, n + 1)
+�
+(293)
+In this gauge the problem reduces to a system with a theory with a q × 1 unit cell with q inequiv-
+alent sites. In momentum space, the (magnetic) Brillouin zone of this system is − π
+q ≤ k1 ≤ π
+q and
+−π ≤ k2 ≤ π. The Fourier transform of the operators c†(r) and c(r) are, respectively, c†(k) and
+c(k) which obey the standard anticommutation relations, {c(k), c(q)} = {c†(k), c†(q)} = 0, and
+{c†(k), c(q)} = δ(k − q). In momentum space the Hamiltonian becomes
+H = q
+� π/q
+−π/q
+dkx
+2π
+� π
+−π
+dky
+2π
+�H(kx, ky)
+(294)
+where
+�H(kx, ky) =1
+q
+q−1
+�
+n=0
+�
+2tx cos(kx + 2πφn) c†(kx + 2πφn, ky) c(kx + 2πφn, ky)
++ ty
+�
+e−iky c†(kx + 2π(n + 1)φ, ky) c(kx + 2πnφ, ky) + eiky c†(kx + 2π(n − 1)φ, ky) c(kx + 2πnφ, ky)
+��
+(295)
+The spectrum of this system was first investigated by Hofstadter [192] consists of q bands which
+for q an odd integer are separated by finite energy gaps.
+For fixed values of kx and ky (in the magnetic Brillouin zone), the Hamiltonian �H(kx, ky) is the
+tight-binding model of a one-dimensional chain on the 1D lattice of q sites located at kx + 2πnφ
+with nearest neighbor hopping. The single-particle (Bloch) states un(k) of this 1D model obeys
+the Schr¨odinger Equation
+2tx cos(kx + 2πnφ)un(k) + ty
+�
+e−iky un−1(k) + eiky un+1(k)
+�
+= Enun
+(296)
+which is known as Harper’s equation. In general this equation does not admit an analytic solution.
+However, the nature qualitative features of the spectrum can be obtained by an expansion either
+on tx/ty or on ty/tx. TKNN used degenerate perturbation theory to show that the spectrum has q
+bands and that the r-th band is characterized by two integers sr and ℓr which are the solution of
+the Diophantine equation
+r = q sr + p ℓr
+(297)
+with ℓ0 = s0 = 0. It isl also obvious that for r = q sq = 1 and ℓq = 0 (for all p).
+Furthermore, TKNN showed that there was an, until then unsuspected, relation between the
+Hall conductivity σxy of a gapped system of electrons in periodic potentials in a uniform magnetic
+field and the Berry connection of the filled bands. This relation implies the quantization of the
+59
+
+Hall conductivity and its computation in terms of a topological invariant, the Chern number of the
+occupied bands. They showed that in this context each band has a non-trivial Berry connection
+A(r)
+j = i⟨ur(k)|∂ j|ur(k⟩ (with ∂ j = ∂kj) of the form of Eq.(281) with a non-vanishing (first) Chern
+number C(r)
+1 , i.e. the flux through the magnetic Brillouin zone.
+The conductivity tensor characterizes the electrical properties of a physical system. Linear
+Response Theory provides a framework for computing the conductivity tensor by perturbing the
+system with a weak external electromagnetic field Aµ and computing the currents that they induce
+[193, 10]. The expectation value of the gauge-invariant current operator Jµ(x) is
+⟨Jµ(x)⟩ = − i
+ℏ
+δ
+δAµ(x) ln Z[Aµ]
+(298)
+where Z[Aµ] is the partition function in the presence of a background (i.e. classical) electro-
+magnetic field Aµ(x). In general spacetime dimension D = d + 1, the lowest order in the vector
+potential Aµ, the induced current is given in terms of the polarization tensor Πµν(x, y),
+ln Z[Aµ] = i
+2
+�
+dDx
+�
+dDy Aµ(x) Πµν(x, y) Aν(y) + O(A3)
+(299)
+Therefore, the induced current is related to the external field by
+⟨Jµ(x)⟩ ≡ Jµ(x) =
+�
+dDy Πµν(x, y) Aν(y)
+(300)
+Expressions of this type are know as a Kubo formula. The polarization tensor can be regarded as
+a generalized susceptibility.
+Although we are using a continuum relativistic notation, these expressions are generally
+valid, even in lattice systems. Gauge invariance requires that the polarization tensor be con-
+served,
+∂µΠµν(x, y) = 0
+(301)
+In general, the (retarded) polarization tensor Πµν(x, y) is related to the the (retarded) current-
+current correlation function DR
+µν(x, y),
+Dµν(x, y) = − i
+ℏΘ(x0 − y0)⟨[Jµ(x), Jν(y)]⟩
+(302)
+by the identity
+ΠR
+µν(x, y) = DR
+µν(x, y) − iℏ
+�δJµ(x)
+δAν(y)
+�
+(303)
+The last term in Eq.(303), usually called a “contact term”. This term vanishes only for a theory
+of relativistic fermions (in the continuum). In all other cases, lattice or continuum, relativistic
+or not, the contact term does not vanish and its form depends on the specific theory. In non-
+relativistic systems, e.g. in a Fermi liquid, this term is the origin of the f-sum rule [193, 194].
+When the external field Aµ represents is a (locally) uniform electric field E, the induced
+current is Ji = σi jE j, where σi j is the conductivity tensor, which can be obtained from the
+polarization tensor as the limit
+σi j = lim
+ω→0
+1
+iω lim
+q→0 Πi j(ω, q)
+(304)
+In a metal, which has a Fermi surface, the order of limits in which ω and q vanish matters and
+only the order of limits specified above is the correct one to take. Ina Dirac systems, that we
+will discuss below, the order does not matter due to relativistic invariance. The other case in
+which the order does not matter is the Chern insulator that we are interested in. In general, in an
+isotropic system, the conductivity tensor has a symmetric part and an antisymmetric part. The
+symmetric part of the conductivity tensor yields the longitudinal conductivity which has all the
+effects of dissipation. The antisymmetric part does not vanish in the presence of a magnetic field
+or, more generally, if time reversal symmetry is broken, and yields the Hall conductivity.
+A Chern insulator is an insulator and as such is a state with an energy gap. In such a state the
+longitudinal conductivity vanishes since there is no dissipation in a gapped state. But in a Chern
+insulator time reversal invariance is broken. In the system that we are discussing is broken by
+60
+
+the magnetic field. The Hall conductivity can be calculated from the antisymmetric part of the
+polarization tensor as the limit
+σxy = lim
+ω→0
+i
+ωΠxy(ω, 0)
+(305)
+In addition, the contact term does not contribute to the Hall conductivity. As a result the hall
+conductivity can be computed in terms of the antisymmetric component of the current-correlation
+function.
+let us now return to the theory of free fermions on a lattice in a (commensurate) magnetic
+field takes. As we saw the electronic states are split into q bands with single particle states
+|ψn(k)⟩ where k takes values on the first magnetic Brillouin zone. We will compute the Hall
+conductivity for this system assuming that the Fermi energy EF lies in the gap between the n-th
+and the n − 1-th bands. Let us label by α the occupied bands and by β the unoccupied bands.
+Hence Eα(k) < EF < Eβ(k). In this case, the Kubo formula for the Hall conductivity becomes
+σxy = ie2
+ℏ
+�
+Eα<EF<Eβ
+�
+BZ
+d2k
+(2π)2
+⟨uα(k)|Jy|uβ(k)⟩⟨uβ(k)|Jx|uα(k)⟩ − ⟨uα(k)|Jx|uβ(k)⟩⟨uβ(k)|Jy|uα(k)⟩
+(Eα(k) − Eβ(k))2
+(306)
+In the lattice model the current operator is simply
+J = e
+ℏ∂k �H(k)
+(307)
+Using this form of the current and the Schr¨odinger equation of Eq.(296) we can rewrite the Kubo
+formula for the Hall conductivity, Eq.(306), as a sum over only the occupied states labeled by α.
+The expression for the Hall conductivity now takes the more compact form
+σxy = e2
+ℏ
+�
+α
+�
+BZ
+d2k
+(2π)2ǫjl∂ j⟨uα(k)|∂k|uα(k)⟩ = e2
+h
+�
+α
+1
+2π
+�
+BZ
+F (α)(k) = e2
+h
+�
+α
+C(α)
+1
+(308)
+Therefore, we conclude that each occupied band contributes to the Hall conductivity an amount
+equal to its first Chern number (multiplied by e2/h). This is the main result of TKNN [179].
+In the specific case of the model that we are discussing here the Chern number is given
+explicitly in terms of the solution of the Diophantine equation, Eq.(297)
+C(r)
+1 = 1
+2π
+�
+Γ
+dk jA(r)
+j (k) = ℓr − ℓr−1
+(309)
+where the integers ℓr and ℓr−1 are the solutions of the Diophantine equation for the bands r and
+r − 1 (A detailed derivation can be found in Ref.[9], chapter 12). The TKNN integers ℓr are
+topological invariants of the bands.
+We conclude, with TKNN, that for a system with r filled bands the Hall conductivity σ(r)
+xy is
+given by the expression
+σ(r)
+xy = e2
+h
+r�
+n=1
+(ℓr − ℓr−1) = e2
+h ℓr
+(310)
+thus showing that this system exhibits and integer quantum Hall effect and that the Hall conduc-
+tivity is (up to the ratio of universal constants e2/h) given by a topological invariant, the first
+Chern number of the band.
+We conclude with two comments. The first is that when all the bands are occupied the
+Hall conductivity vanishes. This obvious physical constraint is automatically satisfied by the
+Diophantine equation. The other comment concerns the case of what happens when q is an even
+integer. In this case it is possible to show that there are band crossings. This is what happens for
+q = 2 which is the π flux lattice. in this case one has a theory of two species of Dirac bi-spinors.
+As we will see in section 10.1.5 what happens depends on how the gapless state is reached.
+10.1.5. The Anomalous Quantum Hall Effect
+We will now discuss a simple and insightful model of a Chern insulator that displays the
+quantum anomalous Hall effect, that is a Hall effect in a system with broken time reversal sym-
+metry in the absence of a uniform magnetic field. In order to do that we will reconsider the
+graphene model with additional terms in the Hamiltonian of Eq.(280) that will open up energy
+gaps in its spectrum. There are several ways to do this. One way requires breaking the symme-
+try between the two sublattices A and B, by assigning them a site energy of ±M, which breaks
+61
+
+the mirror reflection symmetry of the hexagon. This is what happens with graphene on a boron
+nitrade substrate [178] (there are other ways that involve breaking the point group symmetry
+[195]). Another way to open a gap is to break time reversal symmetry which is accomplished by
+complex hopping amplitudes t2 exp(iφ) between nearest neighbor A sites (and between nearest
+neighbor B sites as well). The phase φ is oriented from rA to rA+ai, where a1 = d2−d3, etc. (and
+similarly for the B sites). Physically, the oriented sum of the phases φ on the triangles generated
+by the hopping terms t1 and t2 represent the magnetic fluxes through each triangle such that the
+total flux on the elementary hexagon is zero. This model was considered by Haldane [196]. With
+these changes the one-particle Hamiltonian becomes 2 × 2 Hermitian matrix
+H(k) = h0(k) 1 + h(k) · σ
+(311)
+where I is the 2 × 2 identity matrix and σ = (σ1, σ2, σ3) is a vector made of the three Pauli
+matrices. The functions h0(k) and h(k) = (h1(k), h2(k), h3(k) are, respectively, given by
+h0(k) =2t2 cos φ
+�
+i=1,2,3
+cos(k · ai),
+h1(k) = t1
+�
+i=1,2,3
+cos(k · di),
+h2(k) =t1
+�
+i=1,2,3
+sin(k · di),
+h3(k) = M − 2t2 sin φ
+�
+i=1,2,3
+sin(k · ai)
+(312)
+Provided |t2/t1| < 1/3 the spectrum of the Hamiltonian of Eq.(280) has two bands that do not
+overlap and have a finite energy gap unless they cross at the corners K and K′ the Brillouin zone
+where the gap is smallest. The two bands cross at K and/or K′ if M = ±3
+√
+3t2 sin φ (for K and K′,
+respectively). When either one of these conditions are not met the spectrum is gapped at either K
+or K′ or, more generally, at both. For values of the parameters for which the band gaps are small,
+the low energy states of the Hamiltonian of Eq.(280) is that of a theory of two Dirac fields, that
+we will label as ψ1(x) and ψ2(x), whose one-particle Hamiltonians have the form of Eq.(275) but
+with generally different masses m1 ∝ M − 3
+√
+3t2 sin φ and m2 ∝ M + 3
+√
+3t2 sin φ. So, in general,
+both fields will have non-vanishing masses which can have the same sign or opposite sign.
+We conclude that the effective low-energy theory consist of two species of massive Dirac
+bi-spinors, albeit generally with different masses. To examine the electromagnetic response we
+will couple the lattice model and the effective low energy theory to a slowly varying background
+electromagnetic field Aµ. The Lagrangian of the low energy theory is
+L = ¯ψ1(i/∂ − m1)ψ1 + ¯ψ2(i/∂ − m2)ψ2 − eAµ( ¯ψ1γµψ1 + ¯ψ1γµψ1)
+(313)
+In general the two masses will be different in magnitude and sign.
+Equivalently the Dirac
+fermions may have the same or opposite chirality. We will examine the electromagnetic re-
+sponse for both cases in section 10.1.6 where we will see that in the phase in which the masses of
+both Dirac fermions have the same sign this model describes a topological insulator that exhibits
+the anomalous quantum Hall effect, whereas in the phase where the signs are opposite it is a
+conventional insulator.
+However, by varying the parameters we can tune to special values of these parameters where
+one of the two masses vanishes. In section 10.1.6 we will see that these special values of the
+parameters correspond to quantum phase transitions at which the topological properties of the
+ground state change. We will see that this is a quantum critical point with special properties.
+In particular, time reversal invariance is explicitly broken at this quantum critical point. In this
+regime the heavy Dirac fermion plays the role of the heavy regulating field as in the Pauli-Villars
+regularization of the theory of a single Dirac fermion. In the lattice model this is a manifestation
+of the Nielsen-Ninomiya theorem which requires that there should be an even number of Dirac
+fermions [174, 175, 177]. In this context, the heavy fermion is the “doubler”.
+10.1.6. The Parity Anomaly
+We saw in section 7.3 that Dirac fermions in 1+1 dimensions have a chiral anomaly. That
+statement mean that although the theory of free massless Dirac fermions has a global chiral sym-
+metry, the associated current, which we called j5
+µ, is not conserved when the system is coupled
+to an external gauge field. This effect happened because the right and left moving (Weyl) com-
+ponents of the massless Dirac fermion cannot be separately conserved in the presence of a gauge
+field.
+Analogs of the 1+1-dimensional chiral anomaly exist on 3+1 dimensions (and, more gener-
+ally, in all even spacetime dimensions). The cancelation of these anomalies is a key condition
+62
+
+for the consistency of gauge theories [96]. We will see in section 10 that these anomalies play a
+role on the theory of 3D topological insulators. More generally, anomalies play a central role in
+the theory of symmetry-protected topological (SPT) phases of matter [197, 198, 199].
+Dirac fermions in 2+1 dimensions do not have an analog of the chiral anomaly for the simple
+reason that the chiral symmetry does ot exist. There is, however, an anomaly involving parity
+or, equivalently, time reversal invariance, known as the parity anomaly. The parity anomaly was
+discovered in the computation of the effective action of the electromagnetic field in a theory of
+massive Dirac fermions in 2+1 dimensions by Redlich [200, 201].
+Consider theory on a single massive Dirac fermion (a bi-spinor) ψ(x) in 2+1 dimensions
+coupled to a background U(1) gauge field Aµ. The Lagrangian is
+L[ψ, ¯ψ, Aµ] = ¯ψi /Dψ − m ¯ψψ
+(314)
+where the covariant derivative is Dµ = ∂µ + iAµ. The effective action S eff[Aµ] of the gauge field
+Aµ is formally obtained by computing the path integral
+Z[Aµ] = exp(iS eff[Aµ]) =
+�
+DψD ¯ψ exp(i
+�
+d3x L[ψ, ¯ψ, Aµ]) = Det(i /D[Aµ] − m)
+(315)
+where we used that the theory is free to write the partition function as a functional determinant. In
+1+1 dimensions, in the case of a massless theory, this determinant can be computed exactly using
+the heat kernel (or zeta-function) technique [202, 203]. Aside from important technicalities, the
+reason for why this works is that as a consequence of the chiral anomaly, the effective action
+computed at one loop order in the fermions is exact in 1+1 dimensions. In higher dimensions
+this is no longer true.
+We will sketch the computation of the effective action to one loop order, which means
+quadratic in the gauge field Aµ, which has the form
+S eff[Aµ] = 1
+2
+�
+d3x
+�
+d3y Aµ(x) Πµν(x − y) Aν(y) + . . .
+(316)
+where the ellipsis contains the higher order contributions. The polarization operator Πµν(x − y)
+in momentum space is (here pµ is the momentum transfer) by the one-loop result
+Πµν(p) =
+�
+d3q
+(2π)3 tr �S (q + p/2)γµS (q − p/2)γν�
+(317)
+The symbol tr signifies the trace over the spinor labels and S (p) is the (massive) Feynman prop-
+agator
+S (p) =
+1
+/p − m + iǫ
+(318)
+The polarization tensor Πµν(p) has the explicit decomposition into a parity even and a parity odd
+contributions
+Πµν(p) = (p2gµν − pµpν) Π0(p2) − iǫµνλpλ ΠA(p2)
+(319)
+where p2 = p2
+0 − p2. The polarization tensor is manifestly transverse, pµΠµν(p) = 0 and, hence,
+the one-loop effective action of Eq.(316) is gauge-invariant (as it should be). The parity even
+Π0(p2) and parity odd ΠA(p2) kernels are given by
+Π0(p2) = − |m|
+4πp2 +
+1
+8π
+�
+p2
+�4m2
+p2 + 1
+�
+sinh−1
+
+�
+p2
+�
+4m2 − p2
+
+(320)
+ΠA(p2) = −
+m
+2π
+�
+p2 sinh−1
+
+�
+p2
+�
+4m2 − p2
+
+(321)
+These two results are obtained in the limit in which the regulator scales are sent to infinity leaving
+behind these finite results, which hold provided the Dirac mass m does not vanish.
+In the low energy regime, |p2| ≪ m2, the kernels take the asymptotic values
+Π0(p2) ≃
+1
+16π|m|
+ΠA(p2) ≃ − 1
+4πsgn(m)
+(322)
+This result is the leading contribution to the parity even term (the first term) up to corrections
+of the order of p2/m2. However, the second term, which is parity odd, is the exact contribution
+63
+
+in the limit p → 0 for all non-vanishing values of the Dirac mass. In this regime, the effective
+Lagrangian of the gauge field Aµ takes the local form of the sum of a Maxwell term and a Chern-
+Simons term
+Leff[Aµ] = −
+1
+4π|m|FµνFµν ± 1
+2
+1
+4π ǫµνλ Aµ∂νAλ
+(323)
+where Fµν = ∂νAν − ∂νAµ is the field strength of the gauge field. Clearly, since the Maxwell term
+has two derivatives while the Chern-Simons term has one derivative, in the long distance regime
+the Maxwell term is irrelevant (subdominant). Also, while the prefactor of the Chern-Simons
+term is a pure number, the prefactor of the Maxwell term is proportional to
+1
+|m| (as required by
+dimensional analysis).
+The second term in the Lagrangian of Eq.(323) is a Chern-Simons term for the background
+gauge field Aµ [163, 164]. This term is gauge-invariant (up to boundary terms) but breaks parity
+and time reversal invariance. This phenomenon is called the parity anomaly [200], although
+Witten [199] has argued that it is better to call this phenomenon a time-reversal anomaly. The
+sign of the prefactor of the Chern-Simons term depends of the sign of the Dirac mass m: positive
+for m > 0 and negative for m < 0 or, alternatively for positive and negative chirality of the Dirac
+(bi-spinor) field which fixes the sign of the breaking of time reversal invariance.
+In addition to the sign, the prefactor of the Chern-Simons term is equal to 1
+2, and it is not
+an integer. In section 9.6 we showed that for a dynamical Chern-Simons gauge theory to be
+defined consistently on a closed surface this coefficient must be quantized and should take integer
+values. The fact that the coefficient is half-quantized means that the classical global symmetry of
+a theory of a single Dirac bi-spinor cannot be gauged, which means that the gauge theory cannot
+be quantized. This is an example of an obstruction to the quantization of a gauge theory due to
+an anomaly [96].
+We will now use these results to compute the effective low-energy action for the theories
+of lattice Dirac fermions of section 10.1.2. In those systems the low energy theory is that of
+two Dirac (bi-spinors) with different masses m1 and m2. Since both Dirac fields are coupled
+(minimally) to the same background gauge field Aµ, the effective Lagrangian is just the sum of
+the contribution for each Dirac fermion
+Leff[Aµ] = −
+1
+4π|m|eff
+FµνFµν + 1
+2
+�sgn(m1) + sgn(m2)� 1
+4πǫµνλAµ∂νAλ
+(324)
+where |m|eff is
+1
+|m|eff
+=
+1
+|m1| +
+1
+|m2|
+(325)
+This result implies that, as anticipated in section 10.1.2, this theory has two phases: a parity even
+phase and a parity-odd phase. In the regime in which the signs of the two mass terms are equal
+and opposite, sgn(m1) = −sgn(m2), the coefficient of the Chern-Simons term cancels and the
+low-energy effective Lagrangian is a Maxwell term (generally with an effective speed of light
+much smaller than that in vacuum): this phase is a conventional insulator.
+Conversely, in the phase in which the two masses have the same sign, sgn(m1) = sgn(m2),
+the coefficient of the Chern-Simons terms does not cancel and it is given by ± 1
+4π, where the sign
+is the sign of both mass terms. This phase is also an insulator but one in which time-reversal
+invariance is broken. Moreover, in this phase the induced current jµ by the background field Aµ
+in the long-distance limit is controlled by the Chern-Simons term and it is given by
+jµ = δLeff
+δAµ(x) = ± e2
+2πℏǫµνλ∂νAλ
+(326)
+From this result we see that in the parity-broken anomalous quantum Hall phase the system has
+a correctly quantized and non-vanishing Hall conductivity
+σxy = ±e2
+h
+(327)
+Therefore, the phase with sgn(m1) = sgn(m2) displays the quantum anomalous Hall effect with a
+Hall conductivity whose sign equals the signs of both masses. Notice that this is true regardless
+the magnitudes of the masses m1 and m2, and that only their signs matter. In section 10.1 we will
+identify this phase with a topological phase of matter known as the Chern Insulator.
+We showed that the Hall conductivity of the anomalous quantum Hall phase is, as expected,
+equal to e2/h using the effective low energy Dirac theory. One may wonder if this approximation
+64
+
+may be missing some contributions to the Hall conductivity. Fortunately there is an alternative
+quite elegant way of computing the Hall conductivity as a property of the entire occupied band.
+This approach, originally introduced by Gregory Volovik [204] in the context of superfluid 3He-
+A, and adapted to the theory of the quantum anomalous Hall effect by Viktor Yakovenko [205],
+by Maarten Golterman, Karl Jansen and David Kaplan for a theory of Wilson fermions in odd
+dimensional hypercubic lattices [206], and by Xiao-Liang Qi, Yong-Shi Wu and Shou-Cheng
+Zhang [207], involves the derivation of a topological invariant for a two-band model.
+As we saw above the Hall conductivity is obtained from the xy component of the polarization
+operator as the limit
+σxy = lim
+ω→0
+i
+ω lim
+q→0 Πxy(ω, q)
+(328)
+For a free fermion system Πxy(ω, 0) is given by the current correlator
+Πxy(ω, 0) =
+�
+BZ
+d2k
+(2π)2
+� ∞
+−∞
+dΩ
+2π tr
+�
+Jx(k)G(k, ω + Ω)Jy(k)G(k, Ω)
+�
+(329)
+where G(k, Ω) is the propagator for a two-band free fermion system. A generic two-band free
+fermion system has a one-particle Hamiltonian of the form given in Eq.(311). The (one-body)
+current operator for such a system is obtained as
+Jl(k) = ∂h0(k)
+∂kl
+1 + ∂ha(k)
+∂kl
+σa
+(330)
+The propagator G(k, ω) is the 2 × 2 matrix (in band indices)
+G(k, ω) =
+1
+ω1 − h(k) · σ + iǫ =
+P+(k)
+ω − E+(k) + iǫ +
+P−(k)
+ω − E−(k) + iǫ
+(331)
+where P±(k) are the operators that project onto the (empty) conduction band and the (filled)
+valence band whose energies are, respectively E±(k),
+P±(k) = 1
+2
+�
+1 ± ˆh(k) · σ
+�
+(332)
+where ˆh(k) is the unit vector defined for every momentum k of the first Brillouin zone
+ˆh(k) =
+h(k)
+||h(k)||
+(333)
+Upon performing the frequency integration and the band traces in the expression for Πxy(ω, 0) of
+Eq.(329), we find that the Hall conductivity takes the form
+σxy = e2
+2ℏ
+�
+BZ
+d2k
+(2π)2 ǫabc
+∂ˆha(k)
+∂kx
+∂ˆhb(k)
+∂ky
+ˆhc(k) (n+(k)) − n−(k))
+(334)
+where n±(k) are the Fermi functions (at zero temperature) for the two bands. Since
+E+(k) − E−(k) = 2||h(k)|| = 2
+�
+h2(k) > 0
+(335)
+there is a non-vanishing gap on the entire Brillouin zone between the occupied valence band,
+with n−(k) = 1, and the unoccupied conduction band, with n+(k) = 0. For the insulating state
+the Fermi energy lies inside this gap and the expression for the Hall conductivity reduces to the
+following
+σxy = − e2
+2ℏ
+�
+BZ
+d2k
+(2π)2 ǫabc
+∂ˆha(k)
+∂kx
+∂ˆhb(k)
+∂ky
+ˆhc(k)
+(336)
+In our discussion of quantum antiferromagnets in one space dimensions in section 5.2 we found
+that their effective low-energy action contained a crucial topological term proportional to the
+integer-valued topological invariant Q (the winding number) of Eq.(103) which classifies the
+smooths maps (homotopies) of the 2D surface (say a sphere S 2) onto the target space of a three-
+component unit vector field which also a sphere S 2. These equivalence classes are represented
+by the notation Π2(S 2) ≃ Z. In the case at hand the unit vector ˆh(k) are points on a 2-sphere.
+Hence, ˆh(k) is a map of the first Brillouin zone (which is a 2-torus) to the sphere S 2. Such maps
+65
+
+are also classified by the same integer-valued topological invariant defined in Eq.(103). These
+results imply that the Hall conductivity of the two-band system is
+σxy = e2
+2πℏQ[ˆh]
+(337)
+In other words we have shown that the Hall conductivity is given in terms of a topological in-
+variant of the occupied band (in units of e2/h), the winding number Q. In the two-band model
+the topological invariant Q plays the same role as the Chern number does in the work of TKNN
+[179, 187] When Q � 0, the two-band system exhibits the quantized anomalous quantum Hall
+effect. This si a property of the entire band of occupied states, and not just a consequence of the
+low energy approximation. This result implies that the low energy approximation captures all of
+the topology of the band. It also implies that in these lattice models the Berry curvature is highly
+concentrated near the points in momentum space where the two bands are close in energy.
+.
+We will now consider the problem of the quantum phase transition between the trivial and
+the Chern insulator. To address this problem we will tune the parameters of the lattice model
+discussed in section 10.1.2, the phase φ and the ratio of hopping amplitudes t2/t1, to the point at
+which the mass of one of the two species of Dirac fermions, say the fermion ψ1, is zero, m1 = 0,
+while keeping the fermion ψ2 massive, m2 � 0. In the low energy regime we have a massless
+fermion and a massive fermion. This point in parameter space is a quantum phase transition
+between a trivial insulator and a Chern insulator. In the low energy regime the fermion ψ1 is
+massless. A massless fermion is a scale-invariant system in the sense that the correlators of all
+its observables exhibit power law behavior (free field in this case).
+We will now discuss briefly the electromagnetic response of this system at the quantum phase
+transition where one fermion becomes massless. In particular, it is natural to ask if the coeffi-
+cient of the parity-odd (Chern-Simons) term non-vanishing at the quantum phase transition. To
+answer this question we will look at the behavior of the parity-even kernel Π0(p2) and the parity-
+odd kernel ΠA(p2) in the massless limit for the light fermion, m1 → 0. We find that the total
+contribution of both the light fermion and of the heavy fermion to the polarization kernels is
+(assuming m2/m1 → ∞)
+lim
+m1→0 Π0(p2) =
+i
+16
+�
+p2 ,
+lim
+|m2|→∞ ΠA(p2) = − 1
+4πsgn(m2)
+(338)
+The important conclusion is that at this quantum critical point the parity-even kernel Π0(p2)
+is non-local and that the heavy regulator fermion (the “doubler”) yields the leading finite non-
+vanishing and local contribution to the parity-odd kernel ΠA(p2).
+We can now use these results to compute the conductivity tensor σi j at the quantum critical
+point where m1 → 0. Since the system is spatially isotropic the conductivity tensor has the form
+σi j =
+� σxx
+σxy
+−σxy
+σxx
+�
+(339)
+where we used that σyy = σxx since the system is isotropic. The longitudinal conductivity σxx
+and the Hall conductivity σxy are
+σxx = π
+8
+e2
+h ,
+σxy = ±1
+2
+e2
+h
+(340)
+in other words, at the quantum critical point the system has a finite (and universal) longitudinal
+conductivity. This result may seem surprising as there is no disorder in this model. A finite
+universal longitudinal conductivity is a standard occurrence in 2D systems at a quantum critical
+point. For example, at the superconductor-insulatortransition the conductivity is (conjectured) to
+be) σxx = e2/2h (with σxy = 0). In addition, it also has finite and also universal Hall conductivity.
+The Hall conductivity at the quantum critical point is due to the heavy fermionic “doubler” and is
+equal to 1/2 (in units of e2/h). So, the quantum critical point is not time-reversal invariant since
+this symmetry is broken at the UV (lattice).
+We can examine the massless theory using a more formal approach [199, 208]. In a gauge-
+invariant regularization of the massless theory the partition function of a Dirac fermion coupled to
+a background (unquantized) U(1) gauge field, the partition function is not time-reversal invariant.
+66
+
+It is given by
+Z[Aµ] =
+�
+D ¯ψDψ exp
+�
+i
+�
+d3x ¯ψi /D[Aµ]ψ
+�
+= det(i /D[Aµ]) =
+����det(i /D[Aµ])
+���� exp
+�
+±iπ
+2η[Aµ]
+�
+(341)
+where the sign depends on how time-reversal invariance is broken by the choice of regularization.
+The quantity η[Aµ] that appears in the phase factor is the Atiyah-Patodi-Singer η-invariant [147]
+which we already encountered in the discussion of the fractionally charged solitons in section 8.3.
+With some caveats [199, 208], the phase factor of the partition function of Eq.(341) is commonly
+written in the form of a 1/2-quantized Chern-Simons term
+π
+2η[Aµ] ≡ ±1
+2
+�
+d3x 1
+4πǫµνλAµ∂νAλ
+(342)
+often denoted as a U(1)1/2 Chern-Simons term. This term plays the same role as the contribution
+of the heavy fermion doubler in the lattice theory.
+However, we can also wonder if there is a way to have a time-reversal invariant theory of a
+single massless Dirac fermion, m → 0. This case cannot be realized in a y 2D lattice model but,
+as will see, it can be realized on the surface states of a 3D time-reversal invariant Z2 topological
+insulator, which we will discuss in section 10.1.
+10.2. Three-dimensional Z2 topological insulators
+The classification of topological insulators in terms of a Chern number is only possible in
+even space dimensions in systems with broken time reversal symmetry: in d = 2 the Berry
+connection is abelian and the topological invariant is the first Chern number while in in d = 4 the
+Berry connection in a non-abelian SU(2) gauge field and the topological invariant is the second
+Chern number [135], etc. We will now discuss the time-reversal invariant topological insulators
+[209, 190, 135] and, in particular, those that are invariant under inversion symmetry. Such states
+exist in both two and space dimensional insulator with strong spin-orbit coupling.
+10.2.1. Z2 Topological Invariants
+Let {|un(k)⟩} be the Bloch states. We will represent time reversal by the anti-unitary operator
+Θ that acts on the single particle (Bloch) states by complex conjugating the state and reverses the
+spin. For spin-1/2 fermions Θ = exp(iπσ2)K, where K is the complex conjugation operator. In
+this case Θ2 = −1. Let us assume that we have two occupied Bloch bands for each point k f the
+Brillouin zone. In this case the states form a rank-2 vector bundle over the torus of the Brillouin
+zone. In time reversal invariant systems the anti-unitary time reversal transformation T induces
+an involution in the Brillouin zone that identifies the points k and −k. Time reversal the acts
+on the one-particle (Bloch) Hamiltonian as ΘH(k)Θ−1 = H(−k). The states |un(±k)⟩ are related
+by time reversal as |un(−k)⟩ = Θ|un(k)⟩ which implies that the bundle is real. The condition
+Θ2 = −1 implies that the bundle is real. In algebraic topology these bundles are classified by an
+integer (here the number of occupied bands) and a Z2 index that will allow us to classify these
+states.
+In a periodic lattice there exists a set of points Qi of the Brillouin zone with the property
+that they differ by their images under the action of time reversal by a reciprocal lattice vector,
+−Qi = Qi + G. In d = 2 there four such points and d = 3 there are eight points, and are given
+by Qi = 1
+2
+�
+j n jb j, where n j = 0, 1, j = 1, 2 in d = 2 and j = 1, 2, 3 in d = 3. Here b j are
+the primitive lattice vectors. Kane and Mele [210] defined the 2N × 2N antisymmetric matrix
+�m,n(k) = ⟨um(−k)|Θ|un(k)⟩. They showed that at each time reversal invariant point Qi one can
+define an index δi
+δi =
+�
+det[�(Qi)]
+Pf[�(Qi)]
+= ±1
+(343)
+where det[�] and Pf[�] are, respectively, the determinant and the Pfaffian of the matrix �, and
+det[�] = Pf[�]2. The sign of the quantities δi can be made unambiguous by requiring that the
+Bloch states be continuous. In addition, the quantities δi are gauge-dependent. However the
+products
+(−1)ν =
+4
+�
+i=1
+δi
+(344)
+67
+
+in d = 2, and
+(−1)ν0 =
+8
+�
+i=1
+δi,
+(−1)νk =
+�
+nk=1,nj�k=0,1
+δi(n1, n2, n3)
+(345)
+are gauge and are also topological invariant. The Z2-valued indices ν and ν0 are robust to disorder
+and are called strong topological indices. Furthermore, Fu and Kane showed that if ξn(Qi) = ±1
+are the parity eigenvalues of the occupied parity eigenstates, the quantities δi are given by δi =
+�N
+m=1 ξ2m(Qi). In d = 3 the index ν0 does not rely on the existence of inversion symmetry.
+In the case of a two-band model in d = 2 and in d = 3, the states are four-component spinors
+reflecting the two bands and the two spin components. In the context of these systems with
+strong spin-orbit coupling spin is actually the z-component of the atomic total angular momentum
+J of the electrons with energies close to the Fermi energy. In these systems the one-particle
+Hamiltonian H(k) is a 4 × 4 Hermitian matrix which can be expanded as a linear combination
+of Dirac matrices. A simple and very useful model of systems of this type is the Wilson fermion
+model (with continuous time) [55, 211, 135] of a square (cubic) lattice with 4 states per site
+(parity and spin) whose Hamiltonian three dimensions is
+H(k) = sin k · α + M(k) β
+(346)
+where α = γ0γ and β = γ0 are the conventional 4 × 4 Dirac matrices. The γ matrices satisfy
+the Clifford algebra {γµ, γν} = 2gµν1, where gµν = diag(1, −1, −1, −1) is the metric tensor of
+four-dimensional Minkowski space time. An additional γ matrix of interest is γ5 = iγ0γ1γ2γ3.
+The (Wilson) mass term M(k) in two dimensions is M(k) = M + cos k1 + cos p2 − 2, and in
+three dimensions is M(k) = M + cos k1 + cos k2 + cos k3 − 3. In Eq.(346) we see that, consistent
+with the requirements of the Nielsen-Ninomiya theorem [174, 175], in addition to a possible low
+energy Dirac fermion (if M is small) there are three more massive Dirac fermions (in d = 2) and
+seven other Dirac fermions in d = 3, so that the total number of Dirac fermions is always even, 8
+in this case. With Wilson’s mass term the additional Dirac fermions (the “doublers”) are always
+heavy even if the Dirac fermion near the Γ point Q = 0 is light.
+In the Dirac basis time reversal is the operation Θ = (iσ2 ⊗ I)K, where K is complex con-
+jugation, and parity is P = β. The matrices α and β commute with PΘ. At the time-reversal
+and parity invariant points of the Brillouin zone {Qi} the Hamiltonian only depends of the the
+matrix β, H(Qi) = M(Qi)β. Since the parities of the spinors are the eigenvalues of the matrix
+β, we conclude that in the two-band models the quantities δi are simply equal to the sign of
+the mass of the fermions defined at the time-reversal-invariant points Qi of the BZ [209, 135],
+δi = δ(Qi) = −sgn M(Qi). Using this result it follows that in two dimensions the system is a Z2
+topological insulator with index ν = 1 (mod 2) if 0 < M < 4 while it is trivial for other values
+of M, i.e. ν = 0 mod 2. Similarly, in three dimensions a Z2 (strong) topological insulator exists
+only if 0 < M < 2 (in the other regimes this system is in a weak topological insulator state or in
+a trivial one).
+We conclude that both in two and three dimensions the low energy theory of the Z2 topolog-
+ical insulators consists of a single Dirac fermion whose mass is small compared to that of the
+fermion doublers and has the opposite sign. In both cases there is a quantum phase transition
+between a trivial insulator and the time-reversal invariant topological insulator at the point where
+he mass of the light fermion vanishes.
+10.2.2. The Axial Anomaly and the Effective Action
+We will now look at the electromagnetic response of a Z2 topological insulator in 3+1 dimen-
+sions. This problem can be addressed more easily using the continuum field theory description
+which is valid in the regime where the mass M is weak. In this regime one species of Dirac
+fermions is light (i.e. its mass is small) while the Dirac doublers remain heavy. Much as we did
+in our discussion of the Chern insulators and the parity anomaly in section 10.1.6 we will keep
+in mind that the fermion doublers play the role of heavy regulators, such as in the Pauli-Villars
+scheme, in Quantum Field Theory. Here too, we will see that a field-theoretic anomaly, known
+as the axial anomaly plays a central role in the physics. The analysis is very similar to what we
+did with the chiral anomaly in 1+1 dimensions in section 7.3.
+Let us begin with a theory of a single massive Dirac fermion in 3+1 dimensions. The La-
+grangian of the free massive Dirac theory is
+L = ¯ψ � i/∂ − m� ψ
+(347)
+68
+
+The equation of motion of the spinor field operator ψ(x) (I omit the spinor indices here) is the
+Dirac equation
+�i/∂ − m� ψ = 0
+(348)
+The Dirac Lagrangian has a global gauge symmetry ψ → eiθψ which requires that the Dirac
+current jµ = ¯ψγµψ is locally conserved, ∂µ jµ = 0. The massless Dirac theory can be decomposed
+into a theory of two Weyl bi-spinor fields which obey separate Dirac Lagrangians. Furthermore,
+the massless theory has the additional global symmetry under the transformation ψ → eiθγ5ψ and
+the additional formally locally conserved axial current j5
+µ = i ¯ψγµγ5ψ. However the conservation
+law of the axial current is violated in the massive Dirac theory
+∂µ j5
+µ = −2mi ¯ψγ5ψ
+(349)
+since the two Weyl bi-spinors transmute into each other in the presence of a mass term. This is
+the origin of the phenomenon of neutrino oscillations. In the condensed matter physics context
+there is a similar phenomenon in systems of Weyl semimetals which have crossings between the
+valence and conducting bands at two locations ±Q of the BZ, with each crossing associated with
+each Weyl bi-spinor. A charge density wave with ordering wavevector 2Q mixes the two Weyl
+fermions which become gapped, becoming effectively a single massive Dirac fermion [212].
+This state is often called an “axionic”-charge-density-wave.
+We will now show that the axial symmetry has an anomaly and cannot be gauged. Thus, we
+will consider the problem of a Dirac theory coupled to a background U(1) gauge field Aµ and
+reexamine the putative conservation of the axial current j5
+µ. This question can be addressed in
+different ways. Quite early on Steven Adler [133], and John Bell and Roman Jackiw [134] exam-
+ined this problem by computing a Dirac fermion triangle diagram for the process of a neutral pion
+decaying into two photons, π0 → 2γ. In particle physics the pion is the Goldstone boson of the
+spontaneously broken chiral symmetry. The analog of this problem in condensed matter physics
+is the phase mode of an incommensurate charge density wave. The relativistic Lagrangian for
+this problem is a theory of Dirac fermions coupled to a complex scalar field φ = φ1 +iφ2 through
+two Yukawa couplings
+L = ¯ψi /Dψ + gφ1 ¯ψψ + igφ2 ¯ψγ5ψ − V(φ1, φ2) = ¯ψi /Dψ + g|φ| ¯ψeiγ5θψ − V(|φ|2)
+(350)
+where |φ|2 = φ2
+1 + φ2
+2, tan θ = φ2/φ1, and Dµ = ∂µ + ieAµ is the covariant derivative. Here we are
+regarding the gauge field Aµ as a background probe field.
+The triangle Feynman diagram computes the polarization tensor for the electromagnetic field
+Aµ with an insertion of the coupling to the complex scalar field in an otherwise massless theory.
+Assuming a gauge-invariant regularization of the diagram, this computation finds that the axial
+current j5
+µ is anomalous and is not conserved even in the massless theory [133, 134]
+∂µ j5
+µ = − e2
+16π2 FµνF∗
+µν
+(351)
+where F∗
+µν = 1
+2ǫµνλρFλρ is the dual of the electromagnetic field tensor. This is the axial anomaly.
+To see how the axial anomaly arises we will follow the physically transparent approach of
+Nielsen and Ninomiya [176], that we also employed in section 7.3 in 1+1 dimensions. We will
+consider a theory of free massless dirac fermions coupled to a background electromagnetic field.
+Since the theory is massless, the Dirac equation decouples into an equation to the right handed
+Weyl fermion ψR (with positive chirality γ5ψR = +ψR) and a left handed Weyl fermion ψL (with
+negative chirality, γ5ψL = −ψL). In the gauge A0 = 0, the Dirac equations become
+[i∂0 − (−i∂ − eA) · σ]ψR = 0,
+[i∂0 − (i∂ − eA) · σ]ψL = 0
+(352)
+Let us now consider look at the solutions of the Weyl equation for right-handed fermions ψR,
+Eq.(352). The left-handed fermions ψL are analyzed similarly. Wd will consider a gauge field
+A1 = 0 and A2 = Bx1 representing a uniform static magnetic field of strength B pointing along
+the x3 direction. In this this gauge the eigenstates are plane waves along the directions x2 and
+x3 and harmonic oscillator states along the direction x1. The eigenvalue spectrum consists of
+Landau levels with energies
+E(n, p3, σ3) = ±
+�
+2eB
+�
+n + 1
+2
+�
++ p2
+3 + eBσ3
+�1/2
+(353)
+69
+
+for n = 0, 1, 2, . . ., except for the zero mode with n = 0 and σ3 = −1, for which
+E(n = 0, p3, σ3 = −1) = ±p3
+(354)
+where the + sign holds for ψR and the − sign for ψL. Just as in the case of non-relativistic fermions
+the relativistic Landau levels are degenerate. We will consider a system of Dirac fermions at
+charge neutrality and, hence, EF = 0. The positive and negative energy states are charge conju-
+gate of each other and the negative energy states are filled. For right-handed fermions, the zero
+mode states with p3 < 0 are filled while for right handed states the zero modes with p3 > 0 are
+filled. The density of states of the zero modes is LeB/(4π2) (where L is the linear size of the
+system).
+Let us consider now turning on an external electric field E parallel to the magnetic field B.
+Just as we saw in 1+1 dimensions in section 7.3, the electric field leads to pair creation by shifting
+the Fermi momentum to pF for the zero modes. There is no particle creation for the states with
+n � 0 and they do not contribute to the anomaly. The rate of creation of right-handed fermions
+NR,
+dNR
+dx0
+= 1
+L
+LeB
+4π2
+pF
+dx0
+= e2
+4π2 EB
+(355)
+The annihilation rate of left handed particles is
+dNL
+dx0
+= − 1
+L
+LeB
+4π2
+pF
+dx0
+= − e2
+4π2 EB
+(356)
+and the creation rate of left handed anti-particles is
+d ¯NL
+dx0
+= 1
+L
+LeB
+4π2
+pF
+dx0
+= e2
+4π2 EB
+(357)
+the axial anomaly is the total rate of creation of right handed particles and of left-handed antipar-
+ticles:
+dQ5
+dx0
+= dNR
+dx0
++ d ¯NL
+dx0
+= e2
+2π2 EB
+(358)
+which agrees with the expression of Eq.(351).
+We now turn to the Z2 topological insulator. As we saw this is a system with two species of
+(4 component) Dirac spinors. In the topological phase the sign of the Dirac mass term one of the
+Dirac fermions (the one near the Γ point in the lattice model) is opposite (negative) to the sign
+of the mass term of the of the other Dirac fermion (the fermion doubler). Explicit calculations
+[135, 213, 214] on the lattice model obtain the result that the effective low-energy action for the
+electromagnetic gauge field Aµ in the topological phase is
+S eff[Aµ] =
+�
+d4x
+�
+− 1
+4e2 FµνFµν +
+θ
+32π2e2ǫµνλρFµνFλρ
+�
++ . . .
+(359)
+In a time-reversal invariant system the allowed values of the θ angle of Eq.(359) are restricted to
+be θ = nπ, with n ∈ Z. The case θ = 0 (mod 2π) represents a trivial insulator whereas θ = π (mod
+2π) holds for a Z2 time-reversal invariant topological insulator.
+The second term in the effective action of the electromagnetic gauge field of Eq.(359), known
+as the θ term, has been extensively discussed in the high-energyphysics literature [202, 215, 216].
+The derivation of this term is subtle. As it stands, unless θ is varying in space-time (in which case
+this is known a the axion field) this term is a total derivative. In non-abelian Yang-Mills gauge
+theories this term is proportional to a topological invariant known as the Pontryagin index which
+counts the instanton number of the gauge field configurations [99, 93]. In the context of the
+Lagrangian of Eq.(350) this term is induced by the coupling of the Dirac fermion to the Yukawa
+coupling of the complex scalar field φ to the Dirac and γ5 mass terms. In this case, the lowest
+order contribution is given by the triangle diagram. The fact that this term is exact at lowest order
+reflects the fact that the axial anomaly is in fact a non-perturbative effect which is the same in
+both the weak coupling and the strong coupling regimes [96].
+In the phase where the potential V(|φ|2) in the Lagrangian of Eq.(171) has a minimum at
+|φ0| exp(iθ) the chiral symmetry is spontaneously broken and the phase field θ(x) is the Goldstone
+boson of the spontaneously broken chiral symmetry. In this phase the Dirac fermions become
+massive through the Yukawa couplings to the complex scalar field, and the phase of the field φ
+enters in the effective action as an axion field which couples to the gauge field Aµ through the θ
+term of the effective action. We should note that in a recently studied axionic CDW state of a
+Weyl semimetal [212] the phase of the CDW plays the role of the axion field.
+70
+
+10.2.3. Theta terms, and Domain walls: Anomaly and the Callan-Harvey Effect
+We will now discuss some remarkable behaviors of three-dimensional Z2 topological in-
+sulators. We will begin with the electromagnetic response encoded in the effective action of
+Eq.(359). We will assume that the θ angle is effectively a slowly varying Goldstone mode of the
+spontaneously broken U(1) chiral symmetry (i.e. the axion field) present in the Lagrangian of
+Eq.(350). If the field θ is constant, the θ term is a total derivative and it does not contribute to
+the local equations of motion. We will see below that this term [lays a key role in the physics of
+a domain wall, which we will regard as the interface of a Z2 topological insulator and a trivial
+insulator. In the general case in which θ varies slowly (as Goldstone modes do) its presence leads
+to interesting modification of Maxwell’s equations, known as axion electrodynamics [215]:
+▽ · E =˜ρ − e2 ▽ θ · B,
+▽ × E = − ∂tB
+▽ × B =∂tE + ˜j + e2 (∂tθ + ▽θ × E) ,
+▽ · B =0
+(360)
+where ˜ρ and ˜j are external probe electric charge and current densities. The equations of axion
+electrodynamics have many remarkable properties. Here we will focus on effects: the topological
+magnetoelectric effect [135] and the Witten effect [217].
+Let us consider a Z2 topological insulator with a flat open boundary (the x1 − x2 plane)
+perpendicular to the direction x3. We will assume that the topological insulator lies at x3 < 0.
+This means that for x3 < 0, and far from the surface, θ(x3) → π, while in the trivial vacuum, and
+also far from the surface, θ(x3) → 0. We will assume that the change of θ from 0 to π occurs
+on a short distance ξ. We will call this configuration an axion domain wall. In the region where
+θ(x3) is changing, |x3| ≲ ξ, an applied uniform magnnetic field B induces a uniform electric
+field E parallel to B whose magnitude is proportional to the change in θ. This is the topological
+magnetoelectric effect [135].
+A similar striking effect is obtained by considering the case of a magnetic monopole of mag-
+netic charge 2π/e (as required by Dirac quantization) inside a sphere of the trivial region of radius
+R, with θ = 0, surrounded by a region with θ � 0. The two regions are separated by a thin (axion)
+wall in which θ changes for 0 to π in a narrow shell of thickness ξ ≪ R. The equations of axion
+electrodynamics imply that the magnetic monopole induces an electric charge Qe on the surface
+of the sphere
+Qe = e∆θ
+2π
+(361)
+Thus, a magnetic monopole acquires an electric charge and becomes a dyon. This is the Wit-
+ten effect [217]. In the particular case of a Z2 topological insulator, time reversal invariance
+requires that ∆θ = π, which implies that a monopole with unit magnetic charge has an elec-
+tric charge e/2. A similar argument implies that an external magnetic field perpendicular to the
+open surface of the Z2 topological insulator (which is essentially an axion domain wall) induces
+an electric charge polarization on the surface proportional to the total magnetic flux, which is
+another manifestation of the topological magnetoelectric effect.
+The reader should readily recognize that the result of Eq.(361) for the electric charge induced
+my the magnetic monopole is the same as the Goldstone-Wilczek equation for the fractional
+charge for a one dimensional soliton of Eq.(215). We will now see that this is not just an analogy.
+We will follow here the general approach of Curtis Callan and Jeffrey Harvey [218] who extended
+the earlier work of Goldstone and Wilczek [141]. Let us consider the bulk of a Z2 topological
+insulator and assume that θ(x) is slowly varying. We can use the triangle diagram calculation to
+find that an electromagnetic gauge field Aµ induces a current ⟨Jµ(x) given by
+⟨Jµ(x)⟩ = −i
+e
+16π2 ǫµνλρ
+φ∗(x)∂νφ(x) − φ(x)∂νφ∗(x)
+|φ(x)|2
+Fλρ(x) =
+e
+8π2 ǫµνλρ∂νθ(x)Fλρ(x)
+(362)
+This result implies that, in the case of a domain wall in the x1 − x2 plane, a magnetic field
+perpendicular to the wall induces a current towards the wall and, hence, a charge accumulation
+on the wall. Where does this charge come from? To understand this problem we will consider
+a Z2 topological insulator occupying a slab of macroscopic size L between a wall at x3 = 0 and
+a far way “anti-wall” at x3 = L. In this configuration a magnetic field normal to the wall(s)
+induces a transfer of charge from one wall to the other. Similarly, an electric field parallel to the
+wall induces a current also parallel to the wall and perpendicular to the electric field, i.e. a Hall
+current.
+In 1+1 dimensions we saw that soliton configurations acquire a fractional charge associated
+to states bound to the soliton (which are zero modes when ∆θ = π). We will see that the surfaces
+71
+
+of the 3D Z2 topological insulators also have zero modes which ar Weyl fermions propagating
+on the wall. To see how this works we will consider a 3+1 dimensional Dirac fermion with a
+Dirac mass that changes sign at x3 = 0. The Lagrangian now ill be
+L = ¯ψi/∂ψ + gφ(x) ¯ψψ
+(363)
+where φ(x) is now a real scalar field that has the asymptotic behaviors φ(x3) = φ0 f(x3) such
+that limx3→±∞ f(x3) = ±1. We will assume that f(x3) is a monotonous function of x3, but its
+actual dependence on x3 is immaterial aside from the requirement that it should change sign at
+some point which we will take to be x3 = 0. This is a special case of Eq.(350) with φ1 = φ and
+φ2 = 0. We will now recognize that this just a Dirac fermion with a position-dependent Dirac
+mass m(x3) = gφ(x3). The one-particle Dirac Hamiltonian for this system is
+H = −iα · ▽ + m(x3)β
+(364)
+where α and β are the four 4 × 4 Dirac matrices. By symmetry, this Hamiltonian can be split into
+two Hamiltonians, H = Hwall + H⊥,
+Hwall = −iα1∂1 − iα2∂2,
+H⊥ = −iα3∂3 + m(x3)β
+(365)
+Let the spinor ψ± be and eigenstate of the anti-hermitian Dirac matrix γ3 = βα3 with eigenvalues
+±i, respectively
+γ3ψ± = ±iψ±
+(366)
+We seek a spinor solution ψ± of the Dirac equation Eq.(364) such that H⊥ψ± = 0
+± ∂3ψ± + m(x3)ψ± = 0
+(367)
+which is also a solution of
+(iγ0 − iγ1∂1 − iγ2∂2)ψ± = 0
+(368)
+In other words, it is a solution of the massless Dirac equation in 2+1 dimensions. The require-
+ment that ψ± be an eigenstate of γ3 reduces the number of spinor components from four to two.
+The full solution has the form
+ψ± = η±(x0, x1, x2)F±(x3),
+±∂3F±(x3) = −m(x3)F±(x3)
+(369)
+For a domain wall with limx3→∞ m(x3) = +m, the normalizable solution is F+(x3) is
+F+(x3) = F(0) exp
+�
+−
+� ∞
+0
+dx′
+3 m(x′
+3)
+�
+(370)
+where F(0) is a constant. For the anti-domain wall, for which limx3→∞ m(x3) = −m, the normal-
+izable solution is F−(x3).
+We conclude that there is a 2+1-dimensional massless Dirac theory that describes the quan-
+tum states bound to the wall which propagate along the wall. These states are the generalization
+of the fractionally charged mid-gap states of solitons in one dimension discussed in section 8.3.
+The energy of these states is E(p) = ±|p|, where p = (p1, p2) (where I set the Fermi velocity to
+unity). Experimental evidence for 2+1 dimensional massless Dirac fermions on the surface of the
+3D Z2 topological insulator Bi2Te3 has been found in spin-polarized angle-resolved photoemis-
+sion studies of the surface states which showed that they have a linear energy-momentumrelation
+(expected of massless Dirac fermions) as well as the spin-momentum locking characteristic of
+these spinor states [219].
+Having succeeded in showing that the surface of the Z2 time-reversal invariant 3D topological
+insulator has a two-component massless Dirac spinor we now want to determine its electromag-
+netic response. In fact we have already discussed this problem in our discussion of the parity
+anomaly in section 10.1.6 where we showed that the effective action for the electromagnetic field
+Aµ of Eq.(342) of a single massless Dirac bi-spinor is a Chern-Simons term with a prefactor
+which is 1/2 of the allowed value. In that purely 2+1-dimensional context we saw that time re-
+versal invariance is actually broken. However, the 3D problem is time-reversal invariant so there
+must be a contribution that cancels this time-reversal anomaly. The answer is that the requisite
+cancellation is supplied by the bulk.
+To see how this can happen we return to the bulk effective action of the 3D topological
+insulator of Eq.(359) where we observed that the θ-term is a total derivative. Suppose that the
+72
+
+system has a boundary at x3 = 0 and that the topological insulator exists for x3 > 0 (where the
+fermion mass is negative, m < 0) and trivial for x3 < 0 (where m > 0). Only the region with
+m < 0 contributes to the θ-term. The region of four-dimensional space time occupied by the
+topological insulator is M and in this region the θ-term becomes
+S θ[A] =
+θ
+32π2
+�
+M
+d4x ǫµνλρFµνFρλ =
+θ
+8π2
+�
+M
+d4x ǫµνλρ∂µAν∂λAλ =
+θ
+8π2
+�
+∂M
+d3x ǫµνλAµ∂νAλ
+(371)
+where ∂M ≡ Σ × R is the boundary of the region M, Σ is the surface of the 3D topological
+insulator and R is time.
+Thus we see that the θ-term of the effective action S θ[A] integrates to the boundary where it
+has the form of a 2+1-dimensional Chern-Simons term. Since for a time-reversal-invariant 3D
+topological insulator θ = π, we see that in this case the boundary Chern-Simons term becomes
+S θ=π[A] = 1
+8π
+�
+∂M
+d3x ǫµνλAµ∂νAλ
+(372)
+with a coefficient which is 1/2 of the allowed value. This bulk contribution either cancels the
+parity anomaly of the boundary state, rendering the full system time-reversal invariant. In other
+words, in this system time-reversal symmetry is realized by cancellation of the anomaly between
+the bulk of the topological insulator and the boundary or, equivalently, by an inflow of the parity
+anomaly between the boundary and the bulk of the system. Another way to phrase this result is
+the statement that a single Dirac fermion cannot exist on its own in 2+1 dimensions but it can
+as the boundary state of a 3+1-dimensional system whose anomaly cancels the anomaly of the
+boundary.
+10.3. Chern-Simons Gauge Theory and The Fractional Quantum Hall Effect
+We will now turn to the problem of the quantum Hall effects. This is a problem that revealed
+the existence of profound and far reaching connections between condensed matter physics, quan-
+tum field theory, conformal field theory and topology. In particular, the fractional quantum hall
+effect is the best studied and best understood topological phase of matter. As such, it has become
+the conceptual springboard for its manifold generalizations.
+The integer (IQHE) and fractional (FQHE) quantum Hall effects are fascinating phenomena
+observed in fluids of electrons in two dimensions in strong perpendicular magnetic fields. The
+integer quantum Hall effect was discovered by Klaus von Klitzing in 1980 [220] in transport
+measurements of the longitudinal and Hall resistivity of the surface states of metal oxide field-
+effect transistors (MOSFET) in magnetic field of up to 15 Tesla. The effect that von Klitzing
+discovered (for which he was awarded the 1985 Nobel Prize in Physics) was that in the high
+field regime the measured Hall conductivity showed a series of sharply defined plateaus at which
+took the values σxy = ne2/h, where n is an integer, and the longitudinal conductivity appeared
+to vanish σxx → 0 as the the temperature was lowered down to T ≃ 1.5 K. Remarkably the
+measured value of the Hall conductivity was obtained with a precision of ∼ 10−9. To this date
+the measurement of the Hall conductivity in the IQHE yields the most precise definition of the
+fine structure constant.
+Subsequent transport experiments in ultra-high-purity GaAs-AlAs heterostructures by Dan
+Tsui, Horst St¨ormer and Art Gossard found that, in addition to the IQHE, two-dimensional elec-
+tron fluids in high magnetic fields exhibit the fractional quantum Hall effect meaning that there
+are equally sharply-defined plateaus of the Hall conductivity at the values σxy = p
+q
+e2
+h , where p
+and q are co-prime integers [221]. Much as in the IQHE case, in the FQHE the longitudinal
+conductivity vanishes at low temperatures. It is important to note that both in the IQHE and in
+the FQHE the observed temperature dependence in the highest purity samples of the longitudinal
+conductivity is activated, σxx ∼ exp(−W/T). The observed value of the energy scale W is the
+experimental estimate of an energy gap in the electron fluid in the quantum Hall states.
+In addition to MOSFETS and GaAs-AlAs heterostructures both the IQHE and the FQHE
+have been seen in several other experimental platforms, particularly in graphene and other 2D
+materials [222, 223]. The explanation of these effects and of a panoply of startling consequences
+that were uncovered in the course of understanding this phenomenon is the focus of this section.
+10.3.1. Landau levels and the Integer Hall effect
+At some level the integer quantum Hall effect can be explained by the Landau quantization of
+the energy levels of a free charged moving in two dimensions in a perpendicular magnetic field
+73
+
+[224]. The Hamiltonian for a non-relativistic particle of charge −e and mass M in a perpendicular
+magnetic field B is
+H =
+1
+2M
+�
+−iℏ ▽ +ie
+c A(x)
+�2
+(373)
+for a uniform perpendicular magnetic field B = B ˆez = ▽ × A(x).
+In the circular gauge the vector potential is Ai = − 1
+2 Bǫi jx j. We will assume that the 2D plane
+has linear size L. The total magnetic flux is Φ = BL2 and we will assume that there is an integer
+number Nφ of magnetic flux quanta piercing the plane, φ = Nφφ0, where φ0 = hc/e is the flux
+quantum. In units such that ℏ = e = c = 1 the flux quantum is φ0 = 2π and Φ = 2πNφ.
+In the presence of a magnetic field the components of the canonical momentum operator
+p = −iℏ ▽ − e
+c A do not commute with each other,
+[pi, p j] − ieℏ
+c Bǫi j
+(374)
+This means that translations in two directions do not commute with each other. However, the
+components of the operator k = p(−B) commute with p (and hence with the one-particle Hamil-
+tonian) but do not commute with each other: the commutator is the same as in Eq. (374). Since
+[k, H] = 0 they act as symmetry generators of the group of magnetic translations. For arbitrary
+displacements a and b the translation operators t(a) = exp(ia· k/ℏ) (and similarly with b) satsify
+t(a)t(b) = exp(ia × b · ez/ℓ2
+0)t(b)t(a)
+(375)
+Magnetic translations only commute with each other is the area subtended by a and b contains
+an integer number of magnetic flux quanta.
+Given the rotational symmetry of the circular gauge it is natural to work in complex coordi-
+nates z = x1 + ix2. We will also use the notation ∂z = (∂1 − i∂2)/2 and ∂¯z = (∂1 + i∂2)/2. In this
+gauge, up to a normalization, the eigenstate wave functions have the form
+ψ(z, ¯z) = f(z, ¯z) exp
+− |z|2
+4ℓ2
+0
+
+(376)
+where ℓ0 =
+�
+ℏc
+e|B| is the magnetic length. In this gauge (and in complex coordinates) the angular
+momentum operator Lz = −iℏ(x1∂2 − x2∂1) = ℏ(z∂z − ¯z∂¯z).
+Any analytic function f(z) is an eigenstate with energy E0 =
+1
+2ℏωc. A complete basis of
+analytic functions are the monomials fn(z) = zn and have energy E0 and angular momentum
+Lz = nℏ. This is the lowest Landau level whose wave functions are ψn(z) = zn exp(−|z|2/4ℓ2
+0). On
+the other hand, an anti-analytic function fN = ¯zN is an eigenstate of energy EN = ℏωc
+�
+N + 1
+2
+�
+,
+where ωc =
+e|B|
+Mc is the cyclotron frequency, and angular momentum Lz = −Nℏ. States with
+angular momentum nℏ have the same energy and the degeneracy is equal to the number of flux
+quanta Nφ. For the most part we will be interested in the states in the lowest Landau level.
+In the absence of disorder the Landau levels have an extensive degeneracy equal to the num-
+ber of flux quantum Nφ. If we consider a system of N electrons in a Landau level the natural
+measure of density is not the areal density ρ = Ne
+L2 but the fraction ν = Ne
+Nφ the states in the Landau
+level which are occupied by electrons. The many-body state in which all Nφ states of the lowest
+Landau level has filling fraction ν = 1. The wave function for this state is the Slater determinant
+of the Landau states in the m = 0 level. After some simple algebra the wave function of this state
+is found to be
+Ψν=1(z1, . . . , zNe) =
+�
+i< j
+(zi − zj) exp
+−
+Ne
+�
+i=1
+|zi|2
+4ℓ2
+0
+
+(377)
+This is the ground state of the non-interacting system and it is non-degenerate. It is easy to
+see that this state has a finite energy gap. Indeed, in the Hilbert space with a fixed number
+of electrons, the lowest energy excitation is a particle-hole pair in which the particle is in an
+unoccupied state of the first excited Landau level, with m = 1 and energy E1 = 3
+2ℏωc and a hole
+a single-particle state in the lowest Landau level, with energy E0 = ℏωc. The excitation energy of
+the electron-hole pair is just ℏωc which is finite. However, in the free particle system this excited
+states has a huge degeneracy as the particle and the hole can be in any single particle state of the
+Landau level.
+74
+
+At a very naive level one can estimate the Hall conductivity for a translationally invariant
+system filling up n Landau level by the following simple argument. If n Landau levels are filled,
+the number of electrons N = nNφ and the total charge is then Q = eN = e n Nφ = e n BL2/φ0 =
+ne2BL2/hc. If a weak and uniform in-plane electric field E is applied in the presence of a per-
+pendicular magnetic field B = B ez, then the entire system (its center of mass) moves at the drift
+velocity � such that � × B = −Ec. Thus, there is a Hall current J = Q�. The current density
+is j = J/L2 = Q�/L2. Hence, ji =
+Qc
+BL2 ǫi j E j, which implies that the Hall conductivity is
+σxy = Qc/BL2 = ne2/h.
+While the above argument is formally correct and yields the correct value of the Hall conduc-
+tivity in this highly idealized setting, at a very basic level it is faulty. To begin with, it assumes
+exact translational invariance and, hence, Galilean symmetry, which are not obeyed in any realis-
+tic system. The second objection is that, even in this idealized system, as the number of electrons
+is varied, the chemical potential (and hence the Fermi energy) jumps discontinuously from one
+Landau level to the next resulting in a linearly increasing Hall conductivity (as predicted classi-
+cally) without any of the observed plateaus. Furthermore, this argument does not explain which
+the Hall conductivity is so precisely quantized whereas, a priori, one would expect that being a
+transport coefficient the Hall conductivity would depend on lots of complicated materials details.
+But, it it turns out that it does not!
+Let us first discuss the observed universality of the Hall conductivity, namely its robustness
+and independence of microscopic details. In a remarkable paper Qian Niu, David Thouless and
+Yong-Shi Wu showed that, provided the ground state is separated by a finite energy gap from
+the excited states, the Hall conductivity is actually a topological invariant [225]. The argument
+has a similarity with what we discussed in the case of the Hall effect on a 2D lattice (in section
+10.1.4) but in this more general system one considers the full many-body wave function, rather
+than the one-particle states. To show that this true they considered a system of N electrons
+with periodic boundary conditions, i.e. on a two-dimensional torus, T 2 = S 1 × S 1, with Nφ
+flux quanta going through the torus. The torus is a rectangle of dimensions L1 and L2 with
+opposite ends identified. This is necessary to allow for a Hall current to be globally allowed. The
+electromagnetic gauge field for a system ona torus with uniform non-vanishing flux cannot obey
+periodic boundary conditions but rather accross the torus the gauge fields must differ by (large)
+gauge transformations
+A1(x1, x2 +L2) = A1(x1, x2)+∂1β2(x1, x2),
+A2(x1 +L1, x2) = A2(x1, x2)+∂2β1(x1, x2) (378)
+while the wave functions themselves obey twisted boundary conditions
+Ψ([x(j) + L1e1]) = exp
+−i e
+ℏc
+Ne
+�
+j=1
+β1([x(j)]) + iθ1
+ × Ψ([x(j)])
+Ψ([x(j) + L2e2]) = exp
+−i e
+ℏc
+Ne
+�
+j=1
+β2([x(j)]) + iθ2
+ × Ψ([x(j)])
+(379)
+where e j (with j = 1, 2,) are two unit vectors on the torus, and θ1 and θ2 are two angles that twist
+the boundary conditions of the wave functions.
+In order to have a current on the torus they assumed that, in addition to the uniform flux going
+through the torus, there a weak uniform electric field E on the torus represented by the constant
+in space vector potential δA = E t = ▽[U(x)t] whose circulation on the two non-contractible
+circles Γ1 and Γ2, which wrap around the directions x1 and x2 of the torus T 2, are
+I j =
+�
+Γj
+δA · dx = t
+�
+Γj
+E · dx = itE jL j
+(380)
+Line integrals of a gauge field on non-contractible loops in space (or space-time) are called
+holonomies. By inspection we see that the angles θ1 and θ2 are given by
+θj = e
+ℏcI j
+(381)
+Alternatively, the angles θ = (θ1, θ2) can be interpreted as magnetic fluxes (in units of the flux
+quantum φ0) through the two non-contractible circles of the torus T 2.
+The angles θ are defined mod 2π since they are phase factors that twist the phase of the
+wave functions, and define a 2-torus of boundary conditions. By comparing with what we did in
+75
+
+section 10.1.4 in the case of the single-particle wave functions in the lattice model, we see that
+the many-body wave function Ψθ and the twist angles θ is the same as the relation between the
+phase of the Hofstadter wave functions with the momentum k of the magnetic BZ (which is also
+a 2-torus). Indeed, as it is always the case, while the phase of (in this case) the many-body wave
+function Ψθ is arbitrary, the changes of this phase as the the twist angle θ is changed are not. To
+quantify this dependence, for a given many-body state Ψ(α)
+θ , where α labels the state, we define a
+Berry connection A(α)(θ) on the torus of boundary conditions θ
+A(α)(θ) = i
+�
+Ψ(α)
+θ
+����∂θ
+����Ψ(α)
+θ
+�
+(382)
+Under a redefinition of the phase of the state
+Ψ(α)
+θ
+→ exp(if(θ)) Ψ(α)
+θ
+(383)
+the Berry connection A(α)(θ) changes by a gauge transformation
+A(α)(θ) → A(α)(θ) − ∂θ f(θ)
+(384)
+Niu, Thouless and Wu [225] showed that the expression of the Kubo formula for the Hall
+conductivity of the state Ψ(α)
+θ , averaged over the boundary conditions θ, is given in terms of the
+flux of the Berry connection A(α)(θ) through the 2-torus of boundary conditions, by the gauge-
+invariant quantity
+⟨(σxy)α⟩θ = e2
+ℏ
+� 2π
+0
+dθ1
+2π
+� 2π
+0
+dθ2
+2π
+�
+∂1A(α)
+2
+− ∂2A(α)
+1
+�
+= e2
+h
+1
+2π
+�
+γ
+A(α)(θ) · dθ
+(385)
+where γ is the square contour with corners at (0, 0), (2π, 0), (0, 2π) and (2π, 2π), that defines the
+2-torus of boundary conditions θ. This result is known as the Niu-Thouless-Wu formula.
+Eq.(385) implies that for the Hall conductivity to be non-vanishing the Berry connection
+A(α)(θ) must have a non-vanishing flux through the torus of boundary conditions. For this to be
+true, just as we did in section 10.1.3, we must consider large gauge transformations of the form
+A1(θ + 2πe2) =A1(θ) + ∂1 f2(θ)
+A2(θ + 2πe1) =A2(θ) + ∂2 f1(θ)
+Ψ(α)({x(j)}; θ + 2π e1) = exp(if1(θ)) Ψ(α)({x(j)}; θ)
+Ψ(α)({x(j)}; θ + 2π e2) = exp(if2(θ)) Ψ(α)({x(j)}; θ)
+(386)
+where e1 and e2 are two orthogonal unit vectors on the torus of boundary conditions. We can
+now repeat the analysis we did in section 10.1.3 which, in this context, implies that the wave
+function Ψ(α)
+θ
+cannot be globally well defined on the torus of boundary conditions, where it must
+have zeros, and where it should be defined in patches. The end result is that the wave functions
+labeled by α are classified by a topological invariant, the first-valued Chern number C(α)
+1 , which
+here it is given by
+C(α)
+1
+= 1
+2π
+�
+γ
+A(α)(θ) · dθ
+(387)
+and, hence, that the Hall conductivity in he state α (averaged over the twisted boundary condi-
+tions) exhibits the integer quantum Hall effect,
+⟨(σxy)(α)⟩θ = C1
+e2
+h
+(388)
+Expressing the Hal conductivity in terms of the first Chern number, which is a topological invari-
+ant, proves that the value of the conductivity cannot be changed by local physics effects, such
+as disorder, etc. The topological nature of the Hall conductivity is the reason for the robustness
+and high precision of the measured value of the Hall conductivity. In subsequent work Niu and
+Thouless showed that, provided the state has a finite energy gap to all excitations, in the thermo-
+dynamic limit the Hall conductivity averaged over boundary conditions and for a given boundary
+condition are equal.
+The result of Eq.(388) seemingly implies that there can only be an integer quantum Hall
+effect which, as we see shortly, it is not the case: there is a fractional quantum Hall effect in
+76
+
+which the Hall conductivity is a fraction, σxy = p
+q
+e2
+h , where p and q are co-prime integers. This
+value of the Hall conductivity is also found experimentally to be obeyed with the same precision
+as in the integer quantum Hall effect. In other words, the Hall conductivity in the fractional
+case must also have a topological character. To understand why this can be the case we need an
+implicit assumption that we made in our derivation. In fact, in deriving the result of Eq.(388) we
+assumed (implicitly) that for each value of θ there is only one many body state Ψ(α)
+θ
+(up to gauge
+transformations). As we will see below, in the fractional quantum Hall effect on a torus (and, in
+fact on any closed surface except the sphere) the ground state is degenerate in the thermodynamic
+limit. We will also see that traversing the torus of boundary conditions once maps one degenerate
+state to another one. In general, we will find that on the 2-torus in the thermodynamic limit there
+are m ∈ Z exactly degenerate states. In this case, one returns to the original state after sweeping
+m times the torus of boundary conditions. We will also see below that the degenerate states are
+labeled by quantum numbers related to the anyons that they support.
+While the topological argument proves the robustness (and universality) of the value of the
+Hall conductivity, it does not provide an insight for why there are plateaus in its magnetic field
+dependence. If for some value of the filling fraction ν = N/Nφ the electron fluid is exactly at a
+ground state Ψ(α) upon changing either the number of particles N or the magnetic field B (but not
+both) the fluid will now will be at the state Ψ(α) plus or minus some number of electrons. The
+existence of the plateau in σxy can be understood if the extra electrons (or holes) do not contribute
+to the Hall conductivity. This happens in the presence of disorder as these extra particles will
+be localized by the disorder and localized states, whose wave functions decay exponentially are
+insensitive to boundary conditions and, hence, these states do not contribute to the conductivity.
+This argument was first put forth by Laughlin [226] who build on the result that in a disordered
+system in two dimensions all states are localized [63] but made the key assumption that the
+state at the center of the Landau level (broadened by disorder) is extended and contributes to
+the conductivity if this state is filled. This picture implies that this is actually a quantum phase
+transition from an insulator (dubbed a Hall insulator) to a state with a quantized Hall conductivity.
+This intuitive and appealing proposal has been the focus of research for many years and
+it is largely an unsolved problem. There is in fact a proposed field theory for this quantum
+phase transition based on a a non-linear sigma model (in replica space) with a topological θ-term
+[227, 228], analogous to the theory we discussed in section 5.2 for quantum antiferromagnets in
+1+1 dimensions. in this proposal the (Boltzmann) longitudinal conductivity σxx plays the role of
+the inverse of the coupling constant of the non-linear sigma model while σxy plays the role of the
+coefficient of the θ-term. Although the RG flows suggested by this approach qualitatively agree
+with currently existing experimental data, and actual derivation is still wanting.
+However, aside from the fact that is is still an unsolved problem, there is the problem that this
+explanation ignores the fact that in a Landau level interactions are dominant and are the key to
+the understanding of the fractional case even in the same samples! A symptom of this problem
+is that, at least in the cleaner samples, as we noted above the longitudinal conductivity vanishes
+exponentially with temperature. This dependence means that there is a clean energy gap (and not
+just a mobility gap) which suggests that the state with additional excitations may have some sort
+of at least local order which would provide for a local energy gap. An actual theory based on this
+picture has yet to be developed.
+10.3.2. The Laughlin Wave Function
+We will turn now to the fractional quantum Hall effect (FQHE). The FQHE is observed in
+fractionally filled Landau levels. These states have fractional filling fractions, namely that ν = Ne
+Nφ
+is a fraction. Most of the observed states are in the lowest Landau level, with N = 0, but a few
+states with very intriguing properties are seen in the first excited Landau Level with N = 1.
+Let us consider for now the states in the lowest Landau level, N = 0. We will assume that
+there is no disorder and, for now, we will ignore the effects of boundaries of the sample. Physi-
+cally, the important interaction here is the Coulomb interaction although, in some samples with
+screening layers, short-range interactions may be important as well. Since there is a large mag-
+netic field, the Zeeman interaction is typically a large energy scale and the electros are expected
+to be fully polarized. However, there are situations under which the gyromagnetic factor can be
+made small (and even zero) and spin unpolarized states have to be considered. In some platforms,
+such as graphene, additional quantum numbers are present, such as “flavor” associated with the
+different Dirac states. In our discussion we will not cover these richer possibilities.
+Under these circumstances we have a problem in which all Nφ available one-particle states are
+77
+
+exactly degenerate (the Landau level is “flat”) but only a fraction of them are filled. In this limit,
+this system as as strongly interaction as it gets. Not surprisingly, in a system of this type time-
+honored standard approximations fail, such as Hartree-Fock. Given the macroscopic degeneracy
+and the nature of the states in a magnetic field, a system of this type may harbor many different
+types of actual ground states depending of the types of interactions that are considered.
+Tsui, St¨ormer and Gossard [221] did transport experiments in the two-dimensional electron
+gas (2DEG) in high-purity GaAs-AlAs heterostructures and reported the discovery of a state with
+a highly precise value of the Hall conductivity of σxy = 1
+3
+e2
+h . Such a state cannot be understood
+in any obvious way in terms of the integer quantum Hall effect. The observation of a clean gap
+in the low temperature longitudinal resistivity σxx → 0 as T → 0 implied that the 2DEG is
+incompressible and quite likely uniform.
+The first breakthrough was Laughlin’s unique insight which led to his explanation of the
+experiments in terms of a novel quantum liquid of fully spin polarized electrons at filling fractions
+ν = 1
+m (with m and odd integer) whose wave function he proposed to be
+Ψm(z1, . . ., zN) =
+�
+i< j
+(zi − zj)m × exp
+−
+Ne
+�
+i=1
+|zi|2
+4ℓ2
+0
+
+(389)
+where N = Ne is the number of electrons, {zi} are their (complex) coordinates and m is an odd
+integer (which makes the wave function antisymmetric, as required by Fermi statistics).
+Laughlin extracted the physics encoded in this wave function by considering the probability
+����Ψm(z1, . . ., zN)
+����
+2
+of finding the N electrons in the locations {zi} (with i = 1, . . ., N). The norm of
+the Laughlin wave function is
+����
+����Ψm
+����
+����
+2
+=
+�
+d2z1 . . . d2zN
+����Ψm(z1, . . ., zN)
+����
+2
+(390)
+The Laughlin wave function Ψm(z1, . . ., zN) is an eigenstate of angular momentum with eigen-
+value Lm = 1
+2mN(N − 1). This remarkable wave function has a large overlap (∼ 98%) with the
+exact wave function with V(R) = e2
+εR (Coulomb) interactions in systems of up to eight particles
+(ε is the dielectric constant).
+The norm of the state can ten be interpreted as the classical partition function of a system
+of a system of N particles at the locations {zi} at inverse temperature β = m with the effective
+interaction
+U(z1, . . ., zN) = −2
+�
+1≤i< j≤N
+ln |z1 − zj| +
+1
+2mℓ2
+0
+N
+�
+j=1
+|zj|2
+(391)
+This classical partition function is a one-component plasma, of a gas of particles with unit (neg-
+ative) charge interacting with each other with the repulsive 2D classical (logarithmic!) Coulomb
+interaction, VCG = − ln |zi − zj| (not 1/R!). The last term represents the contribution of a neu-
+tralizing background positive charge (which here it originates in the gaussian factor of the wave
+function). In other words, this is a one-component plasma. We can check that this is correct
+since ▽2
+�
+|z|2
+2mℓ2
+0
+�
+=
+2
+mℓ2
+0 which corresponds to a uniform (areal) charge density ρ0 =
+1
+2πmℓ2
+0 . Clas-
+sical Monte Carlo simulations of this probability distribution showed that this wave function
+describes an incompressible fluid state if m ≲ 7, while for larger values it represents a crystalline
+state. This approach is known as the plasma analogy.
+10.3.3. Quasiholes have fractional charge
+Laughlin considered a vortex-like excitation in the fluid created by the adiabatic insertion of
+an infinitesimally thin solenoid at some coordinate z0 carrying a magnetic flux Φ, a fluxoid. In
+the presence of this solenoid the single particle state zn exp(|z|2/4ℓ2
+0) acquires a branch cut and
+becomes the state zn+α exp(|z|2/4ℓ2
+0) where α = Φ/φ0, with φ0 = hc/e being the flux quantum.
+This analytic structure is the Aharonov-Bohm effect [155].
+However, if the solenoid carries just one flux quantum α = 1 and the net effect is that the state
+becomes zn+1 exp(−|z|2/4ℓ2
+0) which has one more unit of angular momentum. For these reasons,
+in the presence of a solenoid with one flux quantum inserted at z0, the Laughlin wave function
+for the 2DEG in a large magnetic field now becomes
+Ψqh
+m (z0; z1, . . ., zN) =
+N
+�
+i=1
+(zi − z0) × Ψm(z1, . . . , zN)
+(392)
+78
+
+A calculation using the plasma analogy reveals that the net effect of the solenoid is to expel a
+charge equal to −e/m from the vicinity of the location of the solenoid or, equivalently, to create a
+fractionally charged quasihole at z0 with charge Qqh = +e/m. The charge distribution is uniform
+at long distances but is depleted (hence a quasihole) on a length scale ξ. One can check that this
+state costs a finite energy ε0 which depends on the particular interaction.
+On the other hand, if we consider a system in which the solenoid is inserted adiabatically
+so that after a long time t there is a quasihole at z0, an amount of charge equal to −e/m will
+flow radially outwards to the outer edge of the 2DEG. The adiabatic insertion of the solenoid
+generates a azimuthal electric field (an e.m.f.) and a radial Hall current, and a Hall conductivity
+σxy = 1
+m
+e2
+h . Hence, the Laughlin wave function exhibits the fractional quantum Hall effect and
+its excitations are vortices with fractional charge e/m.
+It is important to note the non-local nature of the Laughlin quasihole. The non-locality is
+inherited from the fluxoid (the solenoid) inserted at z0: even though the magnetic field of the
+fluxoid vanishes away from it, B(x) = Φ δ2(x − x0), its vector potential A(x) does not since its
+circulation on any non-contractible loop γ that contains the location of the fluxoid inside must
+be equal to the flux Φ carried by the fluxoid. Although in principle one can choose a singular
+gauge in which A(x) vanishes locally, this cannot be true everywhere as it must leave behind a
+Dirac-like string on a curve Γ ranging form the location z0 of the fluxoid to the boundary of the
+system with A taking a singular value on Γ. This feature has the same form as the 2D vortex that
+we discussed in section 5.1.1. Kivelson and Ro˘cek [229] showed that the existence of this Dirac
+string causes the phase of a quantum state that carries charge e∗ to have a branch cut on the curve
+Γ and to change by exp(i2π(e∗/e)Φ/φ0), as required by the Aharonov-Bohm effect [155]. This
+effect is unobservable for integer-charged states, i.e. electrons.
+10.3.4. The Jain States
+The construction of the Laughlin quasihole motivated Jainendra Jain [230, 231] to rewrite
+the Laughlin wave function in the form
+Ψm(z1, . . . , zN) =
+�
+i< j
+(zi − zj)m−1 × Ψν=1(z1, . . ., zN)
+(393)
+Here the prefactor represents particles each attached with an even number, m − 1, of flux quanta.
+The second factor,Ψν=1(z1, . . ., zN), is the wave function of fermions filling up a lowest Landau
+level, i.e. the Vandermonde determinant of Eq.(377). In this form, the inverse of the filling factor,
+1/ν, which is the number of flux quanta per particle, is written as 1
+ν = (m − 1) + 1 = m. The
+interpretation now is that the (m − 1)φ0 magnetic flux carried by each particle, which partially
+screens the uniform field down to an effective field Beff = B − (m − 1)φ0 corresponding to one
+effective flux quantum per particle, i.e. a filled lowest Landau level of the effective field Beff. In
+other words, the wave function for the ν = 1/m FQH state is reinterpreted as the νeff = 1 integer
+QH state of N composite fermions, each being an electron carrying attached an even integer m−1
+flux quanta.
+This procedure is known as flux attachment. Physically it means that the strong interactions
+between the electrons for other electrons to be further away which, in virtue of being in a strong
+magnetic field, is equivalent to an increase of their relative angular momentum, as if there was a
+local change in the magnetic field. There is no reason to believe that the composite fermions are
+weakly coupled, even though this is frequently stated in the literature without any real supporting
+evidence.
+Jain then generalized this reinterpretation of the Laughlin state to a whole sequence of FQH
+states obtained by filling p levels of these partially screened magnetic fluxes
+Ψm,p(z1, . . . , zN) = PLLL
+�
+i< j
+(zi − zj)m−1Ψν=p(z1, . . ., zN)
+(394)
+where Ψν=p(z1, . . ., zN) is the wave function of p filled Landau levels (of composite fermions),
+and PLL is an operator that projects onto the lowest Landau level. By counting fluxes we see that
+the filling fractions for the Jain states satisfy
+1
+ν(m, p) = m − 1 ± 1
+p
+(395)
+79
+
+where we allowed for the possibility that the external fluxes may be over-screened by the com-
+posite fermions. Alternatively we can write
+ν(m, p) =
+p
+p(m − 1) + ±1
+(396)
+The Jain states with p = 1 are the laughlin states and each is called the primary state of a
+Jain sequence. Almost all the observed FQH states belong to one of these sequences (and its
+generalizations). In particular, the most prominent states (i.e. those with larger plateaus) belong
+to the sequence 1
+3, 2
+5, 3
+7, 4
+9, 5
+11, . . . and to the reversed sequence 1, 2
+3, 3
+5, 4
+7, . . .. For p → ∞, the Jain
+sequences converge to the values of the filling fraction limp→∞ ν(m, p) =
+1
+m−1. In this limit, the
+m − 1 fluxes attached to each particle exactly cancels th external flux leading us to a (possibly)
+Fermi liquid of composite fermions [232].
+10.3.5. Quasiholes have fractional statistics
+The non-locality and topological nature of the Laughlin quasihole implies that this state must
+be regarded as a soliton (or vortex) of the charged quantum fluid. The vortex-like state with wave
+function Ψqh
+m (z0; z1, . . . , zN) is called the Laughlin quasihole. This state represents a composite
+object of a flux quantum and a fractional charge. This structure of this quantum state is strongly
+reminiscent of the flux-charge composite objects considered by Frank Wilczek who showed that
+such objects should be anyons, states that exhibit fractional statistics [156]. Although at a con-
+ceptual level anyons were proposed (almost) prior to the discovery of the FQHE, it is in this
+setting that fractional statistics entered actual physics. The actual experimental observation took
+work by many people, and was only confirmed in 2020 [233].
+Fractional statistics dictates the analytic form of the wave functions of two or more quasiholes
+[234]. Let us consider a laughlin-type wave function for two quasiholes located at (complex)
+coordinates u and �. Naively we expect the wave function for two quasiholes in the ν = 1/m
+Laughlin state to have the form
+Ψ(u, �; z1, . . . , zN) = N(u, �)
+N
+�
+j=1
+(zj − u)(zj − �) Ψm(z1, . . ., zN)
+(397)
+The prefactor N(u, �) must be chosen to account for the fact that theres is an additional change of
+angular momentum due to the additional fluxoid at �. We also expect that as the quasihole with
+charge e/m is adiabatically carried around the the other quasihole (also with charge e/m) there
+will be ac accrued phase in the wave function due to the Aharonov-Bohm effect of the charge
+of one quasihole circling around the flux of the other (or, equivalently, crossing the branch cut)
+[229]. The requirement of translation invariance and analyticity are met by the choice [234]
+(ignoring a multiplicative normalization constant)
+N(u, �) = (u − �)1/m exp
+−
+1
+4ℓ2
+0m(|u|2 + |�|2)
+
+(398)
+which has a branch cut stretching from u to �. A calculation using the plasma analogy represents
+this state as a set of N charge −1 particles interacting with two additional particles of charge −1/m
+at u and � (and a neutralizing background). The branch cut implies that dragging a quasihole
+around the other during a π rotation followed by a translation (i.e. an exchange), induces a
+monodromy in the wave with a jump in its phase of exp(±iπ/m) (with the sign determined by the
+orientation of the monodromy) in such a way the that wave function changes by a phase factor
+Ψ(u, �; z1, . . ., zN) �→ e±i π
+m Ψ(u, �; z1, . . . , zN)
+(399)
+In other words, the quasihole is an anyon with fractional statistics π/m. Daniel Arovas, J. Robert
+Schrieffer and Frank Wilczek [235] further refined this argument by showing that under such
+a slow adiabatic change the wave function of two quasiholes acquires a Berry phase which ac-
+counts for the fractional statistics. An explicit path-integral derivation of this effect, which uses
+the fact that the quasihole states are coherent states can be found in Ref. [9], chapter 13.
+10.3.6. Hydrodynamic Effective Field Theory
+We will now turn to the approaches to the FQHE using the methods of quantum field theory.
+As we will see the concept of flux attachment plays a key role. In section 9.5 we showed that
+80
+
+Chern-Simons gauge theory is in fact a theory of flux attachment. Thus we expect that Chern-
+Simons gauge theory should play a key role as well.
+It is useful to ask first why should Chern-Simons gauge theory play a central role at least in
+the description of the low energy physics. There is a simple, yet powerful, phenomenological
+argument due to J¨org Fr¨ohlich and Anthony Zee that shows why this should be the case [236].
+This in essence a hydrodynamic argument. The fractional quantum Hall effect occurs in a 2DEG
+in the presence of a large magnetic field which breaks explicitly time reversal invariance. In the
+regime of the FQHE the 2DEG is a uniform charged incompressible fluid in which charge is
+conserved. This means that the charge 3-current jµ = (j0, j) (where j0 is the charge density and
+j is the charge current density) must be locally conserved,
+∂µ jµ = 0
+(400)
+which is the continuity equation. The solution of this equation is that the conserved current can
+be written in terms of a dual (in the Hodge sense) vector field Bµsuch that
+jµ = 1
+2πǫµνλ∂νBλ
+(401)
+The factor of
+1
+2π is introduced for later convenience. The current field jµ is not changed by a
+smooth redefinition of the vector field Bµ → Bµ + ∂µΦ, where Φ(x) is a non-singular field. This
+means that the vector field Bµ is a gauge field.
+The effective action of this theory is a functional of the current distribution µ and, hence, of
+the gauge field Bµ. It should be gauge-invariant, and odd under parity and time-reversal. Since
+the fluid is incompressible and uniform, at long distances and low energies the action must be a
+local functional of the gauge field which should be at least Galilean invariant. In addition, the
+coupling of the fluid to an external electromagnetic probe field Aµ must be of the usual form
+−ejµAµ. To lowest orders in derivatives, there the unique local and gauge-invariant Lagrangian
+density which is odd under time reversal (and parity) is the Chern-Simons theory
+Leff[Bµ] = m
+4πǫµνλBµ∂νBλ − 1
+4g2 F 2
+µν − e
+2πAµǫµνλ∂νBλ + . . .
+(402)
+The first term is the Chern-Simons term of the Lagrangian. The coefficient m is a dimensionless
+integer which, as we saw in section 9.3 is required for the theory to be invariant on a closed
+manifold. The second is a Maxwell term. Here Fµν = ∂µBν −∂νAµ is the field strength tensor of
+the gauge field Bµ. By power counting we see that the coupling constant g2 has units of length−1
+and, hence this term is irrelevant at long distances and will be neglected. The third term is the
+coupling of the current jµ to the electromagnetic field Aµ. Upon an integration by parts this term
+can be written as BµJ µ where
+Jµ = − e
+2πǫµνλ∂νAλ
+(403)
+is a current minimally coupled to Bµ.
+The equation of motion of the (dynamical) gauge field Bµ is
+δL
+δB = 0
+(404)
+which yields the relation
+m
+2πǫµνλBµ∂νBλ = Jµ
+(405)
+From the definition of the current Jµ, Eq.(403), we see that the solution of the equation of
+motion of Eq.(405), up to a gauge transformation, is
+mBµ = −eAµ
+(406)
+By plugging this relation back into the Lagrangian for the gauge field Bµ, Eq (402) (which
+si equivalent to integrate out the gauge field Bµ) we find that the effective Lagrangian of the
+electromagnetic field Aµ is just a Chern-Simons term
+Leff[Aµ] =
+e2
+4πmǫµνλAµ∂νAλ + . . .
+(407)
+81
+
+This result allows us to read-off the Hall conductivity of the fluid
+σxy = 1
+m
+e2
+2πℏ
+(408)
+where we restored units such that ℏ is not unity. In other words, this fluid exhibits the fractional
+quantum Hall effect for a fluid at filling fraction ν = 1/m.
+We should note that this heuristic argument applies equally to systems of fermions, for which
+m is odd, as well as to systems of bosons, for which m is even.
+10.3.7. Composite Boson Field Theory
+We will now discuss tow field-theoretic approaches to the FQHE. These approaches use the
+concept of flux attachment as a mapping of fermions to bosons and as a mapping of fermions
+to fermions. These approaches are a form of duality transformation which engineers a form
+of statistical transmutation. We will first show that Chern-Simons theory can be used to effect
+such a mapping. That this is possible should not be surprising since, as we saw in section 9.7,
+Chen-Simons theory is a theory of fractional statistics and, hence, of anyons.
+We should make some important comments on both approaches before we get into a detailed
+description. While the mapping of fermions to composite bosons and fermions to composite
+bosons is correct, their approximate mean field descriptions violate symmetries of the 2DEG in
+a magnetic field. At the root level is the fact that the flux attachment is local in space time and
+approximate descriptions of these theories bring about a large amount of Landau level mixing,
+even in the large field limit. For instance, at the mean field theory level the composite boson
+approach is equivalent to a Bose-Einstein condensate which is invariant under translations and
+under time reversal. In this picture the breaking of time reversal is present in the coupling to a
+Chern-Simons gauge field. Likewise, the mean field theory of the composite fermion theory is a
+problem of composite fermions in a partially screened. While a partially screened magnetic field
+still breaks time reversal, it does not behave properly under magnetic translations. As we will
+see, these problems will be solved, at least in the low energy regime, by quantum fluctuations
+which restore the symmetries. In both cases, in the fractional states one recovers an effective
+topological field theory which captures the universal physics of these states. Needless to say it,
+both approaches do poorly in the computation of non-universal dimensionful quantities such as
+energy gaps, etc. These difficulties become very severe in the compressible states whose low
+energy behavior is not topological.
+Unlike the wave function approaches which project these states into a specific Landau level
+(usually the lowest), the field theoretic approach does not effect such projection. Several papers
+have been written attempting to do a field theory projected into a Landau level. These approaches
+transformed the problem into quantum field theory on a non-commutative plane, which is inher-
+ent the nature of the states in a magnetic field. Although some significant progress has been made
+in this direction, these theories remain poorly understood [237, 238, 239, 240, 241, 242, 243]
+In this section we will focus on the composite boson theory. Let us consider a theory in 2+1
+dimensions with two dynamical abelian gauge fields Aµ and Bµ, whose Lagrangian is a sum of
+a Chern-Simons term at level k and a BF term:
+L = k
+4πǫµνλA µ∂νA λ + 1
+2πǫµνλA µ∂νBλ
+(409)
+The equation of motion for the field Aµ is
+k
+2πǫµνλ∂νA λ + 1
+2πǫµνλ∂νBλ = 0
+(410)
+whose solution (up to a gauge transformation) is kAµ = −Bµ. Plugging this relation into the
+Lagrangian of Eq.(409) we find that the effective Lagrangian for the field Bµ is
+L[Bµ] = − 1
+4πkǫµνλBµ∂νBλ
+(411)
+which states that exchanging Aµ ↔ Bµ is equivalent to the “duality” k ↔ −1/k.
+In section 9.7 we showed that particles coupled to a Chern simons gauge field at level k
+have fractional statistics exp(±iπ/k). Hence, we see that particles coupled to the field Bµ have
+fractional statistics exp(±iπk). In other words, for k odd this coupling amounts to exchanging
+a theory of fermions to a theory of bosons coupled to the gauge field Bµ. This is a form of
+82
+
+bosonization. Alternatively, for k even it maps a theory of fermions to another theory of fermions
+coupled to a gauge field Bµ. These observations have led to distinct, but ultimately equivalent
+description of the quantum Hall effect.
+We will begin the approach of mapping fermions to bosons and use it in the problem of the
+FQHE. This is the Landau-Ginzburg theory (or composite boson theory) of Shoucheng Zhang,
+Hans Hansson and Steven Kivelson [244] (ZHK). In this approach the problem of fermions
+coupled to a magnetic field interacting with each other through a two-body potential (that we
+will assume is ultra with coupling constant λ, for simplicity) becomes the same problem but for
+a theory of bosons which are also coupled to a Chern-Simons gauge field, which we will denote
+by Aµ. The Lagrangian density for this equivalent system of composite bosons is
+LCB = φ∗(x)[iD0 + µ]φ(x) + 1
+2M |Dφ(x)|2 − λ(|φ(z)|4 +
+1
+4πmǫµνλA µ∂νA λ
+(412)
+where x = (x0, x) are the space-time coordinates, µ is the chemical potential, M is the mass of
+the fermions, and m is an arbitrary odd integer. The covariant derivative
+Dµ = ∂µ + i e
+ℏcAµ + iAµ
+(413)
+which effects the minimal coupling of the complex scalar field φ(x) to the background electro-
+magnetic field Aµ and to the Chern-Simons gauge field Aµ, known in this context os known as
+the statistical gauge field.
+ZHK showed that the FQH state can be thought as being closely related to to a phase in which
+the complex scalar field condenses, much as in the case of a superfluid even though the FQH iis
+not a superfluid. To see how this works we will write
+φ(x) =
+�
+ρ(x) exp(iω(x))
+(414)
+The classical equations of motion of the theory of composite bosons, Eq.(412), are
+δLCB
+δφ∗(x) =0
+⇒
+(iD0 + µ)φ(x) − 1
+2M D2φ(x) − 2λ|φ(x)|2φ(x) = 0
+(415)
+LCB
+δA0(x) =0
+⇒
+1
+2πmǫi j∂iA j + |φ(x)|2 = 0
+(416)
+LCB
+δAi(x) =0
+⇒
+1
+2πmǫiαβ∂αA β +
+i
+2M
+�φ∗(x)Diφ(x) − (Diφ(x))∗φ(x)� = 0 (417)
+LCB
+δµ =ρ0L2T
+⇒
+�
+d3x |φ(x)|2 = ρ0L2T
+(418)
+where ρ0 is the areal density of electrons.
+A uniform solution of Eqs. (415)-(418) with constant amplitude of the composite bosons,
+i.e. a uniform and static condensate ¯φ, with uniform statistical gauge field strength ¯
+B, requires
+that the there should be no current in the ground state and that each term of Eq.(417) should be
+zero. This condition implies that the external field is canceled by the average statistical field,
+eB
+ℏc + ¯
+B = 0. This condition together with Eqs.(416) and (418) imply that
+ρ0 = 1
+m
+eB
+ℏc = 1/m
+2πℓ2
+0
+(419)
+and we conclude that the filling fraction is ν = 1
+m, which are the Laughlin states. In addition we
+get that the chemical potential is µ = 2λρ0 and that |¯φ|2 = ρ0, with ρ0 given in Eq.(419).
+However, these mean field results seeming imply that this system is a Bose-Einstein conden-
+sate. To understand why this is incorrect, and to determine what it actually is, we need to go
+beyond the mean field theor that we just described and evaluate the effects of quantum fluctua-
+tions and write
+φ(x) = (ρ0 + δρ(x))1/2 exp(iω(x)),
+eAµ
+ℏc + Aµ = δAµ
+(420)
+The partition function of this theory is a path integral over the fluctuating density fields δρ, the
+Goldstone field ω and the fluctuation of the Chern-Simons gauge field δAµ. Upon integrating out
+83
+
+the massive density fluctuations to lowest (quadratic) order we find that the effective Lagrangian
+for ω and δAµ is
+Leff[ω, δAµ] = κ
+2 (∂0ω − δA0)2 − ρs
+2 (∂iω − δAi)2 +
+1
+4πmǫµνλδA µ∂νδA λ + . . .
+(421)
+The first two terms of Eq.(421) is the effective Lagrangian of the Goldstone mode ω in the
+Bogoliubov theory of superfluidity a compressibility κ and a superfluid stiffness ρs given by
+κ = 1
+2λ,
+ρs = ρ0
+M = ν
+2πℏωc
+(422)
+However, as we see, this is not a superfluid since the phase field is “eaten” by the Chern-Simons
+gauge field δAµ which now acquires a mass term. In this sort of Higgs mechanism the fluctuating
+gauge field has a massive longitudinal mode and a massive transverse mode. Thus, this state doe
+not have any massless modes.
+To see that this theory describes the fractional quantum Hall effect we need to compute the
+linear response to a weak electromagnetic perturbation δAµ(x). This can be accomplished by
+considering the new Lagrangian
+Leff[ω, δAµ, δAµ] = κ
+2
+�
+∂0ω − δA0 − e
+ℏcδA0
+�2
+−ρs
+2
+�
+∂iω − δAi − e
+ℏcδAi
+�2
++ 1
+4πmǫµνλδA µ∂νδA λ+. . .
+(423)
+and integrate out the phase field ω (which can be set to zero in the London/unitary gauge) and
+the fluctuation of the statistical field δAµ to find that effective electromagnetic action is
+S eff[δAµ] =
+1
+4πm
+e2
+ℏ
+�
+d3x ǫµνλδAµ∂νδAλ + . . .
+(424)
+from which we conclude that the Hall conductivity predicted by the composite boson theory is
+σxy = 1
+m
+e2
+h
+(425)
+as it should be for a FQHE at filling fraction ν = 1/m.
+We conclude by looking at vortex states in the composite boson theory. A vortex state is
+a time-independent solution of the equations of motion Eqs.(415)-(418) with the asymptotic
+behavior (in the temporal gauge δA0 = 0)
+lim
+|x|→∞φ(x) = √ρ0 eiϕ(x)
+(426)
+lim
+|x|→∞δAi(x) = ± ∂iϕ(x) = ±ǫi j
+x j
+|x|2
+(427)
+where ϕ(x) is the azimuthal angle on the plane
+ϕ(x) = tan−1
+� x2
+x1
+�
+(428)
+The energy of a neutral vortex (i.e. not coupled to a gauge field) is logarithmically divergent,
+Evortex ≃ ρs
+2 ln(R/a0) where R is the linear size of the system and a0 is a short-distance cutoff
+(see section 5.1.1). However, the situation here is different since the complex scalar field φ(x) is
+coupled to a dynamical Chern-Simons gauge field Aµ. Except for the Chern-Simons nature of
+this gauge field, the problem we are dealing with is similar to that of a superconductor coupled
+to a dynamical gauge field or to the Abelian-Higgs model in quantum field theory. In the case of
+interest here we have finite energy vortex solutions which satisfy the asymptotic condition
+lim
+|x|→∞
+����
+�
+i∂ j − δA j
+�
+φ(x)
+����
+2
+= 0
+(429)
+which is obeyed by configurations that obey Eq.(427). Thus, at long distances, the circulation of
+the Chern-Simons gauge field on a large closed contour Γ that contains the vortex satisfies
+�
+Γ
+δA jdx j = ±2π
+(430)
+84
+
+This vortex carries charge. To see this we compute the local charge density j0(x)
+j0(x) = − δS eff
+δA0(x) = −e δS eff
+δδA0(x) = +e δS CS
+δδA0(x) =
+e
+2πmǫi j∂iδA j(x)
+(431)
+and compute the total charge of the vortex on a larger region Σ whose boundary is Γ and obtain
+Q = e
+�
+d2xj0(x) =
+e
+2πm
+�
+Σ
+d2x ǫi j∂iδA j(x) = e
+m
+1
+2π
+�
+Γ
+dx jδA j(x) = ± e
+m
+(432)
+Therefore, the vortex of the composite boson theory has the same charge as the Laughlin quasi-
+hole.
+To determine the statistics of the vortex we go back to the effective Lagrangian written in the
+form of Eq.(409), (from now on we set δA µ ≡ A µ)
+Leff = κ
+2(∂0ω−A0)2−ρs
+2 (∂ jω−A j)2+ 1
+2πǫµνλA µ∂νBλ− e
+2πǫµνλAµ∂νBλ− m
+4πǫµνλBµ∂νBλ (433)
+where Aµ is the external electromagnetic probe field, and where we used units with ℏ = c = 1.
+The first two terms make the statistical gauge field Aµ massive. In the low energy limit the
+statistical gauge field is frozen to the vortex configurations of the complex scalar field φ or,
+equivalently, to the vortex singularities of its phase field ω. The vorticity current Ωµ is
+Ωµ = ǫµνλ∂νA λ
+(434)
+The effective Lagrangian for the field Bµ is
+Leff[Bµ] = − m
+4πǫµνλBµ∂νBλ − e
+2π Aµǫµνλ∂νBλ + ΩµBµ
+(435)
+which, except for the coupling to the electromagnetic field, has the same form as in Eq.(224),
+and of the effective action introduced in section 10.3.6 on phenomenological grounds.
+The effective topological field theory of Eq.(435) encodes al the universal data of fractional
+quantum Hall fluids. Being topological this effective field theory has no energy scales and de-
+scribes the physics at energies low compared to any excitation energy gap. In addition to yield-
+ing the correct Hall conductivity σxy = νe2/h (with ν = 1/m) and the fractional vortex charge
+Q = e/m, this theory will allow us to draw important additional results about the low energy
+physics.
+By using the results of section 9.7, we conclude that the vortices of the composite boson are
+anyons with fractional statistics exp(±iπ/m), consistent with the conclusions of section 10.3.5.
+In addition, on a 2-torus the ground state of this system is m-fold degenerate. This feature,
+characteristic of systems with topological order, is that the ground state degeneracy depends on
+the topology of the surface on which the 2DEG resides. The ground state on a torus degeneracy
+was shown by Duncan Haldane and Edward Rezayi [245] by an explicit construction of a model
+wave function for the Laughlin states on a torus. On a surface with g handles (i.e. genus g) the
+degeneracy is mg. Xiao-Gang Wen and Qian Niu [246] showed that these results hold for any
+FQH state which are, quite generally, topological quantum fluids.
+Finally, we may ask how many distinct types of vortices does this theory have. The funda-
+mental (Laughlin) vortex has charge e/m and statistics π/m. In principle we could have a vortex
+with any integer topological charge (winding number) n ∈ Z. Such a vortex has charge ne/m and
+statistics πn2/m. However, a vortex with topological charge n = m has charge e and statistics mπ.
+This vortex is indistinguishable from a hole (a missing electron) since it has the same charge and
+statistics. We conclude that a Laughlin state has m distinct vortices with n = 1, 2, . . ., m − 1. We
+notice that this number is the as the number of degenerate states on a torus. In section 10.3.10
+we will see that this fact is closely related to the number of primary fields of a conformal field
+theory associated with the FQH states.
+10.3.8. Composite Fermion Field Theory
+At the beginning of section 10.3.7 we noted that flux attachment can be used to define equiv-
+alent theories: a) by attaching an odd number, m, fluxes to each electron thereby becoming a
+composite boson, and b) by attaching an even number, m − 1, fluxes by which the electrons
+turn into composite fermions, as in Jain’s construction [230]. In this section we discuss the most
+salient features of the field theory approach to composite fermions, introduced by Ana L´opez and
+85
+
+me in 1991[247, 248] as a theory of all the Jain states. This approach was extended in 1993 by
+Bertrand Halperin, Patrick Lee and Nicholas Read [232] to the case of the compressible states.
+By following the same approach that we used in section 10.3.7, but adapted to the case in
+which we map to a theory of a composite Fermi field ψ(x), we find their dynamics is described
+by the effective action [247]
+SCF =
+�
+d3x
+�
+ψ∗(x)[iD0 + µ]ψ(x) +
+1
+2M |Dψ(x)|2
+�
++
+1
+4πn
+�
+d3xǫµνλA µ∂νA λ
+− 1
+2
+�
+d3x
+�
+d3x′ (|ψ(x)|2 − ρ0)V(x − x′)(|ψ(x′)|2 − ρ0)
+(436)
+where n = m − 1 is an even integer. Here V(x − x′) = δ(x0 − x′
+0)V(|x − x′|) is the instantaneous
+electron-electron repulsive interaction, and ρ0 is the average areal neutralizing charge density.
+As before, the covariant derivative is Dµ = ∂µ + ieAµ + Aµ, where Aµ is the electromagnetic field
+(including the uniform magnetic field B), and Aµ is the statistical gauge field. We are using units
+such that ℏ = c = 1.
+We can simplify somewhat the form of the actions for the composite fermions by using the
+fact that the Gauss law of Chern-Simons gauge theory states that the particle density and the
+gauge flux are rigidly tied together, |ψ(x)|2 =
+1
+2πnB ≡ ǫi j∂iA j (this is the flux attachment) as a
+operator identity. Thus, we can write the action as
+SCF =
+�
+d3x
+�
+ψ∗(x)[iD0 + µ]ψ(x) +
+1
+2M |Dψ(x)|2
+�
++
+1
+4πn
+�
+d3x ǫµνλA µ∂νA λ
+− 1
+2
+�
+d3x
+�
+d3x′
+�B(x)
+2πn − ρ0)
+�
+V(x − x′)
+�B(x′)
+2πn − ρ0
+�
+(437)
+Since this action is a quadratic form in the Fermi (Grassmann) fields ψ, we can integrate them
+out to obtain the following effective action for the statistical gauge field Aµ
+S eff[Aµ] = −itr
+�
+iD0 + µ + D2
+2M
+�
++
+1
+4πn
+�
+d3x ǫµνλ (A µ − eAµ)∂ν (A λ − eAλ) + S int[Aµ − eAµ]
+(438)
+where Aµ is a probe external electromagnetic field (with vanishing average) and does not include
+the uniform magnetic field. The interaction term is
+S int[Aµ − eAµ] = −1
+2
+�
+d3x
+�
+d3x′
+�(B(x) − eB(x))
+2πn
+− ρ0)
+�
+V(x − x′)
+�(B(x′) − eB(x′))
+2πn
+− ρ0
+�
+(439)
+Here too B(x) is an external probe with vanishing average.
+We will investigate the properties of the path integral for the statistical gauge field
+Z[Aµ] =
+�
+DAµ exp(iS eff[Aµ, Aµ])
+(440)
+using a saddle point expansion. The saddle-point condition of the effective action of Eq.(438)
+δS eff
+δAµ(x) = 0
+(441)
+leads to the equation of motion for the field Aµ
+⟨jF
+µ (x)⟩ +
+1
+4πnǫµνλ
+�
+F νλ(x) − eFνλ�
+= 0
+(442)
+where ⟨jF
+µ ⟩ is the expectation value of the fermionic current. In addition we need to impose the
+condition for the particle density to be uniform and equal to the neutralizing background charge
+⟨jF
+0 (x)⟩ = ρ0.
+We will assume that the electromagnetic field Aµ describes just a static uniform magnetic
+field of strength B. Under these conditions, the ground state should also be static and uniform.
+This means that the field strength of the statistical gauge field should be constant and uniform
+value ⟨B⟩, and that the current ⟨jF⟩ should vanish in the ground state. On the other hand, the
+Gauss Law of the Chern-Simons gauge field implies that ⟨B⟩ = −2πn ρ0, while the zero current
+condition requires the the (statistical) electric field vanishes, ⟨E ⟩ = 0. Since the composite
+86
+
+fermion couples in the same way to the electromagnetic field Aµ and to the statistical field Aµ we
+conclude that they experience an effective magnetic field
+Beff = B + 1
+e⟨B⟩ = B − 2πn ρ0
+e
+(443)
+If the total number of electrons is N, then ρ0/e = N/L2 where L is the linear size of the system.
+Let us denote by Neff
+φ the total number of flux quanta of the effective magnetic field (in units of
+ℏ = c = 1 in which the flux quantum is φ0 = 2π), and 2πNφ = BL2 is the total flux. Then, the
+total effective flux is
+2πNeff
+φ = 2πNφ − 2πn N
+(444)
+Let ν = N/Nφ be the filling fraction and νeff = N/Neff
+φ
+the effective filling fraction. Eq.(444)
+implies that these filling fractions are related by
+1
+νeff = 1
+ν − n
+(445)
+Recall that n is an even integer. However, the system will be incompressible only if νeff = p ∈ Z.
+In other words, the composite fermions fill an integer number p of the partially screened Landau
+levels, which is Jain’s condition. For this condition to be satisfied the filling fraction ν(p, n) is
+ν±(n, p) =
+p
+np ± 1
+(446)
+which are the Jain fractions. In the special case p = 1 the Jain fractions are the Laughlin fractions,
+with n = m − 1. Similarly we find that Beff is
+Beff = ±
+B
+np ± 1
+(447)
+So we see that the external field B is partially screened to the smaller value Beff which can be
+parallel (screened) to B or anti-parallel (overscreened) to B. For the same reason, at this mean
+field level the cyclotron frequency ωc is reduced by the same amount to ωeff
+c = ωc/(np ± 1).
+We will discuss now the effects of quantum fluctuations for the action of Eq.(438) about the
+saddle-point solutions. We will work to the lowest order (quadratic) in the fluctuations, which
+can be regarded as a semi-classical (or “RPA”) treatment of this theory. The quadratic effective
+action for the fluctuations of statistical gauge field, which we will still denote by Aµ is
+S (2)
+eff[Aµ] =1
+2
+�
+d3x
+�
+d3y A µ(x)ΠCF
+µν (x, y) A ν(y)
++
+1
+4πn
+�
+d3x ǫµνλ(Aµ(x) − eAµ(x))∂ν(Aν(x) − eAν(x)) + S int(Aµ − eAµ)
+(448)
+Here, Πµν
+F (x, y) is the polarization tensor of the composite fermions for p filled effective Landau
+levels. The interaction term of the action is given in Eq.(439).
+As required by gauge invariance, the composite fermion polarization tensor is transverse,
+∂µΠCF
+µν = 0, which fixes the tensorial structure. In momentum and frequency space the com-
+posite fermion polarization tensor ΠCF
+µν (q, ω) depends on three kernels ΠCF
+0 (q, ω), ΠCF
+1 (q, ω) and
+ΠCF
+2 (q, ω), where ΠCF
+0
+and ΠCF
+2
+contain the parity-even response of the composite fermions and
+ΠCF
+1
+is their parity-odd response. Each kernel is given as a series terms representing particle-hole
+processes with simple poles at the excitations energies ωrs = (r − s)ωeff
+c , with r > p (particles)
+and s ≤ p (holes). Each term has a residue which is an integer power of q2 times a Laguerre
+polynomial in q2. Details of these kernels can be found in Ref.[247] and in chapter 13 of Ref.[9].
+However, what will be important here is that, in the gapped states these kernels have a low energy
+and low momentum limit which is local. This will enable us to find a local topological effective
+action only for the gapped states.
+After integrating out the statistical gauge field Aµ we find the effective action for the external
+electromagnetic probe field Aµ, which has the standard form
+S eff[Aµ] = 1
+2
+�
+d3x
+�
+d3y Aµ(x) Πµν(x − y) Aν(y)
+(449)
+which encodes the full electromagnetic response of the system, not just of the composite fermions.
+Once again, gauge invariance requires that the full polarization tensor be transverse, ∂µΠµν = 0.
+87
+
+In section 10.1.4 we showed that there is a relation between the polarization tensor Πµν and
+the current-current correlation function Dµν. There we mentioned the these correlators are re-
+quired (by gauge invariance) to satisfy a Ward identity known as the f-sum rule:
+� ∞
+−∞
+dω
+2π iω DR
+00(ω, q) = ρ0
+M q2
+(450)
+where the (retarded) density-density correlation function is related to the (retarded) polarization
+tensor, DR
+00(ω, q) = ΠR
+00(ω, q). In the limit of low momentum q, at fixed frequency ω, The
+expression for ΠR
+00(ω, q) is
+ΠR
+00(ω, q) ≃ q2ΠR
+0(ω, q) = −ρ0
+M
+q2
+(ω + iǫ)2 − ω2c
+(451)
+where ωc = eB/(Mc) is the cyclotron frequency of the electrons. This expression saturates the
+sum rule. This means that whichever corrections Eq.(451) may have must vanish in the low q
+limit. In other words, in this limit the approximation that we made is actually exact. This is well
+known result in the theory of the Fermi liquid [194].
+In addition, the result of Eq.(451) implies that there is a particle-hole (density) collective
+mode which at low momentum has energy ℏωc, without corrections. This result is known as
+Kohn’s theorem [249] which states that in a Galilean invariant system the 2DEG must have an
+exact eigenstate at the cyclotron frequency. More intuitively, the center of mass of the 2DEG exe-
+cutes a cyclotron motion regardless of whether the particles are free, interacting or not, fermions,
+bosons, or anyons. This is a general result. Notice that the composite fermions do not satisfy
+Kohn’s theorem as they cyclotron mode would be at the effective cyclotron frequency ωeff
+c , and
+that the leading quantum fluctuations have restored the magnetic symmetry.
+In the limit of low energy ω → 0 and low momentum q → 0 of the kernels ΠCF
+0 , ΠCF
+1
+and
+ΠCF
+2
+become
+ΠCF
+0 (0, 0) = p
+2π
+M
+Beff
+≡ εCF,
+ΠCF
+1 (0, 0) = ± p
+2π ≡ σCF
+xy ,
+ΠCF
+2 (0, 0) = − 1
+2π
+p2
+M ≡ −χCF
+(452)
+where εCF is the “dielectric constant” of the composite fermions, χCF is their effective “per-
+meability”, and σCF
+xy is the (integer) composite fermion Hall conductivity. We then find that
+low-energy effective action of the statistical gauge field Aµ is
+S eff[Aµ] =1
+2
+�
+σCF
+xy +
+1
+2πn
+� �
+d3x ǫµνλA µ(x) ∂ν A λ(x)
+− e
+2πn
+�
+d3x ǫµνλA µ(x) ∂ν Aλ(x) + e2
+4πn
+�
+d3x ǫµνλAµ(x) ∂ν Aλ(x)
++
+�
+d3x1
+2
+�
+εCF Ei
+2(x) − χCF B2(x)
+�
+−
+1
+8π2n2
+�
+d3x
+�
+d3y (B(x) − eB(x)) V(x − y) (B(y) − eB(y))
+(453)
+Notice that, except possibly for the last term, the effective action is a local. This is possible
+because the mean field theory state is gapped.
+In what follows we will focus only on the leading terms of the effective action and neglect the
+subleading terms, the least two terms of the effective action which have the form of a Maxwell
+term and an additional (possibly non-local) term. The remaining leading terms have a Chern-
+Simons form and a BF form whose effective Lagrangian is
+Leff[Aµ] = 1
+4π
+�
+±p + 1
+n
+�
+ǫµνλA µ(x) ∂ν A λ(x)− e
+2πn ǫµνλA µ(x) ∂ν Aλ(x)+ e2
+4πn ǫµνλAµ(x) ∂ν Aλ(x)
+(454)
+If now now integrate ot the statistical gauge field Aµ we find the the electromagnetic field Aµ has
+the effective Lagrangian
+Leff[Aµ] = σxy
+2
+ǫµνλAµ(x) ∂ν Aν(x)
+(455)
+from where we find that the Hall conductivity σxy of the 2DEG is
+σxy = e2
+2πℏ
+p
+pn ± 1
+(456)
+88
+
+which is the Hall conductivity of the Jain states (in standard units). Here, as before, n is an even
+integer.
+We will now return to the effective Lagrangian of Eq.(454). As it stands, since the Chern-
+Simons term has fractional level, it is not invariant under large gauge transformations and, as
+such, cannot be defined on a closed manifold. To remedy this problem we can now write this
+Lagrangian in terms of a theory with two dynamical gauge fields Aµ and Bµ as follows:
+Leff[Aµ, Bµ] = p
+4πǫµνλA µ(x) ∂ν A λ(x) − n
+4πǫµνλBµ(x) ∂ν Bλ(x)
++ 1
+2πǫµνλA µ(x) ∂ν Bλ(x) − e
+2πǫµνλBµ(x) ∂νAλ
+(457)
+This is a topological quantum field theory for all the Jain states [250]. This theory can be defined
+on any manifold no matter its topology. It is easy to see that upon integrating out the field Bµ
+we recover the effective field theory for Aµ of Eq.(454). Also, in the special case of the Laughlin
+states, for which p = 1, we can integrate out the field Aµ, and arrive to the effective topological
+field theory for the field Bµ of Eq.(435) that we derived in the composite boson theory in section
+10.3.7.
+Let us rewrite the Lagrangian of Eq.(457) in a more compact yet general form. To this end we
+define a multicomponent gauge field A I
+µ with I = 1, . . ., L, an L-component charge vector t, and
+an L × L second rank symmetric matrix KIJ. Wen and Zee have given a general classification of
+a large class of fractional quantum Hall states, said to be abelian, whose topological field theory
+is defined by the Lagrangian [251, 252]
+L = 1
+4πKIJǫµνλA I
+µ (x) ∂ν A J
+λ (x) − e
+2πtIǫµνλAµ(x) ∂ν A λ
+I (x)
+(458)
+In the case of the theory of Eq.(457) which has two components, L = 2, with A 1
+µ = Aµ and
+A 2
+µ = Bµ. The two-component vector is t = (0, 1) and the matrix 2 × 2 matrix is
+K =
+�−p
+1
+1
+n
+�
+(459)
+Wen and Zee showed that, quite generally, a theory of the K-matrix form has a vacuum degener-
+acy on a torus of |detK| and, on a surface of genus g, the degeneracy is |detK|g. They also showed
+that the Hall conductivity is σxy = ν e2/h where the filling fraction is
+ν =
+L
+�
+I,J=1
+tI K−1
+IJ tJ
+(460)
+The properties of the quasiparticles can be determined by computing the appropriate correla-
+tor. In the composite fermion theory, whose Lagrangian is given in Eq.(437), the gauge-invariant
+composite fermion propagator is
+GCF(x − y; γ(x, y)) =
+�
+ψ†(x) exp(i
+�
+γ(x,y)
+dzµ(eAµ(z) + A µ))ψ(y)
+�
+(461)
+which, as in any gauge theory, it is gauge-invariant but path-dependent. Here γ is an oriented
+open path with endpoints that the space-time locations x and y. The composite fermion propa-
+gator has information about many of the properties of the quasiparticles, including their electric
+charge. To find what is the statistics of the quasiparticles we need to compute a two-particle cor-
+relator (a four-point function) which is defined in terms of two open oriented paths, say γ1(x, y)
+and γ2(u, w), with endpoints at the locations of the two particles, respectively.
+The computation of these correlators is done using a feynam sum over trajectories with dif-
+ferent weights which depend on the gauge field configurations. To do these calculations is, in
+general, a complicated problem. However, in a gapped state, the quasiparticles are massive and
+in the low energy regime these expressions are dominated by a classical trajectory on an oriented
+path ˜γ(x, y) with the same endpoints x and y. The end result reduces to the computation of the
+expectation value of a Wilson loop operator
+W[Γ] =
+�
+exp(i
+�
+Γ
+dzµ(eAµ(z) + A µ(z))
+�
+A µ
+(462)
+89
+
+on the closed oriented path Γ = γ � ˜γ− (where ˜γ−(y, x) has the opposite orientation as ˜γ(x, y) and
+runs from y to x. The explicit dependence on the external electromagnetic field yields the value
+of the charge of the quasiparticle through the Aharanov-Bohm effect. Likewise, the two-particle
+correlator is reduced, in the asymptotic low energy regime, to the computation of an expectation
+value of two Wilson loops in the effective Chern-Simons theory, as was done in section 9.7,
+which yields the fractional statistics of the quasiparticles. The interested reader can find details
+in chapter 13 of Ref.[9].
+A system described by a theory of the form of Eq.(458) has different types of quasiparticles.
+In general we can assign an integer-valued quantum number ℓI ∈ Z as the charge of the quasi-
+particle with respect to the gauge field A µ
+I . The general coupling of a quasiparticle current has
+the form Lqp = jµℓIA µ
+I . Then, the charge Q[ℓ] and the statistics δ[ℓ] of a general quasiparticle
+defined by the integer-valued L-component vector ℓ are
+Q[ℓ] = −etIK−1
+IJ ℓJ,
+δ[ℓ]
+π
+= ℓIK−1
+IJ ℓJ + 1
+(463)
+where the +1 is required to refer the statistics to fermions (not bosons) [250]. These results are
+general and hold for any abelian FQH state [251].
+In the case of the Jain states, the charge vector is t = (0, 1) and the quasiparticle is labeled by
+the vector ℓ = (1, 0) which yield the results for the charge Q and the statistics δ
+Q =
+−e
+np + 1,
+δ = π
+�
+−n
+np + 1 + 1
+�
+(464)
+which are the quantum numbers of the quasiparticles of the Jain states. For the Laughlin state at
+ν = 1/3 we obtains Q = −e/3 and δ = π/3 (as we should), which for the first Jain state at ν = 2/5
+the charge Q and the statistics δ are Q = −e/5 and δ = 3π/5. With some caveats, the predictions
+for the quasiparticle charge in the FQH states with ν = 1/3 and ν = 2/5 have been confirmed in
+noise measurements at a quantum point contact by R. de Picciotto and coworkers [253] and by
+L. Samindayar and coworkers [254]. Fractional statistics has been measured experimentally in
+both FQH states at ν = 1/3 and ν = 2/5 in (Fabry-Perot) interferometers by J. Nakamaura, S.
+Liang, G. Gardner, and M. Manfra [233].
+10.3.9. The Compressible States
+At large values p → ∞ the Jain sequences converge to the even-denominator fractions
+ν∞(n) = 1/n. In this limit, Beff → 0 and ωeff
+c
+→ 0. In other words, at the filling fractions
+ν∞(n) the average effective field experienced by the composite fermions is zero. This mean
+field theory then would predict that this state is essentially a Fermi liquid of composite fermions
+which is a compressible state. In addition, one can compute the effective charge of the composite
+fermion and find that it is e/(np ±1), which also vanishes in the compressible limit. It is straight-
+forward to see that the Fermi sea of the composite fermions is a disk with Fermi momentum
+pF = (2ν∞(n))1/2ℏ/ℓ0 and that the Fermi energy is EF = ν∞(n)ℏ/(Mℓ2
+0). Halperin, Lee, and Read
+[232] did an in-depth analysis of the consequences of this compressible state which are largely
+in agreement with experiment.
+In our discussion of the incompressible FQ states, both in the composite boson and in the
+composite fermion approaches in sections 10.3.7 and 10.3.8 we saw that the mean field approxi-
+mation violated symmetries that were restored by quantum fluctuations. This fact allowed us the
+derive effective filed theories which are asymptotically exact in the low energy IR regime. What
+allowed us to obtain these exact results is the existence of a gap. However, the compressible
+states are gapless and in some sense should be regarded theories of a quantum critical system.
+In fact, if we reexamine the field theory of composite fermions of section 10.3.8 we see
+that, in the compressible limit p → ∞, the action of Eq.(437) describes a system of fermions at
+finite density coupled to the fluctuations Aµ of the statistical gauge field with a Chern-Simons
+term, but without an external uniform field (which has been canceled by the flux of the average
+statistical gauge field). This theory has two salient features. one is that the only trace left of
+the explicitly broken time-reversal symmetry is in the presence of the Chern-Simons term. This
+means that while the mean field theory looks like a Fermi liquid, the consequences of the broken
+time reversal invariance only enters in the quantum fluctuations. In addition, in this regime this
+theory also breaks the particle-hole symmetry of a half-filled Landau level (in the high magnetic
+field regime).
+However, these contributions, already at one-loop order, typically contain IR divergencies.
+These IR divergencies are due to singular forward scattering processes of the composite fermions
+90
+
+by the gauge fields. They are most severe in the computation of the fermion self energy and de-
+pend on the form of the electron-electron interactions. For instance if the interaction potential
+V(R) is short range, the one-loop contribution to the fermion self-energy has an imaginary part
+withe the behavior Σ′′(ω) ∼ ω2/3 which is much larger than the real part Σ′(ω) as ω → 0 and the
+the Fermi energy EF is approached [232]. This behavior, which is found in many metallic sys-
+tems at quantum criticality [255, 256, 69] implies that the width of the quasiparticle is asymptot-
+ically much larger than the energy as the Femi surface is approached. This one-loop result means
+that perturbation theory fails in the IR and that actual behavior may be non-perturbative. This
+problem is present even in the case of unscreened e2
+R (Coulomb) interactions where the singular
+behavior is milder with Σ′′(ω) ∼ ω ln ω (known as Marginal Fermi Liquid scaling [257]).
+At any rate, these IR singular behaviors imply that the quasiparticle picture is inadequate and
+the that in these regimes there may be no quasiparticles. These systems are generally known as
+“Strange Metals”. Many strongly interacting physical systems of interest, ranging form high Tc
+superconductors to (largely conjectured) spin liquids with spinon Fermi surfaces to even quark-
+gluon plasmas share these types of singular IR behaviors [257, 258]. In spite of a large body
+of theoretical work, it has remained largely unsolved problem. Perturbative renormalization
+group methods used to justify the Landau theory of the Fermi liquid [259, 260] generally fail
+in these regimes, although in some cases long range interactions have allowed some degree of
+control [261, 262]. Remarkably, the AdS/CFT Gauge/Gravity Duality [263] has brought new
+insights into strange metal behaviors [264]. We will see below that relativistic dualities have
+given additions insights into the physics of compressible states.
+10.3.10. Fractional Quantum Hall Wave Functions and Conformal Field Theory
+The Laughlin wave function and its generalizations share the remarkable feature of being
+essentially universal. Aside form the dependence on the magnetic length ℓ0 in the gaussian
+factors which ensure that the function is integrable at long distances (a feature inherited from
+the single-particle states in the Landau level), the FQH “ideal” wave functions do not have any
+length scales. In addition, in almost all cases, the ideal wave functions are the exact ground state
+wave functions of some suitable ultra-local Hamiltonian projected onto the lowest Landau level.
+Another feature, closely related to their universality and analyticity, is that these wave functions
+look similar to correlation functions in two-dimensional critical (chiral) critical systems. We will
+now see that these features are not accidental and point to a deep relation between quantum Hall
+states and Conformal Field Theory (CFT).
+Although the relation between the simpler case of the Laughlin state was known to several
+people, such as Sergio Fubini [265], this deep connection with rational CFT was articulated in
+considerable generality in a somewhat earlier paper by Gregory Moore and Nicholas Read [266].
+This approach showed its full power in the formulation of the non-abelian quantum Hall states.
+Let us begin with the simpler case of the Laughlin states with ν = 1/m, where m is odd for
+fermions and even for bosons. We will consider a U(1) Euclidean CFT of a compact boson (a
+scalar field) φ(x) closely related to our discussion of bosonization in 1+1 dimensions in section
+7.4. This theory has been extensively studied in the CFT literature [103, 102, 104] The Euclidean
+action for this theory is
+S = 1
+8π
+�
+d2x
+�
+∂µφ
+�2
+(465)
+This theory is compactified by the condition that the only allowed observables are invariant under
+the discrete symmetry
+φ(x) �→ φ(x) + 2πRn
+(466)
+where R is the compactification radius and n is an arbitrary integer. The observables that satisfy
+this condition are vertex operators of the form
+Vp(x) = exp
+�
+i p
+Rφ(x)
+�
+(467)
+where p is an integer.
+The correlators of this theory are products of holomorphic (right-moving in Minkowski
+spacetime) and antiholomorphic (left-moving in Minkowski spacetime) factors. In the context
+of the FQH states we will be interested in a it chiral theory, whose correlators are holomorphic
+(analytic) functions in complex coordinates z = x1+ix2. Formally, the field φ(z, ¯z) is decomposed
+into a sum of a holomorphic field φ(z) and and antiholomorphic field φ(¯z), so that the propagators
+is
+⟨φ(z, ¯z) φ(�, ¯�)⟩ = − ln(z − �) − ln(¯z − ¯�)
+(468)
+91
+
+In what follows we will focus on the chiral(holomorphic) field φ(z) whose propagator is
+⟨φ(z)φ(�) = − ln(z − �)
+(469)
+The correlator of the chiral current operator
+j(z) = i
+R∂zφ(z)
+(470)
+is given by
+⟨i∂zφ(z) i∂�φ(�)⟩ ∼
+1
+(z − �)2
+(471)
+which tells us that this operator has (chiral) scaling dimension ∆ = 1. Similarly the correlator of
+two chiral vertex operators
+Vp(z) = exp
+�
+i p
+Rφ(z)
+�
+(472)
+is
+⟨Vp(z) V−p(�)⟩ =
+1
+(z − �)p2/R2
+(473)
+whose (chiral) scaling dimension is ∆p = p2/(2R2). These expressions are very similar to the op-
+erators that we used in section 5.1.1 to describe vortices in 2D superfluids (and planar magnets).
+The main difference is that here the fields and their correlators are holomorphic where is section
+5.1.1 the correlators are the product of a holomorphic and antiholormophic factors.
+For general p and R the correlator of Eq.(473) has a branch cut from z to �. This means that
+the monodromies of the operators, i.e. dragging an operator around the other, induces a change
+in the phase of the correlator equal to 2πp2/R2. This behavior is reminiscent of the analytic
+structure the the quasiparticle wave function in the FQH states and of the braiding operation
+between anyons.
+Moore and Read [266] showed that the following correlator in a U(1) compactified boson
+CFT (with compactification radius R = √m) is equal to the Laughlin wave function at filling
+fraction ν = 1/m,
+Ψm(z1, . . . .zN) =
+� 
+N
+�
+i=1
+exp
+�
+i √mφ(zi)
+� exp
+�
+−i
+�
+d2z′ √m ρ0 φ(z′)
+� �
+(474)
+=
+�
+i< j
+(zi − zj)m exp
+−1
+4
+N
+�
+i=1
+|zi|2
+
+(475)
+where we have used units in which the magnetic length is ℓ0 = 1.
+In section 5.1.1 we saw that correlators of vertex operators are non-vanishing only if their
+total charge (vorticity) is zero. The reason for this condition is that the (Euclidean) action is
+invariant under arbitrary uniform shifts of the field φ. Now, in the correlator on the right hand side
+of Eq.(475) we have a product of N vertex operators, each with the same charge √m. Then, under
+an arbitrary shift by α of the field φ(z), the product of vertex operators change by a phase equal
+to N √mα. On the other hand the other operator in the correlator, which describes a continuum
+background charge, changes by a phase − √mρ0L2α, where L2 is the area. Since ρ0 is the areal
+particle density, N/L2, the two changes of the phase cancel each other exactly. Thus, the operator
+in the expectation value of Eq.(475) is charge neutral and has a non-vanishing expectation value.
+The other important feature of the correlator of Eq.(475) is the choice of compactification
+radius, R = √m, and of the vertex operator Vp(z) with p = m:
+Vm(z) = exp(i √mφ(z))
+(476)
+whose correlation function is
+⟨Vm(z) V−m(�)⟩ =
+1
+(z − �)m
+(477)
+Which is invariant under a 2π rotation (since m ∈ Z). However under a rotation by π (i.e. an
+exchange) it changes by (−1)m. Hence, for m odd it changes sign, and the vertex operator Vm
+behaves as a fermion, while for m even the vertex operator behaves as a boson. We will call the
+vertex operator Vm ≡ Ve the electron operator.
+92
+
+In other words, the correlator of Eq.(475) is the expectation value of N vertex operators each
+describing an electron (for m odd) and a neutralizing background charge. Then, an elementary
+calculation yields the expression of the Laughlin wave function.
+In this formulation the ground state wave function for the integer QH state at ν = 1, which
+has the for of a Vandermonde determinant shown in Eq.(377), is interpreted as the correlator of
+N electron operators in the U(1) CFT with compactification radius R = 1. In this simple case,
+the electron operator is the vertex operator V1(z) = exp(iφ(z)), and there are no other operators
+(aside from the current j(z) = i∂zφ). The propagator of this vertex operator is
+⟨V1(z)V−1(�)⟩ =
+1
+z − �
+(478)
+which indeed represents a fermion with electric charge e.
+We will now see that the vertex operator of Eq.(472) with p = 1 (and R = √m)
+V1(z) = exp
+� i√mφ(z)
+�
+(479)
+represents the Laughlin quasihole. The correlator of this vertex operator is
+⟨V1(z) V−1(�)⟩ =
+1
+(z − �)1/m
+(480)
+which has a branch cut stretching from z to � and, as a result, under a rotation by ±π its phase
+changes by ±π/m, just as the Laughlin anyons do. To see that this operator indeed is related to
+the Laughlin quasihole we will compute the effect of inserting the vertex operator V1(z) in the
+correlator of Eq.(475) and find
+�
+exp
+� i√mφ(�)
+�
+N
+�
+i=1
+exp
+�
+i √mφ(zi)
+� exp
+�
+−i
+�
+d2z′ √m ρ0 φ(z′)
+� �
+=
+=
+N
+�
+i=1
+(zi − �)
+�
+i< j
+(zi − zj)m exp
+−1
+4
+N
+�
+i=1
+|zi|2 − 1
+4m|�|2
+
+= Ψm(�; z1, . . ., zN)
+(481)
+which is, indeed, the wave function for the Laughlin quasihole of Eq.(392). The insertion of
+an additional vertex operator V1(u) at u inside the expectation value of Eq.(481) yields the two-
+quasihole wave function of Eq.(397), proposed by Halperin [234] and discussed in section 10.3.5,
+with the same branch cut shown in Eq.(398) and, therefore carry fractional statistics π/m.
+The operator product expansion discussed in section 3.3.2 of the vertex operators of this CFT
+yields new insight on the quasiholes. Indeed the OPE of two vertex operators of charges p and q
+is
+lim
+�→u Vp(�)Vq(u) = lim
+�→u
+1
+(� − u)∆p+∆q−∆[p+q]m V[p+q]m(u)
+(482)
+where [p]m is the integer p modulo a multiple of m, i.e. if p = mr + s (with r a non-negative
+integer and 0 ≤ s < m, then [p]m = s. Eq.(482) is understood in the sense that the contribution
+of all additional operators to the right hand side vanish as � → u. In other words, the vertex
+operators are primary fields of this CFT and there are only m primary fields. The physical process
+described by the OPE is called fusion.
+A CFT with a finite number of primaries is called a rational CFT [103, 102]. This result is,
+of course, the same statement that we made is section 10.3.7 that the FQH state has m distinct
+anyons (vortices). This result also means that a vertex operator of charge m, i.e. an electron, is
+indistinguishable from the identity operator, V0. In this sense, the allowed primaries are fields
+which are local respect to the electron operator Vm. From the point of view of the FQH state, an
+operator that creates or destroys an electron (at fixed filling fraction) has no effect, as the FQH
+state is a fluid made of electrons This also means that all physical operators must braid trivially
+with the electron operator. In a CFT an operator of this type is said to generate and extended
+symmetry algebra [267, 268].
+Furthermore, the OPE of the chiral current j(z) with the vertex operator V1(z) is
+lim
+�→u j(�)V1(u) = 1/m
+z − wV1(u)
+(483)
+93
+
+which implies that the vertex operator represents a state with charge 1/m (in units of the electric
+charge e).
+The CFT approach to construct FQH states has become a very powerful tool. We will see in
+section 10.3.11 that FQH states on an open geometry (e.g. a disk) have edge states which can
+be understood a chiral CFTs. In addition to providing a deeper understanding of more general
+abelian (muliti-component) FQH states with a K-matrix structure, new classes of of FQH states
+with non-abelian (multi-dimensional) representations of the Braid Group has been proposed.
+The anyons of these states are of particular interest since they have been proposed originally
+by Kitaev in 1997 [169] that anyons can be used as physical qubits for topologically-protected
+quantum computation [169, 269, 270, 271, 170].
+So far we discussed the case of abelian anyons. Abelian anyons have the property that the
+fusion of two anyons is another anyon. Similarly, a braiding operation between two anyons is
+equivalent (up to a phase factor) to another anyon. Mathematically this means that abelian anyons
+are one-dimensional representations of the Braid group. The fractional statistics of an anyon is
+then a label of the representation of the Braid group. As we saw, the same holds under fusion.
+However, the Braid group generally admits multi-dimensional representations. In this more
+general case the fusion (or braiding) of two anyons is represented by a linear combination (su-
+perposition) of anyon states. Linear combinations of anyon states are represented by unitary
+matrices of rank greater than 1. Since matrices generally do not commute braiding and/or fusion
+processes correspond to a multiplication of these matrices. Such unitary processes are then re-
+garded as quantum gates. This concept is the physical basis of topological quantum computation.
+It is currently the cutting edge of research in the field.
+We will now discuss the simplest non-abelian FQH states, the Moore-Read (MR) states [266].
+The Moore-Read wavefunctions are
+ΨMR(z1, . . . , zN) = Pf
+�
+1
+zi − zj
+� �
+i< j
+(zi − zj)n exp(− 1
+4ℓ2
+0
+N
+�
+i=1
+|zi|2)
+(484)
+which describes a FQH of electrons at filling fraction ν = 1/n, with n an even integer. Here
+Pf
+�
+1/(zi − zj
+�
+is the Pfaffian of the matrix 1/(zi − zj).
+This state was motivated by the discovery of an unexpected FQH state in the N = 1 Landau
+level with an even denominator filling fraction ν = 5/2 by Willett and coworkers in 1987 [272,
+273, 274]. Until then (and since then) this is the only FQH state with an even denominator
+filling fraction. The Pfaffian factor has a simple pole when two coordinates approach each other.
+However, provided n ≥ 1 the “Laughlin factor” in the MR wavefunction cancels the singularity.
+We will shortly discuss the CFT construction of the Moore-Read state. The poles in the MR
+wavefunction suggest that in this state the electrons can get closer to each other than in a Laughlin
+state. This observation suggests that there may be some form of attraction (or suppression of
+repulsion) between the electrons in the MR state and motivated the notion that the observed
+even-denominator plateau may be physically related to some sort of pairing interaction. Shortly
+after the Moore-Read proposal was made, a paired wave function at ν = 1/2 with a Pfaffian factor
+was also proposed by Greiter, Wen and Wilczek [275, 276, 277] who suggested that it reflects
+electrons pairing in the ℓ = 1 angular momentum channel in time-reversal breaking condensate
+with symmetry px + ipy.
+Although, following the logic behind the Kohn-Luttinger mechanism for paring in angular
+momentum state ℓgeq1 with repulsive interactions [278, 279, 280] it may possible to get a paired
+state even in the lowest Landau level (although there is no evidence for it, so far), the N = 1
+Landau level may be more hospitable to such a state. Indeed, in the N = 1 Landau level the
+single-particle states i have angular momentum greater or equal than 1, and their probability
+distributions are suppressed both at short and long distances; these states look like smoke-rings.
+The matrix elements of the Coulomb potential in the N = 1 Landau level are parametrically
+suppressed at short distances (compared with the states in the lowest, N = 0, Landau level). In
+this scenario, the ν = 5/2 plateau is interpreted as a ν = 1/2 fully polarize state of the N = 1
+landau level, with an N = 0 Landau level filled with electrons with both spin polarizations in
+a state with ν = 2. There is numerical evidence that an instability into a p-wave paired state
+such as the MR state is favored if the short-distance repulsion between the composite fermions is
+softened [281].
+In fact, in 1983 Halperin [282] considered generalizations of the Laughlin state in which
+due to very strong attractive interactions electrons would form clusters which then condense into
+a Laughlin-type state. The simplest example was Laughlin state of bosons (paired electrons)
+94
+
+a paired state at ν = 1/2 whose Laughlin quasihole carries charge e/4. Extensive numerical
+calculations are more compatible with the MR state than with the Halperin state of pairs [283].
+We will see shortly that the Halperin state is an abelian relative of the MR state.
+Moore and Read derived the wave function of Eq.(484) by considering a CFT with two
+sectors: a chiral boson CFT with compactification radius R = √n, and a chiral Majorana fermion
+CFT, i.e. the chiral part of the 2D Ising CFT, discussed in section 4.2. The chiral Ising CFT has
+central charge c = 1/2 and several primary fields: the identity I, the twist field σ, and the
+(chiral) Majorana fermion χ with (chiral) scaling dimensions 0, 1/16, and 1/2, respectively. The
+propagator of the Majorana fermion, which is a free field, is
+⟨χ(z) χ(�)⟩ =
+1
+z − �
+(485)
+The primary field σ, the “twist field”, is non-local with respect to the Majorana fermion χ, and
+changes its boundary conditions from periodic to antiperiodic.
+The fusion rules of the chiral Ising CFT are
+χ ⋆ χ = I,
+σ ⋆ σ = I + χ,
+σ ⋆ χ = χ
+(486)
+What will be important below is that the σ field has two fusion channels. As in the analysis of
+the Laughlin states, the compactified boson n primaries (anyons), the vertex operators Vp(x) =
+exp(ipφ(x)/n), with n = 0, 1, . . .n and there are n types of anyons, and a (charge) current J =
+i√n∂zφ(z).
+The first task is to identify the electron operator which has to be a fermion and has to
+carry electric charge. This means that it is a composite of an operators (with Fermi statis-
+tics) the (neutral) chiral ising CFT and a vertex operator whose charge is an integer multi-
+ple of e.
+The desired electron operator is ψe(z) = χ(z) exp(i √nφ(z)) whose correlator is
+⟨χ(z)Vn(z)χ(�)V−n(z)⟩ = 1/(z − �)1+n, which is a fermion with charge e.
+The n-point function of the chiral Majorana fermions is
+⟨χ(z1) . . .χ(zN)⟩ = Pf
+�
+1
+zi − zj
+�
+(487)
+which follows fro applying Wick’s theorem. Then we see that the MR wave function is the
+expectation value
+ΨMR(z1, . . ., zN) = ⟨χ(z1) . . .χ(zN)⟩ ×
+� 
+N
+�
+i=1
+exp
+�
+i √nφ(zi)
+� exp
+�
+−i
+�
+d2z′ √n ρ0 φ(z′)
+� �
+(488)
+Next we need to identify the primary fields of the tensor product of the chiral Ising CFT, Z2,
+and the chiral U(1) CFT with compactification radius R = √n. The allowed primary field must
+belocal with respect to the electron operator, ψe. This leaves us with four primary fields: a) the
+identity I, b) the non-abelian vortex σ(z) × exp
+�
+i
+2 √nφ(z)
+�
+with charge e/(2n) and non-abelian
+statistics, c) the charge-neutral Majorana fermion χ, and d) the abelian vortex (the Laughlin
+quasihole) exp
+�
+i√nφ(z)
+�
+, with charge e/n and abelian statics δ = π/n.
+The he new feature here is the non-abelian fractional statistics of the non-abelian vortex
+(sometimes called a “half-vortex”. Its non-abelian character is a consequence of the fact that the
+fusions of two twist fields has two channels, see Eq.(486). This implies that the wave function of
+four non-abelian vortices can be expressed as a linear combination of two so-called conformal
+blocks. In other words, this wave function belongs to a degenerate two-dimensional Hilbert
+space, each component labeled by a conformal block [102]. A braiding operation between two
+non-abelian vortices is a monodromy of the wave function that induces a unitary transformation
+U in this Hilbert space
+U =
+1√
+2
+exp
+�
+iπ
+�1
+8 + 1
+4n
+��
+×
+� 1,
+1
+−1
+1
+�
+(489)
+Therefore, the degenerate Hilbert space of four quasiholes provides a two-dimensional represen-
+tation of the Braid Group. This observation is the basis for regarding a state of four non-abelian
+quasiholes as a qubit. Furthermore, Nayak and Wilczek showed that a state of 2p quasiholes
+spans a 2p−1-dimensional degenerate Hilbert space [284]. This means that, for large p, the de-
+generacy per quasihole is
+√
+2. In other words, this degeneracy is not due to a local degree of
+freedom attached to each quasihole but that it is shared in a non-local fashion!
+95
+
+In what sense are the Moore-Read states paired? In an insightful paper Read and Green [285]
+used a BCS theory approach to investigate the pairing properties of a condensate of composite
+fermions in the px + ipy channel. The mean-field BCS Hamiltonian is
+HF =
+�
+d2k
+�
+(ε(k) − µ) ψ†(k)ψ(k) + 1
+2
+�
+∆∗(k)ψ(−k)ψ(k) + ∆(k)ψ†(k)ψ†(−k)
+��
+(490)
+where ψ(k) is the composite fermions field, ε(k) =
+k2
+2M, and ∆(k) is the gap function. For a
+px + ipy condensate the gap function the gap function transforms under rotations as an ℓ = −1
+angular momentum eigenstate, and for k → 0 it behaves as ∆(k) = (kx − iky)∆, where ∆ is a
+constant pairing amplitude.
+The BCS ground state has the form
+|G⟩ =
+�
+k
+�
+u(k) + �(k)ψ†(k)ψ†(−k)
+�
+|0⟩
+(491)
+where |u(k)|2 = |�(k)|2 = 1. The amplitudes u(k) and �(k)| satisfy the Bogoliubov-de Gennes
+Equation (BdG)
+� ξ(k)
+−∆∗(k)
+−∆(k)
+−ξ∗(k)
+� �u(k)
+�(k)
+�
+≡ E(k) ˆn(k) · σ
+�u(k)
+�(k)
+�
+= E(k)
+�u(k)
+�(k)
+�
+(492)
+where ξ(k) = ε(k)−µ, σ = (σx, σy, σz) are the Pauli matrices, and ˆn(k) = (−Re∆(k), Im∆(k), ξ(k))
+is a unit vector defined for every k. The eigenvalues E(k) and the eigenvectors (u(k), �(k) are
+E(k) =
+�
+ξ2(k) + |∆(k)|2,
+�(k)
+u(k) = −E(k) − ξ(k)
+∆∗(k)
+(493)
+The spinor amplitudes are |u(k)|2 = 1
+2
+�
+1 + ξ(k)
+E(k)
+�
+and �(k) = 1
+2
+�
+1 − ξ(k)
+E(k)
+�
+.
+In the low momentum limit and in real space the BdG Equations become
+i∂tu = − µu + ∆∗i(∂x + i∂y)�
+i∂t� =µ� + ∆i(∂x − i∂y)u
+(494)
+which is just the Dirac Equation in 2+1 dimensions, with µ playing the ole of the mass, and with
+the restriction that the spinor (u, �) obeys the Majorana condition
+(u, �)
+�0
+1
+1
+0
+�
+=
+�u
+�
+�
+(495)
+The Majorana condition is obeyed in all superconductors and reflects that fact that in these con-
+densates only the fermion parity (−1)NF (with NF being the number of fermions) is conserved,
+while the fermion number NF is not.
+This is a good BCS state in that it is fully gapped and it is chiral. Assuming that the pair
+fields in the px and py channels are equal, they showed that the BCS ground state |G⟩ has the
+form
+|G⟩ ∼ exp
+
+1
+2
+�
+k
+g(k)ψ†(k)ψ†(−k)
+ |0⟩
+(496)
+Projected onto a state with N fermions with real space coordinates xi the wave function is a
+Pfaffian
+Ψ(x1, . . ., xN) = ⟨x1, . . ., xN|G⟩ = Pf(g(xi − x j))
+(497)
+The long distance behavior of this wave function depends on whether the chemical potential
+µ > 0 (this is the weak-pairing or BCS regime), or µ < 0 (which is the strong pairing BEC-
+like regime). While in the case of an s-wave superconductor these two regimes are smoothly
+connected by a crossover, in the px + ipy case they are separated by a quantum phase transition
+at µ = 0. In the weak pairing regime the function g(k) has the asymptotic long distance form, as
+k → 0,
+g(k) ≃ −
+2µ
+(kx + iky)∆∗
+(498)
+where the pair field behaves as ∆(k) ≃ (kx − iky)∆. In real space (in complex coordinates) the
+function g(z) becomes
+g(z) =
+� iµ
+π∆∗
+� 1
+z
+(499)
+96
+
+which is the form used in the Pfaffian wave function. On the other hand, in the strong pairing
+regime, µ < 0, g(k) has the asymptotic behavior
+g(k) ≃ −A(kx − iky)
+a−2
+0 + k2
+(500)
+where A and a0 are functions of ∆ and µ,
+A =
+2|µ|M∆
+2|µ| + m|∆|2 ,
+a0 =
+1
+2|µ|
+�
+2|µ|
+M + |∆|2
+(501)
+In real space, at distances |x| ≫ a0 the Fourier transform of g(k) decays exponentially, and as
+a0 → ∞, µ → 0 and the Fourier transform g(z) has the power-law behavior
+g(z) ≃
+� i|∆|
+2π∆∗
+�
+1
+z|z|
+(502)
+This means that µ = 0 is a quantum critical point that separates the weak pairing phase from the
+strong pairing phase, which have distinct properties.
+Read and Green then used a topological argument, originally formulated by Grisha Volovik
+[286] in the context of superfluid 3He films. To see this we notice that the three component
+unit vector ˆn(k) takes values on the surface of a sphere S 2. In addition, since �(k) → 0 for
+k → ∞(and, hence u(k) → 1), we can wrap the momentum space k onto a sphere S 2. Thus
+ˆn(k) are smooth functions of S 2 �→ S 2. such maps are classified into the homotopy classes of
+π2(S 2) ≃ Z given in terms of the integer-valued topological invariant
+Q = 1
+4π
+�
+d2k ˆn(k) · ∂kx ˆn(k) × ∂ky ˆn(k)
+(503)
+In the strong pairing phase µ < 0 and ξ(k) > 0 and ˆn(k) takes values only on the northern
+hemisphere of the target space S 2. Such maps can be deformed continuously to the North Pole
+and, hence, are topologically trivial, Q = 0. On the other hand, in the weak pairing phase, µ > 0,
+ξ(k) takes both positive and negative values. In this phase ˆn(k) is topologically non-trivial and
+the topological invariant Q takes non-trivial values ±1.
+The upshot of this analysis is that, in the weak pairing phase, the paired FQH state is, at this
+mean field level, a two-dimensional topological superconductor. Read and Green [285] further
+showed that this pairing state has vortices with an interesting fermionic spectrum. A vortex is a
+state win which at long distances the (complex) amplitude of the px + ipy condensate behaves as
+∆ exp(iϕ), where ϕ is the azimuthal angle, which winds by 2π on a circumference of large radius
+R. The BdG equation, Eq.(494), has zero mode solutions. In polar coordinates (r, ϕ) the BdG
+Equation the zero modes satisfy
+∆ieiϕ �
+∂r + i
+r∂ϕ
+�
+� =µu
+∆ie−iϕ �
+∂r − i
+r∂ϕ
+�
+u = − µ�
+(504)
+The normalizable zero mode spinor solution is
+�u(r, ϕ)
+�(r, ϕ)
+�
+= f(r)
+√r
+
+1√
+ieiϕ/2
+1
+√
+−ie−iϕ/2
+
+(505)
+where f(r) is given by
+f(r) ∼ exp
+�
+−
+� r
+0
+dr′ µ(r′)
+|∆|
+�
+∼ exp(−µr/|∆|)
+(506)
+Under a 2π rotation this spinor solution is double-valued
+(u(r, ϕ + 2π), �(r, ϕ + 2π)) = −(u(r, ϕ), �(r, ϕ))
+(507)
+which follows form the global phase invariance. I fact, in all paired states, topological or not, un-
+der a global transformation of the pair field by a phase θ, the (composite) fermion must transform
+with a phase of θ/2,
+∆(x) �→ eiθ∆(x),
+ψ(x) �→ eiθ/2ψ(x),
+ψ†(x) �→ e−iθ/2ψ†(x)
+(508)
+97
+
+In other words, in a paired state the fermion is non-local to the vortex. Hence, under a 2π phase
+transformation of the pair field, the fermions must change sign, and the spinor zero mode solution
+must be double-valued. This means that this state has a brach cut.
+The structure of the vortex and of the fermion zero mode is closely related to the problem of
+solitons with fractional charge that we discussed in section 8.3. In fact, Roman Jackiw and Paolo
+Rossi [287] investigated a closely related problem of a theory of Dirac field in 2+1 dimensions
+coupled to a charged scalar field through a Yukawa coupling with the form of a Majorana mass.
+They showed that such a theory admits vortex solutions with fermion zero modes. In a subsequent
+paper Erik Weinberg [288] showed that these zero modes are counted by an index theorem which
+relates the number of zero modes to the vorticity. More recently, Nishida, Santos and Chamon
+[289] that the relativistic theory of Jackiw and Rossi in the non-relativistic approximation reduces
+to the theory of a px + ipy superconductor.
+There is, however, a subtle but profound difference between the fermion zero modes of the
+one-dimensional fractionally charged solitons and the fermion zero modes of a superconductor.
+In the case of teh 1D solitons, the zero modes can be either occupied or empty rendering a
+soliton charge of −e/2 or +e/2. However, in the case of the superconductor, the BdG fermions
+are charge-neutral since their charge has gone into the condensate. Thus, in a superconductor
+fermion number is not conserved, and only the fermion parity is conserved. This means that
+the field operator associated with the vortex zero mode, which we will denote by γ, must be a
+self-adjoint operator, γ† = γ.
+Ivanov [290] showed that this behavior is the origin of the non-abelian fractional statistics in
+this system. Indeed, if one considers a configuration with 2n vortices, each will carry a Majorana
+zero mode γi. Since these are fermion operators they satisfy the usual anticommutator algebra,
+{γi, γ j} = 2δi j. We can (arbitrarily) group the 2n Majorana fermion operators into n pairs. For
+each pair we can define a complex Dirac operator ψ j = (γ2 j + iγ2 j+1)/2 (and its adjoint), which
+satisfies the usual fermionic algebra, {ψ j, ψ†
+k} = δ jk. Each Dirac fermion can be either in an
+empty state or in an occupied state. Thus a system of 2n vortices supports a degenerate Hilbert
+space of dimension 2n−1, which agrees with the results of Nayak and Wilczek [284]. However,
+the assignment of Majorana operators to specific pairs is actually arbitrary, which amounts to
+a particular definition of the branch cuts. Changing the assignment of the operators into pairs
+is then equivalent to a rearrangement of the cuts. In 2006 Michael Stone and Suk Bum Chung
+[291] showed that the these vortices obey the braiding and fusion properties of Ising anyons.
+These properties follow from the branch cut configurations which affect the monodromies of the
+vortices. It is important to stress that the vortices with zero modes are the non-abelions. Majorana
+fermions are fermions and, as such, are abelian.
+In addition to the Moore-read Pfaffian state, the CFT approach has led to the formulation of
+new non-trivial FQH states. Read and Rezayi [292] proposed a series of so-called cluster states,
+which generalize the concept of paired states. They investigated a particular type of cluster states
+in which the Pfaffian factor of the MR state is replaced by a correlator of parafermion primary
+fields of a Zk CFT of Zamolodchikov and Fateev [293]. Parafermions were originally introduced
+by Fradkin and Kadanoff [152] (see also Ref.[294]) as a generalization of the fermions of the 2D
+Ising model to the Zk clock model. In this model one can define several types of parafermions
+each consisting of the fusion of a charge operator and a magnetic (disorder) operator. These
+operators obey an algebra of the type AB = exp(ip2π/k)BA, where the integer p depends on the
+electric and magnetic charges of the parafermions, which is reminiscent of fractional statistics. In
+close analogy with the Majorana fermions of the 2D Ising model, Zk CFT describes the behavior
+of these systems their self-dual points.
+The Zk CFT is actually much richer and has more primary fields than the Z2 case of the
+Ising model. As a CFT, the Zk theory is the same as the CFT on the coset SU(2)−k/U(1). A
+coset means that a U(1) sector has been projected out of the spectrum. Here we will focus on a
+special case of the Z3 CFT. This CFT has a parafermion primary field, which we will call ψ, and
+a non-abelian primary that we will call τ. Read and Rezayi [292] proposed to replace the Pfaffian
+factor of the MR state with a correlator of N parafermions of a Z3 CFT and a Laughlin factor
+with an exponent of n + 2/3. Such states require that the number of electrons N be a multiple of
+3. Thus, these states can be viewed as sates in which the electrons cluster in groups of 3. They
+also considered the more general case of the Zk CFT in which case N is a multiple of k. The
+resulting filling fractions are ν = k/(mk + 2) (with m ≥ 0). The Read-Rezayi k = 3 parafermion
+state is a leading candidate for the observed FQH plateau at ν = 12/5 = 2 + 2/5 [295, 296] (the
+compering state being the 2/5 Jain state). There is strong numerical support for the 12/5 state
+98
+
+being a Z3 parafermion [297].
+The quasiholes obtained by inserting the primary fields τ in the correlator of parafermions
+have interesting properties which stem from their basic fusion rule
+τ ⋆ τ = I + τ
+(509)
+Read and Rezayi [292] showed that the number of conformal blocks for 3n quasiholes, i.e. the
+number of degenerate states, in the parafermion theory is the Fibonacci number F3n−2, where
+F1 = 1, F2 = 2, F3 = 3, F4 = 5. In general Fm = Fm−1 + Fm−2. For m → ∞, the rate of
+increase approaches the limit limm→∞ Fm/Fm−1 = (1 +
+√
+5)/2 which is known as the Golden
+Mean. In other words, for large m, the dimension of the Hilbert space increases exponentially
+as ((1 +
+√
+5)/2)m. Aside from these interesting mathematical curiosities, the significance of this
+fusion rule is that these states, regarded as“qubits”, define unitary transformations that cover
+uniformly the Bloch sphere which is required for universal quantum computation [270].
+At the level of topological quantum field theory the Moore-Read and Read-Rezayi states are
+related to the non-abelian Chern-Simons gauge theory with gauge group SU(2) at Chern-Simons
+level k. In the more general case of the gauge group SU(N) on a manifold M action is given by
+S CS[Aµ] = k
+4π
+�
+M
+d3x ǫµνλ
+�
+Aµ
+a∂νAλ
+a + 2
+3 fabcAµ
+aAν
+bAλ
+c
+�
+(510)
+where k ∈ Z is the level, the gauge fields Aµ = Aµ
+ata take values in the algebra of SU(N), with {ta}
+being the N2 − 1 generators, and fabc the structure constants of SU(N). The expectation values
+of Wilson loop operators of this theory compute the Jones polynomials which are topological
+invariants of knot theory [165]. The connection with SU2)k Chern-Simons gauge theory has
+been essential in formulating effective low energy quantum field theories for the non-abelian
+states [298, 299, 300, 301, 302, 303].
+10.3.11. Edge States and Chiral Conformal Field Theory
+We will now discuss the edge states of the FQH states. As we saw the FQH states are
+incompressible in the bulk and all bulk excitations are gapped. The edge of the region occupied
+by the fluid (in many cases that edge of the physical sample) is where the bulk gap collapses and
+hence where the system has low energy excitations.
+The role of edges and their nature can be seen already in the simple case of the integer Hall
+effect treated as a system of free fermions in the lowest Landau level. In this state all single
+particle states are occupied, and the bulk ground state wave function is a Slater state which takes
+the form of the Vandermonde determinant of Eq.(377). The potential that keeps the electrons
+inside the sample increases monotonously near the edge. As we discussed in section 10.3.1, in
+this region the electrons experience an electric field E that pushes them into the sample, and in
+the presence of the perpendicular magnetic field B the electrons move along the edge at the drift
+velocity � = c|E|/|B|. At some spatial location, corresponding to the locus of a single particle
+Landau state, the potential crosses the Fermi energy and in that region potential is essentially a
+linear function of the coordinate and hence of momentum (or angular momentum depending on
+the gauge that is being used). In other words, the low energy states are one branch of a one-
+dimensional chiral excitation, such as in the right-moving states of our discussion of the chiral
+anomaly in section 7.2
+In section 9.5 we showed that a Chern-Simons gauge theory on a manifold with a boundary
+projects onto a chiral CFT on that boundary. In that section we showed that an abelian Chern-
+Simons theory U(1)m integrates to the boundary, the 1+1-dimensional Minkowski spacetime
+S 1 × R, as a chiral U(1)m CFT for a compactified boson with compactification radius R = 1
+whose action is given by
+S [ϕ] =
+�
+S 1×R
+d2x 1
+4π
+�
+∂0φ∂1φ − �(∂1φ2)
+�
+(511)
+where we rescaled the compactified scalar by a factor of √m to that its compactification radius
+R = √m as in section 10.3.10. The only difference between the theory of Eq.(241) and what we
+did in section 10.3.10 is that this theory is in a 1+1-dimensional Minkowski spacetime S 1 × R
+while before we were in Euclidean spacetime.In the case of the integer quantum Hall effect ν = 1
+(and hence m = 1) and the (time-ordered) correlator of the vertex operator, V1(x, t) = exp(iφ(x, t))
+is
+⟨T exp(iφ(x, t)) exp(−iφ(0, 0))⟩ =
+1
+x − �t − iε
+(512)
+99
+
+which is, indeed, the propagator of a chiral free fermion.
+On the other hand, for the Laughlin states at filling fraction ν = 1/m (and compactification
+radius R = √m) the propagator of the electron ψe ∼ exp(i √mφ) is
+⟨T exp(i √mφ(x, t)) exp(−i √mφ(0, 0))⟩ ∼
+1
+(x − �t − iε)m
+(513)
+while the propagator of the quasihole exp
+�
+i√mφ
+�
+is
+�
+T exp
+� i√mφ(x, t)
+�
+exp
+�
+− i√mφ(0, 0)
+��
+∼
+1
+(x − �t − iε)1/m
+(514)
+Therefore, the CFT of the edge is the same as the CFT of the ideal wavefunction with the only
+difference that the former is in Minkowski spacetime while the latter is in Euclidean spacetime.
+In this sense there is a one-to-one correspondence between the bilk and the edge. We can
+also see how that works by considering a fundamental Wilson arc in the bulk along a path γ(x, y)
+where x and y are at the boundary
+⟨W[γ]⟩ = ⟨exp(i
+�
+γ(x,y)
+dxµA µ) ≡ ⟨exp( i√mφ(x)) exp(− i√mφ(y))⟩
+(515)
+where on the right hand side we have rescaled the field by √m, as before. The scaling dimensions
+of the electron, the quasihole and the current are ∆e = m
+2 , ∆qh =
+1
+2m and ∆current = 1. These results
+will be important shortly.
+The same structure applies to the multi-component abelian FQH states. The only difference is
+that in multi-component FQH states the edge consists, in general, of a charge field which couples
+to the electromagnetic field and one or more neutral edge states. An example is the theory of the
+non-abelian Pfaffian states at filling fraction ν = 1/n. In this case the edge theory consists of a
+compactified chiral boson φ of radius R = √n and a chiral Majorana (neutral) fermion χ. At a
+formal level the Lagrangian for the edge state(s) is a sum of two terms
+L = 1
+4π(∂xφ∂tφ − �c(∂xφ)2) + χi(∂t − �n∂x)χ
+(516)
+where �c and �n are the velocities of the (charged) compactified chiral scalar φ and of the neu-
+tral Majorana field χ. Numerically (and experimentally) it is known that the charge mode is
+(substantially) faster than the neutral mode(s), �c > �n, and often by significant factors.
+A superficial reading of this Lagrangian suggests that these degrees of freedom are decou-
+pled. However this is not correct. Only operators from both sectors which are local with respect
+to the electron ψe ∼ χ exp(i √nφ) are physically allowed. Here an operator is local with respect
+to the electron means that the operator braids trivially with the electron. This condition restricts
+the allowed observables. In the case of the fermionic MR state, with n = 2, the allowed operators
+are the non-abelion σ exp(iφ/2
+√
+2) with scaling dimension is ∆ = 1/8 and electric charge is
+Q = 1/4, the Majorana fermion with scaling dimension ∆ = 1/2 and no electric charge Q = 0,
+and the Laughlin quasiparticle with scaling dimension ∆ = 1/4 and charge Q = 1/2. Its electron
+operator has scaling dimension ∆ = 3/2 and charge Q = 1.
+Conformal field theory has an important defining universal quantity called the central charge
+which is closely related to the energy-momentum tensor of the theory [102]. For pedagogical
+introductions to CFT see Refs.[304, 45] and chapter 21 of Ref.[10]. The energy-momentum
+tensor Tµν is a locally conserved current, ∂µTµν = 0. Its local conservation implies the global
+conservation of energy and momentum. As such, the energy-momentum tensor is a fundamental
+observable of any quantum field theory. In a conformal field theory, the energy-momentum is
+also the generator of local scale and conformal transformations. In the special case of 1+1-
+dimensional systems, such as the edge states of the FQH fluids, the energy momentum tensor
+has special and crucially important properties. In addition of being locally conserved, conformal
+invariance requires that Tµν must be traceless, T µ
+ν = 0, since its trace is the generator of dilations.
+In this case, if the theory is chiral, the energy momentum tensor has only one (right-moving)
+component T = (T00 − T11)/2. In complex coordinates of the Euclidean metric, the correlator of
+the energy momentum tensor T = Tzz is [305]
+⟨T(z)T(�)⟩ =
+c/2
+(z − �)4
+(517)
+100
+
+where c is a universal quantity known as the it central charge of the CFT. In the case of the
+compactified boson the central charge is c = 1 whereas for a massless Majorana fermion c = 1/2.
+The central charge of the CFT enters in many observables of fundamental physical impor-
+tance. For example, for a 1+1-dimensional system of length L, such as the edge state of a FQH
+droplet, the ground state energy Egnd for large L has the behavior [306, 307]
+Egnd = ε0L − c π�
+6L + O(1/L2)
+(518)
+where ε0 is the ground state energy density, which is a non-universal quantity, c is the central
+charge of the CFT and � is the velocity of the massless modes. The second term in Eq.(518)
+is known as the Casimir energy. Similarly, the free energy density f(T) of a CFT has the low
+temperature Stephan-Boltzmann behavior (in one dimension)
+f(T) = ε0 + cπT 2
+6� + O(T 3)
+(519)
+where we set the Boltzmann constant to unity. The low-temperature specific heat c(T) is
+c(T) = cπT
+3� + O(T 2)
+(520)
+In a chiral CFT, such as the edge states of the FQH fluids, the central charge enters in the low
+temperature behavior of the (Hall) thermal conductivity κxy, [285]
+κxy = cπT
+6ℏ
+(521)
+Much of what is known about FQH states comes form experiments involving their edge
+states. In a series of exquisite experiments Granger, Eisenstein and Reno [308] measured κxy in
+the edge states of a ν = 1 quantum Hall fluid and showed that the heat transport is indeed chiral,
+i.e. the edge state behaves as a heat “diode”. This effect was confirmed in the FQH states by
+Bid and coworkers [309] who, in addition, where used this effect to detect the neutral modes in
+several Jain states at filling fractions 2/3 and 2/5 and in the non-abelian state at ν = 5/2.
+The simplest experimental probes are quantum point contacts and typically are of two types:inter-
+edge tunneling inside a FQH liquid or tunneling of electrons into the edge state of a FQH liquid.
+Since the bulk is gapped, only the edge states participate in these tunneling processes. In the
+general case we will have two edges and a tunnel process at a point contact. Here the two edges
+can be either the two edges of the same FQH state, in which case the process happens inside the
+FQH liquid and involves tunneling of quasiparticles, or the edges of different liquids, in which
+case this process is external and involves tunneling of electrons. Problems of these types were
+first investigated by Charles Kane and Matthew Fisher [310] which led to a considerable amount
+of work on these problems.
+Let us consider first the case of tunneling into the edge state of a ν = 1/m Laughlin state
+from a Fermi liquid, which we will take to be a QH fluid in the ν = 1 state. The point contact is
+at x = 0 The total Lagrangian is
+L = Ledge + LFL − Γ eiω0t ψ†
+e,edge(0, t) ψe,FL(0, t) + h.c.
+(522)
+where ω0 = eV/ℏ, and V is the bias voltage between the two fluids, and Γ is a tunneling matrix
+element. The local electron spectral density (the density of states) N(ω) at energy ω is
+N(ω) = Im lim
+x→0+
+� ∞
+−∞
+dt Ge(x, t) eiωt = const. |ω|m−1
+(523)
+where Ge(x, t) = ⟨Tψ†
+e(x, t)ψe(0, 0)⟩ is the electron correlator in the FQH edge, shown in Eq.(512).
+This result, combined with Fermi’s Golden Rule for a point contact with voltage bias V, predicts
+a tunneling current from a Fermi liquid (FL) into the chiral Luttinger liquid (CLL) of the edge
+state to be [311]
+I(V) = 2π e
+ℏ|Γ|2
+� 0
+−eV
+dE NCLL(E, T)NFL(E + eV, T) ∝ Vm
+(524)
+and a tunneling differential conductance G(V)
+G(V) = dI
+dV = 2π e
+ℏ|Γ|2NFL(0)NCLL(V, T) ∝ Vm−1
+(525)
+101
+
+which is non-Ohmic and vanishes as V → 0.
+Early experiments by Albert Chang and cowerkers , on a geometry that (most likely) had
+many point contacts, confirmed the predicted CLL behavior [312, 313], although there were
+discrepancies in the measured exponents. The behavior seen in more recent experiments by
+Cohen and coworkers [314], on a point contact in graphene, are consistent with the theoretical
+predictions [311]. Interestingly, these newer class of experiments [315, 314] also show evidence
+of an analog of Andreev reflection predicted to exist near the strong coupling fixed point of the
+theory of Sandler, Chamon and Fradkin [316] as a consequence of electron fractionalization.
+Most experiments are done on a geometry call a Hall bar in which the QH fluid occupies a
+rectangular region with its length L larger than its width W. In this geometry, there are chiral
+edge states at opposite sides of the Hall bar propagating in opposite directions. One type of point
+contact consists in creating a constriction in the quantum Hall fluid by applying a gate, a bias
+potential on a narrow strip accross the Hall bar. This gate repels the electrons in the QH fluid,
+forcing the opposite edges to approach each other in the proximity of the gate. In the absence
+of the gate, momentum is conserved on each edge which forbids tunneling accross the Hall bar
+since the edge states have opposite Fermi momentum. However, the gate breaks translation in-
+variance on both edges and tunneling between them is now allowed. In other words, the gate
+creates a point contact between the opposite propagating edge states leading to a tunneling pro-
+cess between the edges of the QH liquid. Thus, this is internal tunneling as oppose the process
+of tunneling between two different liquids that we discussed above.
+The tunneling Lagrangian for this system has the same form as in Eq.(522) except that now
+the two edges are identical. The theory Kane and Fisher shows that in the factional case the most
+relevant process is the tunneling of FQH quasiparticles. The Fermi Golden Rule argument used
+in Eq.(525) to compute the differential conductance also applies in the present case except that
+now the two densities of states are the densities of states of the (Laughlin) quasiparticles (instead
+of electrons). The local quasiparticle density of states is Nqp(ω) ∼ |ω|
+2
+m −1. Then, the differential
+tunneling conductance G(V) in this case is
+G(v) ∼ V2( 1
+m −1)
+(526)
+This behavior is also non-Ohmic but, unlike the case of electron tunneling of Eq.(522), the differ-
+ential tunneling conductance now diverges as V → 0. This behavior means that the Γ → 0 fixed
+point is unstable and that the point contact flow to a strong coupling fixed point at Γ → ∞. Early
+experiments by Milliken, Umbach and Webb in 1996 [317] showed indications of CLL behavior.
+The predicted behavior was confirmed in the experiments of Roddaro and coworkers [318, 319].
+In the RG language, we can define a dimensionless tunneling amplitude g by Γ = a∆−1g,
+where ∆ is the scaling dimension of the tunneling operator and a is a UV cutoff. The “tree-level”
+beta function is readily found to be
+β(g) = a∂g
+∂a = (1 − ∆) g + O(g2)
+(527)
+which shows that if the scaling dimension of the operator of the tunneling particle is ∆ < 1, then
+this tunneling process is relevant, while is ∆ > 1 it is irrelevant.
+Kane and Fisher argued that there should be a crossover from the IR unstable weak-coupling
+fixed point governed by quasiparticle tunneling to an IR stable strong coupling fixed point gov-
+erned by electron tunneling. This crossover is reminiscent of the the crossover in quantum impu-
+rity problems such as the Kondo problem of a magnetic impurity in a metal [320, 32, 52, 53, 321].
+The main difference is that the impurity coupling is marginal and the crossover scale, the Kondo
+temperature TK, which is related to the coupling constant g as TK ∼ exp(−1/g), while in the FQH
+constriction the crossover scale is TK(g) ∼ g1/(1−∆). Kane and Fisher argued that this crossover
+can be viewed as a process that interpolated between a FQH fluid with a weak constriction to a
+regime in which the Hall bar splits into two pieces with weak electron tunneling between them.
+It turns out that, after some manipulations, the model of the constriction can be mapped into
+a one-dimensional compactified boson ϕ on a semi-infinite line, x ≥ 0, with a vertex operator
+acting at the boundary. The action of this system, known as boundary sine-Gordon is
+S = 1
+8π
+� ∞
+−∞
+dt
+� ∞
+0
+dx (∂µϕ)2 + Γqp
+� ∞
+−∞
+dt cos
+� �
+ν
+2ϕ(0, t)
+�
+(528)
+This theory has two fixed points: a) the IR unstable fixed point at Γqp = 0 where the field
+Neumann boundary conditions at x = 0, ∂xϕ = 0, and b) an IR stable fixed point at Γqp → ∞
+102
+
+where the field has Dirichlet boundary conditions, ϕ = 2π √2/ν n (with n ∈ Z). It turns out that
+this is an integrable field theory. Paul fendley, Andreas Ludwig and Hubert Saleur [322] used the
+thermodynamic Bethe Ansatz to calculate the differential tunneling conductance for the Laughlin
+state at ν = 1/3 as a function of voltage V and temperature T and obtained the full weak to strong
+crossover RG flow. Detailed predictions of this theory have been verified experimentally by the
+work of Roddaro and coworkers [318, 319]. Point contact experiments were performed in the
+more challenging ν = 5/2 FQH state by Miller and coworkers [323] and Radu and coworkers
+[324]. They found that the MR state is the the one that best fits their experimental results.
+The charge of particle can, in principle, be found directly by measuring the noise of a weak
+current. This process is known as shot noise. In the case of the constriction, the quasiparticle
+current operator is Iqp = 2eνΓqp sin
+� √ν/2φ + ω∗
+0t
+�
+, where ω∗
+0 = eνV/ℏ. The noise spectrum,
+S (ω), of the tunneling current qp is [325]
+S (ω) =
+� ∞
+−∞
+dt ⟨{Iqp(t), Iqp(0)}⟩ eiωt
+(529)
+Using the expression of the quasiparticle correlator of Eq.(514), one readily finds that, to leading
+order in Γqp, the noise spectrum is
+S (ω) = eν⟨Iqp⟩
+
+�
+1 − ω
+ω∗
+0
+�2ν−1
++
+�
+1 + ω
+ω∗
+0
+�2ν−1
+(530)
+In the limit of zero frequency the noise spectrum takes the shot noise form
+lim
+ω→0 S (ω) = 2e∗⟨Iqp⟩
+(531)
+where the expectation value of the tunneling current is
+⟨Iqp⟩ =
+2π
+Γ(2ν)eν|Γqp|2 ω∗
+0
+2ν−1
+(532)
+Chamon, Freed and Wen [326] calculated the exact noise spectrum for the case of a ν = 1/2
+bosonic FQH state and were able to investigate the crossover as well, and Fendley, Ludwig and
+Saleur [327] used the Bethe Ansatz to construct a soliton basis to compute the DC shot noise.
+These theoretical predictions were tested in tour-de-force experiments by de Picciotto and
+coworkers [253] and Samindayar and coworkers [254] who were able to measure the fractional
+charge of e/3 in the ν = 1/3 state and, with some caveats, e/5 in the ν = 2/5 state. Dolev and
+coworkers [328] went on to measure the charge from noise experiments in the nonabelian state
+at ν = 5/2 and obtained results consistent with e∗ = e/4 as predicted by theory.
+We will now consider experiments on Hall bars with two quantum point contacts created by
+two narrow gates transversal to the bar and separated at some distance d each other. Quantum
+devices of this type are Fabry-P´erot quantum interferometers. The way they operate is as follows
+[329, 298]. A current is injected in the bottom edge. At the first quantum point contact (QPC)
+part if the current I1 tunnels to the opposite edge and the other part I2 goes on and tunnels at the
+second QPC. The FQH fluid is assumed to occupy uniformly the region between the two QPC’s.
+There is some magnetic flux Φ is that region which also contains a number of localized vortices
+(quasiparticles). When both currents rejoin at the top edge they interfere and the interference has
+information on the charge of the particles that tunnels through the Aharonov-Bohm effect with
+the flux Φ. The interference also has information on the fractional statistics of the quasiparticles
+that tunneled through their braiding with the static vortices. Chamon, Freed, Kivelson, Sondhi
+and Wen proposed a setup of this type to measure the fractional statistics of the quasiparticles for
+the abelian states [329]. They showed that the total current It = I1 + I2 is given by
+It = e∗|Γeff|2 2π
+Γ(2ν) |ω0|2ν−1 sign(ω∗
+0)
+(533)
+where Γeff is give by
+|Γeff|2 = |Γ1|2 + |Γ2|2 + (Γ1Γ∗
+2 + Γ∗
+1Γ2) Fν
+�ω∗
+0d
+�
+�
+(534)
+Here Γ1 and Γ2 are the tunneling amplitudes at the two QPCs, ν = 1/m is the filling fraction, � is
+the velocity of the edge modes, d is the distance between the two QPCs and
+Fν(x) = √πΓ(2ν)
+Γ(ν)
+Jν−1/2(x)
+(2x)ν−1/2
+(535)
+103
+
+where Γ(z) is the Euler Gamma function and Jν−1/2(z) is the Bessel function of the first kind. In
+the presence of Nq localized quasiparticles in the area between the two QPCs, the contribution
+of the phases of the tunneling matrix elements get shifted to
+Γ∗
+1Γ2 = ¯Γ∗
+1 ¯Γ2 exp
+�
+−2πi
+�
+ν Φ
+φ0
+− νNq
+��
+(536)
+where φ0 is the flux quantum. In Eq.(536) the first term in the phase shift is the Aharonov-
+Bohm effect of the tunneling quasiparticles, while the second term in the phase shift 2πνNq is
+the contribution of the fractional statistics of the tunneling quasiparticle as its worldline braids
+with the Nq localized quasiparticles. This means that there is an interference contribution to
+the tunneling current (and also to the transmitted current) which is sensitive to both the charge
+and to the fractional statistics of the quasiparticles. This is this effect that is being measured in
+experiments.
+Early attempts at doing this experiment were made by Camino and coworkers [330] but were
+difficult to interpret partly due to subtle reasons related to the difficulty in controlling the actual
+area of the region comprised between the two QPCs [331]. Technical advances in the fabrication
+of these devices led in 2020 to the first successful measurement of the fractional statistics of the
+quasiholes of the Laughlin state at ν = 1/3 (and also at the Jain state at ν = 2/5) by Nakamura,
+Liang, Gardner and Manfra [233].
+The nonabelian case is more subtle both theoretically and experimentally. The theory of the
+abelian interferometer of Chamon and coworkers was generalized to the nonabelian FQH states
+by Fradkin, Nayak, Tsvelik and Wilczek in 1998 [298]. The structure of the interferometer is the
+same as in the abelian case but the interference effects are different. In addition, in the case of
+the MR state, taside from the nonabelian quasiparticle σ ∼ χ exp(iφ/2
+√
+2) has charge e/4 and
+nonabelian fractional statistics, the MR state has two more abelian anyons, the charge neutral
+Majorana fermion χ and the charge e/2 Laughlin quasiparticle. The number of anyons and their
+properties are very different in different nonabelian FQH states.
+Ignoring for now the existence of more quasiparticles, let us focus now on the fundamen-
+tal anyon which is nonabelian. Fradkin and coworkers showed [298] considered a nonabelian
+quasihole that is injected to the bottom edge, tunnels at the first QPC and arrives at the left end
+of the top edge in state |ψ⟩. If a second such quasiparticle is now injected but now tunnels to
+the top edge at the second QPC., arriving at the left end of the top edge in the state eiαBNq|ψ⟩,
+where α is the Aharonov-Bohm phase determined by the flux Φ piercing the interferometer and
+BNq si the braiding operator os the second quasiparticle that is circling around the Nq localized
+quasiparticles in the interferometer. Then the tunneling conductance measured at the left exit of
+the top edge has an interference contribution
+σxx ∝ |Γ1|2 + |Γ2|2 + Re
+�
+Γ∗
+1Γ2eiα⟨ψ|BNq|ψ⟩
+�
+(537)
+The matrix element ⟨ψ|BNq|ψ⟩ is given in terms of the expectation value of the Wilson loop oper-
+ators of the tunneling quasiparticles braided with the Wilson loops of the Nq localized quasiparti-
+cles in the area of the interferometer. In 1989 Edward Witten [165] showed that this expectation
+value, which is computed in the nonabelian Chern-Simons gauge theory, is equal to a topolog-
+ical invariant of the braid known as the Jones polynomial VNq(e1π/4). Therefore, the oscillatory
+component of the tunneling current (and of the conductance) measures a topological invariant!
+Ref.[298] gave a general algorithm for the computation of this matrix element. However, in
+the particular case of the MR states, the non-abelian gauge theory associated with these states is
+the SU(2)2 Chern-Simons gauge theory. In this case, an explicit calculation Bonderson, Kitaev
+and Shtengel [332] and by Stern and Halperin [333] leads to the result
+σxx ∝|Γ1|2 + |Γ2|2,
+for Nq odd
+σxx ∝|Γ1|2 + |Γ2|2 + 2 |Γ1||Γ2|(−1)Nψ cos
+�
+α + arg
+�Γ2
+Γ1
+�
++ π
+4 Nq
+�
+,
+for Nq even
+(538)
+Here, Nψ = 1 when the Nq quasiparticles fuse into the state ψ and Nψ = 0 otherwise. The
+interference effect is absent for Nq odd since an odd number of σ quasiparticles cannot fuse into
+the identity state I. This simple even-odd effect is special for SU(1)2. In the general case and,
+in particular in the SU2)3 case which applies to the k = 3 Read-Rezayi (“Fibonacci”) state, the
+expressions are more complex.
+104
+
+At any rate, even in the MR state the situation is more complicated for two reasons. One
+is that the charge e/2 abelian Laughlin quasiparticle is always able to tunnel thus spoiling the
+even-odd effect. the other complication is that the nonabelian FQH occurs in the N = 1 Landau
+level and the edge states of the nonabelian FQH state is surrounded by an abelian ν = 2 state,
+which makes accessing the interesting edge states more difficult. Robert Willett has pioneered the
+fabrication and operation of the interferometer for the nonabelian state at ν = 5/2 [334, 335, 336].
+With some significant caveats although there is solid evidence of nonabelian braiding, more work
+remains to be done on this system.
+11. Particle-Vortex Dualities in 2+1 dimensions
+11.1. Electromagnetic Duality
+The oldest form of duality in Physics is, perhaps, Dirac’s observation that in the absence of
+electric charges and currents Maxwell’s equations are invariant under the exchange of electric
+and magnetic fields, E → B and B → −E [114]. This observation led him to conjecture the
+existence of magnetic monopoles. In a relativistic invariant formulation, Maxwell’s equations in
+free space can be expressed in terms of the electromagnetic field tensor Fµν and the dual field
+tensor F∗
+µν
+∂µFµν = 0,
+∂µF∗
+µν = 0
+(539)
+where F∗
+µν = 1
+2ǫµνλρFλρ, and where ǫµνλρ is the fourth-rank antisymmetric Levi-Civita tensor, The
+first Maxwell equation in Eq.(539) is just the wave equation in free Minkowski spacetime. The
+second equation in Eq.(539) is known as the Bianchi identity. The Bianchi identity is a constraint
+which implies that there are no magnetic monopoles and that the second rank antisymmetric field
+strength tensor can be expressed in terms of the vector potential Aµ as Fµν = ∂µAν − ∂νAµ.
+This is an example of what in differential geometry is called Hodge duality, which relates a
+vector or, more generally a tensor field to its Hodge dual. In general in D dimensions the Hodge
+dual of a (fully antisymmetric) tensor of rank p is an antisymmetric tensor of rank D − p. When
+contracted with the oriented infinitesimal element of a p-dimensional hypersurface, dx1 ∧ dx2 ∧
+. . . ∧ dxp, an antisymmetric tensor of rank p, Fµ1µ2...µp, defines a p-differential form, called a p-
+form. Thus, differential forms embody the physically intuitive notions of circulation of a vector
+field, flux of a second rank tensor, etc. We will see below that duality transformations are closely
+related to these geometric notions of duality.
+11.2. Particle-Vortex Duality in 2+1 dimensions
+In this section we will extend the particle-vortex duality discussed in 2D in section 5.1.1 (see
+Eq.(75)) to 3D. In the field theory interpretation, 2D is a 1+1-dimensional Minkowski spacetime
+and the XY model is a representation of a complex scalar field of unit modulus. In the duality, the
+particles are the particle-like excitations of the complex scalar field. The high temperature phase
+of the XY model is viewed as a partition function of a set of oriented loops that carry charge. the
+loops are the worldlines of the particles of the XY model.
+11.2.1. The 3D XY Model
+This picture can be seen as follows. Consider an XY model on a D-dimensional hypercubic
+lattice whose sites are labelled by {r}. At each site there is a periodic variable θ(r) ∈ [0, 2π). The
+partition function is
+ZXY =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+exp
+
+1
+T
+�
+r
+�
+j=1,...,D
+�
+cos ∆jθ(r)
+�
+(540)
+where ∆jθ(r) = θ(r+e j)−θ(r), with j = 1, . . ., D, is the lattice difference, and T is the temperature
+(in the classical statistical mechanical picture). Since the interaction on each link is a periodic
+function of the phase difference ∆jθ(r), we can expand the Gibbs weight (for each link!) in a
+Fourier series or, what is the same, in the integer-valued representations of the group U(1) (which
+is the global symmetry of the XY model) to obtain
+ZXY =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+∞
+�
+ℓj(r)=−∞
+exp
+−
+�
+r, j
+T
+2 ℓ2
+j(r) + i
+�
+r
+θ(r)∆jℓ j(r)
+
+(541)
+105
+
+where we used a Gaussian approximation for the modified Bessel function In(z), and ∆jℓ j(r) =
+�D
+j=1(ℓ j(r)−ℓ j(r−e j)) (where e j is the lattice unit vector along the direction j) denotes the lattice
+divergence. Integrating-out the phase variables θ(r) we obtain
+ZXY =
+�
+r
+∞
+�
+ℓj(r)=−∞
+�
+r
+δ(∆jℓ j(r)) exp
+−
+�
+r
+D
+�
+j=1
+T
+2 ℓ2
+j(r)
+
+(542)
+Therefore, the partition function is given by a sum over loops of conserved currents ℓ j(r) defined
+on the links of the lattice, with a weight on each link ∝ exp(−ℓ2
+j/2β). These loops are the (lattice)
+worldlines of the particles of the complex scalar field. In the phase where this representation
+is convergent, the complex scalar field is massive, the XY model is gapped, and these particles
+have short range interactions. This picture is true in all dimensions, 3D included.
+On the other hand, in 2D the vortices are point-like events in Euclidean spacetime which in-
+teract with each other through long range, logarithmic, interactions. In the field theory language,
+in 2D the point-like vortices are instantons which govern the low temperature phase of the XY
+model. In spacetime dimensions D > 2 the vortices become extended objects: vortex loops (or
+strings) in 3D, closed vortex surfaces in 4D, etc. In 3D the vortex loops are magnetic flux tubes
+which interact with each other through a Biot-Savart type interaction, namely bits of vortices
+interact with each other with a 3D Coulomb interaction much in the same way as with loops of
+current in magnetostatics,
+Z3DXY =
+�
+{mj(˜r}
+�
+˜r
+δ(∆jm j(˜r)) exp
+−2π2
+T
+�
+˜r,˜r′
+3
+�
+j=1
+m j(˜r)G0(˜r − ˜r′)m j(˜r′) − α
+�
+˜r, j
+m2
+j(˜r)
+
+(543)
+where {˜r} labels the sites of the dual (cubic) lattice, the variables m j(˜r) take values on the integers,
+α is a (short-distance) vortex core energy, and G0(˜r − ˜r′) is the 3D lattice propagator (Green
+function) which at long distances has the standard form
+G0(x − x′) =
+1
+4π|x − x′|
+(544)
+In Eq.(543) the vortex loops are represented by the integer-valued conserved currents m j(˜r),
+which are naturally interpreted as the world lines of magnetic charges.
+We could have also reached the same result by following the line of reasoning that we used in
+section 5.1.1. Indeed, let θ(x) be the phase field of a complex scalar field φ(x) deep in its broken
+symmetry state where the amplitude of the field φ(x) can be taken to be approximately fixed. The
+partition function in this phase reduces to
+Z[aµ] =
+�
+Dθ exp
+�
+− 1
+2g
+�
+d3x
+�
+∂µθ − aµ
+�2�
+(545)
+where g is a coupling constant which in the XY model is proportional to the temperature. Here,
+much as in Eq.(70), the gauge field aµ represents the vortices. Indeed, in 3D the vorticity is
+represented by a locally conserved vector field ωµ
+ωµ = ǫµνλ∂νaλ = 2π
+�
+k
+mk
+µδ(x − xk)
+(546)
+where mk
+µ are the integer-valued vortex (magnetic) currents we used above. As before, the pe-
+riodic nature of the phase field θ(x) implies that the vortex currents must be quantized and be
+integer-valued.
+Here too we can perform a Hubbard-Stratonovich transformation in terms of a vector field bµ
+to find a dual theory which now is
+Z(aµ) =
+�
+DbµDθ exp
+�
+−g
+2
+�
+d3x b2
+µ − i
+�
+d3x bµ(∂µθ − aµ)
+�
+(547)
+Upon integrating out the phase field θ, which acts as a Lagrange multiplier which imposes the
+constraint ∂µbµ = 0. This constraint implies that we can write the field bµ in terms of a dual
+gauge field ϑµ
+bµ = ǫµνλ∂νϑλ
+(548)
+106
+
+Therefore, we can rewrite the partition function as
+Z(aµ) =
+�
+Dϑ exp
+�
+−g
+4
+�
+d3x f 2
+µν + i
+�
+d3x ωµϑµ
+�
+(549)
+where fµν = ∂µϑν − ∂νϑν is the field strength of the dual gauge field ϑµ. Notice that the vortex
+current ωµ is minimally coupled to the dual gauge field ϑµ. If we now integrate-out the gauge
+field ϑµ we obtain an expression for the weight in the path integral for a configuration of vortices
+mk
+µ which is identical to the result of Eq.(543).
+What we have shown above is that in 3D the Goldstone phase of a complex scalar field
+with coupling constant g is the dual of a Maxwell gauge theory. with coupling constant 1/g.
+Moreover, the periodicity of the phase field implies that the dual gauge field ϑµ is compact in
+the sense that its fluxes must obey flux quantization. It is straightforward to show that a charge
+operator Vn = exp(inθ) with electric charge n in the scalar field theory is represented in the dual
+gauge theory by a magnetic monopole of magnetic charge n.
+In summary we showed that the 3D XY model (equivalent to the theory of the complex scalar
+field) can be written in terms of two different models of loop configurations. We saw that the
+XY model is a theory of particle (electric) loops in the high temperature phase and of vortex
+(magnetic) loops in the low temperature phase. However the particle loops have short range
+interactions while the vortex loops have long range interactions. Thus, the 3D XY model is not
+self-dual. In spite of having a representation in terms of vortex loops, the 3D XY model is in
+reality has a very different behavior that the 2D system. The Kosterlitz-Thouless theory describes
+the phase transition in terms of a vortex-antivortex unbinding transition upon which the vortices
+proliferate. In addition, in 2D the Goldstone phase is actually a line of fixed points, there is never
+true long range order but, instead, power-law correlations of the physical observables. In the
+Kosterlitz-Thouless theory the disordered phase arises the strong fluctuations of the phase field
+due to the proliferation of vortices while, at the local level, the order parameter still has a finite
+magnitude. A consequence of this behavior is that the superfluid density has a universal jump at
+the Kosterlitz-Thouless transition [337].
+In contrast in the 3D XY model the Goldstone phase is a true phase with long range order.
+The theory has a continuous phase transition (the thermodynamic superfluid transition) at which
+the superfluid density vanishes continuously. This means that the phase transition of the 3D XY
+is better described by the Wilson-Fisher fixed point of a complex scalar field [30, 48], rather than
+by a vortex proliferation picture of the Kosterlitz-Thouless theory [84, 85]. In particular, since
+this theory does not have magnetic monopoles, the vortex loops cannot proliferate as they do in
+2D. Instead, as the phase transition is approached, the vortex loops grow large in size but also
+become fractal-like objects whose effective core size diverges as the transition is approached. In
+fact, qualitative vortex proliferation arguments readily lead to the incorrect conclusion that the
+transition should be (strongly) first order [338].
+In spite of these differences, particle-vortex duality still plays an important role in 3D by
+relating the 3D XY model to another theory which is its dual under electromagnetic duality. Here
+electromagnetic duality is understood as the exchange of the electric worldlines (electric loops)
+and the magnetic loops (vortex loops) while exchanging strong and weak coupling, T ↔ 1/T, in
+this case between different theories. This duality was investigated by Michael Peskin [339], by
+Paul Thomas and Michael Stone [338], and by Chandan Dasgupta and Bertrand Halperin [340].
+11.2.2. Scalar QED in 3D
+These authors considered a theory of a superconductor represented by a complex scalar field
+ϕ(x), coupled to a fluctuating electromagnetic (Maxwell) field, also known as the abelian Higgs
+model, or scalar electrodynamics (scalar QED) aµ, with µ = 1, 2, 3 The partition function of this
+theory is
+ZSC =
+�
+DϕDϕ∗Daµ exp
+�
+−
+�
+d3x L[ϕ, ϕ∗, aµ]
+�
+(550)
+where
+L = |Dµ(a)ϕ|2 + m2|ϕ|2 + u|ϕ|4 + 1
+4e2 f 2
+µν
+(551)
+where the covariant derivative is Dµ(a) = ∂µ + iqaµ, with the integer q being the charge of the
+scalar field (in units of the coupling constant e), and fµν = ∂µaν − ∂νaµ is the field strength. This
+theory is known as (Euclidean) scalar electrodynamics (or, the abelian Higgs model).
+107
+
+The scalar electrodynamics was extensively studied using the perturbative renormalization
+group within the epsilon expansion (near 4 dimensions) which leads to the conclusion that it
+has a weakly (fluctuation-induced) first order phase transition [109, 341]. Scalar QED of an N-
+component scalar field coupled to a Maxwell gauge field has a continuous phase transition with
+a non-trivial fixed point for N ≥ Nc ≃ 183 (!) [342]. Since these results rely on perturbation
+theory (or in the large-N limit) it was long suspected that the physics may be different in D = 3.
+We will see that particle-vortex duality provides the answer to this question [340].
+We begin by writing the lattice version of Eq.(550) which is obtained by coupling the XY
+model of Eq.(540) to a dynamical (Euclidean) Maxwell field
+ZSC =
+�
+r
+� 2π
+0
+dθ(r)
+2π
+� ∞
+−∞
+daµ(r)
+2π
+exp
+�
+−S (θ, aµ)
+�
+(552)
+where the action S (θ, aµ) is
+S (θ, aµ) = − 1
+T
+�
+r,µ
+cos
+�
+∆µθ(r) − qaµ(r)
+�
++ 1
+4e2
+�
+r,µ,ν
+�
+∆µaν(r) − ∆νaµ(r)
+�2
+(553)
+In this action we assumed that the abelian gauge fields do not have monopole configurations.
+The partition function of Eq.(550) admits a representation as a sum over loops. We can then
+used the same line of reasoning that led to Eq.(542) and write the partition function as a sum over
+loops of the worldlines of the particles (charges) of the complex scalar field
+ZSC
+Zgauge
+0
+=
+�
+r
+∞
+�
+ℓµ(r)=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r)
+
+�
+exp
+iq
+�
+r,µ
+ℓµ(r)aµ(r)
+
+�
+a
+(554)
+where ⟨O[a]⟩a is the expectation value of the operator O[a] over the gauge fields aµ, and Zgauge
+0
+is
+the partition function of the free gauge fields. After computing this free-field expectation value
+we obtain the result
+ZSC
+Zgauge
+0
+=
+∞
+�
+{ℓµ(r)}=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r) − q2e2
+2
+�
+r,µ
+�
+r′,ν
+ℓµ(r) Gµν(r − r′) ℓν(r′)
+
+(555)
+where
+Gµν(r − r′) = ⟨aµ(r)aν(r′)⟩
+(556)
+is the Euclidean propagator of the free gauge fields. Since the configurations of the worldlines,
+the currents represented by ℓµ(r), are conserved and satisfy the local constraint ∆µℓµ = 0, the
+second term in the exponent of Eq.(555) is gauge invariant. Hence, we can use the propagator in
+the Feynman gauge
+Gµν(r − r′) = δµνG0(r − r′)
+(557)
+where G0(r− r′) is the 3D lattice propagator which has the same Coulomb form at long distances
+already given in Eq.(544).
+Therefore we can write the partition function of the scalar QED model (the superconductor)
+as a sum over worldline loop configurations of the particles of the scalar field in the equivalent
+form
+ZSC
+Zgauge
+0
+=
+∞
+�
+{ℓµ(r)}=−∞
+δ(∆µℓµ(r)) exp
+−
+�
+r,µ
+T
+2 ℓ2
+µ(r) − q2e2
+2
+�
+r,r′,µ
+ℓµ(r) G0(r − r′) ℓµ(r′)
+
+(558)
+which is the same as the partition function of the 3D XY model in the broken symmetry state,
+given in Eq.(543), with the identification of the particle loops of the abelian Higgs model with the
+vortices of the 3D XY model and a relation between the coupling constants. Hence we obtain the
+duality between the broken symmetry phase of the 3DXY model (“low T”) and the symmetric
+phase of the abelian Higgs model (“high T”)
+ℓµ ↔ mµ,
+q2e2 ↔ 2π2
+T ,
+T
+2 ↔ α
+(559)
+where the quantities on the left hand side of the identifications refer to the 3D abelian Higgs
+model and the quantities on the right hand side to the 3D XY model.
+108
+
+We can now repeat the same analysis but in the broken symmetry of the abelian Higgs model.
+In this phase the gauge field aµ becomes massive by the Higgs mechanism (or, what is the same,
+by the Meissner effect of the superconductor), and in this phase the vortex loops mµ of the
+complex scalar field have short range interactions. This phase then is mapped onto the unbroken
+phase of the 3DXY model with the screened vortex loops of the abelian Higgs model identified
+with the particle loops of the 3D XY model, and with the same identifications between the
+coupling constants of the two theories given in Eq.(559). The identification between the two
+partition functions implies that the phase transitions must be the same. In other words, the
+duality implies that the 3D abelian Higgs model has a continuous phase transition with the same
+fixed point as that of the complex scalar field in 3D. The only difference is that the phases are
+reversed and, for this reason, this is sometimes called an “inverted” 3D XY transition [340].
+11.2.3. The duality mapping
+We can summarize these results with the following identification of two Lagrangians (in real
+time, Minkowski signature)[208]
+|∂µ(B)φ|2 − m2|φ|2 − u|φ|4 ↔ |∂µ(a)ϕ|2 + m2|ϕ|2 − u|ϕ|4 − 1
+4e2 f 2
+µν + 1
+2πǫµνλaµ∂νBλ
+(560)
+The left hand side of this equivalence is the Lagrangian of the 3D complex scalar field φ and Bµ
+is a background electromagnetic gauge field. The right hand side is the Lagrangian of the abelian
+Higgs model with a scalar field ϕ and a U(1) gauge field aµ. Notice that the mass terms of the
+two sides are inverted reflecting the reverse order of their two phases. This identification also
+shows that the current of the scalar field jµ = − i
+2(φ∗∂µφ − φ∂µφ∗) is identified as
+jµ ↔ 1
+2πǫµνλ∂νaλ
+(561)
+The field operator of the complex scalar field creates worldlines (as opposed to loops) of charged
+particles. In the dual abelian Higgs theory this operator is identified with an operator that creates
+a magnetic monopole of the U(1) gauge field aµ with unit magnetic charge. It is easy to see that in
+the broken symmetry phase of the abelian Higgs model a monopole-antimonopole pair creation
+operator decays exponentially with distance due to the flux expulsion effect (the Meissner effect
+of a superconductor). This means that the monopoles are confined with a linear potential. This
+is the same behavior of the complex scalar field φ in its symmetric phase where the propagator
+of the complex scalar field decays exponentially with distance. Conversely, in the broken sym-
+metry phase of the complex scalar field theory, the field operator φ condenses and its propagator
+approaches the value |⟨φ⟩|2 at long distances. This is the same behavior one readily finds for the
+monopole operator of the abelian Higgs model in the symmetric phase .
+11.3. Bosonization in 2+1 dimensions
+The particle-vortex duality that we discussed in section 11.2 essentially has been understood
+since the 1980s. In section 7.4 we discussed in detail the problem of bosonization in 1+1 dimen-
+sions whic is a mapping between a massless Dirac fermion and a massless compactified scalar
+field. As we noted, this mapping is a powerful theoretical tool to investigate non-perturbatively
+the structure of many theories in 1+1 dimensions of great physical interest. This success has
+motivated a sustained effort to extend these concepts to higher dimensions where the problem in
+many ways is more subtle.
+Fermi systems in dimensions higher differ from the 1+1-dimensional case in two significant
+ways. Due to the kinematic restriction of one space dimension a massless relativistic fermion
+and a theory of fermions (relativistic or not) at finite density are mostly equivalent to each other.
+The reason is quite simple. Consider a free massless Dirac theory in 1+1 dimensions. As we
+saw, it is equivalent to a theory of two chiral fermions, the right and left moving component of
+the spinor. If we add a chemical potential µ, the right and left moving states will be filled up to
+µ, which is the Fermi energy. While in space dimensions d > 1 a finite chemical potential leads
+to a finite fermi surface, in d = 1 space dimensions the Fermi “surface” is just two points, for
+the right and left moving fermion modes respectively. The Dirac Hamiltonian at finite chemical
+potential is
+H = ψ†(−i)α∂xψ − µψ†ψ
+(562)
+109
+
+Under a chiral transformation with angle θ(x), the field operators change as ψR → eiθψR and
+ψL → e−iθψL and the Hamiltonian becomes
+H = ψ†(−i)α∂xψ − (µ − ∂xθ) ψ†ψ
+(563)
+Clearly if we choose ∂xθ = µ (or, what is the same, θ = µx) the explicit dependence on the
+chemical potential has been cancelled and the transformed Dirac Hamiltonian is the same as the
+Hamiltonian at zero chemical potential. However under this transformation the mass operators
+transform as ¯ψψ → cos(2µx) ¯ψψ and i ¯ψγ5ψ → sin(2µx) i ¯ψγ5. In the non-relativistic context this
+change amounts to a modulation of the charge density with wave vector Q = 2pF(µ).
+In contrast, in space dimensions d > 1 the massless Dirac system and a system of fermions
+at finite density (relativistic or not) are no longer equivalent to each other as the latter system has
+a Fermi surface (of co-dimension d − 1) wile the former system does not. This difference has
+profound effects on their dynamics: the system of fermions at finite density is, at least in the weak
+coupling regime, is expected to be a Fermi liquid (except for an instability to a superconducting
+state even for infinitesimal attractive interactions) while the massless Dirac theory is stable as all
+local interactions are irrelevant.
+11.3.1. Bosonization of the Dirac theory in 2+1 dimensions
+We will now turn to the problem of Bose-Fermi mappings in relativistic systems in 2+1
+dimensions. This problem is of great interest both in Condensed Matter Physics and in High
+Energy Physics.
+Bosonization in 2+1 dimensions is a more subtle problem than in 1+1 dimensions that we
+discussed in section 7.4. There we saw, in 1+1 dimensions bosonization consists of a set of
+operator identities relating two free field theories, a theory of massless Dirac fermions and a
+compactified massless free scalar field φ(x). The success of that program is largely based on the
+fact that both theories are massless (and hence are scale and conformally invariant), on the chiral
+anomaly, and on the relation between the gauge current of the Dirac theory with the topological
+current of the compactified boson.
+The physics in 2+1 dimensions (and in general) is very different. For instance, instead of
+the chiral anomaly in 2+1 dimensions theories of relativistic fermions have a parity anomaly,
+discussed in section 10.1.6. We will also see that although in 2+1 dimensions there are theories
+of free massless Dirac fermions and free massless compactified bosons they are no longer dual to
+each other. For these and other reasons it has been difficult to extend the bosonization program
+to 2+1 dimensions.
+One of the first bosonization constructions for a theory of massive (relativistic) Dirac fields
+was put forth by Alexander Polyakov in 1988 [168, 167]. For reasons that no longer relevant,
+Polyakov considered a theory of CP1 complex scalar fields (in its unbroken phase) coupled to
+a U(1)1 Chern-Simons gauge theory. Nevertheless, his main argument, with some corrections
+associated with the parity anomaly, still holds correct.
+Here we will follow the work of Hart Goldman and myself [343] and consider instead a
+theory of a single massive complex field coupled to a U(1)1 Chern-Simons gauge theory whose
+Lagrangian density is
+L = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1
+4πǫµνλA µ∂νA λ
+(564)
+In essence, this theory is a relativistic version of what we did in section 10.3.7 where we mapped
+non-relativistic fermions to (also non-relativistic) bosons whereas here the theory is relativistic
+and we are mapping bosons to fermions. In his work Polyakov actually considered the problem
+of the propagator of a free massive scalar field coupled to the Chern-Simons gauge field. In this
+limit, the propagator G(x, x′) is the transition amplitude from x to x′ (in Minkowski spacetime).
+In the case in which φ(x) is a free massive field this propagator can be expressed in the for of a
+Feynman path-integral as a sum over all open oriented paths {Px,x′} with endpoints at x and x′
+which in Euclidean spacetime is
+Gγ(x,x′) =
+�
+{Px,x′ }
+exp(−mL(Px,x′))
+�
+exp
+�
+i
+�
+Px,x′
+dzµA µ(z)
+� �
+U(1)1
+(565)
+where the expectation value is over the Chern-Simons gauge fields Aµ of Eq.(564).
+110
+
+Let us consider now the partition function of the bosons which is a sum over all closed paths
+{P}, which we will assume to be non-intersecting (which requires a short distance repulsion)
+Z =
+�
+{P}
+exp(−mL(P))
+�
+exp(i
+�
+P
+dzµA µ(z))
+�
+U(1)1
+(566)
+The expectation value is given by
+�
+exp
+�
+i
+�
+P
+dzµA µ(z)
+� �
+U(1)1 = exp
+�
+−1
+2
+�
+P
+�
+P
+dxµdyν⟨Aµ(x)Aν(y)
+�
+≡ exp(iπW(P))
+(567)
+where W(P) is the writhe of the closed path P. The computation of the writhe requires a regu-
+larization. One possible regularization is to thicken the path which is equivalent to including a
+Maxwell term for the gauge field in the Lagrangian. In general, the writhe of the path P is
+W(P) = S L(P) − T(P)
+(568)
+where S L(P) is an integer-valued topological invariant called the self-linking number and T(P)
+is the twist (or torsion) of the path. The twist T(P) is a Berry phase which depends on the
+coordinates of the space in which the path P is embedded, and it is not a topological invariant.
+We will now compute the Berry phase T(P). Let ˆe(s) be a unit vector tangent to the path
+P and where we parametrized the closed path P by a coordinate s ∈ [0, L]. The closed path
+P is the boundary of a surface Σ we can take to be a disk. We will write the Berry phase by
+extending ˆe(s) smoothly to the interior of Σ as ˆe(s, u), where 0 ≤ u ≤ 1 and ˆe(s, u = 1) = ˆe(s)
+and ˆe(s, u = 0) = ˆe0 ≡ constant. With these definition the Berry phase is
+T(P) ≡ W(ˆe) = 1
+2π
+� L
+0
+ds
+� 1
+0
+du ˆe · ∂sˆe × ∂uˆe
+(569)
+which is defined modulo an integer.
+In this formulation, in the bosonic theory the amplitude for a path of length L with tangent
+vector ˆe(s) with endpoints at x and x′ is
+G(x − x′) =
+� ∞
+0
+dL
+�
+D ˆe(s) δ
+�
+1 − |ˆe|2�
+δ
+�
+x − x′ −
+� L
+0
+ds ˆe(s)
+�
+exp(−|m|L ± iπW(ˆe)) (570)
+In momentum space we find
+G(p) =
+� ∞
+0
+dL
+�
+Dˆe δ(1 − |ˆe|2) exp(−|m|L ± iπ W(ˆe)) exp(ipµ
+� L
+0
+ds ˆeµ(s))
+(571)
+By inspection of Eq.(569) we see that this is the same expression that we looked in the theory of
+the path integral for spin in section 4.3.2. for spin S = 1/2 particle in a magnetic field bµ = ±2pµ.
+The equation of motion for ˆe is
+∂sˆeµ = ±2ǫµνλˆeλ = i[H, ˆeµ]
+(572)
+where H = ∓pµˆeµ is the Hamiltonian. Upon quantization, ˆeµ satisfies the commutation relations
+[ˆeµ, ˆeν] = 2iǫµνλˆeλ
+(573)
+This means that we should make the identification ˆeµ → σµ where σµ are the three 2 × 2 Pauli
+matrices. Up to rescaling of the mass m → M, upon performing the path integral of Eq.(571) we
+find that G(P) is the (Euclidean) Dirac propagator [168]
+G(p) =
+1
+ipµσµ − M
+(574)
+Using these results the partition function of Eq.(566) becomes
+Zfermion = det[i/∂ − M] =
+�
+DJ δ(∂µJµ) exp �−|m|L[J] − isgn(M)πΦ[J]�
+(575)
+where Φ[J] = W[J], defined in Eq.(568).
+111
+
+We will return to this construction shortly below when we include the effects of the parity
+anomaly.
+Early attempts at deriving a mapping for a theory of Dirac fermions in 2+1 dimensions were
+based on their behavior in the presence of gauge fields and the associated parity anomaly, dis-
+cussed in section 10.1.6. These early theories are essentially a hydrodynamic description of the
+Dirac theory deep in the massive phase. With minor differences, the same results were derived
+independently (and simultaneously) by two groups, Fidel Schaposnik and myself [344] and Cliff
+Burgess and Francisco Quevedo [345]. This approach was reexamined more recently by A. Chan,
+T. Hughes, S. Ryu and myself in 2016 in a derivation of a hydrodynamic effective field theory for
+topological insulators in different dimensions [346, 347]. These derivations are similar in spirit
+to what we discussed in section 10.3.6 for the fractional quantum Hall effect. We will follow the
+approach of Ref.[346] for the special case of a Chern insulator 2+1 dimenxions.
+The Dirac theory, in any dimension, is globally gauge invariant. By Noether’s theorem this
+means that it has a locally conserved current, jµ = ¯ψγµψ which thus satisfies ∂µ jµ = 0. Here
+too, this means that we can write jµ = ǫµνλ∂νbλ, where bµ is a gauge field. We expect that the
+effective action of the gauge field bµ should be local, gauge invariant and, in this case, relativistic
+invariant. In 2+1 dimensions the effective action should break time reversal invariance and hence
+it should have a Chern-Simons term.
+We will consider the free massive Dirac theory coupled to a background probe gauge field Aµ.
+The Lagrangian is L = ¯ψ(i/(D) − M)ψ, where Dµ = ∂µ + iAµ. As we saw in section 10.1.6, gauge
+invariance and locality require that we have an even number of Dirac fermions. Here we will
+assume that one of the Dirac fermions is massive and acts as a regulator. The partition function
+Z[Aµ] =
+�
+D ¯ψDψ exp(iS F[ ¯ψ, ψ, Aµ])
+(576)
+By definition the expectation value of a product of current operators jµ can be obtained by func-
+tional differentiation of the partition function with respect to Aµ.
+The partition function is gauge invariant, i.e.
+Z[Aµ = Z[Aµ + aµ]
+(577)
+where the vector field aµ is a pure gauge, aµ = ∂µφ and fµnu = ∂µa − ν − ∂µaµ = 0. Hence, aµ is
+said to be flat. Therefore, up to a normalization
+Z[Aµ] =
+�
+D[aµ]pure Z[Aµ + aµ]
+=
+�
+Daµ δ(fµν = 0) Z[Aµ + aµ]
+=
+�
+DaµDbµ Z[Aµ + aµ] exp
+�
+− i
+2
+�
+d3x ǫµνλ bµ fνλ
+�
+(578)
+where in the last equality we introduced a representation of the delta function and the vector field
+bµ plays the role of a Lagrange multiplier field. using the invariance of the integration measure
+under aµ → aµ − Aµ we find
+Z[Aµ] =
+�
+DaµDbµZ[aµ] exp
+�
+− i
+2
+�
+d3x ǫµνλ bµ (fνλ − Fνλ)
+�
+(579)
+where Fµν is the field strength of the external field Aµ. From these identities and by differentiation
+of Eq.(579) it follows that a general expectation value of products of currents is
+⟨jµ(x) jν(y) . . .⟩ =
+δ
+δAµ(x)
+δ
+δAν(y) . . .Z[Aµ] = ⟨ǫµαβ∂αbβ ǫνγδ∂γbδ . . .⟩
+(580)
+which means that, at the operator level, we can identify [344]
+jµ(x) ⇔ ǫµνλ∂νbλ
+(581)
+This result is actually general. The only difference is the nature of the field b isn different di-
+mensions: in 1+1 is a pseudo-scalar, in 2+1 is a vector (gauge) field, in 3+1 is an anti-symmetric
+(Kalb-Ramond) tensor field, etc.
+112
+
+Returning to the partition function of Eq.(579) we find, using the result of Eq.(324) applied
+the partition function Z[aµ], that Z[Aµ] is given by
+Z[Amu] =
+�
+DaµDbµ exp(i
+�
+d3x Leff[aµ, bµ, Aµ])
+(582)
+where the effective Lagrangian is
+Leff = −ǫµνλ bµ ∂νaλ + s
+4πǫµνλaµ∂νaλ −
+1
+4geff
+fµν f µν + Aµǫµνλ∂νbλ + . . .
+(583)
+where fµν = ∂µaν − ∂νaµ. Here s = 1 in the Chern insulator, s = 0 in the trivial insulator and
+s = 1/2 at the quantum critical point, and geff an effective coupling constant (with dimensions of
+length−1). Here we included the effects of the parity anomaly. We recognize that the first term of
+the effective Lagrangian of Eq.(583) is a BF term.
+In Ref.[346] a similar result is also derived for a 3+1-dimensional topological insulator. The
+main differences are that there is no parity anomaly but an axial anomaly and that the Lagrange
+multiplier field is a Kalb-Ramond field,
+Leff[aµ, bµν, Aµ] = ǫµνλρ bµν ∂λaρ +
+θ
+8π2 ǫµνλρ∂µaν∂λaρ − 1
+4g2 fµν f µν + Aµǫµνλρ∂νbλρ + . . . (584)
+where θ = π in the topological insulator and θ = 0 in the trivial insulator.
+Although these results are correct, they do not give a full bosonization mapping. In particular,
+these results do not identify a fixed point for the dual bosonic theory and, along with it, a full
+mapping of the observables. Progress on this problem has only been achieved in the past few
+years both in the high energy literature [348, 349, 350, 351, 352, 199, 353, 354], and in the
+condensed matter physics literature [355, 356, 357, 358, 359]. Much of that new insight was
+presented in a 2016 insightful paper by Nathaniel Seiberg, T. Senthil, Chong Wang and Edward
+Witten [208]. Many of the dualities are actually conjectures supported by strong consistency
+checks. A derivation of the basic bosonization duality based on loop models was constructed by
+Hart Goldman and myself [343].
+The basic conjectured bosonization duality is a mapping of a free Dirac fermion to a gauge
+complex scalar field with a Chern-Simons term [352, 208]. We can think of the Dirac theory as
+either being defined entirely in 2+1 dimensions or as a being defined at the boundary of a non-
+trivial 3+1 dimensional topological insulator. In the first scenario we need to take into account
+the contribution of the fermionic doublers (this is what happens in the case of a lattice theory)
+or as the Pauli-Villars heavy fermionic regulators. In both cases, the additional heavy degrees of
+freedom cancel the anomaly of the 2+1-dimensional Dirac fermion, see section 10.1.6. In the
+second scenario, there is only one Dirac fermion at the boundary and the bulk θ-term cancels the
+anomaly of the boundary theory, see section 10.2.3. With these provisos, the Lagrangian LA of
+the free massive (or massless) Dirac fermion in 2+1 dimensions coupled to the electromagnetic
+gauge field Aµ is
+LA = ¯ψ(i /D(Aµ) − M)ψ − 1
+8πǫµνλAµ∂νAλ
+(585)
+where M is the Dirac mass. We will call this Theory A. Here /Dµ(A) = γµ(∂µ − iAµ) and Aµ
+is a background (non-dynamical) gauge field. The last term in Eq.(585) is the Chern-Simons
+term with the 1/2-quantized coefficient, the usual short-hand for the η-invariant term of Eq.(342)
+needed to cancel the parity anomaly.
+From the effective field theories we discussed above we know that the fermionic current
+maps to the curl of a gauge field, see Eq(581). Then, the conjectured bosonic dual is given by the
+Lagrangian of Eq(564) with an extra term for the (dual) coupling to the electromagnetic gauge
+filed. We will call this Theory B whose Lagrangian LB is
+LB = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1
+4πǫµνλA µ∂νA λ + 1
+2πǫµνλAµ∂νA λ
+(586)
+To check the consistence we first observe that both theories are anomaly free. By functional
+differentiation of the two partition functions we check that we get the correct mapping for the
+fermionic current, jµ ↔ ǫµνλ∂νA λ. This mapping implies that the charge density of Theory A
+maps onto the gauge field flux of Theory B, which is electromagnetic duality. We see that this
+bosonization is a relativistic version of flux attachment.
+113
+
+Let us now identify the mapping of the phases of both theories. If the Dirac mass M < 0,
+Theory A describes an anomalous quantum Hall insulator. Integrating out the massive fermions
+we fund that the effective action for the electromagnetic gauge field Aµ is a U(1)1 Chern-Simons
+theory. This means that in this phase σxy = −1/(2π) (in units in which e = ℏ = c = 1). Looking
+now at Theory B we see that if m2 > 0, the scalar field is in the unbroken phase, and it is massive.
+In this phase we set φ = 0 and find that the low energy resulting theory if just a U(1)1 Chern-
+Simons gauge theory coupled to the curl of the electromagnetic field. Upon integrating out the
+Aµ gauge field we find that the effective action of Aµ is also a U(1)1 Chern-Simons term, which
+implies that σxy = −1/(2π).
+Conversely, for M > 0 the effective electromagnetic action of the fermionic theory is a
+Maxwell term (the Chern-Simons term canceled). This phase is a trivial insulator. Looking now
+at Theory B we see that for m2 < 0 this theory is in its Higgs phase where ⟨φ⟩ � 0 and the gauge
+field Aµ now has a mass term, ∝ A 2
+µ . In the low energy limit Aµ → 0, and the Hall conductivity
+vanishes, consistent to what is expected from Theory A.
+On the other hand, for M = 0 Theory A is at a quantum critical point with a very simple CFT
+structure but a non-vanishing Hall conductivity σxy = 1/(4π). It is natural to map this CFT to
+a Wilson-Fisher (WF) fixed point of Theory B. At the WF one sets the (renormalized!) mass
+m2
+R = 0 (not the bare mass). In the absence of the Chern-Simons gauge field this WF fixed point
+in well understood form high quality (five loop!) epsilon expansion calculations (enhanced with
+Borel resummation) [49], and by more recent numerical Conformal Bootstrap methods [360].
+However, not much is known of the gauged version of the CFT of Theory B since it cannot
+be accessed by either methods. If the mapping to Theory A is correct it should have a Hall
+conductivity of σxy = 1/(4π). This conjectured value hast not yet been confirmed.
+Consider now a monopole operator of the gauge field Aµ of Theory B. We will denote this
+operator MA . The Chern-Simons term is not gauge invariant in a monopole background. To put
+it differently it has charge 1 under the Chern-Simons gauge field Aµ. It also has charge 1 under
+the electromagnetic field Aµ. Likewise the scalar field φ has charge 1 under the Chern-Simons
+gauge field Aµ. The operator φ†MA is gauge-invariant under the Chern-Simons gauge field Aµ
+(i.e. it has charge 0). This composite operator has electromagnetic charge 1 (which it inherited
+from the monopole). Moreover, it has spin-1/2 required by the Wu-Yang construction [151].
+In other words, the operator φ†MA has the same quantum numbers as the Dirac fermion and
+are identified by this conjecture. Notice, however, that the free massless Dirac field has scaling
+dimension ∆ψ = 1 in 2+1 dimensions. The conjecture implies the composite operator φ†MA
+should also have scaling dimensions 1 at the (gauged) WF fixed point of Theory B. Many of these
+conjectures have been verified in non-abelian versions of Theory A and Theory B: a theory of
+free massless Dirac fermions coupled to a Chern-Simons gauge theory with gauge group SU(N)k
+and a theory of complex scalars coupled to a non-abelian Chern-Simons gauge theory with gauge
+group SU(k)N, both in the limits N → ∞ and k → ∞ with N/k fixed [348, 349, 350, 351, 352].
+The particle-vortex duality discussed in section 11.2 combined with the fermion-vortex du-
+ality we sketched here provide a web of dualities. These identifications constitute a powerful
+non-perturbative tool which has led to many significant developments. For instance, Goldman
+and myself [361] used the web of dualities to explain the experimentally observed self-duality
+at fractional quantum Hall plateau transitions, which was a long standing puzzle. It has also
+provided a powerful new tool to derive effective field theories of non-abelian fractional quantum
+Hall states [301] and even to propose novel states with a single (Fibonacci) anyon [303].
+11.3.2. Bosonization of the Fermi Surface
+A form of bosonization has been developed for the case of systems of fermions with a Fermi
+surface. Dense Fermi systems at sufficiently weak coupling are well described by the Landau
+theory of the Fermi liquid [13, 3, 194, 260, 259]. In this regime, the system of fermions has a
+collective mode, a bound state of particles and holes, which at long wavelengths is well described
+by the random phase approximation (RPA) [362]. However, in space dimensions d > 1 there is no
+kinematical restriction and quasi-particles and quasi-holes may move in their separate ways. The
+result is that, in addition to the collective modes, there is a low-energy spectrum of renormalized
+but essentially free quasi-particles. Because of the existence of this quasi-particle spectrum the
+collective modes generally (but not always) decay into particle-hole pairs resulting in a finite
+lifetime of the collective modes.
+Superficially the particle-hole collective modes are similar to the scalar field of the bosonized
+theory in d = 1. To an extent it has been possible to “bosonize the Fermi surface” and to treat it
+as a quantum mechanical object [363, 364, 365, 366, 367, 368]. In this approach one essentially
+114
+
+regards each direction normal to the Fermi surface as a one-dimensionalchiral fermion, including
+the current algebra structure, subject to global constraints that cancels the anomalies. This point
+of view ensures that the total fermion number in the Fermi sea is conserved. Here I will only
+present a short summary of the main ideas.
+As in the one-dimensional case one defines a filled Fermi sea state |FS⟩. This is a state of non-
+interacting fermions filling up all one-particle states up to the Fermi energy EF. For concreteness
+we consider a system in two space dimensions. In this case, neglecting lattice effects, the Fermi
+surface is a circumference of radius pF, the Fermi momentum. At fixed fermion number, the
+excitations are particle-hole pairs. The operator nk(q) = c†(k+ q
+2) c(k− q
+2) creates a particle-hole
+pair with relative momentum q with total momentum k. In the low energy regime k is a point on
+the Fermi surface (with |k| = pF) and q is small momentum compared with pF. In what follows
+we will label the point k on the Fermi surface by the angle θ of the arc spanned by k and an
+arbitrary origin on the Fermi surface.
+We will normal-order the particle-hole creation operator with respect to the filled Fermi sea
+and define δ(q, θ) =: nk(q) := nk(q) − ⟨FS|nk(q)|FS⟩. Haldane [363], Houghton and Marston
+[364, 367], and Castro Neto and Fradkin [365, 366] showed that in real space that these normal
+ordered density operators obey the equal-time commutation relations
+[δn(x, θ), δn(x′, θ′)] = − 1
+2π kF(θ) · ▽δ2(x − x′)δ(θ − θ′)
+(587)
+where kF(θ) is a unit vector normal to the Fermi surface at angle θ. This is the algebra of the
+quantum fluctuations of the Fermi surface. We can further define at each point θ on the Fermi
+surface a Bose field ϕ(x, θ) such that
+δn(x, θ) = N(0)�F(θ) · ▽ϕ(x, θ)
+(588)
+where N(0) is the density of one-particle states at the Fermi surface and �F(θ) is the Fermi veloc-
+ity at the location θ. The scalar field ϕ(x, θ) is a chiral boson at θ which parametrizes the quantum
+fluctuations of the Fermi surface. The quantum dynamics of the chiral bosons ϕ(x, θ) is governed
+by the action
+S =1
+2N(0)
+� 2π
+0
+dθ
+2π
+�
+d2x dt
+�
+−∂tϕ(x, θ) �F(θ) · ▽ϕ(x, θ) − (�F(θ) · ▽ϕ(x, θ))2�
++1
+2N(0)
+� 2π
+0
+dθ
+2π
+� 2π
+0
+dθ′
+2π
+�
+d2x d2x′ dt F(x − x′; θ − θ′) �F(θ) · ▽ϕ(x, θ) �F(θ′) · ▽ϕ(x, θ′)
+(589)
+where x = (x, t), and F(x−x′; θ−θ′) are the Fermi liquid parameters that parametrize the effective
+forward scattering interactions among the fermion quasiparticles on the Fermi surface [13]. The
+equation of motion predicted by this (quadratic) action is equivalent to the linearized quantum
+Boltzmann equation familiar from the Landau theory of the Fermi liquid. A non-linear extension
+has been introduced recently by Delacr´etaz and coworkers [368]. Recent work on the role of
+quantum anomalies in dense Fermi systems has yielded new insights on these problems [369].
+In the regime, where the Landau theory of the Fermi liquid is expected to work [13], bosoniza-
+tion of the Fermi surface approach has reproduced the previously known results. There remain
+many open problems in this approach (and others) in the vicinity of quantum phase transitions. In
+spite of intense research using many different approaches, quantum phase transitions in metallic
+systems are not yet fully understood beyond perturbation theory [255, 256]. Non-perturbative
+ideas such as deconfined quantum criticality have been proposed [370] and significant work has
+been done using large-N methods [371, 372]. A particularly important metallic quantum phase
+transition (and perhaps the simplest) occurs near the Pomeranchuk instability of the fermi liquid.
+A quantum phase transition to an electron nematic state [373] has been predicted and studied
+within the Hertz-Millis approach. It has also been studied using higher dimensional bosoniza-
+tion [373, 374, 375]. Quantum Monte Carlo simulations [376] have shown that the vicinity
+of a nematic quantum critical point can trigger a superconducting state. Nematic Fermi fluids
+have been found in many physical systems of interest ranging from high temperature supercon-
+ductors (such as cuprates and iron superconductors), to electron gases in large magnetic fields
+[377, 378, 379, 380].
+115
+
+12. Conclusions
+In this chapter I have attempted to cover the role of Quantum Field Theory in modern Con-
+densed Matter Physics. Its role and influence is vast and deep. In spite of the length of this
+chapter, I have not done justice to its role in many important areas of Condensed Matter. In
+particular, I have not touched on the the theory of quantum spin liquids, fractons, and particu-
+larly topological phases in three space dimensions, among others. An area in which the ideas
+and concepts of QFT have a huge impact , both conceptually and as a tool, is on the problem of
+quantum entanglement. The concept of quantum entanglement as applied to condensed matter
+systems, both in equilibrium and out of equilibrium, is a major area of current development. It
+has become a crucial tool for characterizing systems at quantum criticality as well as topological
+phases of matter. Nevertheless, I hope that the reader will find this chapter both insightful and a
+useful resource.
+Acknowledgments
+This work was supported in part by the US National Science Foundation through grant No.
+DMR 1725401 at the University of Illinois.
+References
+[1] J. R. Schrieffer, Theory of Superconductivity, Frontiers in Physics, Addison-Wesley, New York, New York, 1964.
+[2] Y. Nambu, G. Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconduc-
+tivity. I, Physical Review 122 (1961) 345.
+[3] A. A. Abrikosov, L. P. Gorkov, I. E. Dziyaloshinski, Methods of Quantum Field Theory in Statistical Physics,
+Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1963.
+[4] A. L. Fetter, J. D. Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, New York,
+1971.
+[5] S. Doniach, E. H. Sondheimer, Green’s Functions for Solid State Physicists, Frontiers in Physics, W. A. Benjamin
+Inc., New York, New York, 1974.
+[6] R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, New York, 1965.
+[7] R. P. Feynman, Statistical Mechanics, A Set of Lectures, Frontiers in Physics, W. A. Benjamin Inc., Reading,
+Massachusetts, 1972.
+[8] L. D. Faddeev, Introduction to Functional Methods, in: Methods of Field Theory, Proceedings of the Les Houches
+Summer School 1975, Session XXVIII, R. Balian and J. Zinn-Justin editors, North Holland, Amsterdam, 1976,
+pp. 1–40.
+[9] E. Fradkin, Field Theories of Condensed Matter Physics, Second Edition, Cambridge University Press, Cam-
+bridge, U.K., 2013, (expanded 2nd edition of the originally published book in 1991).
+[10] E. Fradkin, Quantum Field Theory: An Integrated Approach, Princeton University Press, Princeton, N. J., 2021.
+[11] A. Altland, B. D. Simons, Condensed Matter Field Theory, 2nd Edition, Cambridge University Press, Cambridge
+U.K., 2010.
+[12] D. Pines, P. Nozi`eres, The Theory of Quantum Liquids, Vol. I, W. A. Benjamin Inc., New York, New York, 1966.
+[13] G. Baym, C. J. Pethick, Landau Fermi Liquid Theory, John Wiley & Sons, New York, NY, 1991.
+[14] L. P. Kadanoff, G. Baym, Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Non-
+Equilibrium Problems, Frontiers in Physics, W. A. Benjamin Inc., New York, New York, 1962.
+[15] A. Kamenev, Field theory of Non-Equilibrium Systems, 1st Edition, Cambridge University Press, Cambridge,
+U.K., 2011.
+[16] P. C. Martin, Measurements and Correlation Functions, Gordon and Breach Science Publishers Ltd., New
+York/London/Paris, 1968.
+[17] D. C. Mattis, The Theory of Magnetism, Harper & Row, New York, New York, 1965.
+[18] L. D. Landau, On the Theory of Phase Transitions, Zh. Eksp. Teor. Fiz. 7 (1937) 19.
+[19] L. D. Landau, E. M. Lifshitz, Statistical Physics, Vol. 5 of Course of Theoretical Physics, Pergamon Press, Oxford,
+U.K., 1959.
+[20] L. Onsager, Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition, Physical Review
+65 (1944) 117.
+[21] T. D. Schultz, D. C. Mattis., E. H. Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions,
+Rev. Mod. Phys. 36 (1964) 856.
+[22] E. Lieb, T. Schultz, D. C. Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics (N. Y.) 16
+(1961) 407.
+[23] P. Pfeuty, The one-dimensional Ising model with a transverse field, Annals of Physics 57 (1970) 79.
+[24] C. N. Yang, The Spontaneous Magnetization of a Two-Dimensional Ising Model, Physical Review 85 (1952) 808.
+[25] L. P. Kadanoff, Scaling Laws for Ising Models near Tc, Physics 2 (1966) 263.
+[26] L. P. Kadanoff, Spin-spin correlations in the two-dimensional Ising model, Il Nuovo Cimento B 44 (1966) 276.
+[27] E. Efrati, Z. Wang, A. Kolan, L. P. Kadanoff, Real-space renormalization in Statistical Mechanics, Rev. Mod.
+Phys. 86 (2014) 647–667.
+[28] K. G. Wilson, Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scal-
+ing Picture, Physical Review B 4 (1971) 3174–3183.
+[29] K. G. Wilson, Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior,
+Physical Review B 4 (1971) 3184–3205.
+[30] K. G. Wilson, M. E. Fisher, Critical Exponents in 3.99 Dimensions, Phys. Rev. Lett. 28 (1972) 240.
+116
+
+[31] K. G. Wilson, J. B. Kogut, The Renormalization Group and the ǫ Expansion, Physics Reports C 12 (1974) 75.
+[32] K. G. Wilson, The Renormalization Group: Critical phenomena and the Kondo problem, Rev. Mod. Phys. 47
+(1975) 773.
+[33] K. G. Wilson, The Renormalization Group and Critical Phenomena, Rev. Mod. Phys. 55 (1983) 583.
+[34] A. Z. Patashinskii, V. L. Pokrovskii, Behavior of Ordered Systems Near the Transition Point, Soviet Physics JETP
+23 (1966) 292, Zh. Ekzp. i Teor. Fiz. 50, 439 (1966).
+[35] M. E. Fisher, The Theory of Equilibrium Critical Phenomena, Reports on Progress in Physics 30 (1967) 615.
+[36] M. E. Fisher, The renormalization group in the theory of critical behavior, Reviews of Modern Physics 46 (11974)
+597–615.
+[37] L. P. Kadanoff, W. G¨otze, D. Hamblen, R. Hecht, E. A. S. Lewis, V. V. Palciauskas, M. Rayl, J. Swift, D. Aspnes,
+J. Kane, Static Phenomena Near Critical Points: Theory and Experiment, Rev. Mod. Phys. 39 (1967) 395–431.
+[38] M. Gell-Mann, F. E. Low, Quantum Electrodynamics at Small Distances, Physical Review 95 (1954) 1300.
+[39] N. N. Bogoliubov, D. V. Shirkov, Introduction to the Theory of Quantized Fields, Interscience, New York, New
+York, 1959.
+[40] L. P. Kadanoff, Operator Algebra and the Determination of Critical Indices, Phys. Rev. Lett. 23 (1969) 1430.
+[41] K. G. Wilson, Non-Lagrangian Models of Current Algebra, Phys. Rev. 179 (1969) 1499.
+[42] A. M. Polyakov, Conformal Symmetry of Critical Fluctuations, JETP Letters 12 (1970) 381, [ZhETF Pis. red. 12,
+538 (1970)].
+[43] A. M. Polyakov, Non-Hamiltonian approach to quantum field theory, Soviet Physics JETP 39 (1974) 10, [Zh.
+Eksp. Teor. Fiz. 66, 23 (1974)].
+[44] L. P. Kadanoff, H. Ceva, Determination of an Operator Algebra for the Two-Dimensional Ising Model, Phys. Rev.
+B 3 (1971) 3918.
+[45] J. Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge
+University Press, Cambridge, U.K., 1996.
+[46] G. ’t Hooft, M. Veltman, Regularization and Renormalization of Gauge Fields, Nuclear Physics B 44 (1972) 189.
+[47] C. G. Bollini, J. J. Giambiagi, Dimensional Renormalization: The Number of Dimensions as a Regularizing
+Parameter, Il Nuovo Cimento B 12 (1972) 20.
+[48] D. J. Amit, Field Theory, the Renormalization Group and Critical Phenomena, McGraw Hill, New York, NY,
+1980.
+[49] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th Edition, International Series of Monographs
+in Physics, Oxford University Press, Oxford, U.K., 2002.
+[50] D. J. Gross, F. Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys. Rev. Lett. 30 (1973) 1343–
+1346.
+[51] P. W. Anderson, A poor man’s derivation of the scaling laws of the Kondo problem, J. Phys. C: Solid State Phys.
+3 (1970) 2436.
+[52] N. Andrei, Diagonalization of the Kondo Hamiltonian, Physical Review Letters 45 (1980) 379.
+[53] P. B. Wiegmann, Exact solution of the Kondo problem, JETP Letters 31 (1980) 392.
+[54] H. D. Politzer, Reliable Perturbative Results for Strong Interactions?, Phys. Rev. Lett. 30 (1973) 1346.
+[55] K. G. Wilson, Confinement of Quarks, Physical Review D 10 (1974) 2445.
+[56] J. Kogut, L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys. Rev. D 11 (1975) 395.
+[57] M. Creutz, L. Jacobs, C. Rebbi, Monte Carlo computations in lattice gauge theories, Physics Reports 95 (4) (1983)
+201–282.
+[58] A. M. Polyakov, Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive
+Yang-Mills fields, Physics Letters B 59 (1975) 79.
+[59] S. H. Shenker, J. Tobochnik, Monte Carlo renormalization group analysis of the classical Heisenberg model in
+two dimensions, Phys. Rev. B 22 (1980) 4462.
+[60] E. Br´ezin, J. Zinn-Justin, Renormalization of the Nonlinear σ Model in 2 + ǫ Dimensions: Application to the
+Heisenberg Ferromagnets, Phys. Rev. Lett. 36 (1976) 691.
+[61] D. H. Friedan, Nonlinear Models in 2 + ǫ Dimensions, Annals of Physics 163 (1985) 318.
+[62] P. W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492.
+[63] E. Abrahams, P. W. Anderson, D. C. Licciardello, T. V. Ramakrishnan, Scaling Theory of Localization: Absence
+of Quantum Diffusion in Two Dimensions, Phys. Rev. Lett. 42 (1979) 673.
+[64] F. J. Wegner, The mobility edge problem: Continuous symmetry and a conjecture, Zeitschrift f¨ur Physik B Con-
+densed Matter 35 (1979) 207.
+[65] A. J. McKane and M. Stone, Localization as adn Alternative to Goldstone’s Theorem, Annals of Physics 131
+(1981) 36–55.
+[66] Shinobu Hikami, Anderson localization in a non-lienar σ-model representation, Physical Review B 24 (1981)
+2671–2679.
+[67] K. G. Wilson, Quantum Field Theory Models in Less Than 4 Dimensions, Phys. Rev. D 7 (1973) 2911.
+[68] S. L. Sondhi, S. M. Girvin, J. P. Carini, D. Shahar, Continuous quantum phase transitions, Rev. Mod. Phys. 69
+(1997) 315.
+[69] S. Sachdev, Quantum Phase Transitions, 2nd Edition, Cambridge University Press, Cambridge, UK, 2011.
+[70] E. Fradkin, L. Susskind, Order and disorder in gauge systems and magnets, Phys. Rev. D 17 (1978) 2637.
+[71] J. B. Kogut, An introduction to lattice gauge theory and spin systems, Rev. Mod. Phys. 51 (1979) 659.
+[72] P. M. Chaikin, T. C. Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge,
+U.K., 1995.
+[73] E. Ardonne, P. Fendley, E. Fradkin, Topological order and conformal quantum critical points, Annals of Physics
+310 (2004) 493.
+[74] A. Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986.
+[75] E. Fradkin, M. Stone, Topological terms in one- and two-dimensional quantum Heisenberg antiferromagnets,
+Phys. Rev. B 38 (1988) 7215(R).
+[76] P. B. Wiegmann, Multivalued functionals and geometrical approach for quantization of relativistic particles and
+strings, Nuclear Physics B 323 (1989) 311.
+[77] M. V. Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proceedings of the Royal Society of
+London A 392 (1984) 45.
+117
+
+[78] B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett. 51 (1983) 2167.
+[79] D. Dombre, N. Read, Absence of the Hopf invariant in the long-wavelength action of two-dimensional antiferro-
+magnets, Phys. Rev. B 38 (1988) 7181.
+[80] F. D. M. Haldane, O(3) non-linear σ model and the topological distinction between integer and half-integer-spin
+antiferromagnets in two dimensions, Phys. Rev. Lett. 61 (1988) 1029.
+[81] S. Chakravarty, B. I. Halperin, D. R. Nelson, Low-temperature behavior of two-dimensional quantum antiferro-
+magnets, Phys. Rev. Lett. 60 (1988) 1057.
+[82] N. D. Mermin, The topological theory of defects in ordered media, Rev. Mod. Phys. 51 (1979) 591.
+[83] A. A. Abrikosov, On the magnetic properties of superconductors of the second group, Soviet Physics JETP 5
+(1957) 1174, Zh. Ekzp. i Teor. Fiz. 32, 1442 (1957).
+[84] J. M. Kosterlitz, D. J. Thouless, Order, metastability and phase transitions in two-dimensional systems, J. Phys.
+C: Solid State Phys. 6 (1973) 1181.
+[85] J. M. Kosterlitz, The critical properties of the two-dimensional XY model, J. Phys. C: Solid State Phys. 7 (1974)
+1046.
+[86] V. L. Berezinskii, Destruction of Long-Range Order in One-Dimensional and Two-Dimensional Systems having
+a Continuous Symmetry Group I. Classical Systems, Soviet Physics JETP 32 (1971) 493.
+[87] V. L. Berezinskii, Destruction of Long-Range Order in One-Dimensional and Two-Dimensional Systems having
+a Continuous Symmetry Group II. Quantum Systems, Soviet Physics JETP 34 (1972) 610.
+[88] J. V. Jos´e, L. P. Kadanoff, S. Kirkpatrick, D. R. Nelson, Renormalization, vortices, and symmetry-breaking per-
+turbations in the two-dimensional planar model, Phys. Rev. B 16 (1977) 1217.
+[89] B. I. Halperin, D. R. Nelson, Theory of Two-Dimensional Melting, Physical Review Letters 41 (1978) 121.
+[90] A. P. Young, Melting and the vector Coulomb gas in two dimensions, Physical Review B 19 (1979) 1855.
+[91] J. Toner, D. R. Nelson, Smectic, cholesteric, and Rayleigh-Benard order in two dimensions, Physical Review B
+23 (1981) 316.
+[92] D. R. Nelson, J. Toner, Bond-orientational order, dislocation loops, and melting of solids and smectic-A liquid
+crystals, Physical Review B 24 (1981) 363.
+[93] S. Coleman, Aspects of Symmetry, Cambridge University Press, Cambridge U.K., 1985.
+[94] R. Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, The Netherlands, 1985.
+[95] H. B. Nielsen, P. Olesen, Vortex-line models for dual strings, Nuclear Physics B 61 (1973) 45.
+[96] G. ’t Hooft, Symmetry Breaking through Bell-Jackiw Anomalies, Physical Review Letters 37 (1976) 8.
+[97] A. M. Polyakov, Quark confinement and topology of gauge theories, Nucl. Phys. B 120 (1977) 429.
+[98] A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, London, 1987.
+[99] C. G. Callan, R. F. Dashen, D. J. Gross, The structure of the gauge theory vacuum, Physics Letters B 63 (1976)
+334.
+[100] A. O. Caldeira, A. J. Leggett, Quantum Tunneling in a Dissipative System, Annals of Physics 149 (1983) 374–456.
+[101] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. Fisher, A. Garg, W. Zwerger, Dynamics of the dissipatuve two-state
+system, Reviews of Modern Physics 59 (1987) 1–85.
+[102] P. Di Francesco, P. Mathieu, D. S´en´echal, Conformal Field Theory, Springer-Verlag, Berlin, 1997.
+[103] P. Ginsparg, Applied Conformal Field Theory, in: E. Br´ezin, J. Zinn-Justin (Eds.), Champs, Cordes et Ph´enom´enes
+Critiques/ Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School 1988, Session
+XLIX, Elsevier Science Publishers B.V., Amsterdam, The Netherlands, 1989, pp. 1–168.
+[104] J. Polchinski, String Theory, Cambridge University Press, Cambridge U.K., 1998.
+[105] L. P. Kadanoff, Lattice Coulomb gas representations of two-dimensional problems, Journal of Physics A: Mathe-
+matical and Theoretical 11 (1978) 1399.
+[106] B. Nienhuis, Two-dimensional critical phenomena and the Coulomb gas, in: C. Domb, M. Green, J. L. Lebowitz
+(Eds.), Phase Transitions and Critical Phenomena, Vol. 11, Academic Press, London, U. K., 1987, p. 1.
+[107] N. D. Mermin, H. Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional
+Isotropic Heisenberg Models, Phys. Rev. Lett. 17 (1966) 1133–1136.
+[108] P. C. Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys. Rev. 158 (1967) 383–386.
+[109] S. Coleman, There are no Goldstone bosons in two dimensions, Communications in Mathematical Physics 31
+(1973) 259–264.
+[110] F. A. Schaposnik, Pseudoparticles and confinement in the two-dimensional Abelian Higgs model, Physical Review
+D 18 (1978) 1183.
+[111] G. ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys. Rev. D 14 (1976)
+3432–3450.
+[112] H. Georgi, S. Glashow, Unity of all elementary-particle forces, Physical Review Letters 32 (1974) 438.
+[113] A. Polyakov, Compact gauge fields and the infrared catastrophe, Physics Letters B 59 (1975) 82 – 84.
+[114] P. Dirac, Quantised singularities in the electromagnetic field, Proc. Roy. Soc. London 133 (1931) 60.
+[115] F. D. M. Haldane, ‘θ’ physics’ and quantum spin chains, J. Appl. Phys. 57 (1985) 3359.
+[116] H. Bethe, Theory of metals. I. Eigenvalues and eigenfunctions of the linear atomic chain, Z. Physik 71 (1931)
+205.
+[117] C. N. Yang, C. P. Yang, Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function
+Interaction, J. Math. Phys. 10 (1969) 1115.
+[118] I. Affleck, F. D. M. Haldane, Critical theory of quantum spin chains, Phys. Rev. B 36 (1987) 5291.
+[119] E. Witten, Non-Abelian Bosonization in Two Dimensions, Comm. Math. Phys. 92 (1984) 455.
+[120] V. G. Knizhnik, A. B. Zamolodchikov, Current Algebra and Wess-Zumino Model in Two Dimensions, Nuclear
+Physics B 247 (1984) 83.
+[121] I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki, Valence Bond Ground States in Isotropic Quantum Antiferromag-
+nets, Comm. Math. Phys. 115 (1988) 477.
+[122] M. Hagiwara, K. Katsumata, I. Affleck, B. I. Halperin, J. P. Renard, Observation of S = 1/2 degrees of Freedom
+in an S = 1 Linear-Chain Heisenberg Antiferromagnet, Physical Review Letters 65 (1990) 3181.
+[123] H. A. Kramers, G. H. Wannier, Statistics of the Two-Dimensional Ferromagnet. Part I, Phys. Rev. 60 (1941) 252.
+[124] F. J. Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, J. Math.
+Phys. 12 (1971) 2259.
+[125] S. Elitzur, R. B. Pearson, J. Shigemitsu, Phase structure of discrete Abelain spin and gauge systems, Physical
+118
+
+Review D 19 (1979) 3698.
+[126] A. Ukawa, P. Windey, A. H. Guth, Dual variables for lattice gauge theories and the phase structure of Z(N)
+systems, Physical Review D 21 (1980) 1013.
+[127] F. Haldane, Luttinger liquid thepry of one-dimensional quantum fluids. I. Properties of the Luttinger model and
+their extension to the general 1D interacting spinless Fermi gas, J. Phys. C: Solid State Phys. 14 (1981) 2585.
+[128] D. C. Mattis, E. H. Lieb, Exact Solution of a Many-Fermion System and its Associated Boson Field, Journal of
+Mathematical Physics 6 (1965) 304.
+[129] A. Luther, V. J. Emery, Backward Scattering in the One-Dimensional Electron Gas, Physical Review Letters 33
+(1974) 589.
+[130] S. Mandelstam, Soliton operators for the quantized sine-Gordon equation, Physical Review D 11 (1975) 3026.
+[131] S. Coleman, Quantum sine-Gordon equation as the massive Thirring model, Phys. Rev. D 11 (1975) 2088.
+[132] D. J. Gross, A. Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys. Rev. D 10 (1974)
+3235.
+[133] S. E. Adler, The Axial-Vector Vertex in Spinor Electrodynamics, Physical Review 177 (1969) 2426.
+[134] J. S. Bell, R. Jackiw, The PCAC Puzzle: π0 → γγ in the σ-model, Nuovo Cimento A 60 (1969) 47.
+[135] X. L. Qi, T. L. Hughes, S. C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B
+78 (2008) 195424.
+[136] J. Schwinger, Field Theory Commutators, Phys. Rev. Lett. 3 (1959) 296.
+[137] E. Witten, Some Properties of the ( ¯ψψ)2 Model in Two Dimensions, Nucl. Phys. B 142 (1978) 285.
+[138] W. P. Su, J. R. Schrieffer, A. J. Heeger, Solitons in Polyacetylene, Phys. Rev. Lett. 42 (1979) 1698.
+[139] R. Jackiw, J. R. Schrieffer, Solitons with fermion number 1/2 in condensed matter and relativistic field theories,
+Nuclear Physics B 190 (1981) 253.
+[140] R. Jackiw, C. Rebbi, Solitons with fermion number 1/2, Physical Review D 13 (1976) 3398.
+[141] J. Goldstone, F. Wilczek, Fractional Quantum Numbers on Solitons, Phys. Rev. Lett. 47 (1981) 986.
+[142] H. Takayama, Y. R. Lin-Liu, K. Maki, Continuum model for solitons in polyacetylene, Phys. Rev. B 21 (1980)
+2388–2393.
+[143] D. K. Campbell, A. R. Bishop, Solitons in polyacetylene and relativistic-field-theory models, Phys. Rev. B 24
+(1981) 4859–4862.
+[144] D. Campbell, A. Bishop, Soliton excitations in polyacetylene and relativistic field theory models, Nuclear Physics
+B 200 (1982) 297–328.
+[145] J. E. Hirsch, E. Fradkin, Effect of quantum fluctuations on the peierls instability: A monte carlo study, Phys. Rev.
+Lett. 49 (1982) 402–405.
+[146] E. Fradkin, J. E. Hirsch, Phase diagram of one-dimensional electron-phonon systems. I. The Su-Schrieffer-Heeger
+model, Phys. Rev. B 27 (1983) 1680.
+[147] M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral Asymmetry and Riemaniann Geometry. I, Math. Proc. Cambridge
+Philos. Soc. 77 (1975) 43.
+[148] P. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, U.K., 1930.
+[149] L. D. Landau, E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 1st Edition, Vol. Volume 3 of Course
+of Theoretical Physics, Pergamon Press, Oxford, U.K., 1957.
+[150] S. Weinberg, The Quantum Theory Of Fields, 1st Edition, Vol. I-III, Cambridge University Press, Cambridge,
+U.K., 2005.
+[151] T. T. Wu, C. N. Yang, Dirac monopole without strings: Monopole harmonics, Nuclear Physics B 107 (1976) 365.
+[152] E. Fradkin, L. P. Kadanoff, Disorder variables and para-fermions in two-dimensional statistical mechanics, Nu-
+clear Physics B 170 (1980) 1.
+[153] J. M. Leinaas, J. Myrheim, On the theory of identical particles, Il Nuovo Cimento B (1971-1996) 37 (1977) 1–23.
+[154] M. G. G. Laidlaw, C. M. DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys-
+ical Review D 3 (1971) 1375–1378.
+[155] Y. Aharonov, D. Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys. Rev. 115 (1959)
+485–491.
+[156] F. Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys. Rev. Lett. 48 (1982) 1144.
+[157] F. Wilczek, Quantum Mechanics of Fractional-Spin Particles, Phys. Rev. Lett. 49 (1982) 957–959.
+[158] A. S. Goldhaber, Connection of Spin and Statistics for Charge-Monopole Composites, Phys. Rev. Lett. 36 (1976)
+1122–1125.
+[159] D. Finkelstein, J. Rubinstein, Connection between Spin, Statistics, and Kinks, Journal of Mathematical Physics 9
+(1968) 1762–1779.
+[160] T. Skyrme, A Unified Theory of Mesons and Baryons, Nuclear Physics 31 (1962) 556–569.
+[161] Y.-S. Wu, General Theory for Quantum Statistics in Two Dimensions, Phys. Rev. Lett. 52 (1984) 2103–2106.
+[162] F. Wilczek, A. Zee, Linking numbers, spin, and statistics of solitons, Phys. Rev. Lett. 51 (1983) 2250.
+[163] S. Deser, R. Jackiw, S. Templeton, Three-Dimensional Massive Gauge Theories, Physical Review Letters 48
+(1982) 975.
+[164] S. Deser, R. Jackiw, S. templeton, Topologically Massive Gauge Theories, Annals of Physics 140 (1982) 372–411.
+[165] E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351.
+[166] G. T. Horowitz, Exactly Soluble Diffeomorphism Invaraint Theories, Communications in Mathematical Physics
+125 (1989) 417.
+[167] J. Grundberg, T. H. Hansson, A. Karlhede, On Polyakov’s Spin Factors, Nuclear Physics B 347 (1990) 420–440.
+[168] A. M. Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields, Mod. Phys. Lett. A 3 (1988) 325.
+[169] A.
+Y.
+Kitaev,
+Fault-tolerant
+quantum
+computation
+by
+anyons,
+Annals
+of
+Physics
+303
+(2003)
+2,
+(arXiv:quant-ph/9707021).
+[170] S. Das Sarma, M. Freedman, C. Nayak, S. H. Simon, A. Stern, Non-abelian anyons and topological quantum
+computation, Rev. Mod. Phys. 80 (2008) 1083.
+[171] A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, The electronic properties of graphene,
+Rev. Mod. Phys. 81 (2009) 109–162.
+[172] X. L. Qi, S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83 (2011) 1057.
+[173] L. Susskind, Lattice Fermions, Physical Review D 16 (1977) 3031.
+[174] H. B. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice (I). Proof by homotopy theory, Nuclear Physics B
+119
+
+185 (1981) 20–40.
+[175] H. B. Nielsen, M. Ninomiya, Absence of neutrinos on a lattice (II). Intiuitive topological proof, Nuclear Physics
+B 193 (1981) 173–194.
+[176] H. B. Nielsen, M. Ninomiya, The Adler-Bell-Jackiw Anomaly and Weyl fermions in a Crystal, Physics Letters B
+130 (1983) 389.
+[177] D. Friedan, A Proof of the Nielsen-Ninomiya Theorem, Communications in Mathematical Physics 85 (1982)
+481–490.
+[178] G. W. Semenoff, Condensed-Matter Simulation of a Three-Dimensional Anomaly, Physical Review Letters 53
+(1984) 2449.
+[179] D. J. Thouless, M. Kohmoto, M. P. Nightingale, M. den Nijs, Quantized Hall Conductance in a Two-Dimensional
+Periodic Potential, Physical Review Letters 49 (1982) 405.
+[180] I. Affleck, J. B. Marston, Large-N limit of the Heisenberg-Hubbard Model: Implications for High-Tc Supercon-
+ductors, Phys. Rev. B 37 (1988) 3774.
+[181] X. G. Wen, F. Wilczek, A. Zee, Chiral spin states and superconductivity, Phys. Rev. B 39 (1989) 11413–11423.
+[182] F. Bloch, ¨Uber der quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ur physik 52 (1929) 555–600.
+[183] C. Kittel, Introduction to Solid State Physics, 1st Edition, John Wiley & Sons, New York, New York, 1996.
+[184] N. Ashcroft, D. N. Mermin, Solid State Physics, Holt, Reinhart and Winston, New York, New York, 1976.
+[185] B. Bradlyn, L. Elcoro, J. Cano, M. G. Vergniory, Z. Wang, C. Felser, M. I. Aroyo, B. A. Bernevig, Topological
+quantum chemistry, Nature 547 (2017) 298–305.
+[186] J. Cano, B. Bradlyn, Band Representations and Topological Quantum Chemistry, Annual Review of Condensed
+Matter Physics 12 (2021) 225–246.
+[187] M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Annals of Physics 160 (1985)
+343.
+[188] A. P. Schnyder, S. Ryu, A. Furusaki, A. W. W.Ludwig, Classification of topological insulators and superconductors
+in three spatial dimensions, Phys. Rev. B 78 (2008) 195125.
+[189] A. Kitaev, Periodic table for topological insulators and superconductors, in: M. Feigelman (Ed.), Advances in
+Theoretical Physics: Landau Memorial Conference, Vol. 1134, American Institute of Physics, AIP Conference
+Proceedings, College Park, Maryland, 2009, p. 22.
+[190] J. E. Moore, L. Balents, Topological invariants of time-reversal-invariant band structures, Phys. Rev. B 75 (2007)
+121306.
+[191] E. Fradkin, M. Kohmoto, Quantized Hall effect and geometric localization of electrons on lattices, Phys. Rev. B
+35 (1987) 6017–6023.
+[192] D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,
+Physical Review B 14 (1876) 2239.
+[193] P. C. Martin, Measurements and Correlation Functions, Gordon & Breach, Philadelphia, Pennsylvania, 1967.
+[194] P. Nozi`eres, D. Pines, The Theory of Quantum Liquids: Normal Fermi Liquids, W. A. Benjamin Inc., New York,
+New York, 1966.
+[195] S. Ryu, C. Mudry, C.-Y. Hou, C. Chamon, Masses in graphene-like two-dimensional electronic systems: Topo-
+logical defects in order parameters and their fractional exchange statistics, Physical Review B 80 (2009) 205319.
+[196] F. D. M. Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the
+“Parity Anomaly”, Phys. Rev. Lett. 61 (1988) 2015–2018.
+[197] X. Chen, Z.-C. Gu, Z.-X. Liu, X.-G. Wen, Symmetry-Protected Topological Orders in Interacting Bosonic Sys-
+tems, Science 338 (2012) 1604–1606.
+[198] T. Senthil, Symmetry-protected topological phases of quantum matter, Annual Review of Condensed Matter
+Physics 6 (2015) 299–324.
+[199] E. Witten, Fermion path integrals and topological phases, Reviews of Modern Physics 88 (2016) 035001.
+[200] A. N. Redlich, Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions, Physical Re-
+view Letters 52 (1984) 18.
+[201] A. N. Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three dimensions,
+Phys. Rev. D 29 (1984) 2366.
+[202] K. Fujikawa, Path-Integral Measure for gauge-Invariant Fermion Theories, Physical Review Letters 42 (1979)
+1195.
+[203] R. E. Gamboa Sarav´ı, M. A. Muschietti, F. A. Schaposnik, J. E. Solomin, Chiral Symmetry and Functional
+Integral, Annals of Physics 157 (1984) 360.
+[204] G. E. Volovik, Zeros in the fermion spectrum in superfluid systems as diabolical points, Journal of Experimental
+and Theoretical Physics Letters 46 (1987) 98.
+[205] V. M. Yakovenko, Chern-Simons Terms and n Field in Haldane’s Model for the Quantum Hall Effect without
+Landau Levels, Physical Review Letters 65 (1990) 251.
+[206] M. F. Golterman, K. Jansen, D. B. Kaplan, Chern-Simons currents and chiral fermions on the lattice, Physics
+Letters B 301 (1993) 219.
+[207] X.-L. Qi, Y.-S. Wu, S.-C. Zhang, Topological quantization of the spin Hall effect in two-dimensional paramagnetic
+semiconductors, Physical Review B 74 (2006) 085308.
+[208] N. Seiberg, T. Senthil, C. Wang, E. Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics,
+Annals of Physics 374 (2016) 395 – 433.
+[209] L. Fu, C. L. Kane, Topological insulators with inversion symmetry, Phys. Rev. B 76 (2007) 045302.
+[210] C. L. Kane, E. J. Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys. Rev. Lett. 95 (2005)
+146802.
+[211] B. A. Bernevig, T. L. Hughes, S. C. Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe
+Quantum Wells, Science 314 (2006) 1757.
+[212] J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A.
+Bernevig, C. Felser, Axionic charge-density wave in the Weyl semimetal (TaSe4)2I, Nature 575 (2019) 315–319.
+[213] A. M. Essin, J. E. Moore, D. Vanderbilt, Magnetic Polarizability and Axion Electrodynamics in Crystalline Insu-
+lators, Physical Review Letters 102 (2009) 146805.
+[214] P. Hosur, S. Ryu, A. Vishwanath, Chiral topological insulators, superconductors, and other competing orders in
+three diemsnions, Physical Review B 81 (2010) 045120.
+120
+
+[215] F. Wilczek, Two applications of axion electrodynamics, Phys. Rev. Lett. 58 (1987) 1799–1802.
+[216] Michael E. Peskin and Daniel V. Schroeder, An Introduction to Quantum Field Theory, The Advanced Book
+Program, Perseus Books Publishing L.L.C., Reading, Massachusetts, 1995.
+[217] E. Witten, Dyons of charge eθ/2π, Physics Letters B 86 (1979) 283.
+[218] C. G. Callan, J. A. Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl. Phys. B 250
+(1985) 427.
+[219] D. Hsieh, Y. Xia, D. Qian, L. Wray, J. H. Dil, F. Meier, J. Osterwalder, L. Patthey, J. G. Checkelsky, N. P. Ong,
+A. V. Fedorov, H. Lin, A. Bansil, D. Grauer, Y. S. Hor, R. J. Cava, M. Z. Hasan, A tunable topological insulator
+in the spin helical Dirac transport regime, Nature 460 (2009) 1101–1105.
+[220] K. von Klitzing, G. Dorda, M. Pepper, New Method for High-Accuracy Determination of the Fine-Structure
+Constant Based on Quantized Hall Resistance, Physical Review Letters 45 (1980) 494.
+[221] D. C. Tsui, H. L. Stormer, A. C. Gossard, Two-Dimensional Magnetotransport in the Extreme Quantum Limit,
+Physical Review Letters 48 (1982) 1559.
+[222] X. Du, I. Skachko, F. Duerr, A. Luican, E. Y. Andrei, Fractional quantum Hall effect and insulating phase of Dirac
+electrons in graphene, Nature 462 (2009) 192–195.
+[223] A. A. Zibrov, C. Kometter, H. Zhou, E. M. Spanton, T. Taniguchi, K. Watanabe, M. P. Zaletel, A. F. Young,
+Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level, Nature 549 (2017)
+360–364.
+[224] L. Landau, Diamagnetismus der Metalle, Zeitschrift f¨ur Physik 64 (1930) 629–637.
+[225] Q. Niu, D. J. Thouless, Y.-S. Wu, Quantized Hall conductance as a topological invariant, Physical Review B 31
+(1985) 3372.
+[226] R. B. Laughlin, Quantized Hall conductivity in two dimensions, Physical Review B 23 (1981) 5632–5633.
+[227] H. Levine, S. B. Libby, A. Pruisken, Electron Delocalization by a Magnetic Field in Two Dimensions, Physical
+Review Letters 51 (1983) 1915.
+[228] A. Pruisken, On localization in the theory of the quantized hall effect: A two-dimensional realization of the
+θ-vacuum, Nuclear Physics B 235 (1984) 277.
+[229] S. Kievelson, M. Ro˘cek, Consequences of Gauge Invariance for Fractionally Charged Quasi-Particles, Physics
+Letters B 85 (156).
+[230] J. K. Jain, Incompressible quantum Hall states, Physical Review B 40 (1989) 8079.
+[231] J. K. Jain, Composite-fermion approach for the fractional quantum Hall effect, Physical Review Letters 63 (1989)
+199.
+[232] B. I. Halperin, P. A. Lee, N. Read, Theory of the half-filled Landau level, Phys. Rev. B 47 (1993) 7312.
+[233] J. Nakamura, S. Liang, G. C. Gardner, M. J. Manfra, Direct observation of anyonic braiding statistics, Nature
+Physics 16 (2020) 931–936.
+[234] B. I. Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Physical Review
+Letters 52 (1984) 1583.
+[235] D. Arovas, J. R. Schrieffer, F. Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys. Rev. Lett. 53
+(1984) 722.
+[236] J. Fr¨ohlich, A. Zee, Large-scale physics of the quantum Hall Fluid, Nuclear Physics B 364 (1991) 517.
+[237] V. Pasquier, F. Haldane, A dipole interpretation of the ν = 1/2 state, Nuclear Physics B 516 (1998) 719.
+[238] N. Read, Lowest-Landau-level theory of the quantum Hall effect: The Fermi-liquid-like state of bosons at filling
+factor one, Phys. Rev. B 58 (1998) 16262.
+[239] L. Susskind, The Quantum Hall Fluid and Non-Commutative Chern-Simons Theory, unpublished (2001).
+arXiv:arXiv:hep-th/0101029.
+[240] A. P. Polychronakos, Quantum Hall States as matrix Chern-Simons theory, Journal of High Eenergy Physics 2001
+(2001) 011.
+[241] E. Fradkin, V. Jejjala, R. G. Leigh, Non-commutative Chern-Simons for the Quantum Hall system and Duality,
+Nuclear Physics B 642 (2002) 483–500.
+[242] Z. Dong, T. Senthil, Noncommutative field theory and composite Fermi liquids in some quantum Hall systems,
+Phys. Rev. B 102 (2020) 205126.
+[243] H. Goldman, T. Senthil, Lowest Landau level theory of the bosonic Jain states, Phys. Rev. B 105 (2022) 075130.
+[244] S. C. Zhang, T. H. Hansson, S. Kivelson, Effective-Field-Theory Model for the Fractional Quantum Hall Effect,
+Phys. Rev. Lett. 62 (1989) 82.
+[245] F. Haldane, E. H. Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect,
+Physical Review B 31 (1985) 2529.
+[246] X. G. Wen, Q. Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random
+potential and on high-genus Riemann surfaces, Phys. Rev. B 41 (1990) 9377.
+[247] A. L´opez, E. Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories, Phys. Rev. B 44 (1991)
+5246.
+[248] A. L´opez, E. Fradkin, Response Functions and Spectrum of Collective Excitations of Fractional Quantum Hall
+Effect Systems, Physical Review B 47 (1993) 7080.
+[249] W. Kohn, Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas, Physical
+Review 123 (1961) 1242.
+[250] A. L´opez, E. Fradkin, Universal structure of the edge states of the fractional quantum Hall states, Physical Review
+B 59 (1999) 15323.
+[251] X.-G. Wen, A. Zee, Classification of Abelian quantum Hall states and matrix formulation of topological fluids,
+Physical Review B 46 (1992) 2290.
+[252] X. G. Wen, Topological Orders and Edge Excitations in Fractional Quantum Hall States, Advances in Physics 44
+(1995) 405.
+[253] R. de Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin, D. Mahalu, Direct observation of a fractional
+charge, Nature 389 (1997) 162.
+[254] L. Saminadayar, D. C. Glattli, Y. Jin, B. Etienne, Observation of the e/3 Fractionally Charged Laughlin Quasipar-
+ticle, Physical Review Letters 79 (1997) 2526.
+[255] J. A. Hertz, Quantum critical phenomena, Phys. Rev. B 14 (1976) 1165–1184.
+[256] A. J. Millis, Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Physical
+121
+
+Review B 48 (1993) 7183–7196.
+[257] C. M. Varma, P. B. Littlewood, S. Schmitt-Rink, E. Abrahams, A. E. Ruckenstein, Phenomenology of the normal
+state of Cu-O high-temperature superconductors, Phys. Rev. Lett. 63 (1989) 1996–1999.
+[258] J. Polchinski, Low-energy dynamics of the spinon-gauge system, Nuclear Physics B 422 (1994) 617–633.
+[259] R. Shankar, Renormalization-group approach to interacting fermions, Reviews of Modern Physics 66 (1994) 129–
+192.
+[260] J. Polchinski, Effective Field Theory and the Fermi Surface, in: J. A. Harvey, J. Polchinski (Eds.), Theoretical
+Advanced Study Institute (TASI 92): From Black Holes and Strings to Perticles, World-Scientific Publishing Co.,
+Singapore, 1993, pp. 235–276, proceedings of TASI 1992, Boulder (Colorado, USA), June 1-26 (1992).
+[261] C. Nayak, F. Wilczek, Renormalization group approach to low temperature properties of a non-Fermi liquid metal,
+Nuclear Physics B 430 (1994) 534–562.
+[262] C. Nayak, F. Wilczek, Non-Fermi liquid fixed point in 2 + 1 dimensions, Nuclear Physics B 417 (1994) 359–373.
+[263] J. M. Maldacena, Large N limit of superconformal field theories and supergravity, Advances in Theoretical and
+Mathematical Physics 2 (1998) 231.
+[264] T. Faulkner, N. Iqbal, H. Liu, J. McGreevy, D. Vegh, Strange Metal Transport Realized by Gauge/Gravity Duality,
+Science 329 (2010) 1043–1047.
+[265] S. Fubini, Vertex Operators and the Quantum Hall Effect, Modern Physics Letters A 6 (1991) 347–358.
+[266] G. Moore, N. Read, Non-Abelions in the Fractional Quantum Hall Effect, Nuclear Physics B 360 (1991) 362.
+[267] G. Moore, N. Seiberg, Taming the Conformal Zoo, Phys. Lett. B 220 (1989) 422.
+[268] G. Moore, N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177.
+[269] M. H. Freedman, A. Kitaev, Z. Wang, Simulation of topological field theories by quantum computers, Commun.
+Math. Phys. 227 (2002) 587.
+[270] M. H. Freedman, A. Kitaev, M. J. Larsen, Z. Wang, Topological Quantum Computation, Comm. Math. Phys. 227
+(2002) 605.
+[271] J. Preskill, Topological Quantum Computation, Lecture Notes for Physics 219: Quantum Computation, chapter
+9; unpublished; Caltech (2004).
+URL http://www.theory.caltech.edu/~preskill/ph219/topological.pdf
+[272] R. Willett, J. P. Eisenstein, H. L. St¨ormer, D. C. Tsui, A. C. Gossard, J. H. English, Observation of an even-
+denominator quantum number in the fractional quantum Hall effect, Physical Review Letters 59 (1987) 1776.
+[273] J. P. Eisenstein, R. Willett, H. L. St¨ormer, D. C. Tsui, A. C. Gossard, J. H. English, Collapse of the Even-
+Denominator Fractional Quantum Hall Effcet in Tilted Fields, Physical Review Letters 61 (1988) 997.
+[274] W. Pan, J. S. Xia, H. L. St¨ormer, D. C. Tsui, C. Vicente, E. D. Adams, N. S. Sullivan, L. N. Pfeiffer, K. W.
+Baldwin, K. W. West, Experimental studies of the fractional quantum Hall effect in the first excited Landau level,
+Phys. Rev. B 77 (2008) 075307.
+[275] M. Greiter, X.-G. Wen, F. Wilczek, Paired hall State at Half Filling, Physical Review Letters 66 (1991) 3205.
+[276] M. Greiter, X. G. Wen, F. Wilczek, Paired Hall States, Nuclear Physics B 374 (1992) 567–614.
+[277] X. G. Wen, Non-Abelian statistics in the Fractional quantum Hall States, Physical Review Letters 66 (1991) 802.
+[278] W. Kohn, J. M. Luttinger, New Mechanism for Superconductivity, Physical Review Letters 15 (1965) 524.
+[279] A. V. Chubukov, Kohn-Luttinger effect and the instability of a two-dimensional repulsive Fermi liquid at T = 0,
+Physical Review B 48 (1993) 1097.
+[280] S. Raghu, S. A. Kivelson, Superconductivity from repulsive interactions in the two-dimensional electron gas,
+Physical Review B 83 (2011) 094518.
+[281] K. Park, V. Melik-Alaverdian, N. E. Bonesteel, J. K. Jain, Possibility of p-wave pairing of composite fermions at
+ν = 1/2, Physical Review B 58 (1998) R10167.
+[282] B. I. Halperin, Theory of the quantized Hall conductance, hevetica Physica Acta 56 (1983) 75–102.
+[283] R. H. Morf, Transition from quantum Hall to compressible states in the second Landau level: New light on the
+ν = 5/2 enigma, Physical Review Letters 80 (1998) 1505.
+[284] C. Nayak, F. Wilczek, 2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum
+Hall states, Nuclear Physics B 479 (1996) 529–553.
+[285] N. Read, D. Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symme-
+tries and the fractional quantum Hall effect, Physical Review B 61 (2000) 10267.
+[286] G. E. Volovik, An analog of the quantum Hall effect in a superfluid 3He film, Soviet Physics JETP 67 (1988)
+1804.
+[287] R. Jackiw, P. Rossi, Zero modes of the vortex-fermion system, Nuclear Physics B 190 (1981) 681.
+[288] E. J. Weinberg, Index calculations for the fermion-vortex system, Phys. Rev. D 24 (1981) 2669.
+[289] Y. Nishida, L. Santos, C. Chamon, Topological superconductors as nonrelativistic limits of Jackiw-Rossi and
+Jackiw-Rebbi models, Phys. Rev. B 82 (2010) 144513.
+[290] D. A. Ivanov, Non-abelian statistics of half-quantum vortices in p-wave superconductors, Phys. Rev. Lett. 86
+(2001) 268.
+[291] M. Stone, S.-B. Chung, Fusion rules and vortices in px + ipy superconductors, Phys. Rev. B 73 (2006) 014505.
+[292] N. Read, E. Rezayi, Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited
+Landau level, Physical Review B 59 (1999) 8084.
+[293] A. B. Z. V. A. Fateev, Nonlocal(parafermion) currents in two-dimensional conformal quantum field theory and
+self-dual critical points in ZN-symmetric statistical systems, Soviet Physics JETP 62 (1985) 215.
+[294] F. C. Alcaraz, R. K¨oberle, Hidden parafermions in Z(N) theories, Physical Review D 24 (1981) 1652.
+[295] W. Pan, H. L. St¨ormer, D. C. Tsui, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Fractional Quantum Hall Effect of
+Composite Fermions, Physical Review B 90 (2003) 016801.
+[296] J. S. Xia, W. Pan, C. L. Vicente, E. D. Adams, N. S. Sullivan, H. L. Stormer, D. C. Tsui, L. N. Pfeiffer, K. W.
+Baldwin, K. W. West, Electron Correlation in the Second Landau Level: A Competition Between Many Nearly
+Degenerate Quantum Phases, Physical Review Letters 93 (2004) 176809.
+[297] E. H. Rezayi, N. Read, Non-Abelian quantized Hall states of electrons at filling factors 12/5 and 13/5 in the first
+excited Landau level, Physical Review B 79 (2009) 075306.
+[298] E. Fradkin, C. Nayak, A. Tsvelik, F. Wilczek, A Chern-Simons effective field theory for the Pfaffian quantum Hall
+state, Nuclear Physics B 516 (1998) 704.
+[299] E. Fradkin, C. Nayak, K. Schoutens, Landau-Ginzburg theories for non-abelian quantum Hall states, Nuclear
+122
+
+Physics B 546 (1999) 711.
+[300] M. Barkeshli, X.-G. Wen, Effective field theory and projective construction for Zk parafermion fractional quantum
+Hall states, Physical Review B 81 (2010) 155302.
+[301] H. Goldman, R. Sohal, E. Fradkin, Landau-Gizburg theories of non-Abelian quantum Hall states from non-
+Abelian bosonization, Physical Review B 100 (2019) 115111.
+[302] H. Goldman, R. Sohal, E. Fradkin, Non-Abelian fermionization and the landscape of quantum Hall phases, Phys.
+Rev. B 102 (2020) 195151.
+[303] H. Goldman, R. Sohal, E. Fradkin, Composite particle construction of the Fibonacci fractional quantum Hall state,
+Phys. Rev. B 103 (2021) 235118.
+[304] I. Affleck, Field theory methods and quantum critical phenomena, in: Strings, Fields and Critical Phenomena,
+Proceedings of the Les Houches Summer School 1988, Session XLIX, E. Br´ezin and J. Zinn-Justin editors, North-
+Holland, Amsterdam, the Netherlands, 1990, p. 563.
+[305] A. A. Belavin, A. M. Polyakov, A. B. Zamolodchikov, Infinite Conformal Symmetry in two-Dimensional Quan-
+tum Field Theory, Nuclear Physics B 241 (1984) 333.
+[306] I. Affleck, Universal Term in the Free Energy at a Critical Point and the Conformal Anomaly, Phys. Rev. Lett. 56
+(1986) 746.
+[307] H. Bl¨ote, J. L. Cardy, M. P. Nightingale, Conformal Invariance, the Central Charge, and Universal Finite-Size
+Amplitudes at Criticality, Phys. Rev. Lett. 56 (1986) 742.
+[308] G. Granger, J. P. Eisenstein, J. L. Reno, Observation of Chiral Heat Transport in the Quantum Hall Regime,
+Physical Review Letters 102 (2009) 086803.
+[309] A. Bid, N. Ofek, H. Inoue, M. Heiblum, C. L. Kane, V. Umansky, D. Mahalu, Observation of neutral modes in
+the fractional quantum Hall regime, Nature 466 (2010) 585–590.
+[310] C. L. Kane, M. Fisher, Transmission through barriers and resonant tunneling in an interacting one-dimensional
+electron gas, Phys. Rev. B 46 (1992) 15233.
+[311] C. Chamon, E. Fradkin, Distinct universal conductances in tunneling to quantum Hall states: The role of contacts,
+Phys. Rev. B 56 (1997) 2012.
+[312] A. M. Chang, L. N. Pfeiffer, K. W. West, Observation of Chiral Luttinger Behavior in Electron Tunneling into
+Fractional Quantum Hall Edges, Physical Review Letters 77 (1996) 2538.
+[313] A. M. Chang, Chiral Luttinger liquids at the fractional quantum Hall edge, Reviews of Modern Physics 75 (2004)
+1449.
+[314] L. A. Cohen, N. L. Samuelson, T. Wang, T. Taniguchi, K. Watanabe, M. P. Zaletel, A. F. Young, Univer-
+sal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact, unpublished (2022).
+arXiv:\protect\vrulewidth0pt\protect\href{http://arxiv.org/abs/2212.01374}{arXiv:2212.01374}.
+[315] M. Hashisaka, T. Jonckheere, T. Akiho, S. Sasaki, J. Rech, T. martin, K. Muraki, Andreev reflection of fractional
+quantum Hall quasiparticles, Nature Communications 12 (2021) 2794.
+[316] N. P. Sandler, C. Chamon, E. Fradkin, Noise measurements and fractional charge in fractional quantum Hall
+liquids, Phys. Rev. B 59 (1999) 12521.
+[317] F. P. Milliken, C. P. Umbach, R. A. Webb, Indicatiosn of a Luttinger liquid in the fractional quantum Hall regime,
+Solid State Communications 97 (1996) 309–313.
+[318] S. Roddaro, V. Pellegrini, F. Beltram, G.iorgioBiasiol, L. Sorba, Interedge Strong-to-Weak Scattering Evolution
+at a Constriction in the Fractional Quantum Hall Regime, Physical Review Letters 93 (2004) 046801.
+[319] S. Roddaro, V. Pellegrini, F. Beltram, Quasi-particle tunneling at a constriction in a fractional quantum Hall state,
+Solid State Communications 131 (2004) 565.
+[320] P. W. Anderson, G. Yuval, D. R. Hammann, Exact results in the Kondo problem. II. Scaling theory, qualitatively
+correct solution, and some new results on one-dimensional classical statistical models, Phys. Rev. B 1 (1970)
+4464.
+[321] N. Read, D. M. Newns, On the solution of the Coqblin-Schreiffer Hamiltonian by the large-N expansion technique,
+Journal of Physics C: Solid State Physics 16 (1983) 3273.
+[322] P. Fendley, A. W. W. Ludwig, H. Saleur, Exact Conductance through Point Contacts in the ν = 1/3 Fractional
+Quantum Hall Effect, Phys. Rev. Lett. 74 (1995) 3005.
+[323] J. B. Miller, I. P. Radu, D. Z. Zumb¨uhl, E. Levenson-Falk, M. A. Kastner, C. M. Marcus, L. N. Pfeiffer, K. W.
+West, Fractional quantum Hall effect in a quantum point contact at filling fraction 5/2, Nature Physics 3 (2007)
+561.
+[324] I. Radu, J. B. Miller, C. M. Marcus, M. A. Kastner, L. N. Pfeiffer, K. W. Baldwin, K. W. West, Quasi-Particle
+Properties from Tunneling in the ν = 5/2 Fractional Quantum Hall State, Science 320 (2008) 899.
+[325] C. L. Kane, M. Fisher, Nonequilibrium noise and fractional charge in the quantum Hall effect, Phys. Rev. Lett. 72
+(1994) 724.
+[326] C. Chamon, D. E. Freed, X. G. Wen, Nonequilibrium quantum noise in chiral Luttinger liquids, Phys. Rev. B 53
+(1996) 4033.
+[327] P. Fendley, A. Ludwig, H. Saleur, Exact Nonequilibrium dc Shot Noise in Luttinger Liquids and Fractional Quan-
+tum Hall Devices, Phys. Rev. Lett. 75 (1995) 2196.
+[328] M. Dolev, M. Heiblum, V. Umansky, A. Stern, D. Mahalu, Observation of a quarter of an electron charge at the
+ν = 5/2 quantum Hall state, Nature 452 (2008) 829.
+[329] C. Chamon, D. E. Freed, S. A. Kivelson, S. L. Sondhi, X. G. Wen, Two point-contact interferometer for quantum
+Hall systems, Physical Review B 55 (1997) 2331.
+[330] F. E. Camino, W. Zhou, V. J. Goldman, Realization of a Laughlin quasiparticle interferometer: Observation of
+fractional statistics, Physical Review B 72 (2005) 075342.
+[331] B. I. Halperin, A. Stern, I. Neder, B. Rosenow, Theory of the Fabry-P´erot quantum Hall interferometer, Physical
+Review B 83 (2011) 155440.
+[332] P. Bonderson, K. Shtengel, J. K. Slingerland, Probing non-abelian statistics with quasiparticle interferometry,
+Physical Review Letters 97 (2006) 016401.
+[333] A. Stern, B. I. Halperin, Proposed Experiments to Probe the Non-Abelian ν = 5/2 Quantum Hall State, Phys. Rev.
+Lett. 96 (2006) 016802.
+[334] R. L. Willett, L. N. Pfeiffer, K. W. West, Measurement of filling factor 5/2 quasiparticle interference with obser-
+vation of charge e/4 and e/2 period oscillations, Proceedings of the National Academy of Sciences 106 (2009)
+123
+
+8853–8858.
+[335] R. L. Willett, C. Nayak, K. Shtengel, L. N. Pfeiffer, K. W. West, Magnetic-Field-Tuned Aharonov-Bohm Oscilla-
+tions and Evidence for Non-Abelian Anyons at ν = 5/2, Phys. Rev. Lett. 111 (2013) 186401.
+[336] R. L. Willett, K. Shtengel, C. Nayak, L. N. Pfeiffer, Y. J. Chung, M. L. Peabody, K. W. Baldwin, K. W. W.
+West, Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding, (unpublished) (2019).
+arXiv:\protect\vrulewidth0pt\protect\href{http://arxiv.org/abs/1905.10248}{arXiv:1905.10248}.
+[337] D. R. Nelson, J. M. Kosterlitz, Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys.
+Rev. Lett. 39 (1977) 1201–1205.
+[338] P. R. Thomas, M. Stone, Nature of the phase transition in a non-linear O(2)3 model, Nuclear Physics B 144 (2)
+(1978) 513–524.
+[339] M. E. Peskin, Mandelstam-’t Hooft duality in abelian lattice models, Annals of Physics 113 (1978) 122–152.
+[340] C. Dasgupta, B. I. Halperin, Phase Transition in a Lattice Model of Superconductivity, Phys. Rev. Lett. 47 (1981)
+1556–1560.
+[341] B. I. Halperin, T. C. Lubensky, S. keng Ma, First-Order Phase Transitions in Superconductors and Smectic-A
+Liquid Crystals, Physical Review Letters 32 (1974) 292.
+[342] M. Moshe, J. Zinn-Justin, Quantum field theory in the large N limit: a review, Physics Reports 385 (2003) 69–228.
+[343] H. Goldman, E. Fradkin, Loop Models, Modular Invariance, and Three Dimensional Bosonization, Phys. Rev. B
+97 (2018) 195112.
+[344] E. Fradkin, F. A. Schaposnik, The Fermion-Boson Mapping in Three Dimensional Quantum Field Theory, Phys.
+Lett. B 338 (1994) 253.
+[345] C. P. Burgess, F. Quevedo, Bosonization as duality, Nucl. Phys. B 421 (1994) 373.
+[346] A. Chan, T. L. Hughes, S. Ryu, E. Fradkin, Effective field theories for topological insulators by functional
+bosonization, Phys. Rev. B 87 (2013) 085132.
+[347] A. P. O. Chan, T. Kvorning, S. Ryu, E. Fradkin, Effective hydrodynamic field theory and condensation picture of
+topological insulators, Phys. Rev. B 93 (2016) 155122.
+[348] S. Giombi, S. Minwalla, S. Prakash, S. P. Trivedi, S. R. Wadia, X. Yin, Chern-Simons theory with vector fermion
+matter, Eur. Phys. J. C 72 (2012) 1–65.
+[349] S. Jain, S. Minwalla, S. Yokoyama, Chern Simons duality with a fundamental boson and fermion, Journal of High
+Eenergy Physics 2013 (2013) 37.
+[350] O. Aharony, G. Gur-Ari, R. Yacoby, d = 3 bosonic vector models coupled to Chern-Simons gauge theories, J.
+High Energy Phys. 03 (2012) 37.
+[351] O. Aharony, G. Gur-Ari, R. Yacoby, Correlation functions of large N Chern-Simons-Matter theories and bosoniza-
+tion in three dimensions, J. High Energy Phys. 2012 (2012) 28.
+[352] O. Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, J. High Energy Phys. 02 (2016)
+93.
+[353] N. Seiberg, E. Witten, Gapped boundary phases of topological insulators via weak coupling, Progress of Theoret-
+ical and Experimental Physics 2016 (2016) ptw083.
+[354] A. Karch, D. Tong, Particle-Vortex Duality from 3D Bosonization, Phys. Rev. X 6 (2016) 031043.
+[355] D. T. Son, Is the composite fermion a Dirac particle?, Phys. Rev. X 5 (2015) 031027.
+[356] M. A. Metlitski, A. Vishwanath, Particle-vortex duality of two-dimensional dirac fermion from electric-magnetic
+duality of three-dimensional topological insulators, Phys. Rev. B 93 (2016) 245151.
+[357] C. Wang, T. Senthil, Dual Dirac liquid on the surface of the electron topological insulator, Phys. Rev. X 5 (2015)
+041031.
+[358] J.-Y. Chen, J. H. Son, C. Wang, S. Raghu, Exact Boson-Fermion Duality on a 3D Euclidean Lattice, Phys. Rev.
+Lett. 120 (2018) 016602.
+[359] D. F. Mross, J. Alicea, O. I. Motrunich, Explicit Derivation of Duality between a Free Dirac Cone and Quantum
+Electrodynamics in (2 + 1) Dimensions, Physical Review Letters 117 (2016) 016802.
+[360] D. Simmons-Duffin, The Conformal Bootstrap, in: J. Polchinski, P. Vieira, O. DeWolfe (Eds.), TASI 2015: Pro-
+ceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics, New Frontiers in Fields
+and Strings, World Scientific, Singapore, 2017, pp. 1–74.
+[361] H. Goldman, E. Fradkin, Dirac composite fermions and emergent reflection symmetry about even-denominator
+filling fractions, Physical Review B 98 (2018) 165137.
+[362] D. Bohm, D. Pines, A Collective Description of Electron Interactions: III. Coulomb Interactions in a Degenerate
+Electron Gas, Phys. Rev. 92 (1953) 609–625.
+[363] F. D. M. Haldane, Luttinger’s Theorem and Bosonization of the Fermi Surface, in: R. A. Broglia, J. R. Schrieffer
+(Eds.), Proceedings of the International School of Physics “Enrico Fermi”, Course CXXI “Perspectives in Many-
+Particle Physics”, International School of Physics “Enrico Fermi”, North-Holland, Amsterdam, The Netherlands,
+1994, pp. 5–29, arXiv:cond-mat/0505529.
+[364] A. Houghton, J. B. Marston, Bosonization and fermion liquids in dimensions greater than one, Phys. Rev. B 48
+(1993) 7790–7808.
+[365] A. H. Castro Neto, E. Fradkin, Bosonization of the Low Energy Excitations of Fermi Liquids, Phys. Rev. Lett. 72
+(1994) 1393.
+[366] A. H. Castro Neto, E. Fradkin, Bosonization of Fermi Liquids, Phys. Rev. B 49 (1994) 10877.
+[367] A. Houghton, H.-J. Kwon, J. B. Marston, Multidimensional bosonization, Advances in Physics 49 (2000) 141–
+228.
+[368] L. V. Delacr´etaz, Y.-H. Du, U. Mehta, D. T. Son, Nonlinear bosonization of Fermi surfaces: The method of
+coadjoint orbits, Phys. Rev. Research 4 (2022) 033131.
+[369] Z. D. Shi, H. Goldman, D. V. Else, T. Senthil, Gifts from anomalies: Exact results for Landau phase transitions in
+metals, SciPost Phys. 13 (2022) 102.
+[370] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, M. P. A. Fisher, Deconfined Quantum Critical Points, Science
+303 (2004) 1490.
+[371] A. L. Fitzpatrick, S. Kachru, J. Kaplan, S. Raghu, Non-Fermi-liquid fixed point in a Wilsonian theory of quantum
+critical metals, Phys. Rev. B 88 (2013) 125116.
+[372] M. A. Metlitski, D. F. Mross, S. Sachdev, T. Senthil, Cooper pairing in non-Fermi liquids, Phys. Rev. B 91 (2015)
+115111.
+124
+
+[373] V. Oganesyan, S. A. Kivelson, E. Fradkin, Quantum theory of the nematic Fermi liquid, Physical Review B 64
+(2001) 195109.
+[374] M. J. Lawler, D. G. Barci, V. Fern´andez, E. Fradkin, L. Oxman, Non-perturbative behavior of the quantum phase
+transition to a nematic Fermi liquid, Physical Review B 73 (2006) 085101.
+[375] M. J. Lawler, E. Fradkin, Local quantum criticality at the nematic quantum phase transition, Physical Review B
+75 (2007) 0333304.
+[376] S. Lederer, Y. Schattner, E. Berg, S. A. Kivelson, Enhancement of Superconductivity near a Nematic Quantum
+Critical Point, Phys. Rev. Lett. 114 (2015) 097001.
+[377] S. A. Kivelson, E. Fradkin, V. J. Emery, Electronic liquid-crystal phases of a doped Mott insulator, Nature 393
+(1998) 550–553.
+[378] E. Fradkin, S. A. Kivelson, M. J. Lawler, J. P. E. andAndrew P. Mackenzie, Nematic Fermi Fluids in Condensed
+Matter Physics, Annual Review of Condensed Matter Physics 1 (2010) 153–178.
+[379] E. Fradkin, S. A. Kivelson, J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature
+superconductors, Reviews of Modern Physics 87 (2015) 457–482.
+[380] R. M. Fernandes, P. P. Orth, J. Schmalian, Intertwined Vestigial Order in Quantum Materials: Nematicity and
+Beyond, Annual Review of Condensed Matter Physics 10 (2019) 133–154.
+125
+
diff --git a/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/load_file.txt b/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..036197db149f8ebd36a046a6ee3aeb4af1ad0da2
--- /dev/null
+++ b/0dFQT4oBgHgl3EQfDjUj/content/tmp_files/load_file.txt
@@ -0,0 +1,7820 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf,len=7819
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='13234v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='str-el]  30 Jan 2023  Field Theoretic Aspects of Condensed Matter Physics: An  Overview⋆  Eduardo Fradkin  Department of Physics and Institute for Condensed Matter Theory,  University of Illinois at Urbana-Champaign, 1110 West Green St, Urbana Illinois 61801-3080, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Abstract  In this chapter I discuss the impact of concepts of Quantum Field Theory in modern Condensed  Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although the interplay between these two areas is certainly not new, the impact and  mutual cross-fertilization has certainly grown enormously with time, and quantum Field Theory  has become a central conceptual tool in Condensed Matter Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this chapter I cover how  these ideas and tools have influenced our understanding of phase transitions, both classical and  quantum, as well as topological phases of matter, and dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Keywords:  Contents  1  Introduction  3  2  Early Years: Feynman Diagrams and Correlation Functions  3  3  Critical Phenomena  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Classical Critical Phenomena .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Landau-Ginzburg Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  The Renormalization Group  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Scaling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  The Operator Product Expansion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Fixed Points  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4  Universality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5  RG flows .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6  Asymptotic Freedom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9  4  Quantum Criticality  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Dynamic Scaling .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  The Ising Model in a Transverse Field .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Quantum Antiferromagnets and Non-Linear Sigma Models .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Spin coherent states .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  14  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Path integral for a spin-S degree of freedom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Quantum Ferromagnet .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4  Quantum Antiferromagnet .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  16  5  Topological Excitations  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Topological Excitations: Vortices and Magnetic Monopoles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Vortices in two dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Magnetic monopoles in compact electrodynamics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  22  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Topology and open integer-spin chains .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  26  ⋆Work was supported in part by the US National Science Foundation through grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' DMR 1725401 at the Uni-  versity of Illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Preprint submitted to Encyclopedia of Condensed Matter Physics 2e  February 1, 2023  6  Duality in Ising Models  27  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Duality in the 2D Ising Model  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  27  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  The 3D duality: Z2 gauge theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  28  7  Bosonization  31  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Dirac fermions in one space dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  31  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Chiral symmetry and chiral symmetry breaking .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  34  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  The chiral anomaly .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  36  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4  Bosonization, anomalies and duality .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  38  8  Fractional Charge  42  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Solitons in one dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  42  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  Polyacetylene .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  43  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Fractionally charged solitons .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  43  9  Fractional Statistics  45  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  Basics of fractional statistics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  46  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  What is a topological field theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  46  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3  Chern-Simons Gauge Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  47  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4  BF gauge theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  48  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5  Quantization of Abelian Chern-Simons Gauge Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  48  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6  Vacuum degeneracy a torus .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  50  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7  Fractional Statistics and Braids .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  51  10 Topological Phases of Matter  54  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Topological Insulators .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  54  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Dirac fermions in 2+1 dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  54  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 Dirac Fermions and Topological Insulators  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  55  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 Chern invariants  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  56  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 The quantum Hall effect on a lattice .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  58  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 The Anomalous Quantum Hall Effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  61  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 The Parity Anomaly  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  62  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 Three-dimensional Z2 topological insulators .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  67  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Z2 Topological Invariants  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  67  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 The Axial Anomaly and the Effective Action  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  68  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 Theta terms, and Domain walls: Anomaly and the Callan-Harvey Effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  71  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 Chern-Simons Gauge Theory and The Fractional Quantum Hall Effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  73  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Landau levels and the Integer Hall effect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  73  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 The Laughlin Wave Function .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  77  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 Quasiholes have fractional charge .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  78  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 The Jain States .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  79  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 Quasiholes have fractional statistics .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  80  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 Hydrodynamic Effective Field Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  80  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 Composite Boson Field Theory  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  82  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='8 Composite Fermion Field Theory  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  85  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='9 The Compressible States .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  90  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10 Fractional Quantum Hall Wave Functions and Conformal Field Theory  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  91  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='11 Edge States and Chiral Conformal Field Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  99  11 Particle-Vortex Dualities in 2+1 dimensions  105  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Electromagnetic Duality  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 105  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 Particle-Vortex Duality in 2+1 dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 105  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 The 3D XY Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 105  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 Scalar QED in 3D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 107  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 The duality mapping .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 109  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 Bosonization in 2+1 dimensions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 109  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 Bosonization of the Dirac theory in 2+1 dimensions  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 110  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 Bosonization of the Fermi Surface .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 114  12 Conclusions  116  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Introduction  Many (if not most) puzzling problems in Condensed Matter Physics involve systems with  a macroscopically large number of degrees of freedom often in regimes of large fluctuations,  thermal and/or quantum mechanical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The description of the physics of systems of this type  requires the framework provided by Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although quantum field theory has  its origins in high-energy physics, notably in the development of Quantum Electrodynamics, it  has found a nurturing home in Condensed Matter Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is a long history of of cross-fertilization between both fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the 1950’s in many  of the most significant developments in Condensed Matter Physics, Quantum Field Theory has  played a key role if not in the original development but certainly in the eventual understanding the  meaning and the further development of the discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As a result, many of the discoveries and  concepts developed in Condensed Matter have had a reciprocal impact in Quantum Field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  One can already see this interplay in the development of the Bardeen-Cooper-Schrieffer theory  of superconductivity [1] and its implications in the theory of dynamical symmetry breaking in  particle physics by Nambu [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The close and vibrant relationship between both fields has continued to these days, and it  is even stronger today than before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many textbooks have been devoted to teaching these ideas  and concepts to new generations of condensed matter physicists (and field theorists as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The earlier texts focused on Green functions which are computed in perturbation theory using  Feynman diagrams [3] [4] [5], while the more modern ones have a broader scope, use path  integrals and attack non-perturbative problems [6, 7] [8] [9, 10] [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In two recent books I have  discussed many aspects of the interrelation between condensed matter physics and quantum field  theory in more depth than I can do in this chapter [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Early Years: Feynman Diagrams and Correlation Functions  Quantum field theory played a key role in the development of the Theory of the Fermi Liquid  [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The theory of the Fermi liquid was first formulated by Landau using the framework of  hydrodynamics and the quantum Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau’s ideas were later given a micro-  scopic basis using Green functions and Feynman diagrams [3], including the effects of quantum  fluctuations at finite temperature and non-equilibrium behavior [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Linear response theory  was developed which allowed the computation of response functions (such as electrical conduc-  tivities and magnetic susceptibilities) from the computation of correlation functions for a given  microscopic theory [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In turn, correlation functions can be computed in terms of a set of Feyn-  man diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These concepts and tools borrowed many concepts from field theory including  the study of the analytic structure of the generalized susceptibilities and the associated spectral  functions (together with the use of dispersion relations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These developments led to the deriva-  tion of the fluctuation-dissipation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These ideas were widely applied to metals [13] and  superconductors [1], as well as to quantum magnets [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The spectrum of an interacting system has low energy excitations characterized by a set of  quantum numbers associated with the symmetries of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These low energy excitations are  known as quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of the Landau theory of the Fermi liquid the quasiparticle is  a “dressed” electron: it is a low energy excitation with the same quantum numbers (charge and  spin) and an electron but with a renormalized effective mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There are many such quasiparticles  in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The correlation functions (the propagators) of a physical system  has a specific analytic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In momentum (and frequency) space, the quasiparticle spectrum  is given by the poles of the correlators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The role of symmetries and, in particular of gauge invariance, in the structure of correlation  functions was investigated extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A direct consequence of symmetries is the existence of  Ward identities which must be satisfied by all the correlation functions of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ward iden-  tities are exact relations that relate different correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such identities contain a host  of important results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, in a theory with a globally conserved charge, the Hamiltonian  (and the action) have a global U(1) symmetry associated with the transformation of the local field  operator φ(x) (which can be fermionic or bosonic) to a new field φ′(x) = eiθφ(x) (where θ is a  constant phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Theories with a global continuous symmetry have a locally conserved current  (and satisfy a continuity equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Ward identity requires the correlators of these currents  (and densities) to be be transverse (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' they should have vanishing divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the absence  of so-called quantum anomalies (which we will discuss below) global symmetries can be made  local and become gauge symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3  In many circumstances a global symmetry can be spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the global sym-  metry is continuous, then the Ward identities imply the existence of gapless excitations known  as Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example in the case of a superfluid, which has a spontaneously broken  U(1) symmetry the Goldstone boson is the gapless phase mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instead, a the N´eel phase of a  quantum antiferromagnet has two gapless Goldstone bosons, the magnons of the spontaneously  broken SO(3) global symmetry of this state of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Another example is the Ward identity of  quantum electrodynamics, which relates the electron self-energy to the electron-photon vertex  function, which also holds in non-relativistic electron fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, these identities implied  the existence of sum rules that the spectral functions must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' All of these results became part  of the standard toolkit of condensed matter experimentalists in analyzing their data and for the-  orists to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ward identities and sum rules also imply restrictions on the allowed  approximations which are often needed to obtain predictions from a microscopic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Critical Phenomena  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Classical Critical Phenomena  The late 1960s and particularly 1970s brought about an intense back and forth between con-  densed matter physics and field theory in the context of the problem of classical critical phenom-  ena and phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This was going to become a profound revolution on the description of  macroscopic physical systems with large-scale fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The problem of continuous (“sec-  ond order”) phase transitions has a long history going back to the work of Landau [18, 19] who  introduced the concept of an order parameter field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This turned out to be a powerful concept of  broad applicability in many physical systems sometimes quite different from each other at the  microscopic level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A simple example is that of a ferromagnet with uniaxial anisotropy in which the spins of the  atoms in a crystal are strongly favored to be aligned (or anti-aligned) along certain directions of  the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The simplest microscopic model for this problem is the Ising model, a spin system in  which the individual spins are allowed to take only two values, σ = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The partition function  of the Ising model (in any dimension) is  Z =  �  [σ]  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed− J  T  �  ⟨r,r′⟩  σ(r)σ(r′)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (1)  where J is the exchange coupling constant and T is the temperature (measured in energy units);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  here [σ] denotes the sume over the 2N spin configurations (for a lattice with N sites), and ⟨r, r′⟩  are nearest neighbor sites of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The order parameter of the Ising model is the local mag-  netization ⟨σ(r)⟨ which, in the case of a ferromagnet, is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The partition function of the  Ising model can be computed trivially in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The solution of the two-dimensional  Ising model by Onsager constituted a tour-de-force in theoretical physics [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Its actual meaning  remained obscure for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The work of Schultz, Mattis and Lieb [21] evinced a deep con-  nection between Onsager’s solution and the problem if the spectrum of one-dimensional quantum  spin chains [22] (specifically the one-dimensional Ising model in a transverse field [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One  important result was that the Ising model was in fact a theory of (free) fermions which, crudely  speaking, represented the configurations of domain walls of the magnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, even this sim-  ple model cannot be solved exactly in general dimension, and approximate mean field theories  of various sorts were devised over time to understand its physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau-Ginzburg Theory  Landau’s approach assumed that close enough to a phase transition, the important spin con-  figurations are those for which the local magnetization varies slowly on lattice scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  picture the local magnetization, on long enough length scales, becomes an order parameter field  that takes values on the real numbers, and can be positive of negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the order parameter  field is effectively the average of local magnetizations on some scale large compared to the lattice  scale, which we will denote by a real field φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The thermodynamics properties of a system of  this type in d dimensions can be described in terms of a free energy  F[φ] =  �  ddx  �κ  2 (▽φ(x))2 + a(T − Tc)φ2(x) + uφ4(x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  �  (2)  4  which is known as the Ginzburg-Landaufree energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here κ is the stiffness of the order parameter  field, Tc is the (mean-field) critical temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a and u are two (positive) constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  expression make sense if the transition is continuous and hence that the order parameter is small  near the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The energy of the Ising model is invariant under the global symmetry [σ] �→  [−σ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the symmetry of the group Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Likewise, the Ginzburg-Landau free energy has the  global (discrete) symmetry [φ(x)] �→ [−φ(x)], and also has a Z2 global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Landau’s approach, which was a mean field theory, the equilibrium state is the global  minimum of this free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The nature of the equilibrium state depends on whether T > Tc  or T < Tc: for T > Tc the global minimum is the trivial configuration, ¯φ(x) = 0 (this is the  paramagnetic state), whereas for T < Tc the equilibrium state is two fold degenerate, ¯φ(x) =  ±(a(Tc − T)/2)β, with the two degenerate states being related by the Z2 symmetry (this is the  ferromagnetic state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the Landau theory the critical exponent of the magnetization is β = 1/2  and the critical exponent of the correlation length is ν = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in the case of the 2D Ising  model the order parameter exponent is β = 1/8 [24] and the correlation length exponent is ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These (and other) apparent discrepancies led many theorists for much of the 1960s believe that  each model was different and that these behaviors reflected microscopic differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition,  Landau’s theory was regarded as phenomenological and believed to be of questionable validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Renormalization Group  This situation was to change with the development of the Renormalization Group, due pri-  marily to the work of Leo P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff [25, 26, 27] and Kenneth G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson [28, 29, 30, 31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The renormalization Group was going to have (and still has) a profound effect both in Condensed  Matter Physics and in Quantum Field Theory (and beyond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Scaling  Several phenomenological theories were proposed in the 1960s to describe the singular be-  havior of physical observables near a continuous phase transition [34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These early works  argued that in order to explain the singular behavior of the observables the free energy density  had to have a singular part which should be a homogeneous function of the temperature, mag-  netic field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A function f(x) is homogeneous if it satisfies the property that it transforms  irreducibly under dilations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' f(λx) = λk f(x), there λ is a real positive number (a scale) and  k is called the degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These heuristic ideas then implied that the critical exponents should obey  several identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1966 Kadanoff wrote and insightful paper in which he showed that the ho-  mogeneity hypothesis implied that in that regime these systems should obey scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' He showed  that this can be justified by performing a sequence of block-spin transformations in which the  configurations that vary rapidly at the lattice scale a become averaged at the scale of a larger sized  block of length scale ba > a which resulted in a renormalization of the coupling constants from  {K} at scale a to {K′} at the new scale ba [25, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the block spin transformation  amounts to a scale transformation and a renormalization of the couplings (and operators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' From  this condensed matter/statistical physics perspective the important physics is in the long distance  (“infrared”) behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A significant consequence of these ideas was that close enough to a critical point, if the  distance |x − y| between two local observables O(x) and O(y) is large compared to the lattice  spacing a but small compared to the correlation length ξ, their correlation function takes the  form of a power law  ⟨O(x)O(y)⟩ ∼  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  |x − y|2∆O  (3)  where ∆O is a positive real number known as the scaling dimension of the operator O [34, 25, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These conjectures were known to be satisfied in the non-trivial case of the 2D Ising model [26],  as well as in the Landau-Ginzburg theory once the effects of Gaussian fluctuations were included  [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, in the 2D Ising model, the scaling dimensions of the local magnetization σ is  ∆σ = 1/8 and of the energy density ε is ∆ε = 1, which were sufficient to explain all the singular  behaviors known at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The concept of renormalization actually originated earlier in quantum field theory as part  of the development of Quantum Electrodynamics (QED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In QED the notion of renormaliza-  tion was used to “hide” the short distance (“ultraviolet”) divergencies of the Feynman diagrams  needed to compute physical processes involving electrons (and positrons) and photons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' their  strong, divergent, dependence of an artificially introduced short-distance cutoff or regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  particular the sum of the leading diagrams that enter in the electron-photon vertex amounted to  5  a redefinition (renormalization) of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It was observed by Murray Gell-Mann  and Francis Low that this renormalization was equivalent to the solution of a first order differ-  ential equation that governed the infinitesimal change of the coupling, the fine structure constant  α = e2/4π, under an infinitesimal change of the UV cutoff Λ [38]  Λ dα  dΛ ≡ β(α) = 2  3πα2 + O(α3)  (4)  where β(α) is the Gell-Mann-Low beta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Except for the work by Nikolai Bogoliubov  and coworkers [39], this reinterpretation by Gell-Mann and Low was not actively pursued, partly  because it predicted that the renormalized coupling became very large at short distances, α → ∞,  and, conversely, it vanished in the deep long distance regime, α → 0 (if the electron bare mass is  zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other terms, QED is strongly coupled in the UV and trivial in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The same behavior  was found in the case of the theory of a scalar field φ(x) with an φ4 interaction which is relevant  in the theory of phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition to these puzzles, the1960s saw the experimental  development of the physics of hadrons which involve strong interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these reasons, for  much of that decade most high-energy theorists had largely abandoned the use of quantum field  theory, and explored other, phenomenologically motivated, approaches (which led to an early  version of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') At any rate the notion that the physics may depend on the scale was  present as was the notion that in some regimes field theories may exhibit scale-invariance at least  in an approximate form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Operator Product Expansion  The next stage of the development of these ideas was the concept of the operator product  expansion (OPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we denote by {O j(x)} the set of all possible local operators in a theory (a field  theory or a statistical mechanical system near criticality), then the product of two observables on  this list closer to each other than to any other observable (and to the correlation length ξ) obeys  the expansion  lim  x→y O j(x)Ok(y) = lim  x→y  �  l  C jkl  |x − y|∆j+∆k−∆l Ol  � x + y  2  �  (5)  where this equation should be understood as a weak identity, valid inside an expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Remarkably, this concept was derived independently and simultaneously by Leo Kadanoff [40]  (who was working in critical phenomena), by Kenneth Wilson [41] (who was interested in the  short distance singularities arising in Feynman diagrams), and by Alexander Polyakov [42, 43]  (also working in critical phenomena).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (5) {∆j} are the scaling dimensions of the operators  {O j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The coefficients C jkl are (like the dimensions) universal numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a follow up paper  Polyakov showed that if the theory has conformal invariance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' scale invariance augmented by  conformal transformations which preserve angles, then, provided the operators O j are suitably  normalized, the coefficients C jkl of the OPE are determined by a three point correlator  ⟨O j(x)Ok(y)Ol(z)⟩ =  C jkl  |x − y|∆jk|y − z|∆kl|z − x|∆l j  (6)  where ∆jk = ∆j + ∆k − ∆l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These results constitute the beginnings of Conformal Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In a nontrivial check, Kadanoff and Ceva showed that the OPE holds for the local observables of  the 2D Ising model [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fixed Points  The next and crucial step in the development of the renormalization Group was made by  Kenneth Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson was a high-energy theorist who wanted to know how to properly define  a quantum field theory and the physical meaning of renormalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In a Lorentz invariant quantum field theory one is interested in the computation of the ex-  pectation value of time-ordered operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of a self-interacting scalar field φ(x) in  D-dimensional Euclidean space-time, obtained by analytic continuation from Minkowski space-  time to imaginary time, the observables are computed from the functional (or path) integral by  functional differentiation of the partition function  Z =  �  Dφ exp  �  −S (φ, ∂µφ) +  �  dDx J(x)φ(x)  �  (7)  6  with respect to the local sources J(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a scalar field the Euclidean action is  S =  �  dDx  �1  2(∂µφ(x))2 + m2  2 φ2(x) + λ  4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='φ4(x)  �  (8)  which has the same form as the free energy of the Landau-Ginzburg theory of phase transitions  shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is apparent that the Landau-Ginzburg theory is the classical limit of the theory  of the scalar field whose partition function is a sum over all histories of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to  see that en expansion of the partition function (or of a correlator) in powers of the the coupling  constant λ can be cast in the form of a sum of Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To lowest order in λ a typical  Feynman diagram involves a one-loop integral in momentum space of the form  I(p) =  �  dDq  (2π)D  1  (q2 + m2)((q − p)2 + m2)  (9)  As noted by Wilson in his Nobel Lecture [33],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' this integral has large contributions from the IR  region of small momenta q ∼ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' but for any dimension D ≥ 4 has a much larger contribution  form large the UV region of large momenta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which requires the introduction of a UV cutoff Λ  in momentum space (or a lattice spacing a in real space by defining the theory on a hypercubic  lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In quantum field theory one then has to require that somehow one takes the limit a → 0  (or Λ → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To take this limit in the field theory is very much analogous to the definition of  a conventional integral in terms of a limit of a Riemann sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The difference is that this is a  functional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While for a function of bounded variation in a finite interval (a, b) the limit  of a partition of the interval into N steps each of length ∆x, such that N∆x = b − a, exists and  defines the integral of the function  lim  ∆x→0 lim  N→∞  N  �  j=1  f(x j)∆x j =  � b  a  dx f(x),  (10)  the analogous statement does not obviously exists in general for a functional integral, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' an  integral over a space of functions which is what is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, although thousands of  integrals of a function are known to exist, there are extremely few examples for a functional  integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, in order to take the continuum limit the lattice spacing must approach zero,  a → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that physical scales, such as the correlation length ξ, must diverge in lattice  units so that they can be fixed in physical units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But to do that one has to be asymptotically  close to a continuous phase transition!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the problem of defining a quantum field theory is  equivalent to the problem of critical phenomena at a continuous phase transition!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Wilson gave a systematic formulation to the Renormalization Group by generalizing the ear-  lier ideas introduced by Kadanoff and the earlier work by Gell-Mann and Low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson’s key  contribution was the introduction of the concept of a fixed point of the Renormalization Group  transformation [28, 29, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw, the block-spin transformation is a procedure for  coarse graining the degrees of freedom of a physical system resulting in a renormalization of the  coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon the repeated action of the RG transformation its effect can be pictured  as a flow in the space of coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in addition of integrating-out short dis-  tance degrees of freedom one needs to restore the units of length which have changed under that  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This requires a rescaling of lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Once this is done, Wilson showed that the resulting  RG flows necessarily have fixed points, special values of the couplings which are invariant (fixed)  under the action of the RG transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' He then deduced that at a fixed point the theory has  no scales, aside from the linear size L of the system and the microscopic UV cutoff (the lattice  spacing a in a spin system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This analysis means that for length scales long compared to a → 0 but short compared to  L → ∞ the theory acquired a new, emergent, symmetry: scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, at a fixed  point the correlators of all local observables must be homogeneous functions (hence, must scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Universality  A crucial consequence of the concept if the fixed point is that phase transitions can be clas-  sified into universality classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Universality means that a large class of physical systems with  different microscopic properties have fixed points with the same properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the same scaling  dimensions, operator product expansions and correlation functions at long distances independent  on how they are defined microscopically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although the renormalization group transformation is  a transformation scheme that we define and, because of that the location in coupling constant  7  space of the fixed point itself does depends on the scheme we choose, its universal properties are  the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, universality classes depend only on features such as the space (and spacetime)  dimension and the global symmetries of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But the systems themselves may be quite  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This we speak if the Ising universality class in 2D, on the superfluid (or XY) transi-  tion class in 3D, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This concept, which originated in the theory of phase transition, has been  adopted and generalized in the development of conformal field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' RG flows  Combined with the condition that the correlators decay at long separations, homogeneity  implies that the correlators must have the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, this equation also implies  that at a fixed point the operators (the local observables) have certain scaling dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let  us consider a theory close to a fixed point whose action we will denote by S ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let {O j(x)} be  a complete set of local observables whose scaling dimensions are {∆j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The total action of the  theory close to the fixed point then can be expanded as a linear combination of the operators with  dimensionless coupling constants {g j}  S = S ∗ +  �  dDx  �  j  g ja∆j−DO j(x)  (11)  Under a change of length scale x → x′ = bx, with b > 1, the operators (which must transform  homogeneously)change as O j(bx) = b−∆jO j(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the phase space changes as dDx′ = bDddx,  we can keep the form of the action provided the coupling constants also change to compensate  for these changes as g′  j = bD−∆jg j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let b = |x′|/|x| = 1+da/a, where da is an infinitesimal change  of the UV cutoff a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, if we integrate-out the degrees of freedom in the range a < |x| < a+da,  the rate of change of the coupling constants {g j} under this rescaling is  adg j  da ≡ β(g j) = (D − ∆j)g j + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (12)  which we recognize as a Gell-Mann Low beta function for each coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This result says that if the scaling dimension ∆j < D, then the renormalized coupling will  increase as we increase the length scale, g′  j > g j, and along this direction in coupling constant  space the RG flows away from the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conversely, if ∆j > D the renormalized coupling  flows to smaller values, g′  j < g j and the RG flows into the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We then say that an  operator is relevant if its scaling dimension satisfies ∆j < D, and that it is irrelevant if ∆j > D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  If ∆j = D then we say that operator is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To go beyond this simple dimensional analysis one has to include the effects of fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To lowest orders in the couplings one finds [45]  adg j  da = (D − ∆j)g j +  �  k,l  C jkl gkgl + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (13)  where {C jkl} are the coefficients of the OPE shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This expression is the general form  of a perturbative renormalization group and it is valid close enough to a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Wilson and Fisher [30] used a similar approach to analyze how fluctuations alter the results  of the Landau-Ginzburg theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They considered the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (7) with the action  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instead of working in real space they considered the problem in momentum space and  partitioned the field configurations into slow and fast modes  φ(x) = φ<(x) + φ>(x)  (14)  where φ>(x) are configurations whose Fourier components have momenta in the range bΛ < |p| <  Λ, where Λ is a UV momentum cutoff and b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, if we choose b → 1, the fast modes φ >  (x) have components in a thin momentum shell near the UV cutoff Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conversely, the slow modes  φ<(x) have momenta in the range 0 ≤ |p| < bΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One can then use Feynman diagrams to integrate  out the fast modes and derive an effective low-energy action with renormalized couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  case of a free field theory (with λ = 0) the scaling dimension of the φ4 operator is ∆4 = 2(D − 2)  whereas the φ2 operator has dimension ∆2 = D − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon defining a dimensionless mass and  coupling constant by m2 = tΛ2 and λ = gΛ4−D, the beta functions are found to be [10]  β(t) = − Λ dt  dΛ = 2t + g  2 − gt  2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (15)  β(g) = − Λ dg  dΛ = (4 − D)g − 3  2g2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (16)  8  (where we absorbed an uninteresting factor if the definition of the coupling constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The RG flows of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (16) show that the free-field (Gaussian) fixed point at g = 0 is stable for  D > 4 and the asymptotic IR behavior is the same as predicted by the Landau-Ginzburg theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, for D < 4, the free-field fixed point becomes unstable and a new fixed point arises at  g∗ = 2  3ǫ + O(ǫ2), where we have set ǫ = 4 − D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the Wilson-Fisher fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At this  fixed point the correlation length diverges with an exponent ν = 1  2 + ǫ  12 + O(ǫ2), which deviates  from the predictions of the Landau-Ginzburg theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The small parameter of this expansion is ǫ,  and this is known as the ǫ expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Wilson-Fisher (WF) fixed point is an example of a non-trivial fixed point at which the  correlation length is divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It has only one relevant operator, the mass term, which in the IR  flows into the symmetric phase for t > 0 and flows to the broken symmetry for t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conversely,  in the UV it flows into the WF fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these reasons condensed matter physicists say  that this is an IR unstable (or critical) fixed point while high-energy physicists say that it is the  UV fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At this fixed point a non-trivial field theory can be defined with non-trivial  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' UV fixed points also define examples of what in high-energy physics are called  renormalizable field theories and can be used to define a continuum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The D = 4-dimensional theory is special in that the φ4 operator is marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As can be seen  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (16), at D = 4 the beta function for the dimensionless coupling constant g does not have  a linear term and is quadratic in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the operator is marginally irrelevant, and its  beta function has the same behavior as the beta function of Gell-Mann and Low for QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such  theories are said to have a “triviality problem” since, up to logarithmic corrections to scaling,  there are no interactions in the IR and, conversely, become large in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There are also fixed points at which the correlation length ξ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These fixed points are  IR stable (and in a sense trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These stable fixed points are sinks of the IR RG flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such  fixed points define stable phases of matter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the broken symmetry state, the symmetric (or  unbroken state), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However in the UV they are unstable and in high-energy physics such fixed  points correspond to non-renormalizable field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  More sophisticated methods are needed to go beyond the lowest order beta functions of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (13), and the computation of critical exponents beyond the leading non-trivial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pos-  sibly the record high-precision calculations have been done for φ4 theory for which the beta  function is know to O(ǫ5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This has been achieved using the method of dimensional regulariza-  tion [46, 47, 48] (with minimal subtraction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Special resummation methods (Borel-Pad´e) have  been used to do these calculations in D = 3 dimensions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Remarkably, these results are  so precise that in the case of the superfluid transition, which is well described by a φ4 theory  with a complex field, the results could only be tested in the microgravity environment of the  International Space Station!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Asymptotic Freedom  There are several physical systems systems of great interest whose beta function has the form  β(g) = Ag2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (17)  The coupling constant has a different interpretation in each theory and the constant A > 0,  opposite to the sign of the beta function found in QED and φ4 theory in D = 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This beta function means that while the associated operator is marginal, with this sign is actually  marginally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This also means that the fixed point is unstable in the IR but the departure  from the fixed point is logarithmically small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conversely, the in the UV the RG flows into the  fixed point and the effective constant is weak at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the origin of the term  asymptotic freedom [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The paradigmatic examples of theories with a beta function of this  form are the Kondo problem, the 2D non-linear sigma model, and the D = 4 dimensional Yang-  Mills non-abelian gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Kondo problem is the theory of a localized spin-1/2 degree of freedom in a metal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the  electrons of the metal couple to this quantum impurity through an exchange interaction of the  impurity and the magnetic moment density of the mobile electrons in the metal with coupling  constant J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem is actually one dimensional since only the s wave channel of the  mobile (conduction band) electrons actually couple to the localized impurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1970 Philip  Anderson developed a theory of the Kondo problem in terms of the renormalization of the Kondo  coupling constant g as a function of the energy scale [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Anderson used perturbation theory  in J to progressively integrated out the modes of the conduction electrons close to an effective  9  bandwidth Ec and found that the beta function has the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (17)  Ec  dJ  dEc  = −ρJ2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (18)  where ρ is the density of states at the Fermi energy of the conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This work  implied that the free-impurity fixed point is IR unstable and that the effective coupling constant  J increases as the energy cutoff Ec is lowered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' He argued that at some energy scale, the Kondo  scale, perturbation theory breaks down and that there is a crossover to a strong coupling regime  which is not accessible in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Shortly thereafter, in 1973 Wilson developed a numerical renormalization group approach  which showed that the Kondo problem is indeed a crossover from the free impurity fixed point  to the “renormalized” Fermi liquid [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, Wilson use the numerical renormalization  group to examine the approach to the strong coupling fixed point and showed that it is charac-  terized by a Wilson ratio, a universal number obtained from the low temperature specific heat  and the impurity magnetic susceptibility (in suitable units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson’s numerical RG predicted  a number close to 2π for the this ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1980 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Andrei and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wiegmann showed (indepen-  dently) that the Kondo problem is an example of an integrable field theory that can be solved  by the Bethe ansatz [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Their exact result was consistent with Wilson’s RG, including the  numerical value of the Wilson ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In 1972 Gerard ’t Hooft and Martinus Veltman showed that Yang-Mills gauge theory is renor-  malizable [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This groundbreaking result opened the door to use quantum field theory to de-  velop the theory of strong interactions in particle physics known as Quantum Chromodynamics  (QCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1973 David Gross and Frank Wilczek [50] and, independently, David Politzer [54]  computed the renormalization group beta function of Yang-Mills theory with gauge group G and  found it to be of the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (17),  Λ dg  dΛ = − g3  16π2  11  3 C2(G) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (19)  where g is the Yang-Mills coupling constant and Λ is a UV momentum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here C2(G) is the  quadratic Casimir for a gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For S U(3), the case of physical interest, C2(S U(3)) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This result implies that under the RG at large momenta (short distances) the Yang-Mills coupling  constant flows to zero (up to logarithmic corrections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result holds in the presence of quarks  provided the number of quark flavors is less than a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence at short distances the  effective coupling is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gross and Wilczek called this phenomenon asymptotic freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This behavior was consistent with the observation of weakly coupled quarks in deep inelastic  scattering experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the flip side of asymptotic freedom is that at low energies  (long distances) the coupling constant grows without limit, which implies that at low energies  perturbation theory is not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This strong infrared behavior suggested that in QCD quarks  are permanently confined in color neutral bound states (hadrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, unlike the Kondo  problem we just discussed, QCD is not an integrable theory (so far as we know) and to show that  it confines has required the development of Lattice Gauge Theory [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To this date the best  evidence for quark confinement has been obtained using large-scale Monte Carlo simulations in  Lattice Gauge Theory [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We close this subsection with a discussion of an important case: the non-linear sigma models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The O(N) non-linear sigma model is the continuum limit of the classical Heisenberg model for a  spin with N components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Historically, the non-linear sigma model is the effective field theory for  pions in particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will discuss its role in the theory of quantum antiferromagnets in  subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 and especially in the case of quantum antiferromagnetic spin chains in subsection  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The simplest non-linear sigma model is a theory of an N-component scalar field n(x) which  satisfies the unite length local constraint, n2(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Euclidean Lagrangian is  L = 1  2g(∂µn(x))2  (20)  where g is the coupling constant (the temperature in the classical Heisenberg model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At the  classical level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in the broken symmetry phase, where ⟨n⟩ � 0, this model describes the N − 1  massless modes (Goldstone bosons) of the spontaneously broken O(N) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dimensional  analysis shows that the coupling constant has units of ℓD−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, we expect to find marginal  behavior at D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1975 Alexander Polyakov used a momentum shell renormalization group  10  in D = 2 dimensions and showed that the beta function of this model is (here a is the short-  distance cutoff) [58]  β(g) = adg  da = N − 2  2π g2 + O(g3)  (21)  Hence, in D = 2 dimensions also this theory is asymptotically free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As in the other examples we  just discussed, asymptotic freedom here also implies that the coupling constant g grows to large  values in the low-energy (long-distance) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In close analogy with Yang-Mills theory in  D = 4 dimensions, Polyakov conjectured that the O(N) non-linera sigma model also undergoes  dynamical dimensional transmutation [50], that the global O(N) symmetry is restored and that  for all values of the coupling constant g the theory is in a massive with a finite correlation length  ξ ∼ exp((N − 2)/2πg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Extensive numerical simulations, used to construct a renormalization  group using Monte Carlo simulations [59], showed that there is indeed a smooth crossover from  the weak coupling (low temperature) regime to the high temperature regime where the correlation  length is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The non-linear sigma model is a renormalizable field theory in D = 2 dimensions  [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For D > 2 dimensions it can be studied using the 2 + ǫ expansion [60, 49], which predicts  the existence of a nontrivial UV fixed point and a phase transition from a Goldstone phase to a  symmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It turns out that there is a significant number of asymptotically free non-linear sigma models  in D = 2 dimensions, many of physical interest [61], in particular non-linear sigma models  whose target manifold is a coset space, a quotient of a group G and a subgroup H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The O(N) non  linear sigma model is an example since the broken symmetry space leaves the O(N −1) subgroup  unbroken (the manifold of the Goldstone bosons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In that case the quotient is O(N)/O(N − 1)  which is isomorphic to the N −1 dimensional sphere S N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In later sections we will discuss other  examples in which more general non-linear sigma models play an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Models on coset spaces arise in the theory of Anderson localization in D = 2 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Anderson localization is the problem of a fermion (an electron) in a disordered system in which  the electron experiences a random electrostatic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the limit of strong disorder Philip  Anderson showed that all one-particle states are exponentially localized and the diffusion con-  stant (and the conductivity) vanishes[62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There was still the question of when it is possible  for the electron to have a finite diffusion constant (and conductivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In D = 2 dimensions  the conductivity is a dimensionless number which suggests that this may be the critical dimen-  sion for diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abrahams, Anderson, Licciardello and Ramakrishnan used a weak disorder  calculation to construct a scaling theory that implied that in D = 2 dimensions the RG flow  of the conductivity at long distances (large samples) flows to zero and all states are localized  [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shortly thereafter Wegner gave strong arguments that showed that the existence of diffu-  sion implied that there are low-energy “diffusson” modes which behaved as Goldstone modes of  a non-linear sigma model on the quotient manifold O(N+ + N−)/O(N+) × O(N−) in the “replica  limit” N± → 0 [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A field theory approach to this non-linear sigma model was developed by  McKane and Stone [65] and by Hikami [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Criticality  Quantum criticality is the theory of a phase transition of a system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a magnetic system)  at zero temperature that occurs as a coupling constant (or parameter) is varied continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although not necessarily under that name, this question has existed as a conceptual problem for  a long time, In particular, already in 1973 Wilson considered the problem of the behavior of  quantum filed theories blow four spacetime dimensions and their phase transitions [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The modern interest in condensed matter physics stems from discoveries made since the late  1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since that time he behavior of condensed matter systems at a quantum critical point has  emerged as a major focus in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There were several motivations for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One  was (and still is) to understand the behavior of quantum antiferromagnets in the presence of  frustrating interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Frustrating interactions are interactions which favor incompatible types  of antiferromagnetic orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The result is the presence of intermediate non-magnetic “valence  bond” phases that favor the formation of spin singlets between nearby spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' these phases typ-  ically either break the point group symmetry of the lattice or are spin liquids (which will be  discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Another motivation is that doped quantum antiferromagnets typically harbor  superconducting phases (among others) whose high-temperature behavior is a “strange” metal  that violates the basic assumptions (and behaviors) of Fermi liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The most studied version  of this problem is the case of the copper oxide high temperature superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It was con-  jectured that there is a quantum critical point inside the superconducting phase at which the  11  antiferromagnetic order (or other orders) disappears and which may be the reason for the strange  metal behavior above the superconducting critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many of these questions are  discussed in depth by Sondhi and coworkers [68] and in the textbook by Sachdev [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dynamic Scaling  We will consider a general quantum phase transition and assume that it is scale invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, except for the case of relativistic quantum field theories, in condensed matter systems  space and time do not need to scale in the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us assume that the system of interest has  just one coupling constant g and that the system of interest has a quantum phase transition (at zero  temperature) at some critical value gc between two phases, for instance one with a spontaneously  broken symmetry and a symmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the quantum phase transition is continuous then the  correlation length ξ will diverge at gc and so will the correlation time ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However these two  scales are in general different and do not necessarily diverge at the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, in general,  if some physical quantity is measured at the quantum critical point at some momentum p and  frequency ω, the length scale of the measurement is 2π/|p| and at a frequency is ω ∼ |p|z, where  z is the dynamic critical exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us say that we measure the observable O at momentum p  and frequency ω at gc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Scale invariance in both space and time means that at gc the observable  O(p, ω) at momentum p and frequency ω must scale as  O(p, ω) = |p|−∆O ˜O(|p|z/ω)  (22)  where ∆O is the scaling dimension of the observable O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The situation changes at finite temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A quantum field theory at temperature T is  described by a path integral on a manifold which along the imaginary time direction τ is finite  of length 2π/T and periodic for a theory bosonic fields and anti-periodic for fermionic fields  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the imaginary time direction is finite, the behavior for correlation times ξτ < 2π/T  and ξτ > 2π/T must be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, in the first regime the behavior is essentially the same  as at T = 0, while in the second it should be given by the classical theory in the same space  dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='At the quantum critical point gc there is only one time scale ξτ ∼ 2π/T and only one  length scale ξ ∼ (2π/T)1/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Ising Model in a Transverse Field  The prototype of the quantum phase transition is the Ising model in a transverse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  model describes a system of spin-1/2 degrees of freedom with ferromagnetic interactions with  uniaxial anisotropy in the presence of a transverse uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hamiltonian is  H = −J  �  ⟨r,r′⟩  σ3(r)σ3(r′) − h  �  r  σ1(r)  (23)  where J and h are the exchange coupling constant and the strength of the transverse field, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here σ1 and σ3 are the two Pauli matrices defined on the sites {r} of a lattice with  ferromagnetic interactions between spins on nearest neighboring sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hilbert space is the  tensor product of the states of the spins at each site of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At each site there are two nat-  ural bases of states: the eigenstates of σ3, which we denote by | ↑⟩ and | ↓⟩ (whose eigenvalues  are ±1), and the eigenstates of σ1, which we denote by |±⟩ (whose eigenvalues are also ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is well known that the Ising Model in a Transverse Field on a hypercubic lattice in D  dimensions is equivalent to a classical Ising model in D + 1 dimensions [70, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These two  models are related through the transfer matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, a classical Ising model can be regarded  as a path integral representation of the quantum model in one dimension less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For simplicity  we will see how this work for the 2D the classical ferromagnetic Ising model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (1), but the  construction is general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will regard the configuration of spins on a row of the 2D lattice as  a state of a quantum system, and the set of states on all rows as the evolution of the state along  the perpendicular direction that we will regard as a discretized imaginary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The contribution  from two adjacent rows to the partition function defines the matrix element of a matrix between  two arbitrary configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Statistical Physics this matrix is known as the Transfer Matrix ˆT  and the full partition function (with periodic boundary conditions) is  Z = tr ˆT Nτ  (24)  where Nτ is the number of rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the case of the ferromagnetic Ising model (actually, for any  unfrustrated model) the transfer matrix can always be constructed to be hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This property  12  holds in fact for any theory that satisfies a property known as reflection positivity which requires  that all (suitably defined) correlation functions be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For theories of this type, and the Ising  model is an example, the matrix elements of the transfer matrix can be identified with the matrix  element of the evolution operator of a quantum theory for a small imaginary time step [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Also,  the positivity of the correlators is equivalent to the condition of positivity of the norm of states in  the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  For the classical models that satisfy these properties, all directions of the lattice are equiv-  alent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, asymptotically close to the critical point, the behavior of all the correlators  becomes isotropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' invariant under the symmetries of Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that  the arbitrary choice of the direction for the transfer matrix is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Consequently, tin the  quantum model its equal-time correlators behave the same way as its correlation functions in  imaginary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words space and time behave in the same way and the quantum the-  ory is relativistically invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This implies at the quantum critical point the energy ε(p) of its  massless excitations should behave as ε(p) = v |p|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a relativistic theory the dynamical critical  exponent must be z = 1 and the coefficient v is the speed of the excitations (the “speed of light”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We should note that this is not necessarily always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There are in fact classical systems,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' liquid crystals [72], which are spatially anisotropic and map onto quantum mechanical theo-  ries in one less dimension for which the dynamical critical exponent z � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One such example are  the Lifshitz transitions of nematic liquid crystals in three dimensions and the associated quantum  Lifshitz model in D = 2 dimensions, for which the dynamical exponent is z = 2 [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Just as in the classical counterpart in D + 1 dimensions, the quantum model in D ≥ 1 has  two phases: a broken symmetry ferromagnetic phase for J ≫ h and a symmetric paramagnetic  phase for h ≫ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the symmetric phase the ground state is unique (asymptotically is the  eigenstate of σ1 with eigenvalue +1), while in the broken symmetry phase the ground state  is doubly degenerate (and asymptotically is an eigenstate of σ3) and there is a non-vanishing  expectation value of the local order parameter ⟨σ3(r)⟩ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the symmetric phase the correlation  function of the local order parameter decays exponentially with distance with a correlation length  ξ, as does the connected correlation function in the broken symmetry phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The model has  a continuous quantum phase transition at a critical value of the ratio h/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For general space  dimensions D > 1 this model is not exactly solvable and much of what we know about it is due  to large-scale numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This problem was solved exactly in one-dimension [23] using the Jordan-Wigner transfor-  mation that maps a one dimensional quantum spin system to a theory of free fermions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  fermion operators at site j are  χ1(j) = K(j − 1)σ3(j),  χ2(j) = i ˆK(j)σ3(j)  (25)  where K(j) is the kink creation operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the operator that creates a domain wall between  sites j and j + 1 [70], and is given by  K(j) =  �  n≤ j  σ1(n)  (26)  The operators χ1(j) and χ2(j) are hermitian, χ†  j(n) = χ j(n), and obey the anticommutation algebra  {χ j(n), χ j′(n′)} = 2δ j j′δnn′  (27)  Hence, they are fermionic operators are hermitian, anti-commute with each other and square to  the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Operators of this type are called Majorana fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Alternatively, we can use the more conventional (Dirac) fermion operators c(n) and its adjoint  c†(n) which are related to the Majorana fermions as  c(n) = χ1(n) + iχ2(n),  c†(n) = χ1(n) − iχ2(n)  (28)  which obey the standard anticommutation algebra  {c(n), c(n′)} = {c†(n), c†)n′)} = 0,  {c(n), c†(n′)} = δ − nn′  (29)  In this sense, a Majorana fermion is half of a Dirac fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In terms of the Majorana operators the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (23) becomes  H = i  �  j  χ1(j)χ2(j) + ig  �  j  χ2(j)χ1(j + 1)  (30)  13  where we have rescaled the Hamiltonian by a factor of h and the coupling constant is g = J/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Here we have not specified the boundary conditions (which depend on the fermion parity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qual-  itatively, the Majorana fermions can be identified with the domain walls of the classical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the Ising model the number of domain walls on each row is not conserved but their parity is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Likewise, the number of Majorana fermions NF is not conserved either but the fermion parity,  (−1)NF, is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is an elementary excercise to show that the spectrum of this theory has a gap G(g) which  vanishes at gc = 1 as G(g) ∼ |g − gc|ν, with an exponent ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the Hamiltonian of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (30) is quadratic in the Majorana operators, these operators obey linear equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the scaling regime we take the limit of the lattice spacing a → 0 and the coupling constant  g → gc = 1, while keeping the quantity m = (g − gc)/a fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime, the two-component  hermitian spinor field χ = (χ1, χ1), and χ† = χ, satisfies a Dirac equation  (i/∂ − m)χ = 0  (31)  where we set the speed � = 1, and where defined the 2 × 2 Dirac gamma-matrices γ0 = σ2,  γ1 = iσ3, and γ5 = σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon defining ¯χ = χTγ0, we find that the Lagrangian of this field theory  is  L = ¯χi/∂χ − 1  2m ¯χχ  (32)  which indeed becomes massless at the quantum phase transition of the Ising spin chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these  considerations, we say that the phase transition of the Ising model (2D classical or 1D quantum)  is in the universality class of massless Majorana fermions where m → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (32) we have  used the standard Feynman slash notation, /a = γµaµ, where aµ is a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Antiferromagnets and Non-Linear Sigma Models  As we noted above, the discovery of high temperature superconductors in the copper ox-  ide compounds prompted the study of the behavior of these strongly correlated materials at low  temperatures and of possible quantum phase transitions which they may host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The prototypical  cuprate material La2CuO4 is a quasi two-dimensional Mott insulator which exhibits long-range  antiferromagnetic order below a critical temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A simple microscopic model is a spin-S  quantum Heisenberg antiferromagnet on the 2D square lattice of the Cu atoms, whose Hamilto-  nian is  H = 1  2  �  r,r′  J(|r − r′|) S(r) · S(r′)  (33)  where S are the spin-S angular momentum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will consider the case where the ex-  change interaction for nearest neighbors J is dominant and a weaker J′ ≪ J for next nearest  neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this section we do not consider the regime J′ ≃ J in which the interactions com-  pete for incompatible ground states due to frustration effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Spin coherent states  The simplest way to see the physics of this antiferromagnet is to construct a path-integral  representation for a spin-S system using spin coherent states [74, 75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For details see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9]  which we follow here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A coherent state of the (2S + 1-dimensional) spin-S representation of  SU(2) is the state |n⟩, labeled by the spin polarization unit vector n  |n⟩ = eiθ(n0×n)·S |S, S ⟩  (34)  where n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The states of the spin-S representation are spanned by the eigenstates of S 3 and  S2,  S 3 |S, M⟩ = M|S, M⟩,  S2|S, M⟩ = S (S + 1)|S, M⟩  (35)  and |S, S ⟩ is the highest weight state with eigenvalues S and S (S + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (34) n0 is a unit  vector along the axis of quantization (the direction e3), and θ is the colatitude, such that n · n0 =  cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Two spin coherent states, |n1⟩ and |n2⟩, are not orthonormal,  ⟨n1|n2⟩ = eiΦ(n1,n2,n0) S  �1 + n1 · n2)  2  �S  (36)  where Φ(n1, n2, n0) is the area of the spherical triangle of the unit sphere spanned by the unit  vectors n1, n2 and n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, there is an ambiguity in the definition of the area of the spherical  14  triangle since the sphere is a 2-manifold without boundaries: if the “inside” triangle has spherical  area Φ, the complement (“outside”) triangle has area 4π − Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the ambiguity of the phase  prefactor of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (36) is  ei4πS = 1  (37)  since S is an integer or a half-integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, the quantization of the representations of SU(2) makes  the ambiguity unobservable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, the spin coherent states |n⟩ satisfy the resolution of the  identity  I =  �  |n⟩⟨n|  �2S + 1  4π  �  δ(n2 − 1) d3n  (38)  and  ⟨n|S|n⟩ = S n  (39)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Path integral for a spin-S degree of freedom  As an example consider problem of a spin-S degree of freedom coupled to an external mag-  netic field B(t) that varies slowly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The (time-dependent) Hamiltonian is given by the  Zeeman coupling  H(t) = B(t) · S  (40)  As usual, the path-integral is obtained by inserting the (over-complete) set of coherent states at  a large number of intermediate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The resulting path integral is a sum of the histories of the  spin polarization vector n(t)  Z = tr exp  �  i  � T  0  dt H(t)  �  =  �  Dn exp (iS[n])  �  t  δ(n2(t) − 1)  (41)  where the action is  S = S SWZ[n] − S  � T  0  dt B(t) · n(t)  (42)  where SWZ[n] is the Wess-Zumino action  SWZ[n] =  � T  0  dt A[n] · ∂tn  (43)  =  � 1  0  dτ  � T  0  dt n(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' τ) · ∂tn(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' τ) × ∂τn(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' τ)  (44)  where A[n] is the vector potential of a Dirac magnetic monopole (of unit magnetic charge) at  the center of the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The vector potential A[n] has a singularity associated with the  Dirac string of the monopole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can write an equivalent expression which is singularity-free  using Stokes Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We did this in the second line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (44) which required to extend the  circulation of A on the closed path described by n(t) to the flux of the vector potential through  the submanifold Σ of the unit sphere S 2 whose boundary is the history n(t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the area of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The smooth (and arbitrary) extension of configuration n(t) to the interior of Σ is done by defining  n(t, τ) such that n(t, 1) = n0, n(t, 0) = n(t), and n(0, τ) = n(T, τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since SWZ is the area of the  submanifold Σ of the unit sphere S 2, just as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (37), here too there is an ambiguity of 4π in  the definition of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here too, this ambiguity is invisible since the spin S is an integer or a  half-integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The path integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (41) was derived first by Michael Berry [77] (and extended by Barry  Simon [78]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The first term (which we called Wess-Zumino by analogy with its field theoretic  versions) is called the Berry Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The role of this term, which is first order in time derivatives,  is to govern the quantum dynamics of the spin which, in presence of a uniform magnetic field,  executes a precessional motion of the (Bloch) sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is also apparent from this expression that  in the large-S limit, the path integral can be evaluated by means of a semiclassical approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The coherent-state construction shows that this problem is equivalent to the path integral  of a formally massless non-relativistic particle of unit electric charge on the surface of the unit  sphere with a magnetic monopole of magnetic charge S in its interior!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is not surprising  since the Hilbert space of a non-relativistic particle moving on the surface of a sphere with and  radial magnetic field (the field of a magnetic monopole) has a Landau level type spectrum with  a degeneracy given by the flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The condition of a massless particle means that only the lowest  Landau level survives and all other levels have an infinite energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The coherent state approach has been used to derive a path integral formulation for ferromag-  nets and antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A detailed derivation can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Ferromagnet  We will consider first the simpler case of a quantum ferromagnet and in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (33) we will set  J = −|J| < 0 for nearest neighbors and zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The action for the path-integral for the  spin-S quantum Heisenberg ferromagnet on a hypercubic lattice is  S = S  �  r  SWZ[n(r, t)] − |J|S 2  2  �  ⟨r,r′⟩  � T  0  dt �n(r, t) − n(r′, t)�2  (45)  where we have subtracted the classical ground state energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The oder parameter for this theory  is the expectation value of the local magnetization, n = ⟨n(r)⟩, which is constant in space but  points in an arbitrary direction in spin space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the low energy regime the important configurations are slowly varying in space and we  can simply approximate the action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (45) by its continuum version in d space dimensions  S = S  ad  0  �  ddx SWZ[n(x, t)] − |J|S 2  2ad  0  �  ddx  � T  0  dt (▽n(x, t))2  (46)  where a0 is the lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As before, the path integral;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' is done for a field which satisfies ev-  erywhere in space-time the constraint n2(x, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This action can be regarded as non-relativistic  non-linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is straightforward to show that the classical equations of motion for this theory are the  Landau-Lifshitz equations  ∂tn = |J|S a2  0 n × ▽2n  (47)  subject to the constraint n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Due to the constraint, the Landau-Lifshitz equation is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will a decomposition of the field into a longitudinal and two transverse components, σ and  π, respectively  n =  �σ  π  �  (48)  subject to the constraint σ2 +π2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The linearized Landau-Lifshitz equations become (to linear  order in π)  ∂tπ1 ≃ −|J|S a2  0 ▽2 π2,  ∂tπ2 ≃ +|J|S a2  0 ▽2 π1  (49)  The solution to these equations are ferromagnetic spin waves (magnons or Bloch waves) which  satisfy the dispersion relation  ω(p) ≃ |J|S a2  0p2 + O(p4)  (50)  which shows that the dynamic exponent for a ferromagnet is z = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that in this case the  two transverse components are not independent (they are effectively a dynamical pair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  are the Goldstone bosons of a ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Antiferromagnet  Formally, the quantum antiferromagnet has a coherent state path integral whose action is  S = S  �  r  SWZ[n(r, t)] − JS 2  2  �  ⟨r,r′⟩  � T  0  dt n(r, t) · n(r′, t)  (51)  with J > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a bipartite lattice, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the 1D chain, and the square and cubic lattices, the classi-  cal ground state is an antiferromagnet with a N´eel order parameter, the staggered magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let m(r) be the expectation value of the local magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A bipartite lattice is the union of  two interpenetrating sublattices, and the local magnetization is staggered, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it takes values with  opposite signs (with equal values) on the two sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, we make the change of variables,  n(r, t) → (−1)rn(r, t) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (51) and find  S = S  �  r  (−1)rSWZ[n(r, t)] − JS 2  2  �  ⟨r,r′⟩  � T  0  dt (n(r, t) − n(r′, t))2  (52)  We want to obtain the low energy effective action for the field n(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To this end, we decompose  this field into a slowly varying part, that we will call m(r, t), and a small rapidly varying part  l(r, t) (which represents ferromagnetic fluctuations)  n(r, t) = m(r, t) + (−1)ra0l(r, t)  (53)  16  Since n2(r, t) = 1, we will demand that the slowly varying part also obeys the constraint,  m2(r, t) = 1, and require that the two components be orthogonal to each other, m · l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Due to the behavior of the staggered Wess-Zumino terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (52), the resulting continuum  field theory turns out to have subtle but important differences between one dimension and higher  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will state the results for two and higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will discuss in  detail the one-dimensional below when we discuss the role of topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It turns out that if the dimension d > 1, the contribution of the staggered Wess-Zumino terms  for smooth field configurations is [75, 79, 80]  lim  a0→0 S  �  r  (−1)rSWZ[n(r, t)] = S  �  d3x l(x, t) · m(x, t) × ∂tm(x, t)  (54)  The continuum limit of the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (52) in the case of a two-dimensional system is  lim  a0→0  JS 2  2  �  ⟨r,r′⟩  � T  0  dt �n(r, t) − n(r′, t)�2 = a0  JS 2  2  �  d3x  �  (▽m(x, t))2 + 4l2(x, t)  �  (55)  The massive field l[x, t] represents ferromagnetic fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since this is a massive field it can  be integrated-out leading to an effective field theory for the antiferromagnetic fluctuations m(x, t)  whose Lagrangian is that of a non-linear sigma model  L = 1  2g  � 1  vs  (∂tm(x, t) − vs(▽m(x, t))2  �  (56)  where the coupling constant is g = 2/S and the spin-wave velocity is vs = 4a0JS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we to allow  for a weak next-nearest-neighbor interaction J′ > 0, the coupling constant g and the spin wave  velocity vs become renormalized to g′ ≃ g/ √1 − 2J′/J and v′  s ≃ vs  √1 − 2J′/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that that the quantum fluctuations about a N´eel state are well described by a non-  linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Provided the frustration effects of the next-nearest-neighbor interactions are  weak enough, the long-range antiferromagnetic N´eel order should extend up to a critical value  of the coupling constant gc where the RG beta function has a non-trivial zero, which signals a  quantum phase transition to a strong coupling phase without long-range antiferromagnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Motivated by the discovery of high temperature superconductivity in the strongly correlated  quantum antiferromagnet La2CuO4 (at finite hole doping) in 1988 Chakravarty, Halperin and  Nelson [81] utilized a quantum non-linear sigma model to analyze this system and its quantum  phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' La2CuO4 is a quasi-two-dimensional material and so it exhibits strong quantum  and thermal fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The upshot of this analysis is that while at T = 0 the non-linear sigma  model has a quantum phase transition, at T > 0 the long range order is absent in a strictly 2D  system but present in the actual material due to the weak-three-dimensional interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, in  the strict 2D case there is no phase transition but two different crossover regimes: a renormal-  ized classical regime (without long range order), a quantum disordered regime and a quantum  critical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' La2CuO4 has long range N´eel (antiferromagnetic) order at T = 0 and is in the  renormalized classical regime (with long range order due to the weak 3D interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The non-linear sigma model does not describe the nature of the ground state for g > gc  beyond saying that there is no long range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The problem is that, unlike the Ising model in a  transverse field, the microscopic tuning parameter is the next nearest neighbor antiferromagnetic  coupling J′, and to reach the regime g ≃ gc one has to make J′ ≃ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the regime in which  frustration effects become strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime the assumption that the important configurations  are smooth and close to the classical N´eel state is incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The nature of the ground state turns  out to depend on the value of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological Excitations  Topology has come to play a crucial role both in Condensed Matter Physics and in Quan-  tum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological concepts have been used to classify topological excitations such  as vortices and dislocations and to provide a mechanism for phase transitions, quantum num-  ber fractionalization, tunneling processes in field theories, and nonperturbative construction of  vacuum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will discuss a few representative cases of what has become a very vast  subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological Excitations: Vortices and Magnetic Monopoles  In Condensed Matter Physics topological excitations play a central role in the description of  topological defects and on their role in phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here topology integers in the classifi-  cation of the configuration space into equivalence classes characterized by topological invariants  [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The most studied example are vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vortices play a key role in the mixed phase of type  II superconductors in a uniform magnetic field [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vortices also play a key role in the Statistical  Mechanics of 2D superfluids and the the 2D classical XY model [84, 85] [86, 87] [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Disloca-  tions and disclinations play an analogous role in the theory of classical melting [84, 89, 90], and  2D and 3D classical liquid crystals [91, 92, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A similar problem occurs in Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Theories with global symmetries, such  as the two-dimensional O(3) non-linear sigma model discussed above, when formulated in Eu-  clidean space-time have instantons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Typically instantons are finite Euclidean action configu-  rations, which are also classified into equivalence classes (associated with homotopy groups)  labeled by topological invariants [93, 94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instantons play a central role in understanding the  non-perturbative structure of gauge theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gauge theories with a compact gauge group cou-  pled to matter fields have non-trivial vortex [95] and monopole [96, 97, 98] configurations, as  do non-abelian Yang-Mills gauge theories [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instantons have also played a central role in  Condensed Matter Physics as well, notably in Haldane’s work on 1D quantum antiferromag-  nets (discussed below), and in the problem of macroscopic quantum tunneling and coherence  [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vortices in two dimensions  In this section I will focus on the the problem of the superfluid transition in 2D and the  closely related problem of the phase transition of a magnet with an easy-plane anisotropy, the  classical XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A superfluid is described by an order parameter that is a one-component  complex field φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If electromagnetic fluctuations are ignored, this description also applies  to a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The complex field can be written in terms of an amplitude |φ(x)|, whose  square represents the local superfluid density, and a phase θ(x) = arg(φ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Deep in the  superfluid phase the amplitude is essentially constant, that we will set to be a real positive  number φ0, while the phase field θ(x) is periodic with period 2π and can fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Simi-  larly, an easy-plane ferromagnet is described by a two-component real order parameter field  M(x) = (M1(x), M2(x)) = |M(x)|(cosθ(x), sin θ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Deep in the ferromagnetic phase the ampli-  tude |M| is essentially constant but the phase field θ(x) can fluctuate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will assume that we are in a regime where the local superfluid density |φ0|2 is well formed  (or, equivalently that |M| is locally well formed) but that the phase field is fluctuating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  regime the problem at hand is an O(2) ≃ U(1) non-linear sigma model, and its partition function  takes the form  Z =  �  Dθ exp  �  −  �  d2x 1  2g  �  ∂µθ(x)  �2�  (57)  where we defined the coupling constant g = T/J|φ0|2, where T is the temperature, J is an in-  teraction strength, and |φ0|2 is the magnitude (squared) of the amplitude of the order parameter,  which we will take to be constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' κ = J|φ0|2 is the phase stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Except for the requirement that the phase field be locally periodic, θ ≃ θ + 2π, superficially  this seems to be a trivial free (Gaussian) field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see that the periodicity (or, com-  pactification) condition makes this theory non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, configurations of the phase field  that are weak enough that that do not see the periodicity condition, for all practical purposes, can  regarded as being non-compact and ranging from −∞ to +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However there are many configu-  rations for which the periodicity condition is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such configurations are called vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Even in the absence of vortices, the periodic (compact) nature of the phase field is essential  to the physics of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact the only allowed observables must be invariant under  local periodic shifts of the phase field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This implies that the phase field θ itself is not a physical  observable but that exponentials of the phase of the form exp(inθ(x)) are physical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This operator  is just the order parameter field of the XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Conformal Field theory operators of this  type are called vertex operators [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see below that this theory has a dual field ϑ,  associated with vortices, and that there are vertex operators of the dual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In String Theory the  model of a compactified scalar is known as the compactified boson and represents the coordinate  of a string on a compactified space, in this case a circle S 1 [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To picture a vortex consider a large closed curve C on the 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, topologically  a closed curve is isomorphic to a circle, C ≃ S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The phase field θ(x) is equivalent to a unit  18  circle S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore the configuration space are maps of S 1 (the large circle) onto S 1 (the unit  circle of the order parameter space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The configurations can be classified by the number of times  the phase winds on the large circle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The winding number is an integer called the topological  charge of the configuration, the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, a vortex is a configuration of the phase field θ(x)  that winds by 2πm (where m is an integer):  (∆θ)C  2π  = 1  2π  �  C  dx · ▽θ(x) ≡ i  � 2π  0  dϕ  2π eiθ(ϕ)∂ϕe−iθ(ϕ) = m  (58)  where ϕ ∈ (0, 2π] is the azimuthal angle for a vector at the center of the large circle C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here  n is the vorticity or winding number of the configuration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' m > 0 is a vortex and m < 0 is an  anti-vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The vorticity is a topological invariant of the field the configuration θ(x) which does  not change under smooth changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The winding number of a vortex is a topological invariant that classifies the configurations  of the phase field as continuous maps of a large circle S 1 onto the unit circle S 1 defined by the  phase field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Topology such continuous maps are called homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The winding number  classifies these maps into a discrete set of equivalence classes, which form a homotopy group  under the composition of two configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the homotopy group is called Π1(S 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since the equivalence classes are classified by a topological invariant that takes integer values,  the homotopy group Π1(S 1) is isomorphic to the group of integers, Z [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The field jµ(x) = ∂µθ(x) is the superfluid current, and the vorticity ω(x) is the curl of the  current, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ω(x) = ǫµν∂µ jν(x) = ǫµν∂µ∂νθ(x)  (59)  which vanishes unless θ(x) has a branch-cut singularity across which the phase field jumps by  2πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let ω(x) be the vorticity field with singularities at the locations {x j} of vortices with topo-  logical charge m j  ω(x) = 2π  �  j  m jδ2(x − x j)  (60)  which is satisfied by the phase field configuration  θ(x) =  �  j  2πm jIm ln(z − zj)  (61)  where we have used the complex coordinates z = x1 + ix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Away from the singularities {x j},  this configuration obeys the Laplace equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, it has a Cauchy-Riemann dual field ϑ(x)  which satisfies the Cauchy-Riemnann equation  ∂µϑ = ǫµν∂νθ  (62)  which satisfies the Poisson equation  − ▽2ϑ(x) = ω(x)  (63)  whose solution is  ϑ(x) =  �  d2y G(|x − y|) ω(y)  (64)  where G(|x − y|) is the Green function of the 2D Laplacian  − ▽2G(|x − y|) = δ2(x − y)  (65)  In 2D this Green function is  G(|x − y|) = 1  2π ln  �  a  |x − y|  �  (66)  where a is a short distance cutoff (a lattice spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows we will assume that the  Green function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (66) has been cutoff so that G(|x − y|) = 0 for |x − y| ≤ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The energy of a configuration of vortices {n j} with vanishing total vorticity, �  j m j = 0, is  E[θ] = Jφ2  0  2  �  d2x  �  ∂µθ  �2  = Jφ2  0  2  �  d2x  �  d2y ω(x) G(|x − y|) ω(y) = 2πJφ2  0  �  j>k  m jmk ln  �  a  |x j − xk  �  (67)  19  where we used that configurations with non-vanishing vorticity do not contribute to the partition  function since they have infinite energy in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We conclude that, up to an  unimportant prefactor, that the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (57) is the same as the partition function  a gas of charges {m j} (the vortices) with total vanishing vorticity, �  j m j = 0,  Z2DCG =  �  [{mj}]  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−2π Jφ2  0  T  �  j<k  m jmk ln  �|x j − xk|  a  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (68)  which is known as the two-dimensional (neutral) Coulomb gas [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Kosterlitz and Thouless argued that the neutral two dimensional Coulomb gas has a phase  transition a critical temperature TKT between a low temperature phase dielectric phase in which  vortices and anti-vortices are bound into neutral pairs and a high temperature phase in which  pairs unbind, the vortex charges are screened and the vortex gas is in a plasma phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A simple  estimate of the critical temperature is found by computing the free energy of a single vortex  Fvortex = Evortex − TS vortex, where Evortex of a single vortex and S vortex is the vortex entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  2D the energy of a single vortex of unit topological charge, n = 1, is πJφ2  0 ln(L/a), where L is the  linear size of the system, and the vortex entropy is S vortex = ln(L/a)2, where (L/a)2 is the number  of places where a vortex can be located, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the area of the system in units of the spacing a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Vortices unbind (proliferate) when the entropy beats the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the critical point occurs  at a temperature TKT where the vortex free energy vanishes, Fvortex[TKT] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This yields the  estimate  TKT = π  2 Jφ2  0  (69)  which is the Kosterlitz-Thouless critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is an alternative, equivalent way to see this physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let is consider the theory if  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (57) in the background of a U(1) gauge field Aµ(x)  Z[A] =  �  Dθ exp  �  − 1  2g  �  d2x  �  ∂µθ − Aµ  �2�  (70)  where the curl of the background gauge field Aµ is chosen to be equal to the local vorticity of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (59),  ǫµν∂µAν(x) = ω(x)  (71)  In other words, the gauge Aµ field forces the phase field to have vortices at the locations {x j}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Next we will use a Hubbard-Stratonovichtransformation (a Gaussian identity) to write Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (70)  in the equivalent form  Z[A] =  �  Dθ  �  Daµ exp  �  −g  2  �  d2x a2  µ + i  �  d2x aµ(x)  �  ∂µθ(x) − Aµ(x)  ��  (72)  where we neglected an unimportant prefactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (72) the phase field can be integrated-out  (after an integration by parts) to yield the constraint ∂µaµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This constraint is solved by  introducing the dual field ϑ(x) such that  aµ(x) = ǫµν∂νϑ(x)  (73)  Hence, the partition function Z[A] becomes  Z[A] =  �  Dϑ exp  �  −g  2  �  d2x (∂µϑ)2 + i  �  d2x ϑ(x) ω(x)  �  (74)  where we have assumed that the total vorticity is zero,  �  d2x ω(x) = 2π �  j m j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This identity  show that the quantization of the vortex topological charge requires that the dual field ϑ should  also be periodic (compact) with period 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is elementary to see that after performing the gaussian integral over the field ϑ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (74)  and summing over all vortex configurations (with total vanishing vorticity) we reproduce the  partition function of the 2D neutral Coulomb gas [105, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (70) defined  in terms of the phase field θ and the theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (74) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is an example of a  duality transformation which amounts to the exchange of the fields  θ ↔ ϑ,  and  g ↔ 1/g  (75)  20  A more careful analysis done on the lattice shows that the dual field ϑ is defined on the dual of  the square lattice (which is also a square lattice) [88, 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, by differentiating both  the partition function of the phase field of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (70) and the partition function of the dual field  ϑ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (74) with respect to the background field Aµ we obtain the duality identification of the  currents  1  g∂µθ ←→ gǫµν∂νϑ  (76)  as an operator identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This identity has the same form as the duality of forms in geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this case, the dual of the current ∂µθ, which is a 1-form field, is another 1-form field, ∂µϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In both cases the field θ and its dual ϑ are associated with global symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Later we will  encounter similar duality identifications but their content will be different in different dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Moreover, running these identities backwards it is easy to see that the insertion of the order  parameter operator exp(ipθ(x)) (with p and integer), which acts as a source on the phase field θ,  in the partition function is is equivalent to minimally couple the dual field ϑ to a singular gauge  field Bµ(x) whose curl is p at the location x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, charges are dual to vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now use the identity we derived in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (74) to compute the full partition function  which is a sum over all vortex configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Now we will assume that the vortices have a large  core energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the energy associated with the short distance behavior of the Green function of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(66)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a vortex with unit topological charge we will call this energy u, and for a vortex  with charge m the core energy will be um2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The effects of the core energy can be accounted for  in terms of a vortex fugacity z = exp(−un2/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The total partition function now is  Z =  �  {mj}  �  Dϑ exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−g  2  �  d2x (∂µϑ)2 + i  �  j  2πm jϑ(x j) − u  T  �  j  m2  j  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (77)  where, once again, we assumed global charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the limit in which z = exp(−u/T) ≪ 1, only the vortices with the lowest topological  charges m = ±1 contribute and, furthermore, they are dilute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit we can perform over  the configurations with the lowest topological charge and derive the effective field theory of the  dual field ϑ:  ZSG =  �  Dϑ exp  �  −  �  d2x  �g  2(∂µϑ)2 − � cos(2πϑ)  ��  (78)  which is known as the sine-Gordon field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The coupling constant of the sine-Gordon theory  is � = 2 exp(−u/T)/a2, where a is the core size (the lattice spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the cosine operator  of the sine-Gordon theory represents the effects of the vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can now analyze the role of vortices by looking at the renormalization group of the sine-  Gordon theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The first question is what is the scaling dimension of the cosine operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  can be computed by looking at the vortex operator correlator in the theory with � = 0:  ⟨exp(i2πϑ(x)) exp(−i2πϑ(y))⟩ =  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  |x − y|2π/g  (79)  which implies that the vortex scaling dimension is  ∆vortex = π  g  (80)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (12) implies that in D = 2 dimensions the operator is irrelevant if ∆ > 2, relevant for ∆ < 2  and marginal for ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, vortices are marginal if ∆vortex = 2 which happens if g = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This is the same as to say that the system is at the Kosterlitz-Thouless critical temperature T =  TKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, vortices are irrelevant if T < TKT and relevant for T > TKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This RG analysis tells us that T > TKT, when vortices proliferate, the coupling constant ν  flows to strong coupling to a regime where ϑ is pinned to an integer value and the theory is in a  massive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this phase the connected vortex correlator decays exponentially with distance  which is the same as to say that the vortex charge is screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is why in this phase the  vortices proliferate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It can be shown that in this phase the correlator of the spins of the XY  model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the correlator of the vertex operator exp(iθ(x)), decays exponentially with distance  and the systems is in its disordered phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is still the question of the nature of the phase with T < TKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Mermin-Wagner The-  orem [107] (and its generalizations by Hohenberg [108] and Coleman [109]) states that classical  21  statistical mechanical systems with a global continuous symmetry group cannot undergo spon-  taneous symmetry breaking in space dimensions D ≤ 2 (and quantum systems with space-time  dimensions D ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Does this theory violate this the Mermin-Wagner Theorem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The answer is  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to see that in the phase in which the vortices are irrelevant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' for T < TKT, the  correlator of the order parameter operator is always a power law of the distance |x − y|,  ⟨exp(iθ(x)) exp(−iθ(y))⟩ =  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  |x − y|g/2π  (81)  with an exponent that depends on temperature and satisfies  g  2π =  T  2πJφ2  0  ≤ 1  4  (82)  In other words, the entire low temperature phase is not an ordered phase of matter since the cor-  relator is not constant at long distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, the correlators of the order parameter  exp(iθ) and of the vortices exp(iϑ) have a power la behavior for T < TKT, we conclude that in  this temperature range the system is scale invariant and that it has a line of critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In summary, we succeeded in expressing the partition function as a sum over configurations  of the topological excitations, the vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We succeeded in doing that because vortices are  labeled by their coordinates on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, we found a non-trivial phase transition  since the entropy and the energy both scale logarithmically with the linear size of the system  or, equivalently, that vortices became marginally relevant at a critical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the  mechanism behind the Kosterlitz-Thouless transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  One may ask if this construction is generic and the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As an example consider the  Abelian Higgs model in D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This model has a complex scalar field minimally coupled to a  Maxwell gauge field and, hence, its gauge group is U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the classical spontaneously broken  phase, the gauge field becomes massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this phase the long range coherence of the phase field  of the vortices of the scalar field is screened at the scale of the penetration depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As a result  the Euclidean action of the vortices is now finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, on longer scales the interaction  energy between vortices becomes short ranged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, instead of a 2D Coulomb gas now one  has a gas of particles (vortices and anti-vortices) with short range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the  entropy always dominates and the vortices proliferate [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This behavior is quite analogous to  the restoration of symmetry by proliferation of domain walls in one-dimensional classical spin  chains [19] and to the analogous problem of tunneling in the path integral formulation of quantum  mechanical double-well potentials [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As a caveat, we should note that, in spite of the obvious  similarities, the 2D Abelian Higgs model does not describe correctly a 2D superconductor (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a  superconducting film) since the electromagnetic field is not confined to the film and this renders  the electromagnetic action nonlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Magnetic monopoles in compact electrodynamics  Instantons are of great interest in Quantum Field Theory since they provide for a mechanism  to understand the non-perturbative structure of these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For this reasons they have been  used to understand the mechanisms of quark confinement and the role of quantum anomalies  in non-Abelian gauge theories [99, 97, 98, 96, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instantons in non-Abelian gauge theories  are magnetic monopoles and the condensation of monopoles have long been argued to be the  mechanism behind quark confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The simplest non-Abelian gauge theory that has magnetic monopoles is the Georgi-Glashow  [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This model has a three-component real field φ and an SU(2) Yang-Mills gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  its Higgs phase the scalar field φ acquires an expectation value which breaks the gauge symmetry  group SU(2) down to its diagonal U(1) subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since U(1) ⊂ SU(2), this Abelian gauge group  is compact, meaning that its magnetic fluxes are quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov [113] and ’t Hooft [111]  showed that in 2+1 Euclidean dimensions have non-singular instanton solutions which at long  distances resemble the magnetic monopole originally proposed by Dirac in 1931 [114]  Bi(x) = q  2  xi  |x|2 − 2πqδi,3δ(x1)δ(x2)θ(−x3)  (83)  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (83) is the magnetic radial field of a monopole of magnetic charge q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  second term represents an infinitely long infinitesimally thin solenoid ending at the location of  the monopole, x = 0, that supplies the quantized magnetic flux 2πq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This singular term is  known as the Dirac string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The string itself (and its orientation) is physically unobservable to any  22  electrically charged particle that obeys the Dirac quantization condition, qe = 2π (in units where  ℏ = c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the language of a lattice gauge theory [32], a theory with a compact (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' periodic) U(1)  gauge fields on a D = 3 cubic lattice, describing a compact gauge field in 2 + 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This theory should have instantons that resemble magnetic monopoles much in the same way as  a theory with a compact global U(1) symmetry has vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The simplest example is Polyakov’s  compact electrodynamics [97] whose partition function is  Z =  �  x,µ  � 2π  0  dAµ(x)  2π  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  1  4e2  �  x,µ,ν  cos(Fµν(x))  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (84)  where Fµν(x) = ∆µAν(x)−∆νAµ(x) ≡ �  µ Aµ is the magnetic flux through the elementary plaquette  labeled by a site x and a pair of directions, µ and ν, with µ = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This theory is invariant under  local gauge transformations Aµ(x) → Aµ(x) + ∆µΦ(x) and it is also invariant under local periodic  shifts of the gauge fields Aµ(x) → Aµ(x) + 2πℓµ(x), where ℓµ(x) ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The plaquette flux operator  satisfies the lattice version of the Bianchi identity that the product of exponentials of the flux on  the faces of every elementary cube of the lattice is  �  cubefaces  eiFµν(x) = 1  (85)  which says that the theory can have magnetic monopoles of integer magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will analyze this theory following an approach analogous to what we used for vortices in  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='To this end we will consider the partition function  Z[Bµν] =  �  DAµ exp  �  − 1  4e2  �  d3x  �  Fµν(x) − Bµν(x)  �2�  (86)  where Fµν(x) = ∂µAν − ∂νAµ is the field strength of the abelian U(1) Maxwell gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Here Bµν(x) is an (anti-symmetric) two-form background gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The coupling constant of  this theory is e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since Aµ is a connection it has units of length−1, and F2  µν is a dimension 4 field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Then, in D = 3 dimensions, e2 has units of length−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The theory is invariant under two local transformations, namely the usual invariance under  gauge transformations  Aµ(x) → Aµ(x) + ∂µΦ(x),  Bµν(x) → Bµν(x)  (87)  where Φ(x) is an arbitrary smooth function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The presence of the background two-form field  Bµν now requires invariance under one-form gauge transformations  Aµ(x) → Aµ(x) + aµ(x),  Bµν(x) → Bµν(x) + ∂µaν − ∂νaµ  (88)  The two-form gauge field Bµν essentially represents the magnetic monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let {m j} be a  configuration of monopoles of charges m j with coordinates {x j}, with total vanishing monopole  charge, �  j m j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let M(x) be the magnetic monopole density at x,  M(x) = 2π  �  j  m j δ3(x − x j)  (89)  which can be expressed as the curl of the two-form gauge field Bµν,  M(x) = 1  2ǫµνλ∂µBνλ(x)  (90)  We will proceed next much in the same way as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (72) and rewrite the partition function  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (86) in terms of a two-form Hubbard-Stratonovich field bµν(x) such that  Z[B] =  �  DAµ  �  Dbµν exp  �  −e2  4  �  d3x b2  µν(x) + i  �  d3x 1  2bµν(x)  �  Fµν(x) − Bµν(x)  ��  =  �  DAµ  �  Dbµν exp  �  −e2  4  �  d3x b2  µν(x) + i  �  d3x  �  Aµ(x)∂νbµν(x) − 1  2bµν(x)Bµν(x)  ��  (91)  23  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the gauge field Aµ plays the role of a Lagrange multiplier field the enforces the constraint  ∂νbµν(x) = 0  (92)  which is solved in terms of a compact scalar field ϑ(x)  bµν(x) = ǫµνλ∂λϑ(x)  (93)  Using this identity and the definition of the monopole density M(x) we find that the partition  function Z[Bµν] of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (86) becomes  Z[B] =  �  Dϑ exp  �  −  �  d3x  �e2  2 (∂µϑ(x))2 + iM(x)ϑ(x)  ��  =  �  Dϑ exp  \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0−e2  2  �  d3x(∂µϑ(x))2 + 2πi  �  j  m jϑ(x j)  \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  (94)  which requires that the field ϑ obeys the compactification condition ϑ → ϑ + n, where n is an  arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (94) says that the magnetic monopole instantons of the compact U(1) gauge  theory are dual to charges of the dual phase field ϑ, which has a compact U(1) global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The full partition function is obtained by summing over all monopole configurations satis-  fying the total neutrality condition, �  j m j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As in section section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1, we will weigh the  configurations with a coupling u and find  Z =  �  {mj}  Z{m j}  �  Dϑ exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−e2  2  �  d3x (∂µϑ)2 +  �  j  2πim jϑ(x j) − u  �  j  m2  j  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (95)  which is the same theory we found in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (77) except that now we are in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, summing  only over dilute configurations of monopoles and anti-monopoles we find, once again the sine-  Gordon theory but now in D = 3 dimensions:  Z =  �  Dϑ exp  �  −  �  d3x  �e2  2 (∂µϑ)2 − � cos(2πϑ)  ��  (96)  with � = 2 exp(−u)/a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In spite of the similarities between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (96) and the sine-Gordon theory in 2D, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (78), the  physics is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is straightforward to see that, in the limit v = 0, the monopole operator  correlator is  ⟨exp �2πiϑ(x)) exp(−2πiϑ(y)� = exp  �4π2  e2 [G(|x − y|) − G(0)]  �  ≃ exp  � π  2e2  � 1  R − 1  a  ��  (97)  where a is the short-distance cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Unlike the behavior of the correlator of the vortex operators  in 2D found in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (79), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (97) does not show a power-law behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The reason is that at � = 0  the compactified field ϑ should be regarded as the Goldstone boson of a spontaneously broken  U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the cosine operator is now always relevant and the field ϑ is pinned  and its fluctuations are actually massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Looking back at the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (95), we could integrate out the field ϑ and  obtain an expression with the same form as the Coulomb gas of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (68) except that now this is  the three-dimensional neutral Coulomb gas [97]  Z3DCG =  �  [{mj}]  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed− π  2e2  �  j<k  m jmk  �  1  |x j − xk| − 1  a  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (98)  Now the energy of an isolated monopole is finite (in the infrared) while the entropy is still log-  arithmically divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The conclusion is that the monopoles proliferate and the 3D neutral  Coulomb gas is always in a plasma phase withe screened interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These results also imply  that the expectation value of the Wilson loop operator in the U(1) gauge theory decays exponen-  tially with the area of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The conclusion is that in 2+1 dimensions compact electrody-  namics is in a confining phase for all values of the coupling constant e2 [97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Non-Linear Sigma Models and Antiferromagnetic Quantum Spin Chains  We will now discuss the one-dimensional case in which topology plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In one  space dimension, the sites of the chain are labelled by an integer j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', N (with N even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  After some simple manipulations, the continuum limit of the staggered Wess-Zumino terms of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (52) is  lim  a0→0 S  N  �  j=1  (−1) jSWZ[n(j, t)] =S  2  �  d2x m(x, t) · ∂tm(x, t) × ∂xm(x, t)  + S  �  d2x l(x, t) · m(x, t) × ∂tm(x, t)  (99)  The continuum limit of the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (52) is essentially the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  By putting together the results of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (99) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (55),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we find that in the case of a 1D chain  the continuum field theory has the Lagrangian density  L[m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' l] = −2a0JS 2l2 + S l · m × ∂tm − a0JS 2  2  (∂xm)2 + S  2 m · ∂tm × ∂xm  (100)  Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we integrate-out the ferromagnetic fluctuations represented by the massive field l(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' t) to  find that the effective low-energy Lagrangian has the form  L = 1  2g  � 1  vs  (∂tm)2 − vs (∂xm)2  �  + θ  4π m · ∂xm × ∂tm  (101)  which is a non-linear sigma model with N = 3 components (and hence has a global O(3) sym-  metry) with a topological term whose coupling constant is θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result was first derived by  Haldane [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (101) the effective coupling constant g, the spin-wave velocity vs, and the θ-angle are  given by g = 2  S , vs = 2a0JS , and θ = 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As in the 2D case, since g = 2/S , large S implies that  the non-linear sigma model is in the weak coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Provided that vs > 0, one can always  rescale the time and space coordinates without affecting the form of the Lagrangian, including  the coupling constant or, as we will see, the value of the θ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows we will assume  that we have done the rescaling in such a way that we set vs = 1 and, that time and space scale as  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus we will use a relativistic notation and, after an analytic continuation to imaginary  time, we label the time coordinate by x2 and the space direction by x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The partition function  now takes the form  Z =  �  Dmexp  �  − 1  2g  �  Ω  d2x  �  ∂µn  �2 + iθ Q[m]  �  (102)  where Ω is the spacetime manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this notation the first term of the exponent is called the  Euclidean action of the non-linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (102) we denoted by Q[m] the quantity  Q[m] = 1  8π  �  Ω  d2x ǫµνm · ∂µm × ∂νm  (103)  Here we will consider the case in which the spacetime manifold Ω is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular  we will assume that it is a two-sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The quantity Q[m] is the integral of a total derivative  which counts the number of times the field configuration m(x1, x2) wraps around the sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In other words, it yields a non-vanishing result only for “large” configurations which wind (or  wrap) around the sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since Q[n] is an integer, it has the same for all smooth the field  configuration m that can be smoothly deformed into each other and are homotopically equiva-  lent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we demand that the field configurations m have finite Euclidean action, which requires  that at infinity the configurations take the same (but arbitrary) value of m, we have effectively  compactified the x1 − x2 plane into a two sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand the field m is restricted  by the constraint m2 = 1 to take values on a two-sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, the field configurations  m(x1, x2) are smooth maps of the S 2 of the coordinate space to the S 2 of the target space of the  field m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the integer Q[m] classifies the smooth field configurations into a set of equiva-  lence classes each labeled by the integer Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Under composition homotopies form groups, and the  equivalence classes themselves also form a group which, in this case, is isomorphic to the group  of integers, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Topology, these statements are summarized by the notation Π2(S 2) ≃ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  25  configurations with non-zero values of Q are called instantons which, in the quantum problem,  represent tunneling processes of the non-linear sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Another consequence of Q being an integer is that the contribution of its term to the weight of  the path integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (103) is a periodic function of θ angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, since the only  allowed values of the θ are θ = 0 (mod 2π) for S integer, or θ = π (mod 2π) for S a half-integer,  the contribution of the topological invariant Q to the weight of the path integral is  exp(iθ Q[m]) = (−1)2S Q  (104)  Therefore, for spin chains with S integer, the weight is 1 and the topological invariant does not  contribute to the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But, if S is a half-integer, the weight is (−1)Q[m], and it does  contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, its contribution is the same for all half-integer values of S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These results have important consequences for the physics of spin chains which led Haldane  to some startling conclusions [115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the weak coupling regime, g ≪ 1 (equivalently, for  large S ), we can use the perturbative renormalization group and derive the beta function for all  these O(3) non-linear sigma models (with or without topological terms) and find that their beta  functions are the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (21) with N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, for all S , the effective coupling flows  to large values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can make this inference since the topological term yields no contribution for  all configurations which are related by smooth deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, we infer that all spin chains  with S integer are in a massive phase with an exponentially small energy gap ∼ exp(−2πS ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  result is nowadays known as the Haldane gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand, these results also imply that all spin chain with half-integer spin S are  also the same and, in particular, th same as spin-1/2 chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the Hamiltonian of the  quantum spin chain with spin-1/2 degrees of freedom is an example of an integrable system and  its spectrum is known to be gapless from its Bethe Ansatz solution [116, 117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the  spin-1/2 chain is not only gapless but its low energy states are gapless solitons with a relativistic  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The low energy description of a theory of this type must be described by a conformal  field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane concluded that the RG flow for spin-1/2 chains must have an IR stable fixed  point at some finite (and large) value of the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This conjecture was confirmed by  Affleck and Haldane [118] who showed that the spin-1/2 Heisenberg chains are in the universality  class of the SU(2)1 Wess-Zumino-Witten model [119] whose CFT was solved by Knizhnik and  Zamolodchikov [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topology and open integer-spin chains  In the preceding section we showed that integer spin chains have a Haldane gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We did that  by showing that in that case the topological term is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the derivation is correct  provided the spacetime manifold is closed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a sphere, a torus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' What happens if the system  has a boundary?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us denote the boundary of Ω by Γ = ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example we will take Γ to be  along the imaginary time direction and hence that it is a circle of circumference 1/T, where T  is the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The topological term has the same form as the Berry phase term of the path  integral for spin, the “Wess-Zumino” term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (44), but its prefactor is 1/2 as big.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, for  a system with an open boundary the topological term yields a net contribution equal to a Berry  phase with a net prefactor of S/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In other words, the boundary of the integer spin chain behaves as a localized degree of free-  dom whose spin is 1/2 (mod an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Form the periodicity requirement we also see that if  the spin chain is made of odd-integer degrees of freedom, there should be a spin 1/2 degree of  freedom localized at the open boundary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='. Conversely, if the chain is made of even-integer spins,  there is no boundary degree of freedom!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This line of argument implies that an antiferromagnetic  chain with odd-integer spins must have a spin-1/2 degree of freedom at the boundary whereas  a chain of even-integer spins should not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that the “bulk” behavior is the same for both  odd and even integer spin chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The difference is whether or not they have a non-trivial “zero-  mode” state at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, the existence of this state is robust, i,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it cannot  be removed by making smooth changes to the quantum Hamiltonian or, what is the same, the  boundary state is topologically protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A simple system that displays a protected spin-1/2 zero mode at the boundary is a generalized  S = 1 spin chain with Hamiltonian  H = α  N  �  j=1  S(j) · S(j + 1) + β  N  �  j=1  (S(j) · S(j + 1))2  (105)  26  where S(j) = (S x(j), S y(j), S z(j) are the spin 1 matrices at each lattice site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem was  examined in great detail by Affleck, Kennedy, Lieb and Tasaki [121] who showed that at the  special value of the parameters α = 1/2 and β = 1/6 this Hamiltonian takes the form of a sum of  projection operators  H =  �  j  P2(S(j) + S(j + 1))  (106)  where P2 is an operator that projects out the spin 2 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These authors constructed the exact  ground state, known as the AKLT state, of this Hamiltonian by writing each spin 1 degree of  freedom of two spin-1/2 degrees of freedom at each site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They showed that the ground state is a  projected product state in which the “constituent’ spin-1/2 degrees of freedom (each labeled by  + and − respectively) on nearby sites j and j + 1 are in a valence bond singlet state of the form  1√  2(| ↑j,+, ↓j+1,−⟩− ↓j,+, ↑j+1,−⟩ (and symmetrizing at each site to project onto a spin 1 state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This state of the spin-1 chain is translation invariant and gapped and hence agrees with Haldane’s  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, it has a spin-1/2 degree of freedom at each open boundary [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The arguments discussed above show that the AKLT state is a gapped topological state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  gapless spin-1/2 boundary states are an example of spin fractionalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The spin-1/2 boundary  degrees of freedom are an example of an edge state which are present in many, thought not all,  topological phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One may ask what symmetry protects the gaplessness of these edge  degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only perturbation that would give a finite energy gap to the spin-1/2  edge states is an external magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, this perturbation would break the global  SU(2) symmetry of the Hamiltonian as well as time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Duality in Ising Models  Duality plays a significant role of our understanding of statistical physics and of quantum  field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many seemingly unrelated correspondences between different theories have come  to be called dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Duality in the 2D Ising Model  One of the earliest versions of duality transformations was used to relate the high-temperature  expansion of the 2D classical Ising model and its low-temperature expansion [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In all dimen-  sions, for concreteness we will think of a hypercubic lattice, the high temperature expansion is a  representation of the partition function as a sum of contributions of closed loops on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At  temperature T, a configuration of loops γ contributes with a weight which in the Ising model has  the form C(γ) tanhL(γ)(β), where β = 1/T and L(γ) is the length of the loop γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the number  of links on the loop, and C(γ) is an entropic factor that counts the number of allowed loops with  fixed perimeter L(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However not all loop configurations are allowed as in the Ising model they  satisfy constraints such as being non-overlapping, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 we noted that the partition function of the classical Ising model (in any dimen-  sion) can be interpreted as the path-integral of a quantum spin model in one dimension less on a  lattice with a discretized imaginary time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this picture, the loops γ can be regarded as processes  in which pairs of particles are created at some initial (imaginary) time, evolve and eventually are  annihilated at a later (imaginary) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the high temperature phase is a theory of  a massive scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The restrictions on the allowed loop configurations represents interactions  among these particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the temperature range in which the expansion in loops is convergent  the particles are massive as the loops are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As the radius of convergence of the expansion is  approached, longer and increasingly fractal-like loops begin to dominate the partition function,  and concomitantly the mass of the particles decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This process is the signal of the approach  to a continuous phase transition where the particles become massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, right at the critical  point the particle interpretation is lost as the associated fields acquire anomalous dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Returning to the 2D classical Ising model, Kramers and Wannier also considered the low  temperature expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is an expansion around one of the broken symmetry state, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  the state with all spins up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this low temperature regime the partition function is a sum of  configurations of flipped spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A typical configuration is a set of clusters of flipped spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  absence of a uniform field, a configuration of flipped spins has a energy cost only on links of the  lattice with oppositely aligned spins (“broken bonds”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus a cluster of flipped spins costs an  energy equal to 2 (I assumed that I set J = 1) for each broken bond at the boundary of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This boundary is domain wall which is a closed loop on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2D the dual of the  square lattice is the square lattice of the dual sites (the centers of the elementary plaquettes of the  27  2D lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2D the links of the direct lattice pierce the links of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can see  that this analogous to the the geometric duality of forms: sites (“0-forms”) are dual to plaquettes  (“2-forms”) and links (“1-forms”) are dual to links (also “1-forms”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Thus, in 2D the low temperature expansion is an expansion in the loops γ∗ of the dual lattice  that represent the domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can also regard the domain walls (the dual loops) as the  histories of of pairs of particles on he dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the weight of each dual loop is  exp(−2β) per link of γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Except for that, the counting and restrictions on the dual loops γ∗  are the same as those of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that there is a one-to-one correspondence between the  two expansions with the replacement tanh β ↔ exp(−2β∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This mapping means that the dual  of the 2D classical Ising model (with global Z2 symmetry) at inverse temperature β is a dual  Ising model (also with global Z2 symmetry) on the dual lattice at inverse temperature β∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  correspondence is in close analogy with what we discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In particular if one assumes that there is a transition at βc = 1/Tc then there should also be a  transition at β∗  c = −1/2 lntanh βc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, if one further assumes (as Kramers and Wannier did  [123]) that there is a unique transition (correct in the Ising model but not in other cases), then the  critical point must be such that exp(−2βc) = tanh βc, which yields the value Tc = 2/ ln(  √  2 + 1),  which agrees with the Onsager result [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 we showed that the classical 2D Ising model is equivalent to the one-dimensional  Ising model in a transverse field whose Hamiltonian is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The 1D Ising model on  a transverse field has spin degrees of freedom defined on the sites of a one-dimensional chain  labeled by an integer-valued variable n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The 1D Hamiltonian is expressed in terms of local op-  erators, the Pauli matrices σ3(n) and σ1(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The 1D chain has a dual lattice whose sites are the  midpoints of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus in 1D sites (“0-forms”) are dual to links (“1-forms”) and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now see that there is a Hamiltonian version of the Kramers-Wannier duality [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (26) we introduced the kink creation operator which flips are σ3 operators to the left and  including site j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For clarity we will denote the kink creation operator operator as τ3(˜n), where  ˜n is the site of the dual lattice between the sites n and n + 1 of the original lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now  define an operator τ1(˜n) = σ3(n)σ3(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The operators τ1(˜n) and τ3(˜n) satisfy the same algebra  os the Pauli operators σ1(n) and σ3(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, we readily find the Hamiltonian of the dual  theory is the same (up to boundary conditions) as the original Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (23) except  that the dual coupling constant is λ∗ = 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, once again, if we assume that there is a single  (quantum) phase transition we require λ∗  c = λc which is only satisfied by λc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this language  the kink creation operator plays the role of a disorder operator [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The 3D duality: Z2 gauge theory  We will now discuss the role of duality in the 3D Ising model on a cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This case,  and its generalizations to higher dimensions, was considered first by Franz Wegner [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 we discussed the loop representation of the high temperature expansion and  showed that it has the form form in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, in 3D the high temperature expansion  is an expansion in closed loops γ with weight of tanh β per unit link of loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, in 3D as  well, the high temperature phase can be regarded as field theory of a massive scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As in  the 2D case, the weight of a loop configuration is tanh β for each link of the loop γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, the low temperature expansion has a radically different form and physical inter-  pretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Much as in 2D, the low temperature expansion is an expansion in clusters of flipped  spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here too, in the absence of an external field, the only energy cost resides at the boundary  of the clusters of overturned spins, the domain walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But in 3D the clusters occupy volumes  whose boundaries are closed surfaces Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 3D a link (bond) is dual to a plaquette (expressing  the fact that in 3D a 1-form is dual to a 2-form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence the closed domain walls of overturned  spins is dual to a closed surface Σ∗ on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The weight of each configuration of closed  surfaces is exp(−2β) for each plaquette of the surface Σ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These facts mean that the dual of the 3D Ising model is a theory on the dual lattice with  coupling constant β∗, with exp(−2β∗) = tanh β, such that its expansion for small β∗ is a sum over  configurations of closed surfaces σ∗, and for large β∗ is a sum of configurations of closed loops  γ∗ on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The dual theory is the naturally defined on (dual) plaquettes, not on links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To this end, let us define a set of Ising-like degrees of freedom σµ(x) = ±1 located on the links  (x, µ) of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These degrees of freedom are coupled on each plaquette of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since each plaquette has four links, the coupling involves the degrees of freedom on all four links  28  of each plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The partition function of the dual theory is  Z =  �  {σµ(x)}  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8edβ∗  �  plaquettes  σµ(x)σν(x + eµ)σµ(x + eµ + eν)σν(x)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (107)  where the sum in th exponent (the negative of the “action”) runs on all the plaquettes of the dual  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The action of the theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (107) is invariant under the reversal of all six Ising degrees of  freedom on links sharing a given site x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a local symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Unlike the 3D Ising model,  which has a global Z2 symmetry of flipping all spins simultaneously, this theory has local (or  gauge) Z2 symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the simplest example of a lattice gauge theory in which the degrees  of freedom are gauge fields that take values on the Z2 gauge group [124, 32, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We saw that in the spin model the high temperature expansion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the expansion in powers of  tanh β) is a sum over loop configurations which can be interpreted in terms of processes in which  pairs of particles are created, evolve (in imaginary time) and then are destroyed (also in pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The analogous interpretation of the expansion of the Z2 gauge theory in powers of tanh β∗ =  exp(−2β) as a sum over the configurations of closed surfaces is not in terms of the histories of  particles but in terms of the histories of closed strings which, as they evolve, sweep the closed  surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The physics is, however, more complex as the sum over surfaces runs over all surfaces  of arbitrary topology with arbitrary number of handles (or genus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, over (imaginary) time a  closed strong is created, evolves, splits into two closed strings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore, the 3D Ising model, which has a Z2 global symmetry is dual to a Z2 gauge theory  which has a Z2 local symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The duality maps the high temperature (disordered) phase of  the Ising model to the strong coupling (small β∗) phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the gauge theory  language this is the confining phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This can be seen by computing the expectation value of the  Wilson loop operator on the closed loop Γ, which here reads ⟨�  (x,µ)∈Γ σµ(x)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the small β∗  phase this expectation value decays exponentially with the size of the minimal surface bounded  by the loop Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This behavior of the area law of the Wilson loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Under duality the insertion of  this operator is equivalent to an Ising model with a domain wall terminating on the loop Γ, and  the area law of the Wilson loop is the consequence of the fact that if the Ising model has long  range order, the free energy cost of the domain wall scales with its area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, in this phase  of the gauge theory the closed strings are small meaning that the string tension (the energy per  unit length of string) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the Ising model language, the string tension becomes surface  tension of the domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the preceding subsection we discussed a Hamiltonian version of the duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will briefly  do the same in the case of the 2+1 dimensional Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The hamiltonian of the Ising model  in a transverse field on a square lattice is  H2D−TFIM = −  �  r  σ1(r) − λ  �  r, j=1,2  σ3(r)σ3(r + e j)  (108)  where r labels the sites of the square lattice and e j (with j = 1, 2) are the two (orthonormal)  primitive unit vectors of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here too, at each site we have a two-level system (the spins),  σ1 and σ3 are Pauli matrices acting on these states at each site and λ is the coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Just as in the 1D case, this system has two phases: an ordered phase for λ > λc and a disordered  phase for λ < λc, where λc is a critical coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This Hamiltonian is invariant under the global  Z2 symmetry generated by the global spin flip operator Q = �  r σ1(r) which commutes with the  Hamiltonian, [Q, H2D−TFIM] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let is consider now a Z2 gauge theory on the dual of the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now define  a two-dimensional Hilbert space on each link, denoted by (˜r, j) (with j = 1, 2) of the dual lattice  with sites labelled by ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will further define at each link (˜r, j) the Pauli operators σ1  j(˜r) and  σ3  j(˜r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hamiltonian of the Z2 gauge theory is  HZ2gauge = −  �  ˜r, j  σ1  j(˜r) − g  �  ˜r  σ3  1(˜r)σ3  2(˜r + ˜e1)σ3  1(˜r + ˜e1 + ˜e2)σ3  2(˜r)  (109)  where ˜e j = ǫjkek (with j, k = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Unlike the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (108), the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (109) has a local symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let  ˜Q(˜r) be the operator that flips the Z2 gauge degrees of freedom on the four links that share the  site ˜r,  ˜Q(˜r) =  �  j=1,2  σ1  j(˜r)σ1  j(˜r − ˜e j)  (110)  29  These operators commute with each other, [ ˜Q(˜r), ˜Q(˜r′)] = 0, and commute with the Hamiltonian,  [HZ2gauge, ˜Q(˜r)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hilbert space of gauge-invariant states are eigenstates of ˜Q(r) with unit  eigenvalue,  ˜Q(r)|Phys⟩ = |Phys⟩  (111)  (for all r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This constraint is the the Z2 Gauss Law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hamiltonian HZ2gauge has two phases: a  confining phase for g < gc, and a deconfined phase for g > gc (where gc is a critical coupling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The duality transformation is defined so that  σ1(r) =  �  plaquette(r)  σ3  j(˜r)  (112)  is the product of the σ3  j operators of the gauge theory on a dual plaquette centered at the site r,  and  σ1  1(˜r) = σ3(r)σ3(r + e2),  σ1  2(˜r) = σ3(r)σ3(r + e1)  (113)  These identities imply that the gauge theory constraint of the Z2 Gauss Law, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (111), is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This also means that duality is a mapping of the gauge-invariant sector of the Z2 gauge theory  to the Hilbert space of the Ising model in a transverse field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (112) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (113) imply that the Hamiltonians H2D−TFIM and HZ2gauge are equal to each  other with the identification of the coupling constants g = 1/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the ordered phase of the  Ising model, λ > λc, maps onto the confining phase of the Z2 gauge theory and, conversely, the  disordered phase of the Ising model maps onto the deconfined phase of the gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (113) also implies that the operator σ3(r) of the Ising model can be identified with the  operator  σ3(r) =  �  γ(r)  σ1  j(˜r)  (114)  where the product is on the links of the dual lattice pierced by the path γ(r) on the direct lattice  ending at the site r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  While σ3(r) is of course just the order parameter of the Ising model, the dual operator defined  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (114) anti-commutes with the plaquette term of the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (109) and creates an  Z2 flux excitation at the plaquette.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With some abuse of language, this operator can be regarded as  creating a Z2 “magnetic monopole”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since in the confining phase this operator has an expectation  value, we can regard this phase as a magnetic condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now discuss the role of boundary conditions which is different in both theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let  us examine the behavior of the Z2 gauge theory with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Periodic  boundary conditions means that the 2D space is topologically a 2-torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is straightforward to  show that the ground state in the confining phase is unique and insensitive to boundary condi-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The physics of the deconfined phase is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now see that on a 2-torus  it has a four-fold degenerate ground state and that the degeneracy is not due to the spontaneous  breaking of a global symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let γ1 and γ2 be two non-contractible loops on the torus along the directions 1 and 2 re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider the (“electric”) Wilson loop operators along γ1 and γ2, W[γ1] =  �  (r, j)∈γ1 σ3  j(r) and similarly for γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly let us consider the (“magnetic”) ’t Hooft operators  ˜W[˜γ1] and ˜W[γ2] defined on the non-contractible closed paths of the dual lattice ˜γ1 and ˜γ2, such  that ˜W[˜γ1] = �  ˜γ1 σ1  2(˜r) and ˜W[˜γ2] = �  ˜γ2 σ1  1(˜r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to see that the Wilson and’t Hooft  operators satisfy that W[γi]2 = ˜W[˜γ j]2 = I, and that  [W[γi], W[γ j] = [ ˜W[˜γi], ˜W[˜γ j]] = 0,  {W[γ1], ˜W[˜γ2]} = {W[γ2], ˜W[˜γ1]} = 0  (115)  Let us consider the special limit of g → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit the Wilson loop operators W[γi] on  the two non-contractibleloops of the 2-torus commute with the Hamiltonian HZ2gauge of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (109).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore the eigenstates of the Hamiltonian can be chosen to be the eigenstates of these two Wil-  son loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the Wilson loop operators are hermitian and obey W[γi]2 = I, the spectrum of  each loop is two-dimensional | ± 1⟩i (i = 1, 2) with eigenvalues ±1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At g → ∞ these  states are also eigenstates of HZ2gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the ground state of HZ2gauge is four dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  ’t Hooft loops ˜W[˜γi] act as ladder operators in this restricted Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The conclusion is that, on a 2-torus, at least in the limit g → ∞, the ground state of HZ2gauge  is degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, this degeneracy is not due to the spontaneous breaking of a global  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rather, it reflects the topological character of the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this, one can extend  this analysis to a theory to be on a more general two-dimensional and instead of a 2-torus we can  30  consider a surface with g handles (not to be confused with the coupling constant!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' g = 0 for  a sphere (or disk), g = 1 for a 2-torus, g = 2 for a pretzel, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For each of these two-dimensional  surfaces the number different number of non-contractible loops is g, and the Wilson loops defined  on them commute with each other (and with HZ2gauge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, in the limit g → ∞ HZ2gauge has  a ground state degeneracy of 2g which, clearly, depends only on the topology of the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We  conclude that, at least as g → ∞, the Hamiltonian HZ2gauge is in a topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, is the exact degeneracy found in the limit g → ∞ a property of the entire deconfined  phase?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a finite system the expansion in powers of 1/g is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, in  the thermodynamic limit, L → ∞, the expansion has a finite radius of convergence with gc being  an upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider the ground state at g = ∞ with eigenvalue +1 for the Wilson  loop W[γ1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The degenerate state with eigenvalue −1 is created by the ’t Hooft loop ˜W[˜γ2], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  | − 1⟩1 = ˜W[˜γ2] | + 1⟩1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The ’t Hooft operator is a product of σ1 operators on links along the  direction 1 crossed by the path ˜γ2 of the dual lattice, which involves (at least) L links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, it  takes L orders in the expansion in powers of 1/g to mix the state | + 1⟩1 with the state | − 1⟩1,  and this amplitude is of order 1/gL, which is exponentially small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The same argument applies to  the mixing between all four states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a finite but large system of linear size L, the degeneracy  is lifted but the energy splitting is exponentially small in the system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the topological  protection is a feature of the deconfined phase in the thermodynamic limit L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that the deconfined phase of the Z2 lattice gauge theory is a topological phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This result extends to the case of a gauge theory with discrete gauge group Z, the cyclic group  of k elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general spacetime dimension D > 2 the Zk gauge theory also has a confined  and a deconfined phase and if k ≳ 4 it also has an intermediate “Coulomb phase”[125, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 we will see that the low-energy (IR) regime of the deconfined phase is described by  a topological quantum field theory known as the level k BF theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization  The behavior of one-dimensional electronic systems is of great interest in Condensed Matter  Physics for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One is that, even for infinitesimally weak interactions, these one-  dimensional metals violate the basic principles of the Landau Theory of the Fermi Liquid [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A central assumption of Femi Liquid theory is that at low energies the excitation energy of the  fermion (electron) quasiparticle is always much larger than its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, at asymptotically  low energies the quasiparticle excitations become increasingly sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A manifestation of this feature is that the quasiparticle propagator (the “Green function”) has  a pole on the real frequency axis with a finite residue Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In one-dimensional metals this assump-  tion always fails since the residue Z vanishes, the Fermi field acquires a non-trivial anomalous  dimension, and the pole is replaced by a branch cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To this date this is the best example of what  is called a ”non-Fermi liquid”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In one-dimensional metals this non-Fermi liquid is often called a  “Luttinger Liquid” [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A detailed analysis can be found in chapter 6 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9], and in other  books.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Bosonization provides for a powerful tool to understand the physics of these non-trivial sys-  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization is a duality between a massless Dirac field in 1+1 dimensions and a (also  massless) relativistic Bose (scalar) field [128, 129, 130, 131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this context, bosonization is a set  of operator identities relating observables between two different (dual) continuum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These identities have a close resemblance to the Jordan-Wigner transformation which relates op-  erators of a theory of spinless (Dirac) fermions on a one-dimensional lattice to a theory of bosons  with hard cores (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' spins) on the same lattice [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac fermions in one space dimensions  To understand how these operator identities come about and what these fields mean in the  Condensed Matter context we will consider the simple problem of a system of non-interacting  spinless fermions c(n) on a one-dimensional chain of length L (with L → ∞), for simplicity with  periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hamiltonian is  H0 = −t  L  �  n=1  c(n)†c(n + 1) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (116)  In momentum space, and in the thermodynamic limit L → ∞, the Hamiltonian becomes  H0 =  � π  −π  dp  2π (ε0(p) − µ) c†(p)c(p)  (117)  31  where −π ≤ p ≤ π, µ is the chemical potential (which fixes the fermion number) and ε0(p) is the  (free) quasiparticle energy which, in this case, is  ε0(p) = −2t cos p  (118)  This simple system has a global internal symmetry  c′(n) = eiαc(n),  c′(n)† = e−iαc(c)  (119)  where α is a constant phase (with period 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This global symmetry reflects the conservation of  fermion number  NF =  �  n  c†(n)c(n)  (120)  The free fermion Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (116) is also invariant under lattice translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The ground state of this system is obtained by occupying all single particle states with energy  below the chemical potential, E ≤ µ ≡ EF, which defines the Fermi energy EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows  we will redefine the zero of the energy at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the thermodynamic limit the  one-particle states are labeled by momenta defined in the first Brillouin zone [−π, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The ground  state |G⟩ of this system is obtained by filling up all single-particle states with momentum p in  the range [−pF, pF), where pF is the Fermi momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the occupied states have energy  ε0(p) ≤ EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The ground state is  |G⟩ =  �  |p|≤pF  c†(p)|0⟩  (121)  and is called the filled Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will further assume that the fermionic system is dense in  the sense that the occupied states are a finite fraction of the available states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we assume that  |EF| is a finite fraction of the bandwidth W = 4t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The single-particle excitations have low energy if |ε0(p) − EF| ≪ |EF|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This range can only  be accessed by quasiparticles with momenta p close to ±pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For states wit p close to pF we can  a linearized approximation and write  ε0(p) ≃ �F(p − pF) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (122)  and similarly for the states near −pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here �F = ∂ǫ0  ∂p  ���pF = 2t sin pF is the Fermi velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let q  being the momentum measured from pF (or −pF in the other case), This approximation is correct  in a range of momenta −Λ ≤ q ≤ Λ, where 2π  L ≪ Λ ≪ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime we can approximate  the lattice fields c(n) with two continuum right moving ψR(x) and left moving ψL(x) Fermi fields,  such that (with x = na0)  c(n) ∼ eipF xψR(x) + e−ipF xψL(x)  (123)  in terms of which the Hamiltonian takes the continuum form  Hcontinuum =  �  dx  �  ψ†  R(x)(−i�F)∂xψR(x) − ψ†  L(x)(−i�F)∂xψL(x)  �  =  � dq  2πq�F  �  ψ†  R(q)ψR(q) + ψ†  R(q)ψR(q)  �  (124)  In what follows I will rescale time and space in such a way as to set �F = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (124) is the Hamiltonian of a massless Dirac field in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It describes the  effective low energy behavior of the excitations of the fermionic system defined on a lattice by  the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(116).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here low energy means asymptotically close to the Fermi energy  (which we have set to zero) and momenta close to ±pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see that this effective relativistic  field theory gives a complete description of the universal low energy physics encoded in the  microscopic model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (116) up to some subtle issues associated with what are called quantum  anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us denote by ψ(x) the bi-spinor field ψ(x) = (ψR(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ψL(x)) and the 2 × 2 Dirac gamma-  matrices in terms of the three Pauli matrices are  γ0 = σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  γ1 = −iσ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  γ5 = σ3  (125)  which obey the Dirac algebra  {γµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' γν} = 2gµν  (126)  32  where gµν is the metric tensor in 1+1-dimensional Minkowski spacetime  gµν =  �1  0  0  −1  �  (127)  Using the standard notation ¯ψ(x) = ψ†(x)γ0 (where I left the spinor index α = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2 implicit)  the Lagrangian density of the free massless Dirac fermion is  L = ¯ψi/∂ψ  (128)  where we have used the Feynman slash notation which denotes the contraction of a vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  say Aµ with the Dirac gamma matrices with a slash,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aµγµ ≡ /A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Formally, the Dirac lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (128) (and the Dirac Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (124)) are in-  variant under two separate global transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It has a global U(1) (gauge) symmetry trans-  formation  ψ′(x) = eiθψ(x)  (129)  (where θ is constant) under which the two components of the bi-spinor ψ transform in the same  way  ψ′  R(x) = eiθψR(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ψ′  L(x) = eiθψL(x)  (130)  The massless Dirac theory is also invariant under a global gauge U(1) chiral transformation  ψ′(x) = eiθγ5ψ(x)  (131)  which in components it reads  ψ′  R(x) = eiθψR(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ψ′  L(x) = e−iθψL(x)  (132)  In general,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' if a system has a global continuous symmetry it should have a locally conserved  current and a globally conserved charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the content of Noether’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the  massless Dirac theory has these two global symmetries one would expect that it should have two  separately conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The global U(1) symmetry has an associated current jµ = (j0, j1) given by  jµ = ¯ψγµψ  (133)  which is invariant under the global U(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In terms of the right and left moving Dirac  fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ψR and ψL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the components of the current are  j0 = ψ†  RψR + ψ†  LψL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  j1 = ψ†  RψR − ψ†  LψL  (134)  This current is locally conserved and satisfies the continuity equation  ∂µ jµ = 0  (135)  It has an associated conserved global charge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which we will call fermion number  Q =  � ∞  −∞  dxj0(x) =  � ∞  −∞  dx  �  ψ†  RψR + ψ†  LψL  �  (136)  This is the continuum version of the conservation of fermion number NF of the free fermion  lattice model discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since this current is conserved it can be coupled to an electromagnetic field through a term  in the Lagrangian  Lint = −ejµAµ  (137)  where Aµ is the electromagnetic vector potential, which in 1+1 dimensions has only two com-  ponents, Aµ = (A0, A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here e is a coupling constant which is interpreted as the electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This coupling amounts to making the U(1) symmetry a local gauge symmetry under which the  fields transform as follows  ψ′(x) = eiθ(x)ψ(x),  A′  µ(x) = Aµ(x) + 1  e∂µθ(x)  (138)  33  Now the conservation of fermion number becomes the conservation of the total electric charge  Qe = −eQ  (139)  In a continuum non-relativistic model the Fermi field is ψ(x) (ignoring spin), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' an electron  gas in one dimension such as a quantum wire, the analog of decomposition shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (123) of  the electron field ψ(x) becomes  ψ(x) = eipF xψR(x) + e−ipF xψL(x)  (140)  The electron field ψ(x) transforms as ψ′(x) = eiαψ(x) under the global U(1) gauge symmetry of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(130).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, the local density operator ρ(x) = ψ†(x)ψ(x) is invariant under this sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Under both decompositions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (123) and (140), the local density operator becomes  ρ(x) = ψ†  R(x)ψR(x) + ψ†  L(x)ψL(x) + ei2pF xψ†  R(x)ψL(x) + e−i2pF xψ†  L(x)ψR(x)  (141)  Clearly the non-relativistic density operator ρ(x) is invariant under the global U(1) gauge sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The same considerations applies to the lattice version of the density operator (the local  occupation number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Thus, the density operator ρ(x) can be decomposed into a slowly varying part (the first two  terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (141)) and the last two terms which oscillate with wave vectors Q = ±2pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  observations imply that we can express ρ(x) in the form  ρ(x) = ¯ρ + j0(x) + eiQxψQ(x) + e−iQxψ−Q(x)  (142)  This decomposition can be interpreted as a Fourier expansion of the density operator in term of  newly defined slowly-varying fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (142) Q = 2pF and ¯ρ is the average density, where we  assumed that the Dirac density operator j0(x) has vanishing expectation value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it is normal-  ordered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (142) we defined the (bosonic) operators  ψQ(x) = ψ†  R(x)ψL(x),  ψ−Q(x) = ψ†  L(x)ψR(x)  (143)  which characterize the oscillatory component of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The operators ψ±Q(x) are the order  parameters of a charge density wave state in one dimension and Q is the ordering wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Repulsive interactions between the electrons can cause scattering process between the right  and left moving components to become relevant (in the Renormalization Group sense) leading  to the spontaneous breaking of translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The resulting state is known as a charge-  density-wave (CDW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this state the operators ψ±Q(x) (or a linear combination of them) acquire  a non-vanishing expectation value, and the expectation value of the density operator ρ(x) has a  (static) modulated component, and the Dirac Hamiltonian density becomes  H = ψ†  R(x)(−i)∂xψR(x) − ψ†  L(x)(−i)∂xψL(x) + m  �  ψ†  R(x)ψL(x) + ψ†  L(x)ψR(x)  �  (144)  where m is the Dirac mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The operator in the second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (144) mixes the right and left  moving components of the Dirac spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This hermitian operator is known as the Dirac mass  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In relativistic notation this operator is written  ¯ψψ = ψ†  R(x)ψL(x) + ψ†  L(x)ψR(x)  (145)  When m � 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' when ⟨ ¯ψψ⟩ � 0, the fermionic spectrum has a mass gap, and the electronic  states with momenta ±pF have an energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Alternatively we could have considered a state  with the in which the (hermitian) operator that has an expectation value is  i ¯ψγ5ψ = i  �  ψ†  R(x)ψL(x) − ψ†  L(x)ψR(x)  �  (146)  which is known as the γ5 mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a more general CDW state both mass terms can be  present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chiral symmetry and chiral symmetry breaking  The CDW states we introduced have different symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this let us observe that the  operators ψ±Q(x) transform non-trivially under a global U(1) chiral transformation  ψ′  ±Q(x) = e∓2iθψ±Q(x)  (147)  34  Upon substituting this transformation in the expansion of the density ρ(x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (142), we see  that a chiral transformation is equivalent to a uniform displacement of the density operator by  θ/pF,  ρ(x) → ρ (x + θ/pF)  (148)  Under a chiral transformation with arbitrary angle θ, the Dirac and γ5 mass term operators trans-  form as an orthogonal transformation of a two-component vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular for a chiral trans-  formation with θ = π/4,  ¯ψψ �→ i ¯ψγ5ψ,  i ¯ψγ5ψ �→ − ¯ψψ  (149)  which is equivalent to a displacement of the density by 1/4 of the period of the CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  sense these two states are equivalent, as is the state with a more general linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The microscopic lattice model (and the continuum non-relativistic model) are invariant under  the spatial inversion symmetry x ↔ −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This also implies a symmetry under the exchange of  right and left moving components of the Dirac spinor, ψR ↔ ψL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the massless Dirac theory  this operation is equivalent to the multiplication of the spinor by the Pauli matrix σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  massless theory the multiplication by σ2 (followed by a chiral transformation with θ = π/2) has  the same effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These symmetries have an important effect on the fermionic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To understand what  they do let us consider the one-particle Dirac Hamiltonian with a Dirac mass m (jn momentum  space)  h =  �p  m  m  −p  �  (150)  This operator anti-commutes with the Pauli matrix σ2, {h, σ2} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let |E⟩ be an eigenstate of  the Hamiltonian h with energy eigenvalue E =  �  p2 + m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider the state σ2|E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is  also an eigenstate of h but with energy −E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  hσ2|E⟩ = −σ2h|E⟩ = −E|E⟩,  ⇒ σ2|E⟩ = | − E⟩  (151)  Hence if |E⟩ is an eigenstate of energy E, then σ2|E⟩ is an eigenstate of energy −E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means  that the spectrum is invariant under charge conjugation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that under this opera-  tion the spinor transforms as  σ2  �ψR  ψL  �  =  �−iψL  iψR  �  = e−iγ5π/2  �ψL  ψR  �  (152)  In other words, this theory is invariant under charge conjugation C and parity P which, combined,  it implies that it is invariant under time-reversal T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The same consideration applies in the case of a γ5 mass term in which case the one-particle  Dirac Hamiltonian now is  h =  � p  −im5  im5  −p  �  (153)  This Hamiltonian now anti-commutes with the Pauli matrix σ1 which also implies that the same  charge conjugation symmetry C, |E⟩ ↔ |−E⟩, is present in the spectrum of the case of a γ5 mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can also repeat the argument on parity invariance P, which is now multiplication by σ1, Thus  the theory is invariant under time-reversal T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, if the theory has both a Dirac mass m and a γ5 mass m5 these symmetries are  broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, the one-particle Dirac Hamiltonian now is  h =  �  p  m − im5  m + im5  −p  �  = pσ3 + mσ1 + m5σ2  (154)  which no longer has a spectral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case CP is broken and, hence, so it T since  CPT remains unbroken (as it should).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In a lattice system this is a symmetry transformation only if θ = πn/pF is a lattice displace-  ment, which restricts the allowed values of the chiral angle to be discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although this is true  interactions play a significant role in the actual behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, there are physical situations  in which an effective continuous symmetry actually emerges in this he infrared (long-distance)  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is what happens when the CDW is incommensurate and, as we will see in the next  subsection, it slides under the action of an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the case of a half-filled system (with only nearest-neighbor hopping matrix elements) the  Fermi wave vectors are ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the allowed discrete chiral transformation has a chiral  35  angle π/2 under which ¯ψψ �→ − ¯ψψ (and similarly for i¯γ5ψ) corresponding by a translation by one  lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The allowed four fermion operator is ( ¯ψψ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the lattice fermions are spinless  this operator reduces to  ( ¯ψψ)2 = −2 jR(x)jL(x) + lim  y→x ψ†  R(x)ψ†  L(x)ψL(y)ψR(y) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (155)  where introduced the right and left moving (chiral) components of the current operator  jR(x) = 1  2(j0(x) + j1(x)) = ψ†  R(x)ψR(x),  jL(x) = 1  2(j0(x) − j1(x)) = ψ†  L(x)ψL(x)  (156)  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (155) is known as the backscattering interaction and has scaling dimension  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, it is a marginal operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The theory with only the first term is known in Condensed  Matter Physics as the Luttinger model and in High-Energy Physics at the (massless) Thirring  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The second operator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (155) formally violates momentum conservation as its total  momentum is 4pF = 2π which is a reciprocal lattice vector and, as such, it is equivalent to  zero (mod 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such an operator is not formally allowed in a (naive) continuum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  operator is due to a lattice Umklapp process and breaks the formal continuous chiral symmetry  to a discrete Z2 subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although the naive scaling dimension of this operator is 2 (and hence  it is formally marginal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the fermions are spinless, the leading operator actually vanishes and  the leading non-vanishing operator actually has dimension 4, which is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, as  shown above, backscattering processes of the form jR(x) jL(x) are part of this operator and are  exactly marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  If the interaction is strong enough the backscattering interaction can make the Umklapp op-  erator relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' When this happens the fermionic system has a quantum phase transition to an  insulating state with a period 2 (commensurate) CDW state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, for spin 1/2  fermions the operator ( ¯ψψ)2 is allowed and is marginally relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the interactions are re-  pulsive the resulting state is an antiferromagnetic N´eel state at quantum criticality, while for  attractive interactions it is a period 2 CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is what happens in the 1D Hubbard model (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9] for a detailed analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There are two theories in relativistic systems which are closely related to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One  if the Gross-Neveu model [132] which is a theory of N species of massless Dirac spinors with  Lagrangian density  LGN = ¯ψai/∂ψa + g( ¯ψaψa)2  (157)  with a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', N (summation of repeated induces is implies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The spin-1/2 Hubbard model  corresponds to the case N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This Lagrangian is invariant only under the discrete chiral  symmetry ¯ψaψa �→ − ¯ψaψa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a discrete, Z2, symmetry and as such it can be spontaneously  broken in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For N ≥ 2, the resulting state has a (dynamically) broken Z2 chiral  symmetry and that there is a chiral condensate ⟨ ¯ψaψa⟩ � 0 corresponding to a period 2 CDW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The other theory is known as the chiral Gross-Neveu model whose Lagrangian density is  LcGN = ¯ψai/∂ψa + g  �  ( ¯ψaψa)2 − ( ¯ψaγ5ψa)2�  (158)  which has the full continuous chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For N = 1 this theory is equivalent to the Lut-  tinger model (and to the Gross-Neveu model if we ignore the umklapp term).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For N ≥ 2 the chi-  ral symmetry is formally broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If this were true this theory would violate the Mermin-Wagner  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However a detailed study (most easily done using bosonization methods) shows that  instead of long range order the correlator of both mass terms decay as a power law as a function  of distance, consistent with the requirements by this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We close this subsection by noting that if the lattice model is not half filled but its density  is either incommensurate or has a higher degree of commensurability, say p/q, the chiral sym-  metry is actually continuous (in the incommensurate case) or effectively continuous since the  requisite umklapp terms are strongly irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, if the lattice model is not at half filling  charge conjugation symmetry C is broken at the lattice scale (in the UV), where it is equivalent  to particle-hole symmetry, but it is recovered in the low-energy, IR, regime (up to irrelevant op-  erators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this sense, both the continuous chiral symmetry and charge conjugation symmetry  can be regarded as emergent IR symmetries  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The chiral anomaly  In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 we showed that the theory of massless Dirac fermions, in addition to a global  U(1) gauge gauge symmetry, has a second conservation law which we called a global U(1) chiral  36  symmetry, shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(131).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This symmetry implies that there is a locally associated chiral  current j5  µ, given by  j5  µ(x) = ¯ψ(x)γµγ5ψ(x)  (159)  which also satisfies a continuity equation  ∂µ j5  µ = 0  (160)  and there is a globally conserved chiral charge Q5  Q5 =  � ∞  −∞  dx j5  0(x) =  � ∞  −∞  dx  �  ψ†  RψR − ψ†  LψL  �  (161)  It is easy to check that the Dirac current jµ and the chiral current j5  µ are related by  j5  µ = ǫµν jµ  (162)  where ǫµν is the second rank Levi-Civita tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The simultaneous conservation of both currents jµ and j5  µ in the massless Dirac theory implies  that the right and left moving densities jR and jL, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (156), should be separately  conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, if the Dirac theory has a mass term  L = ¯ψi/∂ψ − m ¯ψψ  (163)  it is straightforward to show that  ∂µ j5  µ = 2mi ¯ψγ5ψ  (164)  which means that in the massive theory the axial current is not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is easy to under-  stand since the mass term mixes the right and left moving components of the Dirac spinor and,  hence, the right and left moving densities are not conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  What happens in the massless limit, m → 0, is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem was investigated  in the late 1960’s in 3+1 dimensions by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Adler [133] and by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bell and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw [134]  who were interested in the anomalous decay of a neutral pion into two photons, π0 → 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This process appears at third order of perturbation theory and it involves the computation of a  triangle diagram (a fermion loop).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 3+1 dimensions this process has a UV divergence which  needed to be regulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These authors showed that it is not possible to find a regularization  in which both the Dirac (gauge) current jµ = ¯ψγµψ and the axial current j5  µ = ¯ψγµγ5ψ are  conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, if gauge invariance is preserved then the axial current is not and has  an anomaly and results in a non-conservation of the axial current, ∂µ j5  µ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand,  at least in the case of the physical gauge-invariant regularization, the obtained expression for  the anomaly in the axial current is universal, independent of the value of the UV regulator (the  cutoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sometime later G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ’t Hooft showed that in non-abelian gauge theories instant processes  also lead to anomalies and, furthermore, the result was also universal [96, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the result  is universal and, hence, independent of the UV scale, this led to the concept of anomaly matching  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will examine this problem in 1+1 dimensions (although it plays a key role in the theory of  topological insulators in three space dimensions [135]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we noted above, the lattice model is  gauge-invariant and has only one conserved current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The conserved axial current appeared only  in the low-energy regime in which the lattice model is described by a theory of massless Dirac  fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To understand this problem we will consider the theory of fermions in 1+1 dimen-  sions coupled to a U(1) gauge field field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the presence of a background (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' not quantized)  electromagnetic field in the A0 = 0 gauge the free fermion Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (116) becomes  H[A] = −t  L  �  n=1  c†(n)eieA(n,t)/ℏc(n) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (165)  An uniform and constant electric field is E is represented by a vector potential A = −cEt (with  c being the speed of light).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The net effect of this gauge field is to shift of the momentum of the  fermion quasiparticles p → p + ecEt/ℏ or, what is the same, to displace the fermion dispersion  relation in momentum by the ecEt/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the Fermi points are also shifted by that  amount, pF → pF + ecEt/ℏ and −pF → −pF + ecEt/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the single particle states  between pF and pF +ecEt/ℏ that were empty for E = 0 are now occupied and the states between  37  −pF and −pF + ecEt/ℏ that were occupied for E = 0 are now empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that number  of right-moving fermions is increasing at a rate of ecE/ℏ and that the number of left-moving  fermions decreases at the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This results in a net current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Throughout this process the  total number of fermions is not changed, gauge invariance is satisfied, but the number of right  and left moving fermions are not separately concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that this is an effect that involved  the entire Femi sea but the net effect is at low energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us see now how this plays out in the effective Dirac theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The massless Dirac La-  grangian density in the background of an unquantized electromagnetic field Aµ is  L = ¯ψ(i/∂ − e /A)ψ  (166)  Since there is no mass term the Dirac equation still decouples into two equations, for the right  and left moving components of the Dirac spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the A0 = 0 gauge they are  i∂0ψR = (−i∂1 − A1)ψR,  i∂0ψL = (i∂1 − A1)ψL  (167)  In the temporal gauge, A0 = 0, a uniform electric field E = ∂0A1, and A1 increases linearly with  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As A1 increases, the Fermi momentum pF (which is equal to the Fermi energy EF) also  increases at the rate eE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The density of states of a system of length L is L/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, the rate of  change of the number of right-moving fermions is  dNR  dx0  = e  2πE  (168)  where we defined NR =  � L  0 dx jR and similarly for NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the UV regulator of the theory is  compatible with gauge invariance, then the total fermion number must be conserved and the total  vacuum charge must remain equal to zero,  Q =  � L  0  dx j0(x) = NR + NL = 0  (169)  Thus, if NR increases, then NL must decrease by the same amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Or, equivalently, the electric  field E creates an equal number of particles NR and of antiparticles ¯NL = −NL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand, the chiral charge Q5 = NR − NL must increase at the rate  dQ5  dx0  = dNR  dx0  + d ¯NL  dx0  = e  π E  (170)  Again, the details of the UV regularization do not matter, only that it is gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can  also interpret Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (170) as the rate of particle-antiparticle pair creation by an electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In a relativistic notation these results are expressed as  ∂µ j5  µ = e  2πǫµνFµν  (171)  Hence, the formally conserved current j5  µ has an anomaly and is not conserved due to quantum  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since it is not conserved, we cannot gauge the chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the next subsection  we will see that the the anomaly is closely related with bosonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization, anomalies and duality  We will reexamine the problem at hand from the point of view of the fermionic currents of  the Dirac theory jµ as operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the currents obey the continuity equation, ∂µ jµ = 0 one  expects that this would imply that it may be possible to write them as a curl of a scalar field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  jµ(x) = ǫµν∂νφ(x) where φ(x) should be a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since this should be an operator identity  we will need to understand how the currents act on the physical Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Fermi-Bose mapping in 1D systems is closely related to the chiral anomaly we just  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization of a system of 1+1 dimensional massless Dirac fermions is a set of  operator identities understood as matrix elements of the observables in the physical Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These identities were first derived by Daniel Mattis and Elliott Lieb [128], based on earlier work  by Julian Schwinger [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These identities were rediscovered (and their scope greatly expanded)  by Alan Luther and Victor Emery [129], by Sidney Coleman [131], by Stanley Mandelstam  [130], and by Edward Witten [137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A non-abelian version of bosonization was subsequently  derived by Witten [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  38  The physical Hilbert space is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let |FS⟩ ≡ |0⟩ denote the filled Fermi  sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows we will assume that the physical system is macroscopically large abd that  local operators create the physical states by acting finitely on the filled Fermi sea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Physical  observables, such as the right and left moving densities jR(x) and jL(x), need to be normal-  ordered with respect to the physical vacuum state, the filled Fermi (Dirac) sea |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The normal  ordered densities are : jR(x) : ≡ jR(x) − ⟨0|jR(x)|0⟩ and : jL(x) : ≡ jL(x) − ⟨0|jR(x)|0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since  the densities are products of fermion operators they need to be defined as a limit in which the  operators are separated by a short distance η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Crucial to this construction is that the computation  the expectation values be regularized in such a way that the charge (gauge) current jµ(x) is locally  conserved and satisfies the continuity equation ∂µ jµ = 0 as an operator identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows  all expectation values will refer to the filled Fermi sea state |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The propagators of the right and left moving Fermi fields are given by  ⟨ψ†  R(x0, x1)ψR(0, 0)⟩ =  −i  2π(x0 − x1 + iǫ),  ⟨ψ†  L(x0, x1)ψL(0, 0)⟩ =  i  2π(x0 + x1 + iǫ)  (172)  The expectation value of the currents at a space location x1 are  ⟨jR(x1)⟩ = lim  η→0⟨ψ†  R(x1 + η)ψR(x1 − η)⟩ =  i  4πη,  ⟨jL(x1)⟩ = lim  η→0⟨ψ†  L(x1 + η)ψL(x1 − η)⟩ = −i  4πη  (173)  which are divergent at short distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It follows that the normal-ordered right and left moving  current densities satisfy the equal-time commutation relations  [jR(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' jR(x′  1)] = − i  2π∂1δ(x1 − x′  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [jL(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' jL(x′  1)] = + i  2π∂1δ(x1 − x′  1)  (174)  These identities imply that the normal-ordered space-time components of the current jµ = (j0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j1)  satisfy the equal-time commutation relations  [j0(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j1(x′  1)] = − i  π∂1δ(x1 − x′  1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [j0(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j0(x′  1)] = [ji(x1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j1(x′  1)] = 0  (175)  The non-vanishing right-hand sides of these commutators are known as Schwinger terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  identities define the the U(1) (Kac-Moody) current algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We should note that in a theory of non-relativistic Fermi fields ψ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' t) in all dimensions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the  charge density ρ(x) and the current operators j(x) =  1  2i  �  ψ†(x)▽ψ(x) − ▽ψ†(x)▽ψ(x)  �  satisfy a  similar expression (also at equal times)  [ρ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' jk(x′)] = −i e2  mc2 ∂k  �δ(x − x′)ρ(x)� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [ρ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ρ(x′)] = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [jk(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' jl(x′)] = 0  (176)  In one dimension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and in the regime in which the fermions have a macroscopic density so that  ρ(x) ≃ ⟨ρ(x)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' after a multiplicative rescaling of the operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the non-relativistic identities of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (176) are equivalent to the U(1) current algebra of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (175).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The U(1) current algebra of Eq,(175) is reminiscent of the equal-time canonical commutation  relations of a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, if φ(x) is a scalar field and Π(x) = ∂0φ(x) is its canonically  conjugate momentum, then they obey the equal-time canonical commutation relations  [φ(x1), Π(x′  1)] = iδ(x1 − x′  1)  (177)  We can then identify the charge density j0 and the current density j1 with the scalar field operators  j0(x) =  1√π∂1φ(x),  j1(x) = − 1√πΠ(x) = − 1√π∂0φ(x)  (178)  which obey the U(1) current algebra of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (175) as a consequence of the canonical commutation  relations, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (177).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (177) in the Lorentz covariant form  jµ(x) =  1√πǫµν∂νφ(x)  (179)  which is clearly consistent with the local conservation of the current jµ,  ∂µ jµ(x) = 0  (180)  39  Let us examine now the question of the conservation of the chiral current j5  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (162)  we showed that the gauge current and the chiral current are related by j5  µ = ǫµν jν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore the  divergence of the chiral current is  ∂µ j5  µ = ǫµν∂µ jν =  1√πǫµνǫνλ∂µ∂λφ = − 1√π∂2φ  (181)  where we used the identification of the gauge current in terms of the scalar field φ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (179).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore we conclude that  ∂µ j5  µ = 0 ⇔ ∂2φ = 0  (182)  This equation states that the chiral current as an operator identity is conserved if and only if the  field φ is a free massless scalar field, whose Lagrangian density is  LB = 1  2(∂µφ)2  (183)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (166) we considered the free massless Dirac Lagrangian coupled to a background (not  quantized) gauge field Aµ through the usual minimal coupling which here is Lint = −ejµAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' using  the bosonization identity for the gauge current, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (179) we see that the Lagrangian density of  the bosonized theory now becomes  LB[A] =1  2(∂µφ)2 −  e√πǫµν∂νφ(x) Aµ(x)  ≡1  2(∂µφ)2 + J(x)φ(x)  (184)  where the source J(x) is  J(x) =  e√πǫµν∂νAµ(x) =  e  √  4π  F∗(x)  (185)  where F∗(x) = ǫµνFνµ(x) is the (Hodge) dual of the field strength Fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, F∗ is (essentially)  the source for the scalar field φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This implies that the equation of motion of the scalar field  must be  − ∂2φ(x) = J(x) =  e√πǫµν∂νAµ  (186)  Retracing our steps we find that the chiral current j5  µ obeys  ∂µ j5  µ = − 1√π∂2φ = e  2πǫµνFµν  (187)  which reproduces the the chiral anomaly given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (170).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These results suggest that the theory of a free massless Dirac spinor must be equivalent to  the theory of the free massless scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This statement is known as bosonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However  for this identification to be correct there must be an identification of the Hilbert spaces and of all  the operators of each theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will not do this detailed analysis here but we will highlight the  most significant statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us begin with the fermion number of the Dirac theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Consider a system of fermions  of total length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Using the bosonization identity of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (179) we find that the fermion number  NF ≡ Q is given by  NF =  � L  0  dx1 j0(x0, x1) =  1√π  � L  0  dx1 ∂1φ(x0, x1) =  1√π(φ(x0, L) − φ(x0, 0))  (188)  Thus, the vacuum sector of the Dirac theory, with NF = 0, corresponds to the theory of the scalar  field with periodic boundary conditions, φ(x1 = 0) = φ(x1 = L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, since the fermion  number is quantized, NF ∈ Z, changing the fermion number is the same as twisting the boundary  conditions of the scalar so that  φ(x1 + L) = φ(x1) + √πNF  (189)  In String Theory [104] the scalar field is interpreted as the coordinate of a string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Compactifying  the space where the string lives to be a circle of radius R means that the string coordinate is  defined modulo 2πRn, where n is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We see that the condition imposed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (189)  40  is equivalent to say that the scalar field is compactified and that the compactification radius is  R = 1/  √  4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This identification also imposes the restriction that the allowed operators of the  scalar field must obey the identification  φ(x) ∼ φ(x) + 2πRn  (190)  as an equivalency condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The simplest bosonic operators that obey the compactification condition are the vertex oper-  ators Vα(x),  Vα(x) = exp(iαφ(x))  (191)  The compactification condition then requires that the allowed vertex operators should have α =  n/R =  √  4π n, where n is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the propagator of the scalar field in 1+1-dimensional  (Euclidean) spacetime is  G(x − x′) = − 1  2π ln  �|x − x′|  a  �  (192)  where a is a short-distance cutoff, we find that the scaling dimension of the vertex operator is  ∆α = α2/(4π) = n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see shortly that the vertex operator with α =  √  4π is essentially the  Dirac mass operator (which has scaling dimension 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The free massless scalar field can be decomposed into right and left moving components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' φR  and φL respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  φ = φR + φL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ϑ = −φR + φL  (193)  where  ϑ(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x1) =  � x1  −∞  dx′  1Π(x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x′  1)  (194)  is the Cauchy-Riemann dual of the field φ(x) since they satisfy the Cauchy-Riemann equation  ∂µφ = ǫµν∂νϑ  (195)  The right and left moving component of the Dirac spinor are found to have the bosonized expres-  sion [130]  ψR(x) =  1  √  2πa  : exp(i  √  4πφR(x)) :,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ψL(x) =  1  √  2πa  : exp(−i  √  4πφL(x)) :  (196)  It is easy to check that the propagators of these operators agree with the expressions given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (172), and that they have scaling dimension 1/2 and spin 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  How does a chiral transformation act on the scalar field?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A chiral transformation by an angle  θ, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (132), acts on the right and moving fermions as ψ′  R = exp(iθ)ψR and ψ′  L = exp(−iθ)ψL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (196) we see that the right and left moving components of the scalar field transform as  φ′  R = φR + θ/  √  4π and φ′  L = φL + θ/  √  4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that a chiral transformation by an angle  θ of the Dirac fermion by an angle θ is equivalent to a translation (a shift) of the scalar field  φ′ = φ + 2θ/  √  4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can use the Operator Product Expansion discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 to show that the  fermion mass terms ¯ψψ and i ¯ψγ5ψ are given by the following identifications  ¯ψψ =  1  2πa : cos(  √  4πφ) :,  i ¯ψγ5ψ =: sin(  √  4πφ) :  (197)  These operators have scaling dimension 1 and transform properly under chiral transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These identifications imply that a theory of free massive Dirac fermions  LD = ¯ψi/∂ ψ − m ¯ψψ  (198)  is equivalent to the sine-Gordon field theory [131] whose Lagrangian is  LSG = 1  2(∂µφ)2 − g : cos(  √  4πφ) :  (199)  where g = m/(2πa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Given the central role played by the current algebra identities of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (175) one may wonder  if a similar approach might apply in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schwinger terms in current algebra play  an important role in relativistic field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However in higher dimensions their structure is  41  more complex and does not lead to identities of the type we have discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The reason at the  root of this problem is largely kinematical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Bose (scalar) field φ is qualitatively a bound  state, a collective mode in the language of Condensed Matter Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1+1 dimensions this  collective mode exhausts the spectrum at low energies due to the strong kinematical restriction  on one spatial dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The equivalency between the theory of free massive Dirac fermions and the sine-Gordon  theory is an example of the power of bosonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the Dirac side the mapping the theory is  free and its spectrum is well understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But on the sine-Gordon side the theory is non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In fact in the sine-Gordon theory the fermions are essentially solitons, domain walls of the scalar  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these and many other reasons that we do not have space here bosonization plays a  huge role in understanding the non-perturbative behavior of systems both in Condensed Matter  Physics and in Quantum Field Theory in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see in section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 that to  an extent some of these ideas can and have been extended to relativistic systems and classical  statistical mechanical systems in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractional Charge  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Solitons in one dimensions  We begin by returning to the equivalency between the theory of free massive Dirac fermions  and sine-Gordon theory, in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (188) we showed that the boundary conditions  of the compactified scalar field φ(x) are determined by the fermion number NF of the dual Dirac  theory, and that the vacuum sector of the Dirac theory maps onto the sine-Gordon theory with  periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now examine the sector with one fermion, NF = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  sector of the Dirac theory maps onto the sine-Gordon theory with twisted boundary conditions,  φ(L) − φ(0) = √π  (200)  The Hamiltonian of the sine-Gordon theory is  HSG =  � ∞  −∞  dx  �1  2Π2(x) + 1  2 (∂xφ(x)) + g cos  � √  4πφ(x)  ��  (201)  In the sector with periodic boundary conditions the classical ground states are static and uniform  configurations that minimize the potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the potential energy is a periodic func-  tional of φ(x) the classical minima are at φn(x) = (n + 1/2) √π, where n is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The classical energy of these ground states is extensive and is given by Egnd = −gL where L is  the linear size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The classical ground state in the twisted sector is a domain wall (or soliton) which interpo-  lates between the static and uniform ground states φ(x) = ± √π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The classical ground state in  this sector is the static solution of the Euler-Lagrange equation  d2φ  dx2 = −2g √π sin  �  2 √πφ(x)  �  (202)  such that asymptotically satisfies limx→±∞ = ± √π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The solution is the classical soliton con-  figuration  φ(x) =  2√π tan−1 �  exp(2 √πg(x − x0))  �  −  √π  2  (203)  The soliton solution represents a domain wall between two symmetry-related classical ground  states with φ = ± √π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The energy of the soliton (measured from the energy of the ground state in the trivial sector)  is finite and is given by  Esoliton = 4  �  g  π  (204)  where x0 is a zero mode of the soliton solution and represents its coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By coupling the  bosonized theory to a weak electromagnetic field Aµ, as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (184), it is easy to check that  it has electric charge −e and, in this sense represents the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then the identities of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (196)  can be used that as a quantum state it is indeed a fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  42  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyacetylene  In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 we saw that solitons of a scalar field can be regarded as being equivalent to  electrons, fermions with charge −e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now see that in a theory of fermions coupled to  a domain wall of a scalar field, the soliton carries fractional charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem has been  extensively studied in one-dimensional conductors such as polyacetylene, in particular by the  work of Wu-Pei Su, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Robert Schrieffer and Alan Heeger [138] and by Roman Jackiw and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Robert Schrieffer [139].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In quantum field theory this problem was first discussed by Roman  Jackiw and Claudio Rebbi [140] and by Jeffrey Goldstone and Frank Wilczek [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 we showed that the physics of lattice fermions in one dimension at low energies  is well described by a theory of massless Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a one-dimensional conductor, such as  polyacetylene, the fermions couple to the lattice vibrations (phonons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Su, Schrieffer and Heeger  (SSH) [138] proposed a simple model in which the electrons couple to the lattice vibrations  through a modulation of the hopping amplitude between two consecutive sites n and n + 1,  instead of being a constant t, becomes tn,n+1 = t − g(un+1 − un), where un is the displacement of  the ion (a CH group in polyacetylene) at site n from its classical equilibrium position and g is  the electron-phonon coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In polyacetylene there number of electrons (which are  spin-1/2 fermions) is equal to the number of sites of the lettuce and the electronic band is half-  filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At half filling this simple band structure is invariant under a particle-hole transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  If the coupling to the lattice vibrations is included this symmetry remains respected provided the  displacements change sign un → −un for all lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In polyacetylene the lattice dimerizes  (a process known as a Peierls distortion) and the discrete translation symmetry by one lattice  spacing is spontaneously broken: the system becomes a period 2 CDW on the bonds of the  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The broken symmetry state is still invariant under a particle-hole transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The effective field theory of this system is a theory of two Dirac spinors ψα,σ(x), where  α = 1, 2 denotes right and left-moving fermions, and σ =↑, ↓ are the two spin polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  Lagrangian density of this system is  L = ¯ψσi/∂ψσ − gφ(x) ¯ψσ(x)ψσ(x) − 1  2φ(x)2  (205)  The real scalar field φ(x) represents the distortion field of the polyacetylene chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will  assume that the chains has spontaneously distorted and we will regard the scalar field as static  and classical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a good approximation since the masses of the CH complexes is much  bigger than the electron mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This continuum model is due to Takayama, Lin-Liu and Maki  [142] and further developed by Campbell and Bishop [143, 144].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many of the results fund in  this (adiabatic) approximation remain qualitatively correct upon taking into account the quantum  dynamics of the chain, even in the limit in which the ions are treated as being “light” (provided  the spin of the fermions is taken into account) [145, 146].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the field theory the discrete symmetry of displacements by one lattice spacing becomes  the Z2 symmetry φ → −φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a symmetry of the electron phonon system once combined  with the discrete chiral transformation ψ → γ5ψ under which ¯ψψ → − ¯ψψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The ground state is  two fold degenerate ±φ0 with  φ0 = 2Λ�F  g  exp  �  −π�F  g2  �  (206)  and the Dirac fermion (the electron) has a exponentially small mass, m = gφ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractionally charged solitons  Jackiw and Rebbi showed that the 1+1-dimensional classical φ4 theory has the following  soliton solution which interpolates between the two classically ordered states at ±φ0 [140]  φ(x) = φ0 tanh  � x − x0  ξ  �  (207)  where ξ is the correlation length of φ4 theory, and x0 is the (arbitrary) location of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  They further showed that, when coupled to a theory of massless relativistic fermions through a  Yukawa coupling, as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (205), this soliton carries fractional charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The argument goes as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The one-particle Dirac Hamiltonian for a Dirac fermion with a position-dependent mass  m(x) is  H = −iσ1∂x + m(x)σ3 =  �m(x)  −i∂x  −i∂x  −m(x)  �  (208)  43  where m(x) = gφ(x), with φ(x) being the soliton solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(207).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This Hamiltonian is her-  mitian and real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, this Hamiltonian anti-commutes with the Pauli matrix σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  implies that for every positive-energy state |E⟩ with energy +E there is a negative-energy eigen-  state with energy −E given by σ2|E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the spectrum is particle-hole symmetric (or, what  is the same, charge-conjugation invariant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, and consistent with charge-conjugation  symmetry, the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (208) has state with E = 0, a zero-mode, with a normalizable  spinor wave function  ψ0(x) =  1√  2  �−i  1  �  exp  �  −sgn(m)  � x  0  dx′ m(x′)  �  (209)  which exists for an arbitrary function m(x) which changes sign once at some location (which  we took to be x0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw and Rebbi further showed that the soliton (anti-soliton) carries  fractional charge  Q = ∓e  2  (210)  This result follows from the spectral asymmetry identity of the density of states ρS (E) in the  presence of the soliton  Q = −e  2  � ∞  0  (ρS (E) − ρS (−E)) = −e  2η  (211)  where η is the spectral asymmetry of the Dirac operator in the soliton background, and it is known  as the APS η-invariant of Atiyah-Patodi-Singer [147].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Given the one-to-one correspondence  that exists between positive and negative energy states in the spectrum, the spectral asymmetry  follows from the condition that the zero mode be half-filled, which is required by normal-ordering  or, what is the same, by charge neutrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Another way to understand this result is that adding (or  removing) a fermion of charge −e results in the creation of a soliton-antisoliton pair, with each  topological excitation carrying half of the charge of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, in the dimerized  phase the electron is fractionalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this analysis we ignored the spin of the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we  take it into account the spin degree of freedom the soliton is instead a boson with charge ∓e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is an alternative, complementary, way to think about the charge of the soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gold-  stone and Wilczek [141] considered a theory in which the (massless) Dirac fermion is coupled to  two real scalar fields, ϕ1 and ϕ2, with Lagrangian  L = ¯ψi/∂ψ − gϕ1 ¯ψψ − igϕ2 ¯ψγ5ψ ≡ ¯ψi/∂ψ − g|ϕ| ¯ψ exp(iθγ5)ψ  (212)  where |ϕ|2 = ϕ2  1 + ϕ2  2 and θ = tan−1(ϕ2/ϕ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They considered a soliton in which gϕ1 = m is the  constant (in space) Dirac mass and ϕ2 winds slowly between two values ±ϕ0 for x → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  this theory the one-particle Dirac Hamiltonian is  H = −iσ1∂x + gϕ1σ3 + gϕ2σ2  (213)  which is hermitian and complex and, hence, it violates CP invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A perturbative calculation of the induced (gauge-invariant) current jµ, which is given by the  triangle diagram of a fermion loop with two gauge field insertions and a coupling of the scalar  fields, yields the result (with a = 1, 2)  ⟨jµ(x)⟩ = 1  2πǫµνǫab  ϕa∂νϕb  |ϕ|2  = 1  2πǫµν∂νθ  (214)  which is locally conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice, however, that the induced axial current j5  µ is not conserved,  ∂µ⟨j5  µ⟩ = − 1  2π∂2θ � 0 and, hence, this current is anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can now compute the total  charge accumulated as the soliton is created adiabatically to be given by the Goldstone-Wilczek  formula  Q = −e∆θ  2π  (215)  where ∆θ = θ(+∞) − θ(−∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since limx→±∞ ϕ2(x) = ±ϕ0, we obtain the result  Q = − e  π tan−1 �gϕ0  m  �  (216)  In the limit m = gϕ → 0, where CP (or T) invariance is recovered, we get  lim  m→0 Q = −e  2  (217)  44  which is the Jackiw-Rebbi result for the fractional charge of a soliton of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The results for the fractional charge of the soliton can also be derived using the bosonization  identities of section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, the bosonized expression for the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (212) is  L = 1  2  �  ∂µφ  �2 − g|ϕ|  2πa cos  � √  4πφ − θ  �  (218)  Deep in the phase in which the Bose field φ is massive, the non-linear term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (218) locks this  field to the chiral angle θ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  φ =  1  √  4π  θ  (219)  However, the bosonization identities also tell us that the gauge current jµ is given by the curl of  the scalar field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, in this state the current jµ is  jµ =  1√πǫµν∂νφ = 1  2πǫµν∂νθ  (220)  which is the same as the Goldstone-Wilczek result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(214).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' That these two seemingly differ-  ent approaches yield the same result is not accidental as they both follow from the axial anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractional Statistics  A fundamental axiom of Quantum Mechanics is that identical particles are indistinguish-  able [148, 149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In non-relativistic Quantum Mechanics this leads to the requirement that the  quantum states of a system of identical particles must be eigenstates of the pairwise particle ex-  change operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since two exchanges are equivalent to the identity operation this implies that the  states must be even or odd under pairwise exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result, in turn, implies that particles  can be classified as either being bosons (whose states are invariant under pairwise exchanges)  or fermions (whose states change sign under pairwise particle exchanges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A consequence is  that bosons obey the the Bose-Einstein (and can condense into a single particle state) whereas  fermions obey the Fermi-Dirac distribution and must obey the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is  an implicit assumption of this line of reasoning that all relevant states of a system of identical  particles can be efficiently represented by a (suitably symmetrized or antisymmetrized) product  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This classification is present at an even deeper level in (relativistic) Quantum Field Theory  where locality, unitarity and Lorentz invariance require that the fields be classified as represen-  tations of the Lorentz group and obey the Spin-Statistics Theorem [150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Spin-Statistics  Theorem is actually an axiom of local relativistic Quantum Field theory which requires that  fields that transform with an integer spin representation of the Poincar´e group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' scalars, gauge  fields, gravitational fields, etc) must be bosons while fields that transform with a half-integer  spin representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac spinors, etc) must be fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This spin-statistics connection is  intrinsic to the construction of String Theory [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Given these considerations there was a general consensus that fermions and bosons were  the only possible types of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nevertheless several exceptions to this rule were known  to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One is the construction of the magnetic monopole in 3+1 dimensional gauge theory  by Tai-Tsun Wu and Chen-Ning Yang who showed that a scalar coupled to a Dirac magnetic  monopole behaves as a Dirac spinor [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This was an early example of statistical transmutation  by coupling a matter field to a non-trivial configuration of a gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we will note below,  the construction of anyons (particles with fractional statistics) in 2+1 dimensions has a close  parentage to the Wu-Yang example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Examples of statistical transmutation were known to exist  in 1+1 dimensional theories where a system of hard-core bosons was shown to be equivalent to a  theory of free fermions using the Jordan-Wigner transformation [22] which represents a fermion  as a composite operator of a hard-core boson and an operator that creates a kink (or soliton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This construction also underlies the fermion-boson mapping in 1+1 dimensional field theories  [128, 129, 131, 130] that we discussed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Finally, in the late 1970s it was found that  1+1 dimensional ZN spin systems harbor operators known as parafermions which obey the same  algebra shortly afterwards found to be obeyed by anyons in 2+1 dimensions [152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  45  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Basics of fractional statistics  Jon Magne Leinaas and Jan Myrheim wrote an insightful paper in 1977 in which they exam-  ined the structure of the configuration space of the histories of a system of N identical particles  [153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Using the Feynamn path-integral approach, they showed that if the worldlines of the iden-  tical particles are not allowed to cross, then the configuration space is topologically non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Through a detailed analysis they showed that the three and higher dimensions under a pairwise  particle exchange the states must be either even or odd and hence the particles are either bosons  or fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Leinaas and Myrheim also showed that in one and two space dimensions the wave  functions can change by a phase, nowadays known as the statistical angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In retrospect this re-  sult could have been anticipated (but was not) in an earlier paper by Michael G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Laidlaw and  C´ecile Morette DeWitt [154] who did a similar analysis of the configuration space of identical  particles in the Feynman path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Frank Wilczek generalized the Aharonov-Bohm effect [155] to describe the quantum me-  chanics of composite objects made of electric charge and magnetic flux in two space dimensions  [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek showed that composite objects made of a non-relativistic particle of charge q  bound to a magnetic flux of (magnetic) charge Φ behaves as an object with fractional angu-  lar momentum qΦ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here Φ is measured in units of the flux quantum 2π (in units in which  ℏ = c = e = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, in a subsequent paper Wilczek [157] showed that, upon an  adiabatic process in which the two composites exchange positions without their worldlines co-  inciding, the wave function of two identical flux-charge composites changes by a phase factor  exp(±iqΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance if Φ = π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a half-flux quantum) the acquired phase is equal to  exp(iπ) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' thus if the particle was a boson, the composite becomes a fermion and viceversa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Wilczek coined the term anyon to describe the behavior of an arbitrary charge-flux composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Furthermore, this construction also implies that not only fractional statistics but also fractional  spin, consistent with a generalization of the Spin-Statistics Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other terms, “flux-  attachment” implies fractional statistics (and fractional spin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Clearly Wilczek’s construction  gave an explicit physical grounding to the general arguments of the 1977 paper by Leinaas and  Myrheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is worth note the close analogy between this construction in 2+1 dimensions and the  Wu-Yang construction in 3+1 dimensions (whose consistency with the Spin-Statistics Theorem  had been shown earlier on by Alfred Goldhaber [158]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this description the statistics of the composites (the anyons) enters in the form of complex  weights (phases) given in terms of the linking numbers of the worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the concept of  fractional statistics is intimately related to the theory of knots and of the representations of the  braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These concepts were originally introduced in physics to describe statistical trans-  mutation in the theory of solitons [159] in the context of the Skyrme model [160].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yong-Shi Wu  [161] developed an explicit connection between the work of Leinaas and Myrheim and Wilczek’s  work on anyons in terms of operations acting on the worldlines of the anyons and described by  the Braid Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wu’s work and the somewhat early paper by Wilczek and Zee on the statistics  of solitons [162] marked the definite entry of the theory of knots (and of the braid group) in  physics in general and in condensed matter physics in particular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The classification of anyons  in terms of representations of the braid group labeled by the fractional statistics (determined by  linking numbers) as well as of fractional (or topological) spin (determined by the writhing num-  ber of the worldlines) leads to a rich set of physical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As it will turn out, Wilczek’s  anyons are described one-dimensional representations of the braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we will see below,  additional and intriguing (non-abelian) representations will also play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' What is a topological field theory  We will now consider a special class of gauge theories known as topological field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These theories often (but not always) arise as the low energy limit of more complex gauge theo-  ries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general, one expects that at low energies the phase of a gauge theory be either confining  or deconfined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While confining phases have (from really good reasons!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') attracted much atten-  tion, deconfined phases are often regarded as trivial, in the sense that the general expectation  is that their vacuum states be unique and the spectrum of low lying states is either massive or  massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider a gauge theory whose action on a manifold M with metric tensor tensor  gµν(x) is  S =  �  M  dDx √g L(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aµ)  (221)  46  At the classical level,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the the energy-momentum tensor T µν(x) is the linear response of the action  to an infinitesimal change of the local metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  T µν(x) ≡  δS  δgµν(x)  (222)  That a theory is topological means that depends only on the topology of the space in which is  defined and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it is independent of the local properties that depend on the metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  distances, angles, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, at least at the classical level, the energy-momentum tensor of a  topological field theory must vanish identically,  T µν = 0  (223)  In particular, if the theory is topological, the energy (or Hamiltonian) is also zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, if  the theory is independent of the metric, it is invariant under arbitrary coordinate transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Thus, if the theory is a gauge theory, the expectation values of Wilson loops will be independent  of the size and shape of the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Whether or not a theory of this type can be consistently defined  at the quantum level is a subtle problem which we will briefly touch on below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It turns out that, due to the non-local nature of the observables of a gauge theory, the low  energy regime of a theory in its deconfined phase can have non-trivial global properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what  follows, we will say that a gauge theory is topological if all local excitations are massive (and in  fact we will send their mass gaps to infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The remaining Hilbert space of states is determined  by global properties of the theory, including the topology of the manifold of their space-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  several cases, the effective action of a topological field theory does not depend on the metric of  the space-time, at least at the classical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In all cases, the observables are non-local objects,  Wilson loops and their generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chern-Simons Gauge Theory  Gauge theories play a key role in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2+1 dimensions it is possible to define a special  gauge theory which is odd under time reversal invariance and parity: Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Originally introduced in Quantum Field Theory in1982 by Stanley Deser, Roman Jackiw and  Stephen Templeton [163, 164], Chern-Simons gauge theory can be defined for any compact Lie  group, as well as an extension of Einstein’s gravity in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1989 Edward Witten  [165] showed that Chern-Simons gauge theory computes the expectation values of configurations  of Wilson loops, regarded as the worldlines of heavy particles, in terms of a set of topological  invariants known as the Jones polynomial that classify knots in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will consider the simplest case, the U(1) Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Chern-Simons  action for a U(1) gauge field Aµ in 2+1 dimension is  S [A] = k  4π  �  Ω  d3x ǫµνλAµ∂νAλ +  �  Ω  d3x JµAµ  (224)  where Jµ is a set of conserved currents (representing the worldlines of a set of heavy particles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On a closed 3-manifold Ω (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a sphere, a torus, etc) the Chern-Simons action is gauge invariant  provided the parameter k (known as the level) is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the manifold Ω has a boundary,  the action is not gauge invariant at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gauge invariance is restored by additional  boundary degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This structure is general, and not just a feature of the U(1) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since the Chern-Simons action is first order in derivatives it is not invariant under time re-  versal and under parity (which in 2+1 dimensions is a reflection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These symmetries make this  theory relevant to the description of the fractional quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the absence of exter-  nal sources, Jµ = 0, at the classical level the Chern-Simons action is invariant under arbitrary  changes of coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the theory is, at least classically, a topological field  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  For a general non-abelian gauge group G the Chern-Simons action becomes  S = k  4π  �  M  d3x tr  �  AdA + 2  3A ∧ A ∧ A  �  (225)  Here, the cubic term is shorthand for  tr  �  A ∧ A ∧ A  �  ≡ tr  �  ǫµνλAµAνAλ�  (226)  for a gauge field Aµ that takes values on the algebra of the gauge group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  47  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' BF gauge theory  A closely related (abelian) gauge theory is the so-called BF theory [166] which, in a general  spacetime dimension D (even or odd), is a theory of a vector field Aµ (a one-form) and an an-  tisymmetric tensor field B with D − 2 Lorentz indices (a D2 form), known as a Kalb-Ramond  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2+1 dimensions the action of the BF theory is  S = k  2π  �  M  d3x ǫµνλBλ∂µAν  (227)  where, once again, k is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The BF gauge theory has the same content as the topological sector of a discrete Zk gauge  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this we will consider the theory of compact electrodynamics which is a U(1)  gauge theory minimally coupled to charged scalar field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The will assume that the gauge theory  is defined for a compact U(1) gauge group (meaning that the gauge flux is quantized) and that  the complex scalar field has charge integer k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As usual minimal coupling is implemented by  introducing the covariant derivative Dµ − ∂µ + ikAµ, where Aµ is the U(1) gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Deep in  the phase in which the global U(1) symmetry is spontaneously broken, usually called the Higgs  regime, the amplitude of the scalar field is frozen at its (real) vacuum expectation value φ0 but its  phase ω, representing the Goldstone mode, is unconstrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit the Lagrangian of this  theory becomes  L = |φ0|2 �  ∂µω − kAµ  �2 − 1  4e2 F2  µν  (228)  where e is the coupling constant of the gauge field (the electric charge) and Fµν is the field  strength of the gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general spacetime dimension D the charge e has units of  length(D−4)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit the gauge field becomes massive (this is the Higgs mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  theory has fluxes quantized in units of 2π/k and has only k distinct fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the Zk gauge  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Here we will consider this theory in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will use a gaussian (Hubbard-  Stratonovich) decoupling of the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (228) in terms of a gauge field Cµ to write the  Lagrangian in the equivalent form  L = −  1  4|φ0|2C2  µ + Cµ(∂µω − kAµ) − 1  4e2 F2  µν  (229)  Up to an integration by parts, we see that the phase field ω plays the role of a Lagrange multiplier  field which forces the vector field Bµ to obey the constraint ∂µCµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This constraint is solved  by writing Cµ as  Cµ = 1  2πǫµνλ∂νBλ  (230)  of a 1-form gauge field Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon solving the constraint the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (229) becomes  L = k  2πǫµνλAµ∂νBλ − 1  4e2 Fµν(A)2 −  1  32π2|φ0|2 Fµν(B)2  (231)  For spacetime dimensions D < 4 the IR the Maxwell terms for the fields Aµ and Bµ are irrelevant  in the IR and, in this limit, this theory reduces to the BF theory at level k of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(227).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore,  Zk gauge theory is equivalent to the BF theory at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, this result is essentially valid  in all dimensions with the main difference being that the field Bµ is, in general, a rank D − 2  Kalb-Ramond antisymmetric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantization of Abelian Chern-Simons Gauge Theory  By expanding the action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (224) the Lagrangian density becomes  L = k  4πǫi jAi∂0A j + A0  � k  2π B − J0  �  − JiAi  (232)  where B = ǫi j∂iA j is the local flux, J0 is a local classical density and Ji a local classical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Then, the first term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (232) implies that the spatial components of the gauge field obey  equal-time canonical commutation relations  [A1(x), A2(x′)] = i2π  k δ(x − x′)  (233)  48  The second term of the Lagrangian enforces the Gauss Law which for this theory simply  implies that the states in the physical Hilbert space obey the constraint  B(x) = 2π  k J0(x)  (234)  Thus the Gauss Law requires that a charge density necessarily has a magnetic flux attached to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In other terms, the physical states are charge-flux composites as postulated in Wilczek’s theory  [157].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the theoretical basis to the concept of flux attachment which, as we will see in  section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3, is widely used in the theory of the fractional quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The third term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (232) simply states that the Hamiltonian density is just  H = JiAi  (235)  Hence, in the absence of sources the Hamiltonian vanishes, H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Chern-Simons action is locally gauge-invariant, up to boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this let us  perform a gauge transformation, Aµ → Aµ + ∂µΦ, where Φ(x) is a smooth, twice differentiable  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then,  S [Aµ + ∂µΦ] =  �  M  (Aµ + ∂µΦ)ǫµνλ∂ν(Aλ + ∂λΦ)  =  �  M  d3x ǫµνλAµ∂νAλ +  �  M  d3x ǫµνλ∂µΦ∂νAλ  (236)  Therefore, the change is  S [Aµ + ∂µΦ] − S [Aµ] =  �  M  d3x ∂µΦF∗  µ =  �  M  d3x ∂µ(ΦF∗  µ) −  �  M  d3x Φ∂µF∗  µ  (237)  where F∗  µ = ǫµνλ∂νAλ, is the dual field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in the absence of magnetic monopoles,  this field satisfies the Bianchi identity, ∂µF∗µ = ∂µ(ǫµνλ∂νAλ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, using the Gauss  Theorem, we find that the change of the action is a total derivative and integrates to the boundary  δS =  �  M  d3x ∂µ(ΦF∗  µ) =  �  Σ  dS µΦF∗  µ  (238)  where Σ = ∂M is the boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, if Φ is a non-zero constant function on M,  then the change of the action under such a gauge transformation is  δS = Φ × flux(Σ)  (239)  Hence, the action is not invariant if the manifold has a boundary, and the theory must be supplied  with additional degrees of freedom at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Indeed, the flat connections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the solution of the equations of motion, Fµν = 0, are pure  gauge transformations, Aµ = ∂µφ, and have an action that integrates to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let the  M = D × R where D is a disk in space and R is time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The boundary manifold is Σ = S 1 × R,  where S 1 is a circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, in this case, the boundary manifold Σ is isomorphic to a cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  action of the flat configurations reduces to  S =  �  S 1×R  d2x k  4π∂0φ∂1φ  (240)  This implies that the dynamics on the boundary is that of a scalar field on a circle S 1, and obeys  periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although classically the theory does not depend on the metric, it is invariant under arbitrary  transformations of the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, any gauge fixing condition will automatically break  this large symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance, we can specify a gauge condition at the boundary in the form  of a boundary term of the form Lgauge fixing = A2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, the boundary action of the field ϕ  becomes  S [ϕ] =  �  S 1×R  d2x k  4π  �  ∂0φ∂1φ − (∂1φ2)  �  (241)  The solutions of the equations of motion of this compactified scalar field have the form φ(x1∓x0),  and are right (left) moving chiral fields depending of the sign of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This boundary theory is not  topological but is is conformally invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  49  A similar result is found in non-abelian Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of the  S U(N)k Chern-Simons theory on a manifold D × R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' where D is a disk whose boundary is Γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and  R is time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the action is  S CS[A] =  �  D×R  d3x  � k  8πtr  �  ǫµνλAµ∂νAλ + 2  3ǫµνλAµAνAλ  ��  (242)  This theory integrates to the boundary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Γ×R where it becomes the chiral (right-moving) S U(N)k  Wess-Zumino-Witten model (at level k) at its IR fixed point,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' λ2  c = 4π/k  S WZW[g] =  1  4λ2c  �  Γ×R  d2x tr  �  ∂µg∂µg−1�  +  k  12π  �  B  ǫµνλtr  �  g−1∂µg g−1∂νg g−1∂λg  �  (243)  Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' g ∈ S U(N) parametrizes the flat configurations of the Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  boundary theory is a non-trivial CFT, the chiral Wess-Zumino-Witten CFT [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vacuum degeneracy a torus  We will now construct the quantum version of the U(1) Chern-Simons gauge theory on a  manifold M = T 2 × R, where T 2 is a spatial torus, of linear size L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since this manifold  does not have boundaries, the flat connections, ǫi j∂iA j = 0 do not reduce to local gauge transfor-  mations of the form Ai = ∂iΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, the holonomies of the torus T 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the Wilson loops on  the two non-contractible cycles of the torus Γ1 and Γ2 are gauge-invariant observables:  � L1  0  dx1A1 ≡ ¯a1,  � L2  0  dx1A2 ≡ ¯a2  (244)  where ¯a1 and ¯a2 are time-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the flat connections now are  A1 = ∂1Φ + ¯a1  L1  ,  A2 = ∂2Φ + ¯a2  L2  (245)  whose action is  S = k  4π  �  dx0ǫi j¯ai∂0¯a j  (246)  Therefore, the global degrees of freedom ¯a1 and ¯a2 at the quantum level become operators that  satisfy the commutation relations  [¯a1, ¯a2] = i2π  k  (247)  We find that the flat connections are described by the quantum mechanics of ¯a1 and ¯a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A  representation of this algebra is  ¯a2 ≡ −i2π  k  ∂  ∂¯a1  (248)  Furthermore, the Wilson loops on the two cycles become  W[Γ1] = exp  �  i  � L1  0  A1  �  ≡ ei¯a1,  W[Γ2] = exp  �  i  � L2  0  A2  �  ≡ ei¯a2  (249)  and satisfy the algebra  W[Γ1]W[Γ2] = exp(−i2π/k)W[Γ2]W[Γ1]  (250)  Under large gauge transformations  ¯a1 → ¯a1 + 2π,  ¯a2 → ¯a2 + 2π  (251)  Therefore, invariance under large gauge transformations on the torus implies that ¯a1 and ¯a2 define  a two-torus target space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us define the unitary operators  U1 = exp(ik¯a2),  U2 = exp(−ik¯a1)  (252)  which satisfy the algebra  U1U2 = exp(i2πk)U2U1  (253)  50  The unitary transformations U1 and U2 act as shift operators on ¯a1 and ¯a2 by 2π, and hence  generate the large gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, the unitary operators U1 and U2 leave the  Wilson loop operators on non-contractible cycles invariant,  U−1  1 W[Γ1]U1 = W[Γ1],  U−1  2 W[Γ2]U2 = W[Γ2]  (254)  Let |0⟩ be the eigenstate of W[Γ1] with eigenvalue 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W[Γ1]∥0⟩ = |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The state W[Γ2]|0⟩ is  also an eigenstate of W[Γ1] with eigenvalue exp(−i2π/k), since  W[Γ1]W[Γ2]|0⟩ = ei2π/kW[Γ2]W[Γ1]|0⟩ = e−i2π/kW[γ2|0⟩  (255)  More generally, since  W[Γ1]W p[Γ2]∥0⟩ = e−i2πp/kW p[Γ2]|0⟩  (256)  we find that, provided k ∈ Z, there are k linearly independent vacuum states |p⟩ = W p[Γ2]|0⟩, for  the U(1) Chern-Simons gauge theory at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is denoted as the U(1)k Chern-Simons theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore the finite-dimensional topological space on a two-torus is k-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is trivial  to show that, on a surface of genus g, the degeneracy is kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We see that in the abelian U(1)k Chern-Simons theory the Wilson loops must carry k possible  values of the unit charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This property generalizes to the non-abelian theories, which are tech-  nically more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will only state some important results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, if the gauge group  is SU(2) we expect that the Wilson loops will carry the representation labels of the group SU(2),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' they will be labelled by (j, m), where j = 0, 1  2, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and the 2 j+1 values of m satisfy |m| ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, it turns out SU(2)k Chern-Simons theory has fewer states, and that the values of j are  restricted to the range j = 0, 1  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , k  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractional Statistics and Braids  Another aspect of the topological nature of Chern-Simons theory is the behavior of expec-  tation values of products of Wilson loop operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us compute the expectation value of  a product of two Wilson loop operators on two positively oriented closed contours γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will do this computation in the abelian Chern-Simons theory U(1)k in 2+1-dimensional Eu-  clidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Note that the Euclidean Chern-Simons action is pure imaginary since the action  is first-order in derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The expectation value to be computed is  W[γ1 ∪ γ2] =  �  exp  �  i  �  γ1∪γ2  dxµAµ  ��  CS  (257)  The result changes depending on whether the loops γ1 and γ2 are linked or unlinked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  section we will compute the contribution to this expression for a pair of contours γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here  we will not include the contribution to this expectation value for each contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will return  to this problem in section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 where we discuss the problem of fractional spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The expectation value of a Wilson loop on the union of two contours, as in the present case,  γ can be written as  �  exp  �  i  �  γ  dxµAµ  ��  CS =  �  exp  �  i  �  d3xJµAµ  ��  CS  (258)  where the current Jµ is  Jµ(x) = δ(xµ − zµ(t))dzµ  dt  (259)  Here zµ(t) is a parametrization of the contour γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, the expectation value of the Wilson  loop is [165]  �  exp  �  i  �  γ  dxµAµ  ��  CS ≡ exp(iI[γ]CS) = exp  �  − i  2  �  d3x  �  d3y Jµ(x)Gµν(x − y)Jν(y)  �  (260)  where Gµν(x − y) = ⟨Aµ(x)Aν(y)⟩CS is the propagator of the Chern-Simons gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the  loops are closed, the current Jµ is conserved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ∂µJµ = 0, and the effective action I[γ]CS of the  loop γ is gauge-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Euclidean propagator of Chern-Simons gauge theory (in the Feynman gauge) is  Gµν(x − y) = 2π  k G0(x − y)ǫµνλ∂λδ(x − y)  (261)  51  where G0(x − y) is the propagator of the massless Euclidean scalar field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which satisfies  − ∂2G0(x − y) = δ3(x − y)  (262)  Using these results,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we find the following expression for the effective action  I[γ]CS =π  k  �  d3x  �  d3y Jµ(x)Jν(y)G0(x − y)ǫµνλ∂λδ(x − y)  =π  k  �  γ  dxµ  �  γ  dyνǫµνλ∂λG0(x − y)  (263)  Since the current Jµ is conserved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it can be written as the curl of a vector field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' as  Jµ = ǫµνλ∂νBλ  (264)  In the Lorentz gauge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ∂µBµ = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we can write  Bµ = ǫµνλ∂νφλ  (265)  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Jµ = −∂2φµ  (266)  where  φµ(x) =  �  d3y G0(x − y)Jµ(y)  (267)  Upon substituting this result into the expression for Bµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we find  Bµ =  �  d3yǫµνλ∂νG0(x − y)Jλ(y) =  �  γ  ǫµνλ∂νG0(x − y)dyλ  (268)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the effective action I[γ]CS becomes  I[γ]CS = π  k  �  γ  dxµ  �  γ  dyνǫµνλ∂λG0(x − y) = π  k  �  γ  dxµBµ(x)  (269)  Let Σ be an oriented open surface of the Euclidean three-dimensional space whose boundary is  the oriented loop (or union of loops) γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ∂Σ = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, using Stokes Theorem we write in the  last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (269) as  I[γ]CS = π  k  �  Σ  dS µǫµνλ∂νBλ = π  k  �  Σ  dS µJµ  (270)  The integral in the last line of this equation is the flux of the current Jµ through the surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore, this integral counts the number of times nγ the Wilson loop on γ pierces the surface Σ  (whose boundary is γ), and therefore it is an integer, nγ ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will call this integer the linking  number (or Gauss invariant) of the configuration of loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the expectation value  of the Wilson loop operator is  W[γ]CS = eiπnγ/k  (271)  The linking number is a topological invariant since, being an integer, its value cannot be changed  by smooth deformations of the loops, provided they are not allowed to cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now see that this property of Wilson loops in Chern-Simons gauge theory leads to the  concept of fractional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider a scalar matter field that is massive and charged  under the Chern-Simons gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The excitations of this matter field are particles that couple  minimally to the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will be interested in the case in which these particles are  very heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In that limit, we can focus on states that have a few of this particles which will be in  their non-relativistic regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Consider, for example, a state with two particles which in the remote past, at time t = −T →  −∞, are located at two points A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This initial state will evolve to a final state at time t = T →  ∞, in which the particles either go back to their initial locations (the direct process), or to another  one in which they exchange places, A ↔ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At intermediate times, the particles follow smooth  worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These two processes, direct and exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There we see that the direct process is  equivalent to a history with two unlinked loops (the worldlines of the particles), whereas in the  exchange process the two loops form a link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It follows from the preceding discussion that the  52  two amplitudes differ by the result of the computation of the Wilson loop expectation value for  the loops γ1 and γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us call the first amplitude Wdirect and the second Wexchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The result is  Wexchange = Wdirecte±iπ/k  (272)  where the sign depends on how the two worldlines wind around each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  An equivalent interpretation of this result is that if Ψ[A, B] is the wave function with the two  particles at locations A and B, the wavefunction where their locations are exchanged is  Ψ[B, A] = e±iπ/kΨ[A, B]  (273)  where the sign depends on whether the exchange is done counterclockwise or clockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Clearly,  for k = 1 the wave function is antisymmetric and the particles are fermions, while for k → ∞  they are bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At other values of k the particles obey fractional statistics and are called anyons  [156, 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The phase factor φ = ±π/k is called the statistical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Notice that, while for fermions and bosons the statistical phase ϕ = 0, π is uniquely defined  (mod 2π), for other values of k the statistical angle is specified up to a sign that specifies how  the worldlines wind around each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, mathematically the exchange process is known  as a braid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Processes in which the worldlines wind clock and counterclockwise are braids that  are inverse of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Braids can also be stack sequentially yielding multiples of the phase  ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition to stacking braids, Wilson loops can be fused: seen from some distance, a pair of  particles will behave as a new particle with a well defined behavior under braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This process  of fusion is closely related to the concept of fusion of primary fields in Conformal Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Furthermore, up to regularization subtleties [167], the self-linking terms (those with a = b)  yield a topological spin 1/2k, consistent with the spin-statistics connection [168].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For k = 1 this  means that the flux-charge composites have spin 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  What we have just described is a mathematical structure called the Braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The example  that we worked out using abelian Chern-Simons theory yields one-dimensional representations  of the Braid group with the phase ϕ being the label of the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For U(1)k there are k  types of particles (anyons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' That these representations are abelian means that, in the general case  of U(1)k, acting on a one-dimensional representation p (defined mod k) with a one-dimensional  representation q (also defined mod k) yields the representation one-dimensional p+q (mod k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We  will denote the operation of fusing these representations (particles!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') as [q]mod k × [p]mod k = [q +  p]mod k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These representations are in one-to-one correspondence with the inequivalent charges of  the Wilson loops, and with the vacuum degeneracy of the U(1)k Chern-Simons theory on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A richer structure arises in the case of the non-abelian Chern-Simons theory at level k [165],  such as SU(2)k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, for SU(2)1 the theory has only two representations, both are one-  dimensional, and have statistical angles ϕ = 0, π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, for SU(2)k, the content is more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of SU(2)2 the theory has  a) a trivial representation [0] (the identity, (j, m) = (0, 0)), b) a (spinor) representation [1/2]  ((j, m) = (1/2, ±1/2)), and c) a the representation [1] ((j, m) = (1, m), with m = 0, ±1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  states will fuse obeying the following rules: [0] × [0] = [0], [0] × [1/2] = [1/2], [0] × [1] = [1],  [1/2]×[1/2] = [0]+[1], [1/2]×[1] = [1/2], and [1]×[1] = [0] (note the truncation of the fusion  process!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Of particular interest is the case [1/2] × [1/2] = [0] + [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case we have two fusion  channels, labeled by [0] and [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The braiding operations now will act on a two-dimensional  Hilbert space and are represented by 2 × 2 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is an example of a non-abelian repre-  sentation of the braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These rather abstract concepts have found a physical manifestation  in the physics of the fractional quantum Hall fluids, whose excitations are vortices that carry  fractional charge and anyon (braid) fractional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Why this is interesting can be seen by considering a Chern-Simons gauge theory with four  quasi-static Wilson loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance in the case of the SU(2)2 Chern-Simons theory the Wilson  loops carry the spinor representation, [1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we call the four particles A, B, C and D, we  would expect that their quantum state would be completely determined by the coordinates of  the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This, however, is not the case since, if we fuse A with B, the result is either a  state [0] or a state [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, if the particles were prepared originally in some state, braiding  (and fusion) will lead to a linear superposition of the two states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This braiding process defines  a unitary matrix, a representation of the Braid Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The same is true with the other particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, it turns out that for four particles there are only two linearly independent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  two-fold degenerate Hilbert space of topological origin is called a topological qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  53  Moreover, if we consider a system with N (even) number of such particles, the dimension of  the topologically protected Hilbert space is 2  N  2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, for large N, the entropy per particle  grows as 1  2 ln 2 = ln  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore the qubit is not an “internal” degree of freedom of the  particles but a collective state of topological origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Interestingly, there are physical systems,  known as non-abelian fractional quantum Hall fluids that embody this physics and are accessible  to experiments!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these reasons, the non-abelian case has been proposed as a realization of a  topological qubit [169, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological Phases of Matter  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological Insulators  We will now give a brief discussion of the physics of Topological Insulators from a field-  theoretic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Topological insulators are systems whose electronic states (band struc-  tures) have special topological properties which manifest in the existence of symmetry-protected  edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For this reason these systems are known as symmetry-protected topological states  (or SPTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The simplest example is found in one space dimension where it is related to the fas-  cinating problem of fractionally charged solitons and electron fractionalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In several ways  many of the concepts involved in these 1+1-dimensional systems can and have been extended to  higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac fermions in 2+1 dimensions  We will now see that, in spite of the formal similarities withe the 1+1 dimensional case, this  theory has different symmetries, particularly concerning parity and time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  addition, in 2+1 dimensions it is not possible to define a γ5 Dirac matrix which implies that there  is no chiral symmetry and no chiral anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will consider first a theory of a Dirac field  which in 2+1 dimensions is a bi-spinor (as is in 1+1 dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Lagrangian of the Dirac  theory coupled to a background gauge field Aµ is  L = ¯ψiγµ∂µψ − m ¯ψψ − eAµ ¯ψγµψ ≡ ¯ψiγµDµ(A)ψ − m ¯ψψ  (274)  where ¯ψ = ψ†γ0, ¯ψγµψ ≡ jµ is the gauge-invariant (and conserved) Dirac current, and Dµ(A) =  ∂µ + ieAµ is the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (274), γµ are the three 2 × 2 Dirac matrices which obey the algebra, {γµ, γν} = 2gµν,  where gµν = diag(1, −1, −1) is the metric of 2+1 dimensional Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Dirac  matrices γµ can be written in terms of the three 2 × 2 Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance we can choose  γ0 = σ3, γ1 = iσ2 and γ2 = −iσ1 which satisfy the Dirac algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the three gamma  matrices involve all tree Pauli matrices, it is not possible to define a γ5 matrix which would  anticommute with the gamma matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this sense, it is not possible to define a chirality for  bi-spinors in 2+1 dimensions  We could have also chosen a different set of gamma matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance, we could have  chosen γ0 = σ3, γ1 = iσ2 and γ2 = +iσ1 which also satisfy the Dirac algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These two choices  are equivalent to the 2D parity transformation x1 → x1 and x2 → −x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, we can  choose the three gamma matrices to be defined in terms of a right handed or a left handed frame  (or triad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the choice of handedness of the frame used to define the gamma matrices can  be regarded as a chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The parity transformation is also equivalent to a unitary transformation (a change of basis)  U = γ1 = iσ1 which flips the sign of both γ2 and γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, a parity transformation is equivalent  to a change of the sign of γ0 or, what is the same, as changing the sign of the Dirac mass  m → −m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the Dirac theory is charge conjugation invariant, the parity transformation is  equivalent to time reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to see that in 2+1 dimensions the massive Dirac theory  is not invariant under time reversal since this the single particle Dirac Hamiltonian H(p) for  momentum p involves all three Pauli matrices,  H(p) = α · p + βm  (275)  where, as usual, α = γ0γ and β = γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, time reversal is equivalent to the change of the  sign of the Dirac mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These considerations also apply to the massless Dirac theory coupled to  a background gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  54  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac Fermions and Topological Insulators  In addition systems in one space dimension, discussed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1, in which Dirac fermions  naturally describe the low energy electronic degrees of freedom, Dirac fermions also play a role  in many other system in Condensed Matter Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such systems range from the nodal Bogoli-  ubov fermionic excitations of d-wave superconductors in 2D and p-wave superfluids in 3D, to  Chern and Z2 topological insulators in 2D and Z2 topological insulators and Weyl semimetals in  3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The also play a role in spin liquid phases of frustrated spin-1/2 quantum antiferromagnets,  such as Dirac and chiral spin liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac fermions play also a significant role in our under-  standing of the compressible limit of 2D electron gases (2DEG) in large magnetic fields (see  section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will not cover here all of these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instead we will focus of the 2D  Chern topological insulators (which exhibit the anomalous quantum Hell effect) and on the 3D  Z2 topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Condensed matter Physics the important low energy electronic degrees of freedom belong  to the electronic states close to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In such systems the lattice periodic potential  determines the properties of their band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only significant exception to this general  rule is the case of the 2DEGs in GaAs-AlAs heterostructures (the most commonly used platform  for the study of quantum hall effects) whose electronic densities are low enough so that the lattice  periodic potential can for practical purposes almost always be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In general, Dirac (and Weyl) fermions arise when the conduction and valence bands cross  at isolated points of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In such situations, the near the crossing points the low  energy electronic states locally (in momentum space) look like cones and, ignoring possible  anisotropies, look like the states of a Dirac-Weyl theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these reasons, Dirac fermions play  a central role in the theory of graphene type materials [171] and, more significantly, in the theory  of topological insulators [172].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Dirac fermions play a central role in Quantum Electrodynamics and in the Standard Model of  Particle Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The problem of quark confinement requires an understanding of these theories  in a regime inaccessible to Feynman-diagrammatic perturbation theory and an intrinsically non-  perturbative formulation, known as Lattice Gauge Theory [55, 56, 173], needed to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  For this reason in the 1970s fermionic Hamiltonians with crossings at points became of great  interest in High Energy Physics as a way to describing the short-distance dynamics of quarks in  Lattice Gauge Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This program run into difficulties when extended to the theory of Weak Interactions which  have an odd number of species of chiral (Weyl) Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' All local discretizations of the  Dirac equation yield an even number of species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This fact came to be known as the fermion  doubling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These results are special cases of a general theorem, due to Holger Nielsen  and Masao Ninomiya [174, 175, 176] (and extended by Daniel Friedan [177]), which proves that  for systems whose kinetic energy is local in space (as it must be) there must always be an even  number of crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, it is impossible to write a local theory with an odd number of  chiral fermionic species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the context of Condensed Matter Physics, the simplest example of Dirac fermions is found  in the electronic structure of graphene (a single layer of graphite) discussed by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Castro Neto  and coworkers [171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Graphene is an allotrope of crystalline carbon in which the carbon atoms  are arranged in a 2D hexagonal lattice, which is a lattice with two inequivalent sites in its unit  cell, labeled by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let rA and rB be the sites of the two sublattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Each site rA has three  nearest neighbors on the B sublattice located at ri  B = rA + di, where i = 1, 2, 3 and  d1 =  � 1  2  √  3  , 1  2  �  ,  d2 =  � 1  2  √  3  , −1  2  �  ,  d3 =  �  − 1√  3  , 0  �  (276)  (in units in which the lattice spacing is a = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The nearest neighbor A sites are related by the  vectors  a1 =  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  √  3  2 , −1  2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 ,  a2 = (0, 1),  a3 =  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  √  3  2 , −1  2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (277)  where a1 and a2 are the primitive lattice vector of the hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The nearest neighbor  sites of the B sublattice are also related by these same three vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The low energy electronic degrees of freedom of graphene can be described by a single  fermionic state (ignoring spin) at each carbon atom .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let c†(rA) and c†(rB) be the fermion oper-  ators that creates an electron at the site rA and at the site nearest neighbor sites rB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Hamiltonian is  H = t1  �  rA,i=1,2,3  c†(rA)c(rA + di) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (278)  55  where t1 is the hopping matrix element between nearest neighbor A and B sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Fourier space  c(rA) =  �  BZ  d2k  (2π)2ψA(k) exp(ik · rA),  c(rB) =  �  BZ  d2k  (2π)2 ψB(k) exp(ik · rB)  (279)  where BZ is the first Brillouin zone of the hexagonal lattice, a hexagonal region of reciprocal  space with vertices at ±2π/  √  3(1, 1/  √  3) (which are denoted by K and K′, respectively) and their  two images under 2π/3 rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In momentum space the Hamiltonian is [178]  H = t1  �  BZ  d2k  (2π)2  �  ψA(k)  ψB(k)  � �  0  �  j=1,2,3 exp(ik · d j)  �  j=1,2,3 exp(−ik · d j)  0  � �ψA(k)  ψB(k)  �  (280)  The one-particle states of this Hamiltonian have eigenvalues E(k) = ±  ����  �  j=1,2,3 exp(ik · d j  ����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Clearly E(k) vanishes at the K and K′ points, the corners of the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, there  is a crossing of the two bands at the K and K′ points near which the energy eigenvalues are a two  cones at K and K′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the low energy states of graphene consists of two massless  (gapless) Dirac bi-spinors, with opposite chirality/parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Other examples with similar fermion content are a theory of fermions on a 2D square lattice  with a π magnetic flux (1/2 of the flux quantum) per plaquette which arises, for instance, in the  theory of the integer quantum Hall effect on lattices by David Thouless, Mahito Kohmoto, Peter  Nightingale and Marcel den Nijs [179] (albeit for more general flux per plaquette), and in the  theory of flux phases of 2D spin liquids of Ian Affleck and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Brad Marston [180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It also arises  in the theory of the chiral spin liquid of Xiao-Gang Wen, Frank Wilczek and Anthony Zee [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will discuss these theories in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chern invariants  The single-particle quantum states of free fermions (electrons) in the periodic potential of a  crystal are Bloch wave functions labeled by the lattice momentum k and band indices m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  Bloch states are periodic functions of the lattice momenta whose periods are the Brillouin zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Topologically each Brillouin zone is a torus: in 1D is a 1-torus, in 2D a 2-torus, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The sym-  metries (and shapes) of the Brillouin zone are dictated by the symmetries of the host crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this section we will consider the quantum states of systems of fermions on a 2D lattice with  M bands with eigenvalues {Em(k)} with m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The eigenstates are Bloch states {|um(k)⟩}  such that the wave functions ar ψm(x) = um(k) exp(ik · x), where k is a (quasi) momentum in the  first Brillouin zone [182].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the band spectrum is such that all the bands are  separated from each other by a finite gap for all momenta in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the  eigenvalues obey the inequality |Em(k) − En(k)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the system of interest  has N < M filled bands and, consequently, that there is a gap separating the top-most occupied  band N (the “valence band”) and the lowest unoccupied band N + 1 (the “conduction band”) that  does not close everywhere in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider specifically a 2D system and a Bloch state |um(k)⟩ at a momentum k, and let  ∂k|um(k)⟩ be an infinitesimally close Bloch state with momentum k′ = k+dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The inner product  of these two infinitesimally close Bloch states is the vector field (defined on the first Brillouin  zone for each band m)  A(m)  j (k) = i⟨um(k)|∂ j|um(k)⟩  (281)  where j = 1, 2 are the orthogonal direction in momentum space and we have used the notation  ∂i = ∂/∂ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This vector field is known as the Berry Connection of the electronic states in the mth  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The theory of the quantum states of non-interacting electrons in periodic potentials as devel-  oped by Felix Bloch [182] is the foundation of much of what we know about the physics of many  semiconductors and simple metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This theory is the core subject of many textbooks [183, 184].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This theory is based on the classification of the electronic states as representations of the group  of lattice translations and point (and spatial) group symmetry transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A key unstated  assumption in much of this body of work is that the Bloch states are globally well-defined func-  tions on the Brillouin zone of a given crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is rather remarkable that this assumption was not  stated explicitly since the original work by Bloch in 1929 until the work by Thouless, Kohmoto,  Nightingale and den Nijs on the quantum states of electrons on 2D lattices in the presence of an  uniform magnetic field [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The realization that there is a topological obstruction to define the  56  electronic states globally in the Brillouin zone is the key to the development of the the theory of  topological insulators and, more generally, of the modern theory of band structures [185, 186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In quantum mechanics states are defined up to a phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that Bloch states that  differ by a phase describe the same quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words each quantum state |um(k)⟩ is  a member of a ray (or fiber) of states each labeled by a phase,  |um)(k)⟩ �→ exp(ifm(k)) |um(k)⟩  (282)  Changing the states by phase factors defines a U(1)M unitary transformation of physically equiva-  lent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since we have a fibre at each point k, this amount at defining a fibre bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However,  under this transformation the Berry connection A(m)  j (k) changes by a gradient of the phases  A(m)  j (k) �→ A(m)  j (k) + ∂ j fm(k)  (283)  where we assumed that the phases fm(k) are continuous and differentiable functions on the entire  first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the physical states cannot change by redefinitions of the phases of  the basis states we are led to the condition that only the data that is invariant under the gauge  transformations defined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (282) and (283) is physically meaningful, which is encoded in  the gauge-invariant pseudo-scalar quantity F (m)(k)  F (m)(k) = ǫi j∂iA(m)  j (k)  (284)  which is known as the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In what follows we will be interested in the quantity  C(m)  1  = 1  2π  �  Γ  dk jA(m)  j (k) = 1  2π  �  BZ  d2k F (m)(k)  (285)  where Γ is the boundary of the 1st Brillouin zone (which we denote by BZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now show  that the quantity C(m)  1 , which measure the flux of the Berry connection A(m)  j (k) through the first  Brillouin zone (in units of 2π), is an integer independent of the particular connection A(m)  j (k) that  we have chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This integer-valued quantity is a topological invariant known as the first Chern  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, since the first Brillouin zone is a 2-torus ,which is a closed manifold, a Berry  connection with a net flux cannot obey periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instead, we must allow  for generalized (large) gauge transformations that wrap around the 2-torus of the first Brillouin  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem is similar (and closely related to) the problem of the wave functions of a  charged particle moving in the presence of a magnetic monopole [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will consider a 2D  system whose first Brillouin zone is spanned by two reciprocal lattice vectors b1 and b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' related  by the two primitive lattice vectors a1 and a2 by the relation bi · a j = 2πδi j (with i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2) The  generalized gauge transformations now are  |um(k + b1)⟩ = exp(if (1)  m (k)) |um(k)⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  |um(k + b2)⟩ = exp(if (2)  m (k)) |um(k)⟩  A(m)  j (k + b1) =A(m)  j (k) + ∂ j f (1)  m (k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A(m)  j (k + b2) = A(m)  j (k) + ∂ j f (2)  m (k)  (286)  where f (1)  m (k) and f (2)  m (k) are smooth functions of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now show that the Bloch states cannot be defined globally over the Brillouin zone if  the flux of the Berry connection does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' More specifically let us assume that at some  point k0 ∈ T we know the Bloch state |um(k0)⟩ which satisfies the generalized periodic boundary  conditions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(286).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The question is, can we determine the Bloch state at some other also  arbitrary point k′  0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now that there is a topological obstruction that does not allow it if  the flux of the Berry phase is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The reason is that the phase of the Bloch state cannot be  defined since, in general, the Bloch state will vanish at some point of the Brillouin zone where  the phase is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will prove that this si true by following an elegant construction due to Mahito Kohmoto  which goes as follows [187].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Brillouin zone is a torus, that we will denote by T, can be  regarded as the tensor product of two circles, T ≡ S 1 × S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us split the torus T into two  disjoint subsets (or patches) HI and HII such that T = HI  � HII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that in region  HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us assume that the Bloch state vanishes at some point k0 ∈ TI and that the Bloch state  does not vanish for all points k ∈ HII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that we can choose the Bloch state |um(k)⟩  to be real for all k ∈ HII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand we can always assign some arbitrary phase to the  57  Bloch state at k0 ∈ TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Once we have done that we can extend the definition of the phase to a  neighborhood of k0 wholly included in HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we the assume that there is only one zero, then  the phase can be defined over all of HI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The result is that we now have two different definitions  of the phase of the Bloch state on HI and on HII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let |um(k)⟩I and |um(k)⟩II be the two resulting  definitions of the Bloch state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let the closed curve γ be the common boundary of regions HI  and HII, γ = ∂HI = ∂HII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However on the common boundary γ these the two definitions of the  Bloch state must be a gauge transformation,  |um(k)⟩I = exp(ifm(k)) |um(k)⟩II  (287)  on all points k ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The gauge transformation fm(k), known as the transition function, is a  smooth periodic function of the points k on the closed curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Likewise, the Berry connection  A(m)  j (k) has two definitions on the regions HI and HII which differ by a gauge transformation on  all the points k of the common boundary γ,  A(m)  j (k)  ����I − A(m)  j (k)  ����II = ∂ j fm(k)  (288)  The transition function defines a mapping of the closed curve γ, which is topologically equivalent  to a circle S 1, to the phase of the Bloch state which is defined mod 2π and hence is also a circle  S 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence the transition functions fm(k) are homotopies which can be classified by the homotopy  group Π1(S 1) ≃ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We recognize that the classification of the the transition functions is the same  that we used for the vortices of section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the change of the transition function on a  full revolution of the closed curve γ must be an integer multiple of 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now compute the quantity C(m)  1 , given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (285), which measures the flux of the  Berry connection through the 2-torus T that defines the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Using the partition  of the torus T = TI  � TII we can write  C(m)  1  = 1  2π  �  T  d2k F (m)(k) = 1  2π  ��  HI  d2k F (m)(k) +  �  HII  d2k F (m)(k)  �  (289)  Using Stokes Theorem on the two regions HI and HII we get  C(m)  1  = 1  2π  �  γ  dk j  �  A j(k)  ����I − A j(k)  ����II  �  (290)  Since the two definitions of the Berry connection differ by a gauge transformation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (288), we  can express C(m)  1  in terms of the transition function fm(k) on the curve γ  C(m)  1  = 1  2π  �  γ  dk · ∂fm(k) = 1  2π (∆fm(k))γ = n  (291)  where we used that the transition functions are classified by the integer-valued quantity n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that the flux of the Berry curvature C(m)  1  takes only integer values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is known of  the first Chern number C(m)  1  which is a topological invariant since it cannot change by a smooth  redefinition of the curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since C(m)  1  � 0 implies that the Bloch states must vanish at least  at some point (or points) k ∈ HI, then the integer-valued Chern number can only change if a  redefinition of the curve γ crosses at least one point k at which the Bloch state vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The conclusion of this analysis is that whenever the flux of the Berry curvature through the  Brillouin zone does not vanish the Bloch states cannot be defined globally on the Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A direct consequence is that in this case the band is characterized by a topological invariant  called the first Chern number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This number is a property of a given band and, in general, it is  different for each band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a particular case of a more general topological classification of  the states of free fermions on a lattice which depends on on the dimensionality of the system  [188, 189, 190].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The quantum Hall effect on a lattice  We will now show that 2D free fermion systems with Chern bands, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' bands characterized  by a non-vanishing Chern number, are insulators that have a quantized Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will  do this for the theory of the integer quantum Hall effect on a 2D lattice of Thouless, Kohmoto,  Nightingale and den Nijs (TKNN) [179, 187] (see, also, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [191]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although this problem  played a key role in the development of the theory of topological phases of matter, for many  58  years it was viewed as an academic problem since it would require gigantic magnetic fields to  be in the regime of interest for typical solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The situation changed with the development of  twisted bilayer graphene and similar systems which have unit cells large enough for this physics  to be observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These new materials has allowed to study this problem experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  TKNN considered a system of free fermions on a planar (actually square) lattice in a uniform  magnetic field B with flux 2πp/q per plaquette (in units in which the flux quantum φ0 = eℏ/c = 1)  with p and q two co-prime integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The tight-binding Hamiltonian for this simple problem is  H =  �  r, j=1,2  t j c†(r) eiA j(r) c(r + e j) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (292)  where c(r) and c†(r) are fermion creation and annihilation operators on the lattices sites {r =  (m, n)}, e j are lattice unit vectors, and t j, with j = x, y, is a hopping matrix element between  nearest neighbor sites of the square lattice along the x and y directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On each link of the  lattice we defined a vector potential A j(r), where the oriented sum of the lattice vector potentials  on each plaquette 2πφ, where φ = p  q, is the flux on each plaquette of the square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  axial (Landau) gauge A1(m, n) = 0, A2(m, n) = 2πmφ is a periodic function of m with period q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this gauge the Hamiltonian becomes  H =t  �  m,n  �  c†(m + 1, n)c(m, n) + c†(m, n)c(m + 1, n)  �  +t  �  m,n  �  c†(m, n + 1) ei2πφm c(m, n) + c†(m, n) e−i2πφm c(m, n + 1)  �  (293)  In this gauge the problem reduces to a system with a theory with a q × 1 unit cell with q inequiv-  alent sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In momentum space, the (magnetic) Brillouin zone of this system is − π  q ≤ k1 ≤ π  q and  −π ≤ k2 ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Fourier transform of the operators c†(r) and c(r) are, respectively, c†(k) and  c(k) which obey the standard anticommutation relations, {c(k), c(q)} = {c†(k), c†(q)} = 0, and  {c†(k), c(q)} = δ(k − q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In momentum space the Hamiltonian becomes  H = q  � π/q  −π/q  dkx  2π  � π  −π  dky  2π  �H(kx, ky)  (294)  where  �H(kx, ky) =1  q  q−1  �  n=0  �  2tx cos(kx + 2πφn) c†(kx + 2πφn, ky) c(kx + 2πφn, ky)  + ty  �  e−iky c†(kx + 2π(n + 1)φ, ky) c(kx + 2πnφ, ky) + eiky c†(kx + 2π(n − 1)φ, ky) c(kx + 2πnφ, ky)  ��  (295)  The spectrum of this system was first investigated by Hofstadter [192] consists of q bands which  for q an odd integer are separated by finite energy gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  For fixed values of kx and ky (in the magnetic Brillouin zone), the Hamiltonian �H(kx, ky) is the  tight-binding model of a one-dimensional chain on the 1D lattice of q sites located at kx + 2πnφ  with nearest neighbor hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The single-particle (Bloch) states un(k) of this 1D model obeys  the Schr¨odinger Equation  2tx cos(kx + 2πnφ)un(k) + ty  �  e−iky un−1(k) + eiky un+1(k)  �  = Enun  (296)  which is known as Harper’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general this equation does not admit an analytic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, the nature qualitative features of the spectrum can be obtained by an expansion either  on tx/ty or on ty/tx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' TKNN used degenerate perturbation theory to show that the spectrum has q  bands and that the r-th band is characterized by two integers sr and ℓr which are the solution of  the Diophantine equation  r = q sr + p ℓr  (297)  with ℓ0 = s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It isl also obvious that for r = q sq = 1 and ℓq = 0 (for all p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Furthermore, TKNN showed that there was an, until then unsuspected, relation between the  Hall conductivity σxy of a gapped system of electrons in periodic potentials in a uniform magnetic  field and the Berry connection of the filled bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This relation implies the quantization of the  59  Hall conductivity and its computation in terms of a topological invariant, the Chern number of the  occupied bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They showed that in this context each band has a non-trivial Berry connection  A(r)  j = i⟨ur(k)|∂ j|ur(k⟩ (with ∂ j = ∂kj) of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (281) with a non-vanishing (first) Chern  number C(r)  1 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the flux through the magnetic Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The conductivity tensor characterizes the electrical properties of a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Linear  Response Theory provides a framework for computing the conductivity tensor by perturbing the  system with a weak external electromagnetic field Aµ and computing the currents that they induce  [193, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The expectation value of the gauge-invariant current operator Jµ(x) is  ⟨Jµ(x)⟩ = − i  ℏ  δ  δAµ(x) ln Z[Aµ]  (298)  where Z[Aµ] is the partition function in the presence of a background (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' classical) electro-  magnetic field Aµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general spacetime dimension D = d + 1, the lowest order in the vector  potential Aµ, the induced current is given in terms of the polarization tensor Πµν(x, y),  ln Z[Aµ] = i  2  �  dDx  �  dDy Aµ(x) Πµν(x, y) Aν(y) + O(A3)  (299)  Therefore, the induced current is related to the external field by  ⟨Jµ(x)⟩ ≡ Jµ(x) =  �  dDy Πµν(x, y) Aν(y)  (300)  Expressions of this type are know as a Kubo formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The polarization tensor can be regarded as  a generalized susceptibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although we are using a continuum relativistic notation, these expressions are generally  valid, even in lattice systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gauge invariance requires that the polarization tensor be con-  served,  ∂µΠµν(x, y) = 0  (301)  In general, the (retarded) polarization tensor Πµν(x, y) is related to the the (retarded) current-  current correlation function DR  µν(x, y),  Dµν(x, y) = − i  ℏΘ(x0 − y0)⟨[Jµ(x), Jν(y)]⟩  (302)  by the identity  ΠR  µν(x, y) = DR  µν(x, y) − iℏ  �δJµ(x)  δAν(y)  �  (303)  The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (303), usually called a “contact term”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This term vanishes only for a theory  of relativistic fermions (in the continuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In all other cases, lattice or continuum, relativistic  or not, the contact term does not vanish and its form depends on the specific theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In non-  relativistic systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in a Fermi liquid, this term is the origin of the f-sum rule [193, 194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  When the external field Aµ represents is a (locally) uniform electric field E, the induced  current is Ji = σi jE j, where σi j is the conductivity tensor, which can be obtained from the  polarization tensor as the limit  σi j = lim  ω→0  1  iω lim  q→0 Πi j(ω, q)  (304)  In a metal, which has a Fermi surface, the order of limits in which ω and q vanish matters and  only the order of limits specified above is the correct one to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ina Dirac systems, that we  will discuss below, the order does not matter due to relativistic invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The other case in  which the order does not matter is the Chern insulator that we are interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general, in an  isotropic system, the conductivity tensor has a symmetric part and an antisymmetric part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  symmetric part of the conductivity tensor yields the longitudinal conductivity which has all the  effects of dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The antisymmetric part does not vanish in the presence of a magnetic field  or, more generally, if time reversal symmetry is broken, and yields the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A Chern insulator is an insulator and as such is a state with an energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In such a state the  longitudinal conductivity vanishes since there is no dissipation in a gapped state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' But in a Chern  insulator time reversal invariance is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the system that we are discussing is broken by  60  the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hall conductivity can be calculated from the antisymmetric part of the  polarization tensor as the limit  σxy = lim  ω→0  i  ωΠxy(ω, 0)  (305)  In addition, the contact term does not contribute to the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As a result the hall  conductivity can be computed in terms of the antisymmetric component of the current-correlation  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  let us now return to the theory of free fermions on a lattice in a (commensurate) magnetic  field takes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw the electronic states are split into q bands with single particle states  |ψn(k)⟩ where k takes values on the first magnetic Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will compute the Hall  conductivity for this system assuming that the Fermi energy EF lies in the gap between the n-th  and the n − 1-th bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us label by α the occupied bands and by β the unoccupied bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Hence Eα(k) < EF < Eβ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, the Kubo formula for the Hall conductivity becomes  σxy = ie2  ℏ  �  Eα<EF<Eβ  �  BZ  d2k  (2π)2  ⟨uα(k)|Jy|uβ(k)⟩⟨uβ(k)|Jx|uα(k)⟩ − ⟨uα(k)|Jx|uβ(k)⟩⟨uβ(k)|Jy|uα(k)⟩  (Eα(k) − Eβ(k))2  (306)  In the lattice model the current operator is simply  J = e  ℏ∂k �H(k)  (307)  Using this form of the current and the Schr¨odinger equation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (296) we can rewrite the Kubo  formula for the Hall conductivity, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (306), as a sum over only the occupied states labeled by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The expression for the Hall conductivity now takes the more compact form  σxy = e2  ℏ  �  α  �  BZ  d2k  (2π)2ǫjl∂ j⟨uα(k)|∂k|uα(k)⟩ = e2  h  �  α  1  2π  �  BZ  F (α)(k) = e2  h  �  α  C(α)  1  (308)  Therefore, we conclude that each occupied band contributes to the Hall conductivity an amount  equal to its first Chern number (multiplied by e2/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the main result of TKNN [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the specific case of the model that we are discussing here the Chern number is given  explicitly in terms of the solution of the Diophantine equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (297)  C(r)  1 = 1  2π  �  Γ  dk jA(r)  j (k) = ℓr − ℓr−1  (309)  where the integers ℓr and ℓr−1 are the solutions of the Diophantine equation for the bands r and  r − 1 (A detailed derivation can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9], chapter 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The TKNN integers ℓr are  topological invariants of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude, with TKNN, that for a system with r filled bands the Hall conductivity σ(r)  xy is  given by the expression  σ(r)  xy = e2  h  r�  n=1  (ℓr − ℓr−1) = e2  h ℓr  (310)  thus showing that this system exhibits and integer quantum Hall effect and that the Hall conduc-  tivity is (up to the ratio of universal constants e2/h) given by a topological invariant, the first  Chern number of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude with two comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The first is that when all the bands are occupied the  Hall conductivity vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This obvious physical constraint is automatically satisfied by the  Diophantine equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The other comment concerns the case of what happens when q is an even  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case it is possible to show that there are band crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is what happens for  q = 2 which is the π flux lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in this case one has a theory of two species of Dirac bi-spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As we will see in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 what happens depends on how the gapless state is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Anomalous Quantum Hall Effect  We will now discuss a simple and insightful model of a Chern insulator that displays the  quantum anomalous Hall effect, that is a Hall effect in a system with broken time reversal sym-  metry in the absence of a uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In order to do that we will reconsider the  graphene model with additional terms in the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (280) that will open up energy  gaps in its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There are several ways to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One way requires breaking the symme-  try between the two sublattices A and B, by assigning them a site energy of ±M, which breaks  61  the mirror reflection symmetry of the hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is what happens with graphene on a boron  nitrade substrate [178] (there are other ways that involve breaking the point group symmetry  [195]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Another way to open a gap is to break time reversal symmetry which is accomplished by  complex hopping amplitudes t2 exp(iφ) between nearest neighbor A sites (and between nearest  neighbor B sites as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The phase φ is oriented from rA to rA+ai, where a1 = d2−d3, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (and  similarly for the B sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Physically, the oriented sum of the phases φ on the triangles generated  by the hopping terms t1 and t2 represent the magnetic fluxes through each triangle such that the  total flux on the elementary hexagon is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This model was considered by Haldane [196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With  these changes the one-particle Hamiltonian becomes 2 × 2 Hermitian matrix  H(k) = h0(k) 1 + h(k) · σ  (311)  where I is the 2 × 2 identity matrix and σ = (σ1, σ2, σ3) is a vector made of the three Pauli  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The functions h0(k) and h(k) = (h1(k), h2(k), h3(k) are, respectively, given by  h0(k) =2t2 cos φ  �  i=1,2,3  cos(k · ai),  h1(k) = t1  �  i=1,2,3  cos(k · di),  h2(k) =t1  �  i=1,2,3  sin(k · di),  h3(k) = M − 2t2 sin φ  �  i=1,2,3  sin(k · ai)  (312)  Provided |t2/t1| < 1/3 the spectrum of the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (280) has two bands that do not  overlap and have a finite energy gap unless they cross at the corners K and K′ the Brillouin zone  where the gap is smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The two bands cross at K and/or K′ if M = ±3  √  3t2 sin φ (for K and K′,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' When either one of these conditions are not met the spectrum is gapped at either K  or K′ or, more generally, at both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For values of the parameters for which the band gaps are small,  the low energy states of the Hamiltonian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (280) is that of a theory of two Dirac fields, that  we will label as ψ1(x) and ψ2(x), whose one-particle Hamiltonians have the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (275) but  with generally different masses m1 ∝ M − 3  √  3t2 sin φ and m2 ∝ M + 3  √  3t2 sin φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, in general,  both fields will have non-vanishing masses which can have the same sign or opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that the effective low-energy theory consist of two species of massive Dirac  bi-spinors, albeit generally with different masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To examine the electromagnetic response we  will couple the lattice model and the effective low energy theory to a slowly varying background  electromagnetic field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Lagrangian of the low energy theory is  L = ¯ψ1(i/∂ − m1)ψ1 + ¯ψ2(i/∂ − m2)ψ2 − eAµ( ¯ψ1γµψ1 + ¯ψ1γµψ1)  (313)  In general the two masses will be different in magnitude and sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Equivalently the Dirac  fermions may have the same or opposite chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will examine the electromagnetic re-  sponse for both cases in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 where we will see that in the phase in which the masses of  both Dirac fermions have the same sign this model describes a topological insulator that exhibits  the anomalous quantum Hall effect, whereas in the phase where the signs are opposite it is a  conventional insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, by varying the parameters we can tune to special values of these parameters where  one of the two masses vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 we will see that these special values of the  parameters correspond to quantum phase transitions at which the topological properties of the  ground state change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see that this is a quantum critical point with special properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In particular, time reversal invariance is explicitly broken at this quantum critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  regime the heavy Dirac fermion plays the role of the heavy regulating field as in the Pauli-Villars  regularization of the theory of a single Dirac fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the lattice model this is a manifestation  of the Nielsen-Ninomiya theorem which requires that there should be an even number of Dirac  fermions [174, 175, 177].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this context, the heavy fermion is the “doubler”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Parity Anomaly  We saw in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 that Dirac fermions in 1+1 dimensions have a chiral anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' That  statement mean that although the theory of free massless Dirac fermions has a global chiral sym-  metry, the associated current, which we called j5  µ, is not conserved when the system is coupled  to an external gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This effect happened because the right and left moving (Weyl) com-  ponents of the massless Dirac fermion cannot be separately conserved in the presence of a gauge  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Analogs of the 1+1-dimensional chiral anomaly exist on 3+1 dimensions (and, more gener-  ally, in all even spacetime dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The cancelation of these anomalies is a key condition  62  for the consistency of gauge theories [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see in section 10 that these anomalies play a  role on the theory of 3D topological insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' More generally, anomalies play a central role in  the theory of symmetry-protected topological (SPT) phases of matter [197, 198, 199].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Dirac fermions in 2+1 dimensions do not have an analog of the chiral anomaly for the simple  reason that the chiral symmetry does ot exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is, however, an anomaly involving parity  or, equivalently, time reversal invariance, known as the parity anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The parity anomaly was  discovered in the computation of the effective action of the electromagnetic field in a theory of  massive Dirac fermions in 2+1 dimensions by Redlich [200, 201].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Consider theory on a single massive Dirac fermion (a bi-spinor) ψ(x) in 2+1 dimensions  coupled to a background U(1) gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Lagrangian is  L[ψ, ¯ψ, Aµ] = ¯ψi /Dψ − m ¯ψψ  (314)  where the covariant derivative is Dµ = ∂µ + iAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The effective action S eff[Aµ] of the gauge field  Aµ is formally obtained by computing the path integral  Z[Aµ] = exp(iS eff[Aµ]) =  �  DψD ¯ψ exp(i  �  d3x L[ψ, ¯ψ, Aµ]) = Det(i /D[Aµ] − m)  (315)  where we used that the theory is free to write the partition function as a functional determinant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  1+1 dimensions, in the case of a massless theory, this determinant can be computed exactly using  the heat kernel (or zeta-function) technique [202, 203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aside from important technicalities, the  reason for why this works is that as a consequence of the chiral anomaly, the effective action  computed at one loop order in the fermions is exact in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In higher dimensions  this is no longer true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will sketch the computation of the effective action to one loop order, which means  quadratic in the gauge field Aµ, which has the form  S eff[Aµ] = 1  2  �  d3x  �  d3y Aµ(x) Πµν(x − y) Aν(y) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (316)  where the ellipsis contains the higher order contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The polarization operator Πµν(x − y)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='in momentum space is (here pµ is the momentum transfer) by the one-loop result  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Πµν(p) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='d3q  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(2π)3 tr �S (q + p/2)γµS (q − p/2)γν�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(317)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='The symbol tr signifies the trace over the spinor labels and S (p) is the (massive) Feynman prop-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='agator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='S (p) =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='/p − m + iǫ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(318)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='The polarization tensor Πµν(p) has the explicit decomposition into a parity even and a parity odd  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='contributions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Πµν(p) = (p2gµν − pµpν) Π0(p2) − iǫµνλpλ ΠA(p2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(319)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='where p2 = p2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='0 − p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The polarization tensor is manifestly transverse, pµΠµν(p) = 0 and, hence,  the one-loop effective action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (316) is gauge-invariant (as it should be).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The parity even  Π0(p2) and parity odd ΠA(p2) kernels are given by  Π0(p2) = − |m|  4πp2 +  1  8π  �  p2  �4m2  p2 + 1  �  sinh−1  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  �  p2  �  4m2 − p2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (320)  ΠA(p2) = −  m  2π  �  p2 sinh−1  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  �  p2  �  4m2 − p2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (321)  These two results are obtained in the limit in which the regulator scales are sent to infinity leaving  behind these finite results, which hold provided the Dirac mass m does not vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the low energy regime, |p2| ≪ m2, the kernels take the asymptotic values  Π0(p2) ≃  1  16π|m|  ΠA(p2) ≃ − 1  4πsgn(m)  (322)  This result is the leading contribution to the parity even term (the first term) up to corrections  of the order of p2/m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the second term, which is parity odd, is the exact contribution  63  in the limit p → 0 for all non-vanishing values of the Dirac mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime, the effective  Lagrangian of the gauge field Aµ takes the local form of the sum of a Maxwell term and a Chern-  Simons term  Leff[Aµ] = −  1  4π|m|FµνFµν ± 1  2  1  4π ǫµνλ Aµ∂νAλ  (323)  where Fµν = ∂νAν − ∂νAµ is the field strength of the gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Clearly, since the Maxwell term  has two derivatives while the Chern-Simons term has one derivative, in the long distance regime  the Maxwell term is irrelevant (subdominant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Also, while the prefactor of the Chern-Simons  term is a pure number, the prefactor of the Maxwell term is proportional to  1  |m| (as required by  dimensional analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The second term in the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (323) is a Chern-Simons term for the background  gauge field Aµ [163, 164].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This term is gauge-invariant (up to boundary terms) but breaks parity  and time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This phenomenon is called the parity anomaly [200], although  Witten [199] has argued that it is better to call this phenomenon a time-reversal anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  sign of the prefactor of the Chern-Simons term depends of the sign of the Dirac mass m: positive  for m > 0 and negative for m < 0 or, alternatively for positive and negative chirality of the Dirac  (bi-spinor) field which fixes the sign of the breaking of time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In addition to the sign, the prefactor of the Chern-Simons term is equal to 1  2, and it is not  an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 we showed that for a dynamical Chern-Simons gauge theory to be  defined consistently on a closed surface this coefficient must be quantized and should take integer  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The fact that the coefficient is half-quantized means that the classical global symmetry of  a theory of a single Dirac bi-spinor cannot be gauged, which means that the gauge theory cannot  be quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is an example of an obstruction to the quantization of a gauge theory due to  an anomaly [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now use these results to compute the effective low-energy action for the theories  of lattice Dirac fermions of section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In those systems the low energy theory is that of  two Dirac (bi-spinors) with different masses m1 and m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since both Dirac fields are coupled  (minimally) to the same background gauge field Aµ, the effective Lagrangian is just the sum of  the contribution for each Dirac fermion  Leff[Aµ] = −  1  4π|m|eff  FµνFµν + 1  2  �sgn(m1) + sgn(m2)� 1  4πǫµνλAµ∂νAλ  (324)  where |m|eff is  1  |m|eff  =  1  |m1| +  1  |m2|  (325)  This result implies that, as anticipated in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2, this theory has two phases: a parity even  phase and a parity-odd phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the regime in which the signs of the two mass terms are equal  and opposite, sgn(m1) = −sgn(m2), the coefficient of the Chern-Simons term cancels and the  low-energy effective Lagrangian is a Maxwell term (generally with an effective speed of light  much smaller than that in vacuum): this phase is a conventional insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Conversely, in the phase in which the two masses have the same sign, sgn(m1) = sgn(m2),  the coefficient of the Chern-Simons terms does not cancel and it is given by ± 1  4π, where the sign  is the sign of both mass terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This phase is also an insulator but one in which time-reversal  invariance is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in this phase the induced current jµ by the background field Aµ  in the long-distance limit is controlled by the Chern-Simons term and it is given by  jµ = δLeff  δAµ(x) = ± e2  2πℏǫµνλ∂νAλ  (326)  From this result we see that in the parity-broken anomalous quantum Hall phase the system has  a correctly quantized and non-vanishing Hall conductivity  σxy = ±e2  h  (327)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the phase with sgn(m1) = sgn(m2) displays the quantum anomalous Hall effect with a  Hall conductivity whose sign equals the signs of both masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that this is true regardless  the magnitudes of the masses m1 and m2, and that only their signs matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 we will  identify this phase with a topological phase of matter known as the Chern Insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We showed that the Hall conductivity of the anomalous quantum Hall phase is, as expected,  equal to e2/h using the effective low energy Dirac theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One may wonder if this approximation  64  may be missing some contributions to the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fortunately there is an alternative  quite elegant way of computing the Hall conductivity as a property of the entire occupied band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This approach, originally introduced by Gregory Volovik [204] in the context of superfluid 3He-  A, and adapted to the theory of the quantum anomalous Hall effect by Viktor Yakovenko [205],  by Maarten Golterman, Karl Jansen and David Kaplan for a theory of Wilson fermions in odd  dimensional hypercubic lattices [206], and by Xiao-Liang Qi, Yong-Shi Wu and Shou-Cheng  Zhang [207], involves the derivation of a topological invariant for a two-band model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As we saw above the Hall conductivity is obtained from the xy component of the polarization  operator as the limit  σxy = lim  ω→0  i  ω lim  q→0 Πxy(ω, q)  (328)  For a free fermion system Πxy(ω, 0) is given by the current correlator  Πxy(ω, 0) =  �  BZ  d2k  (2π)2  � ∞  −∞  dΩ  2π tr  �  Jx(k)G(k, ω + Ω)Jy(k)G(k, Ω)  �  (329)  where G(k, Ω) is the propagator for a two-band free fermion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A generic two-band free  fermion system has a one-particle Hamiltonian of the form given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(311).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The (one-body)  current operator for such a system is obtained as  Jl(k) = ∂h0(k)  ∂kl  1 + ∂ha(k)  ∂kl  σa  (330)  The propagator G(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ω) is the 2 × 2 matrix (in band indices)  G(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ω) =  1  ω1 − h(k) · σ + iǫ =  P+(k)  ω − E+(k) + iǫ +  P−(k)  ω − E−(k) + iǫ  (331)  where P±(k) are the operators that project onto the (empty) conduction band and the (filled)  valence band whose energies are,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' respectively E±(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  P±(k) = 1  2  �  1 ± ˆh(k) · σ  �  (332)  where ˆh(k) is the unit vector defined for every momentum k of the first Brillouin zone  ˆh(k) =  h(k)  ||h(k)||  (333)  Upon performing the frequency integration and the band traces in the expression for Πxy(ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 0) of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (329), we find that the Hall conductivity takes the form  σxy = e2  2ℏ  �  BZ  d2k  (2π)2 ǫabc  ∂ˆha(k)  ∂kx  ∂ˆhb(k)  ∂ky  ˆhc(k) (n+(k)) − n−(k))  (334)  where n±(k) are the Fermi functions (at zero temperature) for the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since  E+(k) − E−(k) = 2||h(k)|| = 2  �  h2(k) > 0  (335)  there is a non-vanishing gap on the entire Brillouin zone between the occupied valence band,  with n−(k) = 1, and the unoccupied conduction band, with n+(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the insulating state  the Fermi energy lies inside this gap and the expression for the Hall conductivity reduces to the  following  σxy = − e2  2ℏ  �  BZ  d2k  (2π)2 ǫabc  ∂ˆha(k)  ∂kx  ∂ˆhb(k)  ∂ky  ˆhc(k)  (336)  In our discussion of quantum antiferromagnets in one space dimensions in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 we found  that their effective low-energy action contained a crucial topological term proportional to the  integer-valued topological invariant Q (the winding number) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (103) which classifies the  smooths maps (homotopies) of the 2D surface (say a sphere S 2) onto the target space of a three-  component unit vector field which also a sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These equivalence classes are represented  by the notation Π2(S 2) ≃ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case at hand the unit vector ˆh(k) are points on a 2-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Hence, ˆh(k) is a map of the first Brillouin zone (which is a 2-torus) to the sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such maps  65  are also classified by the same integer-valued topological invariant defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  results imply that the Hall conductivity of the two-band system is  σxy = e2  2πℏQ[ˆh]  (337)  In other words we have shown that the Hall conductivity is given in terms of a topological in-  variant of the occupied band (in units of e2/h), the winding number Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the two-band model  the topological invariant Q plays the same role as the Chern number does in the work of TKNN  [179, 187] When Q � 0, the two-band system exhibits the quantized anomalous quantum Hall  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This si a property of the entire band of occupied states, and not just a consequence of the  low energy approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result implies that the low energy approximation captures all of  the topology of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It also implies that in these lattice models the Berry curvature is highly  concentrated near the points in momentum space where the two bands are close in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now consider the problem of the quantum phase transition between the trivial and  the Chern insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To address this problem we will tune the parameters of the lattice model  discussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2, the phase φ and the ratio of hopping amplitudes t2/t1, to the point at  which the mass of one of the two species of Dirac fermions, say the fermion ψ1, is zero, m1 = 0,  while keeping the fermion ψ2 massive, m2 � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the low energy regime we have a massless  fermion and a massive fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This point in parameter space is a quantum phase transition  between a trivial insulator and a Chern insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the low energy regime the fermion ψ1 is  massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A massless fermion is a scale-invariant system in the sense that the correlators of all  its observables exhibit power law behavior (free field in this case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now discuss briefly the electromagnetic response of this system at the quantum phase  transition where one fermion becomes massless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, it is natural to ask if the coeffi-  cient of the parity-odd (Chern-Simons) term non-vanishing at the quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To  answer this question we will look at the behavior of the parity-even kernel Π0(p2) and the parity-  odd kernel ΠA(p2) in the massless limit for the light fermion, m1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We find that the total  contribution of both the light fermion and of the heavy fermion to the polarization kernels is  (assuming m2/m1 → ∞)  lim  m1→0 Π0(p2) =  i  16  �  p2 ,  lim  |m2|→∞ ΠA(p2) = − 1  4πsgn(m2)  (338)  The important conclusion is that at this quantum critical point the parity-even kernel Π0(p2)  is non-local and that the heavy regulator fermion (the “doubler”) yields the leading finite non-  vanishing and local contribution to the parity-odd kernel ΠA(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can now use these results to compute the conductivity tensor σi j at the quantum critical  point where m1 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the system is spatially isotropic the conductivity tensor has the form  σi j =  � σxx  σxy  −σxy  σxx  �  (339)  where we used that σyy = σxx since the system is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The longitudinal conductivity σxx  and the Hall conductivity σxy are  σxx = π  8  e2  h ,  σxy = ±1  2  e2  h  (340)  in other words, at the quantum critical point the system has a finite (and universal) longitudinal  conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result may seem surprising as there is no disorder in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A finite  universal longitudinal conductivity is a standard occurrence in 2D systems at a quantum critical  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, at the superconductor-insulatortransition the conductivity is (conjectured) to  be) σxx = e2/2h (with σxy = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, it also has finite and also universal Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Hall conductivity at the quantum critical point is due to the heavy fermionic “doubler” and is  equal to 1/2 (in units of e2/h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' So, the quantum critical point is not time-reversal invariant since  this symmetry is broken at the UV (lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can examine the massless theory using a more formal approach [199, 208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a gauge-  invariant regularization of the massless theory the partition function of a Dirac fermion coupled to  a background (unquantized) U(1) gauge field, the partition function is not time-reversal invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  66  It is given by  Z[Aµ] =  �  D ¯ψDψ exp  �  i  �  d3x ¯ψi /D[Aµ]ψ  �  = det(i /D[Aµ]) =  ����det(i /D[Aµ])  ���� exp  �  ±iπ  2η[Aµ]  �  (341)  where the sign depends on how time-reversal invariance is broken by the choice of regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The quantity η[Aµ] that appears in the phase factor is the Atiyah-Patodi-Singer η-invariant [147]  which we already encountered in the discussion of the fractionally charged solitons in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  With some caveats [199, 208], the phase factor of the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (341) is commonly  written in the form of a 1/2-quantized Chern-Simons term  π  2η[Aµ] ≡ ±1  2  �  d3x 1  4πǫµνλAµ∂νAλ  (342)  often denoted as a U(1)1/2 Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This term plays the same role as the contribution  of the heavy fermion doubler in the lattice theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, we can also wonder if there is a way to have a time-reversal invariant theory of a  single massless Dirac fermion, m → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This case cannot be realized in a y 2D lattice model but,  as will see, it can be realized on the surface states of a 3D time-reversal invariant Z2 topological  insulator, which we will discuss in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Three-dimensional Z2 topological insulators  The classification of topological insulators in terms of a Chern number is only possible in  even space dimensions in systems with broken time reversal symmetry: in d = 2 the Berry  connection is abelian and the topological invariant is the first Chern number while in in d = 4 the  Berry connection in a non-abelian SU(2) gauge field and the topological invariant is the second  Chern number [135], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now discuss the time-reversal invariant topological insulators  [209, 190, 135] and, in particular, those that are invariant under inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such states  exist in both two and space dimensional insulator with strong spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Z2 Topological Invariants  Let {|un(k)⟩} be the Bloch states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will represent time reversal by the anti-unitary operator  Θ that acts on the single particle (Bloch) states by complex conjugating the state and reverses the  spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For spin-1/2 fermions Θ = exp(iπσ2)K, where K is the complex conjugation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  this case Θ2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us assume that we have two occupied Bloch bands for each point k f the  Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the states form a rank-2 vector bundle over the torus of the Brillouin  zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In time reversal invariant systems the anti-unitary time reversal transformation T induces  an involution in the Brillouin zone that identifies the points k and −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Time reversal the acts  on the one-particle (Bloch) Hamiltonian as ΘH(k)Θ−1 = H(−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The states |un(±k)⟩ are related  by time reversal as |un(−k)⟩ = Θ|un(k)⟩ which implies that the bundle is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The condition  Θ2 = −1 implies that the bundle is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In algebraic topology these bundles are classified by an  integer (here the number of occupied bands) and a Z2 index that will allow us to classify these  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In a periodic lattice there exists a set of points Qi of the Brillouin zone with the property  that they differ by their images under the action of time reversal by a reciprocal lattice vector,  −Qi = Qi + G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In d = 2 there four such points and d = 3 there are eight points, and are given  by Qi = 1  2  �  j n jb j, where n j = 0, 1, j = 1, 2 in d = 2 and j = 1, 2, 3 in d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here b j are  the primitive lattice vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane and Mele [210] defined the 2N × 2N antisymmetric matrix  �m,n(k) = ⟨um(−k)|Θ|un(k)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They showed that at each time reversal invariant point Qi one can  define an index δi  δi =  �  det[�(Qi)]  Pf[�(Qi)]  = ±1  (343)  where det[�] and Pf[�] are, respectively, the determinant and the Pfaffian of the matrix �, and  det[�] = Pf[�]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The sign of the quantities δi can be made unambiguous by requiring that the  Bloch states be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, the quantities δi are gauge-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However the  products  (−1)ν =  4  �  i=1  δi  (344)  67  in d = 2, and  (−1)ν0 =  8  �  i=1  δi,  (−1)νk =  �  nk=1,nj�k=0,1  δi(n1, n2, n3)  (345)  are gauge and are also topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Z2-valued indices ν and ν0 are robust to disorder  and are called strong topological indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, Fu and Kane showed that if ξn(Qi) = ±1  are the parity eigenvalues of the occupied parity eigenstates, the quantities δi are given by δi =  �N  m=1 ξ2m(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In d = 3 the index ν0 does not rely on the existence of inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the case of a two-band model in d = 2 and in d = 3, the states are four-component spinors  reflecting the two bands and the two spin components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the context of these systems with  strong spin-orbit coupling spin is actually the z-component of the atomic total angular momentum  J of the electrons with energies close to the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In these systems the one-particle  Hamiltonian H(k) is a 4 × 4 Hermitian matrix which can be expanded as a linear combination  of Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A simple and very useful model of systems of this type is the Wilson fermion  model (with continuous time) [55, 211, 135] of a square (cubic) lattice with 4 states per site  (parity and spin) whose Hamiltonian three dimensions is  H(k) = sin k · α + M(k) β  (346)  where α = γ0γ and β = γ0 are the conventional 4 × 4 Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The γ matrices satisfy  the Clifford algebra {γµ, γν} = 2gµν1, where gµν = diag(1, −1, −1, −1) is the metric tensor of  four-dimensional Minkowski space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' An additional γ matrix of interest is γ5 = iγ0γ1γ2γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The (Wilson) mass term M(k) in two dimensions is M(k) = M + cos k1 + cos p2 − 2, and in  three dimensions is M(k) = M + cos k1 + cos k2 + cos k3 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (346) we see that, consistent  with the requirements of the Nielsen-Ninomiya theorem [174, 175], in addition to a possible low  energy Dirac fermion (if M is small) there are three more massive Dirac fermions (in d = 2) and  seven other Dirac fermions in d = 3, so that the total number of Dirac fermions is always even, 8  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With Wilson’s mass term the additional Dirac fermions (the “doublers”) are always  heavy even if the Dirac fermion near the Γ point Q = 0 is light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the Dirac basis time reversal is the operation Θ = (iσ2 ⊗ I)K, where K is complex con-  jugation, and parity is P = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The matrices α and β commute with PΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At the time-reversal  and parity invariant points of the Brillouin zone {Qi} the Hamiltonian only depends of the the  matrix β, H(Qi) = M(Qi)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the parities of the spinors are the eigenvalues of the matrix  β, we conclude that in the two-band models the quantities δi are simply equal to the sign of  the mass of the fermions defined at the time-reversal-invariant points Qi of the BZ [209, 135],  δi = δ(Qi) = −sgn M(Qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Using this result it follows that in two dimensions the system is a Z2  topological insulator with index ν = 1 (mod 2) if 0 < M < 4 while it is trivial for other values  of M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ν = 0 mod 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly, in three dimensions a Z2 (strong) topological insulator exists  only if 0 < M < 2 (in the other regimes this system is in a weak topological insulator state or in  a trivial one).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that both in two and three dimensions the low energy theory of the Z2 topolog-  ical insulators consists of a single Dirac fermion whose mass is small compared to that of the  fermion doublers and has the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In both cases there is a quantum phase transition  between a trivial insulator and the time-reversal invariant topological insulator at the point where  he mass of the light fermion vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Axial Anomaly and the Effective Action  We will now look at the electromagnetic response of a Z2 topological insulator in 3+1 dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem can be addressed more easily using the continuum field theory description  which is valid in the regime where the mass M is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime one species of Dirac  fermions is light (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' its mass is small) while the Dirac doublers remain heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Much as we did  in our discussion of the Chern insulators and the parity anomaly in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 we will keep  in mind that the fermion doublers play the role of heavy regulators, such as in the Pauli-Villars  scheme, in Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here too, we will see that a field-theoretic anomaly, known  as the axial anomaly plays a central role in the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The analysis is very similar to what we  did with the chiral anomaly in 1+1 dimensions in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us begin with a theory of a single massive Dirac fermion in 3+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The La-  grangian of the free massive Dirac theory is  L = ¯ψ � i/∂ − m� ψ  (347)  68  The equation of motion of the spinor field operator ψ(x) (I omit the spinor indices here) is the  Dirac equation  �i/∂ − m� ψ = 0  (348)  The Dirac Lagrangian has a global gauge symmetry ψ → eiθψ which requires that the Dirac  current jµ = ¯ψγµψ is locally conserved, ∂µ jµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The massless Dirac theory can be decomposed  into a theory of two Weyl bi-spinor fields which obey separate Dirac Lagrangians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore,  the massless theory has the additional global symmetry under the transformation ψ → eiθγ5ψ and  the additional formally locally conserved axial current j5  µ = i ¯ψγµγ5ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However the conservation  law of the axial current is violated in the massive Dirac theory  ∂µ j5  µ = −2mi ¯ψγ5ψ  (349)  since the two Weyl bi-spinors transmute into each other in the presence of a mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is  the origin of the phenomenon of neutrino oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the condensed matter physics context  there is a similar phenomenon in systems of Weyl semimetals which have crossings between the  valence and conducting bands at two locations ±Q of the BZ, with each crossing associated with  each Weyl bi-spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A charge density wave with ordering wavevector 2Q mixes the two Weyl  fermions which become gapped, becoming effectively a single massive Dirac fermion [212].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This state is often called an “axionic”-charge-density-wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now show that the axial symmetry has an anomaly and cannot be gauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, we  will consider the problem of a Dirac theory coupled to a background U(1) gauge field Aµ and  reexamine the putative conservation of the axial current j5  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This question can be addressed in  different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quite early on Steven Adler [133], and John Bell and Roman Jackiw [134] exam-  ined this problem by computing a Dirac fermion triangle diagram for the process of a neutral pion  decaying into two photons, π0 → 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particle physics the pion is the Goldstone boson of the  spontaneously broken chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The analog of this problem in condensed matter physics  is the phase mode of an incommensurate charge density wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The relativistic Lagrangian for  this problem is a theory of Dirac fermions coupled to a complex scalar field φ = φ1 +iφ2 through  two Yukawa couplings  L = ¯ψi /Dψ + gφ1 ¯ψψ + igφ2 ¯ψγ5ψ − V(φ1, φ2) = ¯ψi /Dψ + g|φ| ¯ψeiγ5θψ − V(|φ|2)  (350)  where |φ|2 = φ2  1 + φ2  2, tan θ = φ2/φ1, and Dµ = ∂µ + ieAµ is the covariant derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we are  regarding the gauge field Aµ as a background probe field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The triangle Feynman diagram computes the polarization tensor for the electromagnetic field  Aµ with an insertion of the coupling to the complex scalar field in an otherwise massless theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Assuming a gauge-invariant regularization of the diagram, this computation finds that the axial  current j5  µ is anomalous and is not conserved even in the massless theory [133, 134]  ∂µ j5  µ = − e2  16π2 FµνF∗  µν  (351)  where F∗  µν = 1  2ǫµνλρFλρ is the dual of the electromagnetic field tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the axial anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To see how the axial anomaly arises we will follow the physically transparent approach of  Nielsen and Ninomiya [176], that we also employed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will  consider a theory of free massless dirac fermions coupled to a background electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since the theory is massless, the Dirac equation decouples into an equation to the right handed  Weyl fermion ψR (with positive chirality γ5ψR = +ψR) and a left handed Weyl fermion ψL (with  negative chirality, γ5ψL = −ψL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the gauge A0 = 0, the Dirac equations become  [i∂0 − (−i∂ − eA) · σ]ψR = 0,  [i∂0 − (i∂ − eA) · σ]ψL = 0  (352)  Let us now consider look at the solutions of the Weyl equation for right-handed fermions ψR,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(352).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The left-handed fermions ψL are analyzed similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wd will consider a gauge field  A1 = 0 and A2 = Bx1 representing a uniform static magnetic field of strength B pointing along  the x3 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this this gauge the eigenstates are plane waves along the directions x2 and  x3 and harmonic oscillator states along the direction x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The eigenvalue spectrum consists of  Landau levels with energies  E(n, p3, σ3) = ±  �  2eB  �  n + 1  2  �  + p2  3 + eBσ3  �1/2  (353)  69  for n = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', except for the zero mode with n = 0 and σ3 = −1, for which  E(n = 0, p3, σ3 = −1) = ±p3  (354)  where the + sign holds for ψR and the − sign for ψL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Just as in the case of non-relativistic fermions  the relativistic Landau levels are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will consider a system of Dirac fermions at  charge neutrality and, hence, EF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The positive and negative energy states are charge conju-  gate of each other and the negative energy states are filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For right-handed fermions, the zero  mode states with p3 < 0 are filled while for right handed states the zero modes with p3 > 0 are  filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The density of states of the zero modes is LeB/(4π2) (where L is the linear size of the  system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider now turning on an external electric field E parallel to the magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Just as we saw in 1+1 dimensions in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3, the electric field leads to pair creation by shifting  the Fermi momentum to pF for the zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is no particle creation for the states with  n � 0 and they do not contribute to the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The rate of creation of right-handed fermions  NR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  dNR  dx0  = 1  L  LeB  4π2  pF  dx0  = e2  4π2 EB  (355)  The annihilation rate of left handed particles is  dNL  dx0  = − 1  L  LeB  4π2  pF  dx0  = − e2  4π2 EB  (356)  and the creation rate of left handed anti-particles is  d ¯NL  dx0  = 1  L  LeB  4π2  pF  dx0  = e2  4π2 EB  (357)  the axial anomaly is the total rate of creation of right handed particles and of left-handed antipar-  ticles:  dQ5  dx0  = dNR  dx0  + d ¯NL  dx0  = e2  2π2 EB  (358)  which agrees with the expression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (351).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We now turn to the Z2 topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw this is a system with two species of  (4 component) Dirac spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the topological phase the sign of the Dirac mass term one of the  Dirac fermions (the one near the Γ point in the lattice model) is opposite (negative) to the sign  of the mass term of the of the other Dirac fermion (the fermion doubler).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Explicit calculations  [135, 213, 214] on the lattice model obtain the result that the effective low-energy action for the  electromagnetic gauge field Aµ in the topological phase is  S eff[Aµ] =  �  d4x  �  − 1  4e2 FµνFµν +  θ  32π2e2ǫµνλρFµνFλρ  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (359)  In a time-reversal invariant system the allowed values of the θ angle of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (359) are restricted to  be θ = nπ, with n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The case θ = 0 (mod 2π) represents a trivial insulator whereas θ = π (mod  2π) holds for a Z2 time-reversal invariant topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The second term in the effective action of the electromagnetic gauge field of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (359), known  as the θ term, has been extensively discussed in the high-energyphysics literature [202, 215, 216].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The derivation of this term is subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As it stands, unless θ is varying in space-time (in which case  this is known a the axion field) this term is a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In non-abelian Yang-Mills gauge  theories this term is proportional to a topological invariant known as the Pontryagin index which  counts the instanton number of the gauge field configurations [99, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the context of the  Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (350) this term is induced by the coupling of the Dirac fermion to the Yukawa  coupling of the complex scalar field φ to the Dirac and γ5 mass terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, the lowest  order contribution is given by the triangle diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The fact that this term is exact at lowest order  reflects the fact that the axial anomaly is in fact a non-perturbative effect which is the same in  both the weak coupling and the strong coupling regimes [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the phase where the potential V(|φ|2) in the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (171) has a minimum at  |φ0| exp(iθ) the chiral symmetry is spontaneously broken and the phase field θ(x) is the Goldstone  boson of the spontaneously broken chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this phase the Dirac fermions become  massive through the Yukawa couplings to the complex scalar field, and the phase of the field φ  enters in the effective action as an axion field which couples to the gauge field Aµ through the θ  term of the effective action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We should note that in a recently studied axionic CDW state of a  Weyl semimetal [212] the phase of the CDW plays the role of the axion field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  70  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Theta terms, and Domain walls: Anomaly and the Callan-Harvey Effect  We will now discuss some remarkable behaviors of three-dimensional Z2 topological in-  sulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will begin with the electromagnetic response encoded in the effective action of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(359).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the θ angle is effectively a slowly varying Goldstone mode of the  spontaneously broken U(1) chiral symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the axion field) present in the Lagrangian of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(350).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the field θ is constant, the θ term is a total derivative and it does not contribute to  the local equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see below that this term [lays a key role in the physics of  a domain wall, which we will regard as the interface of a Z2 topological insulator and a trivial  insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the general case in which θ varies slowly (as Goldstone modes do) its presence leads  to interesting modification of Maxwell’s equations, known as axion electrodynamics [215]:  ▽ · E =˜ρ − e2 ▽ θ · B,  ▽ × E = − ∂tB  ▽ × B =∂tE + ˜j + e2 (∂tθ + ▽θ × E) ,  ▽ · B =0  (360)  where ˜ρ and ˜j are external probe electric charge and current densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The equations of axion  electrodynamics have many remarkable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will focus on effects: the topological  magnetoelectric effect [135] and the Witten effect [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider a Z2 topological insulator with a flat open boundary (the x1 − x2 plane)  perpendicular to the direction x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the topological insulator lies at x3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This means that for x3 < 0, and far from the surface, θ(x3) → π, while in the trivial vacuum, and  also far from the surface, θ(x3) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the change of θ from 0 to π occurs  on a short distance ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will call this configuration an axion domain wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the region where  θ(x3) is changing, |x3| ≲ ξ, an applied uniform magnnetic field B induces a uniform electric  field E parallel to B whose magnitude is proportional to the change in θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the topological  magnetoelectric effect [135].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A similar striking effect is obtained by considering the case of a magnetic monopole of mag-  netic charge 2π/e (as required by Dirac quantization) inside a sphere of the trivial region of radius  R, with θ = 0, surrounded by a region with θ � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The two regions are separated by a thin (axion)  wall in which θ changes for 0 to π in a narrow shell of thickness ξ ≪ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The equations of axion  electrodynamics imply that the magnetic monopole induces an electric charge Qe on the surface  of the sphere  Qe = e∆θ  2π  (361)  Thus, a magnetic monopole acquires an electric charge and becomes a dyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the Wit-  ten effect [217].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the particular case of a Z2 topological insulator, time reversal invariance  requires that ∆θ = π, which implies that a monopole with unit magnetic charge has an elec-  tric charge e/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A similar argument implies that an external magnetic field perpendicular to the  open surface of the Z2 topological insulator (which is essentially an axion domain wall) induces  an electric charge polarization on the surface proportional to the total magnetic flux, which is  another manifestation of the topological magnetoelectric effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The reader should readily recognize that the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (361) for the electric charge induced  my the magnetic monopole is the same as the Goldstone-Wilczek equation for the fractional  charge for a one dimensional soliton of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now see that this is not just an analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will follow here the general approach of Curtis Callan and Jeffrey Harvey [218] who extended  the earlier work of Goldstone and Wilczek [141].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider the bulk of a Z2 topological  insulator and assume that θ(x) is slowly varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can use the triangle diagram calculation to  find that an electromagnetic gauge field Aµ induces a current ⟨Jµ(x) given by  ⟨Jµ(x)⟩ = −i  e  16π2 ǫµνλρ  φ∗(x)∂νφ(x) − φ(x)∂νφ∗(x)  |φ(x)|2  Fλρ(x) =  e  8π2 ǫµνλρ∂νθ(x)Fλρ(x)  (362)  This result implies that, in the case of a domain wall in the x1 − x2 plane, a magnetic field  perpendicular to the wall induces a current towards the wall and, hence, a charge accumulation  on the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Where does this charge come from?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To understand this problem we will consider  a Z2 topological insulator occupying a slab of macroscopic size L between a wall at x3 = 0 and  a far way “anti-wall” at x3 = L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this configuration a magnetic field normal to the wall(s)  induces a transfer of charge from one wall to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly, an electric field parallel to the  wall induces a current also parallel to the wall and perpendicular to the electric field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a Hall  current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In 1+1 dimensions we saw that soliton configurations acquire a fractional charge associated  to states bound to the soliton (which are zero modes when ∆θ = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see that the surfaces  71  of the 3D Z2 topological insulators also have zero modes which ar Weyl fermions propagating  on the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see how this works we will consider a 3+1 dimensional Dirac fermion with a  Dirac mass that changes sign at x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Lagrangian now ill be  L = ¯ψi/∂ψ + gφ(x) ¯ψψ  (363)  where φ(x) is now a real scalar field that has the asymptotic behaviors φ(x3) = φ0 f(x3) such  that limx3→±∞ f(x3) = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that f(x3) is a monotonous function of x3, but its  actual dependence on x3 is immaterial aside from the requirement that it should change sign at  some point which we will take to be x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a special case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (350) with φ1 = φ and  φ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will now recognize that this just a Dirac fermion with a position-dependent Dirac  mass m(x3) = gφ(x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The one-particle Dirac Hamiltonian for this system is  H = −iα · ▽ + m(x3)β  (364)  where α and β are the four 4 × 4 Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By symmetry, this Hamiltonian can be split into  two Hamiltonians, H = Hwall + H⊥,  Hwall = −iα1∂1 − iα2∂2,  H⊥ = −iα3∂3 + m(x3)β  (365)  Let the spinor ψ± be and eigenstate of the anti-hermitian Dirac matrix γ3 = βα3 with eigenvalues  ±i, respectively  γ3ψ± = ±iψ±  (366)  We seek a spinor solution ψ± of the Dirac equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (364) such that H⊥ψ± = 0  ± ∂3ψ± + m(x3)ψ± = 0  (367)  which is also a solution of  (iγ0 − iγ1∂1 − iγ2∂2)ψ± = 0  (368)  In other words, it is a solution of the massless Dirac equation in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The require-  ment that ψ± be an eigenstate of γ3 reduces the number of spinor components from four to two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The full solution has the form  ψ± = η±(x0, x1, x2)F±(x3),  ±∂3F±(x3) = −m(x3)F±(x3)  (369)  For a domain wall with limx3→∞ m(x3) = +m, the normalizable solution is F+(x3) is  F+(x3) = F(0) exp  �  −  � ∞  0  dx′  3 m(x′  3)  �  (370)  where F(0) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the anti-domain wall, for which limx3→∞ m(x3) = −m, the normal-  izable solution is F−(x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude that there is a 2+1-dimensional massless Dirac theory that describes the quan-  tum states bound to the wall which propagate along the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These states are the generalization  of the fractionally charged mid-gap states of solitons in one dimension discussed in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The energy of these states is E(p) = ±|p|, where p = (p1, p2) (where I set the Fermi velocity to  unity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Experimental evidence for 2+1 dimensional massless Dirac fermions on the surface of the  3D Z2 topological insulator Bi2Te3 has been found in spin-polarized angle-resolved photoemis-  sion studies of the surface states which showed that they have a linear energy-momentumrelation  (expected of massless Dirac fermions) as well as the spin-momentum locking characteristic of  these spinor states [219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Having succeeded in showing that the surface of the Z2 time-reversal invariant 3D topological  insulator has a two-component massless Dirac spinor we now want to determine its electromag-  netic response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact we have already discussed this problem in our discussion of the parity  anomaly in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 where we showed that the effective action for the electromagnetic field  Aµ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (342) of a single massless Dirac bi-spinor is a Chern-Simons term with a prefactor  which is 1/2 of the allowed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In that purely 2+1-dimensional context we saw that time re-  versal invariance is actually broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the 3D problem is time-reversal invariant so there  must be a contribution that cancels this time-reversal anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The answer is that the requisite  cancellation is supplied by the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To see how this can happen we return to the bulk effective action of the 3D topological  insulator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (359) where we observed that the θ-term is a total derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Suppose that the  72  system has a boundary at x3 = 0 and that the topological insulator exists for x3 > 0 (where the  fermion mass is negative, m < 0) and trivial for x3 < 0 (where m > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Only the region with  m < 0 contributes to the θ-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The region of four-dimensional space time occupied by the  topological insulator is M and in this region the θ-term becomes  S θ[A] =  θ  32π2  �  M  d4x ǫµνλρFµνFρλ =  θ  8π2  �  M  d4x ǫµνλρ∂µAν∂λAλ =  θ  8π2  �  ∂M  d3x ǫµνλAµ∂νAλ  (371)  where ∂M ≡ Σ × R is the boundary of the region M, Σ is the surface of the 3D topological  insulator and R is time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Thus we see that the θ-term of the effective action S θ[A] integrates to the boundary where it  has the form of a 2+1-dimensional Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since for a time-reversal-invariant 3D  topological insulator θ = π, we see that in this case the boundary Chern-Simons term becomes  S θ=π[A] = 1  8π  �  ∂M  d3x ǫµνλAµ∂νAλ  (372)  with a coefficient which is 1/2 of the allowed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This bulk contribution either cancels the  parity anomaly of the boundary state, rendering the full system time-reversal invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other  words, in this system time-reversal symmetry is realized by cancellation of the anomaly between  the bulk of the topological insulator and the boundary or, equivalently, by an inflow of the parity  anomaly between the boundary and the bulk of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Another way to phrase this result is  the statement that a single Dirac fermion cannot exist on its own in 2+1 dimensions but it can  as the boundary state of a 3+1-dimensional system whose anomaly cancels the anomaly of the  boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chern-Simons Gauge Theory and The Fractional Quantum Hall Effect  We will now turn to the problem of the quantum Hall effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a problem that revealed  the existence of profound and far reaching connections between condensed matter physics, quan-  tum field theory, conformal field theory and topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, the fractional quantum hall  effect is the best studied and best understood topological phase of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As such, it has become  the conceptual springboard for its manifold generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The integer (IQHE) and fractional (FQHE) quantum Hall effects are fascinating phenomena  observed in fluids of electrons in two dimensions in strong perpendicular magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  integer quantum Hall effect was discovered by Klaus von Klitzing in 1980 [220] in transport  measurements of the longitudinal and Hall resistivity of the surface states of metal oxide field-  effect transistors (MOSFET) in magnetic field of up to 15 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The effect that von Klitzing  discovered (for which he was awarded the 1985 Nobel Prize in Physics) was that in the high  field regime the measured Hall conductivity showed a series of sharply defined plateaus at which  took the values σxy = ne2/h, where n is an integer, and the longitudinal conductivity appeared  to vanish σxx → 0 as the the temperature was lowered down to T ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Remarkably the  measured value of the Hall conductivity was obtained with a precision of ∼ 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To this date  the measurement of the Hall conductivity in the IQHE yields the most precise definition of the  fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Subsequent transport experiments in ultra-high-purity GaAs-AlAs heterostructures by Dan  Tsui, Horst St¨ormer and Art Gossard found that, in addition to the IQHE, two-dimensional elec-  tron fluids in high magnetic fields exhibit the fractional quantum Hall effect meaning that there  are equally sharply-defined plateaus of the Hall conductivity at the values σxy = p  q  e2  h , where p  and q are co-prime integers [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Much as in the IQHE case, in the FQHE the longitudinal  conductivity vanishes at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is important to note that both in the IQHE and in  the FQHE the observed temperature dependence in the highest purity samples of the longitudinal  conductivity is activated, σxx ∼ exp(−W/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The observed value of the energy scale W is the  experimental estimate of an energy gap in the electron fluid in the quantum Hall states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In addition to MOSFETS and GaAs-AlAs heterostructures both the IQHE and the FQHE  have been seen in several other experimental platforms, particularly in graphene and other 2D  materials [222, 223].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The explanation of these effects and of a panoply of startling consequences  that were uncovered in the course of understanding this phenomenon is the focus of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau levels and the Integer Hall effect  At some level the integer quantum Hall effect can be explained by the Landau quantization of  the energy levels of a free charged moving in two dimensions in a perpendicular magnetic field  73  [224].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Hamiltonian for a non-relativistic particle of charge −e and mass M in a perpendicular  magnetic field B is  H =  1  2M  �  −iℏ ▽ +ie  c A(x)  �2  (373)  for a uniform perpendicular magnetic field B = B ˆez = ▽ × A(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the circular gauge the vector potential is Ai = − 1  2 Bǫi jx j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that the 2D plane  has linear size L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The total magnetic flux is Φ = BL2 and we will assume that there is an integer  number Nφ of magnetic flux quanta piercing the plane, φ = Nφφ0, where φ0 = hc/e is the flux  quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In units such that ℏ = e = c = 1 the flux quantum is φ0 = 2π and Φ = 2πNφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the presence of a magnetic field the components of the canonical momentum operator  p = −iℏ ▽ − e  c A do not commute with each other,  [pi, p j] − ieℏ  c Bǫi j  (374)  This means that translations in two directions do not commute with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the  components of the operator k = p(−B) commute with p (and hence with the one-particle Hamil-  tonian) but do not commute with each other: the commutator is the same as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since  [k, H] = 0 they act as symmetry generators of the group of magnetic translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For arbitrary  displacements a and b the translation operators t(a) = exp(ia· k/ℏ) (and similarly with b) satsify  t(a)t(b) = exp(ia × b · ez/ℓ2  0)t(b)t(a)  (375)  Magnetic translations only commute with each other is the area subtended by a and b contains  an integer number of magnetic flux quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Given the rotational symmetry of the circular gauge it is natural to work in complex coordi-  nates z = x1 + ix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will also use the notation ∂z = (∂1 − i∂2)/2 and ∂¯z = (∂1 + i∂2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  gauge, up to a normalization, the eigenstate wave functions have the form  ψ(z, ¯z) = f(z, ¯z) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed− |z|2  4ℓ2  0  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (376)  where ℓ0 =  �  ℏc  e|B| is the magnetic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this gauge (and in complex coordinates) the angular  momentum operator Lz = −iℏ(x1∂2 − x2∂1) = ℏ(z∂z − ¯z∂¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Any analytic function f(z) is an eigenstate with energy E0 =  1  2ℏωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A complete basis of  analytic functions are the monomials fn(z) = zn and have energy E0 and angular momentum  Lz = nℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the lowest Landau level whose wave functions are ψn(z) = zn exp(−|z|2/4ℓ2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On  the other hand, an anti-analytic function fN = ¯zN is an eigenstate of energy EN = ℏωc  �  N + 1  2  �  ,  where ωc =  e|B|  Mc is the cyclotron frequency, and angular momentum Lz = −Nℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' States with  angular momentum nℏ have the same energy and the degeneracy is equal to the number of flux  quanta Nφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the most part we will be interested in the states in the lowest Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the absence of disorder the Landau levels have an extensive degeneracy equal to the num-  ber of flux quantum Nφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we consider a system of N electrons in a Landau level the natural  measure of density is not the areal density ρ = Ne  L2 but the fraction ν = Ne  Nφ the states in the Landau  level which are occupied by electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The many-body state in which all Nφ states of the lowest  Landau level has filling fraction ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The wave function for this state is the Slater determinant  of the Landau states in the m = 0 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' After some simple algebra the wave function of this state  is found to be  Ψν=1(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zNe) =  �  i< j  (zi − zj) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  Ne  �  i=1  |zi|2  4ℓ2  0  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (377)  This is the ground state of the non-interacting system and it is non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to  see that this state has a finite energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, in the Hilbert space with a fixed number  of electrons, the lowest energy excitation is a particle-hole pair in which the particle is in an  unoccupied state of the first excited Landau level, with m = 1 and energy E1 = 3  2ℏωc and a hole  a single-particle state in the lowest Landau level, with energy E0 = ℏωc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The excitation energy of  the electron-hole pair is just ℏωc which is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in the free particle system this excited  states has a huge degeneracy as the particle and the hole can be in any single particle state of the  Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  74  At a very naive level one can estimate the Hall conductivity for a translationally invariant  system filling up n Landau level by the following simple argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If n Landau levels are filled,  the number of electrons N = nNφ and the total charge is then Q = eN = e n Nφ = e n BL2/φ0 =  ne2BL2/hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If a weak and uniform in-plane electric field E is applied in the presence of a per-  pendicular magnetic field B = B ez, then the entire system (its center of mass) moves at the drift  velocity � such that � × B = −Ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, there is a Hall current J = Q�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The current density  is j = J/L2 = Q�/L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, ji =  Qc  BL2 ǫi j E j, which implies that the Hall conductivity is  σxy = Qc/BL2 = ne2/h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  While the above argument is formally correct and yields the correct value of the Hall conduc-  tivity in this highly idealized setting, at a very basic level it is faulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To begin with, it assumes  exact translational invariance and, hence, Galilean symmetry, which are not obeyed in any realis-  tic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The second objection is that, even in this idealized system, as the number of electrons  is varied, the chemical potential (and hence the Fermi energy) jumps discontinuously from one  Landau level to the next resulting in a linearly increasing Hall conductivity (as predicted classi-  cally) without any of the observed plateaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, this argument does not explain which  the Hall conductivity is so precisely quantized whereas, a priori, one would expect that being a  transport coefficient the Hall conductivity would depend on lots of complicated materials details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  But, it it turns out that it does not!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us first discuss the observed universality of the Hall conductivity, namely its robustness  and independence of microscopic details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a remarkable paper Qian Niu, David Thouless and  Yong-Shi Wu showed that, provided the ground state is separated by a finite energy gap from  the excited states, the Hall conductivity is actually a topological invariant [225].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The argument  has a similarity with what we discussed in the case of the Hall effect on a 2D lattice (in section  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4) but in this more general system one considers the full many-body wave function, rather  than the one-particle states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To show that this true they considered a system of N electrons  with periodic boundary conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' on a two-dimensional torus, T 2 = S 1 × S 1, with Nφ  flux quanta going through the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The torus is a rectangle of dimensions L1 and L2 with  opposite ends identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is necessary to allow for a Hall current to be globally allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  electromagnetic gauge field for a system ona torus with uniform non-vanishing flux cannot obey  periodic boundary conditions but rather accross the torus the gauge fields must differ by (large)  gauge transformations  A1(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2 +L2) = A1(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2)+∂1β2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A2(x1 +L1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2) = A2(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2)+∂2β1(x1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' x2) (378)  while the wave functions themselves obey twisted boundary conditions  Ψ([x(j) + L1e1]) = exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−i e  ℏc  Ne  �  j=1  β1([x(j)]) + iθ1  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 × Ψ([x(j)])  Ψ([x(j) + L2e2]) = exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−i e  ℏc  Ne  �  j=1  β2([x(j)]) + iθ2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 × Ψ([x(j)])  (379)  where e j (with j = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') are two unit vectors on the torus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and θ1 and θ2 are two angles that twist  the boundary conditions of the wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In order to have a current on the torus they assumed that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in addition to the uniform flux going  through the torus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' there a weak uniform electric field E on the torus represented by the constant  in space vector potential δA = E t = ▽[U(x)t] whose circulation on the two non-contractible  circles Γ1 and Γ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which wrap around the directions x1 and x2 of the torus T 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' are  I j =  �  Γj  δA · dx = t  �  Γj  E · dx = itE jL j  (380)  Line integrals of a gauge field on non-contractible loops in space (or space-time) are called  holonomies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By inspection we see that the angles θ1 and θ2 are given by  θj = e  ℏcI j  (381)  Alternatively, the angles θ = (θ1, θ2) can be interpreted as magnetic fluxes (in units of the flux  quantum φ0) through the two non-contractible circles of the torus T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The angles θ are defined mod 2π since they are phase factors that twist the phase of the  wave functions, and define a 2-torus of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By comparing with what we did in  75  section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 in the case of the single-particle wave functions in the lattice model, we see that  the many-body wave function Ψθ and the twist angles θ is the same as the relation between the  phase of the Hofstadter wave functions with the momentum k of the magnetic BZ (which is also  a 2-torus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, as it is always the case, while the phase of (in this case) the many-body wave  function Ψθ is arbitrary, the changes of this phase as the the twist angle θ is changed are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To  quantify this dependence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' for a given many-body state Ψ(α)  θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' where α labels the state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we define a  Berry connection A(α)(θ) on the torus of boundary conditions θ  A(α)(θ) = i  �  Ψ(α)  θ  ����∂θ  ����Ψ(α)  θ  �  (382)  Under a redefinition of the phase of the state  Ψ(α)  θ  → exp(if(θ)) Ψ(α)  θ  (383)  the Berry connection A(α)(θ) changes by a gauge transformation  A(α)(θ) → A(α)(θ) − ∂θ f(θ)  (384)  Niu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thouless and Wu [225] showed that the expression of the Kubo formula for the Hall  conductivity of the state Ψ(α)  θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' averaged over the boundary conditions θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' is given in terms of the  flux of the Berry connection A(α)(θ) through the 2-torus of boundary conditions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' by the gauge-  invariant quantity  ⟨(σxy)α⟩θ = e2  ℏ  � 2π  0  dθ1  2π  � 2π  0  dθ2  2π  �  ∂1A(α)  2  − ∂2A(α)  1  �  = e2  h  1  2π  �  γ  A(α)(θ) · dθ  (385)  where γ is the square contour with corners at (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (2π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2π) and (2π,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' that defines the  2-torus of boundary conditions θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result is known as the Niu-Thouless-Wu formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (385) implies that for the Hall conductivity to be non-vanishing the Berry connection  A(α)(θ) must have a non-vanishing flux through the torus of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For this to be  true, just as we did in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3, we must consider large gauge transformations of the form  A1(θ + 2πe2) =A1(θ) + ∂1 f2(θ)  A2(θ + 2πe1) =A2(θ) + ∂2 f1(θ)  Ψ(α)({x(j)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ + 2π e1) = exp(if1(θ)) Ψ(α)({x(j)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ)  Ψ(α)({x(j)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ + 2π e2) = exp(if2(θ)) Ψ(α)({x(j)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ)  (386)  where e1 and e2 are two orthogonal unit vectors on the torus of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can  now repeat the analysis we did in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 which, in this context, implies that the wave  function Ψ(α)  θ  cannot be globally well defined on the torus of boundary conditions, where it must  have zeros, and where it should be defined in patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The end result is that the wave functions  labeled by α are classified by a topological invariant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the first-valued Chern number C(α)  1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which  here it is given by  C(α)  1  = 1  2π  �  γ  A(α)(θ) · dθ  (387)  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' that the Hall conductivity in he state α (averaged over the twisted boundary condi-  tions) exhibits the integer quantum Hall effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ⟨(σxy)(α)⟩θ = C1  e2  h  (388)  Expressing the Hal conductivity in terms of the first Chern number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which is a topological invari-  ant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' proves that the value of the conductivity cannot be changed by local physics effects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' such  as disorder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The topological nature of the Hall conductivity is the reason for the robustness  and high precision of the measured value of the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In subsequent work Niu and  Thouless showed that, provided the state has a finite energy gap to all excitations, in the thermo-  dynamic limit the Hall conductivity averaged over boundary conditions and for a given boundary  condition are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (388) seemingly implies that there can only be an integer quantum Hall  effect which, as we see shortly, it is not the case: there is a fractional quantum Hall effect in  76  which the Hall conductivity is a fraction, σxy = p  q  e2  h , where p and q are co-prime integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  value of the Hall conductivity is also found experimentally to be obeyed with the same precision  as in the integer quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the Hall conductivity in the fractional  case must also have a topological character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To understand why this can be the case we need an  implicit assumption that we made in our derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, in deriving the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (388) we  assumed (implicitly) that for each value of θ there is only one many body state Ψ(α)  θ  (up to gauge  transformations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we will see below, in the fractional quantum Hall effect on a torus (and, in  fact on any closed surface except the sphere) the ground state is degenerate in the thermodynamic  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will also see that traversing the torus of boundary conditions once maps one degenerate  state to another one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general, we will find that on the 2-torus in the thermodynamic limit there  are m ∈ Z exactly degenerate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, one returns to the original state after sweeping  m times the torus of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will also see below that the degenerate states are  labeled by quantum numbers related to the anyons that they support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  While the topological argument proves the robustness (and universality) of the value of the  Hall conductivity, it does not provide an insight for why there are plateaus in its magnetic field  dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If for some value of the filling fraction ν = N/Nφ the electron fluid is exactly at a  ground state Ψ(α) upon changing either the number of particles N or the magnetic field B (but not  both) the fluid will now will be at the state Ψ(α) plus or minus some number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  existence of the plateau in σxy can be understood if the extra electrons (or holes) do not contribute  to the Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This happens in the presence of disorder as these extra particles will  be localized by the disorder and localized states, whose wave functions decay exponentially are  insensitive to boundary conditions and, hence, these states do not contribute to the conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This argument was first put forth by Laughlin [226] who build on the result that in a disordered  system in two dimensions all states are localized [63] but made the key assumption that the  state at the center of the Landau level (broadened by disorder) is extended and contributes to  the conductivity if this state is filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This picture implies that this is actually a quantum phase  transition from an insulator (dubbed a Hall insulator) to a state with a quantized Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This intuitive and appealing proposal has been the focus of research for many years and  it is largely an unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is in fact a proposed field theory for this quantum  phase transition based on a a non-linear sigma model (in replica space) with a topological θ-term  [227, 228], analogous to the theory we discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 for quantum antiferromagnets in  1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in this proposal the (Boltzmann) longitudinal conductivity σxx plays the role of  the inverse of the coupling constant of the non-linear sigma model while σxy plays the role of the  coefficient of the θ-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although the RG flows suggested by this approach qualitatively agree  with currently existing experimental data, and actual derivation is still wanting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, aside from the fact that is is still an unsolved problem, there is the problem that this  explanation ignores the fact that in a Landau level interactions are dominant and are the key to  the understanding of the fractional case even in the same samples!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A symptom of this problem  is that, at least in the cleaner samples, as we noted above the longitudinal conductivity vanishes  exponentially with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This dependence means that there is a clean energy gap (and not  just a mobility gap) which suggests that the state with additional excitations may have some sort  of at least local order which would provide for a local energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' An actual theory based on this  picture has yet to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Laughlin Wave Function  We will turn now to the fractional quantum Hall effect (FQHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The FQHE is observed in  fractionally filled Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These states have fractional filling fractions, namely that ν = Ne  Nφ  is a fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Most of the observed states are in the lowest Landau level, with N = 0, but a few  states with very intriguing properties are seen in the first excited Landau Level with N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider for now the states in the lowest Landau level, N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will assume that  there is no disorder and, for now, we will ignore the effects of boundaries of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Physi-  cally, the important interaction here is the Coulomb interaction although, in some samples with  screening layers, short-range interactions may be important as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since there is a large mag-  netic field, the Zeeman interaction is typically a large energy scale and the electros are expected  to be fully polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, there are situations under which the gyromagnetic factor can be  made small (and even zero) and spin unpolarized states have to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In some platforms,  such as graphene, additional quantum numbers are present, such as “flavor” associated with the  different Dirac states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In our discussion we will not cover these richer possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Under these circumstances we have a problem in which all Nφ available one-particle states are  77  exactly degenerate (the Landau level is “flat”) but only a fraction of them are filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit,  this system as as strongly interaction as it gets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Not surprisingly, in a system of this type time-  honored standard approximations fail, such as Hartree-Fock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Given the macroscopic degeneracy  and the nature of the states in a magnetic field, a system of this type may harbor many different  types of actual ground states depending of the types of interactions that are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Tsui, St¨ormer and Gossard [221] did transport experiments in the two-dimensional electron  gas (2DEG) in high-purity GaAs-AlAs heterostructures and reported the discovery of a state with  a highly precise value of the Hall conductivity of σxy = 1  3  e2  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such a state cannot be understood  in any obvious way in terms of the integer quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The observation of a clean gap  in the low temperature longitudinal resistivity σxx → 0 as T → 0 implied that the 2DEG is  incompressible and quite likely uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The first breakthrough was Laughlin’s unique insight which led to his explanation of the  experiments in terms of a novel quantum liquid of fully spin polarized electrons at filling fractions  ν = 1  m (with m and odd integer) whose wave function he proposed to be  Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) =  �  i< j  (zi − zj)m × exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  Ne  �  i=1  |zi|2  4ℓ2  0  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (389)  where N = Ne is the number of electrons, {zi} are their (complex) coordinates and m is an odd  integer (which makes the wave function antisymmetric, as required by Fermi statistics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Laughlin extracted the physics encoded in this wave function by considering the probability  ����Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  ����  2  of finding the N electrons in the locations {zi} (with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The norm of  the Laughlin wave function is  ����  ����Ψm  ����  ����  2  =  �  d2z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' d2zN  ����Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  ����  2  (390)  The Laughlin wave function Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) is an eigenstate of angular momentum with eigen-  value Lm = 1  2mN(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This remarkable wave function has a large overlap (∼ 98%) with the  exact wave function with V(R) = e2  εR (Coulomb) interactions in systems of up to eight particles  (ε is the dielectric constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The norm of the state can ten be interpreted as the classical partition function of a system  of a system of N particles at the locations {zi} at inverse temperature β = m with the effective  interaction  U(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) = −2  �  1≤i< j≤N  ln |z1 − zj| +  1  2mℓ2  0  N  �  j=1  |zj|2  (391)  This classical partition function is a one-component plasma, of a gas of particles with unit (neg-  ative) charge interacting with each other with the repulsive 2D classical (logarithmic!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') Coulomb  interaction, VCG = − ln |zi − zj| (not 1/R!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The last term represents the contribution of a neu-  tralizing background positive charge (which here it originates in the gaussian factor of the wave  function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, this is a one-component plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can check that this is correct  since ▽2  �  |z|2  2mℓ2  0  �  =  2  mℓ2  0 which corresponds to a uniform (areal) charge density ρ0 =  1  2πmℓ2  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Clas-  sical Monte Carlo simulations of this probability distribution showed that this wave function  describes an incompressible fluid state if m ≲ 7, while for larger values it represents a crystalline  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This approach is known as the plasma analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quasiholes have fractional charge  Laughlin considered a vortex-like excitation in the fluid created by the adiabatic insertion of  an infinitesimally thin solenoid at some coordinate z0 carrying a magnetic flux Φ, a fluxoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  the presence of this solenoid the single particle state zn exp(|z|2/4ℓ2  0) acquires a branch cut and  becomes the state zn+α exp(|z|2/4ℓ2  0) where α = Φ/φ0, with φ0 = hc/e being the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This analytic structure is the Aharonov-Bohm effect [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, if the solenoid carries just one flux quantum α = 1 and the net effect is that the state  becomes zn+1 exp(−|z|2/4ℓ2  0) which has one more unit of angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these reasons,  in the presence of a solenoid with one flux quantum inserted at z0, the Laughlin wave function  for the 2DEG in a large magnetic field now becomes  Ψqh  m (z0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) =  N  �  i=1  (zi − z0) × Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN)  (392)  78  A calculation using the plasma analogy reveals that the net effect of the solenoid is to expel a  charge equal to −e/m from the vicinity of the location of the solenoid or, equivalently, to create a  fractionally charged quasihole at z0 with charge Qqh = +e/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The charge distribution is uniform  at long distances but is depleted (hence a quasihole) on a length scale ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One can check that this  state costs a finite energy ε0 which depends on the particular interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand, if we consider a system in which the solenoid is inserted adiabatically  so that after a long time t there is a quasihole at z0, an amount of charge equal to −e/m will  flow radially outwards to the outer edge of the 2DEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The adiabatic insertion of the solenoid  generates a azimuthal electric field (an e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') and a radial Hall current, and a Hall conductivity  σxy = 1  m  e2  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, the Laughlin wave function exhibits the fractional quantum Hall effect and  its excitations are vortices with fractional charge e/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is important to note the non-local nature of the Laughlin quasihole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The non-locality is  inherited from the fluxoid (the solenoid) inserted at z0: even though the magnetic field of the  fluxoid vanishes away from it, B(x) = Φ δ2(x − x0), its vector potential A(x) does not since its  circulation on any non-contractible loop γ that contains the location of the fluxoid inside must  be equal to the flux Φ carried by the fluxoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although in principle one can choose a singular  gauge in which A(x) vanishes locally, this cannot be true everywhere as it must leave behind a  Dirac-like string on a curve Γ ranging form the location z0 of the fluxoid to the boundary of the  system with A taking a singular value on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This feature has the same form as the 2D vortex that  we discussed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson and Ro˘cek [229] showed that the existence of this Dirac  string causes the phase of a quantum state that carries charge e∗ to have a branch cut on the curve  Γ and to change by exp(i2π(e∗/e)Φ/φ0), as required by the Aharonov-Bohm effect [155].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  effect is unobservable for integer-charged states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Jain States  The construction of the Laughlin quasihole motivated Jainendra Jain [230, 231] to rewrite  the Laughlin wave function in the form  Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN) =  �  i< j  (zi − zj)m−1 × Ψν=1(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  (393)  Here the prefactor represents particles each attached with an even number, m − 1, of flux quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The second factor,Ψν=1(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN), is the wave function of fermions filling up a lowest Landau  level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the Vandermonde determinant of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(377).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this form, the inverse of the filling factor,  1/ν, which is the number of flux quanta per particle, is written as 1  ν = (m − 1) + 1 = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  interpretation now is that the (m − 1)φ0 magnetic flux carried by each particle, which partially  screens the uniform field down to an effective field Beff = B − (m − 1)φ0 corresponding to one  effective flux quantum per particle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a filled lowest Landau level of the effective field Beff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  other words, the wave function for the ν = 1/m FQH state is reinterpreted as the νeff = 1 integer  QH state of N composite fermions, each being an electron carrying attached an even integer m−1  flux quanta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This procedure is known as flux attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Physically it means that the strong interactions  between the electrons for other electrons to be further away which, in virtue of being in a strong  magnetic field, is equivalent to an increase of their relative angular momentum, as if there was a  local change in the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is no reason to believe that the composite fermions are  weakly coupled, even though this is frequently stated in the literature without any real supporting  evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Jain then generalized this reinterpretation of the Laughlin state to a whole sequence of FQH  states obtained by filling p levels of these partially screened magnetic fluxes  Ψm,p(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN) = PLLL  �  i< j  (zi − zj)m−1Ψν=p(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  (394)  where Ψν=p(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) is the wave function of p filled Landau levels (of composite fermions),  and PLL is an operator that projects onto the lowest Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By counting fluxes we see that  the filling fractions for the Jain states satisfy  1  ν(m, p) = m − 1 ± 1  p  (395)  79  where we allowed for the possibility that the external fluxes may be over-screened by the com-  posite fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Alternatively we can write  ν(m, p) =  p  p(m − 1) + ±1  (396)  The Jain states with p = 1 are the laughlin states and each is called the primary state of a  Jain sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Almost all the observed FQH states belong to one of these sequences (and its  generalizations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, the most prominent states (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' those with larger plateaus) belong  to the sequence 1  3, 2  5, 3  7, 4  9, 5  11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and to the reversed sequence 1, 2  3, 3  5, 4  7, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='. For p → ∞, the Jain  sequences converge to the values of the filling fraction limp→∞ ν(m, p) =  1  m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit, the  m − 1 fluxes attached to each particle exactly cancels th external flux leading us to a (possibly)  Fermi liquid of composite fermions [232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quasiholes have fractional statistics  The non-locality and topological nature of the Laughlin quasihole implies that this state must  be regarded as a soliton (or vortex) of the charged quantum fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The vortex-like state with wave  function Ψqh  m (z0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN) is called the Laughlin quasihole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This state represents a composite  object of a flux quantum and a fractional charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This structure of this quantum state is strongly  reminiscent of the flux-charge composite objects considered by Frank Wilczek who showed that  such objects should be anyons, states that exhibit fractional statistics [156].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although at a con-  ceptual level anyons were proposed (almost) prior to the discovery of the FQHE, it is in this  setting that fractional statistics entered actual physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The actual experimental observation took  work by many people, and was only confirmed in 2020 [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Fractional statistics dictates the analytic form of the wave functions of two or more quasiholes  [234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider a laughlin-type wave function for two quasiholes located at (complex)  coordinates u and �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Naively we expect the wave function for two quasiholes in the ν = 1/m  Laughlin state to have the form  Ψ(u, �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN) = N(u, �)  N  �  j=1  (zj − u)(zj − �) Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  (397)  The prefactor N(u, �) must be chosen to account for the fact that theres is an additional change of  angular momentum due to the additional fluxoid at �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We also expect that as the quasihole with  charge e/m is adiabatically carried around the the other quasihole (also with charge e/m) there  will be ac accrued phase in the wave function due to the Aharonov-Bohm effect of the charge  of one quasihole circling around the flux of the other (or, equivalently, crossing the branch cut)  [229].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The requirement of translation invariance and analyticity are met by the choice [234]  (ignoring a multiplicative normalization constant)  N(u, �) = (u − �)1/m exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  1  4ℓ2  0m(|u|2 + |�|2)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (398)  which has a branch cut stretching from u to �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A calculation using the plasma analogy represents  this state as a set of N charge −1 particles interacting with two additional particles of charge −1/m  at u and � (and a neutralizing background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The branch cut implies that dragging a quasihole  around the other during a π rotation followed by a translation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' an exchange), induces a  monodromy in the wave with a jump in its phase of exp(±iπ/m) (with the sign determined by the  orientation of the monodromy) in such a way the that wave function changes by a phase factor  Ψ(u, �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) �→ e±i π  m Ψ(u, �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN)  (399)  In other words, the quasihole is an anyon with fractional statistics π/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Daniel Arovas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Robert  Schrieffer and Frank Wilczek [235] further refined this argument by showing that under such  a slow adiabatic change the wave function of two quasiholes acquires a Berry phase which ac-  counts for the fractional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' An explicit path-integral derivation of this effect, which uses  the fact that the quasihole states are coherent states can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9], chapter 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hydrodynamic Effective Field Theory  We will now turn to the approaches to the FQHE using the methods of quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As we will see the concept of flux attachment plays a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 we showed that  80  Chern-Simons gauge theory is in fact a theory of flux attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus we expect that Chern-  Simons gauge theory should play a key role as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is useful to ask first why should Chern-Simons gauge theory play a central role at least in  the description of the low energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is a simple, yet powerful, phenomenological  argument due to J¨org Fr¨ohlich and Anthony Zee that shows why this should be the case [236].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This in essence a hydrodynamic argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The fractional quantum Hall effect occurs in a 2DEG  in the presence of a large magnetic field which breaks explicitly time reversal invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  regime of the FQHE the 2DEG is a uniform charged incompressible fluid in which charge is  conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the charge 3-current jµ = (j0, j) (where j0 is the charge density and  j is the charge current density) must be locally conserved,  ∂µ jµ = 0  (400)  which is the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The solution of this equation is that the conserved current can  be written in terms of a dual (in the Hodge sense) vector field Bµsuch that  jµ = 1  2πǫµνλ∂νBλ  (401)  The factor of  1  2π is introduced for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The current field jµ is not changed by a  smooth redefinition of the vector field Bµ → Bµ + ∂µΦ, where Φ(x) is a non-singular field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  means that the vector field Bµ is a gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The effective action of this theory is a functional of the current distribution \uf6beµ and, hence, of  the gauge field Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It should be gauge-invariant, and odd under parity and time-reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since  the fluid is incompressible and uniform, at long distances and low energies the action must be a  local functional of the gauge field which should be at least Galilean invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, the  coupling of the fluid to an external electromagnetic probe field Aµ must be of the usual form  −ejµAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To lowest orders in derivatives, there the unique local and gauge-invariant Lagrangian  density which is odd under time reversal (and parity) is the Chern-Simons theory  Leff[Bµ] = m  4πǫµνλBµ∂νBλ − 1  4g2 F 2  µν − e  2πAµǫµνλ∂νBλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (402)  The first term is the Chern-Simons term of the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The coefficient m is a dimensionless  integer which, as we saw in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3 is required for the theory to be invariant on a closed  manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The second is a Maxwell term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here Fµν = ∂µBν −∂νAµ is the field strength tensor of  the gauge field Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By power counting we see that the coupling constant g2 has units of length−1  and, hence this term is irrelevant at long distances and will be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The third term is the  coupling of the current jµ to the electromagnetic field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon an integration by parts this term  can be written as BµJ µ where  Jµ = − e  2πǫµνλ∂νAλ  (403)  is a current minimally coupled to Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The equation of motion of the (dynamical) gauge field Bµ is  δL  δB = 0  (404)  which yields the relation  m  2πǫµνλBµ∂νBλ = Jµ  (405)  From the definition of the current Jµ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (403), we see that the solution of the equation of  motion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (405), up to a gauge transformation, is  mBµ = −eAµ  (406)  By plugging this relation back into the Lagrangian for the gauge field Bµ, Eq (402) (which  si equivalent to integrate out the gauge field Bµ) we find that the effective Lagrangian of the  electromagnetic field Aµ is just a Chern-Simons term  Leff[Aµ] =  e2  4πmǫµνλAµ∂νAλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (407)  81  This result allows us to read-off the Hall conductivity of the fluid  σxy = 1  m  e2  2πℏ  (408)  where we restored units such that ℏ is not unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, this fluid exhibits the fractional  quantum Hall effect for a fluid at filling fraction ν = 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We should note that this heuristic argument applies equally to systems of fermions, for which  m is odd, as well as to systems of bosons, for which m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Composite Boson Field Theory  We will now discuss tow field-theoretic approaches to the FQHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These approaches use the  concept of flux attachment as a mapping of fermions to bosons and as a mapping of fermions  to fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These approaches are a form of duality transformation which engineers a form  of statistical transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will first show that Chern-Simons theory can be used to effect  such a mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' That this is possible should not be surprising since, as we saw in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7,  Chen-Simons theory is a theory of fractional statistics and, hence, of anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We should make some important comments on both approaches before we get into a detailed  description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While the mapping of fermions to composite bosons and fermions to composite  bosons is correct, their approximate mean field descriptions violate symmetries of the 2DEG in  a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At the root level is the fact that the flux attachment is local in space time and  approximate descriptions of these theories bring about a large amount of Landau level mixing,  even in the large field limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance, at the mean field theory level the composite boson  approach is equivalent to a Bose-Einstein condensate which is invariant under translations and  under time reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this picture the breaking of time reversal is present in the coupling to a  Chern-Simons gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Likewise, the mean field theory of the composite fermion theory is a  problem of composite fermions in a partially screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While a partially screened magnetic field  still breaks time reversal, it does not behave properly under magnetic translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we will  see, these problems will be solved, at least in the low energy regime, by quantum fluctuations  which restore the symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In both cases, in the fractional states one recovers an effective  topological field theory which captures the universal physics of these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Needless to say it,  both approaches do poorly in the computation of non-universal dimensionful quantities such as  energy gaps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These difficulties become very severe in the compressible states whose low  energy behavior is not topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Unlike the wave function approaches which project these states into a specific Landau level  (usually the lowest), the field theoretic approach does not effect such projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Several papers  have been written attempting to do a field theory projected into a Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These approaches  transformed the problem into quantum field theory on a non-commutative plane, which is inher-  ent the nature of the states in a magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Although some significant progress has been made  in this direction, these theories remain poorly understood [237, 238, 239, 240, 241, 242, 243]  In this section we will focus on the composite boson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let us consider a theory in 2+1  dimensions with two dynamical abelian gauge fields Aµ and Bµ, whose Lagrangian is a sum of  a Chern-Simons term at level k and a BF term:  L = k  4πǫµνλA µ∂νA λ + 1  2πǫµνλA µ∂νBλ  (409)  The equation of motion for the field Aµ is  k  2πǫµνλ∂νA λ + 1  2πǫµνλ∂νBλ = 0  (410)  whose solution (up to a gauge transformation) is kAµ = −Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Plugging this relation into the  Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (409) we find that the effective Lagrangian for the field Bµ is  L[Bµ] = − 1  4πkǫµνλBµ∂νBλ  (411)  which states that exchanging Aµ ↔ Bµ is equivalent to the “duality” k ↔ −1/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 we showed that particles coupled to a Chern simons gauge field at level k  have fractional statistics exp(±iπ/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, we see that particles coupled to the field Bµ have  fractional statistics exp(±iπk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, for k odd this coupling amounts to exchanging  a theory of fermions to a theory of bosons coupled to the gauge field Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a form of  82  bosonization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Alternatively, for k even it maps a theory of fermions to another theory of fermions  coupled to a gauge field Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These observations have led to distinct, but ultimately equivalent  description of the quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will begin the approach of mapping fermions to bosons and use it in the problem of the  FQHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the Landau-Ginzburg theory (or composite boson theory) of Shoucheng Zhang,  Hans Hansson and Steven Kivelson [244] (ZHK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this approach the problem of fermions  coupled to a magnetic field interacting with each other through a two-body potential (that we  will assume is ultra with coupling constant λ, for simplicity) becomes the same problem but for  a theory of bosons which are also coupled to a Chern-Simons gauge field, which we will denote  by Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Lagrangian density for this equivalent system of composite bosons is  LCB = φ∗(x)[iD0 + µ]φ(x) + 1  2M |Dφ(x)|2 − λ(|φ(z)|4 +  1  4πmǫµνλA µ∂νA λ  (412)  where x = (x0, x) are the space-time coordinates, µ is the chemical potential, M is the mass of  the fermions, and m is an arbitrary odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The covariant derivative  Dµ = ∂µ + i e  ℏcAµ + iAµ  (413)  which effects the minimal coupling of the complex scalar field φ(x) to the background electro-  magnetic field Aµ and to the Chern-Simons gauge field Aµ, known in this context os known as  the statistical gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  ZHK showed that the FQH state can be thought as being closely related to to a phase in which  the complex scalar field condenses, much as in the case of a superfluid even though the FQH iis  not a superfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see how this works we will write  φ(x) =  �  ρ(x) exp(iω(x))  (414)  The classical equations of motion of the theory of composite bosons, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (412), are  δLCB  δφ∗(x) =0  ⇒  (iD0 + µ)φ(x) − 1  2M D2φ(x) − 2λ|φ(x)|2φ(x) = 0  (415)  LCB  δA0(x) =0  ⇒  1  2πmǫi j∂iA j + |φ(x)|2 = 0  (416)  LCB  δAi(x) =0  ⇒  1  2πmǫiαβ∂αA β +  i  2M  �φ∗(x)Diφ(x) − (Diφ(x))∗φ(x)� = 0 (417)  LCB  δµ =ρ0L2T  ⇒  �  d3x |φ(x)|2 = ρ0L2T  (418)  where ρ0 is the areal density of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A uniform solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (415)-(418) with constant amplitude of the composite bosons,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a uniform and static condensate ¯φ, with uniform statistical gauge field strength ¯  B, requires  that the there should be no current in the ground state and that each term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (417) should be  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This condition implies that the external field is canceled by the average statistical field,  eB  ℏc + ¯  B = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This condition together with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (416) and (418) imply that  ρ0 = 1  m  eB  ℏc = 1/m  2πℓ2  0  (419)  and we conclude that the filling fraction is ν = 1  m, which are the Laughlin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition we  get that the chemical potential is µ = 2λρ0 and that |¯φ|2 = ρ0, with ρ0 given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (419).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, these mean field results seeming imply that this system is a Bose-Einstein conden-  sate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To understand why this is incorrect, and to determine what it actually is, we need to go  beyond the mean field theor that we just described and evaluate the effects of quantum fluctua-  tions and write  φ(x) = (ρ0 + δρ(x))1/2 exp(iω(x)),  eAµ  ℏc + Aµ = δAµ  (420)  The partition function of this theory is a path integral over the fluctuating density fields δρ, the  Goldstone field ω and the fluctuation of the Chern-Simons gauge field δAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon integrating out  83  the massive density fluctuations to lowest (quadratic) order we find that the effective Lagrangian  for ω and δAµ is  Leff[ω, δAµ] = κ  2 (∂0ω − δA0)2 − ρs  2 (∂iω − δAi)2 +  1  4πmǫµνλδA µ∂νδA λ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (421)  The first two terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (421) is the effective Lagrangian of the Goldstone mode ω in the  Bogoliubov theory of superfluidity a compressibility κ and a superfluid stiffness ρs given by  κ = 1  2λ,  ρs = ρ0  M = ν  2πℏωc  (422)  However, as we see, this is not a superfluid since the phase field is “eaten” by the Chern-Simons  gauge field δAµ which now acquires a mass term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this sort of Higgs mechanism the fluctuating  gauge field has a massive longitudinal mode and a massive transverse mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, this state doe  not have any massless modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To see that this theory describes the fractional quantum Hall effect we need to compute the  linear response to a weak electromagnetic perturbation δAµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This can be accomplished by  considering the new Lagrangian  Leff[ω, δAµ, δAµ] = κ  2  �  ∂0ω − δA0 − e  ℏcδA0  �2  −ρs  2  �  ∂iω − δAi − e  ℏcδAi  �2  + 1  4πmǫµνλδA µ∂νδA λ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (423)  and integrate out the phase field ω (which can be set to zero in the London/unitary gauge) and  the fluctuation of the statistical field δAµ to find that effective electromagnetic action is  S eff[δAµ] =  1  4πm  e2  ℏ  �  d3x ǫµνλδAµ∂νδAλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (424)  from which we conclude that the Hall conductivity predicted by the composite boson theory is  σxy = 1  m  e2  h  (425)  as it should be for a FQHE at filling fraction ν = 1/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We conclude by looking at vortex states in the composite boson theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A vortex state is  a time-independent solution of the equations of motion Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (415)-(418) with the asymptotic  behavior (in the temporal gauge δA0 = 0)  lim  |x|→∞φ(x) = √ρ0 eiϕ(x)  (426)  lim  |x|→∞δAi(x) = ± ∂iϕ(x) = ±ǫi j  x j  |x|2  (427)  where ϕ(x) is the azimuthal angle on the plane  ϕ(x) = tan−1  � x2  x1  �  (428)  The energy of a neutral vortex (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' not coupled to a gauge field) is logarithmically divergent,  Evortex ≃ ρs  2 ln(R/a0) where R is the linear size of the system and a0 is a short-distance cutoff  (see section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the situation here is different since the complex scalar field φ(x) is  coupled to a dynamical Chern-Simons gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Except for the Chern-Simons nature of  this gauge field, the problem we are dealing with is similar to that of a superconductor coupled  to a dynamical gauge field or to the Abelian-Higgs model in quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of  interest here we have finite energy vortex solutions which satisfy the asymptotic condition  lim  |x|→∞  ����  �  i∂ j − δA j  �  φ(x)  ����  2  = 0  (429)  which is obeyed by configurations that obey Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(427).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, at long distances, the circulation of  the Chern-Simons gauge field on a large closed contour Γ that contains the vortex satisfies  �  Γ  δA jdx j = ±2π  (430)  84  This vortex carries charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this we compute the local charge density j0(x)  j0(x) = − δS eff  δA0(x) = −e δS eff  δδA0(x) = +e δS CS  δδA0(x) =  e  2πmǫi j∂iδA j(x)  (431)  and compute the total charge of the vortex on a larger region Σ whose boundary is Γ and obtain  Q = e  �  d2xj0(x) =  e  2πm  �  Σ  d2x ǫi j∂iδA j(x) = e  m  1  2π  �  Γ  dx jδA j(x) = ± e  m  (432)  Therefore, the vortex of the composite boson theory has the same charge as the Laughlin quasi-  hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  To determine the statistics of the vortex we go back to the effective Lagrangian written in the  form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (409), (from now on we set δA µ ≡ A µ)  Leff = κ  2(∂0ω−A0)2−ρs  2 (∂ jω−A j)2+ 1  2πǫµνλA µ∂νBλ− e  2πǫµνλAµ∂νBλ− m  4πǫµνλBµ∂νBλ (433)  where Aµ is the external electromagnetic probe field, and where we used units with ℏ = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The first two terms make the statistical gauge field Aµ massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the low energy limit the  statistical gauge field is frozen to the vortex configurations of the complex scalar field φ or,  equivalently, to the vortex singularities of its phase field ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The vorticity current Ωµ is  Ωµ = ǫµνλ∂νA λ  (434)  The effective Lagrangian for the field Bµ is  Leff[Bµ] = − m  4πǫµνλBµ∂νBλ − e  2π Aµǫµνλ∂νBλ + ΩµBµ  (435)  which, except for the coupling to the electromagnetic field, has the same form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (224),  and of the effective action introduced in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 on phenomenological grounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The effective topological field theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (435) encodes al the universal data of fractional  quantum Hall fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Being topological this effective field theory has no energy scales and de-  scribes the physics at energies low compared to any excitation energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition to yield-  ing the correct Hall conductivity σxy = νe2/h (with ν = 1/m) and the fractional vortex charge  Q = e/m, this theory will allow us to draw important additional results about the low energy  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  By using the results of section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7, we conclude that the vortices of the composite boson are  anyons with fractional statistics exp(±iπ/m), consistent with the conclusions of section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In addition, on a 2-torus the ground state of this system is m-fold degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This feature,  characteristic of systems with topological order, is that the ground state degeneracy depends on  the topology of the surface on which the 2DEG resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The ground state on a torus degeneracy  was shown by Duncan Haldane and Edward Rezayi [245] by an explicit construction of a model  wave function for the Laughlin states on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On a surface with g handles (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' genus g) the  degeneracy is mg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Xiao-Gang Wen and Qian Niu [246] showed that these results hold for any  FQH state which are, quite generally, topological quantum fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Finally, we may ask how many distinct types of vortices does this theory have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The funda-  mental (Laughlin) vortex has charge e/m and statistics π/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In principle we could have a vortex  with any integer topological charge (winding number) n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such a vortex has charge ne/m and  statistics πn2/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, a vortex with topological charge n = m has charge e and statistics mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This vortex is indistinguishable from a hole (a missing electron) since it has the same charge and  statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We conclude that a Laughlin state has m distinct vortices with n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We  notice that this number is the as the number of degenerate states on a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10  we will see that this fact is closely related to the number of primary fields of a conformal field  theory associated with the FQH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Composite Fermion Field Theory  At the beginning of section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 we noted that flux attachment can be used to define equiv-  alent theories: a) by attaching an odd number, m, fluxes to each electron thereby becoming a  composite boson, and b) by attaching an even number, m − 1, fluxes by which the electrons  turn into composite fermions, as in Jain’s construction [230].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this section we discuss the most  salient features of the field theory approach to composite fermions, introduced by Ana L´opez and  85  me in 1991[247, 248] as a theory of all the Jain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This approach was extended in 1993 by  Bertrand Halperin, Patrick Lee and Nicholas Read [232] to the case of the compressible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  By following the same approach that we used in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7, but adapted to the case in  which we map to a theory of a composite Fermi field ψ(x), we find their dynamics is described  by the effective action [247]  SCF =  �  d3x  �  ψ∗(x)[iD0 + µ]ψ(x) +  1  2M |Dψ(x)|2  �  +  1  4πn  �  d3xǫµνλA µ∂νA λ  − 1  2  �  d3x  �  d3x′ (|ψ(x)|2 − ρ0)V(x − x′)(|ψ(x′)|2 − ρ0)  (436)  where n = m − 1 is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here V(x − x′) = δ(x0 − x′  0)V(|x − x′|) is the instantaneous  electron-electron repulsive interaction, and ρ0 is the average areal neutralizing charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As before, the covariant derivative is Dµ = ∂µ + ieAµ + Aµ, where Aµ is the electromagnetic field  (including the uniform magnetic field B), and Aµ is the statistical gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We are using units  such that ℏ = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We can simplify somewhat the form of the actions for the composite fermions by using the  fact that the Gauss law of Chern-Simons gauge theory states that the particle density and the  gauge flux are rigidly tied together, |ψ(x)|2 =  1  2πnB ≡ ǫi j∂iA j (this is the flux attachment) as a  operator identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we can write the action as  SCF =  �  d3x  �  ψ∗(x)[iD0 + µ]ψ(x) +  1  2M |Dψ(x)|2  �  +  1  4πn  �  d3x ǫµνλA µ∂νA λ  − 1  2  �  d3x  �  d3x′  �B(x)  2πn − ρ0)  �  V(x − x′)  �B(x′)  2πn − ρ0  �  (437)  Since this action is a quadratic form in the Fermi (Grassmann) fields ψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' we can integrate them  out to obtain the following effective action for the statistical gauge field Aµ  S eff[Aµ] = −itr  �  iD0 + µ + D2  2M  �  +  1  4πn  �  d3x ǫµνλ (A µ − eAµ)∂ν (A λ − eAλ) + S int[Aµ − eAµ]  (438)  where Aµ is a probe external electromagnetic field (with vanishing average) and does not include  the uniform magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The interaction term is  S int[Aµ − eAµ] = −1  2  �  d3x  �  d3x′  �(B(x) − eB(x))  2πn  − ρ0)  �  V(x − x′)  �(B(x′) − eB(x′))  2πn  − ρ0  �  (439)  Here too B(x) is an external probe with vanishing average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will investigate the properties of the path integral for the statistical gauge field  Z[Aµ] =  �  DAµ exp(iS eff[Aµ, Aµ])  (440)  using a saddle point expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The saddle-point condition of the effective action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (438)  δS eff  δAµ(x) = 0  (441)  leads to the equation of motion for the field Aµ  ⟨jF  µ (x)⟩ +  1  4πnǫµνλ  �  F νλ(x) − eFνλ�  = 0  (442)  where ⟨jF  µ ⟩ is the expectation value of the fermionic current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition we need to impose the  condition for the particle density to be uniform and equal to the neutralizing background charge  ⟨jF  0 (x)⟩ = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will assume that the electromagnetic field Aµ describes just a static uniform magnetic  field of strength B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Under these conditions, the ground state should also be static and uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This means that the field strength of the statistical gauge field should be constant and uniform  value ⟨B⟩, and that the current ⟨jF⟩ should vanish in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, the  Gauss Law of the Chern-Simons gauge field implies that ⟨B⟩ = −2πn ρ0, while the zero current  condition requires the the (statistical) electric field vanishes, ⟨E ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the composite  86  fermion couples in the same way to the electromagnetic field Aµ and to the statistical field Aµ we  conclude that they experience an effective magnetic field  Beff = B + 1  e⟨B⟩ = B − 2πn ρ0  e  (443)  If the total number of electrons is N, then ρ0/e = N/L2 where L is the linear size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us denote by Neff  φ the total number of flux quanta of the effective magnetic field (in units of  ℏ = c = 1 in which the flux quantum is φ0 = 2π), and 2πNφ = BL2 is the total flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, the  total effective flux is  2πNeff  φ = 2πNφ − 2πn N  (444)  Let ν = N/Nφ be the filling fraction and νeff = N/Neff  φ  the effective filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (444)  implies that these filling fractions are related by  1  νeff = 1  ν − n  (445)  Recall that n is an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the system will be incompressible only if νeff = p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In other words, the composite fermions fill an integer number p of the partially screened Landau  levels, which is Jain’s condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For this condition to be satisfied the filling fraction ν(p, n) is  ν±(n, p) =  p  np ± 1  (446)  which are the Jain fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the special case p = 1 the Jain fractions are the Laughlin fractions,  with n = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly we find that Beff is  Beff = ±  B  np ± 1  (447)  So we see that the external field B is partially screened to the smaller value Beff which can be  parallel (screened) to B or anti-parallel (overscreened) to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the same reason, at this mean  field level the cyclotron frequency ωc is reduced by the same amount to ωeff  c = ωc/(np ± 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will discuss now the effects of quantum fluctuations for the action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (438) about the  saddle-point solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will work to the lowest order (quadratic) in the fluctuations, which  can be regarded as a semi-classical (or “RPA”) treatment of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The quadratic effective  action for the fluctuations of statistical gauge field, which we will still denote by Aµ is  S (2)  eff[Aµ] =1  2  �  d3x  �  d3y A µ(x)ΠCF  µν (x, y) A ν(y)  +  1  4πn  �  d3x ǫµνλ(Aµ(x) − eAµ(x))∂ν(Aν(x) − eAν(x)) + S int(Aµ − eAµ)  (448)  Here, Πµν  F (x, y) is the polarization tensor of the composite fermions for p filled effective Landau  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The interaction term of the action is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (439).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As required by gauge invariance, the composite fermion polarization tensor is transverse,  ∂µΠCF  µν = 0, which fixes the tensorial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In momentum and frequency space the com-  posite fermion polarization tensor ΠCF  µν (q, ω) depends on three kernels ΠCF  0 (q, ω), ΠCF  1 (q, ω) and  ΠCF  2 (q, ω), where ΠCF  0  and ΠCF  2  contain the parity-even response of the composite fermions and  ΠCF  1  is their parity-odd response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Each kernel is given as a series terms representing particle-hole  processes with simple poles at the excitations energies ωrs = (r − s)ωeff  c , with r > p (particles)  and s ≤ p (holes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Each term has a residue which is an integer power of q2 times a Laguerre  polynomial in q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Details of these kernels can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [247] and in chapter 13 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, what will be important here is that, in the gapped states these kernels have a low energy  and low momentum limit which is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This will enable us to find a local topological effective  action only for the gapped states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  After integrating out the statistical gauge field Aµ we find the effective action for the external  electromagnetic probe field Aµ, which has the standard form  S eff[Aµ] = 1  2  �  d3x  �  d3y Aµ(x) Πµν(x − y) Aν(y)  (449)  which encodes the full electromagnetic response of the system, not just of the composite fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Once again, gauge invariance requires that the full polarization tensor be transverse, ∂µΠµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  87  In section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 we showed that there is a relation between the polarization tensor Πµν and  the current-current correlation function Dµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There we mentioned the these correlators are re-  quired (by gauge invariance) to satisfy a Ward identity known as the f-sum rule:  � ∞  −∞  dω  2π iω DR  00(ω, q) = ρ0  M q2  (450)  where the (retarded) density-density correlation function is related to the (retarded) polarization  tensor, DR  00(ω, q) = ΠR  00(ω, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the limit of low momentum q, at fixed frequency ω, The  expression for ΠR  00(ω, q) is  ΠR  00(ω, q) ≃ q2ΠR  0(ω, q) = −ρ0  M  q2  (ω + iǫ)2 − ω2c  (451)  where ωc = eB/(Mc) is the cyclotron frequency of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This expression saturates the  sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that whichever corrections Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (451) may have must vanish in the low q  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, in this limit the approximation that we made is actually exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is well  known result in the theory of the Fermi liquid [194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In addition, the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (451) implies that there is a particle-hole (density) collective  mode which at low momentum has energy ℏωc, without corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result is known as  Kohn’s theorem [249] which states that in a Galilean invariant system the 2DEG must have an  exact eigenstate at the cyclotron frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' More intuitively, the center of mass of the 2DEG exe-  cutes a cyclotron motion regardless of whether the particles are free, interacting or not, fermions,  bosons, or anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that the composite fermions do not satisfy  Kohn’s theorem as they cyclotron mode would be at the effective cyclotron frequency ωeff  c , and  that the leading quantum fluctuations have restored the magnetic symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the limit of low energy ω → 0 and low momentum q → 0 of the kernels ΠCF  0 , ΠCF  1  and  ΠCF  2  become  ΠCF  0 (0, 0) = p  2π  M  Beff  ≡ εCF,  ΠCF  1 (0, 0) = ± p  2π ≡ σCF  xy ,  ΠCF  2 (0, 0) = − 1  2π  p2  M ≡ −χCF  (452)  where εCF is the “dielectric constant” of the composite fermions, χCF is their effective “per-  meability”, and σCF  xy is the (integer) composite fermion Hall conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We then find that  low-energy effective action of the statistical gauge field Aµ is  S eff[Aµ] =1  2  �  σCF  xy +  1  2πn  � �  d3x ǫµνλA µ(x) ∂ν A λ(x)  − e  2πn  �  d3x ǫµνλA µ(x) ∂ν Aλ(x) + e2  4πn  �  d3x ǫµνλAµ(x) ∂ν Aλ(x)  +  �  d3x1  2  �  εCF Ei  2(x) − χCF B2(x)  �  −  1  8π2n2  �  d3x  �  d3y (B(x) − eB(x)) V(x − y) (B(y) − eB(y))  (453)  Notice that, except possibly for the last term, the effective action is a local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is possible  because the mean field theory state is gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In what follows we will focus only on the leading terms of the effective action and neglect the  subleading terms, the least two terms of the effective action which have the form of a Maxwell  term and an additional (possibly non-local) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The remaining leading terms have a Chern-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Simons form and a BF form whose effective Lagrangian is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Leff[Aµ] = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='±p + 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='ǫµνλA µ(x) ∂ν A λ(x)− e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2πn ǫµνλA µ(x) ∂ν Aλ(x)+ e2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4πn ǫµνλAµ(x) ∂ν Aλ(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(454)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='If now now integrate ot the statistical gauge field Aµ we find the the electromagnetic field Aµ has  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='the effective Lagrangian  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Leff[Aµ] = σxy  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='ǫµνλAµ(x) ∂ν Aν(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(455)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='from where we find that the Hall conductivity σxy of the 2DEG is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='σxy = e2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2πℏ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='pn ± 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(456)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='which is the Hall conductivity of the Jain states (in standard units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here, as before, n is an even  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now return to the effective Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(454).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As it stands, since the Chern-  Simons term has fractional level, it is not invariant under large gauge transformations and, as  such, cannot be defined on a closed manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To remedy this problem we can now write this  Lagrangian in terms of a theory with two dynamical gauge fields Aµ and Bµ as follows:  Leff[Aµ, Bµ] = p  4πǫµνλA µ(x) ∂ν A λ(x) − n  4πǫµνλBµ(x) ∂ν Bλ(x)  + 1  2πǫµνλA µ(x) ∂ν Bλ(x) − e  2πǫµνλBµ(x) ∂νAλ  (457)  This is a topological quantum field theory for all the Jain states [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This theory can be defined  on any manifold no matter its topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to see that upon integrating out the field Bµ  we recover the effective field theory for Aµ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(454).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Also, in the special case of the Laughlin  states, for which p = 1, we can integrate out the field Aµ, and arrive to the effective topological  field theory for the field Bµ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (435) that we derived in the composite boson theory in section  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us rewrite the Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (457) in a more compact yet general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To this end we  define a multicomponent gauge field A I  µ with I = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', L, an L-component charge vector t, and  an L × L second rank symmetric matrix KIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen and Zee have given a general classification of  a large class of fractional quantum Hall states, said to be abelian, whose topological field theory  is defined by the Lagrangian [251, 252]  L = 1  4πKIJǫµνλA I  µ (x) ∂ν A J  λ (x) − e  2πtIǫµνλAµ(x) ∂ν A λ  I (x)  (458)  In the case of the theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (457) which has two components, L = 2, with A 1  µ = Aµ and  A 2  µ = Bµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The two-component vector is t = (0, 1) and the matrix 2 × 2 matrix is  K =  �−p  1  1  n  �  (459)  Wen and Zee showed that, quite generally, a theory of the K-matrix form has a vacuum degener-  acy on a torus of |detK| and, on a surface of genus g, the degeneracy is |detK|g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They also showed  that the Hall conductivity is σxy = ν e2/h where the filling fraction is  ν =  L  �  I,J=1  tI K−1  IJ tJ  (460)  The properties of the quasiparticles can be determined by computing the appropriate correla-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the composite fermion theory, whose Lagrangian is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (437), the gauge-invariant  composite fermion propagator is  GCF(x − y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' γ(x, y)) =  �  ψ†(x) exp(i  �  γ(x,y)  dzµ(eAµ(z) + A µ))ψ(y)  �  (461)  which, as in any gauge theory, it is gauge-invariant but path-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here γ is an oriented  open path with endpoints that the space-time locations x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The composite fermion propa-  gator has information about many of the properties of the quasiparticles, including their electric  charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To find what is the statistics of the quasiparticles we need to compute a two-particle cor-  relator (a four-point function) which is defined in terms of two open oriented paths, say γ1(x, y)  and γ2(u, w), with endpoints at the locations of the two particles, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The computation of these correlators is done using a feynam sum over trajectories with dif-  ferent weights which depend on the gauge field configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To do these calculations is, in  general, a complicated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in a gapped state, the quasiparticles are massive and  in the low energy regime these expressions are dominated by a classical trajectory on an oriented  path ˜γ(x, y) with the same endpoints x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The end result reduces to the computation of the  expectation value of a Wilson loop operator  W[Γ] =  �  exp(i  �  Γ  dzµ(eAµ(z) + A µ(z))  �  A µ  (462)  89  on the closed oriented path Γ = γ � ˜γ− (where ˜γ−(y, x) has the opposite orientation as ˜γ(x, y) and  runs from y to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The explicit dependence on the external electromagnetic field yields the value  of the charge of the quasiparticle through the Aharanov-Bohm effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Likewise, the two-particle  correlator is reduced, in the asymptotic low energy regime, to the computation of an expectation  value of two Wilson loops in the effective Chern-Simons theory, as was done in section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7,  which yields the fractional statistics of the quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The interested reader can find details  in chapter 13 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A system described by a theory of the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (458) has different types of quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In general we can assign an integer-valued quantum number ℓI ∈ Z as the charge of the quasi-  particle with respect to the gauge field A µ  I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The general coupling of a quasiparticle current has  the form Lqp = jµℓIA µ  I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, the charge Q[ℓ] and the statistics δ[ℓ] of a general quasiparticle  defined by the integer-valued L-component vector ℓ are  Q[ℓ] = −etIK−1  IJ ℓJ,  δ[ℓ]  π  = ℓIK−1  IJ ℓJ + 1  (463)  where the +1 is required to refer the statistics to fermions (not bosons) [250].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These results are  general and hold for any abelian FQH state [251].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the case of the Jain states, the charge vector is t = (0, 1) and the quasiparticle is labeled by  the vector ℓ = (1, 0) which yield the results for the charge Q and the statistics δ  Q =  −e  np + 1,  δ = π  �  −n  np + 1 + 1  �  (464)  which are the quantum numbers of the quasiparticles of the Jain states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For the Laughlin state at  ν = 1/3 we obtains Q = −e/3 and δ = π/3 (as we should), which for the first Jain state at ν = 2/5  the charge Q and the statistics δ are Q = −e/5 and δ = 3π/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With some caveats, the predictions  for the quasiparticle charge in the FQH states with ν = 1/3 and ν = 2/5 have been confirmed in  noise measurements at a quantum point contact by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' de Picciotto and coworkers [253] and by  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Samindayar and coworkers [254].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractional statistics has been measured experimentally in  both FQH states at ν = 1/3 and ν = 2/5 in (Fabry-Perot) interferometers by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nakamaura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Liang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gardner, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Manfra [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Compressible States  At large values p → ∞ the Jain sequences converge to the even-denominator fractions  ν∞(n) = 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this limit, Beff → 0 and ωeff  c  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, at the filling fractions  ν∞(n) the average effective field experienced by the composite fermions is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This mean  field theory then would predict that this state is essentially a Fermi liquid of composite fermions  which is a compressible state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, one can compute the effective charge of the composite  fermion and find that it is e/(np ±1), which also vanishes in the compressible limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is straight-  forward to see that the Fermi sea of the composite fermions is a disk with Fermi momentum  pF = (2ν∞(n))1/2ℏ/ℓ0 and that the Fermi energy is EF = ν∞(n)ℏ/(Mℓ2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, Lee, and Read  [232] did an in-depth analysis of the consequences of this compressible state which are largely  in agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In our discussion of the incompressible FQ states, both in the composite boson and in the  composite fermion approaches in sections 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='8 we saw that the mean field approxi-  mation violated symmetries that were restored by quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This fact allowed us the  derive effective filed theories which are asymptotically exact in the low energy IR regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' What  allowed us to obtain these exact results is the existence of a gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the compressible  states are gapless and in some sense should be regarded theories of a quantum critical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In fact, if we reexamine the field theory of composite fermions of section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='8 we see  that, in the compressible limit p → ∞, the action of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (437) describes a system of fermions at  finite density coupled to the fluctuations Aµ of the statistical gauge field with a Chern-Simons  term, but without an external uniform field (which has been canceled by the flux of the average  statistical gauge field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This theory has two salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' one is that the only trace left of  the explicitly broken time-reversal symmetry is in the presence of the Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  means that while the mean field theory looks like a Fermi liquid, the consequences of the broken  time reversal invariance only enters in the quantum fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, in this regime this  theory also breaks the particle-hole symmetry of a half-filled Landau level (in the high magnetic  field regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, these contributions, already at one-loop order, typically contain IR divergencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These IR divergencies are due to singular forward scattering processes of the composite fermions  90  by the gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They are most severe in the computation of the fermion self energy and de-  pend on the form of the electron-electron interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance if the interaction potential  V(R) is short range, the one-loop contribution to the fermion self-energy has an imaginary part  withe the behavior Σ′′(ω) ∼ ω2/3 which is much larger than the real part Σ′(ω) as ω → 0 and the  the Fermi energy EF is approached [232].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This behavior, which is found in many metallic sys-  tems at quantum criticality [255, 256, 69] implies that the width of the quasiparticle is asymptot-  ically much larger than the energy as the Femi surface is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This one-loop result means  that perturbation theory fails in the IR and that actual behavior may be non-perturbative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  problem is present even in the case of unscreened e2  R (Coulomb) interactions where the singular  behavior is milder with Σ′′(ω) ∼ ω ln ω (known as Marginal Fermi Liquid scaling [257]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  At any rate, these IR singular behaviors imply that the quasiparticle picture is inadequate and  the that in these regimes there may be no quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These systems are generally known as  “Strange Metals”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many strongly interacting physical systems of interest, ranging form high Tc  superconductors to (largely conjectured) spin liquids with spinon Fermi surfaces to even quark-  gluon plasmas share these types of singular IR behaviors [257, 258].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In spite of a large body  of theoretical work, it has remained largely unsolved problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Perturbative renormalization  group methods used to justify the Landau theory of the Fermi liquid [259, 260] generally fail  in these regimes, although in some cases long range interactions have allowed some degree of  control [261, 262].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Remarkably, the AdS/CFT Gauge/Gravity Duality [263] has brought new  insights into strange metal behaviors [264].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see below that relativistic dualities have  given additions insights into the physics of compressible states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fractional Quantum Hall Wave Functions and Conformal Field Theory  The Laughlin wave function and its generalizations share the remarkable feature of being  essentially universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aside form the dependence on the magnetic length ℓ0 in the gaussian  factors which ensure that the function is integrable at long distances (a feature inherited from  the single-particle states in the Landau level), the FQH “ideal” wave functions do not have any  length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, in almost all cases, the ideal wave functions are the exact ground state  wave functions of some suitable ultra-local Hamiltonian projected onto the lowest Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Another feature, closely related to their universality and analyticity, is that these wave functions  look similar to correlation functions in two-dimensional critical (chiral) critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will  now see that these features are not accidental and point to a deep relation between quantum Hall  states and Conformal Field Theory (CFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although the relation between the simpler case of the Laughlin state was known to several  people, such as Sergio Fubini [265], this deep connection with rational CFT was articulated in  considerable generality in a somewhat earlier paper by Gregory Moore and Nicholas Read [266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This approach showed its full power in the formulation of the non-abelian quantum Hall states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us begin with the simpler case of the Laughlin states with ν = 1/m, where m is odd for  fermions and even for bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will consider a U(1) Euclidean CFT of a compact boson (a  scalar field) φ(x) closely related to our discussion of bosonization in 1+1 dimensions in section  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This theory has been extensively studied in the CFT literature [103, 102, 104] The Euclidean  action for this theory is  S = 1  8π  �  d2x  �  ∂µφ  �2  (465)  This theory is compactified by the condition that the only allowed observables are invariant under  the discrete symmetry  φ(x) �→ φ(x) + 2πRn  (466)  where R is the compactification radius and n is an arbitrary integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The observables that satisfy  this condition are vertex operators of the form  Vp(x) = exp  �  i p  Rφ(x)  �  (467)  where p is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The correlators of this theory are products of holomorphic (right-moving in Minkowski  spacetime) and antiholomorphic (left-moving in Minkowski spacetime) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the context  of the FQH states we will be interested in a it chiral theory, whose correlators are holomorphic  (analytic) functions in complex coordinates z = x1+ix2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Formally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the field φ(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ¯z) is decomposed  into a sum of a holomorphic field φ(z) and and antiholomorphic field φ(¯z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' so that the propagators  is  ⟨φ(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ¯z) φ(�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ¯�)⟩ = − ln(z − �) − ln(¯z − ¯�)  (468)  91  In what follows we will focus on the chiral(holomorphic) field φ(z) whose propagator is  ⟨φ(z)φ(�) = − ln(z − �)  (469)  The correlator of the chiral current operator  j(z) = i  R∂zφ(z)  (470)  is given by  ⟨i∂zφ(z) i∂�φ(�)⟩ ∼  1  (z − �)2  (471)  which tells us that this operator has (chiral) scaling dimension ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly the correlator of  two chiral vertex operators  Vp(z) = exp  �  i p  Rφ(z)  �  (472)  is  ⟨Vp(z) V−p(�)⟩ =  1  (z − �)p2/R2  (473)  whose (chiral) scaling dimension is ∆p = p2/(2R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These expressions are very similar to the op-  erators that we used in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 to describe vortices in 2D superfluids (and planar magnets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The main difference is that here the fields and their correlators are holomorphic where is section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 the correlators are the product of a holomorphic and antiholormophic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  For general p and R the correlator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (473) has a branch cut from z to �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that  the monodromies of the operators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' dragging an operator around the other, induces a change  in the phase of the correlator equal to 2πp2/R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This behavior is reminiscent of the analytic  structure the the quasiparticle wave function in the FQH states and of the braiding operation  between anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Moore and Read [266] showed that the following correlator in a U(1) compactified boson  CFT (with compactification radius R = √m) is equal to the Laughlin wave function at filling  fraction ν = 1/m,  Ψm(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='zN) =  � \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  N  �  i=1  exp  �  i √mφ(zi)  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 exp  �  −i  �  d2z′ √m ρ0 φ(z′)  � �  (474)  =  �  i< j  (zi − zj)m exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−1  4  N  �  i=1  |zi|2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (475)  where we have used units in which the magnetic length is ℓ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 we saw that correlators of vertex operators are non-vanishing only if their  total charge (vorticity) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The reason for this condition is that the (Euclidean) action is  invariant under arbitrary uniform shifts of the field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Now, in the correlator on the right hand side  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (475) we have a product of N vertex operators, each with the same charge √m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, under  an arbitrary shift by α of the field φ(z), the product of vertex operators change by a phase equal  to N √mα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand the other operator in the correlator, which describes a continuum  background charge, changes by a phase − √mρ0L2α, where L2 is the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since ρ0 is the areal  particle density, N/L2, the two changes of the phase cancel each other exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the operator  in the expectation value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (475) is charge neutral and has a non-vanishing expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The other important feature of the correlator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (475) is the choice of compactification  radius, R = √m, and of the vertex operator Vp(z) with p = m:  Vm(z) = exp(i √mφ(z))  (476)  whose correlation function is  ⟨Vm(z) V−m(�)⟩ =  1  (z − �)m  (477)  Which is invariant under a 2π rotation (since m ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However under a rotation by π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' an  exchange) it changes by (−1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, for m odd it changes sign, and the vertex operator Vm  behaves as a fermion, while for m even the vertex operator behaves as a boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will call the  vertex operator Vm ≡ Ve the electron operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  92  In other words, the correlator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (475) is the expectation value of N vertex operators each  describing an electron (for m odd) and a neutralizing background charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, an elementary  calculation yields the expression of the Laughlin wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this formulation the ground state wave function for the integer QH state at ν = 1, which  has the for of a Vandermonde determinant shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (377), is interpreted as the correlator of  N electron operators in the U(1) CFT with compactification radius R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this simple case,  the electron operator is the vertex operator V1(z) = exp(iφ(z)), and there are no other operators  (aside from the current j(z) = i∂zφ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The propagator of this vertex operator is  ⟨V1(z)V−1(�)⟩ =  1  z − �  (478)  which indeed represents a fermion with electric charge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now see that the vertex operator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (472) with p = 1 (and R = √m)  V1(z) = exp  � i√mφ(z)  �  (479)  represents the Laughlin quasihole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The correlator of this vertex operator is  ⟨V1(z) V−1(�)⟩ =  1  (z − �)1/m  (480)  which has a branch cut stretching from z to � and, as a result, under a rotation by ±π its phase  changes by ±π/m, just as the Laughlin anyons do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see that this operator indeed is related to  the Laughlin quasihole we will compute the effect of inserting the vertex operator V1(z) in the  correlator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (475) and find  �  exp  � i√mφ(�)  �\uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  N  �  i=1  exp  �  i √mφ(zi)  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 exp  �  −i  �  d2z′ √m ρ0 φ(z′)  � �  =  =  N  �  i=1  (zi − �)  �  i< j  (zi − zj)m exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−1  4  N  �  i=1  |zi|2 − 1  4m|�|2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  = Ψm(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN)  (481)  which is, indeed, the wave function for the Laughlin quasihole of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(392).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The insertion of  an additional vertex operator V1(u) at u inside the expectation value of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (481) yields the two-  quasihole wave function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (397), proposed by Halperin [234] and discussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5,  with the same branch cut shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (398) and, therefore carry fractional statistics π/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The operator product expansion discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 of the vertex operators of this CFT  yields new insight on the quasiholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed the OPE of two vertex operators of charges p and q  is  lim  �→u Vp(�)Vq(u) = lim  �→u  1  (� − u)∆p+∆q−∆[p+q]m V[p+q]m(u)  (482)  where [p]m is the integer p modulo a multiple of m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' if p = mr + s (with r a non-negative  integer and 0 ≤ s < m, then [p]m = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (482) is understood in the sense that the contribution  of all additional operators to the right hand side vanish as � → u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the vertex  operators are primary fields of this CFT and there are only m primary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The physical process  described by the OPE is called fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A CFT with a finite number of primaries is called a rational CFT [103, 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result is,  of course, the same statement that we made is section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 that the FQH state has m distinct  anyons (vortices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This result also means that a vertex operator of charge m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' an electron, is  indistinguishable from the identity operator, V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this sense, the allowed primaries are fields  which are local respect to the electron operator Vm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' From the point of view of the FQH state, an  operator that creates or destroys an electron (at fixed filling fraction) has no effect, as the FQH  state is a fluid made of electrons This also means that all physical operators must braid trivially  with the electron operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a CFT an operator of this type is said to generate and extended  symmetry algebra [267, 268].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Furthermore, the OPE of the chiral current j(z) with the vertex operator V1(z) is  lim  �→u j(�)V1(u) = 1/m  z − wV1(u)  (483)  93  which implies that the vertex operator represents a state with charge 1/m (in units of the electric  charge e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The CFT approach to construct FQH states has become a very powerful tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see in  section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='11 that FQH states on an open geometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' a disk) have edge states which can  be understood a chiral CFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition to providing a deeper understanding of more general  abelian (muliti-component) FQH states with a K-matrix structure, new classes of of FQH states  with non-abelian (multi-dimensional) representations of the Braid Group has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The anyons of these states are of particular interest since they have been proposed originally  by Kitaev in 1997 [169] that anyons can be used as physical qubits for topologically-protected  quantum computation [169, 269, 270, 271, 170].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  So far we discussed the case of abelian anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abelian anyons have the property that the  fusion of two anyons is another anyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly, a braiding operation between two anyons is  equivalent (up to a phase factor) to another anyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mathematically this means that abelian anyons  are one-dimensional representations of the Braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The fractional statistics of an anyon is  then a label of the representation of the Braid group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw, the same holds under fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, the Braid group generally admits multi-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this more  general case the fusion (or braiding) of two anyons is represented by a linear combination (su-  perposition) of anyon states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Linear combinations of anyon states are represented by unitary  matrices of rank greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since matrices generally do not commute braiding and/or fusion  processes correspond to a multiplication of these matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such unitary processes are then re-  garded as quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This concept is the physical basis of topological quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It is currently the cutting edge of research in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now discuss the simplest non-abelian FQH states, the Moore-Read (MR) states [266].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Moore-Read wavefunctions are  ΨMR(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' , zN) = Pf  �  1  zi − zj  � �  i< j  (zi − zj)n exp(− 1  4ℓ2  0  N  �  i=1  |zi|2)  (484)  which describes a FQH of electrons at filling fraction ν = 1/n, with n an even integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here  Pf  �  1/(zi − zj  �  is the Pfaffian of the matrix 1/(zi − zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This state was motivated by the discovery of an unexpected FQH state in the N = 1 Landau  level with an even denominator filling fraction ν = 5/2 by Willett and coworkers in 1987 [272,  273, 274].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Until then (and since then) this is the only FQH state with an even denominator  filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Pfaffian factor has a simple pole when two coordinates approach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, provided n ≥ 1 the “Laughlin factor” in the MR wavefunction cancels the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will shortly discuss the CFT construction of the Moore-Read state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The poles in the MR  wavefunction suggest that in this state the electrons can get closer to each other than in a Laughlin  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This observation suggests that there may be some form of attraction (or suppression of  repulsion) between the electrons in the MR state and motivated the notion that the observed  even-denominator plateau may be physically related to some sort of pairing interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shortly  after the Moore-Read proposal was made, a paired wave function at ν = 1/2 with a Pfaffian factor  was also proposed by Greiter, Wen and Wilczek [275, 276, 277] who suggested that it reflects  electrons pairing in the ℓ = 1 angular momentum channel in time-reversal breaking condensate  with symmetry px + ipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although, following the logic behind the Kohn-Luttinger mechanism for paring in angular  momentum state ℓgeq1 with repulsive interactions [278, 279, 280] it may possible to get a paired  state even in the lowest Landau level (although there is no evidence for it, so far), the N = 1  Landau level may be more hospitable to such a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, in the N = 1 Landau level the  single-particle states i have angular momentum greater or equal than 1, and their probability  distributions are suppressed both at short and long distances;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' these states look like smoke-rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The matrix elements of the Coulomb potential in the N = 1 Landau level are parametrically  suppressed at short distances (compared with the states in the lowest, N = 0, Landau level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  this scenario, the ν = 5/2 plateau is interpreted as a ν = 1/2 fully polarize state of the N = 1  landau level, with an N = 0 Landau level filled with electrons with both spin polarizations in  a state with ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is numerical evidence that an instability into a p-wave paired state  such as the MR state is favored if the short-distance repulsion between the composite fermions is  softened [281].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In fact, in 1983 Halperin [282] considered generalizations of the Laughlin state in which  due to very strong attractive interactions electrons would form clusters which then condense into  a Laughlin-type state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The simplest example was Laughlin state of bosons (paired electrons)  94  a paired state at ν = 1/2 whose Laughlin quasihole carries charge e/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Extensive numerical  calculations are more compatible with the MR state than with the Halperin state of pairs [283].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will see shortly that the Halperin state is an abelian relative of the MR state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Moore and Read derived the wave function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (484) by considering a CFT with two  sectors: a chiral boson CFT with compactification radius R = √n, and a chiral Majorana fermion  CFT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the chiral part of the 2D Ising CFT, discussed in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The chiral Ising CFT has  central charge c = 1/2 and several primary fields: the identity I, the twist field σ, and the  (chiral) Majorana fermion χ with (chiral) scaling dimensions 0, 1/16, and 1/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  propagator of the Majorana fermion, which is a free field, is  ⟨χ(z) χ(�)⟩ =  1  z − �  (485)  The primary field σ, the “twist field”, is non-local with respect to the Majorana fermion χ, and  changes its boundary conditions from periodic to antiperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The fusion rules of the chiral Ising CFT are  χ ⋆ χ = I,  σ ⋆ σ = I + χ,  σ ⋆ χ = χ  (486)  What will be important below is that the σ field has two fusion channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As in the analysis of  the Laughlin states, the compactified boson n primaries (anyons), the vertex operators Vp(x) =  exp(ipφ(x)/n), with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='n and there are n types of anyons, and a (charge) current J =  i√n∂zφ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The first task is to identify the electron operator which has to be a fermion and has to  carry electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that it is a composite of an operators (with Fermi statis-  tics) the (neutral) chiral ising CFT and a vertex operator whose charge is an integer multi-  ple of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The desired electron operator is ψe(z) = χ(z) exp(i √nφ(z)) whose correlator is  ⟨χ(z)Vn(z)χ(�)V−n(z)⟩ = 1/(z − �)1+n, which is a fermion with charge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The n-point function of the chiral Majorana fermions is  ⟨χ(z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='χ(zN)⟩ = Pf  �  1  zi − zj  �  (487)  which follows fro applying Wick’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then we see that the MR wave function is the  expectation value  ΨMR(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', zN) = ⟨χ(z1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='χ(zN)⟩ ×  � \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  N  �  i=1  exp  �  i √nφ(zi)  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 exp  �  −i  �  d2z′ √n ρ0 φ(z′)  � �  (488)  Next we need to identify the primary fields of the tensor product of the chiral Ising CFT, Z2,  and the chiral U(1) CFT with compactification radius R = √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The allowed primary field must  belocal with respect to the electron operator, ψe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This leaves us with four primary fields: a) the  identity I, b) the non-abelian vortex σ(z) × exp  �  i  2 √nφ(z)  �  with charge e/(2n) and non-abelian  statistics, c) the charge-neutral Majorana fermion χ, and d) the abelian vortex (the Laughlin  quasihole) exp  �  i√nφ(z)  �  , with charge e/n and abelian statics δ = π/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The he new feature here is the non-abelian fractional statistics of the non-abelian vortex  (sometimes called a “half-vortex”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Its non-abelian character is a consequence of the fact that the  fusions of two twist fields has two channels, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(486).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This implies that the wave function of  four non-abelian vortices can be expressed as a linear combination of two so-called conformal  blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, this wave function belongs to a degenerate two-dimensional Hilbert  space, each component labeled by a conformal block [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A braiding operation between two  non-abelian vortices is a monodromy of the wave function that induces a unitary transformation  U in this Hilbert space  U =  1√  2  exp  �  iπ  �1  8 + 1  4n  ��  ×  � 1,  1  −1  1  �  (489)  Therefore, the degenerate Hilbert space of four quasiholes provides a two-dimensional represen-  tation of the Braid Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This observation is the basis for regarding a state of four non-abelian  quasiholes as a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furthermore, Nayak and Wilczek showed that a state of 2p quasiholes  spans a 2p−1-dimensional degenerate Hilbert space [284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that, for large p, the de-  generacy per quasihole is  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, this degeneracy is not due to a local degree of  freedom attached to each quasihole but that it is shared in a non-local fashion!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  95  In what sense are the Moore-Read states paired?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In an insightful paper Read and Green [285]  used a BCS theory approach to investigate the pairing properties of a condensate of composite  fermions in the px + ipy channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The mean-field BCS Hamiltonian is  HF =  �  d2k  �  (ε(k) − µ) ψ†(k)ψ(k) + 1  2  �  ∆∗(k)ψ(−k)ψ(k) + ∆(k)ψ†(k)ψ†(−k)  ��  (490)  where ψ(k) is the composite fermions field, ε(k) =  k2  2M, and ∆(k) is the gap function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For a  px + ipy condensate the gap function the gap function transforms under rotations as an ℓ = −1  angular momentum eigenstate, and for k → 0 it behaves as ∆(k) = (kx − iky)∆, where ∆ is a  constant pairing amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The BCS ground state has the form  |G⟩ =  �  k  �  u(k) + �(k)ψ†(k)ψ†(−k)  �  |0⟩  (491)  where |u(k)|2 = |�(k)|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The amplitudes u(k) and �(k)| satisfy the Bogoliubov-de Gennes  Equation (BdG)  � ξ(k)  −∆∗(k)  −∆(k)  −ξ∗(k)  � �u(k)  �(k)  �  ≡ E(k) ˆn(k) · σ  �u(k)  �(k)  �  = E(k)  �u(k)  �(k)  �  (492)  where ξ(k) = ε(k)−µ, σ = (σx, σy, σz) are the Pauli matrices, and ˆn(k) = (−Re∆(k), Im∆(k), ξ(k))  is a unit vector defined for every k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The eigenvalues E(k) and the eigenvectors (u(k), �(k) are  E(k) =  �  ξ2(k) + |∆(k)|2,  �(k)  u(k) = −E(k) − ξ(k)  ∆∗(k)  (493)  The spinor amplitudes are |u(k)|2 = 1  2  �  1 + ξ(k)  E(k)  �  and �(k) = 1  2  �  1 − ξ(k)  E(k)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the low momentum limit and in real space the BdG Equations become  i∂tu = − µu + ∆∗i(∂x + i∂y)�  i∂t� =µ� + ∆i(∂x − i∂y)u  (494)  which is just the Dirac Equation in 2+1 dimensions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' with µ playing the ole of the mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and with  the restriction that the spinor (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' �) obeys the Majorana condition  (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' �)  �0  1  1  0  �  =  �u  �  �  (495)  The Majorana condition is obeyed in all superconductors and reflects that fact that in these con-  densates only the fermion parity (−1)NF (with NF being the number of fermions) is conserved,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  while the fermion number NF is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This is a good BCS state in that it is fully gapped and it is chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Assuming that the pair  fields in the px and py channels are equal, they showed that the BCS ground state |G⟩ has the  form  |G⟩ ∼ exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  1  2  �  k  g(k)ψ†(k)ψ†(−k)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8 |0⟩  (496)  Projected onto a state with N fermions with real space coordinates xi the wave function is a  Pfaffian  Ψ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', xN) = ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', xN|G⟩ = Pf(g(xi − x j))  (497)  The long distance behavior of this wave function depends on whether the chemical potential  µ > 0 (this is the weak-pairing or BCS regime), or µ < 0 (which is the strong pairing BEC-  like regime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While in the case of an s-wave superconductor these two regimes are smoothly  connected by a crossover, in the px + ipy case they are separated by a quantum phase transition  at µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the weak pairing regime the function g(k) has the asymptotic long distance form, as  k → 0,  g(k) ≃ −  2µ  (kx + iky)∆∗  (498)  where the pair field behaves as ∆(k) ≃ (kx − iky)∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In real space (in complex coordinates) the  function g(z) becomes  g(z) =  � iµ  π∆∗  � 1  z  (499)  96  which is the form used in the Pfaffian wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in the strong pairing  regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' µ < 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' g(k) has the asymptotic behavior  g(k) ≃ −A(kx − iky)  a−2  0 + k2  (500)  where A and a0 are functions of ∆ and µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A =  2|µ|M∆  2|µ| + m|∆|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  a0 =  1  2|µ|  �  2|µ|  M + |∆|2  (501)  In real space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' at distances |x| ≫ a0 the Fourier transform of g(k) decays exponentially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and as  a0 → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' µ → 0 and the Fourier transform g(z) has the power-law behavior  g(z) ≃  � i|∆|  2π∆∗  �  1  z|z|  (502)  This means that µ = 0 is a quantum critical point that separates the weak pairing phase from the  strong pairing phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' which have distinct properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Read and Green then used a topological argument, originally formulated by Grisha Volovik  [286] in the context of superfluid 3He films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To see this we notice that the three component  unit vector ˆn(k) takes values on the surface of a sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, since �(k) → 0 for  k → ∞(and, hence u(k) → 1), we can wrap the momentum space k onto a sphere S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus  ˆn(k) are smooth functions of S 2 �→ S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' such maps are classified into the homotopy classes of  π2(S 2) ≃ Z given in terms of the integer-valued topological invariant  Q = 1  4π  �  d2k ˆn(k) · ∂kx ˆn(k) × ∂ky ˆn(k)  (503)  In the strong pairing phase µ < 0 and ξ(k) > 0 and ˆn(k) takes values only on the northern  hemisphere of the target space S 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such maps can be deformed continuously to the North Pole  and, hence, are topologically trivial, Q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' On the other hand, in the weak pairing phase, µ > 0,  ξ(k) takes both positive and negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this phase ˆn(k) is topologically non-trivial and  the topological invariant Q takes non-trivial values ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The upshot of this analysis is that, in the weak pairing phase, the paired FQH state is, at this  mean field level, a two-dimensional topological superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read and Green [285] further  showed that this pairing state has vortices with an interesting fermionic spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A vortex is a  state win which at long distances the (complex) amplitude of the px + ipy condensate behaves as  ∆ exp(iϕ), where ϕ is the azimuthal angle, which winds by 2π on a circumference of large radius  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The BdG equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (494), has zero mode solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In polar coordinates (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ) the BdG  Equation the zero modes satisfy  ∆ieiϕ �  ∂r + i  r∂ϕ  �  � =µu  ∆ie−iϕ �  ∂r − i  r∂ϕ  �  u = − µ�  (504)  The normalizable zero mode spinor solution is  �u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ)  �(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ)  �  = f(r)  √r  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  1√  ieiϕ/2  1  √  −ie−iϕ/2  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (505)  where f(r) is given by  f(r) ∼ exp  �  −  � r  0  dr′ µ(r′)  |∆|  �  ∼ exp(−µr/|∆|)  (506)  Under a 2π rotation this spinor solution is double-valued  (u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ + 2π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' �(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ + 2π)) = −(u(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' �(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ))  (507)  which follows form the global phase invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I fact, in all paired states, topological or not, un-  der a global transformation of the pair field by a phase θ, the (composite) fermion must transform  with a phase of θ/2,  ∆(x) �→ eiθ∆(x),  ψ(x) �→ eiθ/2ψ(x),  ψ†(x) �→ e−iθ/2ψ†(x)  (508)  97  In other words, in a paired state the fermion is non-local to the vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, under a 2π phase  transformation of the pair field, the fermions must change sign, and the spinor zero mode solution  must be double-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that this state has a brach cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The structure of the vortex and of the fermion zero mode is closely related to the problem of  solitons with fractional charge that we discussed in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In fact, Roman Jackiw and Paolo  Rossi [287] investigated a closely related problem of a theory of Dirac field in 2+1 dimensions  coupled to a charged scalar field through a Yukawa coupling with the form of a Majorana mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  They showed that such a theory admits vortex solutions with fermion zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a subsequent  paper Erik Weinberg [288] showed that these zero modes are counted by an index theorem which  relates the number of zero modes to the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' More recently, Nishida, Santos and Chamon  [289] that the relativistic theory of Jackiw and Rossi in the non-relativistic approximation reduces  to the theory of a px + ipy superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is, however, a subtle but profound difference between the fermion zero modes of the  one-dimensional fractionally charged solitons and the fermion zero modes of a superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the case of teh 1D solitons, the zero modes can be either occupied or empty rendering a  soliton charge of −e/2 or +e/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in the case of the superconductor, the BdG fermions  are charge-neutral since their charge has gone into the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, in a superconductor  fermion number is not conserved, and only the fermion parity is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that  the field operator associated with the vortex zero mode, which we will denote by γ, must be a  self-adjoint operator, γ† = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Ivanov [290] showed that this behavior is the origin of the non-abelian fractional statistics in  this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, if one considers a configuration with 2n vortices, each will carry a Majorana  zero mode γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since these are fermion operators they satisfy the usual anticommutator algebra,  {γi, γ j} = 2δi j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can (arbitrarily) group the 2n Majorana fermion operators into n pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For  each pair we can define a complex Dirac operator ψ j = (γ2 j + iγ2 j+1)/2 (and its adjoint), which  satisfies the usual fermionic algebra, {ψ j, ψ†  k} = δ jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Each Dirac fermion can be either in an  empty state or in an occupied state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus a system of 2n vortices supports a degenerate Hilbert  space of dimension 2n−1, which agrees with the results of Nayak and Wilczek [284].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However,  the assignment of Majorana operators to specific pairs is actually arbitrary, which amounts to  a particular definition of the branch cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Changing the assignment of the operators into pairs  is then equivalent to a rearrangement of the cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2006 Michael Stone and Suk Bum Chung  [291] showed that the these vortices obey the braiding and fusion properties of Ising anyons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These properties follow from the branch cut configurations which affect the monodromies of the  vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is important to stress that the vortices with zero modes are the non-abelions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Majorana  fermions are fermions and, as such, are abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In addition to the Moore-read Pfaffian state, the CFT approach has led to the formulation of  new non-trivial FQH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read and Rezayi [292] proposed a series of so-called cluster states,  which generalize the concept of paired states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They investigated a particular type of cluster states  in which the Pfaffian factor of the MR state is replaced by a correlator of parafermion primary  fields of a Zk CFT of Zamolodchikov and Fateev [293].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Parafermions were originally introduced  by Fradkin and Kadanoff [152] (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [294]) as a generalization of the fermions of the 2D  Ising model to the Zk clock model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this model one can define several types of parafermions  each consisting of the fusion of a charge operator and a magnetic (disorder) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These  operators obey an algebra of the type AB = exp(ip2π/k)BA, where the integer p depends on the  electric and magnetic charges of the parafermions, which is reminiscent of fractional statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  close analogy with the Majorana fermions of the 2D Ising model, Zk CFT describes the behavior  of these systems their self-dual points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Zk CFT is actually much richer and has more primary fields than the Z2 case of the  Ising model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As a CFT, the Zk theory is the same as the CFT on the coset SU(2)−k/U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A  coset means that a U(1) sector has been projected out of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will focus on a  special case of the Z3 CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This CFT has a parafermion primary field, which we will call ψ, and  a non-abelian primary that we will call τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read and Rezayi [292] proposed to replace the Pfaffian  factor of the MR state with a correlator of N parafermions of a Z3 CFT and a Laughlin factor  with an exponent of n + 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Such states require that the number of electrons N be a multiple of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, these states can be viewed as sates in which the electrons cluster in groups of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They  also considered the more general case of the Zk CFT in which case N is a multiple of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  resulting filling fractions are ν = k/(mk + 2) (with m ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Read-Rezayi k = 3 parafermion  state is a leading candidate for the observed FQH plateau at ν = 12/5 = 2 + 2/5 [295, 296] (the  compering state being the 2/5 Jain state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There is strong numerical support for the 12/5 state  98  being a Z3 parafermion [297].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The quasiholes obtained by inserting the primary fields τ in the correlator of parafermions  have interesting properties which stem from their basic fusion rule  τ ⋆ τ = I + τ  (509)  Read and Rezayi [292] showed that the number of conformal blocks for 3n quasiholes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the  number of degenerate states, in the parafermion theory is the Fibonacci number F3n−2, where  F1 = 1, F2 = 2, F3 = 3, F4 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general Fm = Fm−1 + Fm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For m → ∞, the rate of  increase approaches the limit limm→∞ Fm/Fm−1 = (1 +  √  5)/2 which is known as the Golden  Mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, for large m, the dimension of the Hilbert space increases exponentially  as ((1 +  √  5)/2)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aside from these interesting mathematical curiosities, the significance of this  fusion rule is that these states, regarded as“qubits”, define unitary transformations that cover  uniformly the Bloch sphere which is required for universal quantum computation [270].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  At the level of topological quantum field theory the Moore-Read and Read-Rezayi states are  related to the non-abelian Chern-Simons gauge theory with gauge group SU(2) at Chern-Simons  level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the more general case of the gauge group SU(N) on a manifold M action is given by  S CS[Aµ] = k  4π  �  M  d3x ǫµνλ  �  Aµ  a∂νAλ  a + 2  3 fabcAµ  aAν  bAλ  c  �  (510)  where k ∈ Z is the level, the gauge fields Aµ = Aµ  ata take values in the algebra of SU(N), with {ta}  being the N2 − 1 generators, and fabc the structure constants of SU(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The expectation values  of Wilson loop operators of this theory compute the Jones polynomials which are topological  invariants of knot theory [165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The connection with SU2)k Chern-Simons gauge theory has  been essential in formulating effective low energy quantum field theories for the non-abelian  states [298, 299, 300, 301, 302, 303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Edge States and Chiral Conformal Field Theory  We will now discuss the edge states of the FQH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw the FQH states are  incompressible in the bulk and all bulk excitations are gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The edge of the region occupied  by the fluid (in many cases that edge of the physical sample) is where the bulk gap collapses and  hence where the system has low energy excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The role of edges and their nature can be seen already in the simple case of the integer Hall  effect treated as a system of free fermions in the lowest Landau level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this state all single  particle states are occupied, and the bulk ground state wave function is a Slater state which takes  the form of the Vandermonde determinant of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(377).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The potential that keeps the electrons  inside the sample increases monotonously near the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we discussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1, in  this region the electrons experience an electric field E that pushes them into the sample, and in  the presence of the perpendicular magnetic field B the electrons move along the edge at the drift  velocity � = c|E|/|B|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At some spatial location, corresponding to the locus of a single particle  Landau state, the potential crosses the Fermi energy and in that region potential is essentially a  linear function of the coordinate and hence of momentum (or angular momentum depending on  the gauge that is being used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the low energy states are one branch of a one-  dimensional chiral excitation, such as in the right-moving states of our discussion of the chiral  anomaly in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2  In section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='5 we showed that a Chern-Simons gauge theory on a manifold with a boundary  projects onto a chiral CFT on that boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In that section we showed that an abelian Chern-  Simons theory U(1)m integrates to the boundary, the 1+1-dimensional Minkowski spacetime  S 1 × R, as a chiral U(1)m CFT for a compactified boson with compactification radius R = 1  whose action is given by  S [ϕ] =  �  S 1×R  d2x 1  4π  �  ∂0φ∂1φ − �(∂1φ2)  �  (511)  where we rescaled the compactified scalar by a factor of √m to that its compactification radius  R = √m as in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only difference between the theory of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (241) and what we  did in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10 is that this theory is in a 1+1-dimensional Minkowski spacetime S 1 × R  while before we were in Euclidean spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='In the case of the integer quantum Hall effect ν = 1  (and hence m = 1) and the (time-ordered) correlator of the vertex operator, V1(x, t) = exp(iφ(x, t))  is  ⟨T exp(iφ(x, t)) exp(−iφ(0, 0))⟩ =  1  x − �t − iε  (512)  99  which is, indeed, the propagator of a chiral free fermion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' for the Laughlin states at filling fraction ν = 1/m (and compactification  radius R = √m) the propagator of the electron ψe ∼ exp(i √mφ) is  ⟨T exp(i √mφ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' t)) exp(−i √mφ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 0))⟩ ∼  1  (x − �t − iε)m  (513)  while the propagator of the quasihole exp  �  i√mφ  �  is  �  T exp  � i√mφ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' t)  �  exp  �  − i√mφ(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 0)  ��  ∼  1  (x − �t − iε)1/m  (514)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the CFT of the edge is the same as the CFT of the ideal wavefunction with the only  difference that the former is in Minkowski spacetime while the latter is in Euclidean spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this sense there is a one-to-one correspondence between the bilk and the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can  also see how that works by considering a fundamental Wilson arc in the bulk along a path γ(x, y)  where x and y are at the boundary  ⟨W[γ]⟩ = ⟨exp(i  �  γ(x,y)  dxµA µ) ≡ ⟨exp( i√mφ(x)) exp(− i√mφ(y))⟩  (515)  where on the right hand side we have rescaled the field by √m, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The scaling dimensions  of the electron, the quasihole and the current are ∆e = m  2 , ∆qh =  1  2m and ∆current = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These results  will be important shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The same structure applies to the multi-component abelian FQH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only difference is  that in multi-component FQH states the edge consists, in general, of a charge field which couples  to the electromagnetic field and one or more neutral edge states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' An example is the theory of the  non-abelian Pfaffian states at filling fraction ν = 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case the edge theory consists of a  compactified chiral boson φ of radius R = √n and a chiral Majorana (neutral) fermion χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At a  formal level the Lagrangian for the edge state(s) is a sum of two terms  L = 1  4π(∂xφ∂tφ − �c(∂xφ)2) + χi(∂t − �n∂x)χ  (516)  where �c and �n are the velocities of the (charged) compactified chiral scalar φ and of the neu-  tral Majorana field χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Numerically (and experimentally) it is known that the charge mode is  (substantially) faster than the neutral mode(s), �c > �n, and often by significant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A superficial reading of this Lagrangian suggests that these degrees of freedom are decou-  pled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However this is not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Only operators from both sectors which are local with respect  to the electron ψe ∼ χ exp(i √nφ) are physically allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here an operator is local with respect  to the electron means that the operator braids trivially with the electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This condition restricts  the allowed observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of the fermionic MR state, with n = 2, the allowed operators  are the non-abelion σ exp(iφ/2  √  2) with scaling dimension is ∆ = 1/8 and electric charge is  Q = 1/4, the Majorana fermion with scaling dimension ∆ = 1/2 and no electric charge Q = 0,  and the Laughlin quasiparticle with scaling dimension ∆ = 1/4 and charge Q = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Its electron  operator has scaling dimension ∆ = 3/2 and charge Q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Conformal field theory has an important defining universal quantity called the central charge  which is closely related to the energy-momentum tensor of the theory [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For pedagogical  introductions to CFT see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [304, 45] and chapter 21 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The energy-momentum  tensor Tµν is a locally conserved current, ∂µTµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Its local conservation implies the global  conservation of energy and momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As such, the energy-momentum tensor is a fundamental  observable of any quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a conformal field theory, the energy-momentum is  also the generator of local scale and conformal transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the special case of 1+1-  dimensional systems, such as the edge states of the FQH fluids, the energy momentum tensor  has special and crucially important properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition of being locally conserved, conformal  invariance requires that Tµν must be traceless, T µ  ν = 0, since its trace is the generator of dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this case, if the theory is chiral, the energy momentum tensor has only one (right-moving)  component T = (T00 − T11)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In complex coordinates of the Euclidean metric, the correlator of  the energy momentum tensor T = Tzz is [305]  ⟨T(z)T(�)⟩ =  c/2  (z − �)4  (517)  100  where c is a universal quantity known as the it central charge of the CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of the  compactified boson the central charge is c = 1 whereas for a massless Majorana fermion c = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The central charge of the CFT enters in many observables of fundamental physical impor-  tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For example, for a 1+1-dimensional system of length L, such as the edge state of a FQH  droplet, the ground state energy Egnd for large L has the behavior [306, 307]  Egnd = ε0L − c π�  6L + O(1/L2)  (518)  where ε0 is the ground state energy density, which is a non-universal quantity, c is the central  charge of the CFT and � is the velocity of the massless modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (518)  is known as the Casimir energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Similarly, the free energy density f(T) of a CFT has the low  temperature Stephan-Boltzmann behavior (in one dimension)  f(T) = ε0 + cπT 2  6� + O(T 3)  (519)  where we set the Boltzmann constant to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The low-temperature specific heat c(T) is  c(T) = cπT  3� + O(T 2)  (520)  In a chiral CFT, such as the edge states of the FQH fluids, the central charge enters in the low  temperature behavior of the (Hall) thermal conductivity κxy, [285]  κxy = cπT  6ℏ  (521)  Much of what is known about FQH states comes form experiments involving their edge  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a series of exquisite experiments Granger, Eisenstein and Reno [308] measured κxy in  the edge states of a ν = 1 quantum Hall fluid and showed that the heat transport is indeed chiral,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the edge state behaves as a heat “diode”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This effect was confirmed in the FQH states by  Bid and coworkers [309] who, in addition, where used this effect to detect the neutral modes in  several Jain states at filling fractions 2/3 and 2/5 and in the non-abelian state at ν = 5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The simplest experimental probes are quantum point contacts and typically are of two types:inter-  edge tunneling inside a FQH liquid or tunneling of electrons into the edge state of a FQH liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Since the bulk is gapped, only the edge states participate in these tunneling processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  general case we will have two edges and a tunnel process at a point contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here the two edges  can be either the two edges of the same FQH state, in which case the process happens inside the  FQH liquid and involves tunneling of quasiparticles, or the edges of different liquids, in which  case this process is external and involves tunneling of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Problems of these types were  first investigated by Charles Kane and Matthew Fisher [310] which led to a considerable amount  of work on these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Let us consider first the case of tunneling into the edge state of a ν = 1/m Laughlin state  from a Fermi liquid, which we will take to be a QH fluid in the ν = 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The point contact is  at x = 0 The total Lagrangian is  L = Ledge + LFL − Γ eiω0t ψ†  e,edge(0, t) ψe,FL(0, t) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (522)  where ω0 = eV/ℏ, and V is the bias voltage between the two fluids, and Γ is a tunneling matrix  element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The local electron spectral density (the density of states) N(ω) at energy ω is  N(ω) = Im lim  x→0+  � ∞  −∞  dt Ge(x, t) eiωt = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' |ω|m−1  (523)  where Ge(x, t) = ⟨Tψ†  e(x, t)ψe(0, 0)⟩ is the electron correlator in the FQH edge, shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (512).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This result, combined with Fermi’s Golden Rule for a point contact with voltage bias V, predicts  a tunneling current from a Fermi liquid (FL) into the chiral Luttinger liquid (CLL) of the edge  state to be [311]  I(V) = 2π e  ℏ|Γ|2  � 0  −eV  dE NCLL(E, T)NFL(E + eV, T) ∝ Vm  (524)  and a tunneling differential conductance G(V)  G(V) = dI  dV = 2π e  ℏ|Γ|2NFL(0)NCLL(V, T) ∝ Vm−1  (525)  101  which is non-Ohmic and vanishes as V → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Early experiments by Albert Chang and cowerkers , on a geometry that (most likely) had  many point contacts, confirmed the predicted CLL behavior [312, 313], although there were  discrepancies in the measured exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The behavior seen in more recent experiments by  Cohen and coworkers [314], on a point contact in graphene, are consistent with the theoretical  predictions [311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Interestingly, these newer class of experiments [315, 314] also show evidence  of an analog of Andreev reflection predicted to exist near the strong coupling fixed point of the  theory of Sandler, Chamon and Fradkin [316] as a consequence of electron fractionalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Most experiments are done on a geometry call a Hall bar in which the QH fluid occupies a  rectangular region with its length L larger than its width W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this geometry, there are chiral  edge states at opposite sides of the Hall bar propagating in opposite directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One type of point  contact consists in creating a constriction in the quantum Hall fluid by applying a gate, a bias  potential on a narrow strip accross the Hall bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This gate repels the electrons in the QH fluid,  forcing the opposite edges to approach each other in the proximity of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the absence  of the gate, momentum is conserved on each edge which forbids tunneling accross the Hall bar  since the edge states have opposite Fermi momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, the gate breaks translation in-  variance on both edges and tunneling between them is now allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the gate  creates a point contact between the opposite propagating edge states leading to a tunneling pro-  cess between the edges of the QH liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, this is internal tunneling as oppose the process  of tunneling between two different liquids that we discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The tunneling Lagrangian for this system has the same form as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (522) except that now  the two edges are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The theory Kane and Fisher shows that in the factional case the most  relevant process is the tunneling of FQH quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Fermi Golden Rule argument used  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (525) to compute the differential conductance also applies in the present case except that  now the two densities of states are the densities of states of the (Laughlin) quasiparticles (instead  of electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The local quasiparticle density of states is Nqp(ω) ∼ |ω|  2  m −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, the differential  tunneling conductance G(V) in this case is  G(v) ∼ V2( 1  m −1)  (526)  This behavior is also non-Ohmic but, unlike the case of electron tunneling of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (522), the differ-  ential tunneling conductance now diverges as V → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This behavior means that the Γ → 0 fixed  point is unstable and that the point contact flow to a strong coupling fixed point at Γ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Early  experiments by Milliken, Umbach and Webb in 1996 [317] showed indications of CLL behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The predicted behavior was confirmed in the experiments of Roddaro and coworkers [318, 319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the RG language, we can define a dimensionless tunneling amplitude g by Γ = a∆−1g,  where ∆ is the scaling dimension of the tunneling operator and a is a UV cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The “tree-level”  beta function is readily found to be  β(g) = a∂g  ∂a = (1 − ∆) g + O(g2)  (527)  which shows that if the scaling dimension of the operator of the tunneling particle is ∆ < 1, then  this tunneling process is relevant, while is ∆ > 1 it is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Kane and Fisher argued that there should be a crossover from the IR unstable weak-coupling  fixed point governed by quasiparticle tunneling to an IR stable strong coupling fixed point gov-  erned by electron tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This crossover is reminiscent of the the crossover in quantum impu-  rity problems such as the Kondo problem of a magnetic impurity in a metal [320, 32, 52, 53, 321].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The main difference is that the impurity coupling is marginal and the crossover scale, the Kondo  temperature TK, which is related to the coupling constant g as TK ∼ exp(−1/g), while in the FQH  constriction the crossover scale is TK(g) ∼ g1/(1−∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane and Fisher argued that this crossover  can be viewed as a process that interpolated between a FQH fluid with a weak constriction to a  regime in which the Hall bar splits into two pieces with weak electron tunneling between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  It turns out that, after some manipulations, the model of the constriction can be mapped into  a one-dimensional compactified boson ϕ on a semi-infinite line, x ≥ 0, with a vertex operator  acting at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The action of this system, known as boundary sine-Gordon is  S = 1  8π  � ∞  −∞  dt  � ∞  0  dx (∂µϕ)2 + Γqp  � ∞  −∞  dt cos  � �  ν  2ϕ(0, t)  �  (528)  This theory has two fixed points: a) the IR unstable fixed point at Γqp = 0 where the field  Neumann boundary conditions at x = 0, ∂xϕ = 0, and b) an IR stable fixed point at Γqp → ∞  102  where the field has Dirichlet boundary conditions, ϕ = 2π √2/ν n (with n ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It turns out that  this is an integrable field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Paul fendley, Andreas Ludwig and Hubert Saleur [322] used the  thermodynamic Bethe Ansatz to calculate the differential tunneling conductance for the Laughlin  state at ν = 1/3 as a function of voltage V and temperature T and obtained the full weak to strong  crossover RG flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Detailed predictions of this theory have been verified experimentally by the  work of Roddaro and coworkers [318, 319].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Point contact experiments were performed in the  more challenging ν = 5/2 FQH state by Miller and coworkers [323] and Radu and coworkers  [324].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They found that the MR state is the the one that best fits their experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The charge of particle can, in principle, be found directly by measuring the noise of a weak  current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This process is known as shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the case of the constriction, the quasiparticle  current operator is Iqp = 2eνΓqp sin  � √ν/2φ + ω∗  0t  �  , where ω∗  0 = eνV/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The noise spectrum,  S (ω), of the tunneling current qp is [325]  S (ω) =  � ∞  −∞  dt ⟨{Iqp(t), Iqp(0)}⟩ eiωt  (529)  Using the expression of the quasiparticle correlator of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (514),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' one readily finds that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' to leading  order in Γqp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the noise spectrum is  S (ω) = eν⟨Iqp⟩  \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  �  1 − ω  ω∗  0  �2ν−1  +  �  1 + ω  ω∗  0  �2ν−1\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  (530)  In the limit of zero frequency the noise spectrum takes the shot noise form  lim  ω→0 S (ω) = 2e∗⟨Iqp⟩  (531)  where the expectation value of the tunneling current is  ⟨Iqp⟩ =  2π  Γ(2ν)eν|Γqp|2 ω∗  0  2ν−1  (532)  Chamon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freed and Wen [326] calculated the exact noise spectrum for the case of a ν = 1/2  bosonic FQH state and were able to investigate the crossover as well,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and Fendley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ludwig and  Saleur [327] used the Bethe Ansatz to construct a soliton basis to compute the DC shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  These theoretical predictions were tested in tour-de-force experiments by de Picciotto and  coworkers [253] and Samindayar and coworkers [254] who were able to measure the fractional  charge of e/3 in the ν = 1/3 state and, with some caveats, e/5 in the ν = 2/5 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dolev and  coworkers [328] went on to measure the charge from noise experiments in the nonabelian state  at ν = 5/2 and obtained results consistent with e∗ = e/4 as predicted by theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now consider experiments on Hall bars with two quantum point contacts created by  two narrow gates transversal to the bar and separated at some distance d each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum  devices of this type are Fabry-P´erot quantum interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The way they operate is as follows  [329, 298].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A current is injected in the bottom edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At the first quantum point contact (QPC)  part if the current I1 tunnels to the opposite edge and the other part I2 goes on and tunnels at the  second QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The FQH fluid is assumed to occupy uniformly the region between the two QPC’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  There is some magnetic flux Φ is that region which also contains a number of localized vortices  (quasiparticles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' When both currents rejoin at the top edge they interfere and the interference has  information on the charge of the particles that tunnels through the Aharonov-Bohm effect with  the flux Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The interference also has information on the fractional statistics of the quasiparticles  that tunneled through their braiding with the static vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, Freed, Kivelson, Sondhi  and Wen proposed a setup of this type to measure the fractional statistics of the quasiparticles for  the abelian states [329].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' They showed that the total current It = I1 + I2 is given by  It = e∗|Γeff|2 2π  Γ(2ν) |ω0|2ν−1 sign(ω∗  0)  (533)  where Γeff is give by  |Γeff|2 = |Γ1|2 + |Γ2|2 + (Γ1Γ∗  2 + Γ∗  1Γ2) Fν  �ω∗  0d  �  �  (534)  Here Γ1 and Γ2 are the tunneling amplitudes at the two QPCs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ν = 1/m is the filling fraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' � is  the velocity of the edge modes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' d is the distance between the two QPCs and  Fν(x) = √πΓ(2ν)  Γ(ν)  Jν−1/2(x)  (2x)ν−1/2  (535)  103  where Γ(z) is the Euler Gamma function and Jν−1/2(z) is the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  the presence of Nq localized quasiparticles in the area between the two QPCs, the contribution  of the phases of the tunneling matrix elements get shifted to  Γ∗  1Γ2 = ¯Γ∗  1 ¯Γ2 exp  �  −2πi  �  ν Φ  φ0  − νNq  ��  (536)  where φ0 is the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (536) the first term in the phase shift is the Aharonov-  Bohm effect of the tunneling quasiparticles, while the second term in the phase shift 2πνNq is  the contribution of the fractional statistics of the tunneling quasiparticle as its worldline braids  with the Nq localized quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that there is an interference contribution to  the tunneling current (and also to the transmitted current) which is sensitive to both the charge  and to the fractional statistics of the quasiparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is this effect that is being measured in  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Early attempts at doing this experiment were made by Camino and coworkers [330] but were  difficult to interpret partly due to subtle reasons related to the difficulty in controlling the actual  area of the region comprised between the two QPCs [331].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Technical advances in the fabrication  of these devices led in 2020 to the first successful measurement of the fractional statistics of the  quasiholes of the Laughlin state at ν = 1/3 (and also at the Jain state at ν = 2/5) by Nakamura,  Liang, Gardner and Manfra [233].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The nonabelian case is more subtle both theoretically and experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The theory of the  abelian interferometer of Chamon and coworkers was generalized to the nonabelian FQH states  by Fradkin, Nayak, Tsvelik and Wilczek in 1998 [298].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The structure of the interferometer is the  same as in the abelian case but the interference effects are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, in the case of  the MR state, taside from the nonabelian quasiparticle σ ∼ χ exp(iφ/2  √  2) has charge e/4 and  nonabelian fractional statistics, the MR state has two more abelian anyons, the charge neutral  Majorana fermion χ and the charge e/2 Laughlin quasiparticle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The number of anyons and their  properties are very different in different nonabelian FQH states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Ignoring for now the existence of more quasiparticles, let us focus now on the fundamen-  tal anyon which is nonabelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin and coworkers showed [298] considered a nonabelian  quasihole that is injected to the bottom edge, tunnels at the first QPC and arrives at the left end  of the top edge in state |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If a second such quasiparticle is now injected but now tunnels to  the top edge at the second QPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', arriving at the left end of the top edge in the state eiαBNq|ψ⟩,  where α is the Aharonov-Bohm phase determined by the flux Φ piercing the interferometer and  BNq si the braiding operator os the second quasiparticle that is circling around the Nq localized  quasiparticles in the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then the tunneling conductance measured at the left exit of  the top edge has an interference contribution  σxx ∝ |Γ1|2 + |Γ2|2 + Re  �  Γ∗  1Γ2eiα⟨ψ|BNq|ψ⟩  �  (537)  The matrix element ⟨ψ|BNq|ψ⟩ is given in terms of the expectation value of the Wilson loop oper-  ators of the tunneling quasiparticles braided with the Wilson loops of the Nq localized quasiparti-  cles in the area of the interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 1989 Edward Witten [165] showed that this expectation  value, which is computed in the nonabelian Chern-Simons gauge theory, is equal to a topolog-  ical invariant of the braid known as the Jones polynomial VNq(e1π/4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, the oscillatory  component of the tunneling current (and of the conductance) measures a topological invariant!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [298] gave a general algorithm for the computation of this matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in  the particular case of the MR states, the non-abelian gauge theory associated with these states is  the SU(2)2 Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, an explicit calculation Bonderson, Kitaev  and Shtengel [332] and by Stern and Halperin [333] leads to the result  σxx ∝|Γ1|2 + |Γ2|2,  for Nq odd  σxx ∝|Γ1|2 + |Γ2|2 + 2 |Γ1||Γ2|(−1)Nψ cos  �  α + arg  �Γ2  Γ1  �  + π  4 Nq  �  ,  for Nq even  (538)  Here, Nψ = 1 when the Nq quasiparticles fuse into the state ψ and Nψ = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  interference effect is absent for Nq odd since an odd number of σ quasiparticles cannot fuse into  the identity state I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This simple even-odd effect is special for SU(1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the general case and,  in particular in the SU2)3 case which applies to the k = 3 Read-Rezayi (“Fibonacci”) state, the  expressions are more complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  104  At any rate, even in the MR state the situation is more complicated for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One  is that the charge e/2 abelian Laughlin quasiparticle is always able to tunnel thus spoiling the  even-odd effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the other complication is that the nonabelian FQH occurs in the N = 1 Landau  level and the edge states of the nonabelian FQH state is surrounded by an abelian ν = 2 state,  which makes accessing the interesting edge states more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Robert Willett has pioneered the  fabrication and operation of the interferometer for the nonabelian state at ν = 5/2 [334, 335, 336].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  With some significant caveats although there is solid evidence of nonabelian braiding, more work  remains to be done on this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Particle-Vortex Dualities in 2+1 dimensions  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Electromagnetic Duality  The oldest form of duality in Physics is, perhaps, Dirac’s observation that in the absence of  electric charges and currents Maxwell’s equations are invariant under the exchange of electric  and magnetic fields, E → B and B → −E [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This observation led him to conjecture the  existence of magnetic monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In a relativistic invariant formulation, Maxwell’s equations in  free space can be expressed in terms of the electromagnetic field tensor Fµν and the dual field  tensor F∗  µν  ∂µFµν = 0,  ∂µF∗  µν = 0  (539)  where F∗  µν = 1  2ǫµνλρFλρ, and where ǫµνλρ is the fourth-rank antisymmetric Levi-Civita tensor, The  first Maxwell equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (539) is just the wave equation in free Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  second equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (539) is known as the Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Bianchi identity is a constraint  which implies that there are no magnetic monopoles and that the second rank antisymmetric field  strength tensor can be expressed in terms of the vector potential Aµ as Fµν = ∂µAν − ∂νAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  This is an example of what in differential geometry is called Hodge duality, which relates a  vector or, more generally a tensor field to its Hodge dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general in D dimensions the Hodge  dual of a (fully antisymmetric) tensor of rank p is an antisymmetric tensor of rank D − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' When  contracted with the oriented infinitesimal element of a p-dimensional hypersurface, dx1 ∧ dx2 ∧  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ∧ dxp, an antisymmetric tensor of rank p, Fµ1µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='µp, defines a p-differential form, called a p-  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, differential forms embody the physically intuitive notions of circulation of a vector  field, flux of a second rank tensor, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will see below that duality transformations are closely  related to these geometric notions of duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Particle-Vortex Duality in 2+1 dimensions  In this section we will extend the particle-vortex duality discussed in 2D in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1 (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (75)) to 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the field theory interpretation, 2D is a 1+1-dimensional Minkowski spacetime  and the XY model is a representation of a complex scalar field of unit modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the duality, the  particles are the particle-like excitations of the complex scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The high temperature phase  of the XY model is viewed as a partition function of a set of oriented loops that carry charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the  loops are the worldlines of the particles of the XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The 3D XY Model  This picture can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Consider an XY model on a D-dimensional hypercubic  lattice whose sites are labelled by {r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At each site there is a periodic variable θ(r) ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  partition function is  ZXY =  �  r  � 2π  0  dθ(r)  2π  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed  1  T  �  r  �  j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=',D  �  cos ∆jθ(r)  �\uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (540)  where ∆jθ(r) = θ(r+e j)−θ(r), with j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', D, is the lattice difference, and T is the temperature  (in the classical statistical mechanical picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the interaction on each link is a periodic  function of the phase difference ∆jθ(r), we can expand the Gibbs weight (for each link!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') in a  Fourier series or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' what is the same,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' in the integer-valued representations of the group U(1) (which  is the global symmetry of the XY model) to obtain  ZXY =  �  r  � 2π  0  dθ(r)  2π  ∞  �  ℓj(r)=−∞  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  �  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j  T  2 ℓ2  j(r) + i  �  r  θ(r)∆jℓ j(r)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (541)  105  where we used a Gaussian approximation for the modified Bessel function In(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and ∆jℓ j(r) =  �D  j=1(ℓ j(r)−ℓ j(r−e j)) (where e j is the lattice unit vector along the direction j) denotes the lattice  divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Integrating-out the phase variables θ(r) we obtain  ZXY =  �  r  ∞  �  ℓj(r)=−∞  �  r  δ(∆jℓ j(r)) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  �  r  D  �  j=1  T  2 ℓ2  j(r)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (542)  Therefore, the partition function is given by a sum over loops of conserved currents ℓ j(r) defined  on the links of the lattice, with a weight on each link ∝ exp(−ℓ2  j/2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These loops are the (lattice)  worldlines of the particles of the complex scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the phase where this representation  is convergent, the complex scalar field is massive, the XY model is gapped, and these particles  have short range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This picture is true in all dimensions, 3D included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand, in 2D the vortices are point-like events in Euclidean spacetime which in-  teract with each other through long range, logarithmic, interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the field theory language,  in 2D the point-like vortices are instantons which govern the low temperature phase of the XY  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In spacetime dimensions D > 2 the vortices become extended objects: vortex loops (or  strings) in 3D, closed vortex surfaces in 4D, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 3D the vortex loops are magnetic flux tubes  which interact with each other through a Biot-Savart type interaction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' namely bits of vortices  interact with each other with a 3D Coulomb interaction much in the same way as with loops of  current in magnetostatics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Z3DXY =  �  {mj(˜r}  �  ˜r  δ(∆jm j(˜r)) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−2π2  T  �  ˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='˜r′  3  �  j=1  m j(˜r)G0(˜r − ˜r′)m j(˜r′) − α  �  ˜r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' j  m2  j(˜r)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (543)  where {˜r} labels the sites of the dual (cubic) lattice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' the variables m j(˜r) take values on the integers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  α is a (short-distance) vortex core energy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and G0(˜r − ˜r′) is the 3D lattice propagator (Green  function) which at long distances has the standard form  G0(x − x′) =  1  4π|x − x′|  (544)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (543) the vortex loops are represented by the integer-valued conserved currents m j(˜r),  which are naturally interpreted as the world lines of magnetic charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We could have also reached the same result by following the line of reasoning that we used in  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, let θ(x) be the phase field of a complex scalar field φ(x) deep in its broken  symmetry state where the amplitude of the field φ(x) can be taken to be approximately fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  partition function in this phase reduces to  Z[aµ] =  �  Dθ exp  �  − 1  2g  �  d3x  �  ∂µθ − aµ  �2�  (545)  where g is a coupling constant which in the XY model is proportional to the temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here,  much as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (70), the gauge field aµ represents the vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Indeed, in 3D the vorticity is  represented by a locally conserved vector field ωµ  ωµ = ǫµνλ∂νaλ = 2π  �  k  mk  µδ(x − xk)  (546)  where mk  µ are the integer-valued vortex (magnetic) currents we used above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As before, the pe-  riodic nature of the phase field θ(x) implies that the vortex currents must be quantized and be  integer-valued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Here too we can perform a Hubbard-Stratonovich transformation in terms of a vector field bµ  to find a dual theory which now is  Z(aµ) =  �  DbµDθ exp  �  −g  2  �  d3x b2  µ − i  �  d3x bµ(∂µθ − aµ)  �  (547)  Upon integrating out the phase field θ, which acts as a Lagrange multiplier which imposes the  constraint ∂µbµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This constraint implies that we can write the field bµ in terms of a dual  gauge field ϑµ  bµ = ǫµνλ∂νϑλ  (548)  106  Therefore, we can rewrite the partition function as  Z(aµ) =  �  Dϑ exp  �  −g  4  �  d3x f 2  µν + i  �  d3x ωµϑµ  �  (549)  where fµν = ∂µϑν − ∂νϑν is the field strength of the dual gauge field ϑµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that the vortex  current ωµ is minimally coupled to the dual gauge field ϑµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we now integrate-out the gauge  field ϑµ we obtain an expression for the weight in the path integral for a configuration of vortices  mk  µ which is identical to the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (543).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  What we have shown above is that in 3D the Goldstone phase of a complex scalar field  with coupling constant g is the dual of a Maxwell gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' with coupling constant 1/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Moreover, the periodicity of the phase field implies that the dual gauge field ϑµ is compact in  the sense that its fluxes must obey flux quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is straightforward to show that a charge  operator Vn = exp(inθ) with electric charge n in the scalar field theory is represented in the dual  gauge theory by a magnetic monopole of magnetic charge n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In summary we showed that the 3D XY model (equivalent to the theory of the complex scalar  field) can be written in terms of two different models of loop configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We saw that the  XY model is a theory of particle (electric) loops in the high temperature phase and of vortex  (magnetic) loops in the low temperature phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However the particle loops have short range  interactions while the vortex loops have long range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thus, the 3D XY model is not  self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In spite of having a representation in terms of vortex loops, the 3D XY model is in  reality has a very different behavior that the 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Kosterlitz-Thouless theory describes  the phase transition in terms of a vortex-antivortex unbinding transition upon which the vortices  proliferate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In addition, in 2D the Goldstone phase is actually a line of fixed points, there is never  true long range order but, instead, power-law correlations of the physical observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  Kosterlitz-Thouless theory the disordered phase arises the strong fluctuations of the phase field  due to the proliferation of vortices while, at the local level, the order parameter still has a finite  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A consequence of this behavior is that the superfluid density has a universal jump at  the Kosterlitz-Thouless transition [337].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In contrast in the 3D XY model the Goldstone phase is a true phase with long range order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The theory has a continuous phase transition (the thermodynamic superfluid transition) at which  the superfluid density vanishes continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the phase transition of the 3D XY  is better described by the Wilson-Fisher fixed point of a complex scalar field [30, 48], rather than  by a vortex proliferation picture of the Kosterlitz-Thouless theory [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular, since  this theory does not have magnetic monopoles, the vortex loops cannot proliferate as they do in  2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Instead, as the phase transition is approached, the vortex loops grow large in size but also  become fractal-like objects whose effective core size diverges as the transition is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  fact, qualitative vortex proliferation arguments readily lead to the incorrect conclusion that the  transition should be (strongly) first order [338].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In spite of these differences, particle-vortex duality still plays an important role in 3D by  relating the 3D XY model to another theory which is its dual under electromagnetic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here  electromagnetic duality is understood as the exchange of the electric worldlines (electric loops)  and the magnetic loops (vortex loops) while exchanging strong and weak coupling, T ↔ 1/T, in  this case between different theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This duality was investigated by Michael Peskin [339], by  Paul Thomas and Michael Stone [338], and by Chandan Dasgupta and Bertrand Halperin [340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Scalar QED in 3D  These authors considered a theory of a superconductor represented by a complex scalar field  ϕ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' coupled to a fluctuating electromagnetic (Maxwell) field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' also known as the abelian Higgs  model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' or scalar electrodynamics (scalar QED) aµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' with µ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 3 The partition function of this  theory is  ZSC =  �  DϕDϕ∗Daµ exp  �  −  �  d3x L[ϕ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ϕ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' aµ]  �  (550)  where  L = |Dµ(a)ϕ|2 + m2|ϕ|2 + u|ϕ|4 + 1  4e2 f 2  µν  (551)  where the covariant derivative is Dµ(a) = ∂µ + iqaµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' with the integer q being the charge of the  scalar field (in units of the coupling constant e),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' and fµν = ∂µaν − ∂νaµ is the field strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  theory is known as (Euclidean) scalar electrodynamics (or, the abelian Higgs model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  107  The scalar electrodynamics was extensively studied using the perturbative renormalization  group within the epsilon expansion (near 4 dimensions) which leads to the conclusion that it  has a weakly (fluctuation-induced) first order phase transition [109, 341].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Scalar QED of an N-  component scalar field coupled to a Maxwell gauge field has a continuous phase transition with  a non-trivial fixed point for N ≥ Nc ≃ 183 (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') [342].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since these results rely on perturbation  theory (or in the large-N limit) it was long suspected that the physics may be different in D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will see that particle-vortex duality provides the answer to this question [340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We begin by writing the lattice version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (550) which is obtained by coupling the XY  model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (540) to a dynamical (Euclidean) Maxwell field  ZSC =  �  r  � 2π  0  dθ(r)  2π  � ∞  −∞  daµ(r)  2π  exp  �  −S (θ, aµ)  �  (552)  where the action S (θ, aµ) is  S (θ, aµ) = − 1  T  �  r,µ  cos  �  ∆µθ(r) − qaµ(r)  �  + 1  4e2  �  r,µ,ν  �  ∆µaν(r) − ∆νaµ(r)  �2  (553)  In this action we assumed that the abelian gauge fields do not have monopole configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (550) admits a representation as a sum over loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can then  used the same line of reasoning that led to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (542) and write the partition function as a sum over  loops of the worldlines of the particles (charges) of the complex scalar field  ZSC  Zgauge  0  =  �  r  ∞  �  ℓµ(r)=−∞  δ(∆µℓµ(r)) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  �  r,µ  T  2 ℓ2  µ(r)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  �  exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ediq  �  r,µ  ℓµ(r)aµ(r)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  �  a  (554)  where ⟨O[a]⟩a is the expectation value of the operator O[a] over the gauge fields aµ, and Zgauge  0  is  the partition function of the free gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' After computing this free-field expectation value  we obtain the result  ZSC  Zgauge  0  =  ∞  �  {ℓµ(r)}=−∞  δ(∆µℓµ(r)) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  �  r,µ  T  2 ℓ2  µ(r) − q2e2  2  �  r,µ  �  r′,ν  ℓµ(r) Gµν(r − r′) ℓν(r′)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (555)  where  Gµν(r − r′) = ⟨aµ(r)aν(r′)⟩  (556)  is the Euclidean propagator of the free gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Since the configurations of the worldlines,  the currents represented by ℓµ(r), are conserved and satisfy the local constraint ∆µℓµ = 0, the  second term in the exponent of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (555) is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, we can use the propagator in  the Feynman gauge  Gµν(r − r′) = δµνG0(r − r′)  (557)  where G0(r− r′) is the 3D lattice propagator which has the same Coulomb form at long distances  already given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (544).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Therefore we can write the partition function of the scalar QED model (the superconductor)  as a sum over worldline loop configurations of the particles of the scalar field in the equivalent  form  ZSC  Zgauge  0  =  ∞  �  {ℓµ(r)}=−∞  δ(∆µℓµ(r)) exp  \uf8eb\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ec\uf8ed−  �  r,µ  T  2 ℓ2  µ(r) − q2e2  2  �  r,r′,µ  ℓµ(r) G0(r − r′) ℓµ(r′)  \uf8f6\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f7\uf8f8  (558)  which is the same as the partition function of the 3D XY model in the broken symmetry state,  given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (543), with the identification of the particle loops of the abelian Higgs model with the  vortices of the 3D XY model and a relation between the coupling constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence we obtain the  duality between the broken symmetry phase of the 3DXY model (“low T”) and the symmetric  phase of the abelian Higgs model (“high T”)  ℓµ ↔ mµ,  q2e2 ↔ 2π2  T ,  T  2 ↔ α  (559)  where the quantities on the left hand side of the identifications refer to the 3D abelian Higgs  model and the quantities on the right hand side to the 3D XY model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  108  We can now repeat the same analysis but in the broken symmetry of the abelian Higgs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this phase the gauge field aµ becomes massive by the Higgs mechanism (or, what is the same,  by the Meissner effect of the superconductor), and in this phase the vortex loops mµ of the  complex scalar field have short range interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This phase then is mapped onto the unbroken  phase of the 3DXY model with the screened vortex loops of the abelian Higgs model identified  with the particle loops of the 3D XY model, and with the same identifications between the  coupling constants of the two theories given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='(559).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The identification between the two  partition functions implies that the phase transitions must be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In other words, the  duality implies that the 3D abelian Higgs model has a continuous phase transition with the same  fixed point as that of the complex scalar field in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only difference is that the phases are  reversed and, for this reason, this is sometimes called an “inverted” 3D XY transition [340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The duality mapping  We can summarize these results with the following identification of two Lagrangians (in real  time, Minkowski signature)[208]  |∂µ(B)φ|2 − m2|φ|2 − u|φ|4 ↔ |∂µ(a)ϕ|2 + m2|ϕ|2 − u|ϕ|4 − 1  4e2 f 2  µν + 1  2πǫµνλaµ∂νBλ  (560)  The left hand side of this equivalence is the Lagrangian of the 3D complex scalar field φ and Bµ  is a background electromagnetic gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The right hand side is the Lagrangian of the abelian  Higgs model with a scalar field ϕ and a U(1) gauge field aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice that the mass terms of the  two sides are inverted reflecting the reverse order of their two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This identification also  shows that the current of the scalar field jµ = − i  2(φ∗∂µφ − φ∂µφ∗) is identified as  jµ ↔ 1  2πǫµνλ∂νaλ  (561)  The field operator of the complex scalar field creates worldlines (as opposed to loops) of charged  particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the dual abelian Higgs theory this operator is identified with an operator that creates  a magnetic monopole of the U(1) gauge field aµ with unit magnetic charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is easy to see that in  the broken symmetry phase of the abelian Higgs model a monopole-antimonopole pair creation  operator decays exponentially with distance due to the flux expulsion effect (the Meissner effect  of a superconductor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that the monopoles are confined with a linear potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This  is the same behavior of the complex scalar field φ in its symmetric phase where the propagator  of the complex scalar field decays exponentially with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conversely, in the broken sym-  metry phase of the complex scalar field theory, the field operator φ condenses and its propagator  approaches the value |⟨φ⟩|2 at long distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the same behavior one readily finds for the  monopole operator of the abelian Higgs model in the symmetric phase .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization in 2+1 dimensions  The particle-vortex duality that we discussed in section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 essentially has been understood  since the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4 we discussed in detail the problem of bosonization in 1+1 dimen-  sions whic is a mapping between a massless Dirac fermion and a massless compactified scalar  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we noted, this mapping is a powerful theoretical tool to investigate non-perturbatively  the structure of many theories in 1+1 dimensions of great physical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This success has  motivated a sustained effort to extend these concepts to higher dimensions where the problem in  many ways is more subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Fermi systems in dimensions higher differ from the 1+1-dimensional case in two significant  ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Due to the kinematic restriction of one space dimension a massless relativistic fermion  and a theory of fermions (relativistic or not) at finite density are mostly equivalent to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The reason is quite simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Consider a free massless Dirac theory in 1+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we  saw, it is equivalent to a theory of two chiral fermions, the right and left moving component of  the spinor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If we add a chemical potential µ, the right and left moving states will be filled up to  µ, which is the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' While in space dimensions d > 1 a finite chemical potential leads  to a finite fermi surface, in d = 1 space dimensions the Fermi “surface” is just two points, for  the right and left moving fermion modes respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Dirac Hamiltonian at finite chemical  potential is  H = ψ†(−i)α∂xψ − µψ†ψ  (562)  109  Under a chiral transformation with angle θ(x), the field operators change as ψR → eiθψR and  ψL → e−iθψL and the Hamiltonian becomes  H = ψ†(−i)α∂xψ − (µ − ∂xθ) ψ†ψ  (563)  Clearly if we choose ∂xθ = µ (or, what is the same, θ = µx) the explicit dependence on the  chemical potential has been cancelled and the transformed Dirac Hamiltonian is the same as the  Hamiltonian at zero chemical potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However under this transformation the mass operators  transform as ¯ψψ → cos(2µx) ¯ψψ and i ¯ψγ5ψ → sin(2µx) i ¯ψγ5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the non-relativistic context this  change amounts to a modulation of the charge density with wave vector Q = 2pF(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In contrast, in space dimensions d > 1 the massless Dirac system and a system of fermions  at finite density (relativistic or not) are no longer equivalent to each other as the latter system has  a Fermi surface (of co-dimension d − 1) wile the former system does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This difference has  profound effects on their dynamics: the system of fermions at finite density is, at least in the weak  coupling regime, is expected to be a Fermi liquid (except for an instability to a superconducting  state even for infinitesimal attractive interactions) while the massless Dirac theory is stable as all  local interactions are irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization of the Dirac theory in 2+1 dimensions  We will now turn to the problem of Bose-Fermi mappings in relativistic systems in 2+1  dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This problem is of great interest both in Condensed Matter Physics and in High  Energy Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Bosonization in 2+1 dimensions is a more subtle problem than in 1+1 dimensions that we  discussed in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There we saw, in 1+1 dimensions bosonization consists of a set of  operator identities relating two free field theories, a theory of massless Dirac fermions and a  compactified massless free scalar field φ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The success of that program is largely based on the  fact that both theories are massless (and hence are scale and conformally invariant), on the chiral  anomaly, and on the relation between the gauge current of the Dirac theory with the topological  current of the compactified boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The physics in 2+1 dimensions (and in general) is very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance, instead of  the chiral anomaly in 2+1 dimensions theories of relativistic fermions have a parity anomaly,  discussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will also see that although in 2+1 dimensions there are theories  of free massless Dirac fermions and free massless compactified bosons they are no longer dual to  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For these and other reasons it has been difficult to extend the bosonization program  to 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  One of the first bosonization constructions for a theory of massive (relativistic) Dirac fields  was put forth by Alexander Polyakov in 1988 [168, 167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For reasons that no longer relevant,  Polyakov considered a theory of CP1 complex scalar fields (in its unbroken phase) coupled to  a U(1)1 Chern-Simons gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nevertheless, his main argument, with some corrections  associated with the parity anomaly, still holds correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Here we will follow the work of Hart Goldman and myself [343] and consider instead a  theory of a single massive complex field coupled to a U(1)1 Chern-Simons gauge theory whose  Lagrangian density is  L = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1  4πǫµνλA µ∂νA λ  (564)  In essence, this theory is a relativistic version of what we did in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='7 where we mapped  non-relativistic fermions to (also non-relativistic) bosons whereas here the theory is relativistic  and we are mapping bosons to fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In his work Polyakov actually considered the problem  of the propagator of a free massive scalar field coupled to the Chern-Simons gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this  limit, the propagator G(x, x′) is the transition amplitude from x to x′ (in Minkowski spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the case in which φ(x) is a free massive field this propagator can be expressed in the for of a  Feynman path-integral as a sum over all open oriented paths {Px,x′} with endpoints at x and x′  which in Euclidean spacetime is  Gγ(x,x′) =  �  {Px,x′ }  exp(−mL(Px,x′))  �  exp  �  i  �  Px,x′  dzµA µ(z)  � �  U(1)1  (565)  where the expectation value is over the Chern-Simons gauge fields Aµ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (564).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  110  Let us consider now the partition function of the bosons which is a sum over all closed paths  {P}, which we will assume to be non-intersecting (which requires a short distance repulsion)  Z =  �  {P}  exp(−mL(P))  �  exp(i  �  P  dzµA µ(z))  �  U(1)1  (566)  The expectation value is given by  �  exp  �  i  �  P  dzµA µ(z)  � �  U(1)1 = exp  �  −1  2  �  P  �  P  dxµdyν⟨Aµ(x)Aν(y)  �  ≡ exp(iπW(P))  (567)  where W(P) is the writhe of the closed path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The computation of the writhe requires a regu-  larization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' One possible regularization is to thicken the path which is equivalent to including a  Maxwell term for the gauge field in the Lagrangian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In general, the writhe of the path P is  W(P) = S L(P) − T(P)  (568)  where S L(P) is an integer-valued topological invariant called the self-linking number and T(P)  is the twist (or torsion) of the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The twist T(P) is a Berry phase which depends on the  coordinates of the space in which the path P is embedded, and it is not a topological invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will now compute the Berry phase T(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Let ˆe(s) be a unit vector tangent to the path  P and where we parametrized the closed path P by a coordinate s ∈ [0, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The closed path  P is the boundary of a surface Σ we can take to be a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will write the Berry phase by  extending ˆe(s) smoothly to the interior of Σ as ˆe(s, u), where 0 ≤ u ≤ 1 and ˆe(s, u = 1) = ˆe(s)  and ˆe(s, u = 0) = ˆe0 ≡ constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With these definition the Berry phase is  T(P) ≡ W(ˆe) = 1  2π  � L  0  ds  � 1  0  du ˆe · ∂sˆe × ∂uˆe  (569)  which is defined modulo an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this formulation, in the bosonic theory the amplitude for a path of length L with tangent  vector ˆe(s) with endpoints at x and x′ is  G(x − x′) =  � ∞  0  dL  �  D ˆe(s) δ  �  1 − |ˆe|2�  δ  �  x − x′ −  � L  0  ds ˆe(s)  �  exp(−|m|L ± iπW(ˆe)) (570)  In momentum space we find  G(p) =  � ∞  0  dL  �  Dˆe δ(1 − |ˆe|2) exp(−|m|L ± iπ W(ˆe)) exp(ipµ  � L  0  ds ˆeµ(s))  (571)  By inspection of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (569) we see that this is the same expression that we looked in the theory of  the path integral for spin in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' for spin S = 1/2 particle in a magnetic field bµ = ±2pµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The equation of motion for ˆe is  ∂sˆeµ = ±2ǫµνλˆeλ = i[H, ˆeµ]  (572)  where H = ∓pµˆeµ is the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon quantization, ˆeµ satisfies the commutation relations  [ˆeµ, ˆeν] = 2iǫµνλˆeλ  (573)  This means that we should make the identification ˆeµ → σµ where σµ are the three 2 × 2 Pauli  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Up to rescaling of the mass m → M, upon performing the path integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (571) we  find that G(P) is the (Euclidean) Dirac propagator [168]  G(p) =  1  ipµσµ − M  (574)  Using these results the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (566) becomes  Zfermion = det[i/∂ − M] =  �  DJ δ(∂µJµ) exp �−|m|L[J] − isgn(M)πΦ[J]�  (575)  where Φ[J] = W[J], defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (568).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  111  We will return to this construction shortly below when we include the effects of the parity  anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Early attempts at deriving a mapping for a theory of Dirac fermions in 2+1 dimensions were  based on their behavior in the presence of gauge fields and the associated parity anomaly, dis-  cussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These early theories are essentially a hydrodynamic description of the  Dirac theory deep in the massive phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With minor differences, the same results were derived  independently (and simultaneously) by two groups, Fidel Schaposnik and myself [344] and Cliff  Burgess and Francisco Quevedo [345].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This approach was reexamined more recently by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chan,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hughes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu and myself in 2016 in a derivation of a hydrodynamic effective field theory for  topological insulators in different dimensions [346, 347].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These derivations are similar in spirit  to what we discussed in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6 for the fractional quantum Hall effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will follow the  approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [346] for the special case of a Chern insulator 2+1 dimenxions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Dirac theory, in any dimension, is globally gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By Noether’s theorem this  means that it has a locally conserved current, jµ = ¯ψγµψ which thus satisfies ∂µ jµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here  too, this means that we can write jµ = ǫµνλ∂νbλ, where bµ is a gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We expect that the  effective action of the gauge field bµ should be local, gauge invariant and, in this case, relativistic  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In 2+1 dimensions the effective action should break time reversal invariance and hence  it should have a Chern-Simons term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will consider the free massive Dirac theory coupled to a background probe gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The Lagrangian is L = ¯ψ(i/(D) − M)ψ, where Dµ = ∂µ + iAµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' As we saw in section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6, gauge  invariance and locality require that we have an even number of Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we will  assume that one of the Dirac fermions is massive and acts as a regulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The partition function  Z[Aµ] =  �  D ¯ψDψ exp(iS F[ ¯ψ, ψ, Aµ])  (576)  By definition the expectation value of a product of current operators jµ can be obtained by func-  tional differentiation of the partition function with respect to Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The partition function is gauge invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Z[Aµ = Z[Aµ + aµ]  (577)  where the vector field aµ is a pure gauge, aµ = ∂µφ and fµnu = ∂µa − ν − ∂µaµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hence, aµ is  said to be flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Therefore, up to a normalization  Z[Aµ] =  �  D[aµ]pure Z[Aµ + aµ]  =  �  Daµ δ(fµν = 0) Z[Aµ + aµ]  =  �  DaµDbµ Z[Aµ + aµ] exp  �  − i  2  �  d3x ǫµνλ bµ fνλ  �  (578)  where in the last equality we introduced a representation of the delta function and the vector field  bµ plays the role of a Lagrange multiplier field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' using the invariance of the integration measure  under aµ → aµ − Aµ we find  Z[Aµ] =  �  DaµDbµZ[aµ] exp  �  − i  2  �  d3x ǫµνλ bµ (fνλ − Fνλ)  �  (579)  where Fµν is the field strength of the external field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' From these identities and by differentiation  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (579) it follows that a general expectation value of products of currents is  ⟨jµ(x) jν(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='⟩ =  δ  δAµ(x)  δ  δAν(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Z[Aµ] = ⟨ǫµαβ∂αbβ ǫνγδ∂γbδ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='⟩  (580)  which means that, at the operator level, we can identify [344]  jµ(x) ⇔ ǫµνλ∂νbλ  (581)  This result is actually general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The only difference is the nature of the field b isn different di-  mensions: in 1+1 is a pseudo-scalar, in 2+1 is a vector (gauge) field, in 3+1 is an anti-symmetric  (Kalb-Ramond) tensor field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  112  Returning to the partition function of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (579) we find, using the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (324) applied  the partition function Z[aµ], that Z[Aµ] is given by  Z[Amu] =  �  DaµDbµ exp(i  �  d3x Leff[aµ, bµ, Aµ])  (582)  where the effective Lagrangian is  Leff = −ǫµνλ bµ ∂νaλ + s  4πǫµνλaµ∂νaλ −  1  4geff  fµν f µν + Aµǫµνλ∂νbλ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  (583)  where fµν = ∂µaν − ∂νaµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here s = 1 in the Chern insulator, s = 0 in the trivial insulator and  s = 1/2 at the quantum critical point, and geff an effective coupling constant (with dimensions of  length−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here we included the effects of the parity anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We recognize that the first term of  the effective Lagrangian of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (583) is a BF term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' [346] a similar result is also derived for a 3+1-dimensional topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  main differences are that there is no parity anomaly but an axial anomaly and that the Lagrange  multiplier field is a Kalb-Ramond field,  Leff[aµ, bµν, Aµ] = ǫµνλρ bµν ∂λaρ +  θ  8π2 ǫµνλρ∂µaν∂λaρ − 1  4g2 fµν f µν + Aµǫµνλρ∂νbλρ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (584)  where θ = π in the topological insulator and θ = 0 in the trivial insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Although these results are correct, they do not give a full bosonization mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In particular,  these results do not identify a fixed point for the dual bosonic theory and, along with it, a full  mapping of the observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Progress on this problem has only been achieved in the past few  years both in the high energy literature [348, 349, 350, 351, 352, 199, 353, 354], and in the  condensed matter physics literature [355, 356, 357, 358, 359].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Much of that new insight was  presented in a 2016 insightful paper by Nathaniel Seiberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Chong Wang and Edward  Witten [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many of the dualities are actually conjectures supported by strong consistency  checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A derivation of the basic bosonization duality based on loop models was constructed by  Hart Goldman and myself [343].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The basic conjectured bosonization duality is a mapping of a free Dirac fermion to a gauge  complex scalar field with a Chern-Simons term [352, 208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can think of the Dirac theory as  either being defined entirely in 2+1 dimensions or as a being defined at the boundary of a non-  trivial 3+1 dimensional topological insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the first scenario we need to take into account  the contribution of the fermionic doublers (this is what happens in the case of a lattice theory)  or as the Pauli-Villars heavy fermionic regulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In both cases, the additional heavy degrees of  freedom cancel the anomaly of the 2+1-dimensional Dirac fermion, see section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the  second scenario, there is only one Dirac fermion at the boundary and the bulk θ-term cancels the  anomaly of the boundary theory, see section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' With these provisos, the Lagrangian LA of  the free massive (or massless) Dirac fermion in 2+1 dimensions coupled to the electromagnetic  gauge field Aµ is  LA = ¯ψ(i /D(Aµ) − M)ψ − 1  8πǫµνλAµ∂νAλ  (585)  where M is the Dirac mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will call this Theory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here /Dµ(A) = γµ(∂µ − iAµ) and Aµ  is a background (non-dynamical) gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (585) is the Chern-Simons  term with the 1/2-quantized coefficient, the usual short-hand for the η-invariant term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' (342)  needed to cancel the parity anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  From the effective field theories we discussed above we know that the fermionic current  maps to the curl of a gauge field, see Eq(581).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Then, the conjectured bosonic dual is given by the  Lagrangian of Eq(564) with an extra term for the (dual) coupling to the electromagnetic gauge  filed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will call this Theory B whose Lagrangian LB is  LB = |Dµ(Aµ)φ|2 − m2|φ|2 − λ|φ|4 + 1  4πǫµνλA µ∂νA λ + 1  2πǫµνλAµ∂νA λ  (586)  To check the consistence we first observe that both theories are anomaly free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' By functional  differentiation of the two partition functions we check that we get the correct mapping for the  fermionic current, jµ ↔ ǫµνλ∂νA λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This mapping implies that the charge density of Theory A  maps onto the gauge field flux of Theory B, which is electromagnetic duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We see that this  bosonization is a relativistic version of flux attachment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  113  Let us now identify the mapping of the phases of both theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the Dirac mass M < 0,  Theory A describes an anomalous quantum Hall insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Integrating out the massive fermions  we fund that the effective action for the electromagnetic gauge field Aµ is a U(1)1 Chern-Simons  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This means that in this phase σxy = −1/(2π) (in units in which e = ℏ = c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Looking  now at Theory B we see that if m2 > 0, the scalar field is in the unbroken phase, and it is massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In this phase we set φ = 0 and find that the low energy resulting theory if just a U(1)1 Chern-  Simons gauge theory coupled to the curl of the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Upon integrating out the  Aµ gauge field we find that the effective action of Aµ is also a U(1)1 Chern-Simons term, which  implies that σxy = −1/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Conversely, for M > 0 the effective electromagnetic action of the fermionic theory is a  Maxwell term (the Chern-Simons term canceled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This phase is a trivial insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Looking now  at Theory B we see that for m2 < 0 this theory is in its Higgs phase where ⟨φ⟩ � 0 and the gauge  field Aµ now has a mass term, ∝ A 2  µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the low energy limit Aµ → 0, and the Hall conductivity  vanishes, consistent to what is expected from Theory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  On the other hand, for M = 0 Theory A is at a quantum critical point with a very simple CFT  structure but a non-vanishing Hall conductivity σxy = 1/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It is natural to map this CFT to  a Wilson-Fisher (WF) fixed point of Theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At the WF one sets the (renormalized!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') mass  m2  R = 0 (not the bare mass).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the absence of the Chern-Simons gauge field this WF fixed point  in well understood form high quality (five loop!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') epsilon expansion calculations (enhanced with  Borel resummation) [49], and by more recent numerical Conformal Bootstrap methods [360].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  However, not much is known of the gauged version of the CFT of Theory B since it cannot  be accessed by either methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' If the mapping to Theory A is correct it should have a Hall  conductivity of σxy = 1/(4π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This conjectured value hast not yet been confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Consider now a monopole operator of the gauge field Aµ of Theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We will denote this  operator MA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Chern-Simons term is not gauge invariant in a monopole background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To put  it differently it has charge 1 under the Chern-Simons gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It also has charge 1 under  the electromagnetic field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Likewise the scalar field φ has charge 1 under the Chern-Simons  gauge field Aµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The operator φ†MA is gauge-invariant under the Chern-Simons gauge field Aµ  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' it has charge 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This composite operator has electromagnetic charge 1 (which it inherited  from the monopole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moreover, it has spin-1/2 required by the Wu-Yang construction [151].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In other words, the operator φ†MA has the same quantum numbers as the Dirac fermion and  are identified by this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Notice, however, that the free massless Dirac field has scaling  dimension ∆ψ = 1 in 2+1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The conjecture implies the composite operator φ†MA  should also have scaling dimensions 1 at the (gauged) WF fixed point of Theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Many of these  conjectures have been verified in non-abelian versions of Theory A and Theory B: a theory of  free massless Dirac fermions coupled to a Chern-Simons gauge theory with gauge group SU(N)k  and a theory of complex scalars coupled to a non-abelian Chern-Simons gauge theory with gauge  group SU(k)N, both in the limits N → ∞ and k → ∞ with N/k fixed [348, 349, 350, 351, 352].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  The particle-vortex duality discussed in section 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2 combined with the fermion-vortex du-  ality we sketched here provide a web of dualities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' These identifications constitute a powerful  non-perturbative tool which has led to many significant developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For instance, Goldman  and myself [361] used the web of dualities to explain the experimentally observed self-duality  at fractional quantum Hall plateau transitions, which was a long standing puzzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It has also  provided a powerful new tool to derive effective field theories of non-abelian fractional quantum  Hall states [301] and even to propose novel states with a single (Fibonacci) anyon [303].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bosonization of the Fermi Surface  A form of bosonization has been developed for the case of systems of fermions with a Fermi  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dense Fermi systems at sufficiently weak coupling are well described by the Landau  theory of the Fermi liquid [13, 3, 194, 260, 259].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this regime, the system of fermions has a  collective mode, a bound state of particles and holes, which at long wavelengths is well described  by the random phase approximation (RPA) [362].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' However, in space dimensions d > 1 there is no  kinematical restriction and quasi-particles and quasi-holes may move in their separate ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  result is that, in addition to the collective modes, there is a low-energy spectrum of renormalized  but essentially free quasi-particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Because of the existence of this quasi-particle spectrum the  collective modes generally (but not always) decay into particle-hole pairs resulting in a finite  lifetime of the collective modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Superficially the particle-hole collective modes are similar to the scalar field of the bosonized  theory in d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' To an extent it has been possible to “bosonize the Fermi surface” and to treat it  as a quantum mechanical object [363, 364, 365, 366, 367, 368].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this approach one essentially  114  regards each direction normal to the Fermi surface as a one-dimensionalchiral fermion, including  the current algebra structure, subject to global constraints that cancels the anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This point  of view ensures that the total fermion number in the Fermi sea is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Here I will only  present a short summary of the main ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  As in the one-dimensional case one defines a filled Fermi sea state |FS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is a state of non-  interacting fermions filling up all one-particle states up to the Fermi energy EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' For concreteness  we consider a system in two space dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In this case, neglecting lattice effects, the Fermi  surface is a circumference of radius pF, the Fermi momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' At fixed fermion number, the  excitations are particle-hole pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The operator nk(q) = c†(k+ q  2) c(k− q  2) creates a particle-hole  pair with relative momentum q with total momentum k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In the low energy regime k is a point on  the Fermi surface (with |k| = pF) and q is small momentum compared with pF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In what follows  we will label the point k on the Fermi surface by the angle θ of the arc spanned by k and an  arbitrary origin on the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  We will normal-order the particle-hole creation operator with respect to the filled Fermi sea  and define δ(q, θ) =: nk(q) := nk(q) − ⟨FS|nk(q)|FS⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane [363], Houghton and Marston  [364, 367], and Castro Neto and Fradkin [365, 366] showed that in real space that these normal  ordered density operators obey the equal-time commutation relations  [δn(x, θ), δn(x′, θ′)] = − 1  2π kF(θ) · ▽δ2(x − x′)δ(θ − θ′)  (587)  where kF(θ) is a unit vector normal to the Fermi surface at angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' This is the algebra of the  quantum fluctuations of the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' We can further define at each point θ on the Fermi  surface a Bose field ϕ(x, θ) such that  δn(x, θ) = N(0)�F(θ) · ▽ϕ(x, θ)  (588)  where N(0) is the density of one-particle states at the Fermi surface and �F(θ) is the Fermi veloc-  ity at the location θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The scalar field ϕ(x, θ) is a chiral boson at θ which parametrizes the quantum  fluctuations of the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The quantum dynamics of the chiral bosons ϕ(x, θ) is governed  by the action  S =1  2N(0)  � 2π  0  dθ  2π  �  d2x dt  �  −∂tϕ(x, θ) �F(θ) · ▽ϕ(x, θ) − (�F(θ) · ▽ϕ(x, θ))2�  +1  2N(0)  � 2π  0  dθ  2π  � 2π  0  dθ′  2π  �  d2x d2x′ dt F(x − x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ − θ′) �F(θ) · ▽ϕ(x, θ) �F(θ′) · ▽ϕ(x, θ′)  (589)  where x = (x, t), and F(x−x′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' θ−θ′) are the Fermi liquid parameters that parametrize the effective  forward scattering interactions among the fermion quasiparticles on the Fermi surface [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The  equation of motion predicted by this (quadratic) action is equivalent to the linearized quantum  Boltzmann equation familiar from the Landau theory of the Fermi liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A non-linear extension  has been introduced recently by Delacr´etaz and coworkers [368].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Recent work on the role of  quantum anomalies in dense Fermi systems has yielded new insights on these problems [369].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  In the regime, where the Landau theory of the Fermi liquid is expected to work [13], bosoniza-  tion of the Fermi surface approach has reproduced the previously known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' There remain  many open problems in this approach (and others) in the vicinity of quantum phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  spite of intense research using many different approaches, quantum phase transitions in metallic  systems are not yet fully understood beyond perturbation theory [255, 256].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Non-perturbative  ideas such as deconfined quantum criticality have been proposed [370] and significant work has  been done using large-N methods [371, 372].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A particularly important metallic quantum phase  transition (and perhaps the simplest) occurs near the Pomeranchuk instability of the fermi liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  A quantum phase transition to an electron nematic state [373] has been predicted and studied  within the Hertz-Millis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It has also been studied using higher dimensional bosoniza-  tion [373, 374, 375].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Monte Carlo simulations [376] have shown that the vicinity  of a nematic quantum critical point can trigger a superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nematic Fermi fluids  have been found in many physical systems of interest ranging from high temperature supercon-  ductors (such as cuprates and iron superconductors), to electron gases in large magnetic fields  [377, 378, 379, 380].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  115  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Conclusions  In this chapter I have attempted to cover the role of Quantum Field Theory in modern Con-  densed Matter Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Its role and influence is vast and deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In spite of the length of this  chapter, I have not done justice to its role in many important areas of Condensed Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' In  particular, I have not touched on the the theory of quantum spin liquids, fractons, and particu-  larly topological phases in three space dimensions, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' An area in which the ideas  and concepts of QFT have a huge impact , both conceptually and as a tool, is on the problem of  quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The concept of quantum entanglement as applied to condensed matter  systems, both in equilibrium and out of equilibrium, is a major area of current development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' It  has become a crucial tool for characterizing systems at quantum criticality as well as topological  phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nevertheless, I hope that the reader will find this chapter both insightful and a  useful resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Acknowledgments  This work was supported in part by the US National Science Foundation through grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  DMR 1725401 at the University of Illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schrieffer, Theory of Superconductivity, Frontiers in Physics, Addison-Wesley, New York, New York, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nambu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jona-Lasinio, Dynamical Model of Elementary Particles Based on an Analogy with Superconduc-  tivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I, Physical Review 122 (1961) 345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abrikosov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gorkov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dziyaloshinski, Methods of Quantum Field Theory in Statistical Physics,  Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Englewood Cliffs, New Jersey, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fetter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Walecka, Quantum Theory of Many-Particle Systems, McGraw-Hill, New York, New York,  1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Doniach, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sondheimer, Green’s Functions for Solid State Physicists, Frontiers in Physics, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Benjamin  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', New York, New York, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Feynman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hibbs, Quantum Mechanics and Path Integrals, McGraw-Hill, New York, New York, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Feynman, Statistical Mechanics, A Set of Lectures, Frontiers in Physics, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Benjamin Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Reading,  Massachusetts, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Faddeev, Introduction to Functional Methods, in: Methods of Field Theory, Proceedings of the Les Houches  Summer School 1975, Session XXVIII, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Balian and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin editors, North Holland, Amsterdam, 1976,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [9] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Field Theories of Condensed Matter Physics, Second Edition, Cambridge University Press, Cam-  bridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2013, (expanded 2nd edition of the originally published book in 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Quantum Field Theory: An Integrated Approach, Princeton University Press, Princeton, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Altland, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Simons, Condensed Matter Field Theory, 2nd Edition, Cambridge University Press, Cambridge  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pines, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nozi`eres, The Theory of Quantum Liquids, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Benjamin Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', New York, New York, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Baym, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pethick, Landau Fermi Liquid Theory, John Wiley & Sons, New York, NY, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [14] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Baym, Quantum Statistical Mechanics: Green’s Function Methods in Equilibrium and Non-  Equilibrium Problems, Frontiers in Physics, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Benjamin Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', New York, New York, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kamenev, Field theory of Non-Equilibrium Systems, 1st Edition, Cambridge University Press, Cambridge,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Martin, Measurements and Correlation Functions, Gordon and Breach Science Publishers Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', New  York/London/Paris, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mattis, The Theory of Magnetism, Harper & Row, New York, New York, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau, On the Theory of Phase Transitions, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 7 (1937) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lifshitz, Statistical Physics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 5 of Course of Theoretical Physics, Pergamon Press, Oxford,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [20] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Onsager, Crystal Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A Two-Dimensional Model with an Order-Disorder Transition, Physical Review  65 (1944) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [21] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schultz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mattis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lieb, Two-Dimensional Ising Model as a Soluble Problem of Many Fermions,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 36 (1964) 856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [22] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lieb, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schultz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mattis, Two soluble models of an antiferromagnetic chain, Annals of Physics (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=') 16  (1961) 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeuty, The one-dimensional Ising model with a transverse field, Annals of Physics 57 (1970) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [24] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yang, The Spontaneous Magnetization of a Two-Dimensional Ising Model, Physical Review 85 (1952) 808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Scaling Laws for Ising Models near Tc, Physics 2 (1966) 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [26] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Spin-spin correlations in the two-dimensional Ising model, Il Nuovo Cimento B 44 (1966) 276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Efrati, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kolan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Real-space renormalization in Statistical Mechanics, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 86 (2014) 647–667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [28] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, Renormalization Group and Critical Phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Renormalization Group and the Kadanoff Scal-  ing Picture, Physical Review B 4 (1971) 3174–3183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [29] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, Renormalization Group and Critical Phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phase-Space Cell Analysis of Critical Behavior,  Physical Review B 4 (1971) 3184–3205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, Critical Exponents in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='99 Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 28 (1972) 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  116  [31] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kogut, The Renormalization Group and the ǫ Expansion, Physics Reports C 12 (1974) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, The Renormalization Group: Critical phenomena and the Kondo problem, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 47  (1975) 773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [33] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, The Renormalization Group and Critical Phenomena, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 55 (1983) 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Patashinskii, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pokrovskii, Behavior of Ordered Systems Near the Transition Point, Soviet Physics JETP  23 (1966) 292, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ekzp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' i Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 50, 439 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, The Theory of Equilibrium Critical Phenomena, Reports on Progress in Physics 30 (1967) 615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, The renormalization group in the theory of critical behavior, Reviews of Modern Physics 46 (11974)  597–615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [37] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G¨otze, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hamblen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hecht, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lewis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Palciauskas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rayl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Swift, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aspnes,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, Static Phenomena Near Critical Points: Theory and Experiment, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 39 (1967) 395–431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [38] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gell-Mann, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Low, Quantum Electrodynamics at Small Distances, Physical Review 95 (1954) 1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [39] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bogoliubov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shirkov, Introduction to the Theory of Quantized Fields, Interscience, New York, New  York, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Operator Algebra and the Determination of Critical Indices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 23 (1969) 1430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [41] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, Non-Lagrangian Models of Current Algebra, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 179 (1969) 1499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [42] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Conformal Symmetry of Critical Fluctuations, JETP Letters 12 (1970) 381, [ZhETF Pis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 12,  538 (1970)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Non-Hamiltonian approach to quantum field theory, Soviet Physics JETP 39 (1974) 10, [Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 66, 23 (1974)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [44] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ceva, Determination of an Operator Algebra for the Two-Dimensional Ising Model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  B 3 (1971) 3918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [45] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cardy, Scaling and Renormalization in Statistical Physics, Cambridge Lecture Notes in Physics, Cambridge  University Press, Cambridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [46] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ’t Hooft, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Veltman, Regularization and Renormalization of Gauge Fields, Nuclear Physics B 44 (1972) 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [47] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bollini, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Giambiagi, Dimensional Renormalization: The Number of Dimensions as a Regularizing  Parameter, Il Nuovo Cimento B 12 (1972) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [48] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Amit, Field Theory, the Renormalization Group and Critical Phenomena, McGraw Hill, New York, NY,  1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [49] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th Edition, International Series of Monographs  in Physics, Oxford University Press, Oxford, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [50] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gross, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Ultraviolet Behavior of Non-Abelian Gauge Theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 30 (1973) 1343–  1346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [51] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Anderson, A poor man’s derivation of the scaling laws of the Kondo problem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C: Solid State Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  3 (1970) 2436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [52] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Andrei, Diagonalization of the Kondo Hamiltonian, Physical Review Letters 45 (1980) 379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [53] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wiegmann, Exact solution of the Kondo problem, JETP Letters 31 (1980) 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [54] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Politzer, Reliable Perturbative Results for Strong Interactions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 30 (1973) 1346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [55] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, Confinement of Quarks, Physical Review D 10 (1974) 2445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kogut, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 11 (1975) 395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [57] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Creutz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jacobs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rebbi, Monte Carlo computations in lattice gauge theories, Physics Reports 95 (4) (1983)  201–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Interaction of goldstone particles in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Applications to ferromagnets and massive  Yang-Mills fields, Physics Letters B 59 (1975) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [59] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shenker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tobochnik, Monte Carlo renormalization group analysis of the classical Heisenberg model in  two dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 22 (1980) 4462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [60] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Br´ezin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin, Renormalization of the Nonlinear σ Model in 2 + ǫ Dimensions: Application to the  Heisenberg Ferromagnets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 36 (1976) 691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [61] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Friedan, Nonlinear Models in 2 + ǫ Dimensions, Annals of Physics 163 (1985) 318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [62] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Anderson, Absence of Diffusion in Certain Random Lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 109 (1958) 1492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [63] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abrahams, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Anderson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Licciardello, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ramakrishnan, Scaling Theory of Localization: Absence  of Quantum Diffusion in Two Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 42 (1979) 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [64] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wegner, The mobility edge problem: Continuous symmetry and a conjecture, Zeitschrift f¨ur Physik B Con-  densed Matter 35 (1979) 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [65] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' McKane and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stone, Localization as adn Alternative to Goldstone’s Theorem, Annals of Physics 131  (1981) 36–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [66] Shinobu Hikami, Anderson localization in a non-lienar σ-model representation, Physical Review B 24 (1981)  2671–2679.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [67] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilson, Quantum Field Theory Models in Less Than 4 Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 7 (1973) 2911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sondhi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Girvin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Carini, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shahar, Continuous quantum phase transitions, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 69  (1997) 315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sachdev, Quantum Phase Transitions, 2nd Edition, Cambridge University Press, Cambridge, UK, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [70] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Susskind, Order and disorder in gauge systems and magnets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 17 (1978) 2637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [71] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kogut, An introduction to lattice gauge theory and spin systems, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 51 (1979) 659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [72] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chaikin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lubensky, Principles of Condensed Matter Physics, Cambridge University Press, Cambridge,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [73] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ardonne, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fendley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Topological order and conformal quantum critical points, Annals of Physics  310 (2004) 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Perelomov, Generalized Coherent States and Their Applications, Springer-Verlag, Berlin, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [75] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stone, Topological terms in one- and two-dimensional quantum Heisenberg antiferromagnets,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 38 (1988) 7215(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [76] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wiegmann, Multivalued functionals and geometrical approach for quantization of relativistic particles and  strings, Nuclear Physics B 323 (1989) 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [77] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Berry, Quantal Phase Factors Accompanying Adiabatic Changes, Proceedings of the Royal Society of  London A 392 (1984) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  117  [78] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 51 (1983) 2167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [79] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dombre, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, Absence of the Hopf invariant in the long-wavelength action of two-dimensional antiferro-  magnets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 38 (1988) 7181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [80] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, O(3) non-linear σ model and the topological distinction between integer and half-integer-spin  antiferromagnets in two dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 61 (1988) 1029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [81] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chakravarty, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, Low-temperature behavior of two-dimensional quantum antiferro-  magnets, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 60 (1988) 1057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [82] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mermin, The topological theory of defects in ordered media, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 51 (1979) 591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [83] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abrikosov, On the magnetic properties of superconductors of the second group, Soviet Physics JETP 5  (1957) 1174, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ekzp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' i Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 32, 1442 (1957).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [84] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kosterlitz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thouless, Order, metastability and phase transitions in two-dimensional systems, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  C: Solid State Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 6 (1973) 1181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [85] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kosterlitz, The critical properties of the two-dimensional XY model, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C: Solid State Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 7 (1974)  1046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [86] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Berezinskii, Destruction of Long-Range Order in One-Dimensional and Two-Dimensional Systems having  a Continuous Symmetry Group I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Classical Systems, Soviet Physics JETP 32 (1971) 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [87] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Berezinskii, Destruction of Long-Range Order in One-Dimensional and Two-Dimensional Systems having  a Continuous Symmetry Group II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quantum Systems, Soviet Physics JETP 34 (1972) 610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [88] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jos´e, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kirkpatrick, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, Renormalization, vortices, and symmetry-breaking per-  turbations in the two-dimensional planar model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 16 (1977) 1217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [89] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, Theory of Two-Dimensional Melting, Physical Review Letters 41 (1978) 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [90] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Young, Melting and the vector Coulomb gas in two dimensions, Physical Review B 19 (1979) 1855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [91] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Toner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, Smectic, cholesteric, and Rayleigh-Benard order in two dimensions, Physical Review B  23 (1981) 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [92] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Toner, Bond-orientational order, dislocation loops, and melting of solids and smectic-A liquid  crystals, Physical Review B 24 (1981) 363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [93] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Coleman, Aspects of Symmetry, Cambridge University Press, Cambridge U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [94] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rajaraman, Solitons and Instantons, North-Holland, Amsterdam, The Netherlands, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [95] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nielsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Olesen, Vortex-line models for dual strings, Nuclear Physics B 61 (1973) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [96] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ’t Hooft, Symmetry Breaking through Bell-Jackiw Anomalies, Physical Review Letters 37 (1976) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [97] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Quark confinement and topology of gauge theories, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 120 (1977) 429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [98] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, London, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [99] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Callan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dashen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gross, The structure of the gauge theory vacuum, Physics Letters B 63 (1976)  334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [100] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Caldeira, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Leggett, Quantum Tunneling in a Dissipative System, Annals of Physics 149 (1983) 374–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [101] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Leggett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chakravarty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dorsey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Garg, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zwerger, Dynamics of the dissipatuve two-state  system, Reviews of Modern Physics 59 (1987) 1–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [102] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Di Francesco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mathieu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S´en´echal, Conformal Field Theory, Springer-Verlag, Berlin, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [103] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ginsparg, Applied Conformal Field Theory, in: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Br´ezin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), Champs, Cordes et Ph´enom´enes  Critiques/ Fields, Strings and Critical Phenomena, Proceedings of the Les Houches Summer School 1988, Session  XLIX, Elsevier Science Publishers B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Amsterdam, The Netherlands, 1989, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 1–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [104] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polchinski, String Theory, Cambridge University Press, Cambridge U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [105] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Lattice Coulomb gas representations of two-dimensional problems, Journal of Physics A: Mathe-  matical and Theoretical 11 (1978) 1399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [106] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nienhuis, Two-dimensional critical phenomena and the Coulomb gas, in: C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Domb, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Green, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lebowitz  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), Phase Transitions and Critical Phenomena, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 11, Academic Press, London, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1987, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [107] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mermin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wagner, Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional  Isotropic Heisenberg Models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 17 (1966) 1133–1136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [108] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hohenberg, Existence of Long-Range Order in One and Two Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 158 (1967) 383–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [109] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Coleman, There are no Goldstone bosons in two dimensions, Communications in Mathematical Physics 31  (1973) 259–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [110] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schaposnik, Pseudoparticles and confinement in the two-dimensional Abelian Higgs model, Physical Review  D 18 (1978) 1183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [111] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ’t Hooft, Computation of the quantum effects due to a four-dimensional pseudoparticle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 14 (1976)  3432–3450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [112] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Georgi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Glashow, Unity of all elementary-particle forces, Physical Review Letters 32 (1974) 438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [113] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Compact gauge fields and the infrared catastrophe, Physics Letters B 59 (1975) 82 – 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [114] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac, Quantised singularities in the electromagnetic field, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' London 133 (1931) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [115] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, ‘θ’ physics’ and quantum spin chains, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 57 (1985) 3359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [116] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bethe, Theory of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eigenvalues and eigenfunctions of the linear atomic chain, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Physik 71 (1931)  205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [117] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yang, Thermodynamics of a One-Dimensional System of Bosons with Repulsive Delta-Function  Interaction, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 10 (1969) 1115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [118] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, Critical theory of quantum spin chains, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 36 (1987) 5291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [119] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Non-Abelian Bosonization in Two Dimensions, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 92 (1984) 455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [120] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Knizhnik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zamolodchikov, Current Algebra and Wess-Zumino Model in Two Dimensions, Nuclear  Physics B 247 (1984) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [121] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kennedy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lieb, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tasaki, Valence Bond Ground States in Isotropic Quantum Antiferromag-  nets, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 115 (1988) 477.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [122] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hagiwara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Katsumata, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Renard, Observation of S = 1/2 degrees of Freedom  in an S = 1 Linear-Chain Heisenberg Antiferromagnet, Physical Review Letters 65 (1990) 3181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [123] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kramers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wannier, Statistics of the Two-Dimensional Ferromagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Part I, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 60 (1941) 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [124] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wegner, Duality in Generalized Ising Models and Phase Transitions without Local Order Parameters, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 12 (1971) 2259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [125] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Elitzur, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pearson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shigemitsu, Phase structure of discrete Abelain spin and gauge systems, Physical  118  Review D 19 (1979) 3698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [126] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ukawa, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Windey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Guth, Dual variables for lattice gauge theories and the phase structure of Z(N)  systems, Physical Review D 21 (1980) 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [127] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, Luttinger liquid thepry of one-dimensional quantum fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Properties of the Luttinger model and  their extension to the general 1D interacting spinless Fermi gas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C: Solid State Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 14 (1981) 2585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [128] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mattis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lieb, Exact Solution of a Many-Fermion System and its Associated Boson Field, Journal of  Mathematical Physics 6 (1965) 304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [129] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Luther, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Emery, Backward Scattering in the One-Dimensional Electron Gas, Physical Review Letters 33  (1974) 589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [130] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mandelstam, Soliton operators for the quantized sine-Gordon equation, Physical Review D 11 (1975) 3026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [131] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Coleman, Quantum sine-Gordon equation as the massive Thirring model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 11 (1975) 2088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [132] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gross, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Neveu, Dynamical symmetry breaking in asymptotically free field theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 10 (1974)  3235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [133] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Adler, The Axial-Vector Vertex in Spinor Electrodynamics, Physical Review 177 (1969) 2426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [134] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, The PCAC Puzzle: π0 → γγ in the σ-model, Nuovo Cimento A 60 (1969) 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [135] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hughes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhang, Topological field theory of time-reversal invariant insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B  78 (2008) 195424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [136] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schwinger, Field Theory Commutators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 3 (1959) 296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [137] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Some Properties of the ( ¯ψψ)2 Model in Two Dimensions, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 142 (1978) 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [138] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Su, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schrieffer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Heeger, Solitons in Polyacetylene, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 42 (1979) 1698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [139] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schrieffer, Solitons with fermion number 1/2 in condensed matter and relativistic field theories,  Nuclear Physics B 190 (1981) 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [140] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rebbi, Solitons with fermion number 1/2, Physical Review D 13 (1976) 3398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [141] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldstone, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Fractional Quantum Numbers on Solitons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 47 (1981) 986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [142] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Takayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lin-Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Maki, Continuum model for solitons in polyacetylene, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 21 (1980)  2388–2393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [143] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Campbell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bishop, Solitons in polyacetylene and relativistic-field-theory models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 24  (1981) 4859–4862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [144] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Campbell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bishop, Soliton excitations in polyacetylene and relativistic field theory models, Nuclear Physics  B 200 (1982) 297–328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [145] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hirsch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Effect of quantum fluctuations on the peierls instability: A monte carlo study, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 49 (1982) 402–405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [146] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hirsch, Phase diagram of one-dimensional electron-phonon systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' The Su-Schrieffer-Heeger  model, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 27 (1983) 1680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [147] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Atiyah, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Patodi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Singer, Spectral Asymmetry and Riemaniann Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cambridge  Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 77 (1975) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [148] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [149] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 1st Edition, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Volume 3 of Course  of Theoretical Physics, Pergamon Press, Oxford, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [150] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Weinberg, The Quantum Theory Of Fields, 1st Edition, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I-III, Cambridge University Press, Cambridge,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [151] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yang, Dirac monopole without strings: Monopole harmonics, Nuclear Physics B 107 (1976) 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [152] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kadanoff, Disorder variables and para-fermions in two-dimensional statistical mechanics, Nu-  clear Physics B 170 (1980) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [153] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Leinaas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Myrheim, On the theory of identical particles, Il Nuovo Cimento B (1971-1996) 37 (1977) 1–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [154] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Laidlaw, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys-  ical Review D 3 (1971) 1375–1378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [155] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aharonov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bohm, Significance of Electromagnetic Potentials in the Quantum Theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 115 (1959)  485–491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [156] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Magnetic Flux, Angular Momentum, and Statistics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 48 (1982) 1144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [157] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Quantum Mechanics of Fractional-Spin Particles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 49 (1982) 957–959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [158] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldhaber, Connection of Spin and Statistics for Charge-Monopole Composites, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 36 (1976)  1122–1125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [159] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Finkelstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rubinstein, Connection between Spin, Statistics, and Kinks, Journal of Mathematical Physics 9  (1968) 1762–1779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [160] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Skyrme, A Unified Theory of Mesons and Baryons, Nuclear Physics 31 (1962) 556–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [161] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wu, General Theory for Quantum Statistics in Two Dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 52 (1984) 2103–2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [162] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zee, Linking numbers, spin, and statistics of solitons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 51 (1983) 2250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [163] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Deser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Templeton, Three-Dimensional Massive Gauge Theories, Physical Review Letters 48  (1982) 975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [164] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Deser, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' templeton, Topologically Massive Gauge Theories, Annals of Physics 140 (1982) 372–411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [165] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Quantum Field Theory and the Jones Polynomial, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 121 (1989) 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [166] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Horowitz, Exactly Soluble Diffeomorphism Invaraint Theories, Communications in Mathematical Physics  125 (1989) 417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [167] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Grundberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hansson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Karlhede, On Polyakov’s Spin Factors, Nuclear Physics B 347 (1990) 420–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [168] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, Fermi-Bose Transmutations Induced by Gauge Fields, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A 3 (1988) 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [169] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Kitaev,  Fault-tolerant  quantum  computation  by  anyons,  Annals  of  Physics  303  (2003)  2,  (arXiv:quant-ph/9707021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [170] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Das Sarma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freedman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Simon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stern, Non-abelian anyons and topological quantum  computation, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 80 (2008) 1083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [171] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Castro Neto, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Guinea, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Peres, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Novoselov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Geim, The electronic properties of graphene,  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 81 (2009) 109–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [172] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhang, Topological insulators and superconductors, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 83 (2011) 1057.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [173] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Susskind, Lattice Fermions, Physical Review D 16 (1977) 3031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [174] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nielsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ninomiya, Absence of neutrinos on a lattice (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Proof by homotopy theory, Nuclear Physics B  119  185 (1981) 20–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [175] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nielsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ninomiya, Absence of neutrinos on a lattice (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Intiuitive topological proof, Nuclear Physics  B 193 (1981) 173–194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [176] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nielsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ninomiya, The Adler-Bell-Jackiw Anomaly and Weyl fermions in a Crystal, Physics Letters B  130 (1983) 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [177] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Friedan, A Proof of the Nielsen-Ninomiya Theorem, Communications in Mathematical Physics 85 (1982)  481–490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [178] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Semenoff, Condensed-Matter Simulation of a Three-Dimensional Anomaly, Physical Review Letters 53  (1984) 2449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [179] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thouless, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kohmoto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nightingale, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' den Nijs, Quantized Hall Conductance in a Two-Dimensional  Periodic Potential, Physical Review Letters 49 (1982) 405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [180] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Marston, Large-N limit of the Heisenberg-Hubbard Model: Implications for High-Tc Supercon-  ductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 37 (1988) 3774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [181] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zee, Chiral spin states and superconductivity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 39 (1989) 11413–11423.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [182] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bloch, ¨Uber der quantenmechanik der elektronen in kristallgittern, Zeitschrift f¨ur physik 52 (1929) 555–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [183] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kittel, Introduction to Solid State Physics, 1st Edition, John Wiley & Sons, New York, New York, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [184] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ashcroft, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mermin, Solid State Physics, Holt, Reinhart and Winston, New York, New York, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [185] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bradlyn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Elcoro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vergniory, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Felser, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aroyo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bernevig, Topological  quantum chemistry, Nature 547 (2017) 298–305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [186] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cano, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bradlyn, Band Representations and Topological Quantum Chemistry, Annual Review of Condensed  Matter Physics 12 (2021) 225–246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [187] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kohmoto, Topological invariant and the quantization of the Hall conductance, Annals of Physics 160 (1985)  343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [188] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schnyder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Furusaki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='Ludwig, Classification of topological insulators and superconductors  in three spatial dimensions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 78 (2008) 195125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [189] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kitaev, Periodic table for topological insulators and superconductors, in: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Feigelman (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), Advances in  Theoretical Physics: Landau Memorial Conference, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 1134, American Institute of Physics, AIP Conference  Proceedings, College Park, Maryland, 2009, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [190] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moore, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Balents, Topological invariants of time-reversal-invariant band structures, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 75 (2007)  121306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [191] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kohmoto, Quantized Hall effect and geometric localization of electrons on lattices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B  35 (1987) 6017–6023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [192] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,  Physical Review B 14 (1876) 2239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [193] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Martin, Measurements and Correlation Functions, Gordon & Breach, Philadelphia, Pennsylvania, 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [194] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nozi`eres, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pines, The Theory of Quantum Liquids: Normal Fermi Liquids, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Benjamin Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', New York,  New York, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [195] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mudry, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, Masses in graphene-like two-dimensional electronic systems: Topo-  logical defects in order parameters and their fractional exchange statistics, Physical Review B 80 (2009) 205319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [196] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, Model for a Quantum Hall Effect without Landau Levels: Condensed-Matter Realization of the  “Parity Anomaly”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 61 (1988) 2015–2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [197] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Symmetry-Protected Topological Orders in Interacting Bosonic Sys-  tems, Science 338 (2012) 1604–1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [198] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Symmetry-protected topological phases of quantum matter, Annual Review of Condensed Matter  Physics 6 (2015) 299–324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [199] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Fermion path integrals and topological phases, Reviews of Modern Physics 88 (2016) 035001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [200] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Redlich, Gauge Noninvariance and Parity Nonconservation of Three-Dimensional Fermions, Physical Re-  view Letters 52 (1984) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [201] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Redlich, Parity violation and gauge noninvariance of the effective gauge field action in three dimensions,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 29 (1984) 2366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [202] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fujikawa, Path-Integral Measure for gauge-Invariant Fermion Theories, Physical Review Letters 42 (1979)  1195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [203] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gamboa Sarav´ı, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Muschietti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schaposnik, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Solomin, Chiral Symmetry and Functional  Integral, Annals of Physics 157 (1984) 360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [204] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Volovik, Zeros in the fermion spectrum in superfluid systems as diabolical points, Journal of Experimental  and Theoretical Physics Letters 46 (1987) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [205] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yakovenko, Chern-Simons Terms and n Field in Haldane’s Model for the Quantum Hall Effect without  Landau Levels, Physical Review Letters 65 (1990) 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [206] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Golterman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jansen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kaplan, Chern-Simons currents and chiral fermions on the lattice, Physics  Letters B 301 (1993) 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [207] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhang, Topological quantization of the spin Hall effect in two-dimensional paramagnetic  semiconductors, Physical Review B 74 (2006) 085308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [208] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Seiberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, A Duality Web in 2+1 Dimensions and Condensed Matter Physics,  Annals of Physics 374 (2016) 395 – 433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [209] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, Topological insulators with inversion symmetry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 76 (2007) 045302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [210] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mele, Z2 Topological Order and the Quantum Spin Hall Effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 95 (2005)  146802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [211] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bernevig, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hughes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhang, Quantum Spin Hall Effect and Topological Phase Transition in HgTe  Quantum Wells, Science 314 (2006) 1757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [212] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gooth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bradlyn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Honnali, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schindler, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Noky, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shekhar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Bernevig, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Felser, Axionic charge-density wave in the Weyl semimetal (TaSe4)2I, Nature 575 (2019) 315–319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [213] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Essin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vanderbilt, Magnetic Polarizability and Axion Electrodynamics in Crystalline Insu-  lators, Physical Review Letters 102 (2009) 146805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [214] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hosur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vishwanath, Chiral topological insulators, superconductors, and other competing orders in  three diemsnions, Physical Review B 81 (2010) 045120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  120  [215] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Two applications of axion electrodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 58 (1987) 1799–1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [216] Michael E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Peskin and Daniel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schroeder, An Introduction to Quantum Field Theory, The Advanced Book  Program, Perseus Books Publishing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Reading, Massachusetts, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [217] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Dyons of charge eθ/2π, Physics Letters B 86 (1979) 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [218] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Callan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Harvey, Anomalies and Fermion Zero Modes on Strings and Domain Walls, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 250  (1985) 427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [219] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hsieh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Xia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Qian, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wray, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dil, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Meier, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Osterwalder, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Patthey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Checkelsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ong,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fedorov, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bansil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Grauer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hor, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cava, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hasan, A tunable topological insulator  in the spin helical Dirac transport regime, Nature 460 (2009) 1101–1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [220] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' von Klitzing, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dorda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pepper, New Method for High-Accuracy Determination of the Fine-Structure  Constant Based on Quantized Hall Resistance, Physical Review Letters 45 (1980) 494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [221] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stormer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gossard, Two-Dimensional Magnetotransport in the Extreme Quantum Limit,  Physical Review Letters 48 (1982) 1559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [222] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Du, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Skachko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Duerr, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Luican, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Andrei, Fractional quantum Hall effect and insulating phase of Dirac  electrons in graphene, Nature 462 (2009) 192–195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [223] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zibrov, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kometter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Spanton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Watanabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zaletel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Young,  Tunable interacting composite fermion phases in a half-filled bilayer-graphene Landau level, Nature 549 (2017)  360–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [224] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Landau, Diamagnetismus der Metalle, Zeitschrift f¨ur Physik 64 (1930) 629–637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [225] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Niu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thouless, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wu, Quantized Hall conductance as a topological invariant, Physical Review B 31  (1985) 3372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [226] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Laughlin, Quantized Hall conductivity in two dimensions, Physical Review B 23 (1981) 5632–5633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [227] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Levine, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Libby, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pruisken, Electron Delocalization by a Magnetic Field in Two Dimensions, Physical  Review Letters 51 (1983) 1915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [228] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pruisken, On localization in the theory of the quantized hall effect: A two-dimensional realization of the  θ-vacuum, Nuclear Physics B 235 (1984) 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [229] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kievelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ro˘cek, Consequences of Gauge Invariance for Fractionally Charged Quasi-Particles, Physics  Letters B 85 (156).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [230] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jain, Incompressible quantum Hall states, Physical Review B 40 (1989) 8079.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [231] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jain, Composite-fermion approach for the fractional quantum Hall effect, Physical Review Letters 63 (1989)  199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [232] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, Theory of the half-filled Landau level, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 47 (1993) 7312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [233] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nakamura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Liang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gardner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Manfra, Direct observation of anyonic braiding statistics, Nature  Physics 16 (2020) 931–936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [234] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, Statistics of Quasiparticles and the Hierarchy of Fractional Quantized Hall States, Physical Review  Letters 52 (1984) 1583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [235] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Arovas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schrieffer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Fractional Statistics and the Quantum Hall Effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 53  (1984) 722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [236] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fr¨ohlich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zee, Large-scale physics of the quantum Hall Fluid, Nuclear Physics B 364 (1991) 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [237] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pasquier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, A dipole interpretation of the ν = 1/2 state, Nuclear Physics B 516 (1998) 719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [238] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, Lowest-Landau-level theory of the quantum Hall effect: The Fermi-liquid-like state of bosons at filling  factor one, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 58 (1998) 16262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [239] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Susskind, The Quantum Hall Fluid and Non-Commutative Chern-Simons Theory, unpublished (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  arXiv:arXiv:hep-th/0101029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [240] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polychronakos, Quantum Hall States as matrix Chern-Simons theory, Journal of High Eenergy Physics 2001  (2001) 011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [241] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jejjala, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Leigh, Non-commutative Chern-Simons for the Quantum Hall system and Duality,  Nuclear Physics B 642 (2002) 483–500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [242] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Noncommutative field theory and composite Fermi liquids in some quantum Hall systems,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 102 (2020) 205126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [243] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Lowest Landau level theory of the bosonic Jain states, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 105 (2022) 075130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [244] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hansson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, Effective-Field-Theory Model for the Fractional Quantum Hall Effect,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 62 (1989) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [245] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rezayi, Periodic Laughlin-Jastrow wave functions for the fractional quantized Hall effect,  Physical Review B 31 (1985) 2529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [246] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Niu, Ground-state degeneracy of the fractional quantum Hall states in the presence of a random  potential and on high-genus Riemann surfaces, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 41 (1990) 9377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [247] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L´opez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Fractional quantum Hall effect and Chern-Simons gauge theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 44 (1991)  5246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [248] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L´opez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Response Functions and Spectrum of Collective Excitations of Fractional Quantum Hall  Effect Systems, Physical Review B 47 (1993) 7080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [249] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kohn, Cyclotron Resonance and de Haas-van Alphen Oscillations of an Interacting Electron Gas, Physical  Review 123 (1961) 1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [250] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L´opez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Universal structure of the edge states of the fractional quantum Hall states, Physical Review  B 59 (1999) 15323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [251] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zee, Classification of Abelian quantum Hall states and matrix formulation of topological fluids,  Physical Review B 46 (1992) 2290.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [252] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Topological Orders and Edge Excitations in Fractional Quantum Hall States, Advances in Physics 44  (1995) 405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [253] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' de Picciotto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Reznikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Heiblum, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Umansky, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bunin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mahalu, Direct observation of a fractional  charge, Nature 389 (1997) 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [254] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Saminadayar, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Glattli, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Etienne, Observation of the e/3 Fractionally Charged Laughlin Quasipar-  ticle, Physical Review Letters 79 (1997) 2526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [255] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hertz, Quantum critical phenomena, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 14 (1976) 1165–1184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [256] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Millis, Effect of a nonzero temperature on quantum critical points in itinerant fermion systems, Physical  121  Review B 48 (1993) 7183–7196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [257] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Varma, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Littlewood, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schmitt-Rink, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Abrahams, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ruckenstein, Phenomenology of the normal  state of Cu-O high-temperature superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 63 (1989) 1996–1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [258] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polchinski, Low-energy dynamics of the spinon-gauge system, Nuclear Physics B 422 (1994) 617–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [259] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shankar, Renormalization-group approach to interacting fermions, Reviews of Modern Physics 66 (1994) 129–  192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [260] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polchinski, Effective Field Theory and the Fermi Surface, in: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Harvey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polchinski (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), Theoretical  Advanced Study Institute (TASI 92): From Black Holes and Strings to Perticles, World-Scientific Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=',  Singapore, 1993, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 235–276, proceedings of TASI 1992, Boulder (Colorado, USA), June 1-26 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [261] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Renormalization group approach to low temperature properties of a non-Fermi liquid metal,  Nuclear Physics B 430 (1994) 534–562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [262] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Non-Fermi liquid fixed point in 2 + 1 dimensions, Nuclear Physics B 417 (1994) 359–373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [263] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Maldacena, Large N limit of superconformal field theories and supergravity, Advances in Theoretical and  Mathematical Physics 2 (1998) 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [264] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Faulkner, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Iqbal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' McGreevy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vegh, Strange Metal Transport Realized by Gauge/Gravity Duality,  Science 329 (2010) 1043–1047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [265] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fubini, Vertex Operators and the Quantum Hall Effect, Modern Physics Letters A 6 (1991) 347–358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [266] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moore, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, Non-Abelions in the Fractional Quantum Hall Effect, Nuclear Physics B 360 (1991) 362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [267] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moore, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Seiberg, Taming the Conformal Zoo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 220 (1989) 422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [268] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moore, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Seiberg, Classical and Quantum Conformal Field Theory, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 123 (1989) 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [269] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kitaev, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, Simulation of topological field theories by quantum computers, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 227 (2002) 587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [270] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freedman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kitaev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Larsen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, Topological Quantum Computation, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 227  (2002) 605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [271] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Preskill, Topological Quantum Computation, Lecture Notes for Physics 219: Quantum Computation, chapter  9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' unpublished;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Caltech (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  URL http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='edu/~preskill/ph219/topological.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='pdf  [272] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Willett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eisenstein, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' St¨ormer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gossard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' English, Observation of an even-  denominator quantum number in the fractional quantum Hall effect, Physical Review Letters 59 (1987) 1776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [273] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eisenstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Willett, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' St¨ormer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gossard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' English, Collapse of the Even-  Denominator Fractional Quantum Hall Effcet in Tilted Fields, Physical Review Letters 61 (1988) 997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [274] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Xia, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' St¨ormer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vicente, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Adams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sullivan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Baldwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Experimental studies of the fractional quantum Hall effect in the first excited Landau level,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 77 (2008) 075307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [275] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Greiter, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Paired hall State at Half Filling, Physical Review Letters 66 (1991) 3205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [276] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Greiter, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, Paired Hall States, Nuclear Physics B 374 (1992) 567–614.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [277] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Non-Abelian statistics in the Fractional quantum Hall States, Physical Review Letters 66 (1991) 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [278] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kohn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Luttinger, New Mechanism for Superconductivity, Physical Review Letters 15 (1965) 524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [279] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chubukov, Kohn-Luttinger effect and the instability of a two-dimensional repulsive Fermi liquid at T = 0,  Physical Review B 48 (1993) 1097.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [280] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Raghu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, Superconductivity from repulsive interactions in the two-dimensional electron gas,  Physical Review B 83 (2011) 094518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [281] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Park, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Melik-Alaverdian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bonesteel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jain, Possibility of p-wave pairing of composite fermions at  ν = 1/2, Physical Review B 58 (1998) R10167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [282] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, Theory of the quantized Hall conductance, hevetica Physica Acta 56 (1983) 75–102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [283] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Morf, Transition from quantum Hall to compressible states in the second Landau level: New light on the  ν = 5/2 enigma, Physical Review Letters 80 (1998) 1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [284] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, 2n-quasihole states realize 2n−1-dimensional spinor braiding statistics in paired quantum  Hall states, Nuclear Physics B 479 (1996) 529–553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [285] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Green, Paired states of fermions in two dimensions with breaking of parity and time-reversal symme-  tries and the fractional quantum Hall effect, Physical Review B 61 (2000) 10267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [286] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Volovik, An analog of the quantum Hall effect in a superfluid 3He film, Soviet Physics JETP 67 (1988)  1804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [287] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jackiw, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rossi, Zero modes of the vortex-fermion system, Nuclear Physics B 190 (1981) 681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [288] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Weinberg, Index calculations for the fermion-vortex system, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D 24 (1981) 2669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [289] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nishida, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Santos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, Topological superconductors as nonrelativistic limits of Jackiw-Rossi and  Jackiw-Rebbi models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 82 (2010) 144513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [290] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ivanov, Non-abelian statistics of half-quantum vortices in p-wave superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 86  (2001) 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [291] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chung, Fusion rules and vortices in px + ipy superconductors, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 73 (2006) 014505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [292] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rezayi, Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited  Landau level, Physical Review B 59 (1999) 8084.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [293] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fateev, Nonlocal(parafermion) currents in two-dimensional conformal quantum field theory and  self-dual critical points in ZN-symmetric statistical systems, Soviet Physics JETP 62 (1985) 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [294] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Alcaraz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K¨oberle, Hidden parafermions in Z(N) theories, Physical Review D 24 (1981) 1652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [295] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' St¨ormer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Baldwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Fractional Quantum Hall Effect of  Composite Fermions, Physical Review B 90 (2003) 016801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [296] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Xia, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vicente, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Adams, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sullivan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stormer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsui, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Baldwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Electron Correlation in the Second Landau Level: A Competition Between Many Nearly  Degenerate Quantum Phases, Physical Review Letters 93 (2004) 176809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [297] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rezayi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, Non-Abelian quantized Hall states of electrons at filling factors 12/5 and 13/5 in the first  excited Landau level, Physical Review B 79 (2009) 075306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [298] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tsvelik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wilczek, A Chern-Simons effective field theory for the Pfaffian quantum Hall  state, Nuclear Physics B 516 (1998) 704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [299] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schoutens, Landau-Ginzburg theories for non-abelian quantum Hall states, Nuclear  122  Physics B 546 (1999) 711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [300] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Barkeshli, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Effective field theory and projective construction for Zk parafermion fractional quantum  Hall states, Physical Review B 81 (2010) 155302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [301] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sohal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Landau-Gizburg theories of non-Abelian quantum Hall states from non-  Abelian bosonization, Physical Review B 100 (2019) 115111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [302] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sohal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Non-Abelian fermionization and the landscape of quantum Hall phases, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 102 (2020) 195151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [303] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sohal, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Composite particle construction of the Fibonacci fractional quantum Hall state,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 103 (2021) 235118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [304] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, Field theory methods and quantum critical phenomena, in: Strings, Fields and Critical Phenomena,  Proceedings of the Les Houches Summer School 1988, Session XLIX, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Br´ezin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin editors, North-  Holland, Amsterdam, the Netherlands, 1990, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [305] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Belavin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polyakov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zamolodchikov, Infinite Conformal Symmetry in two-Dimensional Quan-  tum Field Theory, Nuclear Physics B 241 (1984) 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [306] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Affleck, Universal Term in the Free Energy at a Critical Point and the Conformal Anomaly, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 56  (1986) 746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [307] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bl¨ote, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cardy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nightingale, Conformal Invariance, the Central Charge, and Universal Finite-Size  Amplitudes at Criticality, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 56 (1986) 742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [308] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Granger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Eisenstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Reno, Observation of Chiral Heat Transport in the Quantum Hall Regime,  Physical Review Letters 102 (2009) 086803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [309] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bid, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ofek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Inoue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Heiblum, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Umansky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mahalu, Observation of neutral modes in  the fractional quantum Hall regime, Nature 466 (2010) 585–590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [310] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, Transmission through barriers and resonant tunneling in an interacting one-dimensional  electron gas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 46 (1992) 15233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [311] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Distinct universal conductances in tunneling to quantum Hall states: The role of contacts,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 56 (1997) 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [312] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Observation of Chiral Luttinger Behavior in Electron Tunneling into  Fractional Quantum Hall Edges, Physical Review Letters 77 (1996) 2538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [313] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chang, Chiral Luttinger liquids at the fractional quantum Hall edge, Reviews of Modern Physics 75 (2004)  1449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [314] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Cohen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Samuelson, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Taniguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Watanabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zaletel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Young, Univer-  sal chiral Luttinger liquid behavior in a graphene fractional quantum Hall point contact, unpublished (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  arXiv:\\protect\\vrulewidth0pt\\protect\\href{http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='01374}{arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='01374}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [315] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hashisaka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jonckheere, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Akiho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sasaki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rech, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' martin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Muraki, Andreev reflection of fractional  quantum Hall quasiparticles, Nature Communications 12 (2021) 2794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [316] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sandler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Noise measurements and fractional charge in fractional quantum Hall  liquids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 59 (1999) 12521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [317] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Milliken, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Umbach, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Webb, Indicatiosn of a Luttinger liquid in the fractional quantum Hall regime,  Solid State Communications 97 (1996) 309–313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [318] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Roddaro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pellegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Beltram, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='iorgioBiasiol, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sorba, Interedge Strong-to-Weak Scattering Evolution  at a Constriction in the Fractional Quantum Hall Regime, Physical Review Letters 93 (2004) 046801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [319] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Roddaro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pellegrini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Beltram, Quasi-particle tunneling at a constriction in a fractional quantum Hall state,  Solid State Communications 131 (2004) 565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [320] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Anderson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yuval, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hammann, Exact results in the Kondo problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Scaling theory, qualitatively  correct solution, and some new results on one-dimensional classical statistical models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 1 (1970)  4464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [321] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Read, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Newns, On the solution of the Coqblin-Schreiffer Hamiltonian by the large-N expansion technique,  Journal of Physics C: Solid State Physics 16 (1983) 3273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [322] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fendley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ludwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Saleur, Exact Conductance through Point Contacts in the ν = 1/3 Fractional  Quantum Hall Effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 74 (1995) 3005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [323] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Miller, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Radu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zumb¨uhl, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Levenson-Falk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kastner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Marcus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  West, Fractional quantum Hall effect in a quantum point contact at filling fraction 5/2, Nature Physics 3 (2007)  561.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [324] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Radu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Miller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Marcus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kastner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Baldwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Quasi-Particle  Properties from Tunneling in the ν = 5/2 Fractional Quantum Hall State, Science 320 (2008) 899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [325] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kane, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, Nonequilibrium noise and fractional charge in the quantum Hall effect, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 72  (1994) 724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [326] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freed, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Nonequilibrium quantum noise in chiral Luttinger liquids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 53  (1996) 4033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [327] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fendley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ludwig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Saleur, Exact Nonequilibrium dc Shot Noise in Luttinger Liquids and Fractional Quan-  tum Hall Devices, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 75 (1995) 2196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [328] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dolev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Heiblum, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Umansky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mahalu, Observation of a quarter of an electron charge at the  ν = 5/2 quantum Hall state, Nature 452 (2008) 829.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [329] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chamon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Freed, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sondhi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wen, Two point-contact interferometer for quantum  Hall systems, Physical Review B 55 (1997) 2331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [330] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Camino, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zhou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, Realization of a Laughlin quasiparticle interferometer: Observation of  fractional statistics, Physical Review B 72 (2005) 075342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [331] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stern, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Neder, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rosenow, Theory of the Fabry-P´erot quantum Hall interferometer, Physical  Review B 83 (2011) 155440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [332] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bonderson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shtengel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Slingerland, Probing non-abelian statistics with quasiparticle interferometry,  Physical Review Letters 97 (2006) 016401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [333] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stern, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, Proposed Experiments to Probe the Non-Abelian ν = 5/2 Quantum Hall State, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 96 (2006) 016802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [334] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Willett, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Measurement of filling factor 5/2 quasiparticle interference with obser-  vation of charge e/4 and e/2 period oscillations, Proceedings of the National Academy of Sciences 106 (2009)  123  8853–8858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [335] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Willett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shtengel, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' West, Magnetic-Field-Tuned Aharonov-Bohm Oscilla-  tions and Evidence for Non-Abelian Anyons at ν = 5/2, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 111 (2013) 186401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [336] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Willett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shtengel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nayak, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pfeiffer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chung, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Peabody, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Baldwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  West, Interference measurements of non-Abelian e/4 & Abelian e/2 quasiparticle braiding, (unpublished) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  arXiv:\\protect\\vrulewidth0pt\\protect\\href{http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='org/abs/1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10248}{arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='10248}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [337] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Nelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kosterlitz, Universal Jump in the Superfluid Density of Two-Dimensional Superfluids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 39 (1977) 1201–1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [338] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Thomas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Stone, Nature of the phase transition in a non-linear O(2)3 model, Nuclear Physics B 144 (2)  (1978) 513–524.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [339] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Peskin, Mandelstam-’t Hooft duality in abelian lattice models, Annals of Physics 113 (1978) 122–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [340] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Dasgupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, Phase Transition in a Lattice Model of Superconductivity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 47 (1981)  1556–1560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [341] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Halperin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lubensky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' keng Ma, First-Order Phase Transitions in Superconductors and Smectic-A  Liquid Crystals, Physical Review Letters 32 (1974) 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [342] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Moshe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Zinn-Justin, Quantum field theory in the large N limit: a review, Physics Reports 385 (2003) 69–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [343] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Loop Models, Modular Invariance, and Three Dimensional Bosonization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B  97 (2018) 195112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [344] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schaposnik, The Fermion-Boson Mapping in Three Dimensional Quantum Field Theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 338 (1994) 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [345] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Burgess, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Quevedo, Bosonization as duality, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 421 (1994) 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [346] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Hughes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Effective field theories for topological insulators by functional  bosonization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 87 (2013) 085132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [347] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kvorning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Ryu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Effective hydrodynamic field theory and condensation picture of  topological insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 93 (2016) 155122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [348] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Giombi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Minwalla, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Prakash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Trivedi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wadia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yin, Chern-Simons theory with vector fermion  matter, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' C 72 (2012) 1–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [349] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Jain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Minwalla, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yokoyama, Chern Simons duality with a fundamental boson and fermion, Journal of High  Eenergy Physics 2013 (2013) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [350] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aharony, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gur-Ari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yacoby, d = 3 bosonic vector models coupled to Chern-Simons gauge theories, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 03 (2012) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [351] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aharony, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Gur-Ari, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Yacoby, Correlation functions of large N Chern-Simons-Matter theories and bosoniza-  tion in three dimensions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 2012 (2012) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [352] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Aharony, Baryons, monopoles and dualities in Chern-Simons-matter theories, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 02 (2016)  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [353] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Seiberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Witten, Gapped boundary phases of topological insulators via weak coupling, Progress of Theoret-  ical and Experimental Physics 2016 (2016) ptw083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [354] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Karch, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tong, Particle-Vortex Duality from 3D Bosonization, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' X 6 (2016) 031043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [355] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Son, Is the composite fermion a Dirac particle?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' X 5 (2015) 031027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [356] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Metlitski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vishwanath, Particle-vortex duality of two-dimensional dirac fermion from electric-magnetic  duality of three-dimensional topological insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 93 (2016) 245151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [357] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Dual Dirac liquid on the surface of the electron topological insulator, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' X 5 (2015)  041031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [358] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Son, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Raghu, Exact Boson-Fermion Duality on a 3D Euclidean Lattice, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 120 (2018) 016602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [359] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mross, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Alicea, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Motrunich, Explicit Derivation of Duality between a Free Dirac Cone and Quantum  Electrodynamics in (2 + 1) Dimensions, Physical Review Letters 117 (2016) 016802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [360] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Simmons-Duffin, The Conformal Bootstrap, in: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Polchinski, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vieira, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' DeWolfe (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), TASI 2015: Pro-  ceedings of the 2015 Theoretical Advanced Study Institute in Elementary Particle Physics, New Frontiers in Fields  and Strings, World Scientific, Singapore, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 1–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [361] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Dirac composite fermions and emergent reflection symmetry about even-denominator  filling fractions, Physical Review B 98 (2018) 165137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [362] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Bohm, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Pines, A Collective Description of Electron Interactions: III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Coulomb Interactions in a Degenerate  Electron Gas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 92 (1953) 609–625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [363] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Haldane, Luttinger’s Theorem and Bosonization of the Fermi Surface, in: R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Broglia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schrieffer  (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' ), Proceedings of the International School of Physics “Enrico Fermi”, Course CXXI “Perspectives in Many-  Particle Physics”, International School of Physics “Enrico Fermi”, North-Holland, Amsterdam, The Netherlands,  1994, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 5–29, arXiv:cond-mat/0505529.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [364] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Houghton, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Marston, Bosonization and fermion liquids in dimensions greater than one, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 48  (1993) 7790–7808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [365] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Castro Neto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Bosonization of the Low Energy Excitations of Fermi Liquids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 72  (1994) 1393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [366] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Castro Neto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Bosonization of Fermi Liquids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 49 (1994) 10877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [367] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Houghton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kwon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Marston, Multidimensional bosonization, Advances in Physics 49 (2000) 141–  228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [368] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Delacr´etaz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Du, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mehta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Son, Nonlinear bosonization of Fermi surfaces: The method of  coadjoint orbits, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Research 4 (2022) 033131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [369] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Shi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Goldman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Else, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Gifts from anomalies: Exact results for Landau phase transitions in  metals, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 13 (2022) 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [370] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Vishwanath, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Balents, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sachdev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fisher, Deconfined Quantum Critical Points, Science  303 (2004) 1490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [371] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fitzpatrick, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kachru, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kaplan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Raghu, Non-Fermi-liquid fixed point in a Wilsonian theory of quantum  critical metals, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 88 (2013) 125116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [372] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Metlitski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mross, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Sachdev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Senthil, Cooper pairing in non-Fermi liquids, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' B 91 (2015)  115111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  124  [373] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Oganesyan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Quantum theory of the nematic Fermi liquid, Physical Review B 64  (2001) 195109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [374] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lawler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Barci, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fern´andez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Oxman, Non-perturbative behavior of the quantum phase  transition to a nematic Fermi liquid, Physical Review B 73 (2006) 085101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [375] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lawler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, Local quantum criticality at the nematic quantum phase transition, Physical Review B  75 (2007) 0333304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [376] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lederer, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schattner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Berg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, Enhancement of Superconductivity near a Nematic Quantum  Critical Point, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' 114 (2015) 097001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [377] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Emery, Electronic liquid-crystal phases of a doped Mott insulator, Nature 393  (1998) 550–553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [378] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Lawler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' andAndrew P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Mackenzie, Nematic Fermi Fluids in Condensed  Matter Physics, Annual Review of Condensed Matter Physics 1 (2010) 153–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [379] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fradkin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Kivelson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Tranquada, Colloquium: Theory of intertwined orders in high temperature  superconductors, Reviews of Modern Physics 87 (2015) 457–482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  [380] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Fernandes, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Orth, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content=' Schmalian, Intertwined Vestigial Order in Quantum Materials: Nematicity and  Beyond, Annual Review of Condensed Matter Physics 10 (2019) 133–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
+page_content='  125' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dFQT4oBgHgl3EQfDjUj/content/2301.13234v1.pdf'}
diff --git a/2tFKT4oBgHgl3EQf7y43/vector_store/index.faiss b/2tFKT4oBgHgl3EQf7y43/vector_store/index.faiss
new file mode 100644
index 0000000000000000000000000000000000000000..6438e676f8d08645c330a0265ed9d67e9abf4df6
--- /dev/null
+++ b/2tFKT4oBgHgl3EQf7y43/vector_store/index.faiss
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0cea52c9bf2245480142f3747ad82d1520896e5c01b54cc7e6e13d038f7b2bf0
+size 2621485
diff --git a/39E0T4oBgHgl3EQfvAGt/content/tmp_files/2301.02613v1.pdf.txt b/39E0T4oBgHgl3EQfvAGt/content/tmp_files/2301.02613v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..15783a14c94888ed7a4cd1e0847848ac2dbb6019
--- /dev/null
+++ b/39E0T4oBgHgl3EQfvAGt/content/tmp_files/2301.02613v1.pdf.txt
@@ -0,0 +1,1843 @@
+1
+Learning Deep MRI Reconstruction Models from
+Scratch in Low-Data Regimes
+Salman Ul Hassan Dar, S¸ aban ¨Ozt¨urk, Muzaffer ¨Ozbey, and Tolga C¸ ukur∗
+Abstract— Magnetic resonance imaging (MRI) is an es-
+sential diagnostic tool that suffers from prolonged scan
+times. Reconstruction methods can alleviate this limitation
+by recovering clinically usable images from accelerated ac-
+quisitions. In particular, learning-based methods promise
+performance leaps by employing deep neural networks as
+data-driven priors. A powerful approach uses scan-specific
+(SS) priors that leverage information regarding the underly-
+ing physical signal model for reconstruction. SS priors are
+learned on each individual test scan without the need for
+a training dataset, albeit they suffer from computationally
+burdening inference with nonlinear networks. An alterna-
+tive approach uses scan-general (SG) priors that instead
+leverage information regarding the latent features of MRI
+images for reconstruction. SG priors are frozen at test
+time for efficiency, albeit they require learning from a large
+training dataset. Here, we introduce a novel parallel-stream
+fusion model (PSFNet) that synergistically fuses SS and
+SG priors for performant MRI reconstruction in low-data
+regimes, while maintaining competitive inference times to
+SG methods. PSFNet implements its SG prior based on a
+nonlinear network, yet it forms its SS prior based on a linear
+network to maintain efficiency. A pervasive framework for
+combining multiple priors in MRI reconstruction is algo-
+rithmic unrolling that uses serially alternated projections,
+causing error propagation under low-data regimes. To alle-
+viate error propagation, PSFNet combines its SS and SG
+priors via a novel parallel-stream architecture with learn-
+able fusion parameters. Demonstrations are performed on
+multi-coil brain MRI for varying amounts of training data.
+PSFNet outperforms SG methods in low-data regimes, and
+surpasses SS methods with few tens of training samples.
+In both supervised and unsupervised setups, PSFNet re-
+quires an order of magnitude lower samples compared to
+SG methods, and enables an order of magnitude faster
+inference compared to SS methods. Thus, the proposed
+model improves deep MRI reconstruction with elevated
+learning and computational efficiency.
+Index Terms— image reconstruction, deep learning, scan
+specific, scan general, low data, supervised, unsupervised.
+This work was supported in part by a TUBA GEBIP 2015 fellowship,
+by a BAGEP 2017 fellowship, and by a TUBITAK 121E488 grant awarded
+to T. C¸ ukur.
+S. UH. Dar, S¸ .
+¨Ozt¨urk, M.
+¨Ozbey, and T. C¸ ukur are with the
+Department of Electrical and Electronics Engineering, and the Na-
+tional Magnetic Resonance Research Center, Bilkent University,
+Ankara, Turkey (e-mails: {salman,muzaffer,cukur}@ee.bilkent.edu.tr,
+saban.ozturk@amasya.edu.tr). S¸ .
+¨Ozt¨urk is also with the Amasya
+University, Amasya, Turkey.
+I. INTRODUCTION
+The unparalleled soft-tissue contrast and non-invasiveness
+of MRI render it a preferred modality in many diagnostic
+applications [1], [2], and downstream imaging tasks such
+as classification [3] and segmentation [4], [5]. However, the
+adverse effects of low spin polarization at mainstream field
+strengths on the signal-to-noise ratio make it slower against
+alternate modalities such as CT [6]. Since long scan durations
+inevitably constrain clinical utility, there is an ever-growing
+interest in accelerated MRI methods to improve scan efficiency.
+Accelerated MRI involves an ill-posed inverse problem with the
+aim of mapping undersampled acquisitions in k-space to high-
+quality images corresponding to fully-sampled acquisitions.
+Conventional frameworks for solving this problem rely on
+parallel imaging (PI) capabilities of receive coil arrays [7],
+[8], in conjunction with hand-constructed MRI priors [9], [10].
+A joint objective is iteratively optimized comprising a data-
+consistency (DC) term based on the physical signal model,
+and a regularization term that enforces the MRI prior [9]. The
+physical model constrains reconstructed data to be consistent
+with acquired data while considering coil sensitivities and
+undersampling patterns [11]. Meanwhile, the regularization
+term, often based on a linear transform where data are assumed
+to be compressible [9], introduces suboptimality when the
+distribution of MRI data diverges from the hand-constructed
+prior.
+Deep learning (DL) methods have recently been adopted as
+a promising framework to improve reconstruction performance
+[12]–[16]. Inspired by traditional methods, a powerful approach
+is based on scan-specific (SS) priors that leverage the physical
+signal model to learn a reconstruction specific to each test
+scan, i.e. undersampled k-space data from a given test subject.
+Similar to autocalibration procedures in PI, a first group of
+SS methods perform training using a fully-sampled calibration
+region and then exercise learned dependencies in broader k-
+space [15]–[18]. Following the deep image prior technique, a
+second group of methods use unconditional CNNs as a native
+MRI prior [19]–[21]. These CNNs map low-dimensional latent
+variables onto MR images, and latents and network weights
+are optimized to ensure consistency to acquired data based on
+the physical signal model. In general, SS priors learned on
+each subject at test time avoid the need for separate training
+datasets, and promise improved reliability against atypical
+anatomy. However, they suffer from long inference times that
+can be prohibitive particularly when nonlinear networks are
+arXiv:2301.02613v1  [eess.IV]  6 Jan 2023
+
+EMB
+NPS
+UFFC
+SignalProcessing Society
+0222
+adopted [22]–[24].
+A fundamental alternative is to employ scan-general (SG)
+priors based on deep nonlinear networks that capture latent fea-
+tures of MR images [12]–[14], [25]–[33]. Numerous successful
+architectures have been reported including perceptrons [34],
+basic convolutional neural networks (CNNs) [35]–[38], residual
+or recurrent CNNs [29], [39]–[41], generative adversarial
+networks (GANs) [42]–[46], transformers [47], [48] and
+diffusion models [49], [50]. Physics-guided unrolled methods
+have received particular attention that combine the physical
+signal model as in traditional frameworks and regularization
+via a deep network serving as an SG prior [13], [27], [51]–[53].
+Reconstruction is achieved via serially alternated projections
+through the physical signal model and the SG prior [38], [40],
+[54]–[56]. However, under low-data regimes, the suboptimally
+trained SG prior introduces errors that are propagated across
+the unrolled architecture, compromising performance [6], [57],
+[58]. Furthermore, learning of SG priors requires large training
+datasets from several tens to hundreds of subjects [28], [59],
+[60], which can limit practicality.
+Here, we propose a novel parallel-stream fusion model
+(PSFNet) that consolidates SS and SG priors to enable data-
+efficient training and computation-efficient inference in deep
+MRI reconstruction1. PSFNet leverages an SS stream to perform
+linear reconstruction based on the physical signal model, and an
+SG stream to perform nonlinear reconstruction based on a deep
+network. Unlike conventional unrolled methods based on serial
+projections, here we propose a parallel-stream architecture with
+learnable fusion of SS and SG priors. Fusion parameters are
+adapted across cascades and training iterations to emphasize
+task-critical information. Comprehensive experiments on brain
+MRI datasets are reported to demonstrate PSFNet under both
+supervised and unsupervised settings [62]–[66]. PSFNet is
+compared against an unrolled SG method [27], two SS methods
+[17], [67], and conventional SPIRiT reconstructions [11].
+Compared to the unrolled model, PSFNet lowers training data
+requirements an order of magnitude. Compared to SS models,
+PSFNet offers significantly faster inference times. Our main
+contributions are summarized below:
+• A novel cascaded network architecture is introduced
+that adaptively fuses SS and SG priors across cascades
+and training iterations to improve learning-based MRI
+reconstruction in low-data regimes.
+• The SS prior facilitates learning of the SG prior with
+limited data, and empowers PSFNet to successfully
+generalize to out-of-domain samples.
+• The SG prior improves performance by capturing nonlin-
+ear residuals, and enhances resilience against suboptimal
+hyperparameter selection in the SS component.
+• Parallel-stream fusion of SS and SG priors yields robust
+performance with limited training data in both supervised
+and unsupervised settings.
+II. THEORY
+1see [61] for a preliminary version of this work presented at ISMRM 2021.
+A. Image Reconstruction in Accelerated MRI
+MRI reconstruction is an inverse problem that aims to recover
+an image from a respective undersampled acquisition:
+MFx = y
+(1)
+where F is the Fourier transform, M is the sampling mask
+defining acquired k-space locations, x is the multi-coil image to
+be reconstructed and y are acquired multi-coil k-space data. To
+improve problem conditioning, additional prior information re-
+garding the expected distribution of MR images is incorporated
+in the form of a regularization term:
+ˆx = arg min
+x
+λ||MFx − y||2
+2 + R(x)
+(2)
+where the first term enforces DC between reconstructed and
+acquired k-space data, R(x) reflects the MRI prior, and λ
+controls the balance between the DC and regularization terms.
+The DC term can be implemented by injecting the acquired
+values of k-space data into the reconstruction [13]. Thus,
+mapping through a DC block is given as:
+fDC(x) = F −1ΛFx +
+λ
+1 + λF −1y
+(3)
+where Λ is a diagonal matrix with diagonal entries set to
+1
+1+λ at
+acquired k-space locations and set to 1 in unacquired locations.
+In traditional methods, the regularization term is based on a
+hand-constructed transform domain where data are assumed to
+have a sparse representation [9]. For improved conformation to
+the distribution of MRI data, recent frameworks instead adopt
+deep network models to capture either SG priors learned from
+a large MRI database with hundreds of subjects, or SS priors
+learned from individual test scans. Learning procedures for the
+two types of priors are discussed below.
+SG priors: In MRI, SG priors are typically adopted to
+suppress aliasing artifacts in the zero-filled reconstruction (i.e.,
+inverse Fourier transform) of undersampled k-space acquisitions
+[27]. A deep network model that performs de-aliasing can be
+learned from a large training dataset of undersampled and
+corresponding fully-sampled k-space acquisitions, and then
+employed to implement R(.) in Eq. 2 during inference. The
+regularization term based on an SG prior is given as:
+RSG (x) = arg min
+x
+||CSG(F −1y; ˆθSG) − x||2
+2
+(4)
+where CSG is an image-domain deep network with learned
+parameters ˆθSG. The formulation in Eq. 4 assumes that CSG
+recovers multi-coil output images provided multi-coil input
+images. The parameters θSG for CSG can be learned based
+on a pixel-wise loss between reconstructed and ground-truth
+images. Training is conducted offline via an empirical risk
+minimization approach based on Monte Carlo sampling [13]:
+LSG(θSG) =
+N
+�
+n=1
+||CSG(F −1yn; θSG) − ˘xn||p
+(5)
+where N is the number of training scans, n is the training scan
+index, ||.||p denotes ℓp norm, ˘xn is the ground-truth multi-coil
+image derived from the fully-sampled acquisition for the nth
+scan, and yn are respective undersampled k-space data.
+
+3
+A common approach to build CSG is based on unrolled
+architectures that perform cascaded projections through CNN
+blocks to regularize the image and DC blocks to ensure
+conformance to the physical signal model [27]. Given a total
+of K cascades with tied CNN parameters across cascades, the
+mapping through the kth cascade is [13], [68], [69]:
+xr
+k = fDC
+�
+fSG
+�
+xr
+k−1; θSG
+��
+(6)
+where xr
+k is the image for the rth scan (that could be a training
+or test scan) at the output of the kth cascade (k ∈ [1, 2, ..., K]),
+and xr
+0 = F −1yr where yr are the acquired undersampled data
+for the rth scan. Meanwhile, fSG is the CNN block embedded
+in the kth cascade with parameters θSG.
+As the parameters of SG priors are trained offline and then
+frozen during inference, deeper network architectures can be
+used for enhanced reconstruction performance along with fast
+inference. However, learning deep networks requires substantial
+training datasets that may be difficult to collect. Moreover,
+since SG priors learn aggregate representations of MRI data
+across training subjects, they may show poor generalization to
+subject-specific variability in anatomy [19].
+SS priors: Unlike SG priors, SS priors are not learned from
+a dedicated training dataset but instead they are learned directly
+for individual test scans to improve generalization [15]. The
+SS prior can also be used to implement R(.) in Eq. 2 with the
+respective regularization term expressed as:
+RSS (x) = arg min
+x
+||CSS(F −1y; ˆθSS) − x||2
+2
+(7)
+where CSS is an image-domain network with parameters ˆθSS.
+In the absence of ground-truth images, the parameters θq
+SS for
+the qth test scan can be learned based on proxy k-space losses
+between reconstructed and acquired undersampled data [22].
+Learning is conducted online to minimize this proxy loss:
+LSS(θq
+SS) = ||MFCSS(F −1yq; θq
+SS) − yq||p
+(8)
+where yq are acquired undersampled k-space data for the qth
+scan. An unrolled architecture can be adopted to build CSS
+by performing cascaded projections through network and DC
+blocks, resulting in the following mapping for the kth cascade:
+xq
+k = fDC
+�
+fSS
+�
+xq
+k−1; θq
+SS
+��
+(9)
+fSS can be operationalized as a linear or nonlinear network [22],
+[23]. As the parameters of SS priors are learned independently
+for each test scan, they promise enhanced generalization to
+subject-specific anatomy. However, since training is performed
+online during inference, SS priors can introduce substantial
+computational burden, particularly when deep nonlinear net-
+works are used that also increase the risk of overfitting [70].
+B. PSFNet
+Here, we propose to combine SS and SG priors to maintain
+a favorable trade-off between generalization performance
+and computational efficiency under low-data regimes. In the
+conventional unrolling framework, this requires computation
+of serially alternated projections through the SS, SG and DC
+blocks:
+xr
+k = fDC
+�
+fSG
+�
+fSS
+�
+xr
+k−1; θr
+SS
+�
+; θSG
+��
+(10)
+The unrolled architecture with K cascades can be learned
+offline using the training set. Note that scarcely-trained SG
+blocks under low-data regimes can perform suboptimally,
+introducing residual errors in their output. In turn, these errors
+will accumulate across serial projections to degrade the overall
+performance.
+To address this limitation, here we introduce a novel
+architecture, PSFNet, that performs parallel-stream fusion of
+SS and SG priors as opposed to the serial combination in
+conventional unrolled methods. PSFNet utilizes a nonlinear SG
+prior for high performance, and a linear SS prior to enhance
+generalization without excessive computational burden. The
+two priors undergo parallel-stream fusion with learnable fusion
+parameters η and γ, as displayed in Figure 1. These parame-
+ters adaptively control the relative weighting of information
+extracted by the SG versus SS streams during the course of
+training in order to alleviate error accumulation. As such, the
+mapping through the kth cascade in PSFNet is:
+xr
+k = ηkfDC(fSS(xr
+k−1; θr
+SS))+γkfDC(fSG(xr
+k−1; θSG))
+(11)
+In Eq. 11, the learnable fusion parameters for the SS and
+SG blocks at the kth cascade are ηk and γk, respectively. To
+enforce fidelity to acquired data, DC projections are performed
+on the outputs of SG and SS blocks. In PSFNet, the SG prior
+is learned collectively from the set of training scans and then
+frozen during inference on test scans. In contrast, the SS prior
+is learned individually for each scan, during both training and
+inference.
+Training: PSFNet involves a training phase to learn model
+parameters for the SG prior as well as its fusion with the SS
+prior. For each individual scan in the training set, PSFNet
+learns a dedicated SS prior for the given scan. Since learning
+of a nonlinear SS prior has substantial computational burden,
+we adopt a linear SS prior in PSFNet. In particular, the SS
+block performs dealiasing via convolution with a linear kernel
+[71]:
+fSS(xn
+k−1; θn
+SS) = F −1{θn
+SS ⊛ Fxn
+k−1}
+(12)
+where θn
+SS ∈ C(z×z×w×w) with n denoting the training scan
+index, z denoting the number of coil elements, and w denoting
+the kernel size in k-space. The SS blocks contain unlearned
+Fourier and inverse Fourier transformation layers as their input
+and output layers, respectively, and convolution is computed
+over the spatial frequency dimensions in k-space. Meanwhile,
+the SG prior is implemented as a deep CNN operating in image
+domain:
+fSG(xn
+k−1; θSG) = CNN(xn
+k−1)
+(13)
+Across the scans in the training set, the training loss for PSFNet
+
+4
+Fig. 1: (a) PSFNet comprises a parallel-stream cascade of sub-networks where each sub-network contains (b) a scan-general
+(SG) block, and (c) a scan-specific (SS) block. The two parallel blocks are each succeeded by (d) a data-consistency (DC)
+block, and their outputs are aggregated with learnable fusion weights, ηk and γk where k is the cascade index. At the end of K
+cascades, coil-combination is performed on multi-coil data using sensitivity maps estimated via ESPIRiT [71]. The SG block is
+implemented as a deep convolutional neural network (CNN) and the SS block was implemented as a linear projection layer.
+can then be expressed in constrained form as:
+LP SF Net(θSG,γγγ,ηηη) =
+N
+�
+n=1
+||ηKfDC(fSS(xn
+K−1; ˆθn
+SS))
++ γKfDC(fSG(xn
+K−1; θSG)) − ˘xn||p
+s.t. ˆθn
+SS = arg min
+θn
+SS
+||F −1W nyn − fSS(F −1W nyn; θn
+SS)||2
+2
+(14)
+The constraint in Eq. 14 corresponds to the scan-specific
+learning of the SS prior ˆθn
+SS, which is then adopted to
+calculate the loss. Assuming that the linear relationships among
+neighboring spatial frequencies are similarly distributed across
+k-space [71], ˆθn
+SS is learned by solving a self-regression
+problem on the subset of fully-sampled data in central k-space,
+where W n is a mask operator that selects data within this
+calibration region.
+Note that, unlike deep reconstruction models purely based
+on SG priors, the SG prior in PSFNet is not directly trained to
+remove artifacts in zero-filled reconstructions of undersampled
+data. Instead, the SG prior is trained to concurrently suppress
+artifacts in reconstructed images along with the SS prior; and
+the relative importance attributed to the two priors is determined
+by the fusion parameters at each cascade. As such, the SS prior
+can be given higher weight during initial training iterations
+where the SG prior is scarcely trained, whereas its weight can
+be relatively reduced during later iterations once the SG prior
+has been sufficiently trained. This adaptive fusion approach
+thereby lowers reliance on the availability of large training
+sets.
+Inference: During inference on the qth test scan, the respec-
+tive SS prior is learned online as:
+ˆθq
+SS = arg min
+θq
+SS
+||F −1W qyq − f q
+SS(F −1W qyq; θq
+SS)||2
+2
+(15)
+Afterwards, the learned ˆθq
+SS is used along with the previously
+trained ˆθSG to perform repeated projections through K cas-
+cades as described in Eq. 11. The multi-coil image recovered
+by PSFNet at the output of the K cascade is:
+ˆxq = ηKfDC(fSS(xq
+K−1; ˆθq
+SS))+γKfDC(fSG(xq
+K−1; ˆθSG))
+(16)
+where ˆxq denotes the recovered image. The final reconstruction
+can be obtained by performing combination across coils:
+ˆxq
+combined = A∗ˆxq
+(17)
+where A are coil sensitivities, and A∗ denotes the conjugate
+of A.
+III. METHODS
+A. Implementation Details
+In each cascade, PSFNet contained two parallel streams
+with SG and SS blocks. The SG blocks comprised an input
+layer followed by a stack of 4 convolutional layers with 64
+channels and 3x3 kernel size each, and an output layer with
+ReLU activation functions. They processed complex images
+with separate channels for real and imaginary components. The
+SS blocks comprised a Fourier layer, 5 projection layers with
+identity activation functions, and an inverse Fourier layer. They
+processed complex images directly without splitting real and
+imaginary components. The linear convolution kernel used in
+the projection layers was learned from the calibration region
+by solving a Tikhonov regularized self-regression problem [11].
+The DC blocks comprised 3 layers respectively to implement
+
+a) Parallel-Stream Fusion Model (PSFNet)
+Reconstructed
+Zero-Filled
+Nk
+Reconstruction
+Image
+Coil
+Combination
+Z
+Z
+: b) Scan-General
+c) Scan-Specific
+d) Data-Consistency:
+ReLU
+ReLU
+Conv
+ReLU
+Conv
+ReLU
+Conv
+LU
+Conv
+Conv
+Rel
+ce
+oni
+eLU
+ReLU
+Conv
+Conv
+Conv
+Conv
+Conv
+Xout-im
+ReLi
+ReLi
+ReL!
+R
+Yin5
+forward Fourier transformation, restoration of acquired k-
+space data and inverse Fourier transformation. PSFNet was
+implemented with 5 cascades, K=5. The weights of SG, SS, and
+DC blocks were tied across cascades to limit model complexity
+[27]. The only exception were fusion coefficients that determine
+the relative weighting of the SG and SS blocks at each
+stage (γ1, .., γk, ..., γ5 η1, ...ηk, ..., η5). These fusion parameters
+were kept distinct across cascades. Coil-combination on the
+recovered multi-coil images was performed using sensitivity
+maps estimated via ESPIRiT [71].
+B. MRI Dataset
+Experimental demonstrations were performed using brain
+MRI scans from the NYU fastMRI database [72]. Here, contrast-
+enhanced T1-weighted (cT1-weighted) and T2-weighted acquisi-
+tions were considered. The fastMRI dataset contains volumetric
+MRI data with varying image and coil dimensionality across
+subjects. Note that a central aim of this work was to sys-
+tematically examine the learning capabilities of models for
+varying number of training samples. To minimize potential
+biases due to across-subject variability in MRI protocols, here
+we selected subjects with matching imaging matrix size and
+number of coils. To do this, we only selected subjects with at
+least 10 cross-sections and only the central 10 cross-sections
+were retained in each subject. We further selected subjects
+with an in-plane matrix size of 256x320 for cT1 acquisitions,
+and of 288x384 for T2 acquisitions. Background regions in
+MRI data with higher dimensions were cropped. Lastly, we
+restricted our sample selection to subjects with at least 5 coil
+elements, and geometric coil compression [73] was applied to
+unify the number of coils to 5 in all subjects.
+Fully-sampled acquisitions were retrospectively undersam-
+pled to achieve acceleration rates of R=4x and 8x. Random
+undersampling patterns were designed via either a bi-variate
+normal density function peaking at the center of k-space, or a
+uniform density function across k-space. The standard deviation
+of the normal density function was adjusted to maintain
+the expected value of R across k-space. The fully-sampled
+calibration region spanned a 40x40 window in central k-space.
+C. Competing Methods
+PSFNet was compared against several state-of-the-art ap-
+proaches including SG methods, SS methods, and traditional
+PI reconstructions. For methods containing SG priors, both
+supervised and unsupervised variants were implemented.
+PSFNet: A supervised variant of PSFNet was trained using
+paired sets of undersampled and fully-sampled acquisitions.
+PSFNetUS: An unsupervised variant of PSFNet was im-
+plemented using self-supervision based on only undersampled
+training data. Acquired data were split into two non-overlapping
+sets where 40% of samples was reserved for evaluating the
+training loss and 60% of samples was reserved to enforce DC
+[64].
+MoDL: A supervised SG methods based on an unrolled
+architecture with tied weights across cascades was used [27].
+MoDL serially interleaves SG and DC blocks. The number of
+cascades and the structure of SG and DC blocks were identical
+to those in PSFNet.
+MoDLUS: An unsupervised variant of MoDL was imple-
+mented using self-supervision. A 40%-60% split was performed
+on acquired data to evaluate the training loss and enforce data
+consistency, respectively [64].
+sRAKI-RNN: An SS method was implemented based on the
+MoDL architecture [67]. Learning was performed to minimize
+DC loss on the fully-sampled calibration region. Calibration
+data were randomly split with 75% of samples used to define
+the training loss and 25% of samples reserved to enforce DC.
+Multiple input-output pairs were produced for a single test
+sample by utilizing this split.
+SPIRiT: A traditional PI reconstruction was performed using
+the SPIRiT method [11]. Reconstruction parameters including
+the regularization weight for kernel estimation (κ), kernel size
+(w), and the number of iterations (Niter) were independently
+optimized for each reconstruction task via cross-validation.
+SPARK: An SS method was used to correct residual errors
+from an initial SPIRiT reconstruction [17]. Learning was
+performed to minimize DC loss on the calibration region. The
+learned SS prior was then used to correct residual errors in
+the remainder of k-space.
+D. Optimization Procedures
+For all methods, hyperparameter selection was performed
+via cross-validation on a three-way split of data across subjects.
+There was no overlap among training, validation and test sets
+in terms of subjects. Data from 10 subjects were reserved for
+validation, and data from a separate set of 40 subjects were
+reserved for testing. The number of subjects in the training
+set was varied from 1 to 50. Hyperparameters that maximized
+peak signal-to-noise ratio (PSNR) on the validation set were
+selected for each method.
+Training was performed via the Adam optimizer with
+learning rate ζ=10−4, β1=0.90 and β2=0.99 [74]. All deep
+learning methods were trained to minimize hybrid ℓ1-ℓ2-
+norm loss between recovered and target data (e.g., between
+reconstructed and ground truth images for PSFNet, between
+recovered and acquired k-space samples for PSFNetUS) [64].
+For PSFNet and MoDL, the selected number of epochs was
+200, batch size was set to 2 for the limited number of training
+samples (Nsamples <10), and to 5 otherwise. In DC blocks,
+λ = ∞ was used to enforce strict data consistency. For PSFNet
+and SPIRiT, the kernel width (w) and regularization parameter
+(κ) values were set as (κ, w) = (10−2, 9) at R= 4 and (10−2,
+9) at R=8 for cT1-weighted reconstructions, and as (100, 17)
+at R=4 and (10−2, 17) at R=8 for T2-weighted reconstructions.
+For SPIRiT, the number of iterations Niter was set as 13
+at R=4 and 27 at R=8 for cT1-weighted reconstructions, 20
+at R=4 and 38 at R=8 for T2-weighted reconstructions. For
+sRAKI-RNN, the selected number of epochs was 500 and
+batch size was set to 32. All other optimization procedures
+were identical to MoDL. For SPARK, network architecture
+and training procedures were adopted from [17], except for the
+number of epochs (Nepoch) and learning rate (ζ) which were
+optimized on the validation set as (Nepoch, ζ)= (100, 10−2) For
+
+6
+Fig. 2: Average PSNR across test subjects for (a) cT1- and (b)
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for SPIRiT, SPARK, sRAKI-RNN,
+MoDL and PSFNet.
+cT1-weighted reconstructions, and (Nepoch, ζ)= (250, 10−3)
+for T2-weighted reconstructions.
+All competing methods were executed on an NVidia RTX
+3090 GPU, and models were coded in Tensorflow except for
+SPARK which was implemented in PyTorch. SPARK was
+implemented using the toolbox at https://github.com/
+YaminArefeen/spark_mrm_2021. The code to imple-
+ment PSFNet will be available publicly at https://github.
+com/icon-lab/PSFNet upon publication.
+E. Performance Metrics
+Performance assessments for reconstruction methods were
+carried out by visual observations and quantitative metrics.
+PSNR and structural similarity index (SSIM) were used
+for quantitative evaluation. For each method, metrics were
+computed on coil-combined images from the reconstruction
+and from the fully-sampled ground truth acquisition. Statistical
+differences between competing methods were examined via
+non-parametric Wilcoxon signed-rank tests.
+F. Experiments
+Several different experiments were conducted to system-
+atically examine the performance of competing methods.
+Assessments aimed to investigate reconstruction performance
+under low training data regimes, generalization performance
+in case of mismatch between training and testing domains,
+contribution of the parallel-stream design to reconstruction per-
+formance, sensitivity to hyperparameter selection, performance
+in unsupervised learning, and computational complexity.
+Performance in low-data regimes: Deep SG methods for
+MRI reconstruction typically suffer from suboptimal perfor-
+mance as the size of the training dataset is constrained. To
+systematically examine reconstruction performance, we trained
+supervised variants of PSFNet and MoDL while the number
+of training samples (Nsamples) was varied in the range [2-
+500] cross sections. To attain a given number of samples,
+sequential selection was performed across subjects and across
+cross-sections within each subject. Thus, the number of unique
+subjects included in the training set roughly corresponded to
+Nsamples/10 (since there were 10 cross-sections per subject).
+SS reconstructions were also performed with sRAKI-RNN,
+SPIRiT and SPARK. In the absence of fully-sampled ground
+truth data to guide the learning of the prior, unsupervised
+training of deep reconstruction models may prove relatively
+more difficult compared to supervised training. In turn, this
+may elevate requirements on training datasets for unsupervised
+models. To examine data efficiency for unsupervised training,
+we compared the reconstruction performance of PSFNetUS and
+MoDLUS as Nsamples was varied in the range of [2-500] cross
+sections. Comparisons were also provided against sRAKI-RNN,
+SPIRiT and SPARK.
+Generalization performance: Deep reconstruction models
+can suffer from suboptimal generalization when the MRI data
+distribution shows substantial variation between the training and
+testing domains. To examine generalizability, PSFNet models
+were trained on data from a source domain and tested on data
+from a different target domain. The domain-transferred models
+were then compared to models trained and tested directly
+in the target domain. Three different factors were altered to
+induce domain variation: tissue contrast, undersampling pattern,
+and acceleration rate. First, the capability to generalize to
+different tissue contrasts was evaluated. Models were trained
+on data from a source contrast and tested on data from
+a different target contrast. Domain-transferred models were
+compared to target-domain models trained on data from the
+target contrast. Next, the capability to generalize to different
+undersampling patterns was assessed. Models were trained on
+data undersampled with variable-density patterns and tested
+on data undersampled with uniform-density patterns. Domain-
+transferred models were compared to target-domain models
+trained on uniformly undersampled data. Lastly, the capability
+to generalize to different acceleration rates was examined.
+Models were trained on acquisitions accelerated at R=4x and
+tested on acquisitions accelerated at R=8x. Domain-transferred
+models were compared to target-domain models trained at
+R=8x.
+Sensitivity to hyperparameters: SS priors are learned
+from individual test scans as opposed to SG priors trained
+on larger training datasets. Thus, SS priors might show
+elevated sensitivity to hyperparameter selection. We assessed
+the reliability of reconstruction performance against suboptimal
+
+Nsubjects
+(r
+X
+40
+39
+PSNR (dB)
+38
+SPIRiT
+37
+SPARK
+36
+SRAKI-RNN
+MoDL
+35
+PSFNet
+34
+Nsamples
+b)
+Nsubjects
+X
+6
+40
+39
+38
+PSNR (dB)
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDL
+34
+PSFNet
+33
+Nsamples7
+Fig. 3: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the
+zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each method
+are shown in the bottom row. MoDL and PSFNet were trained on 10 cross-sections from a single subject. SPIRiT, SPARK and
+sRAKI-RNN directly performed inference on test data without a priori model training. PSFNet shows superior performance to
+competing methods in terms of residual reconstruction errors.
+Fig. 4: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the
+zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each method
+are shown in the bottom row. MoDL and PSFNet were trained on 10 cross-sections from a single subject. SPIRiT, SPARK and
+sRAKI-RNN directly performed inference on test data without a priori model training. PSFNet shows superior performance to
+competing methods in terms of residual reconstruction errors.
+hyperparameter selection for SS priors. For this purpose,
+analyses were conducted on SPIRiT, SPARK and PSFNet
+that embody SS methods to perform linear reconstructions
+in k-space. The set of hyperparameters examined included
+regularization parameters for kernel estimation (κ) and kernel
+size (w). Separate models were trained using κ in range [10-3-
+100] and w in range [5-17].
+Computational complexity: Finally, we assessed the com-
+putational complexity of competing methods. For each method,
+training and inference times were measured for a single subject
+with 10 cross-sections. Each cross-section had an imaging
+matrix size of 256x320 and contained data from 5 coils. For
+all methods including SS priors, hyperparameters optimized
+for cT1-weighted reconstructions at R=4 were used.
+Ablation analysis: To assess the contribution of the parallel-
+stream design in PSFNet, a conventional unrolled variant
+of PSFNet was formed, named as PSFNetSerial. PSFNetSerial
+combined the SG and SS priors via serial projections as
+described in Eq. 10. Modeling procedures and the design of
+SG and SS blocks were kept identical between PSFNet and
+PSFNetSerial for fair comparison. Performance was assessed as
+Nsamples was varied in the range of [2-500] cross sections.
+IV. RESULTS
+A. Performance in Low-Data Regimes
+Common SG methods for MRI reconstruction are based
+on deep networks that require copious amounts of training
+data, so performance can substantially decline on limited
+training sets [28], [59]. In contrast, PSFNet leverages an SG
+prior to concurrently reconstruct an image along with an SS
+prior. Therefore, we reasoned that its performance should scale
+favorably under low-data regimes compared to SG methods. We
+also reasoned that PSFNet should yield elevated performance
+compared to SS methods due to residual corrections from its SG
+prior. To test these predictions, we trained supervised variants
+of PSFNet and MoDL along with SPIRiT, sRAKI-RNN, and
+SPARK while the number of training samples (Nsamples) was
+
+ZF
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDI
+PSFNet
+Reference
+0.12
+Error
+0ZF
+SPIRiT
+SPARK
+SRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+08
+Fig. 5: Weighting of the SG (γ) and SS (η) blocks in the
+final cascade of PSFNet. Weights were averaged across models
+trained for cT1- and T2-weighted reconstructions at R=4x.
+Model training was performed for varying number of training
+samples (Nsamples, lower x-axis) and thereby training subjects
+(Nsubjects, upper x-axis). Both blocks are equally weighted
+with very limited training data. As Nsamples increases, the
+weighting of the SG prior becomes more dominant over the
+weighting of the SS prior.
+systematically varied. Figure 2 displays PSNR performance
+for cT1-weighted and T2-weighted image reconstruction as a
+function of Nsamples. PSFNet outperforms the scan-general
+MoDL method for all values of Nsamples (p < 0.05). As
+expected, performance benefits with PSFNet become more
+prominent towards lower values of Nsamples. PSFNet also
+outperforms traditional SPIRiT and scan-specific sRAKI-RNN
+and SPARK methods broadly across the examined range
+of Nsamples (p < 0.05). Note that while MoDL requires
+Nsamples = 30 (3 subjects) to offer on par performance to
+SS methods, PSFNet yields superior performance with as few
+as Nsamples = 2. Representative reconstructions for cT1- and
+T2-weighted images are depicted in Figures 3 and 4, where
+Nsamples = 10 from a single subject were used for training.
+PSFNet yields lower reconstruction errors compared to all other
+methods in this low-data regime, where competing methods
+either show elevated noise or blurring.
+Naturally, the performance of PSFNet increases as more
+training samples are available. Since the SS prior is inde-
+pendently learned for individual samples, it should not elicit
+systematic performance variations depending on Nsamples.
+Thus, the performance gains can be attributed to improved
+learning of the SG prior. In turn, we predicted that PSFNet
+would put more emphasis on its SG stream as its reliability
+increases. To examine this issue, we inspected the weightings
+of the SG (γ) and SS (η) streams as the training set size was
+varied. Figure 5 displays weightings at the last cascade as a
+function of Nsamples. For lower values of Nsamples where the
+quality of the SG prior is relatively limited, the SG and SS
+priors are almost equally weighted. In contrast, as the learning
+of the SG prior improves with higher Nsamples, the emphasis
+on the SG prior increases while the SS prior is less heavily
+Fig. 6: Average PSNR across test subjects for (a) cT1- and (b)
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for SPIRiT, SPARK, sRAKI-RNN,
+MoDLUS and PSFNetUS.
+weighted.
+We then questioned whether the performance benefits of
+PSFNet are also apparent during unsupervised training of
+deep network models. For this purpose, unsupervised variants
+PSFNetUS and MoDLUS were trained via self-supervision [64].
+PSFNetUS was compared against MoDLUS, SPIRiT, sRAKI-
+RNN, and SPARK while the number of training samples
+(Nsamples) was systematically varied. Figure 6 displays PSNR
+performance for cT1-weighted and T2-weighted image recon-
+struction as a function of Nsamples. Similar to the supervised
+setting, PSFNetUS outperforms MoDLUS for all values of
+Nsamples (p < 0.05), and the performance benefits are more
+noticeable at lower Nsamples. In this case, however, MoDLUS
+is unable to reach the performance of the best performing
+SS method (SPARK) even at Nsamples = 500. In contrast,
+PSFNetUS starts outperforming SPARK with approximately
+Nsamples = 50 (5 subjects). The enhanced reconstruction
+quality with PSFNetUS is corroborated in representative re-
+constructions for cT1- and T2-weighted images depicted in
+Figures 7 and 8, where Nsamples = 100 were used for training.
+Taken together, these results indicate that the data-efficient
+nature of PSFNet facilitates the training of both supervised
+and unsupervised MRI reconstruction models.
+
+Nsubjects
+5
+6
+Y (SG)
+0.60
+n (Ss)
+0.55
+Weighting
+0.50
+0.45
+0.40
+4
+6
+8
+NsamplesNsubjects
+a
+X
+40
+39
+38
+37
+(dB)
+36
+SPIRiT
+35
+PSNR
+34
+SPARK
+SRAKI - RNN
+33
+MoDLus
+32
+PSFNetus
+31
+30
+2
+X
+Nsamples
+b)
+Nsubjects
+5
+6
+40
+39
+38
+37
+(dB)
+36
+.343
+SPIRiT
+PSNR 
+SPARK
+SRAKI - RNN
+MoDLus
+32
+PSFNetus
+31
+30
+X
+Nsamples9
+Fig. 7: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with
+the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each
+method are shown in the bottom row. MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects). SPIRiT,
+SPARK and sRAKI-RNN directly performed inference on test data without a priori model training. PSFNetUS shows superior
+performance to competing methods in terms of residual reconstruction errors.
+Fig. 8: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with
+the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition. Error maps for each
+method are shown in the bottom row. MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects). SPIRiT,
+SPARK and sRAKI-RNN directly performed inference on test data without a priori model training. PSFNetUS shows superior
+performance to competing methods in terms of residual reconstruction errors.
+B. Generalization Performance
+An important advantage of SS priors is that they allow
+model adaptation to individual test samples, thereby promise
+enhanced performance in out-of-domain reconstructions [22].
+Yet, SG priors with fixed parameters might show relatively
+limited generalizability during inference [23], [75]. To assess
+generalization performance, we introduced domain variations
+by altering three experimental factors: tissue contrast, under-
+sampling pattern, and acceleration rate. For methods comprising
+SG components, we built both target-domain models that were
+trained in the target domain, and domain-transferred models
+that were trained in a non-target domain. We then compared
+the reconstruction performances of the two models in the target
+domain.
+First, we examined generalization performance when the
+tissue contrast varied between training and testing domains
+(e.g., trained on cT1, tested on T2). Table I lists performance
+metrics for competing methods with Nsamples = 500. While
+performance losses are incurred for domain-transferred PSFNet-
+DT and MoDL-DT models that contain SG components, these
+losses are modest. On average, MoDL-DT shows a loss of
+0.3dB PSNR and 0.1% SSIM (p < 0.05), and PSFNet-DT
+shows a loss of 0.2dB PSNR and 0.1% SSIM (p < 0.05). Note
+that PSFNet-DT still outperforms the closest competing SS
+method by 2.2dB PSNR and 1.8% SSIM (p < 0.05).
+Second, we examined generalization performance when mod-
+els were trained with variable-density and tested on uniform-
+density undersampling patterns. Table II lists performance
+metrics for competing methods. On average across tissue
+contrasts, MoDL-DT suffers a notable performance loss of
+3.6dB PSNR and 2.5% SSIM (p < 0.05). In contrast, PSFNet-
+DT shows a relatively limited loss of 0.4dB PSNR and 0.2%
+SSIM (p < 0.05). Note that PSFNet-DT again outperforms the
+closest competing SS method by 3.4dB PSNR and 3.7% SSIM
+(p < 0.05).
+Third, we examined generalization performance when models
+were trained at R=4x and tested on R=8x. Table III lists
+
+ZF
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+0ZF
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Reference
+0.12
+Error
+010
+TABLE I: Generalization across tissue contrasts. PSNR and
+SSIM values (mean±standard error) across test subjects.
+Results are shown for scan-specific models (SPIRiT, SPARK,
+sRAKI-RNN), target-domain models (MoDL, PSFNet) and
+domain-transferred models (MoDL-DT, PSFNet-DT) at R=4x.
+The tissue contrast in the target domain is listed in the left-most
+column (cT1 or T2), domain-transferred models were trained
+for the non-target tissue contrast.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+37.6
+37.6
+36.8
+38.5
+38.2
+39.9
+39.4
+±1.5
+±1.5
+±1.3
+± 1.5
+±1.5
+±1.7
+±1.6
+T2
+35.8
+36.5
+35.2
+37.9
+37.5
+39.0
+39.0
+±1.0
+±1.0
+±1.1
+± 1.0
+±1.1
+±1.0
+±0.9
+SSIM
+cT1
+93.1
+93.3
+93.8
+95.1
+94.8
+95.8
+95.6
+±1.5
+±1.4
+±1.0
+±1.0
+±1.1
+±1.0
+±1.0
+T2
+90.8
+93.1
+94.9
+96.2
+96.2
+96.7
+96.8
+±1.2
+±1.0
+±0.6
+±0.5
+±0.5
+±0.4
+±0.4
+TABLE II: Generalization across undersampling patterns. PSNR
+and SSIM values (mean±standard error) across test subjects.
+Results are shown for Results are shown for scan-specific
+models (SPIRiT, SPARK, sRAKI-RNN), target-domain models
+(MoDL, PSFNet) and domain-transferred models (MoDL-DT,
+PSFNet-DT) at R=4x. Domain-transferred models were trained
+with variable-density undersampling, and tested on uniform-
+density undersampling. Target-domain models were trained and
+tested with uniform-density undersampling.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+37.1
+37.1
+33.6
+37.0
+33.6
+40.2
+39.9
+±1.8
+±1.7
+±1.4
+± 1.7
+±1.8
+±1.6
+±1.6
+T2
+35.1
+35.6
+31.6
+37.0
+33.2
+40.2
+39.7
+±1.3
+±1.3
+±1.5
+± 1.1
+±1.2
+±1.1
+±1.2
+SSIM
+cT1
+92.9
+93.0
+91.2
+93.4
+91.2
+95.9
+95.6
+±1.5
+±1.5
+±1.5
+±1.3
+±2.0
+±1.2
+±1.2
+T2
+90.6
+92.1
+91.5
+95.6
+92.7
+97.1
+96.9
+±1.5
+±1.5
+±1.2
+±0.7
+±1.1
+±0.6
+±0.6
+performance metrics for competing methods. On average across
+tissue contrasts, MoDL-DT suffers a notable performance loss
+of 1.0dB PSNR and performs slightly better in SSIM by
+0.2%SSIM (p < 0.05), whereas PSFNet-DT shows a lower loss
+of 0.6dB PSNR (p < 0.05) and performs similarly in SSIM
+(p > 0.05). PSFNet-DT outperforms the closest competing SS
+method by 1.2dB PSNR and 1.9% SSIM (p < 0.05). Taken
+together, these results clearly suggest that the SS prior in
+PSFNet contributes to its improved generalization performance
+over the scan-general MoDL method, while the SG prior in
+PSFNet enables it to outperform competing SS methods.
+C. Sensitivity to Hyperparameters
+Parameters of deep networks that implement SS priors are to
+be learned from a single test sample, so the resultant models can
+show elevated sensitivity to the selection of hyperparameters
+compared to SG priors learned from a collection of training
+TABLE III: Generalization across acceleration rates. PSNR
+and SSIM values (mean±standard error) across test subjects.
+Results are shown for scan-specific models (SPIRiT, SPARK,
+sRAKI-RNN), target-domain models (MoDL, PSFNet) and
+domain-transferred models (MoDL-DT, PSFNet-DT). Domain-
+transferred models were trained at R=4x and tested at R=8x.
+Target-domain models were trained and tested at R=8x.
+SPIRiT
+SPARK
+sRAKI-
+RNN
+MoDL
+MoDL-DT
+PSFNet
+PSFNet-
+DT
+PSNR
+cT1
+34.7
+34.8
+34.3
+35.3
+34.5
+36.5
+36.2
+±1.5
+±1.5
+±1.5
+± 1.4
+±1.7
+±1.5
+±1.5
+T2
+33.6
+33.7
+32.6
+34.6
+33.4
+35.6
+34.6
+±1.0
+±1.0
+±0.9
+± 1.0
+±1.2
+±1.1
+±1.2
+SSIM
+cT1
+89.8
+90.8
+91.4
+92.1
+92.2
+93.3
+93.3
+±1.9
+±1.6
+±1.4
+±1.5
+±1.4
+±1.4
+±1.4
+T2
+89.0
+90.1
+92.7
+93.5
+93.7
+94.6
+94.5
+±1.3
+±1.1
+±0.9
+±0.8
+±0.8
+±0.7
+±0.7
+samples. Thus, we investigated the sensitivity of PSFNet to key
+hyperparameters of its SS prior. SPIRiT, SPARK and PSFNet
+methods all embody a linear k-space reconstruction, so the
+relevant hyperparameters are the regularization weight and
+width for the convolution kernel. Performance was evaluated for
+models were trained in the low-data regime (i.e., Nsamples =
+10, 1 subject) for varying hyperparameter values.
+Figure 9 displays PSNR measurements for SPIRiT, SPARK
+and PSFNet across κ in range (10-3-100). While the perfor-
+mance of SPIRiT and SPARK is notably influenced by κ,
+PSFNet is minimally affected by sub-optimal selection. On
+average across contrasts, the difference between the maximum
+and minimum PSNR values is 8.4dB for SPIRiT, 4.5dB for
+SPARK, and a lower 0.7dB for PSFNet. Note that PSFNet
+outperforms competing methods across the entire range of
+κ (p < 0.05). Figure 10 shows PSNR measurements for
+competing methods across w in range (5-17). In this case,
+all methods show relatively limited sensitivity to the selection
+of w. On average across contrasts, the difference between the
+maximum and minimum PSNR values is 1.5dB for SPIRiT,
+0.5dB for SPARK, and 0.2dB for PSFNet. Again, PSFNet
+outperforms competing methods across the entire range of
+w (p < 0.05). Overall, our results indicate that PSFNet
+yields improved reliability against sub-optimal hyperparameter
+selection than competing SS methods.
+D. Computational Complexity
+Next, we assessed the computational complexity of com-
+peting methods. Table IV lists the training times of methods
+with SG priors, MoDL and PSFNet. Note that the remaining
+SS based methods do not involve a pre-training step. As it
+involves learning of an SS prior on each training sample,
+PSFNet yields elevated training time compared to MoDL. In
+return, however, it offers enhanced generalization performance
+and data-efficient learning. Table IV also lists the inference
+times of SPIRiT, SPARK, sRAKI-RNN, MoDL and PSFNet.
+MoDL and PSFNet that employ SG priors with fixed weights
+during inference offer fast run times. In contrast, SPARK and
+
+11
+Fig. 9: PSNR measurements were performed on recovered cT1-
+and T2-weighted images at R=4x. Bar plots in blue color show
+average PSNR across κ ∈ 10-3-101 (i.e., the regularization
+parameter for kernel estimation). Error bars denote the 90%
+interval across κ. Bar plots in red color show PSNR for methods
+that do not depend on the value of κ.
+Fig. 10: PSNR measurements were performed on recovered
+cT1- and T2-weighted images at R=4x. Bar plots in blue color
+show the average PSNR across w ∈ 5-17 (i.e., the kernel size).
+Error bars denote the 90% interval across w. Bar plots in red
+color show PSNR for methods that do not depend on the value
+of w.
+sRAKI-RNN that involve SS priors learned on individual test
+samples have a high computational burden. Although PSFNet
+also embodies an SS prior, its uses a relatively lightweight
+linear prior as opposed to the nonlinear priors in competing
+SS methods. Therefore, PSFNet benefits from data-efficient
+learning while maintaining computationally-efficient inference.
+E. Ablation Analysis
+To demonstrate the value of the parallel-stream fusion
+strategy in PSFNet over conventional unrolling, PSFNet was
+compared against a variant model PSFNetSerial that combined
+SS and SG priors through serially alternated projections.
+Separate models were trained with number of training samples
+in the range Nsamples=[2-500]. Performance in cT1- and T2
+-weighted image reconstruction is displayed in Figure 11.
+PSFNet significantly improves reconstruction performance over
+PSFNetSerial across the entire range of Nsamples considered
+(p < 0.05), and the benefits grow stronger for smaller training
+sets. On average across contrasts for Nsamples < 10, PSFNet
+TABLE IV: Computational complexity of competing methods.
+Training and inference times for data from a single subject,
+with 10 cross-sections, imaging matrix size 256x320 and 5
+coils. Run times are listed for SPARK, sRAKI-RNN, MoDL,
+and PSFNet.
+SPIRiT
+SPARK
+sRAKI-RNN
+MoDL
+PSFNet
+Training(s)
+-
+-
+-
+132
+337
+Inference(s)
+0.85
+23.35
+285.00
+0.25
+1.13
+Fig. 11: Average (a) PSNR and (b) SSIM values for cT1- and
+T2-weighted image reconstructions at R=4x. Model training was
+performed for varying number of training samples (Nsamples,
+lower x-axis) and thereby training subjects (Nsubjects, upper
+x-axis). Results are shown for PSFNet and PSFNetSerial.
+outperforms PSFNetSerial by 1.8dB PSNR and 0.6% SSIM
+(p < 0.05). These results indicate that the parallel-stream
+fusion of SG and SS priors in PSFNet is superior to the serial
+projections in conventional unrolling.
+V. DISCUSSION AND CONCLUSION
+In this study, we introduced PSFNet for data-efficient training
+of deep reconstruction models in accelerated MRI. PSFNet
+synergistically fuses SS and SG priors in a parallel-stream
+architecture. The linear SS prior improves learning efficiency
+while mataining relatively low computational footprint, whereas
+the nonlinear SG prior enables improved reconstruction per-
+formance. For both supervised and unsupervised training
+setups, the resulting model substantially reduces dependence
+on the availability of large MRI datasets. Furthermore, it
+achieves competitive inference times to SG methods, and
+
+38
+(dB)
+36
+PSNR
+34
+32
+30
+MoDL
+SPIRiT
+SPARK
+PSFNet
+SRAKI - RNN38
+(dB)
+36
+PSNR
+34
+32
+30
+MoDL
+SPIRiT
+SPARK
+PSFNet
+SRAKI - RNNNsubjects
+a
+X
+40
+39
+PSNR (dB)
+38
+37
+PSFNet (cTi)
+36
+PSFNet (T2)
+PSFNetserial (cTi)
+35
+PSFNetserial (T2)
+34
+Nsamples
+b)
+Nsubjects
+5
+X
+6
+97
+96
+(%)
+SSIM
+95
+PSFNet (cTi)
+PSFNet (T2)
+94
+PSFNetserial (cTi)
+PSFNetserial (T2)
+93
+Nsamples12
+reliably generalizes across tissue contrasts, sampling patterns
+and acceleration rates.
+Several prominent approaches have been introduced in
+the literature to address the training requirements of deep
+models based on SG priors. One approach is to pre-train
+models on readily available datasets from a separate source
+domain and then to fine-tune on several tens of samples
+from the target domain [28], [59] or else perform SS fine-
+tuning [76]. This transfer learning approach relaxes the domain
+requirements for training datasets. However, the domain-
+transferred models might be suboptimal when training and
+testing data distributions are divergent. In such cases, additional
+training for domain-alignment might be necessary to mitigate
+performance losses. In contrast, PSFNet contains a SS prior that
+allows it to better generalize to out-of-domain data without
+further training. Another approach is to build unsupervised
+models to alleviate dependency on training datasets with paired
+undersampled, fully-sampled acquisitions. Model training can
+be performed either directly on undersampled acquisitions
+via self-supervision [64] or on unpaired sets of undersampled
+and fully-sampled acquisitions via cycle-consistent learning
+[77]. This approach can prove beneficial when fully-sampled
+acquisitions are costly to collect. Nonetheless, the resulting
+models still require relatively large datasets form tens of
+subjects during training [64]. Note that our experiments on
+self-supervised variants of PSFNet and MoDL suggest that
+unsupervised models can be more demanding for data than their
+supervised counterparts. Therefore, the data-efficiency benefits
+of PSFNet might be particularly useful for unsupervised deep
+MRI reconstruction.
+A fundamentally different framework to lower requirements
+on training datasets while offering improved generalizability
+is based on SS priors. In this case, learning can be performed
+directly on test data and models can be adapted to each scan
+[15], [17]. A group of studies have proposed SS methods based
+on relatively compact nonlinear models to facilitate learning
+during inference [15], [17], [18], [78]. However, because learn-
+ing is performed in central k-space, these methods implicitly
+assume that local relationships among spatial frequency samples
+are largely invariant across k-space. While the SS prior in
+PSFNet also rests on a similar assumption, the SG components
+helps correct residual errors that can be introduced due to this
+assumption. Another group of studies have alternatively adopted
+the deep image prior (DIP) approach to build SS methods
+[19], [20], [22], [23]. In DIP, unconditional deep network
+models that map latent variables onto images are used as
+native priors for MR images. The priors are learned by ensuring
+the consistency of reconstructed and acquired data across the
+entire k-space. Despite improved generalization, these relatively
+more complex models require increased inference times. In
+comparison, PSFNet provides faster inference since the weights
+for its SG prior are fixed, and its SS prior involves a compact
+linear operator that is easier to learn.
+Few independent studies on MRI have proposed approaches
+related to PSFNet by combining nonlinear and linear recon-
+structions [6], [17], [78]. Residual RAKI and SPARK methods
+initially perform a linear reconstruction, and then use an SS
+method to correct residual errors via minimizing a DC loss in
+the calibration region [17], [78]. As local relationships among
+data samples might vary across k-space, the learned SS priors
+might be suboptimal. Moreover, these methods perform online
+learning of nonlinear SS priors that introduces relatively high
+computational burden. In contrast, PSFNet incorporates an SG
+prior to help improve reliability against sub-optimalities in the
+SS prior, and uses a linear SS prior for efficiency. Another
+related method is GrappaNet that improves reconstruction per-
+formance by cascading GRAPPA and network-based nonlinear
+reconstruction steps [6]. While [6] intends to improve image
+quality, the main aim of our study is to improve practicality
+by lowering training data requirements of deep models, and
+improving domain generalizability without elevating inference
+times. Note that GrappaNet follows the conventional unrolling
+approach by performing serially alternated projections through
+linear and nonlinear reconstructions, which can lead to error
+propagation under low-data regimes [79]. In contrast, PSFNet
+maintains linear and nonlinear reconstructions as two parallel
+streams in its architecture, and learns to optimally fuse the
+information from the two streams.
+The proposed method can be improved along several lines
+of technical development. First, to improve the capture of high-
+frequency information by the SG prior, an adversarial loss
+term along with a discriminator subnetwork can be included
+in PSFNet [80]. It remains to be demonstrated whether the
+data-efficiency benefits of PSFNet are apparent for adversarial
+training setups. Second, nonlinear activation functions can
+be included in the SS stream to improve the expressiveness
+of the SS prior [78]. While learning of nonlinear priors can
+elevate inference complexity, generalization performance might
+be further improved. Third, the expressiveness of both SS
+and SG priors might be enhanced by incorporating attention
+mechanisms as proposed in recent transformer models [81].
+Fourth, using multimodal image fusion approaches can improve
+performance in case of having a repository with multimodal
+data [82], [83]. Lastly, the benefits of transfer learning and
+PSFNet can be aggregated by pre-training the SG prior on
+natural images to further lower requirements on training data.
+VI. ACKNOWLEDGMENTS
+This work was supported in part by a TUBA GEBIP 2015
+fellowship, by a BAGEP 2017 fellowship, and by a TUBITAK
+121E488 grant awarded to T. C¸ ukur.
+REFERENCES
+[1] S. Bauer, R. Wiest, L.-P. Nolte, and M. Reyes, “A survey of mri-based
+medical image analysis for brain tumor studies,” Physics in Medicine &
+Biology, vol. 58, no. 13, p. R97, 2013.
+[2] A. Shoeibi, M. Khodatars, M. Jafari, N. Ghassemi, P. Moridian,
+R. Alizadehsani, S. H. Ling, A. Khosravi, H. Alinejad-Rokny, H. Lam,
+M. Fuller-Tyszkiewicz, U. R. Acharya, D. Anderson, Y. Zhang, and
+J. M. Gorriz, “Diagnosis of brain diseases in fusion of neuroimaging
+modalities using deep learning: A review,” Information Fusion, vol. 93,
+pp. 85–117, 2023.
+[3] M. Hu, X. Qian, S. Liu, A. J. Koh, K. Sim, X. Jiang, C. Guan,
+and J. H. Zhou, “Structural and diffusion mri based schizophrenia
+classification using 2d pretrained and 3d naive convolutional neural
+networks,” Schizophrenia Research, vol. 243, pp. 330–341, 2022.
+[4] K. R. M. Fernando and C. P. Tsokos, “Deep and statistical learning in
+biomedical imaging: State of the art in 3d mri brain tumor segmentation,”
+Information Fusion, vol. 92, pp. 450–465, 2023.
+
+13
+[5] Z. Zhu, X. He, G. Qi, Y. Li, B. Cong, and Y. Liu, “Brain tumor
+segmentation based on the fusion of deep semantics and edge information
+in multimodal mri,” Information Fusion, vol. 91, pp. 376–387, 2023.
+[6] A. Sriram, J. Zbontar, T. Murrell, C. L. Zitnick, A. Defazio, and D. K.
+Sodickson, “GrappaNet: Combining parallel imaging with deep learning
+for multi-coil MRI reconstruction,” in Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition (CVPR), June
+2020, pp. 14 303–14 310.
+[7] K. P. Pruessmann, M. Weiger, M. B. Scheidegger, and P. Boesiger,
+“SENSE: sensitivity encoding for fast MRI.” Magnetic Resonance in
+Medicine, vol. 42, no. 5, pp. 952–62, 1999.
+[8] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus,
+J. Wang, B. Kiefer, and A. Haase, “Generalized autocalibrating partially
+parallel acquisitions (GRAPPA),” Magnetic Resonance in Medicine,
+vol. 47, no. 6, pp. 1202–1210, 2002.
+[9] M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application
+of compressed sensing for rapid MR imaging,” Magnetic Resonance in
+Medicine, vol. 58, no. 6, pp. 1182–1195, 2007.
+[10] A. Majumdar, “Improving synthesis and analysis prior blind compressed
+sensing with low-rank constraints for dynamic mri reconstruction,”
+Magnetic resonance imaging, vol. 33, no. 1, pp. 174–179, 2015.
+[11] M. Lustig and J. M. Pauly, “SPIRiT: Iterative self-consistent parallel
+imaging reconstruction from arbitrary k-space.” Magnetic Resonance in
+Medicine, vol. 64, no. 2, pp. 457–71, 2010.
+[12] Y. Yang, J. Sun, H. Li, and Z. Xu, “ADMM-CSNet: A deep learning
+approach for image compressive sensing,” IEEE Transactions on Pattern
+Analysis and Machine Intelligence, vol. 42, no. 3, pp. 521–538, 2020.
+[13] J. Schlemper, J. Caballero, J. V. Hajnal, A. Price, and D. Rueckert,
+“A Deep Cascade of Convolutional Neural Networks for MR Image
+Reconstruction,” in International Conference on Information Processing
+in Medical Imaging, 2017, pp. 647–658.
+[14] K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson,
+T. Pock, and F. Knoll, “Learning a variational network for reconstruction
+of accelerated MRI data,” Magnetic Resonance in Medicine, vol. 79,
+no. 6, pp. 3055–3071, 2017.
+[15] M. Akc¸akaya, S. Moeller, S. Weing¨artner, and K. U˘gurbil, “Scan-
+specific robust artificial-neural-networks for k-space interpolation (RAKI)
+reconstruction: Database-free deep learning for fast imaging,” Magnetic
+Resonance in Medicine, vol. 81, no. 1, pp. 439–453, 2019.
+[16] T. H. Kim, P. Garg, and J. P. Haldar, “LORAKI: Autocalibrated recurrent
+neural networks for autoregressive MRI reconstruction in k-space,” arXiv
+preprint arXiv:1904.09390, 2019.
+[17] Y. Arefeen, O. Beker, J. Cho, H. Yu, E. Adalsteinsson, and B. Bilgic,
+“Scan-specific artifact reduction in k-space (spark) neural networks
+synergize with physics-based reconstruction to accelerate mri,” Magnetic
+Resonance in Medicine, vol. 87, no. 2, pp. 764–780, 2022.
+[18] S. A. H. Hosseini, C. Zhang, S. Weing¨artner, S. Moeller, M. Stuber,
+K. Ugurbil, and M. Akc¸akaya, “Accelerated coronary MRI with sRAKI:
+A database-free self-consistent neural network k-space reconstruction for
+arbitrary undersampling,” PLOS ONE, vol. 15, no. 2, pp. 1–13, 2020.
+[19] Y. Korkmaz, S. U. Dar, M. Yurt, M. ¨Ozbey, and T. Cukur, “Unsupervised
+mri reconstruction via zero-shot learned adversarial transformers,” IEEE
+Transactions on Medical Imaging, vol. 41, no. 7, pp. 1747–1763, 2022.
+[20] S. Arora, V. Roeloffs, and M. Lustig, “Untrained modified deep decoder
+for joint denoising and parallel imaging reconstruction,” in Proceedings
+of the 28th Annual Meeting of the ISMRM, 2020, p. 3585.
+[21] M. Z. Darestani and R. Heckel, “Accelerated mri with un-trained neural
+networks,” IEEE Transactions on Computational Imaging, vol. 7, pp.
+724–733, 2021.
+[22] D. Narnhofer, K. Hammernik, F. Knoll, and T. Pock, “Inverse GANs
+for accelerated MRI reconstruction,” in Proceedings of the SPIE, vol.
+11138, 2019, pp. 381 – 392.
+[23] K. C. Tezcan, C. F. Baumgartner, R. Luechinger, K. P. Pruessmann, and
+E. Konukoglu, “MR image reconstruction using deep density priors,”
+IEEE Transactions on Medical Imaging, vol. 38, no. 7, pp. 1633–1642,
+2019.
+[24] Q. Liu, Q. Yang, H. Cheng, S. Wang, M. Zhang, and D. Liang,
+“Highly undersampled magnetic resonance imaging reconstruction using
+autoencoding priors,” Magnetic Resonance in Medicine, vol. 83, no. 1,
+pp. 322–336, 2020.
+[25] T. Eo, Y. Jun, T. Kim, J. Jang, H.-J. Lee, and D. Hwang, “KIKI-net: cross-
+domain convolutional neural networks for reconstructing undersampled
+magnetic resonance images,” Magnetic Resonance in Medicine, vol. 80,
+no. 5, pp. 2188–2201, 2018.
+[26] M. Mardani, E. Gong, J. Y. Cheng, S. Vasanawala, G. Zaharchuk,
+L. Xing, and J. M. Pauly, “Deep generative adversarial neural networks
+for compressive sensing MRI,” IEEE Transactions on Medical Imaging,
+vol. 38, no. 1, pp. 167–179, 2019.
+[27] H. K. Aggarwal, M. P. Mani, and M. Jacob, “MoDL: Model-Based
+deep learning architecture for inverse problems,” IEEE Transactions on
+Medical Imaging, vol. 38, no. 2, pp. 394–405, 2019.
+[28] S. U. H. Dar, M. ¨Ozbey, A. B. C¸ atlı, and T. C¸ ukur, “A transfer-learning
+approach for accelerated MRI using deep neural networks,” Magnetic
+Resonance in Medicine, vol. 84, no. 2, pp. 663–685, 2020.
+[29] D. Lee, J. Yoo, S. Tak, and J. C. Ye, “Deep residual learning for accel-
+erated MRI using magnitude and phase networks,” IEEE Transactions
+on Biomedical Engineering, vol. 65, no. 9, pp. 1985–1995, 2018.
+[30] P. Guo, J. M. J. Valanarasu, P. Wang, J. Zhou, S. Jiang, and V. M. Patel,
+“Over-and-under complete convolutional rnn for mri reconstruction,” in
+International Conference on Medical Image Computing and Computer-
+Assisted Intervention.
+Springer, 2021, pp. 13–23.
+[31] G. Yiasemis, J.-J. Sonke, C. S´anchez, and J. Teuwen, “Recurrent
+variational network: A deep learning inverse problem solver applied
+to the task of accelerated mri reconstruction,” in Proceedings of the
+IEEE/CVF Conference on Computer Vision and Pattern Recognition,
+2022, pp. 732–741.
+[32] R. Hou and F. Li, “Idpcnn: Iterative denoising and projecting cnn for
+mri reconstruction,” Journal of Computational and Applied Mathematics,
+vol. 406, p. 113973, 2022.
+[33] Z. Ramzi, G. Chaithya, J.-L. Starck, and P. Ciuciu, “Nc-pdnet: a
+density-compensated unrolled network for 2d and 3d non-cartesian mri
+reconstruction,” IEEE Transactions on Medical Imaging, 2022.
+[34] K. Kwon, D. Kim, and H. Park, “A parallel MR imaging method using
+multilayer perceptron,” Medical Physics, vol. 44, no. 12, pp. 6209–6224,
+2017.
+[35] S. Wang, Z. Su, L. Ying, X. Peng, S. Zhu, F. Liang, D. Feng, and
+D. Liang, “Accelerating magnetic resonance imaging via deep learning,”
+in IEEE 13th International Symposium on Biomedical Imaging (ISBI),
+2016, pp. 514–517.
+[36] J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general
+deep learning framework for inverse problems,” SIAM Journal on Imaging
+Sciences, vol. 11, no. 2, pp. 991–1048, 2018.
+[37] J. Yoon, E. Gong, I. Chatnuntawech, B. Bilgic, J. Lee, W. Jung, J. Ko,
+H. Jung, K. Setsompop, G. Zaharchuk, E. Y. Kim, J. Pauly, and J. Lee,
+“Quantitative susceptibility mapping using deep neural network: QSMnet,”
+NeuroImage, vol. 179, pp. 199–206, 2018.
+[38] C. M. Hyun, H. P. Kim, S. M. Lee, S. Lee, and J. K. Seo, “Deep learning
+for undersampled MRI reconstruction,” Physics in Medicine and Biology,
+vol. 63, no. 13, p. 135007, 2018.
+[39] A. Hauptmann, S. Arridge, F. Lucka, V. Muthurangu, and J. A. Steeden,
+“Real-time cardiovascular MR with spatio-temporal artifact suppression
+using deep learning–proof of concept in congenital heart disease,”
+Magnetic Resonance in Medicine, vol. 81, no. 2, pp. 1143–1156, 2019.
+[40] S. A. H. Hosseini, B. Yaman, S. Moeller, M. Hong, and M. Akc¸akaya,
+“Dense recurrent neural networks for accelerated MRI: History-cognizant
+unrolling of optimization algorithms,” IEEE Journal of Selected Topics
+in Signal Processing, vol. 14, no. 6, pp. 1280–1291, 2020.
+[41] C. Qin, J. Schlemper, J. Caballero, A. N. Price, J. V. Hajnal, and
+D. Rueckert, “Convolutional recurrent neural networks for dynamic MR
+image reconstruction,” IEEE Transactions on Medical Imaging, vol. 38,
+no. 1, pp. 280–290, 2019.
+[42] T. M. Quan, T. Nguyen-Duc, and W.-K. Jeong, “Compressed sensing
+MRI reconstruction with cyclic loss in generative adversarial networks,”
+IEEE Transactions on Medical Imaging, vol. 37, no. 6, pp. 1488–1497,
+2018.
+[43] S. U. Dar, M. Yurt, M. Shahdloo, M. E. Ildız, B. Tınaz, and T. C¸ ukur,
+“Prior-guided image reconstruction for accelerated multi-contrast MRI
+via generative adversarial networks,” IEEE Journal of Selected Topics in
+Signal Processing, vol. 14, no. 6, pp. 1072–1087, 2020.
+[44] Y. Chen, D. Firmin, and G. Yang, “Wavelet improved GAN for MRI
+reconstruction,” in Proceedings of SPIE, Medical Imaging 2021: Physics
+of Medical Imaging, vol. 11595, 2021, p. 1159513.
+[45] G. Elmas, S. U. Dar, Y. Korkmaz, E. Ceyani, B. Susam, M. Ozbey,
+S. Avestimehr, and T. C¸ ukur, “Federated learning of generative image
+priors for mri reconstruction,” IEEE Transactions on Medical Imaging,
+2022.
+[46] M. Yaqub, F. Jinchao, S. Ahmed, K. Arshid, M. A. Bilal, M. P. Akhter,
+and M. S. Zia, “Gan-tl: Generative adversarial networks with transfer
+learning for mri reconstruction,” Applied Sciences, vol. 12, no. 17, p.
+8841, 2022.
+[47] Y. Korkmaz, M.
+¨Ozbey, and T. Cukur, “Mri reconstruction with
+conditional adversarial transformers,” in International Workshop on
+
+14
+Machine Learning for Medical Image Reconstruction.
+Springer, 2022,
+pp. 62–71.
+[48] P. Guo, Y. Mei, J. Zhou, S. Jiang, and V. M. Patel, “Reconformer:
+Accelerated mri reconstruction using recurrent transformer,” arXiv
+preprint arXiv:2201.09376, 2022.
+[49] A. G¨ung¨or, S. U. Dar, S¸. ¨Ozt¨urk, Y. Korkmaz, G. Elmas, M. ¨Ozbey, and
+T. C¸ ukur, “Adaptive diffusion priors for accelerated mri reconstruction,”
+arXiv:2207.05876, 2022.
+[50] C. Peng, P. Guo, S. K. Zhou, V. M. Patel, and R. Chellappa, “Towards per-
+formant and reliable undersampled mr reconstruction via diffusion model
+sampling,” in International Conference on Medical Image Computing
+and Computer-Assisted Intervention.
+Springer, 2022, pp. 623–633.
+[51] K. Wang, J. I. Tamir, A. De Goyeneche, U. Wollner, R. Brada, S. X. Yu,
+and M. Lustig, “High fidelity deep learning-based mri reconstruction with
+instance-wise discriminative feature matching loss,” Magnetic Resonance
+in Medicine, vol. 88, no. 1, pp. 476–491, 2022.
+[52] F. Gadjimuradov, T. Benkert, M. D. Nickel, and A. Maier, “Robust partial
+fourier reconstruction for diffusion-weighted imaging using a recurrent
+convolutional neural network,” Magnetic Resonance in Medicine, vol. 87,
+no. 4, pp. 2018–2033, 2022.
+[53] Z. Ramzi, C. G R, J.-L. Starck, and P. Ciuciu, “Nc-pdnet: A density-
+compensated unrolled network for 2d and 3d non-cartesian mri recon-
+struction,” IEEE Transactions on Medical Imaging, vol. 41, no. 7, pp.
+1625–1638, 2022.
+[54] D. Polak, S. Cauley, B. Bilgic, E. Gong, P. Bachert, E. Adalsteinsson, and
+K. Setsompop, “Joint multi-contrast variational network reconstruction
+(jVN) with application to rapid 2D and 3D imaging,” Magnetic Resonance
+in Medicine, vol. 84, no. 3, pp. 1456–1469, 2020.
+[55] J. Y. Cheng, M. Mardani, M. T. Alley, J. M. Pauly, and S. S. Vasanawala,
+“Deepspirit: Generalized parallel imaging using deep convolutional neural
+networks,” in Proceedings of the 26th Annual Meeting of the ISMRM,
+2018, p. 0570.
+[56] K. Pooja, Z. Ramzi, G. Chaithya, and P. Ciuciu, “Mc-pdnet: Deep
+unrolled neural network for multi-contrast mr image reconstruction from
+undersampled k-space data,” in 2022 IEEE 19th International Symposium
+on Biomedical Imaging (ISBI), 2022, pp. 1–5.
+[57] N. Tavaf, A. Torfi, K. Ugurbil, and P.-F. Van de Moortele, “Grappa-gans
+for parallel mri reconstruction,” arXiv preprint arXiv:2101.03135, 2021.
+[58] C. M. Sandino, P. Lai, S. S. Vasanawala, and J. Y. Cheng, “Accelerating
+cardiac cine mri using a deep learning-based espirit reconstruction,”
+Magnetic Resonance in Medicine, vol. 85, no. 1, pp. 152–167, 2021.
+[59] F. Knoll, K. Hammernik, E. Kobler, T. Pock, M. P. Recht, and D. K.
+Sodickson, “Assessment of the generalization of learned image recon-
+struction and the potential for transfer learning,” Magnetic Resonance in
+Medicine, vol. 81, no. 1, pp. 116–128, 2019.
+[60] A. S. Chaudhari, C. M. Sandino, E. K. Cole, D. B. Larson, G. E.
+Gold, S. S. Vasanawala, M. P. Lungren, B. A. Hargreaves, and C. P.
+Langlotz, “Prospective deployment of deep learning in mri: A framework
+for important considerations, challenges, and recommendations for best
+practices,” Journal of Magnetic Resonance Imaging, vol. 54, no. 2, pp.
+357–371, 2021.
+[61] S. U. H. Dar, M. Yurt, and T. C¸ ukur, “A few-shot learning approach for
+accelerated mri via fusion of data-driven and subject-driven priors,” in
+Proceedings of the 29th Annual Meeting of the ISMRM, 2021, p. 1949.
+[62] J. I. Tamir, S. X. Yu, and M. Lustig, “Unsupervised deep basis pursuit:
+Learning reconstruction without ground-truth data,” in Proceedings of
+the 27th Annual Meeting of the ISMRM, 2019, p. 0660.
+[63] E. K. Cole, F. Ong, S. S. Vasanawala, and J. M. Pauly, “Fast unsupervised
+mri reconstruction without fully-sampled ground truth data using genera-
+tive adversarial networks,” in Proceedings of the IEEE/CVF International
+Conference on Computer Vision (ICCV) Workshops, October 2021, pp.
+3988–3997.
+[64] B. Yaman, S. A. H. Hosseini, S. Moeller, J. Ellermann, K. U˘gurbil, and
+M. Akc¸akaya, “Self-supervised learning of physics-guided reconstruc-
+tion neural networks without fully sampled reference data,” Magnetic
+resonance in medicine, vol. 84, no. 6, pp. 3172–3191, 2020.
+[65] J. Liu, Y. Sun, C. Eldeniz, W. Gan, H. An, and U. S. Kamilov, “RARE:
+Image reconstruction using deep priors learned without groundtruth,”
+IEEE Journal of Selected Topics in Signal Processing, vol. 14, no. 6, pp.
+1088–1099, 2020.
+[66] S. Wang, R. Wu, C. Li, J. Zou, Z. Zhang, Q. Liu, Y. Xi, and H. Zheng,
+“PARCEL: Physics-based Unsupervised Contrastive Representation Learn-
+ing for Multi-coil MR Imaging,” arXiv:2202.01494, 2022.
+[67] S. A. H. Hosseini, C. Zhang, K. Uˇgurbil, S. Moeller, and M. Akc¸akaya,
+“sraki-rnn: accelerated mri with scan-specific recurrent neural networks
+using densely connected blocks,” in Wavelets and Sparsity XVIII, vol.
+11138. International Society for Optics and Photonics, 2019, p. 111381B.
+[68] Q. Huang, Y. Xian, D. Yang, H. Qu, J. Yi, P. Wu, and D. N. Metaxas,
+“Dynamic mri reconstruction with end-to-end motion-guided network,”
+Medical Image Analysis, vol. 68, p. 101901, 2021.
+[69] J. Huang, S. Wang, G. Zhou, W. Hu, and G. Yu, “Evaluation on
+the generalization of a learned convolutional neural network for mri
+reconstruction,” Magnetic Resonance Imaging, vol. 87, pp. 38–46, 2022.
+[70] D. Ulyanov, A. Vedaldi, and V. Lempitsky, “Deep image prior,” in
+Proceedings of the IEEE Conference on Computer Vision and Pattern
+Recognition (CVPR), 2018, pp. 9446–9454.
+[71] M. Uecker, P. Lai, M. J. Murphy, P. Virtue, M. Elad, J. M. Pauly,
+S. S. Vasanawala, and M. Lustig, “ESPIRiT-an eigenvalue approach to
+autocalibrating parallel MRI: Where SENSE meets GRAPPA,” Magnetic
+Resonance in Medicine, vol. 71, no. 3, pp. 990–1001, 2014.
+[72] F. Knoll, J. Zbontar, A. Sriram, M. J. Muckley, M. Bruno, A. Defazio,
+M. Parente, K. J. Geras, J. Katsnelson, H. Chandarana, Z. Zhang,
+M. Drozdzalv, A. Romero, M. Rabbat, P. Vincent, J. Pinkerton, D. Wang,
+N. Yakubova, E. Owens, C. L. Zitnick, M. P. Recht, D. K. Sodickson,
+and Y. W. Lui, “fastMRI: A publicly available raw k-space and DICOM
+dataset of knee images for accelerated MR image reconstruction using
+machine learning,” Radiology: Artificial Intelligence, vol. 2, no. 1, p.
+e190007, 2020.
+[73] T. Zhang, J. M. Pauly, S. S. Vasanawala, and M. Lustig, “Coil compression
+for accelerated imaging with Cartesian sampling.” Magnetic Resonance
+in Medicine, vol. 69, no. 2, pp. 571–82, 2013.
+[74] D. P. Kingma and J. L. Ba, “Adam: a Method for Stochastic Optimization,”
+in International Conference on Learning Representations, 2015.
+[75] O. Dalmaz, U. Mirza, G. Elmas, M. ¨Ozbey, S. U. Dar, E. Ceyani,
+S. Avestimehr, and T. C¸ ukur, “One model to unite them all: Personalized
+federated learning of multi-contrast mri synthesis,” arXiv:2207.06509,
+2022.
+[76] S. A. H. Hosseini, B. Yaman, S. Moeller, and M. Akc¸akaya, “High-fidelity
+accelerated mri reconstruction by scan-specific fine-tuning of physics-
+based neural networks,” in 2020 42nd Annual International Conference
+of the IEEE Engineering in Medicine Biology Society (EMBC), 2020,
+pp. 1481–1484.
+[77] K. Lei, M. Mardani, J. M. Pauly, and S. S. Vasanawala, “Wasserstein
+GANs for MR imaging: from paired to unpaired training,” IEEE
+transactions on medical imaging, vol. 40, no. 1, pp. 105–115, 2021.
+[78] C. Zhang, S. A. H. Hosseini, S. Moeller, S. Weing¨artner, K. Ugurbil,
+and M. Akcakaya, “Scan-specific residual convolutional neural networks
+for fast mri using residual raki,” in 2019 53rd Asilomar Conference on
+Signals, Systems, and Computers, 2019, pp. 1476–1480.
+[79] M. Murphy, M. Alley, J. Demmel, K. Keutzer, S. Vasanawala, and
+M. Lustig, “Fast l1-SPIRiT compressed sensing parallel imaging MRI:
+scalable parallel implementation and clinically feasible runtime.” IEEE
+transactions on medical imaging, vol. 31, no. 6, pp. 1250–1262, 2012.
+[80] M. ¨Ozbey, O. Dalmaz, S. U. Dar, H. A. Bedel, S¸. ¨Ozturk, A. G¨ung¨or,
+and T. C¸ ukur, “Unsupervised medical image translation with adversarial
+diffusion models,” arXiv:2207.08208, 2022.
+[81] O. Dalmaz, M. Yurt, and T. C¸ ukur, “ResViT: Residual vision transformers
+for multi-modal medical image synthesis,” IEEE Transactions on Medical
+Imaging, vol. 41, no. 7, pp. 2598–2614, 2022.
+[82] H. Zhou, J. Hou, Y. Zhang, J. Ma, and H. Ling, “Unified gradient- and
+intensity-discriminator generative adversarial network for image fusion,”
+Information Fusion, vol. 88, pp. 184–201, 2022.
+[83] J. Liu, R. Dian, S. Li, and H. Liu, “Sgfusion: A saliency guided deep-
+learning framework for pixel-level image fusion,” Information Fusion,
+vol. 91, pp. 205–214, 2023.
+
diff --git a/39E0T4oBgHgl3EQfvAGt/content/tmp_files/load_file.txt b/39E0T4oBgHgl3EQfvAGt/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..931dd588250d7189fab3542f1267130fe1e829e1
--- /dev/null
+++ b/39E0T4oBgHgl3EQfvAGt/content/tmp_files/load_file.txt
@@ -0,0 +1,1563 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf,len=1562
+page_content='1  Learning Deep MRI Reconstruction Models from  Scratch in Low-Data Regimes  Salman Ul Hassan Dar, S¸ aban ¨Ozt¨urk, Muzaffer ¨Ozbey, and Tolga C¸ ukur∗  Abstract— Magnetic resonance imaging (MRI) is an es-  sential diagnostic tool that suffers from prolonged scan  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Reconstruction methods can alleviate this limitation  by recovering clinically usable images from accelerated ac-  quisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In particular, learning-based methods promise  performance leaps by employing deep neural networks as  data-driven priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A powerful approach uses scan-specific  (SS) priors that leverage information regarding the underly-  ing physical signal model for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SS priors are  learned on each individual test scan without the need for  a training dataset, albeit they suffer from computationally  burdening inference with nonlinear networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' An alterna-  tive approach uses scan-general (SG) priors that instead  leverage information regarding the latent features of MRI  images for reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SG priors are frozen at test  time for efficiency, albeit they require learning from a large  training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Here, we introduce a novel parallel-stream  fusion model (PSFNet) that synergistically fuses SS and  SG priors for performant MRI reconstruction in low-data  regimes, while maintaining competitive inference times to  SG methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet implements its SG prior based on a  nonlinear network, yet it forms its SS prior based on a linear  network to maintain efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A pervasive framework for  combining multiple priors in MRI reconstruction is algo-  rithmic unrolling that uses serially alternated projections,  causing error propagation under low-data regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To alle-  viate error propagation, PSFNet combines its SS and SG  priors via a novel parallel-stream architecture with learn-  able fusion parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Demonstrations are performed on  multi-coil brain MRI for varying amounts of training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNet outperforms SG methods in low-data regimes, and  surpasses SS methods with few tens of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  In both supervised and unsupervised setups, PSFNet re-  quires an order of magnitude lower samples compared to  SG methods, and enables an order of magnitude faster  inference compared to SS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Thus, the proposed  model improves deep MRI reconstruction with elevated  learning and computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Index Terms— image reconstruction, deep learning, scan  specific, scan general, low data, supervised, unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  This work was supported in part by a TUBA GEBIP 2015 fellowship,  by a BAGEP 2017 fellowship, and by a TUBITAK 121E488 grant awarded  to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' UH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, S¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  ¨Ozt¨urk, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  ¨Ozbey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur are with the  Department of Electrical and Electronics Engineering, and the Na-  tional Magnetic Resonance Research Center, Bilkent University,  Ankara, Turkey (e-mails: {salman,muzaffer,cukur}@ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='bilkent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='tr,  saban.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='ozturk@amasya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='tr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  ¨Ozt¨urk is also with the Amasya  University, Amasya, Turkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' INTRODUCTION  The unparalleled soft-tissue contrast and non-invasiveness  of MRI render it a preferred modality in many diagnostic  applications [1], [2], and downstream imaging tasks such  as classification [3] and segmentation [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, the  adverse effects of low spin polarization at mainstream field  strengths on the signal-to-noise ratio make it slower against  alternate modalities such as CT [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Since long scan durations  inevitably constrain clinical utility, there is an ever-growing  interest in accelerated MRI methods to improve scan efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Accelerated MRI involves an ill-posed inverse problem with the  aim of mapping undersampled acquisitions in k-space to high-  quality images corresponding to fully-sampled acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Conventional frameworks for solving this problem rely on  parallel imaging (PI) capabilities of receive coil arrays [7],  [8], in conjunction with hand-constructed MRI priors [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  A joint objective is iteratively optimized comprising a data-  consistency (DC) term based on the physical signal model,  and a regularization term that enforces the MRI prior [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  physical model constrains reconstructed data to be consistent  with acquired data while considering coil sensitivities and  undersampling patterns [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Meanwhile, the regularization  term, often based on a linear transform where data are assumed  to be compressible [9], introduces suboptimality when the  distribution of MRI data diverges from the hand-constructed  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Deep learning (DL) methods have recently been adopted as  a promising framework to improve reconstruction performance  [12]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Inspired by traditional methods, a powerful approach  is based on scan-specific (SS) priors that leverage the physical  signal model to learn a reconstruction specific to each test  scan, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' undersampled k-space data from a given test subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Similar to autocalibration procedures in PI, a first group of  SS methods perform training using a fully-sampled calibration  region and then exercise learned dependencies in broader k-  space [15]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Following the deep image prior technique, a  second group of methods use unconditional CNNs as a native  MRI prior [19]–[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' These CNNs map low-dimensional latent  variables onto MR images, and latents and network weights  are optimized to ensure consistency to acquired data based on  the physical signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In general, SS priors learned on  each subject at test time avoid the need for separate training  datasets, and promise improved reliability against atypical  anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, they suffer from long inference times that  can be prohibitive particularly when nonlinear networks are  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='02613v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='IV]  6 Jan 2023  EMB  NPS  UFFC  SignalProcessing Society  0222  adopted [22]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  A fundamental alternative is to employ scan-general (SG)  priors based on deep nonlinear networks that capture latent fea-  tures of MR images [12]–[14], [25]–[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Numerous successful  architectures have been reported including perceptrons [34],  basic convolutional neural networks (CNNs) [35]–[38], residual  or recurrent CNNs [29], [39]–[41], generative adversarial  networks (GANs) [42]–[46], transformers [47], [48] and  diffusion models [49], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Physics-guided unrolled methods  have received particular attention that combine the physical  signal model as in traditional frameworks and regularization  via a deep network serving as an SG prior [13], [27], [51]–[53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Reconstruction is achieved via serially alternated projections  through the physical signal model and the SG prior [38], [40],  [54]–[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, under low-data regimes, the suboptimally  trained SG prior introduces errors that are propagated across  the unrolled architecture, compromising performance [6], [57],  [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Furthermore, learning of SG priors requires large training  datasets from several tens to hundreds of subjects [28], [59],  [60], which can limit practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Here, we propose a novel parallel-stream fusion model  (PSFNet) that consolidates SS and SG priors to enable data-  efficient training and computation-efficient inference in deep  MRI reconstruction1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet leverages an SS stream to perform  linear reconstruction based on the physical signal model, and an  SG stream to perform nonlinear reconstruction based on a deep  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Unlike conventional unrolled methods based on serial  projections, here we propose a parallel-stream architecture with  learnable fusion of SS and SG priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Fusion parameters are  adapted across cascades and training iterations to emphasize  task-critical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Comprehensive experiments on brain  MRI datasets are reported to demonstrate PSFNet under both  supervised and unsupervised settings [62]–[66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet is  compared against an unrolled SG method [27], two SS methods  [17], [67], and conventional SPIRiT reconstructions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Compared to the unrolled model, PSFNet lowers training data  requirements an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Compared to SS models,  PSFNet offers significantly faster inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Our main  contributions are summarized below:  A novel cascaded network architecture is introduced  that adaptively fuses SS and SG priors across cascades  and training iterations to improve learning-based MRI  reconstruction in low-data regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The SS prior facilitates learning of the SG prior with  limited data, and empowers PSFNet to successfully  generalize to out-of-domain samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The SG prior improves performance by capturing nonlin-  ear residuals, and enhances resilience against suboptimal  hyperparameter selection in the SS component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Parallel-stream fusion of SS and SG priors yields robust  performance with limited training data in both supervised  and unsupervised settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' THEORY  1see [61] for a preliminary version of this work presented at ISMRM 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Image Reconstruction in Accelerated MRI  MRI reconstruction is an inverse problem that aims to recover  an image from a respective undersampled acquisition:  MFx = y  (1)  where F is the Fourier transform, M is the sampling mask  defining acquired k-space locations, x is the multi-coil image to  be reconstructed and y are acquired multi-coil k-space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To  improve problem conditioning, additional prior information re-  garding the expected distribution of MR images is incorporated  in the form of a regularization term:  ˆx = arg min  x  λ||MFx − y||2  2 + R(x)  (2)  where the first term enforces DC between reconstructed and  acquired k-space data, R(x) reflects the MRI prior, and λ  controls the balance between the DC and regularization terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The DC term can be implemented by injecting the acquired  values of k-space data into the reconstruction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Thus,  mapping through a DC block is given as:  fDC(x) = F −1ΛFx +  λ  1 + λF −1y  (3)  where Λ is a diagonal matrix with diagonal entries set to  1  1+λ at  acquired k-space locations and set to 1 in unacquired locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  In traditional methods, the regularization term is based on a  hand-constructed transform domain where data are assumed to  have a sparse representation [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For improved conformation to  the distribution of MRI data, recent frameworks instead adopt  deep network models to capture either SG priors learned from  a large MRI database with hundreds of subjects, or SS priors  learned from individual test scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Learning procedures for the  two types of priors are discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SG priors: In MRI, SG priors are typically adopted to  suppress aliasing artifacts in the zero-filled reconstruction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=',  inverse Fourier transform) of undersampled k-space acquisitions  [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A deep network model that performs de-aliasing can be  learned from a large training dataset of undersampled and  corresponding fully-sampled k-space acquisitions, and then  employed to implement R(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=') in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2 during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  regularization term based on an SG prior is given as:  RSG (x) = arg min  x  ||CSG(F −1y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθSG) − x||2  2  (4)  where CSG is an image-domain deep network with learned  parameters ˆθSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The formulation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 4 assumes that CSG  recovers multi-coil output images provided multi-coil input  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The parameters θSG for CSG can be learned based  on a pixel-wise loss between reconstructed and ground-truth  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Training is conducted offline via an empirical risk  minimization approach based on Monte Carlo sampling [13]:  LSG(θSG) =  N  �  n=1  ||CSG(F −1yn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG) − ˘xn||p  (5)  where N is the number of training scans, n is the training scan  index, ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='||p denotes ℓp norm, ˘xn is the ground-truth multi-coil  image derived from the fully-sampled acquisition for the nth  scan, and yn are respective undersampled k-space data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  3  A common approach to build CSG is based on unrolled  architectures that perform cascaded projections through CNN  blocks to regularize the image and DC blocks to ensure  conformance to the physical signal model [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Given a total  of K cascades with tied CNN parameters across cascades, the  mapping through the kth cascade is [13], [68], [69]:  xr  k = fDC  �  fSG  �  xr  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG  ��  (6)  where xr  k is the image for the rth scan (that could be a training  or test scan) at the output of the kth cascade (k ∈ [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', K]),  and xr  0 = F −1yr where yr are the acquired undersampled data  for the rth scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Meanwhile, fSG is the CNN block embedded  in the kth cascade with parameters θSG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  As the parameters of SG priors are trained offline and then  frozen during inference, deeper network architectures can be  used for enhanced reconstruction performance along with fast  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, learning deep networks requires substantial  training datasets that may be difficult to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moreover,  since SG priors learn aggregate representations of MRI data  across training subjects, they may show poor generalization to  subject-specific variability in anatomy [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SS priors: Unlike SG priors, SS priors are not learned from  a dedicated training dataset but instead they are learned directly  for individual test scans to improve generalization [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  SS prior can also be used to implement R(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=') in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2 with the  respective regularization term expressed as:  RSS (x) = arg min  x  ||CSS(F −1y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθSS) − x||2  2  (7)  where CSS is an image-domain network with parameters ˆθSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  In the absence of ground-truth images, the parameters θq  SS for  the qth test scan can be learned based on proxy k-space losses  between reconstructed and acquired undersampled data [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Learning is conducted online to minimize this proxy loss:  LSS(θq  SS) = ||MFCSS(F −1yq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θq  SS) − yq||p  (8)  where yq are acquired undersampled k-space data for the qth  scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' An unrolled architecture can be adopted to build CSS  by performing cascaded projections through network and DC  blocks, resulting in the following mapping for the kth cascade:  xq  k = fDC  �  fSS  �  xq  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θq  SS  ��  (9)  fSS can be operationalized as a linear or nonlinear network [22],  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As the parameters of SS priors are learned independently  for each test scan, they promise enhanced generalization to  subject-specific anatomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, since training is performed  online during inference, SS priors can introduce substantial  computational burden, particularly when deep nonlinear net-  works are used that also increase the risk of overfitting [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet  Here, we propose to combine SS and SG priors to maintain  a favorable trade-off between generalization performance  and computational efficiency under low-data regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In the  conventional unrolling framework, this requires computation  of serially alternated projections through the SS, SG and DC  blocks:  xr  k = fDC  �  fSG  �  fSS  �  xr  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θr  SS  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG  ��  (10)  The unrolled architecture with K cascades can be learned  offline using the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that scarcely-trained SG  blocks under low-data regimes can perform suboptimally,  introducing residual errors in their output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In turn, these errors  will accumulate across serial projections to degrade the overall  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  To address this limitation, here we introduce a novel  architecture, PSFNet, that performs parallel-stream fusion of  SS and SG priors as opposed to the serial combination in  conventional unrolled methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet utilizes a nonlinear SG  prior for high performance, and a linear SS prior to enhance  generalization without excessive computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  two priors undergo parallel-stream fusion with learnable fusion  parameters η and γ, as displayed in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' These parame-  ters adaptively control the relative weighting of information  extracted by the SG versus SS streams during the course of  training in order to alleviate error accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As such, the  mapping through the kth cascade in PSFNet is:  xr  k = ηkfDC(fSS(xr  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θr  SS))+γkfDC(fSG(xr  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG))  (11)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 11, the learnable fusion parameters for the SS and  SG blocks at the kth cascade are ηk and γk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To  enforce fidelity to acquired data, DC projections are performed  on the outputs of SG and SS blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In PSFNet, the SG prior  is learned collectively from the set of training scans and then  frozen during inference on test scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, the SS prior  is learned individually for each scan, during both training and  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Training: PSFNet involves a training phase to learn model  parameters for the SG prior as well as its fusion with the SS  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For each individual scan in the training set, PSFNet  learns a dedicated SS prior for the given scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Since learning  of a nonlinear SS prior has substantial computational burden,  we adopt a linear SS prior in PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In particular, the SS  block performs dealiasing via convolution with a linear kernel  [71]:  fSS(xn  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θn  SS) = F −1{θn  SS ⊛ Fxn  k−1}  (12)  where θn  SS ∈ C(z×z×w×w) with n denoting the training scan  index, z denoting the number of coil elements, and w denoting  the kernel size in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The SS blocks contain unlearned  Fourier and inverse Fourier transformation layers as their input  and output layers, respectively, and convolution is computed  over the spatial frequency dimensions in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Meanwhile,  the SG prior is implemented as a deep CNN operating in image  domain:  fSG(xn  k−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG) = CNN(xn  k−1)  (13)  Across the scans in the training set, the training loss for PSFNet  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1: (a) PSFNet comprises a parallel-stream cascade of sub-networks where each sub-network contains (b) a scan-general  (SG) block, and (c) a scan-specific (SS) block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The two parallel blocks are each succeeded by (d) a data-consistency (DC)  block, and their outputs are aggregated with learnable fusion weights, ηk and γk where k is the cascade index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' At the end of K  cascades, coil-combination is performed on multi-coil data using sensitivity maps estimated via ESPIRiT [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The SG block is  implemented as a deep convolutional neural network (CNN) and the SS block was implemented as a linear projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  can then be expressed in constrained form as:  LP SF Net(θSG,γγγ,ηηη) =  N  �  n=1  ||ηKfDC(fSS(xn  K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθn  SS))  + γKfDC(fSG(xn  K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θSG)) − ˘xn||p  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθn  SS = arg min  θn  SS  ||F −1W nyn − fSS(F −1W nyn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θn  SS)||2  2  (14)  The constraint in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 14 corresponds to the scan-specific  learning of the SS prior ˆθn  SS, which is then adopted to  calculate the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Assuming that the linear relationships among  neighboring spatial frequencies are similarly distributed across  k-space [71], ˆθn  SS is learned by solving a self-regression  problem on the subset of fully-sampled data in central k-space,  where W n is a mask operator that selects data within this  calibration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Note that, unlike deep reconstruction models purely based  on SG priors, the SG prior in PSFNet is not directly trained to  remove artifacts in zero-filled reconstructions of undersampled  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Instead, the SG prior is trained to concurrently suppress  artifacts in reconstructed images along with the SS prior;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' and  the relative importance attributed to the two priors is determined  by the fusion parameters at each cascade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As such, the SS prior  can be given higher weight during initial training iterations  where the SG prior is scarcely trained, whereas its weight can  be relatively reduced during later iterations once the SG prior  has been sufficiently trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' This adaptive fusion approach  thereby lowers reliance on the availability of large training  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Inference: During inference on the qth test scan, the respec-  tive SS prior is learned online as:  ˆθq  SS = arg min  θq  SS  ||F −1W qyq − f q  SS(F −1W qyq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' θq  SS)||2  2  (15)  Afterwards, the learned ˆθq  SS is used along with the previously  trained ˆθSG to perform repeated projections through K cas-  cades as described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The multi-coil image recovered  by PSFNet at the output of the K cascade is:  ˆxq = ηKfDC(fSS(xq  K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθq  SS))+γKfDC(fSG(xq  K−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ˆθSG))  (16)  where ˆxq denotes the recovered image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The final reconstruction  can be obtained by performing combination across coils:  ˆxq  combined = A∗ˆxq  (17)  where A are coil sensitivities, and A∗ denotes the conjugate  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Implementation Details  In each cascade, PSFNet contained two parallel streams  with SG and SS blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The SG blocks comprised an input  layer followed by a stack of 4 convolutional layers with 64  channels and 3x3 kernel size each, and an output layer with  ReLU activation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' They processed complex images  with separate channels for real and imaginary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  SS blocks comprised a Fourier layer, 5 projection layers with  identity activation functions, and an inverse Fourier layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' They  processed complex images directly without splitting real and  imaginary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The linear convolution kernel used in  the projection layers was learned from the calibration region  by solving a Tikhonov regularized self-regression problem [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The DC blocks comprised 3 layers respectively to implement  a) Parallel-Stream Fusion Model (PSFNet)  Reconstructed  Zero-Filled  Nk  Reconstruction  Image  Coil  Combination  Z  Z  : b) Scan-General  c) Scan-Specific  d) Data-Consistency:  ReLU  ReLU  Conv  ReLU  Conv  ReLU  Conv  LU  Conv  Conv  Rel  ce  oni  eLU  ReLU  Conv  Conv  Conv  Conv  Conv  Xout-im  ReLi  ReLi  ReL!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  R  Yin5  forward Fourier transformation, restoration of acquired k-  space data and inverse Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet was  implemented with 5 cascades, K=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The weights of SG, SS, and  DC blocks were tied across cascades to limit model complexity  [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The only exception were fusion coefficients that determine  the relative weighting of the SG and SS blocks at each  stage (γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='., γk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', γ5 η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='ηk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', η5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' These fusion parameters  were kept distinct across cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Coil-combination on the  recovered multi-coil images was performed using sensitivity  maps estimated via ESPIRiT [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' MRI Dataset  Experimental demonstrations were performed using brain  MRI scans from the NYU fastMRI database [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Here, contrast-  enhanced T1-weighted (cT1-weighted) and T2-weighted acquisi-  tions were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The fastMRI dataset contains volumetric  MRI data with varying image and coil dimensionality across  subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that a central aim of this work was to sys-  tematically examine the learning capabilities of models for  varying number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To minimize potential  biases due to across-subject variability in MRI protocols, here  we selected subjects with matching imaging matrix size and  number of coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To do this, we only selected subjects with at  least 10 cross-sections and only the central 10 cross-sections  were retained in each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' We further selected subjects  with an in-plane matrix size of 256x320 for cT1 acquisitions,  and of 288x384 for T2 acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Background regions in  MRI data with higher dimensions were cropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lastly, we  restricted our sample selection to subjects with at least 5 coil  elements, and geometric coil compression [73] was applied to  unify the number of coils to 5 in all subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Fully-sampled acquisitions were retrospectively undersam-  pled to achieve acceleration rates of R=4x and 8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Random  undersampling patterns were designed via either a bi-variate  normal density function peaking at the center of k-space, or a  uniform density function across k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The standard deviation  of the normal density function was adjusted to maintain  the expected value of R across k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The fully-sampled  calibration region spanned a 40x40 window in central k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Competing Methods  PSFNet was compared against several state-of-the-art ap-  proaches including SG methods, SS methods, and traditional  PI reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For methods containing SG priors, both  supervised and unsupervised variants were implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNet: A supervised variant of PSFNet was trained using  paired sets of undersampled and fully-sampled acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNetUS: An unsupervised variant of PSFNet was im-  plemented using self-supervision based on only undersampled  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Acquired data were split into two non-overlapping  sets where 40% of samples was reserved for evaluating the  training loss and 60% of samples was reserved to enforce DC  [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  MoDL: A supervised SG methods based on an unrolled  architecture with tied weights across cascades was used [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  MoDL serially interleaves SG and DC blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The number of  cascades and the structure of SG and DC blocks were identical  to those in PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  MoDLUS: An unsupervised variant of MoDL was imple-  mented using self-supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A 40%-60% split was performed  on acquired data to evaluate the training loss and enforce data  consistency, respectively [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  sRAKI-RNN: An SS method was implemented based on the  MoDL architecture [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Learning was performed to minimize  DC loss on the fully-sampled calibration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Calibration  data were randomly split with 75% of samples used to define  the training loss and 25% of samples reserved to enforce DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Multiple input-output pairs were produced for a single test  sample by utilizing this split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPIRiT: A traditional PI reconstruction was performed using  the SPIRiT method [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Reconstruction parameters including  the regularization weight for kernel estimation (κ), kernel size  (w), and the number of iterations (Niter) were independently  optimized for each reconstruction task via cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPARK: An SS method was used to correct residual errors  from an initial SPIRiT reconstruction [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Learning was  performed to minimize DC loss on the calibration region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The  learned SS prior was then used to correct residual errors in  the remainder of k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Optimization Procedures  For all methods, hyperparameter selection was performed  via cross-validation on a three-way split of data across subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  There was no overlap among training, validation and test sets  in terms of subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Data from 10 subjects were reserved for  validation, and data from a separate set of 40 subjects were  reserved for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The number of subjects in the training  set was varied from 1 to 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hyperparameters that maximized  peak signal-to-noise ratio (PSNR) on the validation set were  selected for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Training was performed via the Adam optimizer with  learning rate ζ=10−4, β1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='90 and β2=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='99 [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' All deep  learning methods were trained to minimize hybrid ℓ1-ℓ2-  norm loss between recovered and target data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', between  reconstructed and ground truth images for PSFNet, between  recovered and acquired k-space samples for PSFNetUS) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  For PSFNet and MoDL, the selected number of epochs was  200, batch size was set to 2 for the limited number of training  samples (Nsamples <10), and to 5 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In DC blocks,  λ = ∞ was used to enforce strict data consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For PSFNet  and SPIRiT, the kernel width (w) and regularization parameter  (κ) values were set as (κ, w) = (10−2, 9) at R= 4 and (10−2,  9) at R=8 for cT1-weighted reconstructions, and as (100, 17)  at R=4 and (10−2, 17) at R=8 for T2-weighted reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  For SPIRiT, the number of iterations Niter was set as 13  at R=4 and 27 at R=8 for cT1-weighted reconstructions, 20  at R=4 and 38 at R=8 for T2-weighted reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For  sRAKI-RNN, the selected number of epochs was 500 and  batch size was set to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' All other optimization procedures  were identical to MoDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For SPARK, network architecture  and training procedures were adopted from [17], except for the  number of epochs (Nepoch) and learning rate (ζ) which were  optimized on the validation set as (Nepoch, ζ)= (100, 10−2) For  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2: Average PSNR across test subjects for (a) cT1- and (b)  T2-weighted image reconstructions at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Model training was  performed for varying number of training samples (Nsamples,  lower x-axis) and thereby training subjects (Nsubjects, upper  x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Results are shown for SPIRiT, SPARK, sRAKI-RNN,  MoDL and PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  cT1-weighted reconstructions, and (Nepoch, ζ)= (250, 10−3)  for T2-weighted reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  All competing methods were executed on an NVidia RTX  3090 GPU, and models were coded in Tensorflow except for  SPARK which was implemented in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPARK was  implemented using the toolbox at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='com/  YaminArefeen/spark_mrm_2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The code to imple-  ment PSFNet will be available publicly at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  com/icon-lab/PSFNet upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Performance Metrics  Performance assessments for reconstruction methods were  carried out by visual observations and quantitative metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSNR and structural similarity index (SSIM) were used  for quantitative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For each method, metrics were  computed on coil-combined images from the reconstruction  and from the fully-sampled ground truth acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Statistical  differences between competing methods were examined via  non-parametric Wilcoxon signed-rank tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Experiments  Several different experiments were conducted to system-  atically examine the performance of competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Assessments aimed to investigate reconstruction performance  under low training data regimes, generalization performance  in case of mismatch between training and testing domains,  contribution of the parallel-stream design to reconstruction per-  formance, sensitivity to hyperparameter selection, performance  in unsupervised learning, and computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Performance in low-data regimes: Deep SG methods for  MRI reconstruction typically suffer from suboptimal perfor-  mance as the size of the training dataset is constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To  systematically examine reconstruction performance, we trained  supervised variants of PSFNet and MoDL while the number  of training samples (Nsamples) was varied in the range [2-  500] cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To attain a given number of samples,  sequential selection was performed across subjects and across  cross-sections within each subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Thus, the number of unique  subjects included in the training set roughly corresponded to  Nsamples/10 (since there were 10 cross-sections per subject).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SS reconstructions were also performed with sRAKI-RNN,  SPIRiT and SPARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In the absence of fully-sampled ground  truth data to guide the learning of the prior, unsupervised  training of deep reconstruction models may prove relatively  more difficult compared to supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In turn, this  may elevate requirements on training datasets for unsupervised  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To examine data efficiency for unsupervised training,  we compared the reconstruction performance of PSFNetUS and  MoDLUS as Nsamples was varied in the range of [2-500] cross  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Comparisons were also provided against sRAKI-RNN,  SPIRiT and SPARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Generalization performance: Deep reconstruction models  can suffer from suboptimal generalization when the MRI data  distribution shows substantial variation between the training and  testing domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To examine generalizability, PSFNet models  were trained on data from a source domain and tested on data  from a different target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The domain-transferred models  were then compared to models trained and tested directly  in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Three different factors were altered to  induce domain variation: tissue contrast, undersampling pattern,  and acceleration rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' First, the capability to generalize to  different tissue contrasts was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Models were trained  on data from a source contrast and tested on data from  a different target contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Domain-transferred models were  compared to target-domain models trained on data from the  target contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Next, the capability to generalize to different  undersampling patterns was assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Models were trained on  data undersampled with variable-density patterns and tested  on data undersampled with uniform-density patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Domain-  transferred models were compared to target-domain models  trained on uniformly undersampled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lastly, the capability  to generalize to different acceleration rates was examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Models were trained on acquisitions accelerated at R=4x and  tested on acquisitions accelerated at R=8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Domain-transferred  models were compared to target-domain models trained at  R=8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Sensitivity to hyperparameters: SS priors are learned  from individual test scans as opposed to SG priors trained  on larger training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Thus, SS priors might show  elevated sensitivity to hyperparameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' We assessed  the reliability of reconstruction performance against suboptimal  Nsubjects  (r  X  40  39  PSNR (dB)  38  SPIRiT  37  SPARK  36  SRAKI-RNN  MoDL  35  PSFNet  34  Nsamples  b)  Nsubjects  X  6  40  39  38  PSNR (dB)  SPIRiT  SPARK  SRAKI-RNN  MoDL  34  PSFNet  33  Nsamples7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the  zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Error maps for each method  are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' MoDL and PSFNet were trained on 10 cross-sections from a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPIRiT, SPARK and  sRAKI-RNN directly performed inference on test data without a priori model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet shows superior performance to  competing methods in terms of residual reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 4: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDL, and PSFNet along with the  zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Error maps for each method  are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' MoDL and PSFNet were trained on 10 cross-sections from a single subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPIRiT, SPARK and  sRAKI-RNN directly performed inference on test data without a priori model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet shows superior performance to  competing methods in terms of residual reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  hyperparameter selection for SS priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For this purpose,  analyses were conducted on SPIRiT, SPARK and PSFNet  that embody SS methods to perform linear reconstructions  in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The set of hyperparameters examined included  regularization parameters for kernel estimation (κ) and kernel  size (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Separate models were trained using κ in range [10-3-  100] and w in range [5-17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Computational complexity: Finally, we assessed the com-  putational complexity of competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For each method,  training and inference times were measured for a single subject  with 10 cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Each cross-section had an imaging  matrix size of 256x320 and contained data from 5 coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For  all methods including SS priors, hyperparameters optimized  for cT1-weighted reconstructions at R=4 were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Ablation analysis: To assess the contribution of the parallel-  stream design in PSFNet, a conventional unrolled variant  of PSFNet was formed, named as PSFNetSerial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNetSerial  combined the SG and SS priors via serial projections as  described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Modeling procedures and the design of  SG and SS blocks were kept identical between PSFNet and  PSFNetSerial for fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Performance was assessed as  Nsamples was varied in the range of [2-500] cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Performance in Low-Data Regimes  Common SG methods for MRI reconstruction are based  on deep networks that require copious amounts of training  data, so performance can substantially decline on limited  training sets [28], [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, PSFNet leverages an SG  prior to concurrently reconstruct an image along with an SS  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Therefore, we reasoned that its performance should scale  favorably under low-data regimes compared to SG methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' We  also reasoned that PSFNet should yield elevated performance  compared to SS methods due to residual corrections from its SG  prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To test these predictions, we trained supervised variants  of PSFNet and MoDL along with SPIRiT, sRAKI-RNN, and  SPARK while the number of training samples (Nsamples) was  ZF  SPIRiT  SPARK  SRAKI-RNN  MoDI  PSFNet  Reference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='12  Error  0ZF  SPIRiT  SPARK  SRAKI-RNN  MoDL  PSFNet  Reference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='12  Error  08  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 5: Weighting of the SG (γ) and SS (η) blocks in the  final cascade of PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Weights were averaged across models  trained for cT1- and T2-weighted reconstructions at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Model training was performed for varying number of training  samples (Nsamples, lower x-axis) and thereby training subjects  (Nsubjects, upper x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Both blocks are equally weighted  with very limited training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As Nsamples increases, the  weighting of the SG prior becomes more dominant over the  weighting of the SS prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  systematically varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Figure 2 displays PSNR performance  for cT1-weighted and T2-weighted image reconstruction as a  function of Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet outperforms the scan-general  MoDL method for all values of Nsamples (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As  expected, performance benefits with PSFNet become more  prominent towards lower values of Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet also  outperforms traditional SPIRiT and scan-specific sRAKI-RNN  and SPARK methods broadly across the examined range  of Nsamples (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that while MoDL requires  Nsamples = 30 (3 subjects) to offer on par performance to  SS methods, PSFNet yields superior performance with as few  as Nsamples = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Representative reconstructions for cT1- and  T2-weighted images are depicted in Figures 3 and 4, where  Nsamples = 10 from a single subject were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNet yields lower reconstruction errors compared to all other  methods in this low-data regime, where competing methods  either show elevated noise or blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Naturally, the performance of PSFNet increases as more  training samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Since the SS prior is inde-  pendently learned for individual samples, it should not elicit  systematic performance variations depending on Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Thus, the performance gains can be attributed to improved  learning of the SG prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In turn, we predicted that PSFNet  would put more emphasis on its SG stream as its reliability  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To examine this issue, we inspected the weightings  of the SG (γ) and SS (η) streams as the training set size was  varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Figure 5 displays weightings at the last cascade as a  function of Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For lower values of Nsamples where the  quality of the SG prior is relatively limited, the SG and SS  priors are almost equally weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, as the learning  of the SG prior improves with higher Nsamples, the emphasis  on the SG prior increases while the SS prior is less heavily  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6: Average PSNR across test subjects for (a) cT1- and (b)  T2-weighted image reconstructions at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Model training was  performed for varying number of training samples (Nsamples,  lower x-axis) and thereby training subjects (Nsubjects, upper  x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Results are shown for SPIRiT, SPARK, sRAKI-RNN,  MoDLUS and PSFNetUS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  weighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  We then questioned whether the performance benefits of  PSFNet are also apparent during unsupervised training of  deep network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For this purpose, unsupervised variants  PSFNetUS and MoDLUS were trained via self-supervision [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNetUS was compared against MoDLUS, SPIRiT, sRAKI-  RNN, and SPARK while the number of training samples  (Nsamples) was systematically varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Figure 6 displays PSNR  performance for cT1-weighted and T2-weighted image recon-  struction as a function of Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Similar to the supervised  setting, PSFNetUS outperforms MoDLUS for all values of  Nsamples (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05), and the performance benefits are more  noticeable at lower Nsamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In this case, however, MoDLUS  is unable to reach the performance of the best performing  SS method (SPARK) even at Nsamples = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast,  PSFNetUS starts outperforming SPARK with approximately  Nsamples = 50 (5 subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The enhanced reconstruction  quality with PSFNetUS is corroborated in representative re-  constructions for cT1- and T2-weighted images depicted in  Figures 7 and 8, where Nsamples = 100 were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Taken together, these results indicate that the data-efficient  nature of PSFNet facilitates the training of both supervised  and unsupervised MRI reconstruction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Nsubjects  5  6  Y (SG)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='60  n (Ss)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='55  Weighting  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='40  4  6  8  NsamplesNsubjects  a  X  40  39  38  37  (dB)  36  SPIRiT  35  PSNR  34  SPARK  SRAKI - RNN  33  MoDLus  32  PSFNetus  31  30  2  X  Nsamples  b)  Nsubjects  5  6  40  39  38  37  (dB)  36  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='343  SPIRiT  PSNR  SPARK  SRAKI - RNN  MoDLus  32  PSFNetus  31  30  X  Nsamples9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7: cT1-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with  the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Error maps for each  method are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPIRiT,  SPARK and sRAKI-RNN directly performed inference on test data without a priori model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNetUS shows superior  performance to competing methods in terms of residual reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 8: T2-weighted image reconstructions at R=4x via SPIRiT, SPARK, sRAKI-RNN, MoDLUS, and PSFNetUS along with  the zero-filled reconstruction (ZF) and the reference image obtained from the fully-sampled acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Error maps for each  method are shown in the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' MoDLUS and PSFNetUS were trained on 100 cross-sections (from 10 subjects).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPIRiT,  SPARK and sRAKI-RNN directly performed inference on test data without a priori model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNetUS shows superior  performance to competing methods in terms of residual reconstruction errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Generalization Performance  An important advantage of SS priors is that they allow  model adaptation to individual test samples, thereby promise  enhanced performance in out-of-domain reconstructions [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Yet, SG priors with fixed parameters might show relatively  limited generalizability during inference [23], [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' To assess  generalization performance, we introduced domain variations  by altering three experimental factors: tissue contrast, under-  sampling pattern, and acceleration rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For methods comprising  SG components, we built both target-domain models that were  trained in the target domain, and domain-transferred models  that were trained in a non-target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' We then compared  the reconstruction performances of the two models in the target  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  First, we examined generalization performance when the  tissue contrast varied between training and testing domains  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', trained on cT1, tested on T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Table I lists performance  metrics for competing methods with Nsamples = 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' While  performance losses are incurred for domain-transferred PSFNet-  DT and MoDL-DT models that contain SG components, these  losses are modest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On average, MoDL-DT shows a loss of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3dB PSNR and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1% SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05), and PSFNet-DT  shows a loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2dB PSNR and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1% SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note  that PSFNet-DT still outperforms the closest competing SS  method by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2dB PSNR and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8% SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Second, we examined generalization performance when mod-  els were trained with variable-density and tested on uniform-  density undersampling patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Table II lists performance  metrics for competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On average across tissue  contrasts, MoDL-DT suffers a notable performance loss of  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6dB PSNR and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5% SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, PSFNet-  DT shows a relatively limited loss of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4dB PSNR and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2%  SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that PSFNet-DT again outperforms the  closest competing SS method by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4dB PSNR and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7% SSIM  (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Third, we examined generalization performance when models  were trained at R=4x and tested on R=8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Table III lists  ZF  SPIRiT  SPARK  sRAKI-RNN  MoDL  PSFNet  Reference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='12  Error  0ZF  SPIRiT  SPARK  sRAKI-RNN  MoDL  PSFNet  Reference  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='12  Error  010  TABLE I: Generalization across tissue contrasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSNR and  SSIM values (mean±standard error) across test subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Results are shown for scan-specific models (SPIRiT, SPARK,  sRAKI-RNN), target-domain models (MoDL, PSFNet) and  domain-transferred models (MoDL-DT, PSFNet-DT) at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The tissue contrast in the target domain is listed in the left-most  column (cT1 or T2), domain-transferred models were trained  for the non-target tissue contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPIRiT  SPARK  sRAKI-  RNN  MoDL  MoDL-DT  PSFNet  PSFNet-  DT  PSNR  cT1  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  T2  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  SSIM  cT1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  T2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  TABLE II: Generalization across undersampling patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSNR  and SSIM values (mean±standard error) across test subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Results are shown for Results are shown for scan-specific  models (SPIRiT, SPARK, sRAKI-RNN), target-domain models  (MoDL, PSFNet) and domain-transferred models (MoDL-DT,  PSFNet-DT) at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Domain-transferred models were trained  with variable-density undersampling, and tested on uniform-  density undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Target-domain models were trained and  tested with uniform-density undersampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPIRiT  SPARK  sRAKI-  RNN  MoDL  MoDL-DT  PSFNet  PSFNet-  DT  PSNR  cT1  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  T2  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  SSIM  cT1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  T2  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  performance metrics for competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On average across  tissue contrasts, MoDL-DT suffers a notable performance loss  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0dB PSNR and performs slightly better in SSIM by  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2%SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05), whereas PSFNet-DT shows a lower loss  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6dB PSNR (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05) and performs similarly in SSIM  (p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet-DT outperforms the closest competing SS  method by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2dB PSNR and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9% SSIM (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Taken  together, these results clearly suggest that the SS prior in  PSFNet contributes to its improved generalization performance  over the scan-general MoDL method, while the SG prior in  PSFNet enables it to outperform competing SS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sensitivity to Hyperparameters  Parameters of deep networks that implement SS priors are to  be learned from a single test sample, so the resultant models can  show elevated sensitivity to the selection of hyperparameters  compared to SG priors learned from a collection of training  TABLE III: Generalization across acceleration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSNR  and SSIM values (mean±standard error) across test subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Results are shown for scan-specific models (SPIRiT, SPARK,  sRAKI-RNN), target-domain models (MoDL, PSFNet) and  domain-transferred models (MoDL-DT, PSFNet-DT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Domain-  transferred models were trained at R=4x and tested at R=8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Target-domain models were trained and tested at R=8x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPIRiT  SPARK  sRAKI-  RNN  MoDL  MoDL-DT  PSFNet  PSFNet-  DT  PSNR  cT1  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  T2  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  SSIM  cT1  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4  T2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='0  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='3  ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='1  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='9  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Thus, we investigated the sensitivity of PSFNet to key  hyperparameters of its SS prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' SPIRiT, SPARK and PSFNet  methods all embody a linear k-space reconstruction, so the  relevant hyperparameters are the regularization weight and  width for the convolution kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Performance was evaluated for  models were trained in the low-data regime (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', Nsamples =  10, 1 subject) for varying hyperparameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Figure 9 displays PSNR measurements for SPIRiT, SPARK  and PSFNet across κ in range (10-3-100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' While the perfor-  mance of SPIRiT and SPARK is notably influenced by κ,  PSFNet is minimally affected by sub-optimal selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On  average across contrasts, the difference between the maximum  and minimum PSNR values is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='4dB for SPIRiT, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5dB for  SPARK, and a lower 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='7dB for PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that PSFNet  outperforms competing methods across the entire range of  κ (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Figure 10 shows PSNR measurements for  competing methods across w in range (5-17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In this case,  all methods show relatively limited sensitivity to the selection  of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On average across contrasts, the difference between the  maximum and minimum PSNR values is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5dB for SPIRiT,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='5dB for SPARK, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='2dB for PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Again, PSFNet  outperforms competing methods across the entire range of  w (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Overall, our results indicate that PSFNet  yields improved reliability against sub-optimal hyperparameter  selection than competing SS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Computational Complexity  Next, we assessed the computational complexity of com-  peting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Table IV lists the training times of methods  with SG priors, MoDL and PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that the remaining  SS based methods do not involve a pre-training step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As it  involves learning of an SS prior on each training sample,  PSFNet yields elevated training time compared to MoDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In  return, however, it offers enhanced generalization performance  and data-efficient learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Table IV also lists the inference  times of SPIRiT, SPARK, sRAKI-RNN, MoDL and PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  MoDL and PSFNet that employ SG priors with fixed weights  during inference offer fast run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, SPARK and  11  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 9: PSNR measurements were performed on recovered cT1-  and T2-weighted images at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bar plots in blue color show  average PSNR across κ ∈ 10-3-101 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', the regularization  parameter for kernel estimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Error bars denote the 90%  interval across κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bar plots in red color show PSNR for methods  that do not depend on the value of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 10: PSNR measurements were performed on recovered  cT1- and T2-weighted images at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bar plots in blue color  show the average PSNR across w ∈ 5-17 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=', the kernel size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Error bars denote the 90% interval across w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bar plots in red  color show PSNR for methods that do not depend on the value  of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  sRAKI-RNN that involve SS priors learned on individual test  samples have a high computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Although PSFNet  also embodies an SS prior, its uses a relatively lightweight  linear prior as opposed to the nonlinear priors in competing  SS methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Therefore, PSFNet benefits from data-efficient  learning while maintaining computationally-efficient inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ablation Analysis  To demonstrate the value of the parallel-stream fusion  strategy in PSFNet over conventional unrolling, PSFNet was  compared against a variant model PSFNetSerial that combined  SS and SG priors through serially alternated projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Separate models were trained with number of training samples  in the range Nsamples=[2-500].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Performance in cT1- and T2  weighted image reconstruction is displayed in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  PSFNet significantly improves reconstruction performance over  PSFNetSerial across the entire range of Nsamples considered  (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05), and the benefits grow stronger for smaller training  sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' On average across contrasts for Nsamples < 10, PSFNet  TABLE IV: Computational complexity of competing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Training and inference times for data from a single subject,  with 10 cross-sections, imaging matrix size 256x320 and 5  coils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Run times are listed for SPARK, sRAKI-RNN, MoDL,  and PSFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  SPIRiT  SPARK  sRAKI-RNN  MoDL  PSFNet  Training(s) 132  337  Inference(s)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='85  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='35  285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 11: Average (a) PSNR and (b) SSIM values for cT1- and  T2-weighted image reconstructions at R=4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Model training was  performed for varying number of training samples (Nsamples,  lower x-axis) and thereby training subjects (Nsubjects, upper  x-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Results are shown for PSFNet and PSFNetSerial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  outperforms PSFNetSerial by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='8dB PSNR and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='6% SSIM  (p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' These results indicate that the parallel-stream  fusion of SG and SS priors in PSFNet is superior to the serial  projections in conventional unrolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' DISCUSSION AND CONCLUSION  In this study, we introduced PSFNet for data-efficient training  of deep reconstruction models in accelerated MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' PSFNet  synergistically fuses SS and SG priors in a parallel-stream  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The linear SS prior improves learning efficiency  while mataining relatively low computational footprint, whereas  the nonlinear SG prior enables improved reconstruction per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' For both supervised and unsupervised training  setups, the resulting model substantially reduces dependence  on the availability of large MRI datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' it  achieves competitive inference times to SG methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' and  38  (dB)  36  PSNR  34  32  30  MoDL  SPIRiT  SPARK  PSFNet  SRAKI - RNN38  (dB)  36  PSNR  34  32  30  MoDL  SPIRiT  SPARK  PSFNet  SRAKI - RNNNsubjects  a  X  40  39  PSNR (dB)  38  37  PSFNet (cTi)  36  PSFNet (T2)  PSFNetserial (cTi)  35  PSFNetserial (T2)  34  Nsamples  b)  Nsubjects  5  X  6  97  96  (%)  SSIM  95  PSFNet (cTi)  PSFNet (T2)  94  PSFNetserial (cTi)  PSFNetserial (T2)  93  Nsamples12  reliably generalizes across tissue contrasts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' sampling patterns  and acceleration rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Several prominent approaches have been introduced in  the literature to address the training requirements of deep  models based on SG priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' One approach is to pre-train  models on readily available datasets from a separate source  domain and then to fine-tune on several tens of samples  from the target domain [28], [59] or else perform SS fine-  tuning [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' This transfer learning approach relaxes the domain  requirements for training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, the domain-  transferred models might be suboptimal when training and  testing data distributions are divergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In such cases, additional  training for domain-alignment might be necessary to mitigate  performance losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, PSFNet contains a SS prior that  allows it to better generalize to out-of-domain data without  further training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Another approach is to build unsupervised  models to alleviate dependency on training datasets with paired  undersampled, fully-sampled acquisitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Model training can  be performed either directly on undersampled acquisitions  via self-supervision [64] or on unpaired sets of undersampled  and fully-sampled acquisitions via cycle-consistent learning  [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' This approach can prove beneficial when fully-sampled  acquisitions are costly to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Nonetheless, the resulting  models still require relatively large datasets form tens of  subjects during training [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that our experiments on  self-supervised variants of PSFNet and MoDL suggest that  unsupervised models can be more demanding for data than their  supervised counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Therefore, the data-efficiency benefits  of PSFNet might be particularly useful for unsupervised deep  MRI reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  A fundamentally different framework to lower requirements  on training datasets while offering improved generalizability  is based on SS priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In this case, learning can be performed  directly on test data and models can be adapted to each scan  [15], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A group of studies have proposed SS methods based  on relatively compact nonlinear models to facilitate learning  during inference [15], [17], [18], [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' However, because learn-  ing is performed in central k-space, these methods implicitly  assume that local relationships among spatial frequency samples  are largely invariant across k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' While the SS prior in  PSFNet also rests on a similar assumption, the SG components  helps correct residual errors that can be introduced due to this  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Another group of studies have alternatively adopted  the deep image prior (DIP) approach to build SS methods  [19], [20], [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In DIP, unconditional deep network  models that map latent variables onto images are used as  native priors for MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' The priors are learned by ensuring  the consistency of reconstructed and acquired data across the  entire k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Despite improved generalization, these relatively  more complex models require increased inference times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In  comparison, PSFNet provides faster inference since the weights  for its SG prior are fixed, and its SS prior involves a compact  linear operator that is easier to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Few independent studies on MRI have proposed approaches  related to PSFNet by combining nonlinear and linear recon-  structions [6], [17], [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Residual RAKI and SPARK methods  initially perform a linear reconstruction, and then use an SS  method to correct residual errors via minimizing a DC loss in  the calibration region [17], [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' As local relationships among  data samples might vary across k-space, the learned SS priors  might be suboptimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moreover, these methods perform online  learning of nonlinear SS priors that introduces relatively high  computational burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, PSFNet incorporates an SG  prior to help improve reliability against sub-optimalities in the  SS prior, and uses a linear SS prior for efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Another  related method is GrappaNet that improves reconstruction per-  formance by cascading GRAPPA and network-based nonlinear  reconstruction steps [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' While [6] intends to improve image  quality, the main aim of our study is to improve practicality  by lowering training data requirements of deep models, and  improving domain generalizability without elevating inference  times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Note that GrappaNet follows the conventional unrolling  approach by performing serially alternated projections through  linear and nonlinear reconstructions, which can lead to error  propagation under low-data regimes [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' In contrast, PSFNet  maintains linear and nonlinear reconstructions as two parallel  streams in its architecture, and learns to optimally fuse the  information from the two streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  The proposed method can be improved along several lines  of technical development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' First, to improve the capture of high-  frequency information by the SG prior, an adversarial loss  term along with a discriminator subnetwork can be included  in PSFNet [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' It remains to be demonstrated whether the  data-efficiency benefits of PSFNet are apparent for adversarial  training setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Second, nonlinear activation functions can  be included in the SS stream to improve the expressiveness  of the SS prior [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' While learning of nonlinear priors can  elevate inference complexity, generalization performance might  be further improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Third, the expressiveness of both SS  and SG priors might be enhanced by incorporating attention  mechanisms as proposed in recent transformer models [81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Fourth, using multimodal image fusion approaches can improve  performance in case of having a repository with multimodal  data [82], [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lastly, the benefits of transfer learning and  PSFNet can be aggregated by pre-training the SG prior on  natural images to further lower requirements on training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ACKNOWLEDGMENTS  This work was supported in part by a TUBA GEBIP 2015  fellowship, by a BAGEP 2017 fellowship, and by a TUBITAK  121E488 grant awarded to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bauer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wiest, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Nolte, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Reyes, “A survey of mri-based  medical image analysis for brain tumor studies,” Physics in Medicine &  Biology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' R97, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Shoeibi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Khodatars, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jafari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ghassemi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moridian,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Alizadehsani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ling, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Khosravi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Alinejad-Rokny, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lam,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Fuller-Tyszkiewicz, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Acharya, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Anderson, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gorriz, “Diagnosis of brain diseases in fusion of neuroimaging  modalities using deep learning: A review,” Information Fusion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 93,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 85–117, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Qian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Koh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sim, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Guan,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, “Structural and diffusion mri based schizophrenia  classification using 2d pretrained and 3d naive convolutional neural  networks,” Schizophrenia Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 243, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 330–341, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Fernando and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tsokos, “Deep and statistical learning in  biomedical imaging: State of the art in 3d mri brain tumor segmentation,”  Information Fusion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 92, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 450–465, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  13  [5] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Qi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cong, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, “Brain tumor  segmentation based on the fusion of deep semantics and edge information  in multimodal mri,” Information Fusion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 91, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 376–387, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sriram, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zbontar, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Murrell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zitnick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Defazio, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Sodickson, “GrappaNet: Combining parallel imaging with deep learning  for multi-coil MRI reconstruction,” in Proceedings of the IEEE/CVF  Conference on Computer Vision and Pattern Recognition (CVPR), June  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 14 303–14 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pruessmann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Weiger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Scheidegger, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Boesiger,  “SENSE: sensitivity encoding for fast MRI.” Magnetic Resonance in  Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 952–62, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Griswold, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jakob, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Heidemann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Nittka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jellus,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kiefer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Haase, “Generalized autocalibrating partially  parallel acquisitions (GRAPPA),” Magnetic Resonance in Medicine,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1202–1210, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Donoho, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, “Sparse MRI: The application  of compressed sensing for rapid MR imaging,” Magnetic Resonance in  Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1182–1195, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Majumdar, “Improving synthesis and analysis prior blind compressed  sensing with low-rank constraints for dynamic mri reconstruction,”  Magnetic resonance imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 174–179, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, “SPIRiT: Iterative self-consistent parallel  imaging reconstruction from arbitrary k-space.” Magnetic Resonance in  Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 457–71, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [12] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Li, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Xu, “ADMM-CSNet: A deep learning  approach for image compressive sensing,” IEEE Transactions on Pattern  Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 521–538, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Schlemper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Caballero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hajnal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Price, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Rueckert,  “A Deep Cascade of Convolutional Neural Networks for MR Image  Reconstruction,” in International Conference on Information Processing  in Medical Imaging, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 647–658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hammernik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Klatzer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kobler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Recht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sodickson,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pock, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Knoll, “Learning a variational network for reconstruction  of accelerated MRI data,” Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 79,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3055–3071, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Weing¨artner, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U˘gurbil, “Scan-  specific robust artificial-neural-networks for k-space interpolation (RAKI)  reconstruction: Database-free deep learning for fast imaging,” Magnetic  Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 439–453, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Garg, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Haldar, “LORAKI: Autocalibrated recurrent  neural networks for autoregressive MRI reconstruction in k-space,” arXiv  preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='09390, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Arefeen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Beker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cho, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Adalsteinsson, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bilgic,  “Scan-specific artifact reduction in k-space (spark) neural networks  synergize with physics-based reconstruction to accelerate mri,” Magnetic  Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 87, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 764–780, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Weing¨artner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Stuber,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ugurbil, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya, “Accelerated coronary MRI with sRAKI:  A database-free self-consistent neural network k-space reconstruction for  arbitrary undersampling,” PLOS ONE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 15, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1–13, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Korkmaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yurt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozbey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cukur, “Unsupervised  mri reconstruction via zero-shot learned adversarial transformers,” IEEE  Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1747–1763, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Arora, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Roeloffs, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “Untrained modified deep decoder  for joint denoising and parallel imaging reconstruction,” in Proceedings  of the 28th Annual Meeting of the ISMRM, 2020, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Darestani and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Heckel, “Accelerated mri with un-trained neural  networks,” IEEE Transactions on Computational Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  724–733, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Narnhofer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hammernik, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Knoll, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pock, “Inverse GANs  for accelerated MRI reconstruction,” in Proceedings of the SPIE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  11138, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 381 – 392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [23] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tezcan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Baumgartner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Luechinger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pruessmann, and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Konukoglu, “MR image reconstruction using deep density priors,”  IEEE Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1633–1642,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [24] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liang,  “Highly undersampled magnetic resonance imaging reconstruction using  autoencoding priors,” Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 83, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 322–336, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Eo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hwang, “KIKI-net: cross-  domain convolutional neural networks for reconstructing undersampled  magnetic resonance images,” Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 80,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2188–2201, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mardani, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zaharchuk,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Xing, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, “Deep generative adversarial neural networks  for compressive sensing MRI,” IEEE Transactions on Medical Imaging,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 167–179, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [27] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Aggarwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mani, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jacob, “MoDL: Model-Based  deep learning architecture for inverse problems,” IEEE Transactions on  Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 394–405, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozbey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ atlı, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “A transfer-learning  approach for accelerated MRI using deep neural networks,” Magnetic  Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 663–685, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yoo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tak, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ye, “Deep residual learning for accel-  erated MRI using magnitude and phase networks,” IEEE Transactions  on Biomedical Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 65, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1985–1995, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Valanarasu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jiang, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Patel,  “Over-and-under complete convolutional rnn for mri reconstruction,” in  International Conference on Medical Image Computing and Computer-  Assisted Intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Springer, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 13–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [31] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yiasemis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sonke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S´anchez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Teuwen, “Recurrent  variational network: A deep learning inverse problem solver applied  to the task of accelerated mri reconstruction,” in Proceedings of the  IEEE/CVF Conference on Computer Vision and Pattern Recognition,  2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 732–741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hou and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Li, “Idpcnn: Iterative denoising and projecting cnn for  mri reconstruction,” Journal of Computational and Applied Mathematics,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 406, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 113973, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [33] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ramzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chaithya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Starck, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ciuciu, “Nc-pdnet: a  density-compensated unrolled network for 2d and 3d non-cartesian mri  reconstruction,” IEEE Transactions on Medical Imaging, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [34] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kwon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kim, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Park, “A parallel MR imaging method using  multilayer perceptron,” Medical Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 44, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6209–6224,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Su, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ying, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Feng, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liang, “Accelerating magnetic resonance imaging via deep learning,”  in IEEE 13th International Symposium on Biomedical Imaging (ISBI),  2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 514–517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ye, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Han, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cha, “Deep convolutional framelets: A general  deep learning framework for inverse problems,” SIAM Journal on Imaging  Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 11, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 991–1048, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [37] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yoon, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gong, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chatnuntawech, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bilgic, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ko,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jung, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Setsompop, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zaharchuk, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee,  “Quantitative susceptibility mapping using deep neural network: QSMnet,”  NeuroImage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 179, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 199–206, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [38] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hyun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Seo, “Deep learning  for undersampled MRI reconstruction,” Physics in Medicine and Biology,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 63, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 13, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 135007, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hauptmann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Arridge, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lucka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Muthurangu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Steeden,  “Real-time cardiovascular MR with spatio-temporal artifact suppression  using deep learning–proof of concept in congenital heart disease,”  Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1143–1156, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [40] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yaman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hong, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya,  “Dense recurrent neural networks for accelerated MRI: History-cognizant  unrolling of optimization algorithms,” IEEE Journal of Selected Topics  in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1280–1291, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Qin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Schlemper, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Caballero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Price, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hajnal, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Rueckert, “Convolutional recurrent neural networks for dynamic MR  image reconstruction,” IEEE Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 38,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 280–290, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [42] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Quan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Nguyen-Duc, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jeong, “Compressed sensing  MRI reconstruction with cyclic loss in generative adversarial networks,”  IEEE Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1488–1497,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yurt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Shahdloo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ildız, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tınaz, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur,  “Prior-guided image reconstruction for accelerated multi-contrast MRI  via generative adversarial networks,” IEEE Journal of Selected Topics in  Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1072–1087, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [44] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Firmin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yang, “Wavelet improved GAN for MRI  reconstruction,” in Proceedings of SPIE, Medical Imaging 2021: Physics  of Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 11595, 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1159513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [45] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Elmas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Korkmaz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ceyani, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Susam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ozbey,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Avestimehr, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “Federated learning of generative image  priors for mri reconstruction,” IEEE Transactions on Medical Imaging,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yaqub, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jinchao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ahmed, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Arshid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bilal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akhter,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zia, “Gan-tl: Generative adversarial networks with transfer  learning for mri reconstruction,” Applied Sciences, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 17, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  8841, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [47] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Korkmaz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  ¨Ozbey, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cukur, “Mri reconstruction with  conditional adversarial transformers,” in International Workshop on  14  Machine Learning for Medical Image Reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Springer, 2022,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 62–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [48] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Jiang, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Patel, “Reconformer:  Accelerated mri reconstruction using recurrent transformer,” arXiv  preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='09376, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [49] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' G¨ung¨or, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozt¨urk, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Korkmaz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Elmas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozbey, and  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “Adaptive diffusion priors for accelerated mri reconstruction,”  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='05876, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Peng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Guo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Patel, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chellappa, “Towards per-  formant and reliable undersampled mr reconstruction via diffusion model  sampling,” in International Conference on Medical Image Computing  and Computer-Assisted Intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Springer, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 623–633.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [51] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tamir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' De Goyeneche, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wollner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Brada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yu,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “High fidelity deep learning-based mri reconstruction with  instance-wise discriminative feature matching loss,” Magnetic Resonance  in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 88, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 476–491, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [52] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gadjimuradov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Benkert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Nickel, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Maier, “Robust partial  fourier reconstruction for diffusion-weighted imaging using a recurrent  convolutional neural network,” Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 87,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2018–2033, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [53] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ramzi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' G R, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Starck, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ciuciu, “Nc-pdnet: A density-  compensated unrolled network for 2d and 3d non-cartesian mri recon-  struction,” IEEE Transactions on Medical Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  1625–1638, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [54] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Polak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cauley, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bilgic, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bachert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Adalsteinsson, and  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Setsompop, “Joint multi-contrast variational network reconstruction  (jVN) with application to rapid 2D and 3D imaging,” Magnetic Resonance  in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1456–1469, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [55] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mardani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Alley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala,  “Deepspirit: Generalized parallel imaging using deep convolutional neural  networks,” in Proceedings of the 26th Annual Meeting of the ISMRM,  2018, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 0570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [56] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pooja, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ramzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chaithya, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ciuciu, “Mc-pdnet: Deep  unrolled neural network for multi-contrast mr image reconstruction from  undersampled k-space data,” in 2022 IEEE 19th International Symposium  on Biomedical Imaging (ISBI), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [57] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tavaf, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Torfi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ugurbil, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Van de Moortele, “Grappa-gans  for parallel mri reconstruction,” arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='03135, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [58] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sandino, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cheng, “Accelerating  cardiac cine mri using a deep learning-based espirit reconstruction,”  Magnetic Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 85, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 152–167, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [59] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Knoll, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hammernik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kobler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pock, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Recht, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Sodickson, “Assessment of the generalization of learned image recon-  struction and the potential for transfer learning,” Magnetic Resonance in  Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 116–128, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [60] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chaudhari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sandino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cole, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Larson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Gold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lungren, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hargreaves, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  Langlotz, “Prospective deployment of deep learning in mri: A framework  for important considerations, challenges, and recommendations for best  practices,” Journal of Magnetic Resonance Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  357–371, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [61] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yurt, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “A few-shot learning approach for  accelerated mri via fusion of data-driven and subject-driven priors,” in  Proceedings of the 29th Annual Meeting of the ISMRM, 2021, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Tamir, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “Unsupervised deep basis pursuit:  Learning reconstruction without ground-truth data,” in Proceedings of  the 27th Annual Meeting of the ISMRM, 2019, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 0660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [63] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Cole, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, “Fast unsupervised  mri reconstruction without fully-sampled ground truth data using genera-  tive adversarial networks,” in Proceedings of the IEEE/CVF International  Conference on Computer Vision (ICCV) Workshops, October 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  3988–3997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [64] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yaman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ellermann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U˘gurbil, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya, “Self-supervised learning of physics-guided reconstruc-  tion neural networks without fully sampled reference data,” Magnetic  resonance in medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3172–3191, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [65] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Eldeniz, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Gan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' An, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kamilov, “RARE:  Image reconstruction using deep priors learned without groundtruth,”  IEEE Journal of Selected Topics in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  1088–1099, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [66] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Xi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zheng,  “PARCEL: Physics-based Unsupervised Contrastive Representation Learn-  ing for Multi-coil MR Imaging,” arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='01494, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [67] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Uˇgurbil, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya,  “sraki-rnn: accelerated mri with scan-specific recurrent neural networks  using densely connected blocks,” in Wavelets and Sparsity XVIII, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  11138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' International Society for Optics and Photonics, 2019, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 111381B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [68] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Huang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Xian, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Qu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Metaxas,  “Dynamic mri reconstruction with end-to-end motion-guided network,”  Medical Image Analysis, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 68, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 101901, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [69] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hu, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yu, “Evaluation on  the generalization of a learned convolutional neural network for mri  reconstruction,” Magnetic Resonance Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 87, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 38–46, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [70] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ulyanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vedaldi, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lempitsky, “Deep image prior,” in  Proceedings of the IEEE Conference on Computer Vision and Pattern  Recognition (CVPR), 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 9446–9454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [71] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Uecker, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Murphy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Virtue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Elad, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “ESPIRiT-an eigenvalue approach to  autocalibrating parallel MRI: Where SENSE meets GRAPPA,” Magnetic  Resonance in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 71, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 990–1001, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [72] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Knoll, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zbontar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sriram, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Muckley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bruno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Defazio,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Parente, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Geras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Katsnelson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Chandarana, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Drozdzalv, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Romero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Rabbat, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vincent, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pinkerton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Wang,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yakubova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Owens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zitnick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Recht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Sodickson,  and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lui, “fastMRI: A publicly available raw k-space and DICOM  dataset of knee images for accelerated MR image reconstruction using  machine learning,” Radiology: Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  e190007, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [73] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “Coil compression  for accelerated imaging with Cartesian sampling.” Magnetic Resonance  in Medicine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 69, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 571–82, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [74] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ba, “Adam: a Method for Stochastic Optimization,”  in International Conference on Learning Representations, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [75] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dalmaz, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mirza, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Elmas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozbey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ceyani,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Avestimehr, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “One model to unite them all: Personalized  federated learning of multi-contrast mri synthesis,” arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='06509,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [76] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yaman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akc¸akaya, “High-fidelity  accelerated mri reconstruction by scan-specific fine-tuning of physics-  based neural networks,” in 2020 42nd Annual International Conference  of the IEEE Engineering in Medicine Biology Society (EMBC), 2020,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1481–1484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [77] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Mardani, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Pauly, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, “Wasserstein  GANs for MR imaging: from paired to unpaired training,” IEEE  transactions on medical imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 105–115, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [78] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hosseini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Moeller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Weing¨artner, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ugurbil,  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Akcakaya, “Scan-specific residual convolutional neural networks  for fast mri using residual raki,” in 2019 53rd Asilomar Conference on  Signals, Systems, and Computers, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1476–1480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [79] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Murphy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Alley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Demmel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Keutzer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Vasanawala, and  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Lustig, “Fast l1-SPIRiT compressed sensing parallel imaging MRI:  scalable parallel implementation and clinically feasible runtime.” IEEE  transactions on medical imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 31, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 1250–1262, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [80] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozbey, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dalmaz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dar, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Bedel, S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' ¨Ozturk, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' G¨ung¨or,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “Unsupervised medical image translation with adversarial  diffusion models,” arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='08208, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [81] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dalmaz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Yurt, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' C¸ ukur, “ResViT: Residual vision transformers  for multi-modal medical image synthesis,” IEEE Transactions on Medical  Imaging, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 41, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 2598–2614, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [82] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhou, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Hou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ma, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Ling, “Unified gradient- and  intensity-discriminator generative adversarial network for image fusion,”  Information Fusion, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 88, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 184–201, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content='  [83] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Dian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Li, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' Liu, “Sgfusion: A saliency guided deep-  learning framework for pixel-level image fusion,” Information Fusion,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 91, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
+page_content=' 205–214, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E0T4oBgHgl3EQfvAGt/content/2301.02613v1.pdf'}
diff --git a/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/2301.02388v1.pdf.txt b/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/2301.02388v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..20e1a54f2f9fbc3e6a2312be53c3606c90f24b21
--- /dev/null
+++ b/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/2301.02388v1.pdf.txt
@@ -0,0 +1,717 @@
+Generating corneal panoramic images from contact specular microscope images
+Yusuke Nagira1†, Yuzuha Hara2, Satoru Hiwa2
+Naoki Okumura3, Noriko Koizumi3 and Tomoyuki Hiroyasu2
+1Graduate School of Life and Medical Sciences, Doshisha University, Japan
+2Department of Biomedical Sciences and Informatics, Doshisha University, Japan
+3Department of Biomedical Engineering, Faculty of Life and Medical Sciences, Doshisha University, Japan
+(Tel: +81-774-65-6020, E-mail: tomo@is.doshisha.ac.jp)
+Abstract: The contact specular microscope has a wider angle of view than that of the non-contact specular microscope but still
+cannot capture an image of the entire cornea. To obtain such an image, it is necessary to prepare film on the parts of the image
+captured sequentially and combine them to create a complete image. This study proposes a framework to automatically generate
+an entire corneal image from videos captured using a contact specular microscope. Relatively focused images were extracted
+from the videos and panoramic compositing was performed. If an entire image can be generated, it is possible to detect guttae
+from the image and examine the extent of their presence. The system was implemented and the effectiveness of the proposed
+framework was examined. The system was implemented using custom-made composite software, Image Composite Software
+(ICS, K.I. Technology Co., Ltd., Japan, internal algorithms not disclosed), and a supervised learning model using U-Net was
+used for guttae detection. Several images were correctly synthesized when the constructed system was applied to 94 different
+corneal videos obtained from Fuchs endothelial corneal dystrophy (FECD) mouse model. The implementation and application
+of the method to the data in this study confirmed its effectiveness. Owing to the minimal quantitative evaluation performed, such
+as accuracy with implementation, it may pose some limitations for future investigations.
+Keywords: U-Net, Semantic Segmentation, Fuchs Endothelial Corneal Dystrophy, Corneal Endothelial Cell
+1. INTRODUCTION
+Fuchs endothelial corneal dystrophy (FECD) is a bilat-
+eral disease, wherein corneal endothelial cells are unable to
+maintain their hexagonal shape. It is characterized by the
+accelerated loss of corneal endothelial cells with changes in
+Descemet’s membrane, resulting in the formation of an ex-
+tracellular matrix called guttae [1]. In the United States, it
+is estimated that 4% of people over 40 years of age are af-
+fected by the disease, occurring more commonly in women
+and more frequently in people in their 40s and 50s. Corneal
+endothelial pump function is lost as the disease progresses,
+causing corneal edema [2]. Presently, corneal transplanta-
+tion is the only reliable treatment, and FECD accounts for
+39% of all corneal transplants performed, making it the most
+common cause of corneal transplantation worldwide [3].
+Rho kinase inhibitors have been reported to promote cell
+proliferation and adhesion to substrates, inhibit corneal en-
+dothelial cell apoptosis, and promote wound healing. There-
+fore, using Rho kinase inhibitor eye drops is a potential
+novel treatment approach alternative to corneal transplanta-
+tion [4][5].
+In drug discovery research for FECD, the state of the
+corneal endothelium, such as the guttae, is observed before
+and after the drug use and evaluated based on the increase
+or decrease in the number of cells. Doctors and researchers
+widely use specular microscopes to observe the state of the
+corneal endothelium.
+However, the range of the micro-
+scope’s imaging capability is limited and the current practice
+estimates the state of the entire cornea from its center. The
+endothelial cell density (ECD) is essential for understanding
+† Yusuke Nagira is the presenter of this paper.
+the pathogenesis of FECD. However, ECD cannot be mea-
+sured accurately owing to the presence of guttae; hence, the
+number of cells is measured manually [6].
+Using a mouse model to demonstrate the pathogenesis of
+FECD, studies have been conducted on the segmentation of
+guttae using U-Net and the calculation of cell density in areas
+excluding guttae [7][8]. In previous studies on the panoramic
+synthesis of the corneal endothelium, focused images were
+extracted manually and stitched together. Panoramic com-
+positing of the entire cornea is yet to be performed because
+the images were localized panoramic images and did not rep-
+resent the entire cornea [9]. Conventional panorama synthe-
+sis software such as AutoStitch [10] does not consider the
+order of the input images used for synthesis. Instead, it ex-
+tracts the image features using Scale-Invariant Feature Trans-
+form (SIFT) [11] and stitches the matching features together.
+When pasting an image, it is deformed and scaled. For im-
+ages with similar characteristics, such as corneal endothelial
+cells, the position of the image to be pasted may be incor-
+rect or the shape of the cells may be deformed owing to the
+deformation of the image. In this case, accurate values can-
+not be obtained when calculating the area of the cells or the
+percentage of guttae. Therefore, the original image needed
+to have as minimal deformations as possible in the combined
+image. Alternatively, it is necessary to provide a mechanism
+to access the original version of the image of interest.
+This study proposes a framework for a system that gen-
+erates images of the entire corneal endothelium from videos
+obtained using a contact specular microscope. In the pro-
+posed framework, the focused images are extracted from the
+video images, feature extraction is performed, and the im-
+ages are synthesized. During synthesis, the system reduces
+arXiv:2301.02388v1  [eess.IV]  6 Jan 2023
+
+or deforms the original image to the minimum and adds a
+feature that allows access to the original image of the area
+of interest. This study examined the proposed framework by
+building a system using two implementation methods. Fur-
+ther, we added a function for automatically detecting guttae
+using U-Net, a type of Deep Learning. This study presents a
+proposed framework and an example of its implementation.
+Quantitative evaluation of whole corneal endothelial images
+and gut detection is insufficient, which poses a limitation that
+must be addressed in future studies.
+2. FRAMEWORK OF CORNEAL PANORAMIC
+IMAGE GENERATION FROM CONTACT
+SPECULAR MICROSCOPE IMAGES
+2.1. Overview of the proposed framework
+Fig.1 presents an overview of the proposed framework.
+An image of the entire cornea was generated from a video
+frame of the cornea captured using a contact-type specu-
+lar microscope. First, a focused still image was extracted
+from the target video image (dataset 1). Second, the features
+of still images were extracted. Third, the matching feature
+points were combined to create a panoramic image. Next,
+the panoramic image was divided into grid regions and the
+most focused image was selected from each region. Guttae
+were detected in the combined panoramic images using deep
+learning. Additionally, still images of the corneal endothe-
+lium containing the guttae and mask images exhibiting the
+location of the guttae were used for the model generation
+(dataset 2).
+2.2. Extraction of in-focus images
+A group of still images was extracted from the captured
+videos. The entire corneal endothelium was captured and
+converted to a single frame-by-frame image. If there were
+N frames in the video, N images were the output in total
+that were then divided into five groups in chronological order.
+The image with the highest in-focus evaluation index was
+selected from each group.
+2.3. Creating panoramic images
+The algorithm for generating a single panoramic image is
+accomplished through the following steps. First, characteris-
+tic points in the images were extracted. Subsequently, a curve
+connecting the characteristic points was obtained. Here, the
+optimal curve connecting the extracted characteristic points
+was obtained. This curve was then used to enlarge or interpo-
+late the image. A panoramic image was generated. In the fol-
+lowing experiments, two algorithms were prepared and the
+results were compared.
+2.4. Image sharpening process
+The synthesized panoramic images were mostly over-
+lapped images. Additionally, the synthesized image is often
+blurred because of the transparency and brightness of the im-
+age change. Therefore, a method was developed to obtain
+clearer images. The implementation of the sharpening pro-
+cess is described in the next section.
+2.5. Creating the Guttae Classifier by U-Net
+The U-Net is a neural network commonly used for image
+segmentation. U-Net uses a convolutional neural network to
+encode an input image as a feature map, subsequently decod-
+ing that feature map to separate specific objects in the input
+image. It is possible to prepare a dataset of corneal images
+and train U-Net using this dataset to generate a model for
+extracting the guttae.
+3. SYSTEM IMPLEMENTATION AND DATA
+APPLICATION
+3.1. Outline
+This study implemented a system to confirm the effec-
+tiveness of the proposed framework.
+Corneal videos ob-
+tained from the FECD mouse model were processed to obtain
+panoramic images of the cornea. Tenengrad was used to ob-
+tain the in-focus images. Finally, two different applications
+were used to generate panoramic images.
+3.2. Mouse Model of FECD
+This study used images of whole corneal endothelial cells
+from the FECD pathology mouse model. A single nucleotide
+mutation in COL8A2 generated these genes, and it has been
+reported that guttae increase over time. The Tissue Engineer-
+ing Laboratory, Graduate School of Biomedical Sciences,
+Doshisha University, provided the images.
+The images
+were taken using a prototype KSSP slit-scanning wide-field
+contact specular microscope (Konan Medical, Inc., Nishi-
+nomiya, Japan), had a resolution of 1620 × 1080 [pixels] at
+a frame rate of 29.9 [fps] in MOV file format. The images
+were taken in the following order: 1) starting from the center
+of the cornea; 2) moving to the top of the cornea; 3) mov-
+ing to the left; 4) filming from the top to the bottom of the
+cornea; 5) moving around the right side; 6) filming from the
+bottom to the center of the cornea. In this study, 94 videos
+were prepared and used.
+3.3. Extraction of in-focus images by Tenengrad
+The focus evaluation index was calculated as follows:
+Tenengrad [12][13] value, which is the gradient of the im-
+age based on the pixel value, is calculated for each region,
+where Gx and Gy are the convolved values of the Sobel op-
+erator of the pixel values in the x-direction and y-direction,
+respectively.
+Φx,y =
+�
+(i,j)∈Ω(x,y)
+(Gx(i, j)2 + Gy(i, j)2)
+(1)
+The Sobel operator is expressed as
+Kx =
+�
+�
+−1
+0
+1
+−2
+0
+2
+−1
+0
+1
+�
+� , Ky =
+�
+�
+−1
+−2
+−1
+0
+0
+0
+1
+2
+1
+�
+�
+The highest value in the quadratic area of a single image
+was considered the focal value for that image. When this
+value was calculated for the entire image, the gradient was
+smaller in the area containing the edge of the corneal en-
+dothelium. In contrast, the gradient increased in the area
+containing corneal endothelial cells. Thus, as the area of
+
+Image integration for 
+panoramic image of the entire cornea
+Learning Phase
+…
+U-Net
+Corneal endothelial
+images including Guttae
+Mask image annotated 
+with Guttae locations
+Dataset 2
+Prediction Phase
+Modeled U-Net
+Video of the entire 
+cornea
+Dataset 1
+• Pre-processing
+• Focused image 
+extraction
+• Feature extraction
+• Image integration
+• Sharpen process of 
+panorama images
+Final panoramic image
+Guttae prediction
+Fig. 1. Overview of the proposed framework
+the rim increases, the Tenengrad value for the entire image
+becomes smaller. This procedure prevents the corneal en-
+dothelium from being excluded from the image even if it is
+appropriately captured.
+3.4. Creating the idealized panoramic artificial CECs
+image data
+To confirm the effectiveness of the panorama synthesis
+software, a set of idealized panoramic artificial corneal en-
+dothelial cell images were created with no blurring or focus
+mismatch on the extracted images. These images were ob-
+tained using the GNU Image Manipulation Program (GIMP).
+These artificial images were created based on the synthesis
+results obtained using the panorama synthesis software de-
+scribed below. A layer was added to the composite image,
+the cell membrane of the corneal endothelium was traced,
+and the areas considered to be guttae were painted black. The
+color of the surrounding endothelial cells was extracted from
+the layer depicting the cell membrane and guttae using the
+color picker function. The layers are filled with the same
+color. This process was applied to the entire cornea to create
+artificial images. For areas where the cell membrane was not
+visible owing to issues such as focus mismatch or blurring,
+the cell membrane was depicted by referring to another im-
+age in which the cell membrane could be observed appropri-
+ately. Idealized panoramic artificial corneal endothelial cell
+images were created that mimicked the distribution and size
+of the guttae, as well as the size and shape of the corneal en-
+dothelial cells. The created image was 1870 × 1080 [pixel]
+in size and was cropped and stored by moving approximately
+10 [pixels] from the center to the top, counterclockwise from
+the top, and counterclockwise from top to bottom, mimick-
+ing the movement of a motion camera.
+3.5. Creating panoramic images
+In the composite process, two types of applications were
+used; the Image Composite Software (ICS) and the panorama
+synthesis algorithm implemented in OpenCV. The algo-
+rithms are explained as follows.
+3.5.1. Image Composite Software (ICS)
+Image Composite Software (K.I. Technology CO., LTD.,
+Yokohama, Kanagawa, Japan) is a panorama compositive
+software created with specifications suitable for image com-
+positing corneal endothelial images. This application was
+used for the composite process and was custom-made. Since
+this is a commercial application, we cannot explain the de-
+tails of its contents due to copyright. The original image is
+not reduced or enlarged when the images are superimposed.
+Additionally, an API to access the original image is provided,
+allowing quick access to the original image of the area of in-
+terest. A flowchart of the panorama compositing process is
+shown in Algorithm 1, where the first image is used as the
+reference image, and the regions that match the first image
+are searched in order of image number. If no match is ob-
+served, the image merged with the reference image is used
+as the reference for the subsequent image, and the process is
+repeated. The image merging is terminated when ten consec-
+utive images are not observed to match the reference image.
+3.5.2. Panorama synthesis algorithm implemented in OpenCV
+Because the details of the process in ICS are not pub-
+licly available, we implemented an algorithm similar to Al-
+gorithm 2 that mimics the ICS process, using OpenCV, an
+open-source computer vision library. The Kth image was the
+closest to the end of the shooting. The jth image and the i+1
+image are matched for SIFT features, and if the two images
+have many similar features, the degree of change between
+the images is calculated and added to the list. The ith image
+is matched to the i + 1 image. If the two images have few
+
+Algorithm 1 Image Composite Software processing details
+1:
+i = 0, j = 0
+2:
+while i + j + 1 ≤ N do
+3:
+A = Images[i]
+4:
+j = 0
+5:
+while j ≤ 10 do
+6:
+B = Images[i+j+1]
+7:
+if Find matching area with A then
+8:
+A = Stitch B onto A
+9:
+i = i + 1, j = 0
+10:
+else
+11:
+j = j + 1
+Algorithm 2 Calculate the difference in coordinates between
+images
+1:
+i = 0
+2:
+coord = [], usedImages = []
+3:
+while j ≤ N do
+4:
+j = i
+5:
+error = 0
+6:
+flag = True
+7:
+while flag do
+8:
+Extract
+SIFT
+features
+of
+Image[j]
+and
+Image[i+1]
+9:
+if Two images could be feature matched then
+10:
+C = cal coord diff(Image[j], Image[i+1])
+11:
+coord.append(C)
+12:
+usedImages.append(Image[j], Image[i+1])
+13:
+i += 1
+14:
+flag = False
+15:
+else
+16:
+error += 1
+17:
+if error ≥ 10 then
+18:
+j += 1
+19:
+else
+20:
+i += 1
+21: usedImages = list(set(usedImages)
+22: return coord, usedImages
+feature points and cannot be matched, add 1 to the values of
+i and f and perform the SIFT feature extraction again. If this
+process fails ten times, add 1 to the value of j and perform
+the process again taking j equivalent to i, that is, j = i. Us-
+ing the above algorithm, the degree of change in coordinates
+between the images used for composition and the images can
+be calculated. The global coordinates of the entire composite
+coordinates can be obtained by setting the smallest values of
+the x and y coordinates to zero and calculating the cumula-
+tive sum till that point.
+3.6. Image sharpening process
+We divided the combined panoramic image into 64 × 64
+[pixel] grid regions. The image number of each composite
+image and the coordinates of the constituent images were
+obtained from the coordinates of the panoramic image. The
+coordinates of the constituent images in the upper-left corner
+of each grid region were obtained. Tenengrad values were
+calculated for the cropped images. The image with the high-
+est Tenengrad value among the cropped images was pasted
+onto a newly created blank image of the same size as the
+panoramic image with the exact extracted coordinates. These
+processes were performed in all regions. The image with the
+highest Tenengrad value among the multiple overlapping im-
+ages was pasted to obtain a clear image.
+3.7. Creating the Guttae Classifier
+3.7.1. Creating Dataset
+A large amount of data is required for training using net-
+works. However, since the amount of data provided in this
+study was small, it was necessary to augment the data. Ad-
+ditionally, due to the large size of the image data, it was nec-
+essary to reduce the size of the images to use them in the
+dataset. Because image resizing results in a loss of infor-
+mation, we developed a new data augmentation method for
+small and large image data. The corneal endothelial cell im-
+age and mask image showing the location of the guttae were
+divided into a grid of 64 × 64 [pixels]. The image was clipped
+three times by 16 [pixels] to the right and three times by 16
+[pixels] to the bottom, thereby shifting the image area. Thus,
+the images were expanded 16 times for a single-grid region.
+Images that did not contain guttae were excluded from the
+dataset.
+3.7.2. Training and Prediction in CNN
+Twenty-one corneal endothelial cell images with a mask
+image showing the location of the guttae were prepared
+(Fig.1 Dataset 2). The Segmentation model PyTorch, which
+is a Python library for implementing segmentation-specific
+CNNs, was used in the experiments. The best encoder back-
+bone is determined using ResNet18, ResNet34, ResNet50,
+VGG11, and VGG16. Twenty-one images were divided into
+two sets of eighteen and three images; eighteen were used
+to develop this model and three were used to determine the
+backbone. The node weights of ImageNet were transferred
+to this model, and fine-tuning was performed. The network
+of U-Net [14] encoders modified to ResNet50 [15] is the best
+backbone. Adam [16] was used as the optimization function
+with an initial learning rate of 1e-4, wherein the loss func-
+tion is the least squares error. Predictions were made on a 64
+× 64 [pixels] image extracted during the sharpening process,
+pasted to the original position, and the location of the guttae
+was predicted for the panoramic image.
+4. RESULTS AND DISCUSSION
+4.1. Comparison of algorithms implemented in ICS and
+OpenCV
+An example of a mouse corneal endothelial cell is shown
+in Fig.2A. The images synthesized using ICS were more cir-
+cular than those synthesized using OpenCV. This result indi-
+cates that the synthesis was performed correctly. On the other
+hand, images synthesized using the OpenCV algorithm were
+not circular and were often pasted in incorrect locations.
+An example of the synthesis result of the algorithm im-
+plemented in OpenCV for artificial cornea data is shown in
+
+(a) Panoramic images by Image Composite Software
+(b) Panoramic images by OpenCV
+A.
+(a) Truth (Artificial cornea image)
+(b) Panoramic image by OpenCV
+B.
+Fig. 2.
+A: Comparison of results with Image Composite Software and software implemented in OpenCV. B: Results with
+OpenCV implemented software on artificial cornea data.(a) shows the composite result and (b) shows the overlap of the
+component images.
+Fig.2B. This is the synthesis result when the focus is per-
+fectly aligned and there are no blurred images. The result is
+almost the same as the correct data, where there is no unnat-
+ural overlap between the composite images, indicating that
+compositing was performed correctly. If the in-focus images
+are well extracted, they can be integrated well using OpenCV.
+Overall, ICS is a more robust method.
+4.2. Synthesis results with ICS
+As previously mentioned, when taking moving images of
+the mouse cornea with the contact specular microscope, the
+images are taken in an upward direction from the center of
+the cornea, then leftward along the edge of the cornea, and
+downward once around the edge. Next, it was photographed
+clockwise and then down to the center. For the 94 videos, the
+left- and right-rotated portions were split, and for each case,
+an integrated image was created using the ICS. The results
+are presented in Table1. An image was classified as ”image
+dropout” if the center of the image was missing, ”distorted
+shape” if the shape was distorted instead of being circular,
+and ”unnatural paste” if the image was pasted in an unnat-
+ural location.
+On the other hand, an image in which the
+shape of the cornea in the combined panoramic image was
+kept circular, the center was not missing, and there were no
+unnaturally pasted parts was classified as a ”success”. The
+number of images in which either the right- or left-rotated
+part of the image was correctly merged was 75. Fig.3 com-
+pares the results of the manual and ICS syntheses. Among
+the videos that failed to be synthesized using ICS, those with
+distorted shapes or unnatural pasting were verified to contain
+frames that deviated from the cornea owing to contamina-
+tion on the specular microscope or a shaking camera. It is
+considered that the stains themselves became feature points,
+and the pasting of the composite part failed. Additionally,
+the position of the image was significantly changed because
+of significant blurring, resulting in an unnatural position for
+Table 1. Synthesis results with ICS
+Result
+Left
+Right
+image dropout
+15
+20
+distorted shape
+6
+12
+unnatural paste
+10
+6
+success
+63
+56
+pasting and distorting the overall shape of the image.
+The results of the ICS and manually composited images
+were almost identical, suggesting that there was no problem
+with the image-compositing algorithm and that the mouse
+cornea was not captured in the first place when the specular
+microscope was used. The image was taken from the cen-
+ter of the mouse cornea in an upward direction, followed by
+leftward rotation along the edge of the cornea, and then a
+downward direction was taken at the point where the image
+had gone around the edge. If there is an area in the center of
+the cornea that has not been photographed at this time, the
+image will be missing.
+4.3. Segmentation of guttae in panoramic images
+The model that detects the guttae location was applied
+to the panoramic images of the videos (Fig.1, Dataset 1).
+Fig.4 shows two prediction examples of the guttae position
+in a panoramic image: (c) and (d) are the prediction results
+in (a) and (b), respectively. The original panoramic images
+were combined using ICS, and the edges of the corneas were
+cropped after sharpening.
+Segmentation was performed on the images synthesized
+from the entire cornea using ICS. Currently, 21 still images
+and a mask image annotated with guttae are used for segmen-
+tation into training and validation datasets. The model was
+evaluated by comparing the validation data with predicted re-
+sults. Fig.4 shows that while the segmentation of the likely
+
+(c) Panoramic image by manually 
+(Panoramic image was integrated Successfully.)
+(d) Panoramic image by manually 
+(There is a hole in the center.)
+(a) Panoramic image by Image Composite Software 
+(Panoramic image was integrated Successfully.)
+(b) Panoramic image by Image Composite Software 
+(There is a hole in the center.)
+Fig. 3.
+Integrated image results by Image Composite Software and manual composite.The left side of each represents the
+composite result, and the right side represents the overlap of the component images.
+(a) Panoramic image with
+corneal rim removed
+(b) Prediction results for 
+guttae location
+(c) Panoramic image with 
+corneal rim removed
+(d) Prediction results for 
+guttae location
+Fig. 4. Two examples of panoramic images and guttae loca-
+tions.
+guttae is successful, it also partially predicts the guttae at the
+edges of the cornea. Further quantitative evaluation is es-
+sential; however, it may pose some limitations for future re-
+search. For this purpose, it is necessary to prepare an image
+to which the Grand Truth of the guttae is assigned.
+4.4. Discussion
+It was observed that the algorithm implemented in
+OpenCV could not correctly synthesize results using mouse
+corneal endothelial cell images. The results showed that the
+images were pasted in unnatural positions compared to those
+obtained using artificial cornea data. The significant differ-
+ence between the mouse corneal endothelial cell image data
+and the artificial cornea data is that, with the artificial cornea
+data, all images are in focus and the images themselves are
+not blurred. For the image data of mouse corneal endothe-
+lial cells, the process of extracting images in focus involved
+extracting images at equal intervals in chronological order,
+which resulted in the extraction of out-of-focus images. In
+contrast, the ICS-based method synthesized the images more
+accurately than the OpenCV-based synthesis software did be-
+cause ICS uses a feature extraction method appropriate for
+corneal endothelial cells.
+By contrast, OpenCV synthesis uses SIFT for feature ex-
+traction. This synthesis is considered to be progressing well.
+Until now, it has not been possible to obtain images of the
+entire cornea because of the narrow imaging range of spec-
+ular microscopy. This study suggests that it is possible to
+obtain a composite image of the entire cornea by extracting
+a relatively focused image from a video of the entire cornea.
+5. CONCLUSIONS AND FUTURE WORK
+The status of the entire cornea is currently inferred from
+the center of diagnosis and observation of corneal endothe-
+lial cells using specular microscopy. If images of the entire
+cornea could be obtained, more studies would be possible.
+In this study, we proposed a framework for generating im-
+ages of the entire cornea from videos captured using contact
+specular microscopy. Focused images were extracted from
+the video and a panoramic composite image was generated.
+Furthermore, we constructed a learning model, U-Net, to ex-
+tract the guttae from the entire image. To study the effec-
+tiveness of the proposed framework, we implemented it and
+applied it to corneal data from a mouse model of FECD. The
+panorama synthesis application used in the implementation
+was our custom-built ICS and the OpenCV algorithm, which
+is an open-source software. Artificial corneal images were
+synthesized with no unnatural aspects in the results. How-
+
+ever, some of the extracted images were not correctly syn-
+thesized if they contained blurred images, and many images
+were correctly synthesized using ICS.
+After the panorama was merged, the image was divided
+into a grid. Majority of the in-focus images were extracted
+and pasted, resulting in a sharper image than the previous
+output obtained using ICS. Using the extracted images within
+the region, we could also predict the guttae location. Al-
+though the implementation and application of the method to
+the data in this study confirmed its effectiveness, few quanti-
+tative evaluations have been performed. Quantitative evalua-
+tion, such as the accuracy of implementation, is an issue for
+the future.
+REFERENCES
+[1]
+Allen O Eghrari, S Amer Riazuddin, and John D
+Gottsch. Fuchs corneal dystrophy. Progress in molecu-
+lar biology and translational science, 134:79–97, 2015.
+[2]
+Gargi Gouranga Nanda and Debasmita Pankaj Alone.
+Current understanding of the pathogenesis of fuchs’ en-
+dothelial corneal dystrophy. Molecular vision, 25:295,
+2019.
+[3]
+Philippe Gain, R´emy Jullienne, Zhiguo He, Mansour
+Aldossary, Sophie Acquart, Fabrice Cognasse, and
+Gilles Thuret. Global survey of corneal transplantation
+and eye banking. JAMA ophthalmology, 134(2):167–
+173, 2016.
+[4]
+Naoki Okumura,
+Shigeru Kinoshita,
+and Noriko
+Koizumi. Application of rho kinase inhibitors for the
+treatment of corneal endothelial diseases. Journal of
+ophthalmology, 2017, 2017.
+[5]
+Naoki Okumura,
+Shigeru Kinoshita,
+and Noriko
+Koizumi. The role of rho kinase inhibitors in corneal
+endothelial dysfunction. Current Pharmaceutical De-
+sign, 23(4):660–666, 2017.
+[6]
+Jay W McLaren, Lori A Bachman, Katrina M Kane,
+and Sanjay V Patel. Objective assessment of the corneal
+endothelium in fuchs’ endothelial dystrophy. Investiga-
+tive ophthalmology & visual science, 55(2):1184–1190,
+2014.
+[7]
+Naoki Okumura, Shohei Yamada, Takeru Nishikawa,
+Kaito Narimoto, Kengo Okamura, Ayaka Izumi, Satoru
+Hiwa, Tomoyuki Hiroyasu, and Noriko Koizumi. U-
+net convolutional neural network for segmenting the
+corneal endothelium in a mouse model of fuchs en-
+dothelial corneal dystrophy. Cornea, 2021.
+[8]
+Takeru Nishikawa, Naoki Okumura, Kaito Narimoto,
+Shohei Yamada, Kengo Okamura, Ayaka Izumi, and
+Noriko Koizumi.
+Deep neural network for the anal-
+ysis of guttae via semi-supervised learning in a fuchs
+endothelial corneal dystrophy mouse model. Investiga-
+tive Ophthalmology & Visual Science, 62(8):826–826,
+2021.
+[9]
+Hiroshi Tanaka, Naoki Okumura, Noriko Koizumi,
+Chie Sotozono, Yasuhiro Sumii, and Shigeru Kinoshita.
+Panoramic view of human corneal endothelial cell layer
+observed by a prototype slit-scanning wide-field con-
+tact specular microscope. British Journal of Ophthal-
+mology, 101(5):655–659, 2017.
+[10] Matthew Brown and David G Lowe.
+Automatic
+panoramic image stitching using invariant features. In-
+ternational journal of computer vision, 74(1):59–73,
+2007.
+[11] David G Lowe. Distinctive image features from scale-
+invariant keypoints. International journal of computer
+vision, 60(2):91–110, 2004.
+[12] Eric Krotkov. Focusing. International Journal of Com-
+puter Vision, 1(3):223–237, 1988.
+[13] Said Pertuz, Domenec Puig, and Miguel Angel Garcia.
+Analysis of focus measure operators for shape-from-
+focus. Pattern Recognition, 46(5):1415–1432, 2013.
+[14] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.
+U-net: Convolutional networks for biomedical image
+segmentation. In International Conference on Medical
+image computing and computer-assisted intervention,
+pages 234–241. Springer, 2015.
+[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian
+Sun. Deep residual learning for image recognition. In
+Proceedings of the IEEE conference on computer vision
+and pattern recognition, pages 770–778, 2016.
+[16] Diederik P Kingma and Jimmy Ba.
+Adam:
+A
+method for stochastic optimization.
+arXiv preprint
+arXiv:1412.6980, 2014.
+
diff --git a/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/load_file.txt b/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a6dd7805d42957b6d4a60bc41bdafc7acbb45881
--- /dev/null
+++ b/4NE0T4oBgHgl3EQfeQCD/content/tmp_files/load_file.txt
@@ -0,0 +1,347 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf,len=346
+page_content='Generating corneal panoramic images from contact specular microscope images  Yusuke Nagira1†, Yuzuha Hara2, Satoru Hiwa2  Naoki Okumura3, Noriko Koizumi3 and Tomoyuki Hiroyasu2  1Graduate School of Life and Medical Sciences, Doshisha University, Japan  2Department of Biomedical Sciences and Informatics, Doshisha University, Japan  3Department of Biomedical Engineering, Faculty of Life and Medical Sciences, Doshisha University, Japan  (Tel: +81-774-65-6020, E-mail: tomo@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='doshisha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='jp)  Abstract: The contact specular microscope has a wider angle of view than that of the non-contact specular microscope but still  cannot capture an image of the entire cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' To obtain such an image, it is necessary to prepare film on the parts of the image  captured sequentially and combine them to create a complete image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study proposes a framework to automatically generate  an entire corneal image from videos captured using a contact specular microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Relatively focused images were extracted  from the videos and panoramic compositing was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If an entire image can be generated, it is possible to detect guttae  from the image and examine the extent of their presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The system was implemented and the effectiveness of the proposed  framework was examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The system was implemented using custom-made composite software, Image Composite Software  (ICS, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Technology Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Japan, internal algorithms not disclosed), and a supervised learning model using U-Net was  used for guttae detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Several images were correctly synthesized when the constructed system was applied to 94 different  corneal videos obtained from Fuchs endothelial corneal dystrophy (FECD) mouse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The implementation and application  of the method to the data in this study confirmed its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Owing to the minimal quantitative evaluation performed, such  as accuracy with implementation, it may pose some limitations for future investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Keywords: U-Net, Semantic Segmentation, Fuchs Endothelial Corneal Dystrophy, Corneal Endothelial Cell  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' INTRODUCTION  Fuchs endothelial corneal dystrophy (FECD) is a bilat-  eral disease, wherein corneal endothelial cells are unable to  maintain their hexagonal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is characterized by the  accelerated loss of corneal endothelial cells with changes in  Descemet’s membrane, resulting in the formation of an ex-  tracellular matrix called guttae [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the United States, it  is estimated that 4% of people over 40 years of age are af-  fected by the disease, occurring more commonly in women  and more frequently in people in their 40s and 50s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Corneal  endothelial pump function is lost as the disease progresses,  causing corneal edema [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Presently, corneal transplanta-  tion is the only reliable treatment, and FECD accounts for  39% of all corneal transplants performed, making it the most  common cause of corneal transplantation worldwide [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Rho kinase inhibitors have been reported to promote cell  proliferation and adhesion to substrates, inhibit corneal en-  dothelial cell apoptosis, and promote wound healing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' There-  fore, using Rho kinase inhibitor eye drops is a potential  novel treatment approach alternative to corneal transplanta-  tion [4][5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  In drug discovery research for FECD, the state of the  corneal endothelium, such as the guttae, is observed before  and after the drug use and evaluated based on the increase  or decrease in the number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Doctors and researchers  widely use specular microscopes to observe the state of the  corneal endothelium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  However, the range of the micro-  scope’s imaging capability is limited and the current practice  estimates the state of the entire cornea from its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  endothelial cell density (ECD) is essential for understanding  † Yusuke Nagira is the presenter of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  the pathogenesis of FECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' However, ECD cannot be mea-  sured accurately owing to the presence of guttae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' hence, the  number of cells is measured manually [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Using a mouse model to demonstrate the pathogenesis of  FECD, studies have been conducted on the segmentation of  guttae using U-Net and the calculation of cell density in areas  excluding guttae [7][8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In previous studies on the panoramic  synthesis of the corneal endothelium, focused images were  extracted manually and stitched together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Panoramic com-  positing of the entire cornea is yet to be performed because  the images were localized panoramic images and did not rep-  resent the entire cornea [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Conventional panorama synthe-  sis software such as AutoStitch [10] does not consider the  order of the input images used for synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Instead, it ex-  tracts the image features using Scale-Invariant Feature Trans-  form (SIFT) [11] and stitches the matching features together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  When pasting an image, it is deformed and scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For im-  ages with similar characteristics, such as corneal endothelial  cells, the position of the image to be pasted may be incor-  rect or the shape of the cells may be deformed owing to the  deformation of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In this case, accurate values can-  not be obtained when calculating the area of the cells or the  percentage of guttae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Therefore, the original image needed  to have as minimal deformations as possible in the combined  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Alternatively, it is necessary to provide a mechanism  to access the original version of the image of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  This study proposes a framework for a system that gen-  erates images of the entire corneal endothelium from videos  obtained using a contact specular microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the pro-  posed framework, the focused images are extracted from the  video images, feature extraction is performed, and the im-  ages are synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' During synthesis, the system reduces  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='02388v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='IV]  6 Jan 2023  or deforms the original image to the minimum and adds a  feature that allows access to the original image of the area  of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study examined the proposed framework by  building a system using two implementation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fur-  ther, we added a function for automatically detecting guttae  using U-Net, a type of Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study presents a  proposed framework and an example of its implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Quantitative evaluation of whole corneal endothelial images  and gut detection is insufficient, which poses a limitation that  must be addressed in future studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' FRAMEWORK OF CORNEAL PANORAMIC  IMAGE GENERATION FROM CONTACT  SPECULAR MICROSCOPE IMAGES  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Overview of the proposed framework  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1 presents an overview of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  An image of the entire cornea was generated from a video  frame of the cornea captured using a contact-type specu-  lar microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' First, a focused still image was extracted  from the target video image (dataset 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Second, the features  of still images were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Third, the matching feature  points were combined to create a panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Next,  the panoramic image was divided into grid regions and the  most focused image was selected from each region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Guttae  were detected in the combined panoramic images using deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally, still images of the corneal endothe-  lium containing the guttae and mask images exhibiting the  location of the guttae were used for the model generation  (dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Extraction of in-focus images  A group of still images was extracted from the captured  videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The entire corneal endothelium was captured and  converted to a single frame-by-frame image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If there were  N frames in the video, N images were the output in total  that were then divided into five groups in chronological order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The image with the highest in-focus evaluation index was  selected from each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating panoramic images  The algorithm for generating a single panoramic image is  accomplished through the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' First, characteris-  tic points in the images were extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Subsequently, a curve  connecting the characteristic points was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Here, the  optimal curve connecting the extracted characteristic points  was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This curve was then used to enlarge or interpo-  late the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A panoramic image was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In the fol-  lowing experiments, two algorithms were prepared and the  results were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image sharpening process  The synthesized panoramic images were mostly over-  lapped images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally, the synthesized image is often  blurred because of the transparency and brightness of the im-  age change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Therefore, a method was developed to obtain  clearer images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The implementation of the sharpening pro-  cess is described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the Guttae Classifier by U-Net  The U-Net is a neural network commonly used for image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' U-Net uses a convolutional neural network to  encode an input image as a feature map, subsequently decod-  ing that feature map to separate specific objects in the input  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is possible to prepare a dataset of corneal images  and train U-Net using this dataset to generate a model for  extracting the guttae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' SYSTEM IMPLEMENTATION AND DATA  APPLICATION  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Outline  This study implemented a system to confirm the effec-  tiveness of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Corneal videos ob-  tained from the FECD mouse model were processed to obtain  panoramic images of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Tenengrad was used to ob-  tain the in-focus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Finally, two different applications  were used to generate panoramic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Mouse Model of FECD  This study used images of whole corneal endothelial cells  from the FECD pathology mouse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A single nucleotide  mutation in COL8A2 generated these genes, and it has been  reported that guttae increase over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Tissue Engineer-  ing Laboratory, Graduate School of Biomedical Sciences,  Doshisha University, provided the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The images  were taken using a prototype KSSP slit-scanning wide-field  contact specular microscope (Konan Medical, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', Nishi-  nomiya, Japan), had a resolution of 1620 × 1080 [pixels] at  a frame rate of 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='9 [fps] in MOV file format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The images  were taken in the following order: 1) starting from the center  of the cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 2) moving to the top of the cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 3) mov-  ing to the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 4) filming from the top to the bottom of the  cornea;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 5) moving around the right side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 6) filming from the  bottom to the center of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In this study, 94 videos  were prepared and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Extraction of in-focus images by Tenengrad  The focus evaluation index was calculated as follows:  Tenengrad [12][13] value, which is the gradient of the im-  age based on the pixel value, is calculated for each region,  where Gx and Gy are the convolved values of the Sobel op-  erator of the pixel values in the x-direction and y-direction,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Φx,y =  �  (i,j)∈Ω(x,y)  (Gx(i, j)2 + Gy(i, j)2)  (1)  The Sobel operator is expressed as  Kx =  �  �  −1  0  1  −2  0  2  −1  0  1  �  � , Ky =  �  �  −1  −2  −1  0  0  0  1  2  1  �  �  The highest value in the quadratic area of a single image  was considered the focal value for that image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' When this  value was calculated for the entire image, the gradient was  smaller in the area containing the edge of the corneal en-  dothelium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In contrast, the gradient increased in the area  containing corneal endothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Thus, as the area of  Image integration for  panoramic image of the entire cornea  Learning Phase  …  U-Net  Corneal endothelial  images including Guttae  Mask image annotated  with Guttae locations  Dataset 2  Prediction Phase  Modeled U-Net  Video of the entire  cornea  Dataset 1  Pre-processing  Focused image  extraction  Feature extraction  Image integration  Sharpen process of  panorama images  Final panoramic image  Guttae prediction  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Overview of the proposed framework  the rim increases, the Tenengrad value for the entire image  becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This procedure prevents the corneal en-  dothelium from being excluded from the image even if it is  appropriately captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the idealized panoramic artificial CECs  image data  To confirm the effectiveness of the panorama synthesis  software, a set of idealized panoramic artificial corneal en-  dothelial cell images were created with no blurring or focus  mismatch on the extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' These images were ob-  tained using the GNU Image Manipulation Program (GIMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  These artificial images were created based on the synthesis  results obtained using the panorama synthesis software de-  scribed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A layer was added to the composite image,  the cell membrane of the corneal endothelium was traced,  and the areas considered to be guttae were painted black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  color of the surrounding endothelial cells was extracted from  the layer depicting the cell membrane and guttae using the  color picker function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The layers are filled with the same  color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This process was applied to the entire cornea to create  artificial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For areas where the cell membrane was not  visible owing to issues such as focus mismatch or blurring,  the cell membrane was depicted by referring to another im-  age in which the cell membrane could be observed appropri-  ately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Idealized panoramic artificial corneal endothelial cell  images were created that mimicked the distribution and size  of the guttae, as well as the size and shape of the corneal en-  dothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The created image was 1870 × 1080 [pixel]  in size and was cropped and stored by moving approximately  10 [pixels] from the center to the top, counterclockwise from  the top, and counterclockwise from top to bottom, mimick-  ing the movement of a motion camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating panoramic images  In the composite process, two types of applications were  used;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' the Image Composite Software (ICS) and the panorama  synthesis algorithm implemented in OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The algo-  rithms are explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image Composite Software (ICS)  Image Composite Software (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Technology CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=', LTD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=',  Yokohama, Kanagawa, Japan) is a panorama compositive  software created with specifications suitable for image com-  positing corneal endothelial images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This application was  used for the composite process and was custom-made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Since  this is a commercial application, we cannot explain the de-  tails of its contents due to copyright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The original image is  not reduced or enlarged when the images are superimposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Additionally, an API to access the original image is provided,  allowing quick access to the original image of the area of in-  terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' A flowchart of the panorama compositing process is  shown in Algorithm 1, where the first image is used as the  reference image, and the regions that match the first image  are searched in order of image number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If no match is ob-  served, the image merged with the reference image is used  as the reference for the subsequent image, and the process is  repeated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image merging is terminated when ten consec-  utive images are not observed to match the reference image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Panorama synthesis algorithm implemented in OpenCV  Because the details of the process in ICS are not pub-  licly available, we implemented an algorithm similar to Al-  gorithm 2 that mimics the ICS process, using OpenCV, an  open-source computer vision library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Kth image was the  closest to the end of the shooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The jth image and the i+1  image are matched for SIFT features, and if the two images  have many similar features, the degree of change between  the images is calculated and added to the list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The ith image  is matched to the i + 1 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If the two images have few  Algorithm 1 Image Composite Software processing details  1:  i = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' j = 0  2:  while i + j + 1 ≤ N do  3:  A = Images[i]  4:  j = 0  5:  while j ≤ 10 do  6:  B = Images[i+j+1]  7:  if Find matching area with A then  8:  A = Stitch B onto A  9:  i = i + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' j = 0  10:  else  11:  j = j + 1  Algorithm 2 Calculate the difference in coordinates between  images  1:  i = 0  2:  coord = [],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' usedImages = []  3:  while j ≤ N do  4:  j = i  5:  error = 0  6:  flag = True  7:  while flag do  8:  Extract  SIFT  features  of  Image[j]  and  Image[i+1]  9:  if Two images could be feature matched then  10:  C = cal coord diff(Image[j],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image[i+1])  11:  coord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='append(C)  12:  usedImages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='append(Image[j], Image[i+1])  13:  i += 1  14:  flag = False  15:  else  16:  error += 1  17:  if error ≥ 10 then  18:  j += 1  19:  else  20:  i += 1  21: usedImages = list(set(usedImages)  22: return coord, usedImages  feature points and cannot be matched, add 1 to the values of  i and f and perform the SIFT feature extraction again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If this  process fails ten times, add 1 to the value of j and perform  the process again taking j equivalent to i, that is, j = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Us-  ing the above algorithm, the degree of change in coordinates  between the images used for composition and the images can  be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The global coordinates of the entire composite  coordinates can be obtained by setting the smallest values of  the x and y coordinates to zero and calculating the cumula-  tive sum till that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Image sharpening process  We divided the combined panoramic image into 64 × 64  [pixel] grid regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image number of each composite  image and the coordinates of the constituent images were  obtained from the coordinates of the panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  coordinates of the constituent images in the upper-left corner  of each grid region were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Tenengrad values were  calculated for the cropped images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image with the high-  est Tenengrad value among the cropped images was pasted  onto a newly created blank image of the same size as the  panoramic image with the exact extracted coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' These  processes were performed in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image with the  highest Tenengrad value among the multiple overlapping im-  ages was pasted to obtain a clear image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating the Guttae Classifier  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Creating Dataset  A large amount of data is required for training using net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' However, since the amount of data provided in this  study was small, it was necessary to augment the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Ad-  ditionally, due to the large size of the image data, it was nec-  essary to reduce the size of the images to use them in the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Because image resizing results in a loss of infor-  mation, we developed a new data augmentation method for  small and large image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The corneal endothelial cell im-  age and mask image showing the location of the guttae were  divided into a grid of 64 × 64 [pixels].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image was clipped  three times by 16 [pixels] to the right and three times by 16  [pixels] to the bottom, thereby shifting the image area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Thus,  the images were expanded 16 times for a single-grid region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Images that did not contain guttae were excluded from the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Training and Prediction in CNN  Twenty-one corneal endothelial cell images with a mask  image showing the location of the guttae were prepared  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1 Dataset 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The Segmentation model PyTorch, which  is a Python library for implementing segmentation-specific  CNNs, was used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The best encoder back-  bone is determined using ResNet18, ResNet34, ResNet50,  VGG11, and VGG16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Twenty-one images were divided into  two sets of eighteen and three images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' eighteen were used  to develop this model and three were used to determine the  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The node weights of ImageNet were transferred  to this model, and fine-tuning was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The network  of U-Net [14] encoders modified to ResNet50 [15] is the best  backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Adam [16] was used as the optimization function  with an initial learning rate of 1e-4, wherein the loss func-  tion is the least squares error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Predictions were made on a 64  × 64 [pixels] image extracted during the sharpening process,  pasted to the original position, and the location of the guttae  was predicted for the panoramic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Comparison of algorithms implemented in ICS and  OpenCV  An example of a mouse corneal endothelial cell is shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The images synthesized using ICS were more cir-  cular than those synthesized using OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This result indi-  cates that the synthesis was performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' On the other  hand, images synthesized using the OpenCV algorithm were  not circular and were often pasted in incorrect locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  An example of the synthesis result of the algorithm im-  plemented in OpenCV for artificial cornea data is shown in  (a) Panoramic images by Image Composite Software  (b) Panoramic images by OpenCV  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  (a) Truth (Artificial cornea image)  (b) Panoramic image by OpenCV  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  A: Comparison of results with Image Composite Software and software implemented in OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' B: Results with  OpenCV implemented software on artificial cornea data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' (a) shows the composite result and (b) shows the overlap of the  component images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This is the synthesis result when the focus is per-  fectly aligned and there are no blurred images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The result is  almost the same as the correct data, where there is no unnat-  ural overlap between the composite images, indicating that  compositing was performed correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If the in-focus images  are well extracted, they can be integrated well using OpenCV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Overall, ICS is a more robust method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Synthesis results with ICS  As previously mentioned, when taking moving images of  the mouse cornea with the contact specular microscope, the  images are taken in an upward direction from the center of  the cornea, then leftward along the edge of the cornea, and  downward once around the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Next, it was photographed  clockwise and then down to the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For the 94 videos, the  left- and right-rotated portions were split, and for each case,  an integrated image was created using the ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The results  are presented in Table1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' An image was classified as ”image  dropout” if the center of the image was missing, ”distorted  shape” if the shape was distorted instead of being circular,  and ”unnatural paste” if the image was pasted in an unnat-  ural location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  On the other hand, an image in which the  shape of the cornea in the combined panoramic image was  kept circular, the center was not missing, and there were no  unnaturally pasted parts was classified as a ”success”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  number of images in which either the right- or left-rotated  part of the image was correctly merged was 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3 com-  pares the results of the manual and ICS syntheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Among  the videos that failed to be synthesized using ICS, those with  distorted shapes or unnatural pasting were verified to contain  frames that deviated from the cornea owing to contamina-  tion on the specular microscope or a shaking camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' It is  considered that the stains themselves became feature points,  and the pasting of the composite part failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Additionally,  the position of the image was significantly changed because  of significant blurring, resulting in an unnatural position for  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Synthesis results with ICS  Result  Left  Right  image dropout  15  20  distorted shape  6  12  unnatural paste  10  6  success  63  56  pasting and distorting the overall shape of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  The results of the ICS and manually composited images  were almost identical, suggesting that there was no problem  with the image-compositing algorithm and that the mouse  cornea was not captured in the first place when the specular  microscope was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The image was taken from the cen-  ter of the mouse cornea in an upward direction, followed by  leftward rotation along the edge of the cornea, and then a  downward direction was taken at the point where the image  had gone around the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If there is an area in the center of  the cornea that has not been photographed at this time, the  image will be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Segmentation of guttae in panoramic images  The model that detects the guttae location was applied  to the panoramic images of the videos (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='1, Dataset 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4 shows two prediction examples of the guttae position  in a panoramic image: (c) and (d) are the prediction results  in (a) and (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The original panoramic images  were combined using ICS, and the edges of the corneas were  cropped after sharpening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Segmentation was performed on the images synthesized  from the entire cornea using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Currently, 21 still images  and a mask image annotated with guttae are used for segmen-  tation into training and validation datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The model was  evaluated by comparing the validation data with predicted re-  sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4 shows that while the segmentation of the likely  (c) Panoramic image by manually  (Panoramic image was integrated Successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (d) Panoramic image by manually  (There is a hole in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (a) Panoramic image by Image Composite Software  (Panoramic image was integrated Successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  (b) Panoramic image by Image Composite Software  (There is a hole in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=')  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Integrated image results by Image Composite Software and manual composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='The left side of each represents the  composite result, and the right side represents the overlap of the component images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  (a) Panoramic image with  corneal rim removed  (b) Prediction results for  guttae location  (c) Panoramic image with  corneal rim removed  (d) Prediction results for  guttae location  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Two examples of panoramic images and guttae loca-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  guttae is successful, it also partially predicts the guttae at the  edges of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Further quantitative evaluation is es-  sential;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' however, it may pose some limitations for future re-  search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For this purpose, it is necessary to prepare an image  to which the Grand Truth of the guttae is assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Discussion  It was observed that the algorithm implemented in  OpenCV could not correctly synthesize results using mouse  corneal endothelial cell images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The results showed that the  images were pasted in unnatural positions compared to those  obtained using artificial cornea data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The significant differ-  ence between the mouse corneal endothelial cell image data  and the artificial cornea data is that, with the artificial cornea  data, all images are in focus and the images themselves are  not blurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' For the image data of mouse corneal endothe-  lial cells, the process of extracting images in focus involved  extracting images at equal intervals in chronological order,  which resulted in the extraction of out-of-focus images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In  contrast, the ICS-based method synthesized the images more  accurately than the OpenCV-based synthesis software did be-  cause ICS uses a feature extraction method appropriate for  corneal endothelial cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  By contrast, OpenCV synthesis uses SIFT for feature ex-  traction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This synthesis is considered to be progressing well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Until now, it has not been possible to obtain images of the  entire cornea because of the narrow imaging range of spec-  ular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' This study suggests that it is possible to  obtain a composite image of the entire cornea by extracting  a relatively focused image from a video of the entire cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' CONCLUSIONS AND FUTURE WORK  The status of the entire cornea is currently inferred from  the center of diagnosis and observation of corneal endothe-  lial cells using specular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' If images of the entire  cornea could be obtained, more studies would be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  In this study, we proposed a framework for generating im-  ages of the entire cornea from videos captured using contact  specular microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Focused images were extracted from  the video and a panoramic composite image was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Furthermore, we constructed a learning model, U-Net, to ex-  tract the guttae from the entire image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' To study the effec-  tiveness of the proposed framework, we implemented it and  applied it to corneal data from a mouse model of FECD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The  panorama synthesis application used in the implementation  was our custom-built ICS and the OpenCV algorithm, which  is an open-source software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Artificial corneal images were  synthesized with no unnatural aspects in the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' How-  ever, some of the extracted images were not correctly syn-  thesized if they contained blurred images, and many images  were correctly synthesized using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  After the panorama was merged, the image was divided  into a grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Majority of the in-focus images were extracted  and pasted, resulting in a sharper image than the previous  output obtained using ICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Using the extracted images within  the region, we could also predict the guttae location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Al-  though the implementation and application of the method to  the data in this study confirmed its effectiveness, few quanti-  tative evaluations have been performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Quantitative evalua-  tion, such as the accuracy of implementation, is an issue for  the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  REFERENCES  [1]  Allen O Eghrari, S Amer Riazuddin, and John D  Gottsch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Fuchs corneal dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Progress in molecu-  lar biology and translational science, 134:79–97, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [2]  Gargi Gouranga Nanda and Debasmita Pankaj Alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Current understanding of the pathogenesis of fuchs’ en-  dothelial corneal dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Molecular vision, 25:295,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [3]  Philippe Gain, R´emy Jullienne, Zhiguo He, Mansour  Aldossary, Sophie Acquart, Fabrice Cognasse, and  Gilles Thuret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Global survey of corneal transplantation  and eye banking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' JAMA ophthalmology, 134(2):167–  173, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [4]  Naoki Okumura,  Shigeru Kinoshita,  and Noriko  Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Application of rho kinase inhibitors for the  treatment of corneal endothelial diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Journal of  ophthalmology, 2017, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [5]  Naoki Okumura,  Shigeru Kinoshita,  and Noriko  Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' The role of rho kinase inhibitors in corneal  endothelial dysfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Current Pharmaceutical De-  sign, 23(4):660–666, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [6]  Jay W McLaren, Lori A Bachman, Katrina M Kane,  and Sanjay V Patel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Objective assessment of the corneal  endothelium in fuchs’ endothelial dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Investiga-  tive ophthalmology & visual science, 55(2):1184–1190,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [7]  Naoki Okumura, Shohei Yamada, Takeru Nishikawa,  Kaito Narimoto, Kengo Okamura, Ayaka Izumi, Satoru  Hiwa, Tomoyuki Hiroyasu, and Noriko Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' U-  net convolutional neural network for segmenting the  corneal endothelium in a mouse model of fuchs en-  dothelial corneal dystrophy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Cornea, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [8]  Takeru Nishikawa, Naoki Okumura, Kaito Narimoto,  Shohei Yamada, Kengo Okamura, Ayaka Izumi, and  Noriko Koizumi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Deep neural network for the anal-  ysis of guttae via semi-supervised learning in a fuchs  endothelial corneal dystrophy mouse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Investiga-  tive Ophthalmology & Visual Science, 62(8):826–826,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [9]  Hiroshi Tanaka, Naoki Okumura, Noriko Koizumi,  Chie Sotozono, Yasuhiro Sumii, and Shigeru Kinoshita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Panoramic view of human corneal endothelial cell layer  observed by a prototype slit-scanning wide-field con-  tact specular microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' British Journal of Ophthal-  mology, 101(5):655–659, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [10] Matthew Brown and David G Lowe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Automatic  panoramic image stitching using invariant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In-  ternational journal of computer vision, 74(1):59–73,  2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [11] David G Lowe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Distinctive image features from scale-  invariant keypoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' International journal of computer  vision, 60(2):91–110, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [12] Eric Krotkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' International Journal of Com-  puter Vision, 1(3):223–237, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [13] Said Pertuz, Domenec Puig, and Miguel Angel Garcia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Analysis of focus measure operators for shape-from-  focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Pattern Recognition, 46(5):1415–1432, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [14] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  U-net: Convolutional networks for biomedical image  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In International Conference on Medical  image computing and computer-assisted intervention,  pages 234–241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content=' In  Proceedings of the IEEE conference on computer vision  and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  [16] Diederik P Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  Adam:  A  method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='  arXiv preprint  arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE0T4oBgHgl3EQfeQCD/content/2301.02388v1.pdf'}
diff --git a/4tE0T4oBgHgl3EQfegAX/content/tmp_files/2301.02390v1.pdf.txt b/4tE0T4oBgHgl3EQfegAX/content/tmp_files/2301.02390v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..32e03da448f9b1bb7bfb2e326a13d99792a6cdb1
--- /dev/null
+++ b/4tE0T4oBgHgl3EQfegAX/content/tmp_files/2301.02390v1.pdf.txt
@@ -0,0 +1,491 @@
+arXiv:2301.02390v1  [eess.IV]  6 Jan 2023
+Deep-learning models in medical image analysis:
+Detection of esophagitis from the Kvasir Dataset
+Kyoka Yoshioka1†, Kensuke Tanioka2, Satoru Hiwa2 and Tomoyuki Hiroyasu2
+1Graduate School of Life and Medical Sciences, Doshisha University, Kyoto, Japan
+2Department of Biomedical Sciences and Informatics, Doshisha University, Kyoto, Japan
+(Tel: +81-774-65-6020; E-mail: tomo@is.doshisha.ac.jp)
+Abstract: Early detection of esophagitis is important because this condition can progress to cancer if left untreated. However,
+the accuracies of different deep learning models in detecting esophagitis have yet to be compared. Thus, this study aimed to
+compare the accuracies of convolutional neural network models (GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3) in
+detecting esophagitis from the open Kvasir dataset of endoscopic images. Results showed that among the models, GoogLeNet
+achieved the highest F1-scores. Based on the average of true positive rate, MobileNet V3 predicted esophagitis more confidently
+than the other models. The results obtained using the models were also compared with those obtained using SHapley Additive
+exPlanations and Gradient-weighted Class Activation Mapping.
+Keywords: Kvasir dataset, Deep Learning, Convolutional Neural Networks, Gradient-Weighted Class Activation Mapping,
+SHAP, SHapley Additive exPlanation
+1. INTRODUCTION
+With the development of artificial intelligence (AI), sev-
+eral studies have focused on the application of this technol-
+ogy in the medical field.
+In gastroenterology, AI is used
+to detect inflammation, polyps, and stomach cancer and de-
+velop systems that can automatically determine the severity
+of symptoms [1] [2] [3] [4]. AI models are expected to im-
+prove diagnostic accuracy and reduce medical costs by pre-
+venting misdiagnosis by humans.
+Various deep learning and AI models, including deep
+learning convolutional neural network (CNN) models, have
+been proposed and used for medical image recognition and
+analysis. However, these models differ in accuracy, and com-
+paring this aspect is important to identify which model is
+suitable for a specific application in endoscopic imaging.
+The z-line is an anatomic landmark located posterior to
+the stomach and esophagus. Esophagitis is an inflammation
+of the esophagus that appears as a break in the esophageal
+mucosa relative to the z-line [5]. The z-line and esophagitis
+can be described as normal and diseased conditions, respec-
+tively. Early detection of esophagitis is necessary because
+this condition can cause complications (e.g., esophageal ul-
+cer, bleeding, and stricture) and progress to cancer if left
+untreated. Therefore, distinguishing between the z-line and
+esophagitis is necessary. However, this procedure is difficult
+[6]. In addition, the accuracies of various models in detecting
+esophagitis have yet to be compared.
+Thus, this study aimed to compare the accuracies of sev-
+eral CNN models, including GoogLeNet [7], ResNet-50 [8],
+MobileNet V2 [9], and MobileNet V3 [10], in identifying
+z-lines and esophagitis in endoscopic images from the open
+Kvasir dataset. These models have received considerable at-
+tention in recent years after winning in the ImageNet Large
+Scale Visual Recognition Challenge (ILSVRC), a competi-
+tion using a large image recognition dataset. The results ob-
+† Kyoka Yoshioka is the presenter of this paper.
+tained by the four CNN models were compared. The training
+models were also compared with the explainable artificial in-
+telligence (XAI) methods Gradient-weighted Class Activa-
+tion Mapping (Grad-CAM) [11] and SHapley Additive ex-
+Planations (SHAP) [12].
+2. DEEP LEARNING IN MEDICAL IMAGE
+ANALYSIS
+2.1. Typical architecture for image classification
+CNN is a deep learning method specialized for image
+recognition. It is widely used for identifying lesion sites in
+medical images. It combines a convolutional layer with a
+pooling layer and finally iterates through all the combined
+layers to generate the results. In this study, we compared
+the results of different CNN models used for site identifi-
+cation in medical images. The CNN models used included
+GoogleNet and ResNet, the successive winning models of
+ILSVRC, and MobileNet V2 and MobileNet V3, which have
+attracted considerable attention in recent years because of
+their small computational and memory.
+2.1.1. GoogLeNet
+GoogLeNet was the winning model at ILSVRC in 2014
+The model consists of an Inception module, 1×1 convolu-
+tion, auxiliary loss, and global average pooling. GoogLeNet
+can be multi-layered using the Inception module, but 1×1
+convolution is performed before each convolution calcula-
+tion to reduce dimensionality resulting from the large num-
+ber of parameters. The Inception module helps process data
+using multiple filters in parallel. The fully connected layer
+is removed to increase the width and depth of the network,
+average pooling is used instead of the fully connected layer
+to avoid gradient loss, and class classification is performed
+on sub-networks branched from the middle of the network
+by auxiliary loss [7].
+
+2.1.2. ResNet
+ResNet was the winning model at the ILSVRC in 2015.
+The problem of learning not progressing due to gradient loss
+and degradation problems was solved using a method called
+Residual Block, which uses 152 very deep layers to solve the
+problem. The key features of this model are residual block
+and batch normalization using shortcut connection. ResNet
+has several models with different layer depths. ResNet-50
+shows higher accuracy than GoogLeNet in ImageNet clas-
+sification [8]. However, ResNet-50 requires about twice as
+many parameters as GoogLeNet.
+2.1.3. MobileNet V2
+MobileNet is a small computationally and memory model
+that can adjust the trade-off between accuracy and compu-
+tational load. Depthwise separable convolution decomposes
+the convolution layer into depthwise and pointwise convolu-
+tion for computation. This mechanism reduces the compu-
+tation cost. Furthermore, V2 introduces expand/projection
+layers and inverted residual blocks. Expand/projection lay-
+ers rapidly increase or decrease the number of channels. Mo-
+bileNet V2 achieves comparable accuracy to GoogLeNet and
+ResNet-50 in ImageNet classification while significantly re-
+ducing the number of parameters [9].
+2.1.4. MobileNet V3
+MobileNet V3 is an improved version of MobileNet V2,
+introducing a squeeze-and-excite structure (SE-block) in the
+inverted residual block, one of the features of MobileNet
+V2. SE-block improves the expressiveness of the model by
+weighting information in the channel direction [13]. Com-
+pared with V2, MobileNet V3 shows more accurate Im-
+ageNet classification while shortening total inference time
+[10].
+2.2. Explainable AI (XAI)
+The CNN models were compared with XAI methods
+Grad-CAM and SHAP. The Discussion section explains the
+results obtained using these techniques.
+2.2.1. Grad CAM
+Grad-CAM displays a color map of the area the CNN is
+gazing at for classification [11]. It is based on the fact that
+variables with large gradients in the output values of the pre-
+dicted class are essential for classification prediction. The
+gradient of each input image pixel with respect to the output
+value of the prediction class in the last convolution layer is
+used.
+2.2.2. SHAP
+SHAP calculates, for each predicted value, how each char-
+acteristic variable affects that prediction [12]. This analysis
+allows us to visualize the impact of an increase or decrease
+in the value of a given characteristic variable.
+3. MATERIALS AND METHODS
+CNN models GoogLeNet, ResNet-50, MobileNet V2, and
+MobileNet V3 were employed to detect esophagitis from the
+open Kvasir dataset of endoscopic images, and their results
+were compared.
+3.1. Kvasir dataset
+The Kvasir dataset is a collection of endoscopic images of
+the gastrointestinal tract. It was annotated and validated by
+certified endoscopists. The dataset was made available in the
+fall of 2017 through the Medical Multimedia Challenge pro-
+vided by MediaEval. It includes anatomical landmarks (py-
+lorus, z-line, and cecum), disease states (esophagitis, ulcera-
+tive colitis, and polyps), and medical procedures (dyed lifted
+polyps and dyed resection margins). The resolution of the
+images from the Kvasir dataset with these eight classes varies
+from 720×576 pixels to 1920×1072 pixels. Each image has
+a different shooting angle, resolution, brightness, magnifica-
+tion, and center point.
+3.2. Prepossessing
+Image prepossessing was performed before training the
+models. Edge artifacts and annotations that interfere with
+learning during the analysis of medical images were re-
+moved. A mask image was created, where pixels with lu-
+minance values below a certain threshold were set to 0. The
+opening process was applied to the mask image to remove the
+annotations. The image was cropped using this final mask
+image to obtain the target area. This process was performed
+on all data.
+Each image in the dataset has a different resolution. All
+images were resized to 224×224 pixels by bilinear comple-
+tion and optimized for deep learning input. In addition to
+these processes, data augmentation was performed on the
+data used for learning. We applied two types of data aug-
+mentation: horizontal and vertical flip.
+3.3. Cross Validation
+A total of 1000 image data sets containing z-lines and
+esophagitis were partitioned into test, training, and validation
+data. First, 25% (n = 250) of the total data were randomly se-
+lected to generate test data. Of the remaining data (75%, n =
+750), 50% (n = 500) was used for training and 25% (n = 250)
+for validation.
+The inner loop consisted of training and validation data.
+The model was trained using the training data, and parame-
+ters such as the optimal number of epochs were determined
+using the validation data. Thus, four training models were
+generated. The test data of each model were evaluated, and
+the average of discrimination accuracy of the four times was
+used as the evaluation value of the CNN model. The test,
+training, and validation data were each partitioned to main-
+tain the class proportions.
+3.4. CNN models
+PyTorch was used for the implementation of GoogLeNet,
+ResNet-50, MobileNet V2, and MobileNet V3.
+The ini-
+tial values of all model parameters were pre-trained by Ima-
+geNet, and the models were trained by fine tuning.
+For all models, the Adam optimizer was used for training.
+The batch size was five, and the maximum number of epochs
+
+was 100. The cross-entropy error shown in equation (1) was
+used as the loss function.
+E(x)
+=
+−
+N
+�
+n=1
+K
+�
+k=1
+dnk log yk(xn; w)
+(1)
+3.5. Evaluation Function
+Five evaluation indices were used in this experiment: ac-
+curacy, precision, recall, specificity, and F1-score. These
+metrics were calculated using the confusion matrix shown
+in Table 1.
+Table 1. Confusion matrix for a two-class problem
+Predicted Class
+(Positive Class)
+Predicted Class
+(Negative Class)
+Actual Class
+(Positive Class)
+True Positive
+False Negative
+Actual Class
+(Negative Class)
+False Positive
+True Negative
+In this experiment, the z-line and esophagitis were judged
+as the negative and positive classes, respectively. In other
+words, data judged to be esophagitis and z-line by the learn-
+ing model were designated true positive (TP) and false neg-
+ative (FN), respectively.
+Meanwhile, data determined to
+be esophagitis and z-line by the training model were des-
+ignated false positive (FP) and true negative (TN), respec-
+tively. Based on the values of TP, FP, TN, and FN obtained
+from the confusion matrix, the accuracy, precision, recall,
+specificity, and F1-score of the models were calculated using
+Equations(2) to (6).
+Accuracy =
+T P + T N
+T P + FP + FN + T N
+(2)
+Precision =
+T P
+T P + FP
+(3)
+Recall =
+T P
+T P + FN
+(4)
+Specificity =
+T N
+T N + FP
+(5)
+F1 score =
+2T N
+2T P + FP + FN
+(6)
+4. RESULTS AND DISCUSSIONS
+4.1. Performance comparison between different archi-
+tecture
+The evaluation indices obtained from the experiments are
+shown in Table 2.
+The F1-score results in Table 2 show that GoogLeNet was
+the best among the four models. In other words, GoogLeNet
+was more reliable in predicting esophagitis than the other
+models. Meanwhile, MobileNet V3 showed the highest pre-
+cision and specificity. In other words, MobileNet V3 was
+the most accurate among the tested models for z-line predic-
+tion. From a medical point of view, an ideal model should be
+Table 2. Performance comparison between different
+architecture
+Model
+ACC
+PREC
+REC
+SPEC
+F1
+GoogLeNet
+0.846
+0.859
+0.830
+0.862
+0.843
+MobileNet V3
+0.842
+0.901
+0.776
+0.908
+0.831
+ResNet-50
+0.833
+0.865
+0.792
+0.874
+0.826
+MobileNet V2
+0.830
+0.852
+0.800
+0.860
+0.825
+likely to distinguish esophagitis with severe symptoms from
+the z-line.
+The average of TP rate were 0.950, 0.923, 0.892, and
+0.841 for MobileNet V3, MobileNet V2, GoogLeNet, and
+ResNet-50, respectively. MobileNet V3 predicted esophagi-
+tis with more confidence than the other models.
+4.2. GoogLeNet analysis
+Grad-CAM and SHAP were applied to the learned model,
+and what kind of the model was created was discussed.
+Fig.1 shows an example of the image results in the case of
+TP predicted by GoogLeNet. In the Grad-CAM results, red
+indicates the most potent activation, and blue indicates the
+weakest activation. In the SHAP results, the SHAP values
+of the patches were computed and rendered in a color map:
+a positive SHHAP value (red) indicates that the class is sup-
+ported. By contrast, a negative SHAP value (blue) indicates
+that the class is rejected.
+Tearing the esophageal mucosa against the z-line is a
+feature of esophagitis.
+According to Fig.1, the results of
+Grad-CAM and SHAP showed that the learned model of
+GoogLeNet can makes predictions focusing on the clinically
+significant aspects of esophagitis images. The GoogLeNet
+model learned the findings that are important for diagnosing
+esophagitis. Comparison results showed that SHAP captured
+the location of multiple mucosal tears in the image more ac-
+curately than Grad-CAM.
+Fig.2 shows the results of applying Grad-CAM and SHAP
+in the FN case. The following can be observed from the re-
+sults of Grad-CAM and SHAP for Fig.2, respectively. In the
+Grad-CAM results, most areas in the image are shown as
+activated regions. Areas that provide the basis for the pre-
+diction are difficult to identify because of the gradient satu-
+ration in the Grad-CAM calculation. In the SHAP results,
+the inflammatory areas of the input image are indicated by
+blue pixels. Blue pixels indicate features that have a negative
+contribution to the prediction. In other words, although the
+model incorrectly identified esophagitis as a z-line, the model
+recognized that areas in the image negatively contributed to
+the z-line decision.
+4.3. MobileNet V3 analysis
+One hundred images were determined to be TP in the
+MobileNet V3 model.
+The SHAP results for the images
+judged to have the highest and lowest probabilities of being
+esophagitis are shown in Fig.3.
+As shown in Fig.3, in cases with a high prediction proba-
+bility, some features may have a negative contribution to the
+
+(a) Raw image
+(b) Grad-CAM
+(c) SHAP
+Fig. 1. True Positive Pattern
+(a) Raw image
+(b) Grad-CAM
+(c) SHAP
+Fig. 2. False Negative Pattern
+Fig. 3. First image predicted positive with 1.000 probability, and second image predicted positive with 0.524 probability.
+prediction. Many features showing negative contributions
+can be identified in the images with low prediction proba-
+bility for Fig.3. In this case, the prediction probability may
+be low.
+5. CONCLUSIONS
+We compared the accuracies of CNN models, including
+GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3,
+in identifying z-line and esophagitis in endoscopic images
+from the open Kvasir dataset.
+Among the four models,
+GoogLeNet had the highest F1-score, and MobileNet V3
+had the highest average TP rate. These results suggest that
+GoogLeNet performs better than state-of-the-art CNN mod-
+els in medical image recognition. In addition, MoblieNet V3
+is a cost-effective model because of its low memory and short
+training time. Each model was analyzed and compared with
+Grad-CAM, and SHAP. Other models, datasets, and model
+analyses are warranted for verification.
+REFERENCES
+[1]
+Peng-Jen Chen, Meng-Chiung Lin, Mei-Ju Lai, Jung-
+Chun Lin, Henry Horng-Shing Lu, and Vincent S
+Tseng. Accurate classification of diminutive colorectal
+polyps using computer-aided analysis. Gastroenterol-
+ogy, 154(3):568–575, 2018.
+[2]
+Toshiaki Hirasawa, Kazuharu Aoyama, Tetsuya Tan-
+imoto, Soichiro Ishihara, Satoki Shichijo, Tsuyoshi
+Ozawa, Tatsuya Ohnishi, Mitsuhiro Fujishiro, Keigo
+Matsuo, Junko Fujisaki, et al. Application of artificial
+intelligence using a convolutional neural network for
+detecting gastric cancer in endoscopic images. Gastric
+Cancer, 21(4):653–660, 2018.
+
+0.0010
+0.0005
+0.0000
+-0.0005
+0.00100.002
+0.001
+0.000
+0.001
+0.002E00'0
+0.002
+0.001
+0000-
+0.001
+0.002
+E00'0-0.004
+E000
+0.002
+0.001
+0.000
+0.001
+0.002
+0.003
+0.004[3]
+Pedro Guimar˜aes, Andreas Keller, Tobias Fehlmann,
+Frank Lammert, and Markus Casper.
+Deep-learning
+based detection of gastric precancerous conditions.
+Gut, 69(1):4–6, 2020.
+[4]
+Yaqiong Zhang, Fengxia Li, Fuqiang Yuan, Kai Zhang,
+Lijuan Huo, Zichen Dong, Yiming Lang, Yapeng
+Zhang, Meihong Wang, Zenghui Gao, et al. Diagnosing
+chronic atrophic gastritis by gastroscopy using artificial
+intelligence. Digestive and Liver Disease, 52(5):566–
+572, 2020.
+[5]
+Konstantin
+Pogorelov,
+Kristin
+Ranheim
+Randel,
+Carsten Griwodz, Sigrun Losada Eskeland, Thomas
+de Lange,
+Dag Johansen,
+Concetto Spampinato,
+Duc-Tien Dang-Nguyen, Mathias Lux, Peter Thelin
+Schmidt, et al. Kvasir: A multi-class image dataset for
+computer aided gastrointestinal disease detection. In
+Proceedings of the 8th ACM on Multimedia Systems
+Conference, pages 164–169, 2017.
+[6]
+Timothy Cogan, Maribeth Cogan, and Lakshman
+Tamil.
+Mapgi:
+accurate identification of anatomi-
+cal landmarks and diseased tissue in gastrointestinal
+tract using deep learning. Computers in biology and
+medicine, 111:103351, 2019.
+[7]
+Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Ser-
+manet, Scott Reed, Dragomir Anguelov, Dumitru Er-
+han, Vincent Vanhoucke, and Andrew Rabinovich. Go-
+ing deeper with convolutions. In Proceedings of the
+IEEE conference on computer vision and pattern recog-
+nition, pages 1–9, 2015.
+[8]
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian
+Sun. Deep residual learning for image recognition. In
+Proceedings of the IEEE conference on computer vision
+and pattern recognition, pages 770–778, 2016.
+[9]
+Mark Sandler, Andrew Howard, Menglong Zhu, An-
+drey Zhmoginov, and Liang-Chieh Chen. Mobilenetv2:
+Inverted residuals and linear bottlenecks. In Proceed-
+ings of the IEEE conference on computer vision and
+pattern recognition, pages 4510–4520, 2018.
+[10] Andrew Howard, Mark Sandler, Grace Chu, Liang-
+Chieh Chen, Bo Chen, Mingxing Tan, Weijun Wang,
+Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al.
+Searching for mobilenetv3.
+In Proceedings of the
+IEEE/CVF international conference on computer vi-
+sion, pages 1314–1324, 2019.
+[11] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek
+Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv
+Batra. Grad-cam: Visual explanations from deep net-
+works via gradient-based localization. In Proceedings
+of the IEEE international conference on computer vi-
+sion, pages 618–626, 2017.
+[12] Scott M Lundberg and Su-In Lee. A unified approach
+to interpreting model predictions. Advances in neural
+information processing systems, 30, 2017.
+[13] Jie Hu, Li Shen, and Gang Sun. Squeeze-and-excitation
+networks. In Proceedings of the IEEE conference on
+computer vision and pattern recognition, pages 7132–
+7141, 2018.
+
diff --git a/4tE0T4oBgHgl3EQfegAX/content/tmp_files/load_file.txt b/4tE0T4oBgHgl3EQfegAX/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d5a361939920e95e6d887574be4016bdc485dba3
--- /dev/null
+++ b/4tE0T4oBgHgl3EQfegAX/content/tmp_files/load_file.txt
@@ -0,0 +1,281 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf,len=280
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='02390v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='IV]  6 Jan 2023  Deep-learning models in medical image analysis:  Detection of esophagitis from the Kvasir Dataset  Kyoka Yoshioka1†, Kensuke Tanioka2, Satoru Hiwa2 and Tomoyuki Hiroyasu2  1Graduate School of Life and Medical Sciences, Doshisha University, Kyoto, Japan  2Department of Biomedical Sciences and Informatics, Doshisha University, Kyoto, Japan  (Tel: +81-774-65-6020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' E-mail: tomo@is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='doshisha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='jp)  Abstract: Early detection of esophagitis is important because this condition can progress to cancer if left untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However,  the accuracies of different deep learning models in detecting esophagitis have yet to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Thus, this study aimed to  compare the accuracies of convolutional neural network models (GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3) in  detecting esophagitis from the open Kvasir dataset of endoscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Results showed that among the models, GoogLeNet  achieved the highest F1-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Based on the average of true positive rate, MobileNet V3 predicted esophagitis more confidently  than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The results obtained using the models were also compared with those obtained using SHapley Additive  exPlanations and Gradient-weighted Class Activation Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Keywords: Kvasir dataset, Deep Learning, Convolutional Neural Networks, Gradient-Weighted Class Activation Mapping,  SHAP, SHapley Additive exPlanation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' INTRODUCTION  With the development of artificial intelligence (AI), sev-  eral studies have focused on the application of this technol-  ogy in the medical field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  In gastroenterology, AI is used  to detect inflammation, polyps, and stomach cancer and de-  velop systems that can automatically determine the severity  of symptoms [1] [2] [3] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' AI models are expected to im-  prove diagnostic accuracy and reduce medical costs by pre-  venting misdiagnosis by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Various deep learning and AI models, including deep  learning convolutional neural network (CNN) models, have  been proposed and used for medical image recognition and  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, these models differ in accuracy, and com-  paring this aspect is important to identify which model is  suitable for a specific application in endoscopic imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The z-line is an anatomic landmark located posterior to  the stomach and esophagus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Esophagitis is an inflammation  of the esophagus that appears as a break in the esophageal  mucosa relative to the z-line [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The z-line and esophagitis  can be described as normal and diseased conditions, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Early detection of esophagitis is necessary because  this condition can cause complications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=', esophageal ul-  cer, bleeding, and stricture) and progress to cancer if left  untreated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Therefore, distinguishing between the z-line and  esophagitis is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, this procedure is difficult  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition, the accuracies of various models in detecting  esophagitis have yet to be compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Thus, this study aimed to compare the accuracies of sev-  eral CNN models, including GoogLeNet [7], ResNet-50 [8],  MobileNet V2 [9], and MobileNet V3 [10], in identifying  z-lines and esophagitis in endoscopic images from the open  Kvasir dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These models have received considerable at-  tention in recent years after winning in the ImageNet Large  Scale Visual Recognition Challenge (ILSVRC), a competi-  tion using a large image recognition dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The results ob-  † Kyoka Yoshioka is the presenter of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  tained by the four CNN models were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The training  models were also compared with the explainable artificial in-  telligence (XAI) methods Gradient-weighted Class Activa-  tion Mapping (Grad-CAM) [11] and SHapley Additive ex-  Planations (SHAP) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' DEEP LEARNING IN MEDICAL IMAGE  ANALYSIS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Typical architecture for image classification  CNN is a deep learning method specialized for image  recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It is widely used for identifying lesion sites in  medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It combines a convolutional layer with a  pooling layer and finally iterates through all the combined  layers to generate the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In this study, we compared  the results of different CNN models used for site identifi-  cation in medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The CNN models used included  GoogleNet and ResNet, the successive winning models of  ILSVRC, and MobileNet V2 and MobileNet V3, which have  attracted considerable attention in recent years because of  their small computational and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet  GoogLeNet was the winning model at ILSVRC in 2014  The model consists of an Inception module, 1×1 convolu-  tion, auxiliary loss, and global average pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet  can be multi-layered using the Inception module, but 1×1  convolution is performed before each convolution calcula-  tion to reduce dimensionality resulting from the large num-  ber of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The Inception module helps process data  using multiple filters in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The fully connected layer  is removed to increase the width and depth of the network,  average pooling is used instead of the fully connected layer  to avoid gradient loss, and class classification is performed  on sub-networks branched from the middle of the network  by auxiliary loss [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet  ResNet was the winning model at the ILSVRC in 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The problem of learning not progressing due to gradient loss  and degradation problems was solved using a method called  Residual Block, which uses 152 very deep layers to solve the  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The key features of this model are residual block  and batch normalization using shortcut connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet  has several models with different layer depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' ResNet-50  shows higher accuracy than GoogLeNet in ImageNet clas-  sification [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' However, ResNet-50 requires about twice as  many parameters as GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V2  MobileNet is a small computationally and memory model  that can adjust the trade-off between accuracy and compu-  tational load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Depthwise separable convolution decomposes  the convolution layer into depthwise and pointwise convolu-  tion for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This mechanism reduces the compu-  tation cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Furthermore, V2 introduces expand/projection  layers and inverted residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Expand/projection lay-  ers rapidly increase or decrease the number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Mo-  bileNet V2 achieves comparable accuracy to GoogLeNet and  ResNet-50 in ImageNet classification while significantly re-  ducing the number of parameters [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3  MobileNet V3 is an improved version of MobileNet V2,  introducing a squeeze-and-excite structure (SE-block) in the  inverted residual block, one of the features of MobileNet  V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' SE-block improves the expressiveness of the model by  weighting information in the channel direction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Com-  pared with V2, MobileNet V3 shows more accurate Im-  ageNet classification while shortening total inference time  [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Explainable AI (XAI)  The CNN models were compared with XAI methods  Grad-CAM and SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The Discussion section explains the  results obtained using these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Grad CAM  Grad-CAM displays a color map of the area the CNN is  gazing at for classification [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It is based on the fact that  variables with large gradients in the output values of the pre-  dicted class are essential for classification prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The  gradient of each input image pixel with respect to the output  value of the prediction class in the last convolution layer is  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' SHAP  SHAP calculates, for each predicted value, how each char-  acteristic variable affects that prediction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This analysis  allows us to visualize the impact of an increase or decrease  in the value of a given characteristic variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MATERIALS AND METHODS  CNN models GoogLeNet, ResNet-50, MobileNet V2, and  MobileNet V3 were employed to detect esophagitis from the  open Kvasir dataset of endoscopic images, and their results  were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Kvasir dataset  The Kvasir dataset is a collection of endoscopic images of  the gastrointestinal tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It was annotated and validated by  certified endoscopists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The dataset was made available in the  fall of 2017 through the Medical Multimedia Challenge pro-  vided by MediaEval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' It includes anatomical landmarks (py-  lorus, z-line, and cecum), disease states (esophagitis, ulcera-  tive colitis, and polyps), and medical procedures (dyed lifted  polyps and dyed resection margins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The resolution of the  images from the Kvasir dataset with these eight classes varies  from 720×576 pixels to 1920×1072 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Each image has  a different shooting angle, resolution, brightness, magnifica-  tion, and center point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Prepossessing  Image prepossessing was performed before training the  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Edge artifacts and annotations that interfere with  learning during the analysis of medical images were re-  moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' A mask image was created, where pixels with lu-  minance values below a certain threshold were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The  opening process was applied to the mask image to remove the  annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The image was cropped using this final mask  image to obtain the target area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' This process was performed  on all data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Each image in the dataset has a different resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' All  images were resized to 224×224 pixels by bilinear comple-  tion and optimized for deep learning input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition to  these processes, data augmentation was performed on the  data used for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' We applied two types of data aug-  mentation: horizontal and vertical flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Cross Validation  A total of 1000 image data sets containing z-lines and  esophagitis were partitioned into test, training, and validation  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' First, 25% (n = 250) of the total data were randomly se-  lected to generate test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Of the remaining data (75%, n =  750), 50% (n = 500) was used for training and 25% (n = 250)  for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The inner loop consisted of training and validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The model was trained using the training data, and parame-  ters such as the optimal number of epochs were determined  using the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Thus, four training models were  generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The test data of each model were evaluated, and  the average of discrimination accuracy of the four times was  used as the evaluation value of the CNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The test,  training, and validation data were each partitioned to main-  tain the class proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' CNN models  PyTorch was used for the implementation of GoogLeNet,  ResNet-50, MobileNet V2, and MobileNet V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The ini-  tial values of all model parameters were pre-trained by Ima-  geNet, and the models were trained by fine tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  For all models, the Adam optimizer was used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The batch size was five, and the maximum number of epochs  was 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The cross-entropy error shown in equation (1) was  used as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  E(x)  =  −  N  �  n=1  K  �  k=1  dnk log yk(xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' w)  (1)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Evaluation Function  Five evaluation indices were used in this experiment: ac-  curacy, precision, recall, specificity, and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These  metrics were calculated using the confusion matrix shown  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Confusion matrix for a two-class problem  Predicted Class  (Positive Class)  Predicted Class  (Negative Class)  Actual Class  (Positive Class)  True Positive  False Negative  Actual Class  (Negative Class)  False Positive  True Negative  In this experiment, the z-line and esophagitis were judged  as the negative and positive classes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other  words, data judged to be esophagitis and z-line by the learn-  ing model were designated true positive (TP) and false neg-  ative (FN), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Meanwhile, data determined to  be esophagitis and z-line by the training model were des-  ignated false positive (FP) and true negative (TN), respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Based on the values of TP, FP, TN, and FN obtained  from the confusion matrix, the accuracy, precision, recall,  specificity, and F1-score of the models were calculated using  Equations(2) to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Accuracy =  T P + T N  T P + FP + FN + T N  (2)  Precision =  T P  T P + FP  (3)  Recall =  T P  T P + FN  (4)  Specificity =  T N  T N + FP  (5)  F1 score =  2T N  2T P + FP + FN  (6)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' RESULTS AND DISCUSSIONS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Performance comparison between different archi-  tecture  The evaluation indices obtained from the experiments are  shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The F1-score results in Table 2 show that GoogLeNet was  the best among the four models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, GoogLeNet  was more reliable in predicting esophagitis than the other  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Meanwhile, MobileNet V3 showed the highest pre-  cision and specificity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, MobileNet V3 was  the most accurate among the tested models for z-line predic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' From a medical point of view, an ideal model should be  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Performance comparison between different  architecture  Model  ACC  PREC  REC  SPEC  F1  GoogLeNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='846  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='859  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='862  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='843  MobileNet V3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='842  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='901  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='776  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='908  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='831  ResNet-50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='833  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='865  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='792  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='874  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='826  MobileNet V2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='852  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='860  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='825  likely to distinguish esophagitis with severe symptoms from  the z-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The average of TP rate were 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='950, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='923, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='892, and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='841 for MobileNet V3, MobileNet V2, GoogLeNet, and  ResNet-50, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3 predicted esophagi-  tis with more confidence than the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' GoogLeNet analysis  Grad-CAM and SHAP were applied to the learned model,  and what kind of the model was created was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1 shows an example of the image results in the case of  TP predicted by GoogLeNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the Grad-CAM results, red  indicates the most potent activation, and blue indicates the  weakest activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the SHAP results, the SHAP values  of the patches were computed and rendered in a color map:  a positive SHHAP value (red) indicates that the class is sup-  ported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' By contrast, a negative SHAP value (blue) indicates  that the class is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Tearing the esophageal mucosa against the z-line is a  feature of esophagitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='1, the results of  Grad-CAM and SHAP showed that the learned model of  GoogLeNet can makes predictions focusing on the clinically  significant aspects of esophagitis images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The GoogLeNet  model learned the findings that are important for diagnosing  esophagitis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Comparison results showed that SHAP captured  the location of multiple mucosal tears in the image more ac-  curately than Grad-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2 shows the results of applying Grad-CAM and SHAP  in the FN case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' The following can be observed from the re-  sults of Grad-CAM and SHAP for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the  Grad-CAM results, most areas in the image are shown as  activated regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Areas that provide the basis for the pre-  diction are difficult to identify because of the gradient satu-  ration in the Grad-CAM calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In the SHAP results,  the inflammatory areas of the input image are indicated by  blue pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Blue pixels indicate features that have a negative  contribution to the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In other words, although the  model incorrectly identified esophagitis as a z-line, the model  recognized that areas in the image negatively contributed to  the z-line decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' MobileNet V3 analysis  One hundred images were determined to be TP in the  MobileNet V3 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  The SHAP results for the images  judged to have the highest and lowest probabilities of being  esophagitis are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3, in cases with a high prediction proba-  bility, some features may have a negative contribution to the  (a) Raw image  (b) Grad-CAM  (c) SHAP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' True Positive Pattern  (a) Raw image  (b) Grad-CAM  (c) SHAP  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' False Negative Pattern  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' First image predicted positive with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='000 probability, and second image predicted positive with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='524 probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Many features showing negative contributions  can be identified in the images with low prediction proba-  bility for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In this case, the prediction probability may  be low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' CONCLUSIONS  We compared the accuracies of CNN models, including  GoogLeNet, ResNet-50, MobileNet V2, and MobileNet V3,  in identifying z-line and esophagitis in endoscopic images  from the open Kvasir dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Among the four models,  GoogLeNet had the highest F1-score, and MobileNet V3  had the highest average TP rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' These results suggest that  GoogLeNet performs better than state-of-the-art CNN mod-  els in medical image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In addition, MoblieNet V3  is a cost-effective model because of its low memory and short  training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Each model was analyzed and compared with  Grad-CAM, and SHAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Other models, datasets, and model  analyses are warranted for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  REFERENCES  [1]  Peng-Jen Chen, Meng-Chiung Lin, Mei-Ju Lai, Jung-  Chun Lin, Henry Horng-Shing Lu, and Vincent S  Tseng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Accurate classification of diminutive colorectal  polyps using computer-aided analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Gastroenterol-  ogy, 154(3):568–575, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [2]  Toshiaki Hirasawa, Kazuharu Aoyama, Tetsuya Tan-  imoto, Soichiro Ishihara, Satoki Shichijo, Tsuyoshi  Ozawa, Tatsuya Ohnishi, Mitsuhiro Fujishiro, Keigo  Matsuo, Junko Fujisaki, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Application of artificial  intelligence using a convolutional neural network for  detecting gastric cancer in endoscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Gastric  Cancer, 21(4):653–660, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='00100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content="002E00'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0000-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content="002  E00'0-0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='004  E000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='004[3]  Pedro Guimar˜aes, Andreas Keller, Tobias Fehlmann,  Frank Lammert, and Markus Casper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Deep-learning  based detection of gastric precancerous conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Gut, 69(1):4–6, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [4]  Yaqiong Zhang, Fengxia Li, Fuqiang Yuan, Kai Zhang,  Lijuan Huo, Zichen Dong, Yiming Lang, Yapeng  Zhang, Meihong Wang, Zenghui Gao, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Diagnosing  chronic atrophic gastritis by gastroscopy using artificial  intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Digestive and Liver Disease, 52(5):566–  572, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [5]  Konstantin  Pogorelov,  Kristin  Ranheim  Randel,  Carsten Griwodz, Sigrun Losada Eskeland, Thomas  de Lange,  Dag Johansen,  Concetto Spampinato,  Duc-Tien Dang-Nguyen, Mathias Lux, Peter Thelin  Schmidt, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Kvasir: A multi-class image dataset for  computer aided gastrointestinal disease detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In  Proceedings of the 8th ACM on Multimedia Systems  Conference, pages 164–169, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [6]  Timothy Cogan, Maribeth Cogan, and Lakshman  Tamil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Mapgi:  accurate identification of anatomi-  cal landmarks and diseased tissue in gastrointestinal  tract using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Computers in biology and  medicine, 111:103351, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [7]  Christian Szegedy, Wei Liu, Yangqing Jia, Pierre Ser-  manet, Scott Reed, Dragomir Anguelov, Dumitru Er-  han, Vincent Vanhoucke, and Andrew Rabinovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Go-  ing deeper with convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings of the  IEEE conference on computer vision and pattern recog-  nition, pages 1–9, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [8]  Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian  Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Deep residual learning for image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In  Proceedings of the IEEE conference on computer vision  and pattern recognition, pages 770–778, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [9]  Mark Sandler, Andrew Howard, Menglong Zhu, An-  drey Zhmoginov, and Liang-Chieh Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Mobilenetv2:  Inverted residuals and linear bottlenecks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceed-  ings of the IEEE conference on computer vision and  pattern recognition, pages 4510–4520, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [10] Andrew Howard, Mark Sandler, Grace Chu, Liang-  Chieh Chen, Bo Chen, Mingxing Tan, Weijun Wang,  Yukun Zhu, Ruoming Pang, Vijay Vasudevan, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  Searching for mobilenetv3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  In Proceedings of the  IEEE/CVF international conference on computer vi-  sion, pages 1314–1324, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [11] Ramprasaath R Selvaraju, Michael Cogswell, Abhishek  Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv  Batra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Grad-cam: Visual explanations from deep net-  works via gradient-based localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings  of the IEEE international conference on computer vi-  sion, pages 618–626, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [12] Scott M Lundberg and Su-In Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' A unified approach  to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Advances in neural  information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content='  [13] Jie Hu, Li Shen, and Gang Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' Squeeze-and-excitation  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
+page_content=' In Proceedings of the IEEE conference on  computer vision and pattern recognition, pages 7132–  7141, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4tE0T4oBgHgl3EQfegAX/content/2301.02390v1.pdf'}
diff --git a/6tAzT4oBgHgl3EQfEvoZ/content/tmp_files/2301.00997v1.pdf.txt b/6tAzT4oBgHgl3EQfEvoZ/content/tmp_files/2301.00997v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..737a4221152ffbf3cb2f8ccff49c9d6417f55e17
--- /dev/null
+++ b/6tAzT4oBgHgl3EQfEvoZ/content/tmp_files/2301.00997v1.pdf.txt
@@ -0,0 +1,848 @@
+Detector and physics simulation using heavy ion collisions at
+NICA-SPD
+I. Denisenko1,a) and R. Pandey2,b)
+1Joint Institute for Nuclear Research, Joliot-Curie 6, Dubna-141980,
+Moscow Region, Russia.
+2Graduate Engineer Trainee, Larsen & Toubro Limited, Faridabad,
+Haryana, India.
+a)iden@jinr.ru
+b)rishav160999@gmail.com
+Abstract
+The space-time picture of hadron formation in high-energy collisions with nuclear
+targets is still poorly known.
+The tests of hadron formation was suggested for the
+first stage of SPD running.
+They will require measuring charged pion and proton
+spectra with the precision better than 10%. A research has been carried out to check
+feasibility of such studies at SPD. In this work, 12C − 12C and 40Ca − 40Ca heavy ion
+collisions at center of mass energy of 11 GeV/nucleon were simulated using the SMASH
+event generator. Firstly, the generator-level events were studied. The distribution of
+track multiplicities and momentum distributions of different types of charged particles
+were obtained. Secondly, the generated events passed through the full reconstruction
+using the SpdRoot framework.
+At this stage particles were identified using dE/dx
+measurement and time-of-flight information. It allowed us to estimate charge track
+multiplicities in the tracking system and purities of charge particles spectra. The results
+on multiplicity are important to estimate occupancies in the tracking system, while the
+results on the pion and proton momentum spectra show that particle identification
+should be acceptable for validation of hadron formation models. This is the first study
+of moderate ion collisions for the SPD Collaboration.
+Keywords:
+Hadron formation effects, Heavy ion collision, SMASH, NICA-SPD, Rapidity,
+Charged track multiplicity, Particle physics event generator.
+1
+arXiv:2301.00997v1  [physics.ins-det]  3 Jan 2023
+
+1
+INTRODUCTION
+The SPD detector is primarily optimized to study spin dependent gluon structure of proton and
+deuteron using open charm production, charmonia production and prompt photons. At the same
+time, its physics program includes studies of various aspects of QCD. The work is devoted to studies
+of hadron formation in nuclear collisions proposed in Ref. [1].
+Hadrons produced in hadron collisions emerge in the form of prehadrons, which interact with
+nucleons with reduced strength. This suppression is poorly known and is described in model de-
+pendent way. This suppression results in different spectra of final particles as is illustrated in Fig.1
+for rapidity distributions (in a similar way it affects the pT spectrum). Naturally, these spectra can
+be used to study hadron formation effects. The required precision of such measurements is 10%.
+The aim of this work is to evaluate feasibility of such measurements with MC simulation. Here,
+ion collisions of 12C − 12C and 40Ca − 40Ca at √s = 11AGeV were generated using the SMASH
+(Simulating Many Accelerated Strongly-interacting Hadrons) event generator. Afterwards, the the
+full simulation and reconstruction was performed using the SpdRoot framework.
+Figure 1: Rapidity spectra of protons and charged pions in 12C − 12C and 40Ca − 40Ca collisions.
+2
+
+12C + 12C, s= 11 GeV
+40Ca + 40Ca, sN = 11 GeV
+105
+106
+Protons
+Protons
+w/oformation
+w/oformation
+default
+default
+QDM
+QDM
+do/dy, mb
+104
+peut = 2 GeV/c -
+ Peut = 2 GeV/c -
+Peut = 1 GeV/c
+Peut = 1 GeV/c
+103
+104
+102
+103
+4 -3 -2 -1
+0
+1
+2
+3
+4
+4
+-3 -2 -1
+0
+1
+2
+3
+4
+y
+y
+12C + 12C, sN= 11 GeV
+40Ca + 40Ca, sNR = 11 GeV
+104
+105
+do/dy, mb
+do/dy, mb
+103
+104
+w/oformation
+w/oformation
+default
+default
+QDM
+QDM
+Peut = 2 GeV/c
+Peut = 2 GeV/c
+Pcut = 1 GeV/c
+pcut = 1 GeV/c
+102
+103
+4 -3 -2 
+-1
+0
+1
+2
+3
+4
+4 -3 -2 -1
+0
+1
+2
+3
+4
+y
+y2
+NICA FACILITY
+The NICA (Nuclotron based Ion Collider fAcility) collider at Joint Institute for Nuclear Research
+in Dubna is is being built to provide beams for two experiments. The first experiment, MPD (Multi
+Purpose Detector), will study properties of dense baryonic matter (matter present at extreme high
+density in QCD phase diagram) like Quark Gluon Plasma. The second experiment, SPD (Spin
+Physics Detector), is devoted to study of spin related phonomena and QCD. Once the NICA collider
+will be operational, scientists will be able to create a special state of matter in laboratory which
+existed for very short interval of time (˜20µ sec) just after the big bang. This special state is called
+as QGP (Quark Gluon Plasma) and it filled the entire universe shortly after the big bang.
+The main parts of NICA facility consists of two independent injector complex (injector for light
+ions, and injector for heavy ions-KRION 6T), Light Ion Linear Accelerator (LU20) for accelerating
+light ions like protons (H+), deutrons, and α-particles upto 5 MeV of K.E, then Heavy Ion Linear
+Accelerator (HILAC) to accelerate heavy ions upto Au to a maximum K.E of 3.2 MeV/n, then a
+Super Conducting (SC) Booster Synchrotron to create ultra high vacuum and to provide complete
+stripping of heavy ions, then a SC Heavy Ion Synchrotron Nuclotron to accelerate both light and
+heavy ions to required beam energy. The accelerated beams will collide at two different locations
+where MPD detector and SPD detector are being built. The schematic view of NICA complex is
+shown in Fig.2.
+Figure 2: Schematic view of NICA complex.
+3
+SPD DETECTOR
+The Spin Physics Detector [2,3] is a 4π universal detector optimized to study spin-related phenomena
+via open charm, charmonia and promopt photons in the collisions of polarized p-p or d-d beams
+with √sNN up to 27 GeV. However, at first stage of NICA-SPD, the expected collision energy
+will be from 3.4 up to 10 GeV, and later on after first upgrade, it is expected to reach upto 27
+GeV. The general layout depicting isometric projection of SPD setup is shown in Fig.3. The main
+parts involved in advanced tracking and particle identification capabilities have been shown. (i)
+The beam pipe passes through the center of the detector, carries the accelerated beams of ions. (ii)
+The MicroMegas detector is to improves the momentum resolution and tracking efficiency of the
+tracking system. (iii) The Straw Tracker (ST) detector is for the reconstruction of the primary and
+3
+
+BM@N Detector
+SPD
+Transport Channel
+HILAC
+Collider
+LU20
+Booster
+-MPD
+Nuclotronsecondary particle tracks and for determination of their momenta. (iv) The Time Of Flight (TOF)
+detector, is a part of Particle Identification (PID) system, and is used for identification of particles
+like π, k, and p with long trajectories. (v) The magnet system shown by red color provides 1T
+of magnetic field along the beam axis. This setup is limited to first stage of SPD operation, and
+will be considered only for the identification of stable charged particles. Neutral particles, like n0,
+photons will be detected at later stages. The main parts of SPD first stage have been explained in
+detail below. There is a possibility to have TOF system for the first stage studies.
+Figure 3: Layout of the SPD setup proposed for first stage at NICA-SPD.
+3.1
+CENTRAL TRACKER
+The innermost detector of SPD consists of a MicroMegas-based Central Tracker (MCT). Its purpose
+is to identify the primary vertex coordinate and to improve momentum resolution and tracking
+efficiency.
+It is based on MicroMegas (Micro Mesh Gaseous Structure) technology and detects
+charged particle by amplifying the charges produced due to ionization of the gas molecules present
+in detector volume. When an ionizing particle track passes through detector volume, it ionizes the
+gas molecules and creates few hundreds of e−-ion pair. Electrons are accelerated opposite to the
+direction of applied electric field of 600 V/cm in ionization gap, while ions are attracted towards
+cathode. When the e− crosses micromesh, it faces intense electric field (> 30 KV/cm) and gains
+enough energy to ionize other gas molecules in its path. During this process an avalanche of e−-ion
+pair is produced (1e− produces 104 e−-ion pairs) which is significant to create an electronic signal
+which is read out by readout electrodes.
+3.2
+STRAW TRACKER
+ST is mainly for the reconstruction of primary and secondary particle tracks and measuring their
+momenta, but also participates in identification of π, K, and p on via energy deposit (dE/dx)
+measurements. It consists of two major parts - barrel (covers radius from 270 to 850 mm) and two
+end-caps. The barrel is divided into 8 modules enclosed in a carbon fiber capsule. Each module has
+30 double layers of straw tubes (dia 1cm) which runs parallel (long straw tubes) and perpendicular
+4
+
+Straw tracker
+Magnet
+Range system
+MicroMegas Endcap
+RangesystemEndcap
+MicroMegas
+Beam-beamcounter
+Beam pipe
+Strawtracker Endcap
+zoomx4
+Zero degree calorimeter(short straw tubes) to the beam axis and contains 1500 and 6000 parallel and perpendicular straw
+tubes respectively. Straw tubes are made of polyethylene terephthalate and outer surface is coated
+with very thin layer of Cu and Au. Carbon capsule is meant to protect the outer surface of these
+tubes from humidity. One side and two opposite ends of capsule are provided with small holes
+where end plugs are fixed. FEE are connected to these end plugs to read the detector signal. Any
+particle which passes through the long straws will send detector signal to both opposite ends while a
+particle passing through short straw will send detector signal to any one side of capsule where FEE
+is attached. Thus, long straws will be read from two opposite ends while short straws will be read
+from one side. The end-caps of ST are divided into 3 modules and each module has 4 hexadecimal
+cameras (U, V, X, Y) to record the four coordinates of any physical quantity like four-momentum.
+The FEE to be used can be similar to the one used at NA64 experiment (for the search of dark
+matter), or DUNE experiment (to detect and study properties of neutrino).
+3.3
+TIME OF FLIGHT DETECTOR
+TOF detector is the part of PID system. Similar to ST, the TOF provides identification of π, k, and
+p by measuring their flight time. The energy loss data registered by ST can be used together with the
+data from TOF for correct identification of particle tracks. The TOF distinguishes charged particles
+(mainly π and k) in the momentum range up to 1.5 GeV. The major parts of TOF comprises of a
+barrel and two end-caps. For the first stage of NICA-SPD, two different designs of TOF has been
+suggested. First one is TOF based on multigap timing Resistive Plate Chambers (mRPC), which
+will consist 220 rectangular plate chambers (160 for the barrel and 30 each for end-caps). Second
+one is based on Plastic Scintillator Tiles and will comprise 10.1K small scintillator tiles (7.4K for
+barrel and 1.4K for each end-caps). Scintillator has a property of emitting light in visible region
+when an ionizing radiation passes through it. So, in this design when a particle passes through
+TOF, scintillated photons are produced which are detected by four Si Photo Multipliers (SiPMs)
+present at each sensor board attached at two extreme ends of scintillator tile.
+4
+EVENT GENERATION
+12C −12C and 40Ca−40Ca heavy ion collisions at √s = 11 AGeV with maximum impact parameter
+set to 8 fm for C-C and 11 fm for Ca-Ca were simulated using SMASH. The fermi motion was
+assumed to be “frozen” and 100K events were generated for each heavy ion collision. The SMASH
+input file for C-C collision is shown below.
+*********** SMASH INPUT ************
+config.yaml file for C-C collision.
+Logging:
+default: INFO
+General:
+Modus:
+Collider
+Time_Step_Mode: Fixed
+Delta_Time:
+0.1
+End_Time:
+200.0
+Randomseed:
+-1
+Nevents:
+100000
+5
+
+Output:
+Output_Interval: 10.0
+Particles:
+Format:
+["Oscar2013"]
+Modi:
+Collider:
+Projectile:
+Particles: {2212: 6, 2112: 6} #C-12
+Target:
+Particles: {2212: 6, 2112: 6} #C-12
+Sqrtsnn: 11.0
+Impact:
+Sample: "quadratic"
+Range: [0.0, 8.0]
+Fermi_Motion: "frozen"
+************************************
+Multiplicity of generated charged particles for C − C and Ca − Ca collisions are shown in
+Fig. 4. The peaks at 12 for 12C+12C collisions and at 40 for 40Ca+40Ca collisions correspond to
+events where no interaction occurred. The rapidity distributions are shown in Fig. 5. The spectra
+obtained from SMASH output show qualitative agreement with the ones in Fig. 1. Peaks for protons
+correspond to particles moving close to the initial beam direction. Moreover, fractions of different
+particle types can be estimated. It can be seen that for |y| < 2 (i.e. within the acceptance of the
+detector) charge particles are dominated by pions. Apart from p±, π±, & K±, marginal numbers
+of sigmas, cascades, and omegas were also generated. The PID efficiency depends on the particle
+momentum.
+The momentum spectra for protons, pions and kaons are shown in Fig. 6 in the
+midrapidity region (|y| < 0.5 for which theoretical predictions has been given) Most of the pions
+have momentum below 0.8 GeV and protons - below 1 GeV. It means that types of these particles
+should be well resolved by dE/dx measurements. When studying pion or proton spectra, there is
+high probability of kaon/pion misidentification, but fraction of such events is strongly suppressed
+by small initial kaon numbers.
+6
+
+(a)
+(b)
+Figure 4: Generator-level multiplicity of charged particles for 12C − 12C collision (a) and 40Ca − 40Ca collisions (b).
+(a)
+(b)
+Figure 5: Rapidity distribution of charged particles in 12C − 12C (a) and 40Ca − 40Ca (b) collision.
+7
+
+Total Multiplicity of Charged Particles, C-12 + C-12
+104
+No. of events
+103
+102
+10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+No. of charged particlesTotal Multiplicity of Charged Particles, Ca-40 + Ca-40
+104
+No. of events
+103
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+No. of charged particlesRapidity distribution of charged particles, C-12 + C-12
+- protons
+105
+.. pions
+. kaons
+104
+No. of charged particles
+103
+102
+10
+3
+2
+3
+5
+Rapidity of charaed particles (yRapidity distribution of charged particles, Ca-40 + Ca-40
+106
+protons
+pions
+105
+kaons
+No. of charged particles
+104
+103
+102
+10
+5
+3
+2
+Y
+Rapidity of charged particles (y)(a) p distribution of p± in 12C − 12C collision.
+(b) p distribution of p± in 40Ca − 40Ca collision.
+(c) p distribution of π± in 12C − 12C collision.
+(d) p distribution of π± in 40Ca − 40Ca collision.
+(e) p distribution of k± in 12C − 12C collision.
+(f) p distribution of k± in 40Ca − 40Ca collision.
+Figure 6: Total momentum distribution of protons, pions, and kaons at generator level in 12C − 12C and 40Ca− 40Ca collision.
+8
+
+Total momentum distribution of pions, C-12 + C-12
+16000
+14000
+12000
+pions
+10000
+8000
+No.
+6000
+4000
+2000
+0
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of pions (p)Total momentum distribution of pions, Ca-40 + Ca-40
+60000
+50000
+40000
+ON
+30000
+20000
+10000
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of pions (p)Total momentum distribution of kaons, C-12 + C-12
+1000
+800
+ of kaons
+600
+No.
+400
+200
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+Total momentum of kaons (p)Total momentum distribution of kaons, Ca-40 + Ca-40
+4000
+3500
+3000
+kaons
+2500
+2000
+No.
+1500
+1000
+500
+0
+0
+0.2
+0.4
+0.6
+0.8
+1.4
+1.6
+1.8
+2
+Total momentum of kaons (p)Total momentum distribution of protons, C-12 + C-12
+1600
+1400
+1200
+protons
+1000
+800
+ON
+600
+400
+200
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+2.2
+Total momentum of protons (p)Total momentum distribution of protons, Ca-40 + Ca-40
+6000
+5000
+ of protons
+4000
+No.
+3000
+2000
+1000
+0.2
+0.4
+0.6
+0.8
+1.4
+1.6
+1.8
+2
+2.2
+Total momentum of protons (p)5
+DETECTOR SIMULATION AND EVENT RECONSTRUCTION
+The detector simulation and reconstruction was performed with the SpdRoot framework. To read
+SMASH generated events the SpdRoot code was modified and additional C++ class was added.
+During the simulation stage the particles were transported through the detector geometrical model
+using Geant4.
+At the reconstruction stage, Geant4 tracks and vertices were reconstructed and
+particle identification with dE/dx and time of flight measurements was performed. For the PID
+three hypotheses were considered: pion, kaon and proton. The reconstructed ionization energy
+losses and “measured” time of flight were used to construct conditional probabilities (e.g. p(t|pid),
+where t is the measured time and pid is a particle type hypothesis). Out of 100K events generated
+by SMASH, first 1K events were considered for detector simulation due to slow data processing in
+SpdRoot.
+6
+ANALYSIS
+A physical analysis was performed using C++ codes and ROOT library based on SpdRoot output.
+All tracks reconstructed in the detector with measured momentum were accepted. For the particle
+type the one that gives the largest conditional probability is adopted.
+Multiplicity, as well as
+kinematic distributions for pions, kaons and protons are studied. For particle momentum spectra
+there are no notable differences between C − C and for Ca − Ca collisions, so only the first ones
+will be considered.
+6.1
+CHARGED TRACK MULTIPLICITY
+The SPD detector set-up is optimized for p − p and d − d collisions. Thus knowing charged track
+multiplicities for ion collisions is important to estimate CT and ST occupancies and feasibility of
+such studies. Fig. 7 shows the total multiplicity of charged particles reconstructed by the tracking
+system in 12C − 12C and 40Ca − 40Ca collisions. The numbers of reconstructed tracks are much
+lower compared to generator-level studies. It is because the geometry of the tracking system is such
+that, tracks with polar angle, θ < 10◦ or > 170◦ do not hit the tracker and passes along the beam
+pipe itself, so such tracks are ignored. Also, there were events without nuclei interactions which
+resulted in no track reconstruction. So, to avoid a large peak at zero due to mentioned reasons, the
+X-axis count starts from 1.
+6.2
+PION MOMENTUM SPECTRUM (12C − 12C)
+The spectra of particles identified as pions separately by ionization losses and by TOF are shown in
+Fig. 8 separately. The spectra show resemblance with the generator plot of pion momentum distri-
+bution. Based pn MC-truth information backround from misidentification other charged particles
+(K±, p±, e±, & µ±) is studied. The obtained distribution for “pions identified as pions” only slightly
+deviates from distribution of all selected pion candidates. The estimated relative contamination of
+the pion spectra is shown in Fig. 9. It can seen that purity above 90% can be obtained up to
+1.2 GeV using either dE/dx or TOF measurements.
+9
+
+Figure 7: Charged track multiplicity reconstructed by in 12C − 12C (left), 40Ca − 40Ca (right) collisions (shown by red) and
+number of particles for which TOF information is available (shown by blue).
+(a) Total momentum distribution of reconstructed charged particles
+identified as π± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as π± by TOF.
+Figure 8: Total momentum distribution of reconstructed π± candidates in 12C − 12C collision (Detector level).
+Figure 9: Purity of the selected pion candidates as a function of their momentum.
+10
+
+Total multiplicity of charged particles passing through tracking system, C12-C12
+60
+TOF
+50
+ST
+40
+events
+30
+NO.
+20
+10
+一
+10
+20
+30
+40
+50
+60
+70
+80
+90
+10090
+TOF
+80
+ST
+70
+events
+60
+50
+No.
+40
+30
+20
+10
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100Charged Particles ldentified as Pions by ST, C12-C12
+350
+Pions identified as pions
+Kaons identified as pions
+Protons identified as pions
+300
+Electrons identified as pions
+Muons identified as pions
+250
+Chargedparticlesidentifiedaspions
+200
+150
+100
+50
+0
+.°
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)Charged Particles ldentified as Pions by TOF, C12-C12
+Pions identified as pions
+300
+Kaons identified as pions
+Protons identified as pions
+250
+Electrons identified as pions
+Muons identified as pions
+Charged particles identified as pions
+200
+Counts
+150
+100
+50
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)Pion spectra precision, C12-C12
+0.8
+0.6
+Counts
+0.4
+0.2
+Precision recordedbyTOF
+Precision recorded by ST
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+p(GeV/c)6.3
+KAON MOMENTUM SPECTRUM (12C − 12C)
+The kaon momentum spectrum was explicitly mentioned among observables to study hadron for-
+mation effects in nuclei. Nevertheless, kaon production may be interesting for the reasons. The
+obtained spectra of kaon candidates is shown in Fig. 10 separately for ionization losses and TOF.
+First of all, the shown data lack statistics. Secondly, it can bee seen that there is a huge contamina-
+tion from misidentified pions. This is explained by very small fraction of generated kaons and the
+fact that probability to select misidentified particle is proportional to their number. The relative
+fraction of correctly identified kaons in shown in Fig. 11.
+(a) Total momentum distribution of reconstructed charged particles
+identified as K± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as K± by TOF.
+Figure 10: Total momentum distribution of reconstructed K± candidates in 12C − 12C collision (Detector level).
+Figure 11: Purity of the selected kaon candidates as a function of their momentum.
+11
+
+Charged Particles ldentified as Kaons by ST, C12-C12
+Kaons identified as kaons
+60
+Pions identified as kaons
+Protons identified as kaons
+Electrons identified as kaons
+50
+Muons identified as kaons
+Charged particles identified askaons
+40
+Counts
+30
+20
+10
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+0
+1.4
+1.6
+1.8
+2
+p(GeV/c)Charged Particles ldentified as Kaons by TOF, C12-C12
+25
+Kaons identified as kaons
+Pions identified as kaons
+Protons identifiedaskaons
+20
+Electrons identified as kaons
+Muons identified as kaons
+Charged particles identified as kaons
+15
+Counts
+10
+5
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+0
+2
+p(GeV/c)Kaon spectra precision, C12-C12
+Precision recorded by TOF
+0.9
+Precision recorded by ST
+0.8
+0.7
+0.6
+unts
+0.5
+Col
+0.4
+0.3
+0.2
+0.1
+0
+0.2
+0.4
+0.6
+0.8
+1.2
+1.4
+1.6
+1.8
+2
+p(GeV/c)6.4
+PROTON MOMENTUM SPECTRUM (12C − 12C)
+Finally, proton momentum spectra have been considered. In this study protons and antiprotons were
+considered together, but the fraction of produced antiprotons is negligible. The proton candidate
+distributions and the contributions from misidentification are shown in Fig. 12. The purity of the
+selected samples is shown in Fig. 13. It can be seen dE/dx measurements alone will not allow
+precise determination of proton spectrum. The reasonably good results can be expected only in
+case of combined identification by ionization losses and TOF system.
+(a) Total momentum distribution of reconstructed charged particles
+identified as p± by ionization losses.
+(b) Total momentum distribution of reconstructed charged particles
+identified as p± by TOF.
+Figure 12: Total momentum distribution of reconstructed p± candidates in 12C − 12C collision (Detector level).
+Figure 13: Purity of the selected proton candidates as a function of their momentum.
+12
+
+Charged Particles ldentified as Protons by ST, C12-C12
+90
+Protons identified as protons
+Kaons identified as protons
+80
+Pions identified as protons
+Electrons identified as protons
+Muons identified as protons
+70
+Charged particles identified as protons
+60
+Counts
+50
+40
+30
+20
+10
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+0
+5
+p(GeV/c)Charged Particles ldentified as Protons by TOF, C12-C12
+90
+Protons identified as protons
+Kaons identified as protons
+80
+Pions identified as protons
+Electrons identified asprotons
+70
+Muons identified as protons
+Charged particles identified as protons
+60
+Counts
+50
+40
+30
+20
+10
+0.5
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+0
+5
+p(GeV/c)Proton spectra precision, C12-C12
+0.8
+Counts
+0.6
+0.4
+0.2
+Precision recorded by TOF
+Precision recorded by ST
+0.5
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+p(GeV/c)7
+SUMMARY
+The goal of this work was to check the feasibility of hadron formation effects studies at the first
+stage of SPD operation. For this purpose an analysis of 12C − 12C and 40Ca − 40Ca collisions were
+performed at the generator level and then the full event reconstruction was done at detector level.
+The multiplicity distributions indicate that occupancies of tracking detectors should be checks.
+Part of the events with high number of charged tracks may not be fully reconstructed. Particle
+identification with ionization losses and TOF was considered separately (for future dE/dx only or
+their combination can be expected). The purity of the measured charged pion distribution for both
+types of ion collisions using dE/dx only is rather good and meets mentioned before requirements. In
+case of combination of information from ionization losses and time of flight system purity of proton
+distribution may be improved.
+References
+[1] V. V. Abramov, A. Aleshko, V. A. Baskov, E. Boos, V. Bunichev, O. D. Dalkarov, R. El-Kholy,
+A. Galoyan, A. V. Guskov and V. T. Kim, et al. Phys. Part. Nucl. 52 (2021) no.6, 1044-1119
+doi:10.1134/S1063779621060022 [arXiv:2102.08477 [hep-ph]].
+[2] V. M. Abazov et al. [SPD proto], [arXiv:2102.00442 [hep-ex]].
+[3] SPD TDR [unpublished].
+13
+
diff --git a/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/2301.03539v1.pdf.txt b/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/2301.03539v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..316bf0f58d7b6968141f04601cb812f0c7d55b6c
--- /dev/null
+++ b/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/2301.03539v1.pdf.txt
@@ -0,0 +1,1782 @@
+1
+Federated Coded Matrix Inversion
+Neophytos Charalambidesµ, Mert Pilanciσ, and Alfred O. Hero IIIµ
+.µEECS Department University of Michigan .σEE Department Stanford University
+Email: neochara@umich.edu, pilanci@stanford.edu, hero@umich.edu
+Abstract
+Federated learning (FL) is a decentralized model for training data distributed across client devices. Coded computing (CC) is
+a method for mitigating straggling workers in a centralized computing network, by using erasure-coding techniques. In this work
+we propose approximating the inverse of a data matrix, where the data is generated by clients; similar to the FL paradigm, while
+also being resilient to stragglers. To do so, we propose a CC method based on gradient coding. We modify this method so that the
+coordinator does not need to have access to the local data, the network we consider is not centralized, and the communications which
+take place are secure against potential eavesdroppers.
+I. INTRODUCTION AND RELATED WORK
+Inverting a matrix is one of the most important operations in numerous applications, such as, signal processing, machine
+learning, and scientific computing [2], [3]. A common way of inverting a matrix is to perform Gaussian elimination, which
+requires O(N 3) operations for square matrices of order N. In high-dimensional applications, this can be cumbersome. Over the
+past few years the machine learning (ML) community has made much progress on federated learning (FL), focusing on iterative
+methods.
+The objective of FL is to leverage computation, communication and storage resources to perform distributed computations for
+ML models, where the data of each federated worker is never shared with the coordinator of the network; that aggregates local
+computations in order to update the model parameters. In FL applications it is important that the data is kept private and secure.
+Distributed computations in the presence of stragglers (workers who fail to compute their task or have longer response time than
+others) must account for the effect of non-responsive workers. Coding-theoretic approaches have been adopted for this purpose
+[4], [5], and fall under the framework of coded computing (CC). Data security is also an increasingly important issue in CC [6].
+Despite the fact that multiplication algorithms imply inversion algorithms and vice versa, in the context of CC; matrix inversion
+has not been studied as extensively as coded matrix multiplication (CMM) [7]. The main reason for this is the fact that the latter
+is non-linear and non-parallelizable as an operator. We point out that distributed inversion algorithms do exist, though these make
+assumptions on the matrix, are specific for distributed and parallel computing platforms, and require a matrix factorization; or
+heavy and multiple communication instances between the servers and the coordinator.
+In [1] a CC method1 was proposed based on gradient coding (GC) [8], which approximates the inverse of a matrix A. In order
+to overcome the obstacle of non-linearity, the columns of A−1 are approximated. When assuming floating-point arithmetic, this
+CCM introduces no numerical nor approximation errors. Note that GC and not CMM was utilized, as the latter does not require
+the encoding to be done locally by the workers.
+Though the two areas of FL and CC seem to be closely related, on the surface they appear incompatible. For instance, in CC
+one often assumes there is a master server that distributes the data and may perform the encoding (encoding by the master server
+is done in CMM, but not in GC), while in FL the central coordinator never has access to the distributed local training data; which
+are located at different client nodes or workers.
+There are a few recent works that leverage CC in order to devise secure FL methods for distributed regression and iterative
+optimization [9]–[14]. In this work, we combine optimization and CC, using erasure coding to protect against stragglers as in
+CC and locally approximating the inverse without revealing the data to the coordinator, to design a FL scheme. Our approach,
+is based on the coded matrix inversion method (CMIM) we develop, which utilizes balanced Reed-Solomon (BRS) codes [15],
+[16]. This results in an efficient decoding in terms of the threshold number of responsive workers needed to perform an error free
+computation. We show that the general class of maximum distance separable (MDS) generator matrices could be used to generate
+a suitable erasure code (Theorem 6). The focus is on BRS codes, which have the following advantages:
+(i) minimum redundancy per job across the network,
+(ii) they optimize communication from workers to the master,
+(iii) we can efficiently decode the resulting method.
+Our CMIM can also be used to compute the Moore–Penrose pseudoinverse Y† of a data matrix Y ∈ RM×N for M ≫ N,
+which is more general than inverting a square matrix. By using the fact that Y† = (Y⊤Y)−1Y⊤, the bottleneck is computing
+the inverse of A = Y⊤Y. In addition, two more matrix multiplications need to take place distributively: computing A before
+the inversion; and �
+A−1Y⊤ after the inverse has been approximated. The matrix products can be computed distributively using
+A preliminary version also considers approximating A† [1], in the CC setting. This work was partially supported by grants ARO W911NF-15-1-0479 and Dept
+of Energy DE-NA0003921.
+1We abbreviate ‘coded computing method/methods’ to CCM/CCMs.
+arXiv:2301.03539v1  [cs.IT]  9 Jan 2023
+
+2
+various CCMS, e.g. we can use a modification of the coded FL approaches of [11] and a CMM from [17]; both of which are based
+on GC. For the remainder of the paper, we focus on the generic problem of inverting a square matrix A.
+The proposed FL approach applies to general linear regression problems. Compared to previous FL iterative approaches [18],
+the difference is that for Yθ = p; with p the label vector and θ the model parameters, the pseudoinverse-regularized regression
+solution is ˆθ = �
+Y†p. Unlike conventional FL methods, this regularized regression can be computed non-iteratively. The non-
+iterative nature of the proposed approach is advantageous in settings such as Kalman filtering, where the matrix inverse must be
+updated in real time as measurements come in.
+The paper is organized as follows. In II we recall basic facts on matrix inversion, least squares approximation and finite fields.
+In III we review BRS codes, and prove two key lemmas regarding their generator matrices. In IV we present the matrix inverse
+approximation algorithm we utilize in our CCM. The main contribution is presented in V. Our federated approach is split into four
+phases, which we group in pairs of two. First, we discuss information sharing from the coordinator to the workers (we consider
+all the clients’ servers as the network’s workers), and then information sharing between the workers. Second, we show how our
+inversion algorithm can be incorporated in linear CCMs, and describe how this fits into the FL paradigm. Concluding remarks and
+future work are presented in VI.
+A. Overview of the Coded Matrix Inversion Method
+In CC the computational network is centralized, and is comprised of a master server who communicates with n workers. The
+idea behind our approximation is that the workers use a least squares solver to approximate multiple columns of A−1, resulting
+in a set of local approximations to submatrices of �
+A−1, which we refer to as blocks. We present approximation guarantees and
+simulation results for steepest descent (SD) and conjugate gradient (CG) iterative optimization methods. By locally approximating
+the columns in this way, the workers can linearly encode the blocks of �
+A−1. The clients have a block of data {Aι}k
+ι=1, which
+constitute the data matrix A =
+�
+A1 · · · Ak
+�
+. To simplify our presentation, we assume that each local data block is of the same
+size; i.e. Aι ∈ RN×T for T = N/k, and that client i has ni servers. Therefore, the total number of servers is n = �k
+j=1 nj.
+We assume the blocks are of the same size, so that the encodings carried out by the clients are consistent. In V, we show that this
+assumption is not necessary. Moreover, for the CCM, it is not required that the number of blocks equal the number of clients. For
+a given natural number γ, assume that γ divides T; denoted γ | T (each local data block Aι is further divided into γ sub-blocks).
+In the case where k ∤ N or γ ∤ T, we can pad the blocks of �
+A−1 so that these assumptions are met.
+A limitation of our proposed CMIM, is the fact that each server needs to have full knowledge of A, in order to estimate columns
+of A−1 through a least squares solver. The sensitivity of Gaussian elimination and matrix inversion also requires that all clients
+have knowledge of each others’ data [1]. This limitation is shared by other coded federated learning methods, e.g. CodedPaddedFL
+[11]. In contrast to CC and GC; where a master server has access to all the data, in FL the data is inherently distributed across
+devices, thus GC cannot be applied directly. We also assume that the coordinator does not intercept the communication between
+the clients, otherwise she could recover the local data. Also, we trust that the coordinator will not invert �
+A−1, to approximate A
+— this would be computationally difficult, for N large.
+Before broadcasting the data amongst themselves, the clients encode their block Ai, which guarantees security from outside
+eavesdroppers. When the clients receive the encoded data, they can decrypt and recover A. Then, their servers act as the workers
+of the proposed CMIM and carry out their assigned computations, and directly communicate their computations back to the
+coordinator. Once the recovery threshold (the minimum number of responses needed to recover the computation) is met, the
+approximation �
+A−1 is recoverable.
+B. Coded Federated Learning
+There are few works that leverage CC to devise secure FL schemes. Most of these works have focused on distributed regression
+and iterative methods, which is the primary application for FL [9]–[13]. Below, we describe and compare these approaches to our
+work.
+The authors of [9] proposed coded federated learning, in which they utilize a CMM scheme. Their security relies on the use
+of random linear codes, to define the parity data. Computations are carried out locally on the systematic data, and only the parity
+data is sent to the coordinator. The main drawback compared to our scheme is that each worker has to generate a random encoding
+matrix and apply a matrix multiplication for the encodings, while we use the same BRS generator matrix across the network,
+based on GC, to linearly encode the local computations. The drawback in our case, is that the workers need to securely share their
+data with each other. This is an artifact of the operation (inversion) we are approximating, and is inevitable in the general case
+where A has no structure. Under the FL setting we are considering, where the data is gathered or generated locally and is not
+i.i.d., we cannot make any assumptions on the structure of A.
+In [11], two methods were proposed. CodedPaddedFL combines one-time-padding with GC to carry out the FL task. Some
+of its disadvantages are that a one-time-pad (OTP) needs to be generated by each worker, and that the OTPs are shared with the
+coordinator, which means that if she gets hold of the encrypted data, she can decrypt it, compromising security. Furthermore,
+there is a heavy communication load and the coordinator needs to store all the pads in order to recover the computed gradients. In
+
+3
+the proposed CMIM, the coordinator generates a set of interpolation points, and shares them with the clients. If the coordinator
+can intercept the communication between the workers, she can decrypt the encrypted data blocks. The second method proposed
+in [11], CodedSecAgg, relies on Shamir’s secret sharing (SSS); which is based on polynomial interpolation over finite fields. In
+contrast, our CMIM relies on GC and Lagrange interpolation.
+Lastly, we discuss the method proposed in [13], which is based on the McEliece cryptosystem, and moderate-density parity-
+check codes. This scheme considers a communication delay model which defines stragglers as the workers who respond slower
+than the fastest worker, and time out after a predetermined amount of time ∆. As the iterative SD process carries on, such workers
+are continuously disregarded. Due to this, there is a data sharing step at each iteration, at which the new stragglers communicate
+encrypted versions of their data to the active workers. Our scheme is non-iterative, and has a fixed recovery threshold. Unlike
+some of the works previously mentioned, which guarantee information-theoretic security, the McEliece based systems and our
+approach have computational privacy guarantees.
+C. Lagrange Interpolation Coded Computing Methods
+While there is extensive literature on matrix-vector and matrix-matrix multiplication, and computing the gradient in the presence
+of stragglers, there is limited work on computing or approximating the inverse of a matrix [19]. The non-linearity of matrix
+inversion prohibits linear or polynomial encoding of the data before the computations are to be performed. Consequently, most
+CCMs cannot be directly utilized. GC is the appropriate CC set up to consider [20], precisely because the encoding takes place
+once the computation has been completed, in contrast to most CMM methods where the encoding is done by the master, before
+the data is distributed.
+Here, we give a brief overview of the GC on which our CMIM is based. We also review “Lagrange Coded Computing” (LCC),
+which has relations to our approach. Then, we give a summary of our proposed CMIM. All these rely on Lagrange interpolation
+over finite fields.
+Gradient codes are a class of codes designed to mitigate the effect of stragglers in data centers, by recovering the gradient of
+differentiable and additively separable objective functions in distributed first order methods [20]. The proposed CMIM utilizes
+BRS generator matrices constructed for GC [8]. The main difference from our work is that in GC the objective is to construct an
+encoding matrix G and decoding vectors aI ∈ Ck, such that a⊤
+I G = ⃗1 for any set of non-straggling workers indexed by I. To
+do so, the decomposition of the BRS generator matrices GI = HIP is exploited, where HI is a Vandermonde matrix; and the
+first row of P is equal to ⃗1. Subsequently a⊤
+I is extracted as the first row of H−1
+I .
+In the proposed CMIM framework, the objective is to design an encoding-decoding pair ( ˜G, ˜DI) for which ˜DI ˜G = IN, for
+all I ⊊ Nn of size k. The essential reason for requiring this condition, as opposed to that of GC, is that the empirical gradient of a
+given dataset is the sum of each individual gradients, while in our scenario if the columns of �
+A−1 are summed; they cannot then
+be recovered.
+The state-of-the art CC framework is LCC, which is used to compute arbitrary multivariate polynomials of a given dataset
+[5], [21]. This approach is based on Lagrange interpolation, and it achieves the optimal trade-off between resiliency, security,
+and privacy. The problem we are considering is not a multivariate polynomial in terms of A. To securely communicate A to
+the workers, we encode it through Lagrange interpolation. Though similar ideas appear in LCC, the purpose and application of
+the interpolation is different. Furthermore, LCC is a point-based approach [22] and requires additional interpolation and linear
+combination steps after the decoding takes place.
+Recall that the workers in the CMIM must compute blocks of �
+A−1. Once they complete their computations, they encode them
+by computing a linear combination with coefficients determined by a sparsest-balanced MDS generator matrix. Referring to the
+advantages claimed for CMIM in Section I, working with MDS generator matrices allows us to meet points (i) and (ii), while BRS
+generator matrices also help us satisfy (iii). Once the recovery threshold is met, the coordinator can recover the approximation
+�
+A−1. The structure of sparsest-balanced generator matrices is also leveraged to optimally allocate tasks to the workers, while
+linear encoding is what allows minimal communication load from the workers to the master. Security against eavesdroppers is
+guaranteed by encoding the local data through a modified Lagrange interpolation polynomial, before it is shared by the clients.
+This CMIM also extends to approximating A† [1].
+II. PRELIMINARY BACKGROUND
+The set of N ×N non-singular matrices is denoted by GLN(R). Recall that A ∈ GLN(R) has a unique inverse A−1, such that
+AA−1 = A−1A = IN. The simplest way of computing A−1 is by performing Gaussian elimination on
+�
+A|IN
+�
+, which gives
+�
+IN
+��A−1] in O(N 3) operations. In Algorithm 1, we approximate A−1 column-by-column. We denote the ith row and column
+of A respectively by A(i) and A(i). The condition number of A is κ2 = ∥A∥2∥A−1∥2. For I an index subset of the rows of a
+matrix M, the matrix consisting only of the rows indexed by I, is denoted by MI.
+In the proposed algorithm we approximate N instances of the least squares minimization problem
+θ⋆
+ls = arg min
+θ∈RM
+�
+∥Aθ − y∥2
+2
+�
+(1)
+
+4
+for A ∈ RN×M and y ∈ RN. In many applications N ≫ M, where the rows represent the feature vectors of a dataset. This has
+the closed-form solution θ⋆
+ls = A†y.
+Computing A† to solve (1) is intractable for large M, as it requires computing the inverse of A⊤A. Instead, we use gradient
+methods to get approximate solutions, e.g. SD or CG, which require less operations, and can be done distributively. One could use
+second-order methods; e.g. Newton–Raphson, Gauss-Newton, Quasi-Newton, BFGS, or Krylov subspace methods instead. This
+would be worthwhile future work.
+When considering a minimization problem with a convex differentiable objective function ψ: Θ → R over an open constrained
+set Θ ⊆ RM, as in (1), the SD procedure selects an initial θ[0] ∈ Θ, and then updates θ according to:
+θ[t+1] = θ[t] − ξt · ∇θψ(θ[t])
+for t = 1, 2, 3, ...
+until a termination criterion is met, for ξt the step-size. The CG method is the most used and prominent iterative procedure for
+numerically solving systems of positive-definite equations.
+Our proposed coding scheme is defined over the finite field of q elements, Fq. We denote its cyclic multiplicative subgroup
+by F×
+q = Fq\{0Fq}. For implementation purposes, we identify finite fields with their realization in C as a subgroup of the circle
+group, since we assume our data is over R. All operations can therefore be carried out over C. Specifically, for β ∈ F×
+q a generator,
+we identify βj with e2πij/q, and 0Fq with 1. The set of integers between 1 and ν is denoted by Nν.
+III. BALANCED REED-SOLOMON CODES
+A Reed-Solomon code RSq[n, k] over Fq for q > n > k, is the encoding of polynomials of degree at most k − 1, for k the
+message length and n the code length. It represents our message over the defining set of points A = {αi}n
+i=1 ⊂ Fq
+RSq[n, k] =
+��
+f(α1), f(α2), · · · , f(αn)
+� ���
+f(X) ∈ Fq[X] of degree ⩽ k − 1
+�
+where αi = αi, for α a primitive root of Fq. Hence, each αi is distinct. A natural interpretation of RSq[n, k] is through its encoding
+map. Each message (m0, ..., mk−1) ∈ Fk
+q is interpreted as f(x) = �k−1
+i=0 mixi ∈ Fq[x], and f is evaluated at each point of A.
+From this, RSq[n, k] can be defined through the generator matrix
+G =
+�
+�
+�
+�
+�
+1
+α1
+α2
+1
+. . .
+αk−1
+1
+1
+α2
+α2
+2
+. . .
+αk−1
+2
+...
+...
+...
+...
+...
+1
+αn
+α2
+n
+. . .
+αk−1
+n
+�
+�
+�
+�
+� ∈ Fn×k
+q
+,
+thus, RS codes are linear codes over Fq. Furthermore, they attain the Singleton bound, i.e. d = n − k + 1, where d is the code’s
+distance, which implies that they are MDS.
+Balanced Reed-Solomon codes [15], [16] are a family of linear MDS error-correcting codes with generator matrices G ∈ Fn×k
+q
+that are:
+• sparsest: each column has the least possible number of nonzero entries
+• balanced: each row contains the same number of nonzero entries
+for the given code parameters k and n. The design of these generators are suitable for our purposes, as:
+1) we have balanced loads across homogeneous workers,
+2) sparse generator matrices reduce the computation tasks across the network,
+3) the MDS property permits an efficient decoding step,
+4) linear codes produce a compressed representation of the encoded blocks.
+A. Balanced Reed-Solomon Codes for CC
+In the proposed CMIM, we leverage BRS generator matrices to approximate A−1. For simplicity, we will consider the case
+where d = s + 1 = nw
+k is a positive integer2, for n the number of workers and s the number of stragglers. Furthermore, d is
+the distance of the code and ∥G(j)∥0 = d for all j ∈ Nk; ∥G(i)∥0 = w for all i ∈ Nn, and d > w since n > k. For decoding
+purposes, we require that at least k = n − s workers respond. Consequently, d = s + 1 implies that n − d = k − 1. For simplicity,
+we also assume d ⩾ n/2. In our setting, each column of G corresponds to a computation task of �
+A−1; which we will denote by
+ˆ
+Ai, and each row corresponds to a worker.
+2The case where nw
+k
+∈ Q+\Z+ is analysed in [8], and also applies to our approach. We restrict our discussion to the case where nw
+k
+∈ Z+.
+
+5
+Our choice of such a generator matrix G ∈ Fn×k
+q
+, solves
+arg
+min
+G∈Fn×k
+q
+�
+nnzr(G)
+�
+s.t.
+∥G(i)∥0 = w, ∀i ∈ Nn
+∥G(j)∥0 = d, ∀i ∈ Nk
+rank(GI) = k, ∀I : |I| = k
+(2)
+which determines an optimal task allocation among the workers of the proposed CMIM.
+Under the above assumptions, the entries of the generator matrix of a BRSq[n, k] code meet the following:
+• each column is sparsest, with exactly d nonzero entries
+• each row is balanced, with w = dk
+n nonzero entries
+where d equals to the number of workers who are tasked to compute each block, and w is the number of blocks that are computed
+by each worker.
+Each column G(j) corresponds to a polynomial pj(x), whose entries are the evaluation of the polynomial we define in (3) at
+each of the points of the defining set A, i.e. Gij = pj(αi) for (i, j) ∈ Nn × Nk. To construct the polynomials {pj(x)}k
+j=1, for
+which deg(pj) ⩽ nnzr(G(j)) = n − d = k − 1, we first need to determine a sparsest and balanced mask matrix M ∈ {0, 1}n×k,
+which is ρ-sparse for ρ = d
+n; i.e. nnzr(G) = ρnk. We use the construction from [8], though it is fairly easy to construct more
+general such matrices, by using the Gale-Ryser Theorem [23], [24]. Furthermore, deterministic constructions resemble generator
+matrices of cyclic codes.
+For our purposes we use B as our defining set of points, where each point corresponds to the worker with the same index. The
+objective now is to devise the polynomials pj(x), for which pj(βi) = 0 if and only if Mij = 0. Therefore:
+(I) Mij = 0
+=⇒
+(x − βi) | pj(x)
+(II) Mij ̸= 0
+=⇒
+pj(βi) ∈ F×
+q
+for all pairs (i, j).
+The construction of BRS[n, k]q from [15] is based on what the authors called scaled polynomials. Below, we summarize the
+polynomial construction based on Lagrange interpolation [8]. We then prove a simple but important result that allows us to
+efficiently perform the decoding step.
+The univariate polynomials corresponding to each column G(j), are defined as:
+pj(x) :=
+�
+i:Mij=0
+� x − βi
+βj − βi
+�
+=
+k
+�
+ι=1
+pj,ι · xι−1 ∈ Fq[x]
+(3)
+which satisfy (I) and (II). By the BCH bound [25, Chapter 9], it follows that deg(pj) ⩾ n − d = k − 1 for all j ∈ Nk. Since
+each pj(x) is the product of n − d monomials, we conclude that the bound on the degree is satisfied and met with equality, hence
+pj,ι ∈ F×
+q for all coefficients.
+By construction, both G and GI are decomposable into a Vandermonde matrix H ∈ Bn×k and a matrix comprised of the
+polynomial coefficients H ∈ (F×
+q )k×k [8]. Specifically, G = HP where Hij = βj−1
+i
+= βi(j−1) and Pij = pj,i are the
+coefficients from (3). This can be interpreted as P(j) defining the polynomial pj(x), and H(i) is comprised of the first k positive
+powers of βi in ascending order, therefore
+pj(βi) =
+k
+�
+ι=1
+pj,ι · βι−1
+i
+= ⟨H(i), P(j)⟩.
+The following lemmas will help us respectively establish in our CC setting the efficiency of our decoding step and the optimality
+of the allocated tasks to the workers. For Lemma 1, recall that efficient matrix multiplication algorithms have complexity O(N ω),
+for ω < 2.373 the matrix multiplication exponent [26].
+Lemma 1. The restriction GI ∈ Fk×k
+q
+of G to any of its k rows indexed by I ⊊ Nn, is an invertible matrix. Moreover, its inverse
+can be computed online in O(k2 + kω) operations.
+Proof. The matrices H and P are of size n × k and k × k respectively. The restricted matrix GI is then equal to HIP, where
+HI ∈ Fk×k
+q
+is a square Vandermonde matrix, which is invertible in O(k2) time [27]. Specifically
+HI =
+�
+�
+�
+�
+�
+1
+βI1
+β2
+I1
+. . .
+βk−1
+I1
+1
+βI2
+β2
+I2
+. . .
+βk−1
+I2
+...
+...
+...
+...
+...
+1
+βIk
+β2
+Ik
+. . .
+βk−1
+Ik
+�
+�
+�
+�
+� ∈ Fk×k
+q
+.
+
+6
+It follows that
+det(HI) =
+�
+{i<j}⊆I
+(βj − βi)
+which is nonzero, since β is primitive. Therefore, HI is invertible. By [15, Lemma 1] and the BCH bound, we conclude that P is
+also invertible. Hence, GI is invertible for any set I.
+Note that the inverse of P can computed a priori by the master before we deploy our CCM. Therefore, computing G−1
+I
+online
+with knowledge of P−1, requires an inversion of HI which takes O(k2) operations; and then multiplying it by P−1. Thus, it
+requires O(k2 + kω) operations in total.
+■
+Lemma 2. The generator matrix G ∈ Fn×k
+q
+of a BRSq[n, k] MDS code defined by the polynomials pj(x) of (3), solves the
+optimization problem (2).
+Proof. The first two constraints are satisfied by the definition of G, which meets the sparsest and balanced constraints with
+equality; for the given parameters. The last constraint is implied by the MDS theorem, which states that every set of k rows of G
+is linearly independent.
+The sparsity constraints of (2) imply that nnzr(G) ⩾ max{nw, kd}, and for our parameters we have nw = kd. This condition
+is met with equality for the chosen G, as
+nnzr(G) =
+�
+j∈Nk
+nnzr(G(j))
+=
+�
+j∈Nk
+#
+�
+pj(βi) ̸= 0 : βi ∈ B
+�
+=
+�
+j∈Nk
+n −
+�
+i : Mij = 0
+�
+=
+�
+j∈Nk
+n − (n − d)
+= kd
+and the proof is complete.
+■
+IV. INVERSE APPROXIMATION ALGORITHM
+Our goal is to estimate A−1 =
+�
+b1 · · · bN
+�
+, for A a square matrix of order N. A key property to note is
+AA−1 = A
+�
+b1 · · · bN
+�
+=
+�
+Ab1 · · · AbN
+�
+= IN
+which implies that Abi = ei for all i ∈ NN, where ei are the standard basis column vectors. Assume for now that we use any
+black-box least squares solver to estimate
+ˆbi ≈ arg min
+b∈RN
+�
+fi(b) := ∥Ab − ei∥2
+2
+�
+(4)
+which we call N times, to recover �
+A−1 :=
+�ˆb1 · · · ˆbN
+�
+. This approach may be viewed as approximating
+�
+A−1 ≈ arg
+min
+B∈RN×N
+�
+∥AB − IN∥2
+F
+�
+.
+Alternatively, one could estimate the rows of A−1. Algorithm 1 shows how this can be performed by a single server.
+Algorithm 1: Estimating A−1
+Input: A ∈ GLN(R)
+for i=1 to N do
+approximate ˆbi ≈ arg minb∈RN
+�
+∥Ab − ei∥2
+2
+�
+end
+return �
+A−1 ←
+�ˆb1 · · · ˆbN
+�
+In the case where SD is used to approximate ˆbi from (4), the overall operation count is O(TiN 2); for Ti the total number of
+descent iterations used. An upper bound on the number of iterations can be determined by the underlying termination criterion,
+e.g. the criterion fi(ˆb[t]) − fi(b⋆
+ls) ⩽ ϵ is guaranteed to be satisfied after T = O(log(1/ϵ)) iterations [28]. The overall error of
+�
+A−1 may be quantified as
+
+7
+• errℓ2( �
+A−1) := ∥ �
+A−1 − A−1∥2
+• errF ( �
+A−1) := ∥ �
+A−1 − A−1∥F
+• errrF ( �
+A−1) := ∥ �
+A−1−A−1∥F
+∥A−1∥F
+=
+N
+�
+i=1
+∥Aˆbi−ei∥2
+∥A−1∥F
+.
+To approximate A−1 distributively, each of the n servers are asked to estimate τ-many ˆbi’s in parallel. When using SD, the
+worst-case runtime by the workers is O(τTmaxN 2), for Tmax the maximum number of iterations of SD among the workers. If CG
+is used, each worker needs no more than a total of Nτ CG steps to exactly compute its task, i.e. O(τNκ2) operations; as each
+instance of (4) is expected to converge in O(κ2) iterations, which is the worst case runtime [29], [30].
+In order to bound errrF ( �
+A−1) = ∥ �
+A−1−A−1∥F
+∥A−1∥F
+, we first upper bound the numerator and then lower bound the denominator.
+Since ∥A−1 − �
+A−1∥2
+F = �N
+i=1 ∥A−1ei − ˆbi∥2
+2, bounding the numerator reduces to bounding ∥A−1ei − ˆbi∥2
+2 for all i ∈ NN.
+This is straightforward
+∥A−1ei − ˆbi∥2
+2
+♦
+⩽ 2
+�
+∥A−1ei∥2
+2 + ∥ˆbi∥2
+2
+�
+$
+⩽ 2
+�
+∥A−1∥2
+2 · ∥ei∥2
+2 + ∥ˆbi∥2
+2
+�
+= 2
+�
+1/σmin(A)2 + ∥ˆbi∥2
+2
+�
+(5)
+where in ♦ we use the fact that ∥u − v∥2
+2 ⩽ 2(∥u∥2
+2 + ∥v∥2
+2), and in $ the submultiplicativity of the ℓ2-norm is invoked. For the
+denominator, by the definition of the Frobenius norm
+∥A−1∥2
+F =
+N
+�
+i=1
+1
+σi(A)2 ⩾
+N
+σmax(A)2 .
+(6)
+By combining (5) and (6) we get
+errrF ( �
+A−1) ⩽
+√
+2
+�
+N/σmin(A)2 + �N
+i=1 ∥ˆbi∥2
+2
+N/σmax(A)2
+�1/2
+=
+√
+2
+�
+κ2
+2 + σmax(A)2
+N
+·
+N
+�
+i=1
+∥ˆbi∥2
+2
+�1/2
+.
+This is an additive error bound in terms of the problem’s condition number, which also shows a dependency on the estimates
+{ˆbi}N
+i=1. Propositions 3 and 4 give error bounds when using SD and CG as the subroutine of Algorithm 1 respectively.
+Proposition 3. For A ∈ GLN(R), we have errF ( �
+A−1) ⩽
+ϵ√
+N/2
+σmin(A)2 and errrF ( �
+A−1) ⩽
+ϵ√
+N/2
+σmin(A), when using SD to solve (4) with
+termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for each i.
+Proof. Recall that for a strongly-convex function with strong-convexity parameter µ, we have the following optimization gap [28,
+Section 9.1.2]
+fi(b) − fi(b⋆
+ls) ⩽ 1
+2µ · ∥∇fi(b)∥2
+2 .
+(7)
+For A ∈ GLN(R) in (4), the constant is µ = σmin(A)2. By fixing ϵ =
+�
+2σmin(A)2η, we have η = 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+. Thus, by (7)
+and our termination criterion:
+∥∇fi(b)∥2 ⩽
+�
+2σmin(A)2η
+=⇒
+fi(b) − fi(b⋆
+ls) ⩽ η ,
+so when solving (4) we get
+fi(b) − fi(b⋆
+ls) = fi(b) − 0 = ∥Aˆbi − ei∥2
+2 ,
+hence
+∥Aˆbi − ei∥2
+2 ⩽ 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+(8)
+
+8
+for all i ∈ NN. We want an upper bound for each summand ∥A−1ei − ˆbi∥2
+2 of the numerator of errrF ( �
+A−1)2:
+∥A−1ei − ˆbi∥2
+2 = ∥A−1(ei − Aˆbi)∥2
+2
+⩽ ∥A−1∥2
+2 · ∥ei − Aˆbi∥2
+2
+♯
+⩽ ∥A−1∥2
+2 · 1
+2 ·
+�
+ϵ
+σmin(A)
+�2
+(9)
+=
+ϵ2
+2σmin(A)4
+(10)
+where ♯ follows from (8), thus errF ( �
+A−1)2 ⩽
+Nϵ2
+2σmin(A)4 . Substituting (9) into the definition of errrF ( �
+A−1) gives us
+errrF ( �
+A−1)2 ⩽ ∥A−1∥2
+2
+∥A−1∥2
+F
+· N
+2 ·
+�
+ϵ
+σmin(A)
+�2 ‡
+⩽
+Nϵ2/2
+σmin(A)2
+where ‡ follows from the fact that ∥A−1∥2
+2 ⩽ ∥A−1∥2
+F .
+■
+In the experiments of Subsection IV-A, we verify that Proposition 3 holds for Gaussian random matrices. The dependence on
+1/σmin(A) is an artifact of using gradient methods to solve the underlying problems (4), since the error will be multiplied by
+∥A−1∥2
+2. In theory, this can be annihilated if one runs the algorithm on pA for p ≈ 1/σmin(A), followed by multiplication of
+the final result by p. This is a way of preconditioning SD. In practice, the scalar p should not be selected to be much larger than
+1/σmin(A), as it could result in �
+A−1 ≈ 0N×N.
+Proposition 4. Assume Algorithm 1 uses CG to solve (4). Then, in O
+�
+N√κ2 ln(1/ϵ)
+�
+iterations, we have errF ( �
+A−1) ⩽ Nϵ.
+Moreover, if A⊤A has ˜N distinct eigenvalues, it converges in at most ˜NN steps.
+Proof. By [29, Section 10] and [31, Section 2], we know that for each subroutine (4) of Algorithm 3, CG requires at most
+O(√κ2 ln(1/ϵ)) iterations in order to attain an ϵ-optimal point, for each ˆbi. Hence, considering all approximate columns {ˆbi}N
+i=1,
+we conclude that the total error in terms of the Frobenius norm of �
+A−1, is at most Nϵ.
+Recall that in order to solve (1) with the CG method in the case where A is neither symmetric, positive-definite, nor square, we
+apply the CG iteration to the normal equations
+A⊤Ay = A⊤θ .
+This follows by setting the derivative of (1) to zero. In our scenario, we are assuming that A ∈ GLN(R), hence A⊤A is full-rank
+and symmetric, thus CG in its simplest form can be used to solve the minimization problems of Algorithm 1. By [30, Theorem
+38.4], it follows that each instance of (4) converges in at most ˜N steps.
+■
+Even though Proposition 4 guarantees convergence in at most ˜NN steps, it does not assume floating-point arithmetic. Therefore,
+this does not hold in practical settings. Our experiments though show that after significantly less steps, we achieve approximations
+of negligible error, which is sufficient for ML and FL applications.
+A. Numerical Experiments
+The accuracy of the proposed algorithms was tested on randomly generated matrices, using both SD and CG for the subroutine
+optimization problems. The depicted results are averages of 20 runs, with termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for SD and
+∥b[t]
+i − b[t−1]
+i
+∥2 ⩽ ϵ for CG, for the given ϵ accuracy parameters. We considered A ∈ R100×100. The error subscripts represent
+A = {ℓ2, F, rF}, N = {ℓ2, F}. We note that significantly fewer iterations took place when CG was used for the same ϵ, though
+this depends heavily on the choice of the step-size. The errors observed in the case of CG, are due to floating-point arithmetic.
+Therefore, there is a trade-off between accuracy and speed when using SD vs. CG.
+Average �
+A−1 errors, for A ∼ 50 · N(0, 1) — SD
+ϵ
+10−1
+10−2
+10−3
+10−4
+10−5
+errA
+O(10−2)
+O(10−5)
+O(10−7)
+O(10−9)
+O(10−12)
+Average �
+A−1 errors, for A ∼ 50 · N(0, 1) — CG
+ϵ
+10−3
+10−4
+10−5
+10−6
+10−7
+errN
+O(10−3)
+O(10−5)
+O(10−8)
+O(10−11) O(10−12)
+errrF
+O(10−3)
+O(10−5)
+O(10−7)
+O(10−10) O(10−12)
+We utilized Algorithm 1 in Newton’s method, for classifying images of four and nine from MNIST, by solving a regularized
+logistic regression minimization problem. For Algorithm 1, we used CG with a fixed number of iteration per column estimation.
+It is clear from Figure 4 that we require no more than 18 iterations per column estimate, for N = 785, to attain the optimal
+classification rate. With more than 18 CG iterations, the same classification rate was obtained.
+
+9
+14
+15
+16
+17
+18
+19
+CG iterarions per column
+0
+0.05
+0.1
+0.2
+0.3
+0.4
+0.45
+Classification error %
+Classification Error
+inversion with CG
+exact inversion
+Fig. 1. MNIST classification error, where Algorithm 1 is used in Newton’s method. In red, we depict the error when exact inversion was used.
+V. FEDERATED CODED MATRIX INVERSION
+In this section, we describe the proposed CMIM (also presented in [1]) which makes Algorithm 1 resilient to stragglers, and
+show how it can be applied to the FL scenario described in the introduction. The CMIM workflow is depicted in Figure 2.
+Our FL-scheme can be broken up in to four phases: (a) the coordinator shares elements β, H of a finite field with all the clients,
+(b) the clients each generate a pseudorandom permutation (PRP) σι, encrypt their corresponding data block Aι through a matrix
+polynomial fι(x), and broadcast {fι(x), σ−1
+ι
+} to the other clients, (c) the clients recover A, compute and encode their assigned
+task Wι, which is communicated to the coordinator, (d) the coordinator decodes once sufficiently many servers respond. It is also
+possible that β, H are determined collectively by the clients, or by a single client, which makes the data sharing secure against a
+curious and dishonest coordinator.
+Fig. 2. Algorithmic workflow of the CMIM, as proposed in [1]. The master shares f(x), an encoding analogous to (12), along with β, {η−1
+j
+}k
+j=1. The workers
+then recover A, compute their assigned tasks, and encode them according to G. Once k encodings Wι are sent back, �
+A−1 can be recovered.
+In our proposed FL approach, we assume there is a trustworthy coordinator who shares certain parameters to each of the k
+clients which constitute the network; e.g. hospitals in a health care network, each of which are comprised of multiple servers.
+What we present works for the case where the clients have local datasets of different sizes, {Ni}k
+i=1. This would result in the
+encoding functions fι(x) having different degrees, or their matrix coefficients being of a different size. In our setting we assume
+the servers are homogeneous, i.e. they have the same computational power. Therefore, equal computational loads are assigned to
+each of them. In order to keep the notation and size of the communication loads consistent, we assume w.l.o.g. that Aι ∈ RN×T
+for all ι ∈ Nk. If this is not the case, before fι(x) are determined, the clients could perform a data exchange phase (e.g. [13]), so
+that Ni = Nj for all i ̸= j. By this, it follows that the number of blocks does not have to be equal to the number of clients. The
+example we describe, is simply a motivation. A flowchart of our approach is presented in Figure 3.
+Moreover, in the case where M > N; for M = �k
+i=1 Ni, we can select a subset of features and/or samples, so that the resulting
+data matrix we consider is square. This can be interpreted as using the surrogate ˜A = SA, where S ∈ RN×M is an appropriate
+(sparse) sketching matrix for matrix inversion [32], which the workers agree on.
+First, in V-A we argue why all of A needs to be known by each of the workers, in order to recover entries or columns of its
+inverse. Then, in V-B we focus on phases (a) and (b), where we utilize Lagrange interpolation to securely share A among the
+workers. We discuss the computation tasks the workers are requested to compute, which are blocks of �
+A−1; and collectively
+correspond to the subroutine problems of Algorithm 1. In V-C we focus on (c) and (d), where we show how the servers encode
+their computations, and describe the coordinator’s decoding step. Optimality of BRS generator matrices in terms of the encoded
+communication loads is established in V-D.
+When assuming floating-point arithmetic, our approach introduces no numerical nor approximation errors. The errors are a
+consequence of using iterative solvers to estimate (4), which we utilize to linearly separate the computations. Therefore, if the
+workers can recover the optimal solutions to the underlying minimization problems, our scheme would be exact.
+
+>A-i =[Ai ... Ak
+W1
+X
+Wn
+X
+f(×)
+f(x)
+f(x)10
+Fig. 3. Flowchart of our proposal, where k = ni = 4 for all i ∈ N4.
+A. Knowledge of A is necessary
+A bottleneck when computing the inverse of a matrix; or estimating its columns, is that the entire matrix needs to be known.
+A single change in the matrix’s entries may result in a non-singular matrix, which conveys how sensitive Gaussian elimination is.
+Such problems are extensively studied in conditioning and stability of numerical analysis [30], and in perturbation theory. This is
+not a focus of our work.
+In the case where only one column is not known, one can determine the subspace in which the missing column lies, but without
+the knowledge of at least one entry of that column, it would be impossible to recover that column. Even with such an approach or
+a matrix completion algorithm, the entire A is determined before we proceed to inverting A; or performing linear regression to
+approximate Ab = ei as in (4).
+Another example, relating to our FL set up, is the case where one of the blocks is different. This could lead to drastic
+miscalculations. In the following example, we consider n = k = 2 and N = 4, where the second server sends a different
+block, which are indicated by a different color and font:
+A1 =
+�
+�
+�
+6
+2
+2
+-5
+0
+−1
+2
+0
+−5
+6
+-1
+-3
+5
+−3
+-4
+3
+�
+�
+�
+A2 =
+�
+�
+�
+6
+2
+−1
+−3
+0
+−1
+5
+6
+−5
+6
+3
+−2
+5
+−3
+1
+6
+�
+�
+� .
+It follows that ∥A−1
+1 ∥F ≈ 90.45, ∥A−1
+2 ∥F ≈ 1, and ∥A−1
+1
+− A−1
+2 ∥0 = 16; i.e. no entries of A−1
+1
+and A−1
+2
+are equal.
+Furthermore, by the data processing inequality [33, Corollary pg.35], the above imply that no less than N 2 information symbols
+can be known by each server, while hoping to approximate a column of A−1. Hence, all clients need full knowledge of each others
+information, and cannot communicate less than NT symbols to each other. This is a consequence of the fact that a dense vector is
+not recoverable from underdetermined linear measurements. They can however send an encoded version of their respective block
+Aι ∈ RN×T to the other clients consisting of NT symbols, determined by a modified Lagrange polynomial, which guarantees
+security against eavesdroppers.
+Similar cryptographic protocols date back to the SSS algorithm [34], which is also based on RS codes. This idea has extensively
+been exploited in LCC [5], yet differs from our approach.
+B. Phases (a), (b) — Data Encryption and Sharing
+Let k, γ ∈ Z+ be factors of N and T respectively, so that T = N
+k and Γ = T
+γ .3 The coordinator constructs a set of distinct
+interpolation points B = {βj}n
+j=1 ⊊ F×
+q , for q > n ⩾ γ.4 To construct this set, it suffices to sample β ∈ F×
+q ; any one of the
+φ(q − 1) primitive roots of Fq (φ is Euler’s totient function), which is a generator of the multiplicative group (F×
+q , ·), and define
+3If γ ∤ T, append 0T ×1 to the end of the first ˜γ = T(modγ) blocks which are each comprised of ˜Γ = ⌊ T
+γ ⌋ columns of Aι, while the remaining γ − ˜γ
+blocks are comprised of ˜Γ + 1 columns. Now, each block is of size T × (˜Γ + 1).
+4For the encodings of the Aι’s, γ points suffice, and we only need to require q > γ. We select B of cardinality n and require q > n ⩾ γ, in order to reuse B
+in our CCM.
+
+β,H
+β,H
+B,H
+β,H
+A
+A
+88
+fi(x),q1
+f2(x),02
+Clients share their corre
+sponding fi(×) and o,-1
+Clients recover A = Ai A2 A3 A4
+Servers carry out computations, and each
+client sends back W, to the coordinator
+=
+Once the threshold is met,11
+each point as βj = βj. Then, a random multiset H = {ηj}γ
+j=1 ∈ 2F×
+q of size γ is generated, i.e. repetitions in H are allowed,
+which will be used to remove the structure of the Lagrange coefficients, as the adversaries could exploit their structure to reveal
+β.
+The element β and set H, are broadcasted securely to all the workers through a public-key cryptosystem, e.g. RSA or McEliece.
+Matrices Aι are partitioned into γ blocks
+Aι =
+�
+A1
+ι · · · Aγ
+ι
+�
+where Ai
+ι ∈ RN×Γ, ∀i ∈ Nγ,
+(11)
+and each client generates a PRP σι ∈ Sγ. The blocks {Aι}k
+ι=1 are encrypted locally through the univariate polynomials
+fι(x) =
+γ
+�
+j=1
+Aj
+ι · ησι(j)
+�
+��
+l̸=j
+x − βl
+βj − βl
+�
+�
+(12)
+for which fι(βj) = ησι(j)Aj
+ι.
+The clients then broadcast {fι(x), σ−1
+ι
+} to each other, and their servers can then recover all Aι’s as follows:
+Aι =
+�
+ησ−1
+ι
+(1)f(β1) · · · ησ−1
+ι
+(γ)f(βγ)
+�
+∈ RN×T .
+(13)
+The coefficients of fι(x) are comprised of NΓ symbols, thus, each polynomial consists of a total of NT symbols, which is the
+minimum number of symbols needed to be communicated. The PRP σι is generated locally by the clients, to ensure that each
+fι(x) differs by more than just the matrix partitions.
+We assume Kerckhoffs’ principle, which states that everyone has knowledge of the system, including the messages fι(x). For
+the proposed CMIM, as long as {β, H} and σ−1
+ι
+are securely communicated, even if fι(x) is revealed, the block Aι is secure
+against polynomial-bounded adversaries (this is the security level assumed by the cryptosystems used for the communication).
+Proposition 5. The encryptions of Aι through fι(x), are as secure against eavesdroppers as the public-key cryptosystems which
+are used when broadcasting {β, H} and σ−1
+ι
+. To recover Aι, an adversary needs to intercept both communications, and break
+both cryptosystems.
+Proof. We prove this by contradiction. Assume that an adversary was able to reverse the encoding fι(x) of Aι. This implies that
+he was able to reveal β and σι(H) := {ησι(j)}γ
+j=1. The only way to reveal these elements, is he was able to both intercept and
+decipher the public-key cryptosystem used by the coordinator, which contradicts the security of the cryptosystem.
+In order to invert the multiplications of σι(H) for each of the evaluations of fι(x), both H and σ−1
+ι
+need to be known. To do so,
+the adversary needs to intercept both the communication between the coordinator and the clients, and the communication between
+the clients, as well as breaking both the cryptosystems used to securely carry out these communications.
+■
+C. Phases (c), (d) — Computations, Encoding and Decoding
+At this stage, the workers have knowledge of everything they need in order to recover A, before they carry out their computation
+tasks. By (13), the recovery is straightforward.
+For Algorithm 1, any CCM in which the workers compute an encoding of partitions of the resulting computation E =
+�
+E1 · · · Ek
+�
+could be utilized. It is crucial that the encoding takes place on the computed tasks {Ei}k
+i=1 in the scheme, and
+not the assigned data or partitions of the matrices that are being computed over (such CMM leverage the linearity of matrix
+multiplication), otherwise the algorithm could potentially not return the correct approximation. This also means that utilizing such
+encryption approaches (e.g. [5]) for guaranteeing security against the workers, is not an option. We face these restrictions due to
+the fact that matrix inversion is a non-linear operator.
+The computation tasks Ei correspond to a partitioning �
+A−1 =
+� ˆ
+A1 · · ·
+ˆ
+Ak
+�
+, of our approximation from Algorithm 1. We
+propose a linear encoding of the computed blocks { ˆ
+Ai}k
+i=1 based on generators satisfying (2). Along with the proposed decoding
+step, we have a MDS-based CCM for matrix inversion.
+We consider the same parameters as in V-B, in order to reuse B in the proposed CMIM. Each ˆ
+Ai is comprised of T distinct but
+consecutive approximations of (4), i.e.
+ˆ
+Ai =
+�ˆb(i−1)T +1 · · · ˆbiT
+�
+∈ RN×T
+∀i ∈ Nk,
+which could also be approximated by iteratively solving
+ˆ
+Ai ≈ arg min
+B∈RN×T
+���AB −
+�
+e(i−1)T +1 · · · eiT
+���2
+F
+�
+.
+Without loss of generality, we assume that the workers use the same algorithms and parameters for estimating the columns
+{ˆbi}N
+i=1. Therefore, workers allocated the same tasks are expected to get equal approximations in the same amount of time.
+
+12
+For our CCM, we leverage BRS generator matrices for both the encoding and decoding steps. We adapt the GC framework,
+so we need an analogous condition to a⊤
+I G = ⃗1 for the CMIM; in order to invoke Algorithm 1. The condition we require is
+˜DI ˜G = IN, for an encoding-decoding pair ( ˜G, ˜DI).
+From our discussion on BRS codes in III-A, we set ˜G = IT ⊗ G and ˜DI = IT ⊗ (GI)−1 for any given set of k responsive
+servers indexed by I. The index set of blocks requested from the ιth worker to compute is denoted by Jι, and has cardinality w.
+The workers’ encoding steps correspond to
+˜G · ( �
+A−1)⊤ = (IT ⊗ G) ·
+�
+��
+ˆ
+A⊤
+1
+...
+ˆ
+A⊤
+k
+�
+�� =
+�
+�
+�
+�
+�
+�
+�
+j∈J1
+pj(β1) · ˆ
+A⊤
+j
+...
+�
+j∈Jn
+pj(βn) · ˆ
+A⊤
+j
+�
+�
+�
+�
+�
+�
+(14)
+which are carried out locally, once they have computed their assigned tasks. We denote the encoding of the ιth worker by Wι ∈
+CT ×N, i.e. Wι = �
+j∈Jι pj(βι) · ˆ
+A⊤
+j , which are sent to the coordinator. The received encoded computations by any distinct k
+servers indexed by I, constitute ˜GI · ( �
+A−1)⊤.
+Lemma 1 implies that as long as k workers respond, the approximation �
+A−1 is recoverable. Moreover, the decoding step
+reduces to a matrix multiplication of k × k matrices. Applying HI to a square matrix can be done in O(k2 log k), through the
+FFT algorithm. The prevailing computation in our decoding, is applying P−1. The decoding step is
+˜DI ·
+�
+˜GI · ( �
+A−1)⊤�
+=
+�
+IT ⊗ (GI)−1�
+·
+�
+IT ⊗ GI
+�
+· ( �
+A−1)⊤
+= (IT · IT ) ⊗
+�
+(GI)−1 · GI
+�
+· ( �
+A−1)⊤
+= IT ⊗ Ik · ( �
+A−1)⊤
+= ( �
+A−1)⊤
+and our scheme is valid.
+The above CCM therefore has a linear encoding done locally by the servers (14), is MDS since s = d − 1, and its decoding
+step reduces to computing and applying G−1
+I
+(Lemma 1). The security of the encodings rely on the secrecy of B, which were
+sent from the coordinator to the workers. For an additional security layer, the interpolation points of B could instead be defined as
+βj = βπ(j), for π ∈ Sn a PRP. In this case, π−1 would also need to be securely broadcasted.
+0
+1
+2
+3
+4
+5
+6
+7
+8
+106
+0
+20
+40
+60
+80
+100
+120
+4.5
+5
+5.5
+6
+6.5
+7
+0
+20
+40
+60
+80
+100
+120
+Fig. 4. Comparison of decoding complexity, when naive matrix inversion is used (so O(k3)) compared to the decoding step implied by Lemma 1, for n = 200
+and varying s. We also provide a logarithmic scale comparison.
+With the above framework, any sparsest-balanced generator MDS matrix [23] would suffice, as long as it satisfies the MDS
+theorem [35]. By Lemma 1, if we set k = Ω(
+√
+N) (similar to [7]), the decoding step could then be done in O(N ω/2) = o(N 1.187),
+which is close to linear in terms of N.
+Theorem 6. Let G ∈ Fn×k be a generator matrix of any MDS code over F, for which ∥G(j)∥0 = n − k + 1 and ∥G(i)∥0 = w
+for all (i, j) ∈ Nn × Nk. By utilizing Algorithm 1, we can devise a linear MDS coded matrix inversion scheme; through the
+encoding-decoding pair ( ˜G, ˜DI).
+Proof. The encoding coefficients applied locally by each of the n workers correspond to a row of G. The encodings of all the
+workers then correspond to ˜G · ( �
+A−1)⊤, for ˜G = IT ⊗ G, as in (14). Consider any set of responsive workers I of size k, whose
+encodings constitute ˜GI ·( �
+A−1)⊤. By the MDS theorem, GI is invertible. Hence, the decoding step reduces to inverting GI; i.e.
+˜DI = IT ⊗ (GI)−1, and is performed online.
+■
+Constructions based on cyclic MDS codes, which have been used to devise GC schemes [36], can also be considered. These
+encoding matrices are not sparsest-balanced, which makes them suitable when considering heterogeneous workers.
+Proposition 7. Any cyclic [n, k] MDS code C over F ∈ {R, C} can be used to devise a coded matrix inversion encoding-decoding
+pair ( ˜G, ˜DI).
+
+13
+Proof. Consider a cyclic [n, n − s] MDS code C over F ∈ {R, C}. Recall that from our assumptions, we have s = n − k. By [36,
+Lemma 8], there exists a codeword g1 ∈ C of support d = s + 1, i.e. ∥g1∥0 = d. Since C is cyclic, it follows that the cyclic shifts
+of g1 also lie in C. Denote the n − 1 consecutive cyclic shifts of g1 by {gi}n
+i=2 ⊊ C ⊊ F1×n, which are all distinct. Define the
+cyclic matrix
+¯G :=
+�
+�
+|
+|
+|
+g⊤
+1
+g⊤
+2
+. . .
+g⊤
+n
+|
+|
+|
+�
+� ∈ Fn×n.
+Since ∥gi∥0 = d and gi is a cyclic shift of gi−1 for all i > 1, it follows that ∥ ¯G(i)∥0 = ∥ ¯G(j)∥0 = d for all i, j ∈ Nn, i.e. ¯G
+is sparsest and balanced. If we erase any s = n − k columns of ¯G, we get G ∈ Fn×k. By erasing arbitrary columns of ¯G, the
+resulting G is not balanced, i.e. we have ∥G(i)∥0 ̸= ∥G(j)∥0 for some pairs i, j ∈ Nn. Similar to our construction based on BRS
+generator matrices, we define the encoding matrix to be ˜G = IT ⊗ G. The local encodings are then analogous to (14).
+Consider an arbitrary set of k non-straggling workers I ⊊ Nn, and the corresponding matrix GI ∈ Fk×k. By [36, Lemma 12,
+B4.], GI is invertible. The decoding matrix is then ˜DI = IT ⊗ (GI)−1, and the condition ˜DI ˜G = IN is met.
+■
+D. Optimality of MDS BRS Codes
+Under the assumption that k = n − s, by utilizing the BRSq[n, k] generator matrices, we achieved the minimum possible
+communication load from the workers to the coordinator. From our discussion in V-A, we cannot hope to receive an encoding
+of less than N 2/k symbols; when we require that k workers respond with the same amount of information symbols in order
+to recover �
+A−1 ∈ RN×N, unless we make further assumptions on the structure of A and A−1. Each encoding Wι consists
+of NT = N 2/k symbols, so we have achieved the lower bound on the minimum amount of information needed to be sent to
+the coordinator. Hence, Wι ∈ CT ×N for any sparsest-balance generator MDS matrix. This also holds true for other generator
+matrices which can be used in Theorem 6, as the encodings are linear (e.g. Proposition 7).
+We also require the workers to estimate the least possible number of columns for the given recovery threshold k. For our choice
+of parameters, the bound of [20, Theorem 1] is met with equality. That is, for all i ∈ Nn:
+∥G(i)∥0 = w = k
+n · d = k
+n · (n − k + 1) ,
+which means that for homogeneous workers, we cannot get a sparser generator matrix. This, along with the requirement that GI
+should be invertible for all possible I, are what we considered in (2).
+VI. CONCLUSION AND FUTURE WORK
+In this paper, we addressed the problem of approximate computation of the inverse of a matrix distributively in a FL setting,
+under the possible presence of straggling workers. We provided approximation error bounds for our approach, as well as security
+and recovery guarantees. We also provided numerical experiments that validated our proposed approach.
+There are several interesting future directions. One is looking into the issue of numerical stability of the BRS approach, and
+exploring other suitable generator matrices, e.g. circulant permutation and rotation matrices [37]. Another direction, is leveraging
+approximate CCMs. The techniques of [22], [38] suggest that carefully selecting interpolation points may lead to more efficient
+(approximate) schemes. In terms of coding-theory, it would be interesting to see if it is possible to reduce the complexity of our
+decoding step. Specifically, could well-known RS decoding algorithms such as the Berlekamp-Welch algorithm be exploited?
+Another important extension is to reduce the communication rounds when computing the pseudoinverse through our approach.
+This depends on the CMM which is being utilized, though using different ones for each of the two multiplications may also be
+beneficial.
+Tribute to Alex Vardy: As this is a special issue dedicated to the memory Alexander Vardy, we mention how this paper relates
+to some of his work. Even though Alex had not worked on CC, his contributions to RS codes are immense. A focus of this paper
+is to reduce the decoding complexity of the proposed BRS-based CCM, while in [39] it was shown that ML decoding of RS
+codes is NP-hard. Another highly innovative work of Vardy’s is [40], in which the ‘Parvaresh-Vardy codes’ were introduced;
+and the associated list-decoding algorithm was shown to yield an improvement over the Guruswami–Sudan algorithm. This was
+subsequently improved by Guruswami and Rudra [41], whose techniques were exploited in [42] to introduce list-decoding in CC.
+APPENDIX A
+ADDITIONAL MATERIAL AND BACKGROUND
+In this appendix, we include material and background which was used in our derivations. First, we recall what an ϵ-optimal
+solution/point is, which was used in the proof of Proposition 4. Next, we state the MDS Theorem and the BCH Bound. We
+then give a brief overview of the GC scheme from [8], to show how it differs from our coded matrix inversion scheme. We also
+explicitly give their construction of a balanced mask matrix M ∈ {0, 1}n×k, which we use for the construction of the BRS
+generator matrices. Lastly, we illustrate a simple example of the encoding matrix.
+
+14
+Definition 8 ( [43]). A point ¯x is said to be an ϵ-optimal solution/point to a minimization problem with objective function f(x),
+if for any x, it holds that f(x) ⩾ f(¯x) − ϵ, where ϵ ⩾ 0. When ϵ = 0, an ϵ-optimal solution is an exact minimizer.
+Theorem 9 (MDS Theorem — [35]). Let C be a linear [n, k, d] code over Fq, with G, H the generator and parity-check matrices.
+Then, the following are equivalent:
+1) C is a MDS code, i.e. d = n − k + 1
+2) every set of n − k columns of H is linearly independent
+3) every set of k columns of G is linearly independent
+4) C⊥ is a MDS code.
+Theorem 10 (BCH Bound — [15], [25]). Let p(x) ∈ Fq[x]\{0} with t cyclically consecutive roots, i.e. p(αj+ι) = 0 for all
+ι ∈ Nt. Then, at least t + 1 coefficients of p(x) are nonzero.
+Algorithm 2: MaskMatrix(n, k, d) [8]
+Input: n, k, d ∈ Z+ s.t. n > d, k and w = kd
+n
+Output: row-balanced mask matrix M ∈ {0, 1}n×k
+M ← 0n×k
+for j = 0 to k − 1 do
+for i = 0 to d − 1 do
+ι ← (i + jd + 1) mod n
+Mr,ι ← 1
+end
+end
+return M
+Even though this was not pointed out in [8], Algorithm 2 does not always produce a mask matrix of the given parameters when
+we select d < n/2. This is why in our work we require d ⩾ n/2.
+The decomposition G = HP is utilized in the GC scheme of [8]. Each column of G corresponds to a partition of the data
+whose partial gradient is to be computed. The polynomials are judiciously constructed in this scheme, such that the constant term
+of each polynomial is 1 for all polynomials, thus P(1) = ⃗1. By this, the decoding vector a⊤
+I is the first row of G−1
+I , for which
+a⊤
+I GI = e⊤
+1 . A direct consequence of this is that a⊤
+I BI = e⊤
+1 T = T(1) = ⃗1, which is the objective for constructing a GC
+scheme.
+A. Generator Matrix Example
+As an example, consider the case where n = 9, k = 6 and d = 6, thus w = kd
+n = 4. Then, Algorithm 2 produces
+M =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∈ {0, 1}9×6 .
+For our CCM, this means that the ith worker computes the blocks indexed by supp(M(i)), e.g. supp(M(1)) = {1, 2, 4, 5}. We
+denote the indices of the respective task allocations by Ji = supp(M(i)). The entries of the generator matrix G are the evaluations
+of the constructed polynomials (3) at each of the evaluation points B = {βi}n
+i=1, i.e. Gij = pj(αi). This results in:
+G =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+p1(β1)
+p2(β1)
+0
+p4(β1)
+p5(β1)
+0
+p1(β2)
+p2(β2)
+0
+p4(β2)
+p5(β2)
+0
+p1(β3)
+p2(β3)
+0
+p4(β3)
+p5(β3)
+0
+p1(β4)
+0
+p3(β4)
+p4(β4)
+0
+p6(β4)
+p1(β5)
+0
+p3(β5)
+p4(β5)
+0
+p6(β5)
+p1(β6)
+0
+p3(β6)
+p4(β6)
+0
+p6(β6)
+0
+p2(β7)
+p3(β7)
+0
+p5(β7)
+p6(β7)
+0
+p2(β8)
+p3(β8)
+0
+p5(β8)
+p6(β8)
+0
+p2(β9)
+p3(β9)
+0
+p5(β9)
+p6(β9)
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+
+15
+APPENDIX B
+DISTRIBUTED PSEUDOINVERSE
+For full-rank rectangular matrices A ∈ RN×M where N > M, one resorts to the left Moore–Penrose pseudoinverse A† ∈
+RM×N, for which A†A = IM. In Algorithm 3, we present how to approximate the left pseudoinverse of A, by using the fact that
+A† = (A⊤A)−1A⊤; since A⊤A ∈ GLN(R). The right pseudoinverse A† = A⊤(AA⊤)−1 of A ∈ RM×N where M < N,
+can be obtained by a modification of Algorithm 3.
+Just like the inverse, the pseudoinverse of a matrix also appears in a variety of applications. Computing the pseudoinverse of
+A ∈ RN×M for N > M is even more cumbersome, as it requires inverting the Gram matrix A⊤A. For this subsection, we
+consider a full-rank matrix A.
+One could naively attempt to modify Algorithm 1 in order to retrieve A† such that A†A = IM, by approximating the rows
+of A†. This would not work, as the underlying optimization problems would not be strictly convex. Instead, we use Algorithm
+3 to estimate the rows of B−1 := (A⊤A)−1, and then multiply the estimate �
+B−1 by A⊤. This gives us the approximation
+�
+A† = �
+B−1 · A⊤.
+The drawback of Algorithm 3 is that it requires two additional matrix multiplications, A⊤A and �
+B−1A⊤. We overcome this
+barrier by using a CMM scheme twice, to recover �
+A† in a two or three-round communication CC approach. These are discussed
+in below.
+Bounds on errF ( �
+A−1) and errrF ( �
+A−1) can be established for both algorithms, specific to the black-box least squares algorithm
+being utilized. This is left for future work.
+Algorithm 3: Estimating A†
+Input: full-rank A ∈ RN×M where N > M
+B ← A⊤A
+for i=1 to M do
+ˆci = arg minc∈R1×M
+�
+gi(c) := ∥cB − e⊤
+i ∥2
+2
+�
+ˆbi ← ˆci · A⊤
+end
+return �
+A† ←
+�
+ˆb⊤
+1 · · · ˆb⊤
+M
+�⊤
+▷ �
+A†(i) = ˆbi
+Corollary 11. For full-rank A ∈ RN×M with N > M, we have errF (�
+A†) ⩽
+√
+Mϵ·κ2
+√
+2σmin(A)3 and errrF (�
+A†) ⩽
+√
+Mϵ·κ2
+√
+2σmin(A)2 when
+using SD to solve the subroutine optimization problems of Algorithm 3, with termination criteria ∥∇gi(c[t])∥2 ⩽ ϵ.
+Proof. From (10), it follows that
+∥B−1ei − ˆc⊤
+i ∥2 ⩽
+ϵ/
+√
+2
+σmin(B)2 =
+ϵ/
+√
+2
+σmin(A)4 =: δ .
+The above bound implies that for each summand of the Frobenius error; ∥ˆbi − A†
+(i)∥2 = ∥ˆciA⊤ − e⊤
+i · B−1A⊤∥2, we have
+∥ˆbi − A†
+(i)∥2 ⩽ δ∥A⊤∥2. Summing the right hand side M times, we get that
+errF (�
+A†)2 ⩽ M · (δ∥A⊤∥2)2
+= Mϵ2 · σmax(A)2
+σmin(A)8
+= Mϵ2 · κ2
+2
+σmin(A)6 .
+By taking the square root, we have shown the first claim.
+Since 1/σmin(A) = ∥A†∥2 ⩽ ∥A†∥F , it then follows that
+errrF (�
+A†) = errF (�
+A†)
+∥A†∥F
+⩽ errF (�
+A†)
+∥A†∥2
+=⩽
+√
+Mϵ · κ2
+√
+2σmin(A)2 ,
+which completes the proof.
+■
+A. Pseudoinverse from Polynomial CMM
+One approach to leverage Algorithm 3 in a two-round communication scheme is to first compute B = A⊤A through a CMM
+scheme, then share B with all the workers who estimate the rows of �
+B−1, and finally use another CMM to locally encode the
+
+16
+estimated columns with blocks of A⊤; to recover �
+A† = �
+B−1 · A⊤. Even though there are only two rounds of communication, the
+fact that we have a local encoding by the workers results in a higher communication load overall. An alternative approach which
+circumvents this issue, uses three-rounds of communication.
+For this approach, we use the polynomial CMM scheme from [7] twice, along with our coded matrix inversion scheme. This
+CMM has a reduced communication load, and minimal computation is required by the workers. To have a consistent recovery
+threshold across our communication rounds, we partition A as in (11) into ¯k = √n − s =
+√
+k blocks. Each block is of size
+N × ¯T, for ¯T = M
+k . The encodings from [7] of the partitions {Aj}¯k
+j=1 for carefully selected parameters a, b ∈ Z+ and distinct
+elements γi ∈ Fq, are
+˜Aa
+i =
+k
+�
+j=1
+Ajγ(j−1)a
+i
+and
+˜Ab
+i =
+k
+�
+j=1
+Ajγ(j−1)b
+i
+for each worker indexed by i. Thus, each encoding is comprised of N ¯T symbols. The workers compute the product of their
+respective encodings ( ˜Aa
+i )⊤ · ˜Ab
+i. The decoding step corresponds to an interpolation step, which is achievable when ¯k2 = k many
+workers respond5, which is the optimal recovery threshold for CMM. Any fast polynomial interpolation or RS decoding algorithm
+can be used for this step, to recover B.
+Next, the master shares B with all the workers (from V-A, this is necessary), who are requested to estimate the column-blocks
+of �
+B−1
+�
+B−1 =
+�
+¯B1 · · · ¯Bk
+�
+where ¯Bj ∈ RM× ¯T ∀j ∈ Nk
+(15)
+according to Algorithm 1. We can then recover �
+B−1 by our BRS based scheme, once k workers send their encoding.
+For the final round, we encode �
+B−1 as
+˜Ba
+i =
+k
+�
+j=1
+¯Bjγ(j−1)a
+i
+which are sent to the respective workers. The workers already have in their possession the encodings ˜Ab
+i. We then carry out the
+polynomial CMM where each worker is requested to send back ( ˜Ba
+i )⊤ · ˜Ab
+i. The master server can then recover �
+A†.
+Theorem 12. Consider G ∈ Fn×k as in Theorem 6. By using any CMM, we can devise a matrix pseudoinverse CCM by utilizing
+Algorithm 3, in two-rounds of communication. By using polynomial CMM [7], we achieve this with a reduced communication
+load and minimal computation, in three-rounds of communication.
+REFERENCES
+[1] N. Charalambides, M. Pilanci, and A. O. Hero, “Secure Linear MDS Coded Matrix Inversion,” in 2022 58th Annual Allerton Conference on Communication,
+Control, and Computing (Allerton), 2022, pp. 1–8.
+[2] B. G. Greenberg and A. E. Sarhan, “Matrix inversion, its interest and application in analysis of data,” Journal of the American Statistical Association, vol. 54,
+no. 288, pp. 755–766, 1959.
+[3] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed.
+USA: Society for Industrial and Applied Mathematics, 2002.
+[4] K. Lee, M. Lam, R. Pedarsani, D. Papailiopoulos, and K. Ramchandran, “Speeding up distributed machine learning using codes,” IEEE Transactions on
+Information Theory, vol. 64, no. 3, pp. 1514–1529, 2017.
+[5] Q. Yu, S. Li, N. Raviv, S. M. M. Kalan, M. Soltanolkotabi, and S. Avestimehr, “Lagrange Coded Computing: Optimal design for resiliency, security and
+privacy,” arXiv preprint arXiv:1806.00939, 2018.
+[6] S. Li and S. Avestimehr, “Coded Computing,” Foundations and Trends® in Communications and Information Theory, vol. 17, no. 1, 2020.
+[7] Q. Yu, M. Maddah-Ali, and S. Avestimehr, “Polynomial codes: an optimal design for high-dimensional coded matrix multiplication,” in Advances in Neural
+Information Processing Systems, 2017, pp. 4403–4413.
+[8] W. Halbawi, N. Azizan, F. Salehi, and B. Hassibi, “Improving Distributed Gradient Descent Using Reed-Solomon Codes,” in 2018 IEEE International
+Symposium on Information Theory (ISIT).
+IEEE, 2018, pp. 2027–2031.
+[9] S. Dhakal, S. Prakash, Y. Yona, S. Talwar, and N. Himayat, “Coded Federated Learning,” in 2019 IEEE Globecom Workshops (GC Wkshps).
+IEEE, 2019,
+pp. 1–6.
+[10] S. Prakash, S. Dhakal, M. R. Akdeniz, Y. Yona, S. Talwar, S. Avestimehr, and N. Himayat, “Coded Computing for Low-Latency Federated Learning over
+Wireless Edge Networks,” IEEE Journal on Selected Areas in Communications, vol. 39, no. 1, pp. 233–250, 2020.
+[11] R. Schlegel, S. Kumar, E. Rosnes, and A. G. i. Amat, “CodedPaddedFL and CodedSecAgg: Straggler Mitigation and Secure Aggregation in Federated
+Learning,” arXiv e-prints, pp. arXiv–2112, 2021.
+[12] S. Kumar, R. Schlegel, E. Rosnes, and A. G. i. Amat, “Coding for Straggler Mitigation in Federated Learning,” arXiv preprint arXiv:2109.15226, 2021.
+[13] M. Xhemrishi, A. G. i. Amat, E. Rosnes, and A. Wachter-Zeh, “Computational Code-Based Privacy in Coded Federated Learning,” arXiv preprint
+arXiv:2202.13798, 2022.
+[14] S. Ha, J. Zhang, O. Simeone, and J. Kang, “Coded Federated Computing in Wireless Networks with Straggling Devices and Imperfect CSI,” in 2019 IEEE
+International Symposium on Information Theory (ISIT), 2019, pp. 2649–2653.
+[15] W. Halbawi, Z. Liu, and B. Hassibi, “Balanced Reed-Solomon Codes,” in 2016 IEEE International Symposium on Information Theory (ISIT).
+IEEE, 2016,
+pp. 935–939.
+[16] ——, “Balanced Reed-Solomon Codes for all parameters,” in 2016 IEEE Information Theory Workshop (ITW).
+IEEE, 2016, pp. 409–413.
+[17] N. Charalambides, H. Mahdavifar, and A. O. Hero, “Numerically Stable Binary Gradient Coding,” arXiv preprint arXiv:2001.11449, 2020.
+[18] J. Koneˇcn`y, H. B. McMahan, D. Ramage, and P. Richtárik, “Federated optimization: Distributed machine learning for on-device intelligence,” arXiv preprint
+arXiv:1610.02527, 2016.
+5We select ¯k =
+√
+k in the partitioning of A in (11) when deploying this CMM, to attain the same recovery threshold as our inversion scheme.
+
+17
+[19] Y. Yang, P. Grover, and S. Kar, “Coded distributed computing for inverse problems,” in Advances in Neural Information Processing Systems, vol. 30. Curran
+Associates, Inc., 2017, pp. 709–719.
+[20] R. Tandon, Q. Lei, A. G. Dimakis, and N. Karampatziakis, “Gradient coding: Avoiding stragglers in distributed learning,” in International Conference on
+Machine Learning, 2017, pp. 3368–3376.
+[21] M. Soleymani, H. Mahdavifar, and A. S. Avestimehr, “Analog Lagrange Coded Computing,” IEEE Journal on Selected Areas in Information Theory, vol. 2,
+no. 1, pp. 283–295, 2021.
+[22] S. Kiani and S. C. Draper, “Successive Approximation Coding for Distributed Matrix Multiplication,” arXiv preprint arXiv:2201.03486, 2022.
+[23] S. H. Dau, W. Song, Z. Dong, and C. Yuen, “Balanced Sparsest Generator Matrices for MDS Codes,” in 2013 IEEE International Symposium on Information
+Theory, 2013, pp. 1889–1893.
+[24] M. Krause, “A Simple Proof of the Gale-Ryser Theorem,” The American Mathematical Monthly, vol. 103, no. 4, pp. 335–337, 1996.
+[25] R. J. McEliece, Theory of Information and Coding, 2nd ed.
+USA: Cambridge University Press, 2001.
+[26] J. Alman and V. V. Williams, “A refined laser method and faster matrix multiplication,” arXiv preprint arXiv:2010.05846, 2020.
+[27] Å. Björck and V. Pereyra, “Solution of Vandermonde Systems of Equations,” Mathematics of Computation, vol. 24, pp. 893–903, 1970.
+[28] S. P. Boyd and L. Vandenberghe, Convex optimization.
+Cambridge university press, 2004.
+[29] J. R. Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain,” 1994.
+[30] L. N. Trefethen and D. Bau III, Numerical linear algebra.
+Siam, 1997, vol. 50.
+[31] S. Bubeck, “Convex optimization: Algorithms and complexity,” Foundations and Trends® in Machine Learning, vol. 8, no. 3-4, pp. 231–357, 2015.
+[Online]. Available: http://dx.doi.org/10.1561/2200000050
+[32] R. M. Gower, “Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices,” arXiv preprint arXiv:1612.06013, 2016.
+[33] T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing).
+USA: Wiley-Interscience,
+2006.
+[34] A. Shamir, “How to Share a Secret,” Communications of the ACM, vol. 22, no. 11, pp. 612–613, 1979.
+[35] S. Ling and C. Xing, Coding Theory: A First Course.
+Cambridge University Press, 2004.
+[36] N. Raviv, I. Tamo, R. Tandon, and A. G. Dimakis, “Gradient Coding from Cyclic MDS Codes and Expander Graphs,” IEEE Transactions on Information
+Theory, vol. 66, no. 12, pp. 7475–7489, 2020.
+[37] A. Ramamoorthy and L. Tang, “Numerically stable coded matrix computations via circulant and rotation matrix embeddings,” IEEE Transactions on
+Information Theory, vol. 68, no. 4, pp. 2684–2703, 2021.
+[38] H. Jeong, A. Devulapalli, V. R. Cadambe, and F. P. Calmon, “ϵ-Approximate Coded Matrix Multiplication Is Nearly Twice as Efficient as Exact
+Multiplication,” IEEE Journal on Selected Areas in Information Theory, vol. 2, no. 3, pp. 845–854, 2021.
+[39] V. Guruswami and A. Vardy, “Maximum-Likelihood Decoding of Reed-Solomon Codes is NP-hard,” IEEE Transactions on Information Theory, vol. 51,
+no. 7, pp. 2249–2256, 2005.
+[40] F. Parvaresh and A. Vardy, “Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time,” in 46th Annual IEEE Symposium on Foundations
+of Computer Science (FOCS’05).
+IEEE, 2005, pp. 285–294.
+[41] V. Guruswami and A. Rudra, “Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy,” IEEE Transactions on Information
+Theory, vol. 54, no. 1, pp. 135–150, 2008.
+[42] M. Soleymani, R. E. Ali, H. Mahdavifar, and A. S. Avestimehr, “List-decodable coded computing: Breaking the adversarial toleration barrier,” IEEE Journal
+on Selected Areas in Information Theory, vol. 2, no. 3, pp. 867–878, 2021.
+[43] F. Bai, Z. Wu, and D. Zhu, “Sequential Lagrange multiplier condition for ϵ-optimal solution in convex programming,” Optimization, vol. 57, no. 5, pp.
+669–680, 2008.
+
diff --git a/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/load_file.txt b/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f38c0be2c1c3ddf561645409831fbe6bd94c48f4
--- /dev/null
+++ b/9NE1T4oBgHgl3EQf7wX0/content/tmp_files/load_file.txt
@@ -0,0 +1,996 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf,len=995
+page_content='1  Federated Coded Matrix Inversion  Neophytos Charalambidesµ, Mert Pilanciσ, and Alfred O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hero IIIµ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='µEECS Department University of Michigan .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='σEE Department Stanford University  Email: neochara@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='edu, pilanci@stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='edu, hero@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='edu  Abstract  Federated learning (FL) is a decentralized model for training data distributed across client devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Coded computing (CC) is  a method for mitigating straggling workers in a centralized computing network, by using erasure-coding techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In this work  we propose approximating the inverse of a data matrix, where the data is generated by clients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' similar to the FL paradigm, while  also being resilient to stragglers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To do so, we propose a CC method based on gradient coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We modify this method so that the  coordinator does not need to have access to the local data, the network we consider is not centralized, and the communications which  take place are secure against potential eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' INTRODUCTION AND RELATED WORK  Inverting a matrix is one of the most important operations in numerous applications, such as, signal processing, machine  learning, and scientific computing [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A common way of inverting a matrix is to perform Gaussian elimination, which  requires O(N 3) operations for square matrices of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In high-dimensional applications, this can be cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Over the  past few years the machine learning (ML) community has made much progress on federated learning (FL), focusing on iterative  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The objective of FL is to leverage computation, communication and storage resources to perform distributed computations for  ML models, where the data of each federated worker is never shared with the coordinator of the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' that aggregates local  computations in order to update the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In FL applications it is important that the data is kept private and secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Distributed computations in the presence of stragglers (workers who fail to compute their task or have longer response time than  others) must account for the effect of non-responsive workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Coding-theoretic approaches have been adopted for this purpose  [4], [5], and fall under the framework of coded computing (CC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Data security is also an increasingly important issue in CC [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Despite the fact that multiplication algorithms imply inversion algorithms and vice versa, in the context of CC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' matrix inversion  has not been studied as extensively as coded matrix multiplication (CMM) [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The main reason for this is the fact that the latter  is non-linear and non-parallelizable as an operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We point out that distributed inversion algorithms do exist, though these make  assumptions on the matrix, are specific for distributed and parallel computing platforms, and require a matrix factorization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' or  heavy and multiple communication instances between the servers and the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In [1] a CC method1 was proposed based on gradient coding (GC) [8], which approximates the inverse of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In order  to overcome the obstacle of non-linearity, the columns of A−1 are approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' When assuming floating-point arithmetic, this  CCM introduces no numerical nor approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Note that GC and not CMM was utilized, as the latter does not require  the encoding to be done locally by the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Though the two areas of FL and CC seem to be closely related, on the surface they appear incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For instance, in CC  one often assumes there is a master server that distributes the data and may perform the encoding (encoding by the master server  is done in CMM, but not in GC), while in FL the central coordinator never has access to the distributed local training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' which  are located at different client nodes or workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  There are a few recent works that leverage CC in order to devise secure FL methods for distributed regression and iterative  optimization [9]–[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In this work, we combine optimization and CC, using erasure coding to protect against stragglers as in  CC and locally approximating the inverse without revealing the data to the coordinator, to design a FL scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Our approach,  is based on the coded matrix inversion method (CMIM) we develop, which utilizes balanced Reed-Solomon (BRS) codes [15],  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This results in an efficient decoding in terms of the threshold number of responsive workers needed to perform an error free  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We show that the general class of maximum distance separable (MDS) generator matrices could be used to generate  a suitable erasure code (Theorem 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The focus is on BRS codes, which have the following advantages:  (i) minimum redundancy per job across the network,  (ii) they optimize communication from workers to the master,  (iii) we can efficiently decode the resulting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Our CMIM can also be used to compute the Moore–Penrose pseudoinverse Y† of a data matrix Y ∈ RM×N for M ≫ N,  which is more general than inverting a square matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By using the fact that Y† = (Y⊤Y)−1Y⊤, the bottleneck is computing  the inverse of A = Y⊤Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In addition, two more matrix multiplications need to take place distributively: computing A before  the inversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' and �  A−1Y⊤ after the inverse has been approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The matrix products can be computed distributively using  A preliminary version also considers approximating A† [1], in the CC setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This work was partially supported by grants ARO W911NF-15-1-0479 and Dept  of Energy DE-NA0003921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  1We abbreviate ‘coded computing method/methods’ to CCM/CCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='03539v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='IT]  9 Jan 2023  2  various CCMS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' we can use a modification of the coded FL approaches of [11] and a CMM from [17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' both of which are based  on GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For the remainder of the paper, we focus on the generic problem of inverting a square matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The proposed FL approach applies to general linear regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Compared to previous FL iterative approaches [18],  the difference is that for Yθ = p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' with p the label vector and θ the model parameters, the pseudoinverse-regularized regression  solution is ˆθ = �  Y†p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Unlike conventional FL methods, this regularized regression can be computed non-iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The non-  iterative nature of the proposed approach is advantageous in settings such as Kalman filtering, where the matrix inverse must be  updated in real time as measurements come in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In II we recall basic facts on matrix inversion, least squares approximation and finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In III we review BRS codes, and prove two key lemmas regarding their generator matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In IV we present the matrix inverse  approximation algorithm we utilize in our CCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The main contribution is presented in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Our federated approach is split into four  phases, which we group in pairs of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' First, we discuss information sharing from the coordinator to the workers (we consider  all the clients’ servers as the network’s workers), and then information sharing between the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Second, we show how our  inversion algorithm can be incorporated in linear CCMs, and describe how this fits into the FL paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Concluding remarks and  future work are presented in VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Overview of the Coded Matrix Inversion Method  In CC the computational network is centralized, and is comprised of a master server who communicates with n workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The  idea behind our approximation is that the workers use a least squares solver to approximate multiple columns of A−1, resulting  in a set of local approximations to submatrices of �  A−1, which we refer to as blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We present approximation guarantees and  simulation results for steepest descent (SD) and conjugate gradient (CG) iterative optimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By locally approximating  the columns in this way, the workers can linearly encode the blocks of �  A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The clients have a block of data {Aι}k  ι=1, which  constitute the data matrix A =  �  A1 · · · Ak  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To simplify our presentation, we assume that each local data block is of the same  size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Aι ∈ RN×T for T = N/k, and that client i has ni servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, the total number of servers is n = �k  j=1 nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  We assume the blocks are of the same size, so that the encodings carried out by the clients are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In V, we show that this  assumption is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Moreover, for the CCM, it is not required that the number of blocks equal the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For  a given natural number γ, assume that γ divides T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' denoted γ | T (each local data block Aι is further divided into γ sub-blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In the case where k ∤ N or γ ∤ T, we can pad the blocks of �  A−1 so that these assumptions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A limitation of our proposed CMIM, is the fact that each server needs to have full knowledge of A, in order to estimate columns  of A−1 through a least squares solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The sensitivity of Gaussian elimination and matrix inversion also requires that all clients  have knowledge of each others’ data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This limitation is shared by other coded federated learning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' CodedPaddedFL  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In contrast to CC and GC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' where a master server has access to all the data, in FL the data is inherently distributed across  devices, thus GC cannot be applied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We also assume that the coordinator does not intercept the communication between  the clients, otherwise she could recover the local data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Also, we trust that the coordinator will not invert �  A−1, to approximate A  — this would be computationally difficult, for N large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Before broadcasting the data amongst themselves, the clients encode their block Ai, which guarantees security from outside  eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' When the clients receive the encoded data, they can decrypt and recover A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, their servers act as the workers  of the proposed CMIM and carry out their assigned computations, and directly communicate their computations back to the  coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Once the recovery threshold (the minimum number of responses needed to recover the computation) is met, the  approximation �  A−1 is recoverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Coded Federated Learning  There are few works that leverage CC to devise secure FL schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Most of these works have focused on distributed regression  and iterative methods, which is the primary application for FL [9]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Below, we describe and compare these approaches to our  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The authors of [9] proposed coded federated learning, in which they utilize a CMM scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Their security relies on the use  of random linear codes, to define the parity data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Computations are carried out locally on the systematic data, and only the parity  data is sent to the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The main drawback compared to our scheme is that each worker has to generate a random encoding  matrix and apply a matrix multiplication for the encodings, while we use the same BRS generator matrix across the network,  based on GC, to linearly encode the local computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The drawback in our case, is that the workers need to securely share their  data with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is an artifact of the operation (inversion) we are approximating, and is inevitable in the general case  where A has no structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Under the FL setting we are considering, where the data is gathered or generated locally and is not  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=', we cannot make any assumptions on the structure of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In [11], two methods were proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' CodedPaddedFL combines one-time-padding with GC to carry out the FL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Some  of its disadvantages are that a one-time-pad (OTP) needs to be generated by each worker, and that the OTPs are shared with the  coordinator, which means that if she gets hold of the encrypted data, she can decrypt it, compromising security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Furthermore,  there is a heavy communication load and the coordinator needs to store all the pads in order to recover the computed gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In  3  the proposed CMIM, the coordinator generates a set of interpolation points, and shares them with the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' If the coordinator  can intercept the communication between the workers, she can decrypt the encrypted data blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The second method proposed  in [11], CodedSecAgg, relies on Shamir’s secret sharing (SSS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' which is based on polynomial interpolation over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In  contrast, our CMIM relies on GC and Lagrange interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Lastly, we discuss the method proposed in [13], which is based on the McEliece cryptosystem, and moderate-density parity-  check codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This scheme considers a communication delay model which defines stragglers as the workers who respond slower  than the fastest worker, and time out after a predetermined amount of time ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' As the iterative SD process carries on, such workers  are continuously disregarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Due to this, there is a data sharing step at each iteration, at which the new stragglers communicate  encrypted versions of their data to the active workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Our scheme is non-iterative, and has a fixed recovery threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Unlike  some of the works previously mentioned, which guarantee information-theoretic security, the McEliece based systems and our  approach have computational privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Lagrange Interpolation Coded Computing Methods  While there is extensive literature on matrix-vector and matrix-matrix multiplication, and computing the gradient in the presence  of stragglers, there is limited work on computing or approximating the inverse of a matrix [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The non-linearity of matrix  inversion prohibits linear or polynomial encoding of the data before the computations are to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Consequently, most  CCMs cannot be directly utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' GC is the appropriate CC set up to consider [20], precisely because the encoding takes place  once the computation has been completed, in contrast to most CMM methods where the encoding is done by the master, before  the data is distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Here, we give a brief overview of the GC on which our CMIM is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We also review “Lagrange Coded Computing” (LCC),  which has relations to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, we give a summary of our proposed CMIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' All these rely on Lagrange interpolation  over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Gradient codes are a class of codes designed to mitigate the effect of stragglers in data centers, by recovering the gradient of  differentiable and additively separable objective functions in distributed first order methods [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The proposed CMIM utilizes  BRS generator matrices constructed for GC [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The main difference from our work is that in GC the objective is to construct an  encoding matrix G and decoding vectors aI ∈ Ck, such that a⊤  I G = ⃗1 for any set of non-straggling workers indexed by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To  do so, the decomposition of the BRS generator matrices GI = HIP is exploited, where HI is a Vandermonde matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' and the  first row of P is equal to ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Subsequently a⊤  I is extracted as the first row of H−1  I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In the proposed CMIM framework, the objective is to design an encoding-decoding pair ( ˜G, ˜DI) for which ˜DI ˜G = IN, for  all I ⊊ Nn of size k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The essential reason for requiring this condition, as opposed to that of GC, is that the empirical gradient of a  given dataset is the sum of each individual gradients, while in our scenario if the columns of �  A−1 are summed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' they cannot then  be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The state-of-the art CC framework is LCC, which is used to compute arbitrary multivariate polynomials of a given dataset  [5], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This approach is based on Lagrange interpolation, and it achieves the optimal trade-off between resiliency, security,  and privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The problem we are considering is not a multivariate polynomial in terms of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To securely communicate A to  the workers, we encode it through Lagrange interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Though similar ideas appear in LCC, the purpose and application of  the interpolation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Furthermore, LCC is a point-based approach [22] and requires additional interpolation and linear  combination steps after the decoding takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Recall that the workers in the CMIM must compute blocks of �  A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Once they complete their computations, they encode them  by computing a linear combination with coefficients determined by a sparsest-balanced MDS generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Referring to the  advantages claimed for CMIM in Section I, working with MDS generator matrices allows us to meet points (i) and (ii), while BRS  generator matrices also help us satisfy (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Once the recovery threshold is met, the coordinator can recover the approximation  �  A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The structure of sparsest-balanced generator matrices is also leveraged to optimally allocate tasks to the workers, while  linear encoding is what allows minimal communication load from the workers to the master.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Security against eavesdroppers is  guaranteed by encoding the local data through a modified Lagrange interpolation polynomial, before it is shared by the clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  This CMIM also extends to approximating A† [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' PRELIMINARY BACKGROUND  The set of N ×N non-singular matrices is denoted by GLN(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Recall that A ∈ GLN(R) has a unique inverse A−1, such that  AA−1 = A−1A = IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The simplest way of computing A−1 is by performing Gaussian elimination on  �  A|IN  �  , which gives  �  IN  ��A−1] in O(N 3) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In Algorithm 1, we approximate A−1 column-by-column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We denote the ith row and column  of A respectively by A(i) and A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The condition number of A is κ2 = ∥A∥2∥A−1∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For I an index subset of the rows of a  matrix M, the matrix consisting only of the rows indexed by I, is denoted by MI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In the proposed algorithm we approximate N instances of the least squares minimization problem  θ⋆  ls = arg min  θ∈RM  �  ∥Aθ − y∥2  2  �  (1)  4  for A ∈ RN×M and y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In many applications N ≫ M, where the rows represent the feature vectors of a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This has  the closed-form solution θ⋆  ls = A†y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Computing A† to solve (1) is intractable for large M, as it requires computing the inverse of A⊤A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Instead, we use gradient  methods to get approximate solutions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' SD or CG, which require less operations, and can be done distributively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' One could use  second-order methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Newton–Raphson, Gauss-Newton, Quasi-Newton, BFGS, or Krylov subspace methods instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This  would be worthwhile future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  When considering a minimization problem with a convex differentiable objective function ψ: Θ → R over an open constrained  set Θ ⊆ RM, as in (1), the SD procedure selects an initial θ[0] ∈ Θ, and then updates θ according to:  θ[t+1] = θ[t] − ξt · ∇θψ(θ[t])  for t = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  until a termination criterion is met, for ξt the step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The CG method is the most used and prominent iterative procedure for  numerically solving systems of positive-definite equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Our proposed coding scheme is defined over the finite field of q elements, Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We denote its cyclic multiplicative subgroup  by F×  q = Fq\\{0Fq}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For implementation purposes, we identify finite fields with their realization in C as a subgroup of the circle  group, since we assume our data is over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' All operations can therefore be carried out over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Specifically, for β ∈ F×  q a generator,  we identify βj with e2πij/q, and 0Fq with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The set of integers between 1 and ν is denoted by Nν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' BALANCED REED-SOLOMON CODES  A Reed-Solomon code RSq[n, k] over Fq for q > n > k, is the encoding of polynomials of degree at most k − 1, for k the  message length and n the code length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' It represents our message over the defining set of points A = {αi}n  i=1 ⊂ Fq  RSq[n, k] =  ��  f(α1), f(α2), · · · , f(αn)  � ���  f(X) ∈ Fq[X] of degree ⩽ k − 1  �  where αi = αi, for α a primitive root of Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, each αi is distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A natural interpretation of RSq[n, k] is through its encoding  map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Each message (m0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=', mk−1) ∈ Fk  q is interpreted as f(x) = �k−1  i=0 mixi ∈ Fq[x], and f is evaluated at each point of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  From this, RSq[n, k] can be defined through the generator matrix  G =  �  �  �  �  �  1  α1  α2  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  αk−1  1  1  α2  α2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  αk−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  1  αn  α2  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  αk−1  n  �  �  �  �  � ∈ Fn×k  q  ,  thus, RS codes are linear codes over Fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Furthermore, they attain the Singleton bound, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' d = n − k + 1, where d is the code’s  distance, which implies that they are MDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Balanced Reed-Solomon codes [15], [16] are a family of linear MDS error-correcting codes with generator matrices G ∈ Fn×k  q  that are:  sparsest: each column has the least possible number of nonzero entries  balanced: each row contains the same number of nonzero entries  for the given code parameters k and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The design of these generators are suitable for our purposes, as:  1) we have balanced loads across homogeneous workers,  2) sparse generator matrices reduce the computation tasks across the network,  3) the MDS property permits an efficient decoding step,  4) linear codes produce a compressed representation of the encoded blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Balanced Reed-Solomon Codes for CC  In the proposed CMIM, we leverage BRS generator matrices to approximate A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For simplicity, we will consider the case  where d = s + 1 = nw  k is a positive integer2, for n the number of workers and s the number of stragglers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Furthermore, d is  the distance of the code and ∥G(j)∥0 = d for all j ∈ Nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' ∥G(i)∥0 = w for all i ∈ Nn, and d > w since n > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For decoding  purposes, we require that at least k = n − s workers respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Consequently, d = s + 1 implies that n − d = k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For simplicity,  we also assume d ⩾ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In our setting, each column of G corresponds to a computation task of �  A−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' which we will denote by  ˆ  Ai, and each row corresponds to a worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  2The case where nw  k  ∈ Q+\\Z+ is analysed in [8], and also applies to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We restrict our discussion to the case where nw  k  ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  5  Our choice of such a generator matrix G ∈ Fn×k  q  , solves  arg  min  G∈Fn×k  q  �  nnzr(G)  �  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ∥G(i)∥0 = w, ∀i ∈ Nn  ∥G(j)∥0 = d, ∀i ∈ Nk  rank(GI) = k, ∀I : |I| = k  (2)  which determines an optimal task allocation among the workers of the proposed CMIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Under the above assumptions, the entries of the generator matrix of a BRSq[n, k] code meet the following:  each column is sparsest, with exactly d nonzero entries  each row is balanced, with w = dk  n nonzero entries  where d equals to the number of workers who are tasked to compute each block, and w is the number of blocks that are computed  by each worker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Each column G(j) corresponds to a polynomial pj(x), whose entries are the evaluation of the polynomial we define in (3) at  each of the points of the defining set A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Gij = pj(αi) for (i, j) ∈ Nn × Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To construct the polynomials {pj(x)}k  j=1, for  which deg(pj) ⩽ nnzr(G(j)) = n − d = k − 1, we first need to determine a sparsest and balanced mask matrix M ∈ {0, 1}n×k,  which is ρ-sparse for ρ = d  n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' nnzr(G) = ρnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We use the construction from [8], though it is fairly easy to construct more  general such matrices, by using the Gale-Ryser Theorem [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Furthermore, deterministic constructions resemble generator  matrices of cyclic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  For our purposes we use B as our defining set of points, where each point corresponds to the worker with the same index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The  objective now is to devise the polynomials pj(x), for which pj(βi) = 0 if and only if Mij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore:  (I) Mij = 0  =⇒  (x − βi) | pj(x)  (II) Mij ̸= 0  =⇒  pj(βi) ∈ F×  q  for all pairs (i, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The construction of BRS[n, k]q from [15] is based on what the authors called scaled polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Below, we summarize the  polynomial construction based on Lagrange interpolation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We then prove a simple but important result that allows us to  efficiently perform the decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The univariate polynomials corresponding to each column G(j), are defined as:  pj(x) :=  �  i:Mij=0  � x − βi  βj − βi  �  =  k  �  ι=1  pj,ι · xι−1 ∈ Fq[x]  (3)  which satisfy (I) and (II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By the BCH bound [25, Chapter 9], it follows that deg(pj) ⩾ n − d = k − 1 for all j ∈ Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Since  each pj(x) is the product of n − d monomials, we conclude that the bound on the degree is satisfied and met with equality, hence  pj,ι ∈ F×  q for all coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  By construction, both G and GI are decomposable into a Vandermonde matrix H ∈ Bn×k and a matrix comprised of the  polynomial coefficients H ∈ (F×  q )k×k [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Specifically, G = HP where Hij = βj−1  i  = βi(j−1) and Pij = pj,i are the  coefficients from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This can be interpreted as P(j) defining the polynomial pj(x), and H(i) is comprised of the first k positive  powers of βi in ascending order, therefore  pj(βi) =  k  �  ι=1  pj,ι · βι−1  i  = ⟨H(i), P(j)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The following lemmas will help us respectively establish in our CC setting the efficiency of our decoding step and the optimality  of the allocated tasks to the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For Lemma 1, recall that efficient matrix multiplication algorithms have complexity O(N ω),  for ω < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='373 the matrix multiplication exponent [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The restriction GI ∈ Fk×k  q  of G to any of its k rows indexed by I ⊊ Nn, is an invertible matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Moreover, its inverse  can be computed online in O(k2 + kω) operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The matrices H and P are of size n × k and k × k respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The restricted matrix GI is then equal to HIP, where  HI ∈ Fk×k  q  is a square Vandermonde matrix, which is invertible in O(k2) time [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Specifically  HI =  �  �  �  �  �  1  βI1  β2  I1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  βk−1  I1  1  βI2  β2  I2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  βk−1  I2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  1  βIk  β2  Ik  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  βk−1  Ik  �  �  �  �  � ∈ Fk×k  q  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  6  It follows that  det(HI) =  �  {i<j}⊆I  (βj − βi)  which is nonzero, since β is primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, HI is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By [15, Lemma 1] and the BCH bound, we conclude that P is  also invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, GI is invertible for any set I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Note that the inverse of P can computed a priori by the master before we deploy our CCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, computing G−1  I  online  with knowledge of P−1, requires an inversion of HI which takes O(k2) operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' and then multiplying it by P−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Thus, it  requires O(k2 + kω) operations in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The generator matrix G ∈ Fn×k  q  of a BRSq[n, k] MDS code defined by the polynomials pj(x) of (3), solves the  optimization problem (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The first two constraints are satisfied by the definition of G, which meets the sparsest and balanced constraints with  equality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' for the given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The last constraint is implied by the MDS theorem, which states that every set of k rows of G  is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The sparsity constraints of (2) imply that nnzr(G) ⩾ max{nw, kd}, and for our parameters we have nw = kd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This condition  is met with equality for the chosen G, as  nnzr(G) =  �  j∈Nk  nnzr(G(j))  =  �  j∈Nk  #  �  pj(βi) ̸= 0 : βi ∈ B  �  =  �  j∈Nk  n −  �  i : Mij = 0  �  =  �  j∈Nk  n − (n − d)  = kd  and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' INVERSE APPROXIMATION ALGORITHM  Our goal is to estimate A−1 =  �  b1 · · · bN  �  , for A a square matrix of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A key property to note is  AA−1 = A  �  b1 · · · bN  �  =  �  Ab1 · · · AbN  �  = IN  which implies that Abi = ei for all i ∈ NN, where ei are the standard basis column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Assume for now that we use any  black-box least squares solver to estimate  ˆbi ≈ arg min  b∈RN  �  fi(b) := ∥Ab − ei∥2  2  �  (4)  which we call N times, to recover �  A−1 :=  �ˆb1 · · · ˆbN  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This approach may be viewed as approximating  �  A−1 ≈ arg  min  B∈RN×N  �  ∥AB − IN∥2  F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Alternatively, one could estimate the rows of A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Algorithm 1 shows how this can be performed by a single server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Algorithm 1: Estimating A−1  Input: A ∈ GLN(R)  for i=1 to N do  approximate ˆbi ≈ arg minb∈RN  �  ∥Ab − ei∥2  2  �  end  return �  A−1 ←  �ˆb1 · · · ˆbN  �  In the case where SD is used to approximate ˆbi from (4), the overall operation count is O(TiN 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' for Ti the total number of  descent iterations used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' An upper bound on the number of iterations can be determined by the underlying termination criterion,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' the criterion fi(ˆb[t]) − fi(b⋆  ls) ⩽ ϵ is guaranteed to be satisfied after T = O(log(1/ϵ)) iterations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The overall error of  �  A−1 may be quantified as  7  errℓ2( �  A−1) := ∥ �  A−1 − A−1∥2  errF ( �  A−1) := ∥ �  A−1 − A−1∥F  errrF ( �  A−1) := ∥ �  A−1−A−1∥F  ∥A−1∥F  =  N  �  i=1  ∥Aˆbi−ei∥2  ∥A−1∥F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  To approximate A−1 distributively, each of the n servers are asked to estimate τ-many ˆbi’s in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' When using SD, the  worst-case runtime by the workers is O(τTmaxN 2), for Tmax the maximum number of iterations of SD among the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' If CG  is used, each worker needs no more than a total of Nτ CG steps to exactly compute its task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' O(τNκ2) operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' as each  instance of (4) is expected to converge in O(κ2) iterations, which is the worst case runtime [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In order to bound errrF ( �  A−1) = ∥ �  A−1−A−1∥F  ∥A−1∥F  , we first upper bound the numerator and then lower bound the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Since ∥A−1 − �  A−1∥2  F = �N  i=1 ∥A−1ei − ˆbi∥2  2, bounding the numerator reduces to bounding ∥A−1ei − ˆbi∥2  2 for all i ∈ NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  This is straightforward  ∥A−1ei − ˆbi∥2  2  ♦  ⩽ 2  �  ∥A−1ei∥2  2 + ∥ˆbi∥2  2  �  $  ⩽ 2  �  ∥A−1∥2  2 · ∥ei∥2  2 + ∥ˆbi∥2  2  �  = 2  �  1/σmin(A)2 + ∥ˆbi∥2  2  �  (5)  where in ♦ we use the fact that ∥u − v∥2  2 ⩽ 2(∥u∥2  2 + ∥v∥2  2), and in $ the submultiplicativity of the ℓ2-norm is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For the  denominator, by the definition of the Frobenius norm  ∥A−1∥2  F =  N  �  i=1  1  σi(A)2 ⩾  N  σmax(A)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  (6)  By combining (5) and (6) we get  errrF ( �  A−1) ⩽  √  2  �  N/σmin(A)2 + �N  i=1 ∥ˆbi∥2  2  N/σmax(A)2  �1/2  =  √  2  �  κ2  2 + σmax(A)2  N N  �  i=1  ∥ˆbi∥2  2  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  This is an additive error bound in terms of the problem’s condition number, which also shows a dependency on the estimates  {ˆbi}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Propositions 3 and 4 give error bounds when using SD and CG as the subroutine of Algorithm 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For A ∈ GLN(R), we have errF ( �  A−1) ⩽  ϵ√  N/2  σmin(A)2 and errrF ( �  A−1) ⩽  ϵ√  N/2  σmin(A), when using SD to solve (4) with  termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Recall that for a strongly-convex function with strong-convexity parameter µ, we have the following optimization gap [28,  Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='2]  fi(b) − fi(b⋆  ls) ⩽ 1  2µ · ∥∇fi(b)∥2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  (7)  For A ∈ GLN(R) in (4), the constant is µ = σmin(A)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By fixing ϵ =  �  2σmin(A)2η, we have η = 1  2 ·  �  ϵ  σmin(A)  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Thus, by (7)  and our termination criterion:  ∥∇fi(b)∥2 ⩽  �  2σmin(A)2η  =⇒  fi(b) − fi(b⋆  ls) ⩽ η ,  so when solving (4) we get  fi(b) − fi(b⋆  ls) = fi(b) − 0 = ∥Aˆbi − ei∥2  2 ,  hence  ∥Aˆbi − ei∥2  2 ⩽ 1  2 ·  �  ϵ  σmin(A)  �2  (8)  8  for all i ∈ NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We want an upper bound for each summand ∥A−1ei − ˆbi∥2  2 of the numerator of errrF ( �  A−1)2:  ∥A−1ei − ˆbi∥2  2 = ∥A−1(ei − Aˆbi)∥2  2  ⩽ ∥A−1∥2  2 · ∥ei − Aˆbi∥2  2  ♯  ⩽ ∥A−1∥2  2 · 1  2 ·  �  ϵ  σmin(A)  �2  (9)  =  ϵ2  2σmin(A)4  (10)  where ♯ follows from (8), thus errF ( �  A−1)2 ⩽  Nϵ2  2σmin(A)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Substituting (9) into the definition of errrF ( �  A−1) gives us  errrF ( �  A−1)2 ⩽ ∥A−1∥2  2  ∥A−1∥2  F  N  2 ·  �  ϵ  σmin(A)  �2 ‡  ⩽  Nϵ2/2  σmin(A)2  where ‡ follows from the fact that ∥A−1∥2  2 ⩽ ∥A−1∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  In the experiments of Subsection IV-A, we verify that Proposition 3 holds for Gaussian random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The dependence on  1/σmin(A) is an artifact of using gradient methods to solve the underlying problems (4), since the error will be multiplied by  ∥A−1∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In theory, this can be annihilated if one runs the algorithm on pA for p ≈ 1/σmin(A), followed by multiplication of  the final result by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is a way of preconditioning SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In practice, the scalar p should not be selected to be much larger than  1/σmin(A), as it could result in �  A−1 ≈ 0N×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Assume Algorithm 1 uses CG to solve (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, in O  �  N√κ2 ln(1/ϵ)  �  iterations, we have errF ( �  A−1) ⩽ Nϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Moreover, if A⊤A has ˜N distinct eigenvalues, it converges in at most ˜NN steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By [29, Section 10] and [31, Section 2], we know that for each subroutine (4) of Algorithm 3, CG requires at most  O(√κ2 ln(1/ϵ)) iterations in order to attain an ϵ-optimal point, for each ˆbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, considering all approximate columns {ˆbi}N  i=1,  we conclude that the total error in terms of the Frobenius norm of �  A−1, is at most Nϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Recall that in order to solve (1) with the CG method in the case where A is neither symmetric, positive-definite, nor square, we  apply the CG iteration to the normal equations  A⊤Ay = A⊤θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  This follows by setting the derivative of (1) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In our scenario, we are assuming that A ∈ GLN(R), hence A⊤A is full-rank  and symmetric, thus CG in its simplest form can be used to solve the minimization problems of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By [30, Theorem  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='4], it follows that each instance of (4) converges in at most ˜N steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  Even though Proposition 4 guarantees convergence in at most ˜NN steps, it does not assume floating-point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore,  this does not hold in practical settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Our experiments though show that after significantly less steps, we achieve approximations  of negligible error, which is sufficient for ML and FL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Numerical Experiments  The accuracy of the proposed algorithms was tested on randomly generated matrices, using both SD and CG for the subroutine  optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The depicted results are averages of 20 runs, with termination criteria ∥∇fi(b[t])∥2 ⩽ ϵ for SD and  ∥b[t]  i − b[t−1]  i  ∥2 ⩽ ϵ for CG, for the given ϵ accuracy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We considered A ∈ R100×100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The error subscripts represent  A = {ℓ2, F, rF}, N = {ℓ2, F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We note that significantly fewer iterations took place when CG was used for the same ϵ, though  this depends heavily on the choice of the step-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The errors observed in the case of CG, are due to floating-point arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Therefore, there is a trade-off between accuracy and speed when using SD vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Average �  A−1 errors, for A ∼ 50 · N(0, 1) — SD  ϵ  10−1  10−2  10−3  10−4  10−5  errA  O(10−2)  O(10−5)  O(10−7)  O(10−9)  O(10−12)  Average �  A−1 errors, for A ∼ 50 · N(0, 1) — CG  ϵ  10−3  10−4  10−5  10−6  10−7  errN  O(10−3)  O(10−5)  O(10−8)  O(10−11) O(10−12)  errrF  O(10−3)  O(10−5)  O(10−7)  O(10−10) O(10−12)  We utilized Algorithm 1 in Newton’s method, for classifying images of four and nine from MNIST, by solving a regularized  logistic regression minimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For Algorithm 1, we used CG with a fixed number of iteration per column estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  It is clear from Figure 4 that we require no more than 18 iterations per column estimate, for N = 785, to attain the optimal  classification rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' With more than 18 CG iterations, the same classification rate was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  9  14  15  16  17  18  19  CG iterarions per column  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='45  Classification error %  Classification Error  inversion with CG  exact inversion  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' MNIST classification error, where Algorithm 1 is used in Newton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In red, we depict the error when exact inversion was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' FEDERATED CODED MATRIX INVERSION  In this section, we describe the proposed CMIM (also presented in [1]) which makes Algorithm 1 resilient to stragglers, and  show how it can be applied to the FL scenario described in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The CMIM workflow is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Our FL-scheme can be broken up in to four phases: (a) the coordinator shares elements β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' H of a finite field with all the clients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  (b) the clients each generate a pseudorandom permutation (PRP) σι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' encrypt their corresponding data block Aι through a matrix  polynomial fι(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' and broadcast {fι(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' σ−1  ι  } to the other clients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' (c) the clients recover A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' compute and encode their assigned  task Wι,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' which is communicated to the coordinator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' (d) the coordinator decodes once sufficiently many servers respond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' It is also  possible that β, H are determined collectively by the clients, or by a single client, which makes the data sharing secure against a  curious and dishonest coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Algorithmic workflow of the CMIM, as proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The master shares f(x), an encoding analogous to (12), along with β, {η−1  j  }k  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The workers  then recover A, compute their assigned tasks, and encode them according to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Once k encodings Wι are sent back, �  A−1 can be recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In our proposed FL approach, we assume there is a trustworthy coordinator who shares certain parameters to each of the k  clients which constitute the network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' hospitals in a health care network, each of which are comprised of multiple servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  What we present works for the case where the clients have local datasets of different sizes, {Ni}k  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This would result in the  encoding functions fι(x) having different degrees, or their matrix coefficients being of a different size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In our setting we assume  the servers are homogeneous, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' they have the same computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, equal computational loads are assigned to  each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In order to keep the notation and size of the communication loads consistent, we assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' that Aι ∈ RN×T  for all ι ∈ Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' If this is not the case, before fι(x) are determined, the clients could perform a data exchange phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' [13]), so  that Ni = Nj for all i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By this, it follows that the number of blocks does not have to be equal to the number of clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The  example we describe, is simply a motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A flowchart of our approach is presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Moreover, in the case where M > N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' for M = �k  i=1 Ni, we can select a subset of features and/or samples, so that the resulting  data matrix we consider is square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This can be interpreted as using the surrogate ˜A = SA, where S ∈ RN×M is an appropriate  (sparse) sketching matrix for matrix inversion [32], which the workers agree on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  First, in V-A we argue why all of A needs to be known by each of the workers, in order to recover entries or columns of its  inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, in V-B we focus on phases (a) and (b), where we utilize Lagrange interpolation to securely share A among the  workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We discuss the computation tasks the workers are requested to compute, which are blocks of �  A−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' and collectively  correspond to the subroutine problems of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In V-C we focus on (c) and (d), where we show how the servers encode  their computations, and describe the coordinator’s decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Optimality of BRS generator matrices in terms of the encoded  communication loads is established in V-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  When assuming floating-point arithmetic, our approach introduces no numerical nor approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The errors are a  consequence of using iterative solvers to estimate (4), which we utilize to linearly separate the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, if the  workers can recover the optimal solutions to the underlying minimization problems, our scheme would be exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  >A-i =[Ai .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ak  W1  X  Wn  X  f(×)  f(x)  f(x)10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Flowchart of our proposal, where k = ni = 4 for all i ∈ N4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Knowledge of A is necessary  A bottleneck when computing the inverse of a matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' or estimating its columns, is that the entire matrix needs to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A single change in the matrix’s entries may result in a non-singular matrix, which conveys how sensitive Gaussian elimination is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Such problems are extensively studied in conditioning and stability of numerical analysis [30], and in perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is  not a focus of our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In the case where only one column is not known, one can determine the subspace in which the missing column lies, but without  the knowledge of at least one entry of that column, it would be impossible to recover that column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Even with such an approach or  a matrix completion algorithm, the entire A is determined before we proceed to inverting A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' or performing linear regression to  approximate Ab = ei as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Another example, relating to our FL set up, is the case where one of the blocks is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This could lead to drastic  miscalculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In the following example, we consider n = k = 2 and N = 4, where the second server sends a different  block, which are indicated by a different color and font:  A1 =  �  �  �  6  2  2  5  0  −1  2  0  −5  6  1  3  5  −3  4  3  �  �  �  A2 =  �  �  �  6  2  −1  −3  0  −1  5  6  −5  6  3  −2  5  −3  1  6  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  It follows that ∥A−1  1 ∥F ≈ 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='45, ∥A−1  2 ∥F ≈ 1, and ∥A−1  1  − A−1  2 ∥0 = 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' no entries of A−1  1  and A−1  2  are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Furthermore, by the data processing inequality [33, Corollary pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='35], the above imply that no less than N 2 information symbols  can be known by each server, while hoping to approximate a column of A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, all clients need full knowledge of each others  information, and cannot communicate less than NT symbols to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is a consequence of the fact that a dense vector is  not recoverable from underdetermined linear measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' They can however send an encoded version of their respective block  Aι ∈ RN×T to the other clients consisting of NT symbols, determined by a modified Lagrange polynomial, which guarantees  security against eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Similar cryptographic protocols date back to the SSS algorithm [34], which is also based on RS codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This idea has extensively  been exploited in LCC [5], yet differs from our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Phases (a), (b) — Data Encryption and Sharing  Let k, γ ∈ Z+ be factors of N and T respectively, so that T = N  k and Γ = T  γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='3 The coordinator constructs a set of distinct  interpolation points B = {βj}n  j=1 ⊊ F×  q , for q > n ⩾ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='4 To construct this set, it suffices to sample β ∈ F×  q ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' any one of the  φ(q − 1) primitive roots of Fq (φ is Euler’s totient function), which is a generator of the multiplicative group (F×  q , ·), and define  3If γ ∤ T, append 0T ×1 to the end of the first ˜γ = T(modγ) blocks which are each comprised of ˜Γ = ⌊ T  γ ⌋ columns of Aι, while the remaining γ − ˜γ  blocks are comprised of ˜Γ + 1 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Now, each block is of size T × (˜Γ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  4For the encodings of the Aι’s, γ points suffice, and we only need to require q > γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We select B of cardinality n and require q > n ⩾ γ, in order to reuse B  in our CCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  β,H  β,H  B,H  β,H  A  A  88  fi(x),q1  f2(x),02  Clients share their corre  sponding fi(×) and o,-1  Clients recover A = Ai A2 A3 A4  Servers carry out computations, and each  client sends back W, to the coordinator  =  Once the threshold is met,11  each point as βj = βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, a random multiset H = {ηj}γ  j=1 ∈ 2F×  q of size γ is generated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' repetitions in H are allowed,  which will be used to remove the structure of the Lagrange coefficients, as the adversaries could exploit their structure to reveal  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The element β and set H, are broadcasted securely to all the workers through a public-key cryptosystem, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' RSA or McEliece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Matrices Aι are partitioned into γ blocks  Aι =  �  A1  ι · · · Aγ  ι  �  where Ai  ι ∈ RN×Γ, ∀i ∈ Nγ,  (11)  and each client generates a PRP σι ∈ Sγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The blocks {Aι}k  ι=1 are encrypted locally through the univariate polynomials  fι(x) =  γ  �  j=1  Aj  ι · ησι(j)  �  ��  l̸=j  x − βl  βj − βl  �  �  (12)  for which fι(βj) = ησι(j)Aj  ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The clients then broadcast {fι(x), σ−1  ι  } to each other, and their servers can then recover all Aι’s as follows:  Aι =  �  ησ−1  ι  (1)f(β1) · · · ησ−1  ι  (γ)f(βγ)  �  ∈ RN×T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  (13)  The coefficients of fι(x) are comprised of NΓ symbols, thus, each polynomial consists of a total of NT symbols, which is the  minimum number of symbols needed to be communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The PRP σι is generated locally by the clients, to ensure that each  fι(x) differs by more than just the matrix partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  We assume Kerckhoffs’ principle, which states that everyone has knowledge of the system, including the messages fι(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For  the proposed CMIM, as long as {β, H} and σ−1  ι  are securely communicated, even if fι(x) is revealed, the block Aι is secure  against polynomial-bounded adversaries (this is the security level assumed by the cryptosystems used for the communication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The encryptions of Aι through fι(x), are as secure against eavesdroppers as the public-key cryptosystems which  are used when broadcasting {β, H} and σ−1  ι  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To recover Aι, an adversary needs to intercept both communications, and break  both cryptosystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We prove this by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Assume that an adversary was able to reverse the encoding fι(x) of Aι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This implies that  he was able to reveal β and σι(H) := {ησι(j)}γ  j=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The only way to reveal these elements, is he was able to both intercept and  decipher the public-key cryptosystem used by the coordinator, which contradicts the security of the cryptosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  In order to invert the multiplications of σι(H) for each of the evaluations of fι(x), both H and σ−1  ι  need to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To do so,  the adversary needs to intercept both the communication between the coordinator and the clients, and the communication between  the clients, as well as breaking both the cryptosystems used to securely carry out these communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Phases (c), (d) — Computations, Encoding and Decoding  At this stage, the workers have knowledge of everything they need in order to recover A, before they carry out their computation  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By (13), the recovery is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  For Algorithm 1, any CCM in which the workers compute an encoding of partitions of the resulting computation E =  �  E1 · · · Ek  �  could be utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' It is crucial that the encoding takes place on the computed tasks {Ei}k  i=1 in the scheme, and  not the assigned data or partitions of the matrices that are being computed over (such CMM leverage the linearity of matrix  multiplication), otherwise the algorithm could potentially not return the correct approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This also means that utilizing such  encryption approaches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' [5]) for guaranteeing security against the workers, is not an option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We face these restrictions due to  the fact that matrix inversion is a non-linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The computation tasks Ei correspond to a partitioning �  A−1 =  � ˆ  A1 · · ·  ˆ  Ak  �  , of our approximation from Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We  propose a linear encoding of the computed blocks { ˆ  Ai}k  i=1 based on generators satisfying (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Along with the proposed decoding  step, we have a MDS-based CCM for matrix inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  We consider the same parameters as in V-B, in order to reuse B in the proposed CMIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Each ˆ  Ai is comprised of T distinct but  consecutive approximations of (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ˆ  Ai =  �ˆb(i−1)T +1 · · · ˆbiT  �  ∈ RN×T  ∀i ∈ Nk,  which could also be approximated by iteratively solving  ˆ  Ai ≈ arg min  B∈RN×T  ���AB −  �  e(i−1)T +1 · · · eiT  ���2  F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Without loss of generality, we assume that the workers use the same algorithms and parameters for estimating the columns  {ˆbi}N  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Therefore, workers allocated the same tasks are expected to get equal approximations in the same amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  12  For our CCM, we leverage BRS generator matrices for both the encoding and decoding steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We adapt the GC framework,  so we need an analogous condition to a⊤  I G = ⃗1 for the CMIM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' in order to invoke Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The condition we require is  ˜DI ˜G = IN, for an encoding-decoding pair ( ˜G, ˜DI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  From our discussion on BRS codes in III-A, we set ˜G = IT ⊗ G and ˜DI = IT ⊗ (GI)−1 for any given set of k responsive  servers indexed by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The index set of blocks requested from the ιth worker to compute is denoted by Jι, and has cardinality w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The workers’ encoding steps correspond to  ˜G · ( �  A−1)⊤ = (IT ⊗ G) ·  �  ��  ˆ  A⊤  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ˆ  A⊤  k  �  �� =  �  �  �  �  �  �  �  j∈J1  pj(β1) · ˆ  A⊤  j  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  �  j∈Jn  pj(βn) · ˆ  A⊤  j  �  �  �  �  �  �  (14)  which are carried out locally, once they have computed their assigned tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We denote the encoding of the ιth worker by Wι ∈  CT ×N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Wι = �  j∈Jι pj(βι) · ˆ  A⊤  j , which are sent to the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The received encoded computations by any distinct k  servers indexed by I, constitute ˜GI · ( �  A−1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Lemma 1 implies that as long as k workers respond, the approximation �  A−1 is recoverable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Moreover, the decoding step  reduces to a matrix multiplication of k × k matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Applying HI to a square matrix can be done in O(k2 log k), through the  FFT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The prevailing computation in our decoding, is applying P−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The decoding step is  ˜DI ·  �  ˜GI · ( �  A−1)⊤�  =  �  IT ⊗ (GI)−1� �  IT ⊗ GI  �  ( �  A−1)⊤  = (IT · IT ) ⊗  �  (GI)−1 · GI  �  ( �  A−1)⊤  = IT ⊗ Ik · ( �  A−1)⊤  = ( �  A−1)⊤  and our scheme is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The above CCM therefore has a linear encoding done locally by the servers (14), is MDS since s = d − 1, and its decoding  step reduces to computing and applying G−1  I  (Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The security of the encodings rely on the secrecy of B, which were  sent from the coordinator to the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For an additional security layer, the interpolation points of B could instead be defined as  βj = βπ(j), for π ∈ Sn a PRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In this case, π−1 would also need to be securely broadcasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  8  106  0  20  40  60  80  100  120  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='5  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='5  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='5  7  0  20  40  60  80  100  120  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Comparison of decoding complexity, when naive matrix inversion is used (so O(k3)) compared to the decoding step implied by Lemma 1, for n = 200  and varying s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We also provide a logarithmic scale comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  With the above framework, any sparsest-balanced generator MDS matrix [23] would suffice, as long as it satisfies the MDS  theorem [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By Lemma 1, if we set k = Ω(  √  N) (similar to [7]), the decoding step could then be done in O(N ω/2) = o(N 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='187),  which is close to linear in terms of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Let G ∈ Fn×k be a generator matrix of any MDS code over F, for which ∥G(j)∥0 = n − k + 1 and ∥G(i)∥0 = w  for all (i, j) ∈ Nn × Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By utilizing Algorithm 1, we can devise a linear MDS coded matrix inversion scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' through the  encoding-decoding pair ( ˜G, ˜DI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The encoding coefficients applied locally by each of the n workers correspond to a row of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The encodings of all the  workers then correspond to ˜G · ( �  A−1)⊤, for ˜G = IT ⊗ G, as in (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Consider any set of responsive workers I of size k, whose  encodings constitute ˜GI ·( �  A−1)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By the MDS theorem, GI is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, the decoding step reduces to inverting GI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ˜DI = IT ⊗ (GI)−1, and is performed online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  Constructions based on cyclic MDS codes, which have been used to devise GC schemes [36], can also be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' These  encoding matrices are not sparsest-balanced, which makes them suitable when considering heterogeneous workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Any cyclic [n, k] MDS code C over F ∈ {R, C} can be used to devise a coded matrix inversion encoding-decoding  pair ( ˜G, ˜DI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Consider a cyclic [n, n − s] MDS code C over F ∈ {R, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Recall that from our assumptions, we have s = n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By [36,  Lemma 8], there exists a codeword g1 ∈ C of support d = s + 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' ∥g1∥0 = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Since C is cyclic, it follows that the cyclic shifts  of g1 also lie in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Denote the n − 1 consecutive cyclic shifts of g1 by {gi}n  i=2 ⊊ C ⊊ F1×n, which are all distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Define the  cyclic matrix  ¯G :=  �  �  |  |  |  g⊤  1  g⊤  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  g⊤  n  |  |  |  �  � ∈ Fn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Since ∥gi∥0 = d and gi is a cyclic shift of gi−1 for all i > 1, it follows that ∥ ¯G(i)∥0 = ∥ ¯G(j)∥0 = d for all i, j ∈ Nn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' ¯G  is sparsest and balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' If we erase any s = n − k columns of ¯G, we get G ∈ Fn×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By erasing arbitrary columns of ¯G, the  resulting G is not balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' we have ∥G(i)∥0 ̸= ∥G(j)∥0 for some pairs i, j ∈ Nn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Similar to our construction based on BRS  generator matrices, we define the encoding matrix to be ˜G = IT ⊗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The local encodings are then analogous to (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Consider an arbitrary set of k non-straggling workers I ⊊ Nn, and the corresponding matrix GI ∈ Fk×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By [36, Lemma 12,  B4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' ], GI is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The decoding matrix is then ˜DI = IT ⊗ (GI)−1, and the condition ˜DI ˜G = IN is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Optimality of MDS BRS Codes  Under the assumption that k = n − s, by utilizing the BRSq[n, k] generator matrices, we achieved the minimum possible  communication load from the workers to the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' From our discussion in V-A, we cannot hope to receive an encoding  of less than N 2/k symbols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' when we require that k workers respond with the same amount of information symbols in order  to recover �  A−1 ∈ RN×N, unless we make further assumptions on the structure of A and A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Each encoding Wι consists  of NT = N 2/k symbols, so we have achieved the lower bound on the minimum amount of information needed to be sent to  the coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hence, Wι ∈ CT ×N for any sparsest-balance generator MDS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This also holds true for other generator  matrices which can be used in Theorem 6, as the encodings are linear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Proposition 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  We also require the workers to estimate the least possible number of columns for the given recovery threshold k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For our choice  of parameters, the bound of [20, Theorem 1] is met with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' That is, for all i ∈ Nn:  ∥G(i)∥0 = w = k  n · d = k  n · (n − k + 1) ,  which means that for homogeneous workers, we cannot get a sparser generator matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This, along with the requirement that GI  should be invertible for all possible I, are what we considered in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this paper, we addressed the problem of approximate computation of the inverse of a matrix distributively in a FL setting,  under the possible presence of straggling workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We provided approximation error bounds for our approach, as well as security  and recovery guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We also provided numerical experiments that validated our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  There are several interesting future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' One is looking into the issue of numerical stability of the BRS approach, and  exploring other suitable generator matrices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' circulant permutation and rotation matrices [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Another direction, is leveraging  approximate CCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The techniques of [22], [38] suggest that carefully selecting interpolation points may lead to more efficient  (approximate) schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In terms of coding-theory, it would be interesting to see if it is possible to reduce the complexity of our  decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Specifically, could well-known RS decoding algorithms such as the Berlekamp-Welch algorithm be exploited?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Another important extension is to reduce the communication rounds when computing the pseudoinverse through our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  This depends on the CMM which is being utilized, though using different ones for each of the two multiplications may also be  beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Tribute to Alex Vardy: As this is a special issue dedicated to the memory Alexander Vardy, we mention how this paper relates  to some of his work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Even though Alex had not worked on CC, his contributions to RS codes are immense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A focus of this paper  is to reduce the decoding complexity of the proposed BRS-based CCM, while in [39] it was shown that ML decoding of RS  codes is NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Another highly innovative work of Vardy’s is [40], in which the ‘Parvaresh-Vardy codes’ were introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  and the associated list-decoding algorithm was shown to yield an improvement over the Guruswami–Sudan algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This was  subsequently improved by Guruswami and Rudra [41], whose techniques were exploited in [42] to introduce list-decoding in CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  APPENDIX A  ADDITIONAL MATERIAL AND BACKGROUND  In this appendix, we include material and background which was used in our derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' First, we recall what an ϵ-optimal  solution/point is, which was used in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Next, we state the MDS Theorem and the BCH Bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We  then give a brief overview of the GC scheme from [8], to show how it differs from our coded matrix inversion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We also  explicitly give their construction of a balanced mask matrix M ∈ {0, 1}n×k, which we use for the construction of the BRS  generator matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Lastly, we illustrate a simple example of the encoding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  14  Definition 8 ( [43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A point ¯x is said to be an ϵ-optimal solution/point to a minimization problem with objective function f(x),  if for any x, it holds that f(x) ⩾ f(¯x) − ϵ, where ϵ ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' When ϵ = 0, an ϵ-optimal solution is an exact minimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Theorem 9 (MDS Theorem — [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Let C be a linear [n, k, d] code over Fq, with G, H the generator and parity-check matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Then, the following are equivalent:  1) C is a MDS code, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' d = n − k + 1  2) every set of n − k columns of H is linearly independent  3) every set of k columns of G is linearly independent  4) C⊥ is a MDS code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Theorem 10 (BCH Bound — [15], [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Let p(x) ∈ Fq[x]\\{0} with t cyclically consecutive roots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' p(αj+ι) = 0 for all  ι ∈ Nt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, at least t + 1 coefficients of p(x) are nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Algorithm 2: MaskMatrix(n, k, d) [8]  Input: n, k, d ∈ Z+ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' n > d, k and w = kd  n  Output: row-balanced mask matrix M ∈ {0, 1}n×k  M ← 0n×k  for j = 0 to k − 1 do  for i = 0 to d − 1 do  ι ← (i + jd + 1) mod n  Mr,ι ← 1  end  end  return M  Even though this was not pointed out in [8], Algorithm 2 does not always produce a mask matrix of the given parameters when  we select d < n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is why in our work we require d ⩾ n/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The decomposition G = HP is utilized in the GC scheme of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Each column of G corresponds to a partition of the data  whose partial gradient is to be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The polynomials are judiciously constructed in this scheme, such that the constant term  of each polynomial is 1 for all polynomials, thus P(1) = ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By this, the decoding vector a⊤  I is the first row of G−1  I , for which  a⊤  I GI = e⊤  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A direct consequence of this is that a⊤  I BI = e⊤  1 T = T(1) = ⃗1, which is the objective for constructing a GC  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Generator Matrix Example  As an example, consider the case where n = 9, k = 6 and d = 6, thus w = kd  n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Then, Algorithm 2 produces  M =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  1  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  1  0  1  1  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ∈ {0, 1}9×6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  For our CCM, this means that the ith worker computes the blocks indexed by supp(M(i)), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' supp(M(1)) = {1, 2, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We  denote the indices of the respective task allocations by Ji = supp(M(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The entries of the generator matrix G are the evaluations  of the constructed polynomials (3) at each of the evaluation points B = {βi}n  i=1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Gij = pj(αi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This results in:  G =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  p1(β1)  p2(β1)  0  p4(β1)  p5(β1)  0  p1(β2)  p2(β2)  0  p4(β2)  p5(β2)  0  p1(β3)  p2(β3)  0  p4(β3)  p5(β3)  0  p1(β4)  0  p3(β4)  p4(β4)  0  p6(β4)  p1(β5)  0  p3(β5)  p4(β5)  0  p6(β5)  p1(β6)  0  p3(β6)  p4(β6)  0  p6(β6)  0  p2(β7)  p3(β7)  0  p5(β7)  p6(β7)  0  p2(β8)  p3(β8)  0  p5(β8)  p6(β8)  0  p2(β9)  p3(β9)  0  p5(β9)  p6(β9)  �  �  �  �  �  �  �  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  15  APPENDIX B  DISTRIBUTED PSEUDOINVERSE  For full-rank rectangular matrices A ∈ RN×M where N > M, one resorts to the left Moore–Penrose pseudoinverse A† ∈  RM×N, for which A†A = IM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' In Algorithm 3, we present how to approximate the left pseudoinverse of A, by using the fact that  A† = (A⊤A)−1A⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' since A⊤A ∈ GLN(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The right pseudoinverse A† = A⊤(AA⊤)−1 of A ∈ RM×N where M < N,  can be obtained by a modification of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Just like the inverse, the pseudoinverse of a matrix also appears in a variety of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Computing the pseudoinverse of  A ∈ RN×M for N > M is even more cumbersome, as it requires inverting the Gram matrix A⊤A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For this subsection, we  consider a full-rank matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  One could naively attempt to modify Algorithm 1 in order to retrieve A† such that A†A = IM, by approximating the rows  of A†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This would not work, as the underlying optimization problems would not be strictly convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Instead, we use Algorithm  3 to estimate the rows of B−1 := (A⊤A)−1, and then multiply the estimate �  B−1 by A⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This gives us the approximation  �  A† = �  B−1 · A⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The drawback of Algorithm 3 is that it requires two additional matrix multiplications, A⊤A and �  B−1A⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We overcome this  barrier by using a CMM scheme twice, to recover �  A† in a two or three-round communication CC approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' These are discussed  in below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Bounds on errF ( �  A−1) and errrF ( �  A−1) can be established for both algorithms, specific to the black-box least squares algorithm  being utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Algorithm 3: Estimating A†  Input: full-rank A ∈ RN×M where N > M  B ← A⊤A  for i=1 to M do  ˆci = arg minc∈R1×M  �  gi(c) := ∥cB − e⊤  i ∥2  2  �  ˆbi ← ˆci · A⊤  end  return �  A† ←  �  ˆb⊤  1 · · · ˆb⊤  M  �⊤  ▷ �  A†(i) = ˆbi  Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' For full-rank A ∈ RN×M with N > M, we have errF (�  A†) ⩽  √  Mϵ·κ2  √  2σmin(A)3 and errrF (�  A†) ⩽  √  Mϵ·κ2  √  2σmin(A)2 when  using SD to solve the subroutine optimization problems of Algorithm 3, with termination criteria ∥∇gi(c[t])∥2 ⩽ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' From (10), it follows that  ∥B−1ei − ˆc⊤  i ∥2 ⩽  ϵ/  √  2  σmin(B)2 =  ϵ/  √  2  σmin(A)4 =: δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  The above bound implies that for each summand of the Frobenius error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' ∥ˆbi − A†  (i)∥2 = ∥ˆciA⊤ − e⊤  i · B−1A⊤∥2, we have  ∥ˆbi − A†  (i)∥2 ⩽ δ∥A⊤∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Summing the right hand side M times, we get that  errF (�  A†)2 ⩽ M · (δ∥A⊤∥2)2  = Mϵ2 · σmax(A)2  σmin(A)8  = Mϵ2 · κ2  2  σmin(A)6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  By taking the square root, we have shown the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Since 1/σmin(A) = ∥A†∥2 ⩽ ∥A†∥F , it then follows that  errrF (�  A†) = errF (�  A†)  ∥A†∥F  ⩽ errF (�  A†)  ∥A†∥2  =⩽  √  Mϵ · κ2  √  2σmin(A)2 ,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  ■  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Pseudoinverse from Polynomial CMM  One approach to leverage Algorithm 3 in a two-round communication scheme is to first compute B = A⊤A through a CMM  scheme, then share B with all the workers who estimate the rows of �  B−1, and finally use another CMM to locally encode the  16  estimated columns with blocks of A⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' to recover �  A† = �  B−1 · A⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Even though there are only two rounds of communication, the  fact that we have a local encoding by the workers results in a higher communication load overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' An alternative approach which  circumvents this issue, uses three-rounds of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  For this approach, we use the polynomial CMM scheme from [7] twice, along with our coded matrix inversion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' This  CMM has a reduced communication load, and minimal computation is required by the workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' To have a consistent recovery  threshold across our communication rounds, we partition A as in (11) into ¯k = √n − s =  √  k blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Each block is of size  N × ¯T, for ¯T = M  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The encodings from [7] of the partitions {Aj}¯k  j=1 for carefully selected parameters a, b ∈ Z+ and distinct  elements γi ∈ Fq, are  ˜Aa  i =  k  �  j=1  Ajγ(j−1)a  i  and  ˜Ab  i =  k  �  j=1  Ajγ(j−1)b  i  for each worker indexed by i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Thus, each encoding is comprised of N ¯T symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The workers compute the product of their  respective encodings ( ˜Aa  i )⊤ · ˜Ab  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The decoding step corresponds to an interpolation step, which is achievable when ¯k2 = k many  workers respond5, which is the optimal recovery threshold for CMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Any fast polynomial interpolation or RS decoding algorithm  can be used for this step, to recover B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Next, the master shares B with all the workers (from V-A, this is necessary), who are requested to estimate the column-blocks  of �  B−1  �  B−1 =  �  ¯B1 · · · ¯Bk  �  where ¯Bj ∈ RM× ¯T ∀j ∈ Nk  (15)  according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We can then recover �  B−1 by our BRS based scheme, once k workers send their encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  For the final round, we encode �  B−1 as  ˜Ba  i =  k  �  j=1  ¯Bjγ(j−1)a  i  which are sent to the respective workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The workers already have in their possession the encodings ˜Ab  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' We then carry out the  polynomial CMM where each worker is requested to send back ( ˜Ba  i )⊤ · ˜Ab  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' The master server can then recover �  A†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Consider G ∈ Fn×k as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By using any CMM, we can devise a matrix pseudoinverse CCM by utilizing  Algorithm 3, in two-rounds of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' By using polynomial CMM [7], we achieve this with a reduced communication  load and minimal computation, in three-rounds of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  REFERENCES  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Charalambides, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Pilanci, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hero, “Secure Linear MDS Coded Matrix Inversion,” in 2022 58th Annual Allerton Conference on Communication,  Control, and Computing (Allerton), 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [2] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Greenberg and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Sarhan, “Matrix inversion, its interest and application in analysis of data,” Journal of the American Statistical Association, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 54,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 288, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 755–766, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [3] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  USA: Society for Industrial and Applied Mathematics, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Lam, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Pedarsani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Papailiopoulos, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ramchandran, “Speeding up distributed machine learning using codes,” IEEE Transactions on  Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1514–1529, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [5] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Raviv, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kalan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Soltanolkotabi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, “Lagrange Coded Computing: Optimal design for resiliency, security and  privacy,” arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='00939, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Li and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, “Coded Computing,” Foundations and Trends® in Communications and Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 17, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [7] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Maddah-Ali, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, “Polynomial codes: an optimal design for high-dimensional coded matrix multiplication,” in Advances in Neural  Information Processing Systems, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 4403–4413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Halbawi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Azizan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Salehi, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hassibi, “Improving Distributed Gradient Descent Using Reed-Solomon Codes,” in 2018 IEEE International  Symposium on Information Theory (ISIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2027–2031.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dhakal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Prakash, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yona, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Talwar, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Himayat, “Coded Federated Learning,” in 2019 IEEE Globecom Workshops (GC Wkshps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  IEEE, 2019,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Prakash, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dhakal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Akdeniz, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yona, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Talwar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Himayat, “Coded Computing for Low-Latency Federated Learning over  Wireless Edge Networks,” IEEE Journal on Selected Areas in Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 39, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 233–250, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Schlegel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kumar, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Rosnes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Amat, “CodedPaddedFL and CodedSecAgg: Straggler Mitigation and Secure Aggregation in Federated  Learning,” arXiv e-prints, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' arXiv–2112, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kumar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Schlegel, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Rosnes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Amat, “Coding for Straggler Mitigation in Federated Learning,” arXiv preprint arXiv:2109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='15226, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Xhemrishi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Amat, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Rosnes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Wachter-Zeh, “Computational Code-Based Privacy in Coded Federated Learning,” arXiv preprint  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='13798, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ha, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Zhang, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Simeone, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kang, “Coded Federated Computing in Wireless Networks with Straggling Devices and Imperfect CSI,” in 2019 IEEE  International Symposium on Information Theory (ISIT), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2649–2653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Halbawi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Liu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hassibi, “Balanced Reed-Solomon Codes,” in 2016 IEEE International Symposium on Information Theory (ISIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  IEEE, 2016,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 935–939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [16] ——, “Balanced Reed-Solomon Codes for all parameters,” in 2016 IEEE Information Theory Workshop (ITW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  IEEE, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 409–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Charalambides, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Mahdavifar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Hero, “Numerically Stable Binary Gradient Coding,” arXiv preprint arXiv:2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='11449, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Koneˇcn`y, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' McMahan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ramage, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Richtárik, “Federated optimization: Distributed machine learning for on-device intelligence,” arXiv preprint  arXiv:1610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='02527, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  5We select ¯k =  √  k in the partitioning of A in (11) when deploying this CMM, to attain the same recovery threshold as our inversion scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  17  [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Grover, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kar, “Coded distributed computing for inverse problems,” in Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Curran  Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=', 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 709–719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Tandon, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Lei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dimakis, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Karampatziakis, “Gradient coding: Avoiding stragglers in distributed learning,” in International Conference on  Machine Learning, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3368–3376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Soleymani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Mahdavifar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, “Analog Lagrange Coded Computing,” IEEE Journal on Selected Areas in Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 283–295, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Kiani and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Draper, “Successive Approximation Coding for Distributed Matrix Multiplication,” arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='03486, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dau, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Song, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dong, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Yuen, “Balanced Sparsest Generator Matrices for MDS Codes,” in 2013 IEEE International Symposium on Information  Theory, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1889–1893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [24] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Krause, “A Simple Proof of the Gale-Ryser Theorem,” The American Mathematical Monthly, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 103, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 335–337, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [25] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' McEliece, Theory of Information and Coding, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  USA: Cambridge University Press, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Alman and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Williams, “A refined laser method and faster matrix multiplication,” arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='05846, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [27] Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Björck and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Pereyra, “Solution of Vandermonde Systems of Equations,” Mathematics of Computation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 24, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 893–903, 1970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [28] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Boyd and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Vandenberghe, Convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Cambridge university press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Shewchuk, “An Introduction to the Conjugate Gradient Method Without the Agonizing Pain,” 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [30] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Trefethen and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Bau III, Numerical linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Siam, 1997, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Bubeck, “Convex optimization: Algorithms and complexity,” Foundations and Trends® in Machine Learning, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 8, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3-4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 231–357, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Available: http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='1561/2200000050  [32] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Gower, “Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices,” arXiv preprint arXiv:1612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='06013, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [33] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Cover and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  USA: Wiley-Interscience,  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Shamir, “How to Share a Secret,” Communications of the ACM, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 612–613, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [35] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ling and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Xing, Coding Theory: A First Course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  Cambridge University Press, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [36] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Raviv, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Tamo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Tandon, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Dimakis, “Gradient Coding from Cyclic MDS Codes and Expander Graphs,” IEEE Transactions on Information  Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 7475–7489, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ramamoorthy and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Tang, “Numerically stable coded matrix computations via circulant and rotation matrix embeddings,” IEEE Transactions on  Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 68, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2684–2703, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Jeong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Devulapalli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Cadambe, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Calmon, “ϵ-Approximate Coded Matrix Multiplication Is Nearly Twice as Efficient as Exact  Multiplication,” IEEE Journal on Selected Areas in Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 845–854, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Guruswami and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Vardy, “Maximum-Likelihood Decoding of Reed-Solomon Codes is NP-hard,” IEEE Transactions on Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 51,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2249–2256, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [40] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Parvaresh and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Vardy, “Correcting Errors Beyond the Guruswami-Sudan Radius in Polynomial Time,” in 46th Annual IEEE Symposium on Foundations  of Computer Science (FOCS’05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  IEEE, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 285–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [41] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Guruswami and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Rudra, “Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy,” IEEE Transactions on Information  Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 135–150, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Soleymani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Ali, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Mahdavifar, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Avestimehr, “List-decodable coded computing: Breaking the adversarial toleration barrier,” IEEE Journal  on Selected Areas in Information Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 867–878, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  [43] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Bai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Wu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' Zhu, “Sequential Lagrange multiplier condition for ϵ-optimal solution in convex programming,” Optimization, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
+page_content='  669–680, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE1T4oBgHgl3EQf7wX0/content/2301.03539v1.pdf'}
diff --git a/9NE2T4oBgHgl3EQflwcz/content/tmp_files/2301.03991v1.pdf.txt b/9NE2T4oBgHgl3EQflwcz/content/tmp_files/2301.03991v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a699a083eafb4a3d9523c295ac29626359623102
--- /dev/null
+++ b/9NE2T4oBgHgl3EQflwcz/content/tmp_files/2301.03991v1.pdf.txt
@@ -0,0 +1,1075 @@
+Constraining cosmological parameters from
+N-body simulations with Variational Bayesian
+Neural Networks
+H´ector J. Hort´ua 1,2,∗, Luz ´Angela Garc´ıa 3 and Leonardo Casta˜neda C. 4
+1 Grupo Signos, Departamento de Matem´aticas, Universidad el Bosque, Bogot´a,
+Colombia.
+2 Maestr´ıa en Ciencia de Datos, Universidad Escuela Colombiana de Ingenier´ıa
+Julio Garavito Bogot´a, Colombia.
+3 Universidad ECCI. Cra. 19 No. 49-20, Bogot´a, Colombia, C´odigo Postal 111311.
+4Observatorio Astron´omico Nacional, Universidad Nacional de Colombia, Bogot´a,
+Colombia.
+Correspondence*:
+-
+hjhortuao@unal.edu.co
+ABSTRACT
+Methods based on Deep Learning have recently applied on astrophysical parameter recovery
+thanks to their ability to capture information from complex data. One of these methods are the
+approximate Bayesian Neural Networks (BNNs) which have demonstrated to yield consistent
+posterior distribution into the parameter space, helpful for uncertainty quantification. However,
+as any modern neural networks, they tend to produce overly confident uncertainty estimates,
+and can introduce bias when BNNs are applied to data. In this work, we implement multiplicative
+normalizing flows (MNFs), a family of approximate posteriors for the parameters of BNNs with the
+purpose of enhancing the flexibility of the variational posterior distribution, to extract Ωm, h, and
+σ8 from the QUIJOTE simulations. We have compared this method with respect to the standard
+BNNs, and the flipout estimator. We found that MNFs combined with BNNs outperform the other
+models obtaining predictive performance with almost one order of magnitude larger that standard
+BNNs, σ8 extracted with high accuracy (r2 = 0.99), and precise uncertainty estimates. The
+latter implies that MNFs provide more realistic predictive distribution closer to the true posterior
+mitigating the bias introduced by the variational approximation, and allowing to work with well
+calibrated networks.
+Keywords: cosmology, N-body simulations, parameter estimation, artificial intelligence, deep neural networks
+1
+INTRODUCTION
+Cosmological simulations offer one of the most powerful ways to understand the initial conditions of
+the Universe and improve our knowledge on fundamental physics (1). They open also the possibility to
+fully explore the growth of structure in both the linear and non-linear regime. Currently, the concordance
+cosmological model, Λ-CDM, gives an accurate description of most of the observations from early to late
+stages of the Universe using a set of few parameters (2). Recent observations from Cosmic Microwave
+Background (CMB) have provided such accurate estimation for the cosmological parameters, and prompted
+a tension with respect to local scales measurement, along with a well-known degeneracy on the total
+non-relativistic matter density parameters (3, 4, 5). Conventionally, the way to capture information from
+1
+arXiv:2301.03991v1  [astro-ph.IM]  9 Jan 2023
+
+Hort´ua et al.
+Parameter estimation via BNNs
+astronomical observations is to compare summary statistics from data against theory predictions. However,
+two major difficulties arise: First, it is not well understood what kind of estimator, or at which degree of
+approximation of order statistic should be better to extract the maximum information from observations. In
+fact, the most common choice is the power spectrum(PS) which has shown to be a powerful tool for making
+inference (2). However, It is well known that PS is not able to fully characterize the statistical properties
+of non-Gaussian density fields, yielding that it would not be suitable for upcoming Large Scale Structure
+(LSS) or 21-cm signals which are highly non-Gaussian (6, 7, 8). Then, PS will miss relevant information if
+only this statistic is used for parameter recovery (9). Second, Cosmologists will require to store and process
+a large number of data, which can be very expensive. Clearly, sophisticated computational tools along with
+new perspectives on data collection, storage, and analysis must be developed in order to interpret these
+observations (10).
+In recent years, artificial intelligence (AI), and Deep Neural Networks (DNNs) have emerged as promising
+tools to tackle the aforementioned difficulties in the cosmological context due to its capability for
+learning relationships between variables in complex data, outperform traditional estimators, and handle
+the demanding computational needs in Astrophysics and Cosmology (10). These standard DNNs have
+been used on a variety of tasks because of their potential for solving inverse problems. However, they
+are prone to overfitting due to the excessive number of parameters to be adjusted, and the lack of
+explanations of their predictions for given instances (11). The latter is crucial for cosmological analysis
+where assessing robustness and reliability of the model predictions are imperative. This problem can be
+addressed by endowing DNNs with probabilistic properties that permit quantifying posterior distributions
+on their outcomes, and provide them with predictive uncertainties. One of these approaches is the
+use of Bayesian Neural Networks (BNNs) comprised of probabilistic layers that capture uncertainty
+over the network parameters (weights), and trained using Bayesian inference (12). Several works have
+utilized BNNs in cosmological scenarios where the combination of DNNs (through Convolutional Neural
+Networks, CNNs) and probabilistic properties, allow to build models adapted to non-Gaussian data
+without requiring a priori choice summary statistic (9, 13, 14, 15), along with quantifying predictive
+uncertainties (16, 17, 18, 19, 20, 21). Indeed, BNNs permit to infer posterior distributions instead of point
+estimates for the weights. These distributions capture the parameter uncertainty, and by subsequently
+integrating over them, we acquire uncertainties related to the network outputs. Nevertheless, obtaining the
+posterior distributions is an intractable task, and approximate techniques such as a Variational Inference(VI)
+must be used in order to put them into practice (22). Despite the approximate posterior distribution over
+the weights employed in VI clearly providing fast computations for inference tasks, they can also introduce
+a degree of bias depending on how complex(or simple) the choice of the approximate distribution family
+is (23). This issue yields overconfident uncertainty predictions and an unsatisfactory closeness measurement
+with respect to the true posterior. In (17, 18), the authors included normalizing flows on the top of BNNs to
+give the joint parameter distribution more flexibility. However, that approach is not implemented into the
+Bayesian framework, preserving the bias.
+In this paper, we attempt to enhance the flexibility of the approximate posterior distribution over the
+weights of the network by employing multiplicative normalizing flows, resulting in accurate and precise
+uncertainty estimates provided by BNNs. We apply this approach to N-body simulations taken from
+QUIJOTE dataset (24) in order to show how BNNs can take not only advantage of non-Gaussian signals
+without requiring a specifying the summary statistic (such as PS) but also, increase the posterior complexity,
+as they yield much larger performance improvements. This paper is organized as follows. Section 2 offers a
+summary of the BNNs framework and a detailed description of Normalizing flow implementation. Section 3
+describes the dataset and analysis tools used in this paper. Numerical implementation and configuration for
+2
+
+Hort´ua et al.
+Parameter estimation via BNNs
+BNNs are described in Section 4. Section 5 presents the results we obtained by training BNNs taking into
+account different approaches and we display the inference of cosmological parameters. It also outlines the
+calibration diagrams to determine the accuracy of the uncertainty estimates. Finally, Section 6 draws the
+main conclusions of this work and possible further directions to the use of BNNs in Cosmology.
+2
+VARIATIONAL BAYESIAN NEURAL NETWORKS
+Here we go into detail about Bayesian Neural Networks (BNNs), and their implementation to perform
+parameter inference. We start with a brief introduction, before focusing on improving the variational
+approximation. We remind the reader to refer to (25, 26, 22) for further details.
+2.1
+Approximate BNNs
+The goal of BNNs is to infer the posterior distribution p(w|D) over the weights w of the network after
+observing the data D = (X, Y ). This posterior can be obtained from Bayes law: p(w|D) ∼ p(D|w)p(w),
+given a likelihood function p(D|w), and a prior on the weights p(w). Once the posterior has been computed,
+the probability distribution on a new test example x∗ is given by
+p(y∗|x∗, D) =
+�
+w
+p(y∗|x∗, w)p(w|D)dw,
+(1)
+where p(y∗|x∗, w) is the predictive distribution for a given value of the weights. For neural networks,
+however, computing the exact posterior is intractable, so one must resort to approximate BNNs for
+inference (26). A popular method to approximate the posterior is variational inference(VI) (22). Let
+q(w|θ) be a family of simple distributions parameterized by θ. So, the goal of VI is to select a distribution
+q(w|θ∗) such that θ∗ minimizes KL
+�
+q(w|θ)
+��p(w|D)
+�
+, being KL[·∥·] the Kullback-Leibler divergence.
+This minimization is equivalent to maximizing the evidence lower bound (ELBO) (26)
+ELBO(θ) = Eq(w|θ)
+�
+log p(Y |X, w)
+�
+− KL
+�
+q(w|θ)
+��p(w)
+�
+,
+(2)
+where Eq(w|θ)[log p(Y |X, w)] is the expected log-likelihood with respect to the variational posterior and
+KL[q(w|θ)||p(w)] is the divergence of the variational posterior from the prior. We can observe from Eq. 2
+that the KL divergence acts as a regularizer that encourages the variational posterior moves towards
+the modes of the prior. A common choice for the variational posterior is a product of independent (i.e.,
+mean-field) Gaussian distributions, one distribution for each parameter w in the network (25)
+q(w|θ) =
+�
+ij
+N(w; µij, σ2
+ij)
+(3)
+being i and j the indices of the neurons from the previous layer and the current layer respectively. Applying
+the reparametrization trick we arrive at wij = µij + σij ∗ ϵij, where ϵij is drawn from a standard normal
+distribution. Furthermore, if the prior is also a product of independent Gaussians, the KL divergence
+between the prior and the variational posterior be computed analytically, which makes this approach
+computationally efficient.
+2.1.1
+Flipout
+In case where sampling from q(w|θ) is not fully independently for different examples in a mini-batch, we
+well obtain gradient estimates with high variance. Flipout method provides an alternative to decorrelate the
+3
+
+Hort´ua et al.
+Parameter estimation via BNNs
+gradients within a mini batch by implicitly sampling pseudo-independent weights for each example (27).
+The method requires two assumptions about the properties of q(w|θ): symmetric with respect to zero,
+and the weights of the network are independent. Under these assumptions, the distribution is invariant to
+element wise multiplication by a random sign matrix ˆr, i.e., ˆw = w◦ˆr, implies that w ∼ q(w) ≈ ˆw ∼ ˆq( ˆw).
+Therefore, the marginal distribution over gradients computed for individual examples will be identical to
+the distribution computed using shared weights samples. Hence, Flipout achieves much lower variance
+updates when averaging over a mini batch. We validate this approach experimentally by comparing against
+Multiplicative normalizing flows.
+2.2
+Uncertainty in BNNs
+BNNs offer a groundwork to incorporate from the posterior distribution both, the uncertainty inherent to
+the data (aleatoric uncertainty), and the uncertainty in the model parameters due to a limited amount of
+training data (epistemic uncertainty) (28). Following (16), assuming that the top of the BNNs consist of a
+mean vector µ ∈ RN and a covariance matrix Σ ∈ RN(N+1)/21, and for a given fixed input x∗, T forward
+passes of the network are computed, obtaining for each of their mean µt and covariance matrix Σt. Then,
+an estimator for approximate the predictive covariance can be written as
+�
+Cov(y∗, y∗|x∗) ≈ 1
+T
+T
+�
+t=1
+Σt
+�
+��
+�
+Aleatoric
++ 1
+T
+T
+�
+t=1
+(µt − µ)(µt − µ)T
+�
+��
+�
+Epistemic
+,
+(4)
+with µ = 1
+T
+�T
+t=1 µt. Notice that in case Σ is diagonal, and σ2 = diag(Σ), the last equation reduces to the
+results obtained in (29, 30)
+�
+Var(y∗|x∗) ≈ 1
+T
+T
+�
+t=1
+σ2
+t
+�
+��
+�
+Aleatoric
++ 1
+T
+T
+�
+t=1
+(µt − ¯µ)2
+�
+��
+�
+Epistemic
+.
+(5)
+In this scenario, BNNs can be used to learn the correlations between the the targets and produce estimates
+of their uncertainties. Unfortunately, the uncertainty computed from Eqs. 4, 5, tends to be miscalibrated, i.e.,
+the predicted uncertainty (taking into account both epistemic and aleatoric uncertainty) is underestimated
+and does not allow robust detection of uncertain predictions at inference. Therefore, calibration diagrams
+along with methods to jointly calibrate aleatoric and epistemic uncertainties, must be employed before
+inferring predictions from BNNs (31). We come back to this point in Section 5.
+2.3
+Multiplicative normalizing flows
+As mentioned previously, the most common family for the variational posterior used in BNNs is the
+mean-field Gaussian distributions defined in Eq. 3. This simple distribution is unable to capture the
+complexity of the true posterior. Therefore, we expect that increasing the complexity of the variational
+posterior, BNNs achieve significant performance gains since we are now able to sample from a complicate
+distribution that more closely resembles the true posterior. Certainly, transforming the variational posterior
+must be followed with fast computations and still being numerically tractable. We now describe in detail
+the Multiplicative Normalizing Flows (MNFs) method that provides flexible posterior distributions in an
+1 Where the targets y ∈ RN.
+4
+
+Hort´ua et al.
+Parameter estimation via BNNs
+efficient way by employing auxiliary random variables and normalizing flows proposed by (32). MNFs
+propose that the variational posterior can be expressed as an infinite mixture of distributions
+q(w|θ) =
+�
+q(w|z, θ)q(z|θ)dz
+(6)
+where θ is the learnable posterior parameter, and z ∼ q(z|θ) ≡ q(z)2 is a vector with the same
+dimension on the input layer, which plays the role of an auxiliary latent variable. Moreover, allowing local
+reparametrizations, the variational posterior for fully connected layers become a modification of Eq. 3
+written as
+w ∼ q(w|z) =
+�
+ij
+N(w; ziµij, σ2
+ij).
+(7)
+Notice that by enhancing the complexity of q(z), we can increase the flexibility of the variational posterior.
+This can be done using Normalizing Flows since the dimensionality of z is much lower compared to the
+weights. Starting from samples z0 ∼ q(z0) from fully factorized Gaussian Eq. 3, a rich distribution q(zK)
+can be obtained by applying a successively invertible K-transformations fK on z0
+zK = NF(z0) = fK ◦ · · · ◦ f1(z0);
+log q(zK) = log q(z0) −
+K
+�
+k=1
+log
+����det ∂fk
+∂zk−1
+���� .
+(8)
+Unfortunately, the KL divergence in Eq. 2 becomes generally intractable as the posterior q(w) is an
+infinite mixture as shown in Eq. 6. This is addressed also in (33) by evoking Bayes law q(zK)q(w|zK) =
+q(w)q(zK|w) and introducing an auxiliary distribution r(zK|w, φ) parameterized by φ, with the purpose
+of approximating the posterior distribution of the original variational parameters q(zK|w) to further lower
+bound the KL divergence term. Therefore, KL divergence term can be bounded as follows
+− KL
+�
+q(w)
+��p(w)
+�
+= −Eq(w)
+�
+log
+�q(w)
+p(w)
+��
+≥ −Eq(w)
+�
+log
+�q(w)
+p(w)
+�
++ KL
+�
+q(zK|w)
+��r(zK|w, φ)
+��
+= −Eq(w)
+�
+log
+�q(w)
+p(w)
+�
++ Eq(zK|w)
+�
+log
+� q(zK|w)
+r(zK|w, φ)
+���
+= −Eq(w)
+�
+Eq(zK|w)
+�
+log
+�q(w)
+p(w)
+��
++ Eq(zK|w)
+�
+log
+� q(zK|w)
+r(zK|w, φ)
+���
+= −Eq(w,zK)
+�
+log
+�q(w)
+p(w)
+�
++ log
+� q(zK|w)
+r(zK|w, φ)
+��
+= Eq(w,zK) [− log (q(w)q(zK|w)) + log r(zK|w, φ) + log p(w)] ⇒
+− KL
+�
+q(w)
+��p(w)
+�
+≥ Eq(w,zK)
+�
+− KL
+�
+q(w|zK)
+��p(w)
+�
++ log q(zK) + log r(zK|w, φ)
+�
+,
+(9)
+where we have taken into account that KL[P∥Q] ≥ 0, and the equality is satisfied iff P = Q. In the last
+line, the first term can be analytically computed since it will be the KL divergence between two Gaussian
+distributions, while the second term is given by the Normalizing flow generated by fK as we observe in
+2 The parameter θ will be omitted in this section for clarity of notation.
+5
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Eq. 8. Finally, the auxiliary posterior term is parameterized by inverse normalizing flows as follows (34)
+z0 = NF−1(zK) = g−1
+1
+◦ · · · ◦ g−1
+K (zK);
+log r(zK|w, φ) = log r(z0|w, φ) +
+K
+�
+k=1
+log
+�����det ∂g−1
+k
+∂zk
+����� , (10)
+where one can parameterize g−1
+K as another normalizing flow. In the paper (32), the authors also propose a
+flexible parametrization of the auxiliary posterior as
+z0 ∼ r(zK|w, φ) =
+�
+i
+N(z0; ˜µi(w, φ), ˜σ2
+i (w, φ)).
+(11)
+We will use the parameterization of the mean ˜µ, and the variance ˜σ2 as in the original paper as well as the
+masked RealNVP (35) as choice of Normalizing flows.
+3
+N-BODY SIMULATIONS DATASET
+In this work, we leverage 2000 hypercubes simulation taken from The Quijote project (24). They
+have been run using the TreePM code Gadget-III (36), and their initial conditions were generated at
+z = 127 using 2LPT (37). The set chosen for this work is made of standard simulations with different
+random seeds with the intention of emulating the cosmic variance. Each instance corresponds to a three-
+dimensional distribution of the density field with size 643. The cosmological parameters vary according
+to Ωm ∈ [0.1, 0.5], Ωb ∈ [0.03, 0.07], h ∈ [0.5, 0.9], ns ∈ [0.8, 1.2], σ8 ∈ [0.6, 1.0], while neutrino mass
+(Mν = 0eV) and the equation of state parameter (w = −1) are kept fixed. The dataset was split into
+training(70%), validation (10%), and test (20%), while hypercubes were logarithmic transformed and the
+cosmological parameters normalized between 0 and 1. In this paper we will build BNNs with the ability to
+predict three out of five aforementioned parameters, Ωm, σ8 and h.
+4
+BNNS IMPLEMENTATION
+We will consider three different BNNs architectures based on the discussion presented in Section 2: standard
+BNNs (prior and variational posterior defined as a mean-field Normal distributions) [sBNNs]; BNNs with
+Flipout estimator [FlipoutBNNs]; and Multiplicative normalizing flows [VBNNs]. The experiments were
+implemented using the TensorFlow v:2.9 and TensorFlow-probability v:0.19 (38). All BNNs designed in this
+paper are comprised of three parts. First, all experiments start with a 643-voxel input layer corresponding to
+the normalised 3D density field followed by the fully-convolutional ResNet-18 backbone as it is presented
+schematically in table 1. All the Resblock are fully pre-activated and their representation can be seen
+in figure. 1. The repository Classification models 3D was used to build the backbone of BNNs (39).
+Subsequently, the second part of BNNs represents the stochasticity of the network. This is comprised
+of just one layer and it depends on the type of BNN used. For sBNNs, we employ the dense variational
+layer which uses variational inference to fit an approximate posterior to the distribution over both the
+kernel matrix and the bias terms. Here, we use as posterior and prior(no-trainable) Normal distributions.
+Experiments with FlipoutBNNs for instance, are made via Flipout dense layer where the mean field normal
+distribution are also utilized to parameterize the distributions. These two layers are already implemented in
+the package TF-probability (38). On the other hand, for VBNNs we have adapted the class DenseMNF
+implemented in the repositories TF-MNF, MNF-VBNN (32) to our model. Here, we use 50 layers for the
+masked RealNVP NF, and the maximum variance for layer weights is around the unity. Finally, the last
+6
+
+Hort´ua et al.
+Parameter estimation via BNNs
+ResNet-18 backbone
+Layer Name
+Input Shape
+Output Shape
+Batch Norm
+(Nbatch, 64,64,64,3)
+(Nbatch, 64,64,64,3)
+3D Convolutional
+(Nbatch, 70,70,70,3)
+(Nbatch, 32,32,32,64)
+Batch Norm+ReLU
+(Nbatch, 32,32,32,64)
+(Nbatch, 32,32,32,64)
+Max Pooling 3D
+(Nbatch, 34,34,34,64)
+(Nbatch, 16,16,16,64)
+Batch Norm+ReLU
+(Nbatch, 16,16,16,64)
+(Nbatch, 16,16,16,64)
+Resblock 1
+�
+(Nbatch, 16, 16, 16, 64)
+(Nbatch, 16, 16, 16, 64)
+�
+(Nbatch, 16,16,16,64)
+Batch Norm+ReLU
+(Nbatch, 16,16,16,64)
+(Nbatch, 16,16,16,64)
+Resblock 2
+�
+(Nbatch, 16, 16, 16, 64)
+(Nbatch, 8, 8, 8, 128)
+�
+(Nbatch, 8,8,8,128)
+Batch Norm+ReLU
+(Nbatch, 8,8,8,128 )
+(Nbatch, 8,8,8,128)
+Resblock 3
+�
+(Nbatch, 8, 8, 8, 128)
+(Nbatch, 4, 4, 4, 256)
+�
+(Nbatch, 4,4,4,256)
+Batch Norm+ReLU
+(Nbatch, 4,4,4,256 )
+(Nbatch, 4,4,4,256)
+Resblock 4
+�
+(Nbatch, 4, 4, 4, 256)
+(Nbatch, 2, 2, 2, 512)
+�
+(Nbatch, 2,2,2,512)
+Batch Norm+ReLU
+(Nbatch, 2,2,2,512 )
+(Nbatch, 2,2,2,512)
+Global Avg Pooling
+(Nbatch, 2,2,2,512)
+(Nbatch, 512)
+Table 1. Configuration of the backbone BNNs used for all experiments presented in this paper.
+part of all BNNs account for the output of the network, which is dependent on the aleatoric uncertainty
+parameterization. We use a 3D multivariate Gaussian distribution with nine parameters to be learnt (three
+means µ for the cosmological parameters, and six elements for the covariance matrix Σ).
+The loss function to be optimized during training is given by the ELBO 2 where the second term is
+associated to the negative log-likelihood (NLL)
+− NLL ∼ 1
+2 log |s · Σ| + 1
+2(y − µ)⊤ (s · Σ)−1 (y − µ),
+(12)
+averaged over the mini-batch. The scalar variable s is equal to one during the training process, and it
+becomes a trainable variable during post-training to recalibrate the probability density function (16, 31).
+The algorithm used to minimize the objective function is the Adam optimizer with first and second moment
+exponential decay rates of 0.9 and 0.999, respectively (40). The learning rate starts from 10−3 and it
+will be reduced by a factor of 0.8 in case that any improvement has not been observed after 10 epochs.
+Furthermore, we have applied warm-up period for which the model turns on progressively the KL term
+in Eq. 2. This is achieved by introducing a β variable in the ELBO, i.e., β · KL
+�
+q(w|θ)
+��p(w)
+�
+, so, this
+parameter starts being equal to 0 and grows linearly to 1 during 10 epochs (41). BNNs were trained with
+32 batches and early stopping callback for avoiding over-fitting. The infrastructure used was the Google
+Cloud Platform (GCP) using a nvidia-tesla-t4 of 16 GB GDDR6 in a N1 machine series shared-core.
+7
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Figure 1a. Illustration of the first skip connection in
+a residual block.
+Figure 1b. Illustration of the second skip connection
+in the residual block.
+Figure 1. Each Resblock includes both skip connection configurations. (A) The Resblock starts with this
+configuration applied to the input tensor. (B) The output of the previous configuration is fed into this
+connection.
+4.1
+Metrics
+We compare all BNN results in terms of performance, i.e., the precision of their predictions for the
+cosmological parameters quantified through Mean Square Error (MSE), ELBO, and plotting the true vs
+predicted values with its coefficient of determination. Also, it is important to quantify the quality of the
+uncertainty estimates. One of the ways to diagnostic the quality of the uncertainty estimates is through
+reliability diagrams. Following (31, 11), we can define perfect calibration of regression uncertainty as
+Eˆσ2
+����
+E[(y − µ)2]
+�� ˆσ2 = α2�
+− α2���
+∀
+�
+α2 ∈ R
+�� α2 ≥ 0
+�
+.
+(13)
+Hence, the predicted uncertainty ˆσ2 is partitioned into K bins with equal width, and the variance per bin is
+defined as
+var(Bk) :=
+1
+��Bk
+��
+�
+i∈Bm
+1
+N
+N
+�
+n=1
+(µi,n − yi)2,
+(14)
+with N stochastic forward passes. On the other hand, the uncertainty per bin is defined as
+uncert(Bk) :=
+1
+|Bk|
+�
+i∈Bk
+ˆσ2
+i .
+(15)
+With these two quantities, we can generate reliability diagrams to assess the quality of the estimated
+uncertainty via plotting var(Bk) vs. uncert(Bk). In addition, we can compute the expected uncertainty
+8
+
+skip connection
+(identity)
+F() +
+Conv3D
+BN+ReLU
+Conv3D
++ (。)
+add
+output
+F()skip connection
+(identity)
+F() + 
+BN+ReLU
+Conv3D
+BN+ReLU
+Conv3D
++ (,)
+ppe
+output
+F()Hort´ua et al.
+Parameter estimation via BNNs
+Metrics
+FlipoutBNNs
+VBNNs
+sBNNs
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+Ωm
+σ8
+h
+Ωmh2
+σ8Ω0.25
+m
+MSE
+0.063
+0.057
+0.190
+ELBO
+20.85
+19.71
+31.57
+r2
+0.82
+0.98
+0.2
+0.03
+0.93
+0.85
+0.99
+0.4
+0.56
+0.95
+0.75
+0.85
+0.01
+0.23
+0.80
+UCE
+0.109
+8.10
+0.26
+0.0008
+0.0008
+0.010
+>1.0
+Table 2. Metrics test set results for all BNNs architectures. High UCE values indicate miscalibration. MSE
+and ELBO are computed only over the cosmological parameters.
+calibration error (UCE) in order to quantify the miscalibration
+UCE :=
+K
+�
+k=1
+|Bk|
+m
+��var(Bk) − uncert(Bk)
+��,
+(16)
+with number of inputs m and set of indices Bk of inputs, for which the uncertainty falls into the bin k. A
+more general approach proposed in (16) consists in computing the expected coverage probabilities defined
+as the x% of samples for which the true value of the parameters falls in the x%-confidence region defined
+by the joint posterior. Clearly, this option is more precise since it captures higher-order statistics through
+the full posterior distribution. However, for simplicity, we will follow the UCE approach.
+5
+ANALYSIS AND RESULTS OF PARAMETER INFERENCE WITH BNNS
+In this section we discuss the results obtained by comparing three different versions of BNNs, the one
+with MNFs, the standard BNN, and the third one using Flipout as estimator. The results reported in this
+section were computed on the Test dataset. Table 2. shows the metrics obtained for each BNN approach.
+As mentioned, MSE, ELBO and r2 provide well estimates for determining the precision of the model,
+while UCE measures the miscalibration. Here, we can observe that VBNNs outperform all experiments,
+not only taking into account the average error, but also the precision for each cosmological parameter along
+with a good calibration in its uncertainty predictions. Followed by VBBNs, we have the FlipoutBNNs,
+however, although this approach yields good cosmological parameter estimation, it understimates their
+uncertainties. Therefore, VBNNs avoids indeed the application of an extra post training step in the Machine
+Learning pipeline related to calibration. Notice that in all experiments, h becomes hardly predicted for all
+model. Figure 2 displays the predicted against true values for Ωm, ωm (instead of h), σ8 and the degeneracy
+direction defined as σ8Ω0.25
+m . Error bars report the epistemic plus aleatoric uncertainties predicted by BNNs,
+which illustrates the advantages of these probabilistic models where the certainty prediction of the model is
+captured instead of traditional DNNs where only point estimates are present. This uncertainty was taken
+from the diagonal part of the covariance matrix.
+5.1
+Calibration metrics
+In figure 3, we analyze the quality of our uncertainty measurement using calibration diagrams. We show
+the predicted uncertainty vs observed uncertainty from our model on the Test dataset. Better performing
+uncertainty estimates should correlate more accurately with the dashed lines. We can see that estimating
+uncertainty from VBNNs reflect better the real uncertainty. Furthermore, the scale for VBNNs is two
+orders of magnitude lower than FlipoutBNN, which also implies how reliable is this models according
+to their predictions. Notice that the even if we partitioned the variance into K = 10 bins with equal
+width, FlipoutBNNs and sBNNs yield underestimate uncertainties (many examples concentrates in lower
+bin values), for this reason we see that while VBNNs supply all ten samples in the calibration plots, for
+9
+
+Hort´ua et al.
+Parameter estimation via BNNs
+Figure 2. Plots of True vs Predicted values provided by the best experiment VBNNs, for Ωm, σ8, and
+some derivative parameters. Points are the mean of the predicted distributions, and error bars stand for the
+heteroscedastic uncertainty associated to epistemic plus aleatoric uncertainty at 1σ.
+the others we have just 3-4 of them. Next, we employed the σ-scaling methodology for calibrating the
+FlipoutBNNs predictions (31). For doing so, we optimize uniquely the loss function described in Eq. 12
+where all parameters related to the BNNs where frozen, i.e., the only trainable parameter was s. After
+training, we got s ∼ 0.723, reducing UCE only up to 10%, and the number of samples in the calibration
+diagrams enlarged to 4-5. This minor performance enhancement means that σ-scaling is not suitable to
+calibrate all BNNs, and alternative re-calibration techniques must be taken into account in order to build
+reliable intervals. At this point, we have noticed the advantages of working with methods that leading with
+networks already well-calibrated after the training step (17).
+5.2
+Joint analysis for Cosmological parameters
+In order to show the parameter intervals and contours from the N-body simulations, we choose randomly
+an example from the test set with true values shown in table 3. The two-dimensional posterior distribution
+of the cosmological parameters are shown in figure 4 and the parameter 95% intervals are reported in
+table 3. We can observe that VBNNs provides considerably tighter and well constraints on all parameters
+10
+
+0.6
+perfectmatch
+1.1
+perfectmatch
+0.5
+1.0
+8
+0.9
+Predicted
+Predicted
+0.3
+0.8
+0.2
+0.7
+0.1
+0.6
+0.2
+0.3
+0.4
+0.7
+0.8
+0.9
+True08
+perfect match
+perfect match
+0.40
+0.8
+0.35
+0.7
+ywu
+0.30
+0.25
+Predicted
+0.6
+0.20
+0.5
+0.15
+0.10
+0.4
+0.05
+0.4
+0.5
+0.6
+0.7
+0.8
+0.1
+0.2
+0.3
+True Ωmh?Hort´ua et al.
+Parameter estimation via BNNs
+Figure 3. Calibration diagrams for the best experiments, VBNNs and FlipoutBNNs. The lower is the
+UCE value, the higher is the calibration of the model. Dashes lines stand for the perfect calibration, so, the
+discrepancy to this identity curve reveals miscalibration.
+with respect to the sBNNs (18). Most important, this technique offers also the correlation among parameters
+and the measurement about how reliable the model in their predictions.
+6
+CONCLUSIONS
+N-body simulations offer one of the most powerful ways to understand the initial conditions of the
+Universe and improve our knowledge on fundamental physics. In this paper we used QUIJOTE dataset, in
+order to show how convolutional DNNs capture non-Gaussian patters without requiring a specifying the
+summary statistic (such as PS). Additionally, we have show how we can build probabilistic DNNs to obtain
+uncertainties which account for the reliability in their predictions. One of the main goals of this paper was
+11
+
+CalibrationforQm withVBNN
+Calibrationforo:withVBNN
+1e-3
+1e-3
+UCE=0.0008
+UCE=0.0008
+4
+2
+m
+2
+1
+0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Expecteduncertainty
+1e-3
+Expecteduncertainty
+1e-3
+le-2 Calibration for h with VBNN
+Calibration for Qm with FlipoutBNNs
+0.3
+rtainty
+2.5
+UCE=0.0105
+Observed uncertainty
+UCE=0.1095
+2.0
+uncer
+0.2
+1.5
+Observed
+0.1
+1.0
+0.5
+0.0
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Expected uncertainty
+1e-2
+Expecteduncertainty
+CalibrationforO: withFlipoutBNNs
+CalibrationforhwithFlipoutBNNs
+Observed uncertainty
+20
+UCE=8.099
+UCE=0.2595
+15
+4
+3
+10
+2
+5
+1
+0
+5
+10
+15
+20
+0
+1
+2
+3
+4
+5
+Expected uncertainty
+ExpecteduncertaintyHort´ua et al.
+Parameter estimation via BNNs
+Figure 4. 68% and 95% parameter constraint contours from one example of Quijote test dataset using
+VBNNs and FlipoutBNNs. The diagonal plots are the marginalized parameter constraints, the dashed lines
+stand for the the true values. This plot was made using Getdist (42).
+also reporting how improves these BNNs when we integrate them with techniques such as a Multiplicative
+normalizing flows to enhance the variational posterior complexity. We found that VBNNs not only provides
+considerably tighter and well constraints on all cosmological parameters as we observed in figure 4, but
+also yields with well-calibrated estimate uncertainties as it was shown in figure 3. Nevertheless, some
+limitations in this research includes simple prior assumptions (mean-field approximations), lower resolution
+in the simulations, and absence of additional calibration techniques. These restrictions will be analysed in
+detail in a future paper.
+12
+
+0.6
+D
+0.5
+m
+0.4
+0.3
+0.8
+0o 0.7
+b
+0.6
+1.0
+60.8
+0.6
+0.65
+.25
+0.60
+0.55
+0.50
+0.6
+2
+0.4
+0.2
+0.3
+0.4
+0.5
+0.6
+0.6
+0.7
+0.8
+0.6
+0.8
+1.0
+0.50 0.55 0.60 0.65
+0.2
+0.4
+0.6
+Qm
+h
+0:Q0.25
+Qmh2
+m
+VBNNs
+FlipoutBNNsHort´ua et al.
+Parameter estimation via BNNs
+Parameter
+95% limits VBNNs
+95% limits FlipoutBNNs
+True Value
+Ωm
+0.47+0.10
+−0.10
+0.45+0.11
+−0.11
+0.495
+σ8
+0.697+0.038
+−0.038
+0.699+0.059
+−0.060
+0.699
+h
+0.81+0.17
+−0.17
+0.78+0.20
+−0.19
+0.800
+σ8Ω0.25
+m
+0.577+0.051
+−0.052
+0.573+0.063
+−0.064
+0.587
+Ωmh2
+0.31+0.19
+−0.18
+0.573+0.063
+−0.064
+0.317
+Table 3. Parameter 95% intervals taken from the parameter constraint contours (figure 4) from one example
+of Quijote test dataset using VBNN and FlipoutBNN.
+ACKNOWLEDGMENTS
+This paper is based upon work supported by the Google Cloud Research Credits program with the award
+GCP19980904.
+Leonardo Casta˜neda was supported by patrimonio aut´onomo fondo Nacional de financiamiento para
+la ciencia y la tecnolog´ıa y la innovacion Francisco Jos´e de Caldas (Minciencias Colombia) grant No
+110685269447 RC-80740-465-2020 projects 69723. H. J. Hort´ua acknowledges the support from cr´editos
+educaci´on de doctorados nacionales y en el exterior- colciencias, and the grant provided by the Google
+Cloud Research Credits program.
+REFERENCES
+1 .Stefano B, Kravtsov A. Cosmological simulations of galaxy clusters. Advanced Science Letters 4
+(2011) 204–227. doi:10.1166/asl.2011.1209.
+2 .Dodelson S. Modern Cosmology (Academic Press, Elsevier Science) (2003).
+3 .Planck C, Aghanim N, Akrami Y, Ashdown M, Aumont J, Baccigalupi C, et al. Planck 2018 results.
+Astronomy and Astrophysics 641 (2020) A6. doi:10.1051/0004-6361/201833910.
+4 .Tinker JL, Sheldon ES, Wechsler RH, Becker MR, Rozo E, Zu Y, et al.
+COSMOLOGICAL
+CONSTRAINTS FROM GALAXY CLUSTERING AND THE MASS-TO-NUMBER RATIO OF
+GALAXY CLUSTERS. The Astrophysical Journal 745 (2011) 16. doi:10.1088/0004-637x/745/1/16.
+5 .Yusofi E, Ramzanpour MA.
+Cosmological constant problem and h0 tension in void-dominated
+cosmology (2022). doi:10.48550/ARXIV.2204.12180.
+6 .Mesinger A, Furlanetto S, Cen R. 21cmfast: a fast, seminumerical simulation of the high-redshift
+21-cm signal. Monthly Notices of the Royal Astronomical Society 411 (2011) 955–972. doi:10.1111/j.
+1365-2966.2010.17731.x.
+7 .Hamann J, Hannestad S, Lesgourgues J, Rampf C, Wong YY. Cosmological parameters from large
+scale structure - geometric versus shape information. Journal of Cosmology and Astroparticle Physics
+2010 (2010) 022–022. doi:10.1088/1475-7516/2010/07/022.
+8 .Abdalla E, Abell’an GF, Aboubrahim A, Agnello A, Akarsu O, Akrami Y, et al. Cosmology intertwined:
+A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions
+and anomalies. Journal of High Energy Astrophysics (2022).
+9 .Gillet N, Mesinger A, Greig B, Liu A, Ucci G. Deep learning from 21-cm tomography of the cosmic
+dawn and reionization. Monthly Notices of the Royal Astronomical Society 484 (2019) 282–293.
+doi:10.1093/mnras/stz010.
+10 .Dvorkin C, Mishra-Sharma S, Nord B, Villar VA, Avestruz C, Bechtol K, et al. Machine learning and
+cosmology (2022). doi:10.48550/ARXIV.2203.08056.
+13
+
+Hort´ua et al.
+Parameter estimation via BNNs
+11 .Guo C, Pleiss G, Sun Y, Weinberger KQ. On calibration of modern neural networks. Proceedings of
+the 34th International Conference on Machine Learning - Volume 70 (JMLR.org) (2017), ICML 17,
+1321–1330.
+12 .Chang DT. Bayesian neural networks: Essentials (2021). doi:10.48550/ARXIV.2106.13594.
+13 .Ravanbakhsh S, Oliva J, Fromenteau S, Price LC, Ho S, Schneider J, et al. Estimating cosmological
+parameters from the dark matter distribution (2017). doi:10.48550/ARXIV.1711.02033.
+14 .Lazanu A. Extracting cosmological parameters from n-body simulations using machine learning
+techniques. Journal of Cosmology and Astroparticle Physics 2021 (2021) 039. doi:10.1088/1475-7516/
+2021/09/039.
+15 .Wang BY, Pisani A, Villaescusa-Navarro F, Wandelt BD. Machine learning cosmology from void
+properties (2022). doi:10.48550/ARXIV.2212.06860.
+16 .Hort´ua HJ, Volpi R, Marinelli D, Malag`o L. Parameter estimation for the cosmic microwave background
+with bayesian neural networks. Physical Review D 102 (2020). doi:10.1103/physrevd.102.103509.
+17 .Hort´ua HJ, Malag`o L, Volpi R. Constraining the reionization history using bayesian normalizing flows.
+Machine Learning: Science and Technology 1 (2020) 035014. doi:10.1088/2632-2153/aba6f1.
+18 .Hortua HJ. Constraining cosmological parameters from n-body simulations with bayesian neural
+networks (2021). doi:10.48550/ARXIV.2112.11865.
+19 .Mancarella M, Kennedy J, Bose B, Lombriser L. Seeking new physics in cosmology with bayesian
+neural networks: Dark energy and modified gravity. Phys. Rev. D 105 (2022) 023531. doi:10.1103/
+PhysRevD.105.023531.
+20 .List F, Rodd NL, Lewis GF, Bhat I. Galactic center excess in a new light: Disentangling the gamma-
+ray sky with bayesian graph convolutional neural networks. Physical Review Letters 125 (2020).
+doi:10.1103/physrevlett.125.241102.
+21 .Wagner-Carena S, Park JW, Birrer S, Marshall PJ, Roodman A, Wechsler RH. Hierarchical inference
+with bayesian neural networks: An application to strong gravitational lensing. The Astrophysical
+Journal 909 (2021) 187. doi:10.3847/1538-4357/abdf59.
+22 .Graves A, editor. Practical Variational Inference for Neural Networks, vol. 24 (Curran Associates,
+Inc.) (2011).
+23 .Charnock T, Perreault-Levasseur L, Lanusse F. Bayesian neural networks (2020). doi:10.48550/ARXIV.
+2006.01490.
+24 .Villaescusa-Navarro F, Hahn C, Massara E, Banerjee A, Delgado AM, Ramanah DK, et al. The quijote
+simulations. The Astrophysical Journal Supplement Series 250 (2020) 2. doi:10.3847/1538-4365/
+ab9d82.
+25 .Abdar M, Pourpanah F, Hussain S, Rezazadegan D, Liu L, Ghavamzadeh M, et al. A review of
+uncertainty quantification in deep learning: Techniques, applications and challenges. Information
+Fusion 76 (2021) 243–297. doi:10.1016/j.inffus.2021.05.008.
+26 .Gal Y. Uncertainty in Deep Learning. Ph.D. thesis, University of Cambridge (2016).
+27 .Wen Y, Vicol P, Ba J, Tran D, Grosse R. Flipout: Efficient pseudo-independent weight perturbations on
+mini-batches (2018). doi:10.48550/ARXIV.1803.04386.
+28 .Kiureghian AD, Ditlevsen O. Aleatory or epistemic? does it matter? Structural Safety 31 (2009) 105 –
+112. doi:https://doi.org/10.1016/j.strusafe.2008.06.020. Risk Acceptance and Risk Communication.
+29 .Kendall A, Gal Y. What uncertainties do we need in bayesian deep learning for computer vision?
+(2017).
+30 .Kwon Y, Won JH, Joon Kim B, Paik M. Ininternational conference on medical imaging with deep
+learning (2018) 13.
+14
+
+Hort´ua et al.
+Parameter estimation via BNNs
+31 .Laves MH, Ihler S, Fast JF, Kahrs LA, Ortmaier T. Well-calibrated regression uncertainty in medical
+imaging with deep learning. Medical Imaging with Deep Learning (2020).
+32 .Louizos C, Welling M. Multiplicative normalizing flows for variational bayesian neural networks.
+Proceedings of the 34th International Conference on Machine Learning - Volume 70 (JMLR.org)
+(2017), ICML’17, 2218–2227.
+33 .Ranganath R, Tran D, Blei DM. Hierarchical variational models. Proceedings of the 33rd International
+Conference on International Conference on Machine Learning - Volume 48 (JMLR.org) (2016),
+ICML’16, 2568–2577.
+34 .Touati A, Satija H, Romoff J, Pineau J, Vincent P. Randomized value functions via multiplicative
+normalizing flows (2018). doi:10.48550/ARXIV.1806.02315.
+35 .Dinh L, Sohl-Dickstein J, Bengio S. Density estimation using real NVP. International Conference on
+Learning Representations (2017).
+36 .Springel V. The cosmological simulation code gadget-2. Monthly Notices of the Royal Astronomical
+Society 364 (2005) 1105–1134. doi:10.1111/j.1365-2966.2005.09655.x.
+37 .Scoccimarro R. Transients from initial conditions: a perturbative analysis. Monthly Notices of the
+Royal Astronomical Society 299 (1998) 1097–1118. doi:10.1046/j.1365-8711.1998.01845.x.
+38 .Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, et al. TensorFlow: Large-scale machine
+learning on heterogeneous systems (2015). Software available from tensorflow.org.
+39 .Solovyev R, Kalinin AA, Gabruseva T. 3d convolutional neural networks for stalled brain capillary
+detection. Computers in Biology and Medicine 141 (2022) 105089. doi:10.1016/j.compbiomed.2021.
+105089.
+40 .Kingma DP, Ba J. Adam: A method for stochastic optimization (2014). doi:10.48550/ARXIV.1412.
+6980.
+41 .Sønderby CK, Raiko T, Maaløe L, Sønderby SK, Winther O.
+Ladder variational autoencoders.
+Proceedings of the 30th International Conference on Neural Information Processing Systems (Red
+Hook, NY, USA: Curran Associates Inc.) (2016), NIPS’16, 3745–3753.
+42 .Lewis A. GetDist: a Python package for analysing Monte Carlo samples (2019).
+15
+
diff --git a/9NE2T4oBgHgl3EQflwcz/content/tmp_files/load_file.txt b/9NE2T4oBgHgl3EQflwcz/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ec7392dc63f6d96300b4fc4b4163fa19e3147c0c
--- /dev/null
+++ b/9NE2T4oBgHgl3EQflwcz/content/tmp_files/load_file.txt
@@ -0,0 +1,880 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf,len=879
+page_content='Constraining cosmological parameters from  N-body simulations with Variational Bayesian  Neural Networks  H´ector J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Hort´ua 1,2,∗, Luz ´Angela Garc´ıa 3 and Leonardo Casta˜neda C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4  1 Grupo Signos, Departamento de Matem´aticas, Universidad el Bosque, Bogot´a,  Colombia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2 Maestr´ıa en Ciencia de Datos, Universidad Escuela Colombiana de Ingenier´ıa  Julio Garavito Bogot´a, Colombia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  3 Universidad ECCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Cra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 19 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 49-20, Bogot´a, Colombia, C´odigo Postal 111311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  4Observatorio Astron´omico Nacional, Universidad Nacional de Colombia, Bogot´a,  Colombia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Correspondence*: hjhortuao@unal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='co  ABSTRACT  Methods based on Deep Learning have recently applied on astrophysical parameter recovery  thanks to their ability to capture information from complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' One of these methods are the  approximate Bayesian Neural Networks (BNNs) which have demonstrated to yield consistent  posterior distribution into the parameter space, helpful for uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However,  as any modern neural networks, they tend to produce overly confident uncertainty estimates,  and can introduce bias when BNNs are applied to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In this work, we implement multiplicative  normalizing flows (MNFs), a family of approximate posteriors for the parameters of BNNs with the  purpose of enhancing the flexibility of the variational posterior distribution, to extract Ωm, h, and  σ8 from the QUIJOTE simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We have compared this method with respect to the standard  BNNs, and the flipout estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We found that MNFs combined with BNNs outperform the other  models obtaining predictive performance with almost one order of magnitude larger that standard  BNNs, σ8 extracted with high accuracy (r2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='99), and precise uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The  latter implies that MNFs provide more realistic predictive distribution closer to the true posterior  mitigating the bias introduced by the variational approximation, and allowing to work with well  calibrated networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Keywords: cosmology, N-body simulations, parameter estimation, artificial intelligence, deep neural networks  1  INTRODUCTION  Cosmological simulations offer one of the most powerful ways to understand the initial conditions of  the Universe and improve our knowledge on fundamental physics (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' They open also the possibility to  fully explore the growth of structure in both the linear and non-linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Currently, the concordance  cosmological model, Λ-CDM, gives an accurate description of most of the observations from early to late  stages of the Universe using a set of few parameters (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Recent observations from Cosmic Microwave  Background (CMB) have provided such accurate estimation for the cosmological parameters, and prompted  a tension with respect to local scales measurement, along with a well-known degeneracy on the total  non-relativistic matter density parameters (3, 4, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Conventionally, the way to capture information from  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='03991v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='IM]  9 Jan 2023  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  astronomical observations is to compare summary statistics from data against theory predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However,  two major difficulties arise: First, it is not well understood what kind of estimator, or at which degree of  approximation of order statistic should be better to extract the maximum information from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In  fact, the most common choice is the power spectrum(PS) which has shown to be a powerful tool for making  inference (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However, It is well known that PS is not able to fully characterize the statistical properties  of non-Gaussian density fields, yielding that it would not be suitable for upcoming Large Scale Structure  (LSS) or 21-cm signals which are highly non-Gaussian (6, 7, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Then, PS will miss relevant information if  only this statistic is used for parameter recovery (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Second, Cosmologists will require to store and process  a large number of data, which can be very expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Clearly, sophisticated computational tools along with  new perspectives on data collection, storage, and analysis must be developed in order to interpret these  observations (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  In recent years, artificial intelligence (AI), and Deep Neural Networks (DNNs) have emerged as promising  tools to tackle the aforementioned difficulties in the cosmological context due to its capability for  learning relationships between variables in complex data, outperform traditional estimators, and handle  the demanding computational needs in Astrophysics and Cosmology (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' These standard DNNs have  been used on a variety of tasks because of their potential for solving inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However, they  are prone to overfitting due to the excessive number of parameters to be adjusted, and the lack of  explanations of their predictions for given instances (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The latter is crucial for cosmological analysis  where assessing robustness and reliability of the model predictions are imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This problem can be  addressed by endowing DNNs with probabilistic properties that permit quantifying posterior distributions  on their outcomes, and provide them with predictive uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' One of these approaches is the  use of Bayesian Neural Networks (BNNs) comprised of probabilistic layers that capture uncertainty  over the network parameters (weights), and trained using Bayesian inference (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Several works have  utilized BNNs in cosmological scenarios where the combination of DNNs (through Convolutional Neural  Networks, CNNs) and probabilistic properties, allow to build models adapted to non-Gaussian data  without requiring a priori choice summary statistic (9, 13, 14, 15), along with quantifying predictive  uncertainties (16, 17, 18, 19, 20, 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Indeed, BNNs permit to infer posterior distributions instead of point  estimates for the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' These distributions capture the parameter uncertainty, and by subsequently  integrating over them, we acquire uncertainties related to the network outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Nevertheless, obtaining the  posterior distributions is an intractable task, and approximate techniques such as a Variational Inference(VI)  must be used in order to put them into practice (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Despite the approximate posterior distribution over  the weights employed in VI clearly providing fast computations for inference tasks, they can also introduce  a degree of bias depending on how complex(or simple) the choice of the approximate distribution family  is (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This issue yields overconfident uncertainty predictions and an unsatisfactory closeness measurement  with respect to the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In (17, 18), the authors included normalizing flows on the top of BNNs to  give the joint parameter distribution more flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However, that approach is not implemented into the  Bayesian framework, preserving the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  In this paper, we attempt to enhance the flexibility of the approximate posterior distribution over the  weights of the network by employing multiplicative normalizing flows, resulting in accurate and precise  uncertainty estimates provided by BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We apply this approach to N-body simulations taken from  QUIJOTE dataset (24) in order to show how BNNs can take not only advantage of non-Gaussian signals  without requiring a specifying the summary statistic (such as PS) but also, increase the posterior complexity,  as they yield much larger performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Section 2 offers a  summary of the BNNs framework and a detailed description of Normalizing flow implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Section 3  describes the dataset and analysis tools used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Numerical implementation and configuration for  2  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  BNNs are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Section 5 presents the results we obtained by training BNNs taking into  account different approaches and we display the inference of cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' It also outlines the  calibration diagrams to determine the accuracy of the uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Finally, Section 6 draws the  main conclusions of this work and possible further directions to the use of BNNs in Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2  VARIATIONAL BAYESIAN NEURAL NETWORKS  Here we go into detail about Bayesian Neural Networks (BNNs), and their implementation to perform  parameter inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We start with a brief introduction, before focusing on improving the variational  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We remind the reader to refer to (25, 26, 22) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  Approximate BNNs  The goal of BNNs is to infer the posterior distribution p(w|D) over the weights w of the network after  observing the data D = (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This posterior can be obtained from Bayes law: p(w|D) ∼ p(D|w)p(w),  given a likelihood function p(D|w), and a prior on the weights p(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Once the posterior has been computed,  the probability distribution on a new test example x∗ is given by  p(y∗|x∗, D) =  �  w  p(y∗|x∗, w)p(w|D)dw,  (1)  where p(y∗|x∗, w) is the predictive distribution for a given value of the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' For neural networks,  however, computing the exact posterior is intractable, so one must resort to approximate BNNs for  inference (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' A popular method to approximate the posterior is variational inference(VI) (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Let  q(w|θ) be a family of simple distributions parameterized by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' So, the goal of VI is to select a distribution  q(w|θ∗) such that θ∗ minimizes KL  �  q(w|θ)  ��p(w|D)  �  , being KL[·∥·] the Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  This minimization is equivalent to maximizing the evidence lower bound (ELBO) (26)  ELBO(θ) = Eq(w|θ)  �  log p(Y |X, w)  �  − KL  �  q(w|θ)  ��p(w)  �  ,  (2)  where Eq(w|θ)[log p(Y |X, w)] is the expected log-likelihood with respect to the variational posterior and  KL[q(w|θ)||p(w)] is the divergence of the variational posterior from the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We can observe from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2  that the KL divergence acts as a regularizer that encourages the variational posterior moves towards  the modes of the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' A common choice for the variational posterior is a product of independent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=',  mean-field) Gaussian distributions, one distribution for each parameter w in the network (25)  q(w|θ) =  �  ij  N(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' µij, σ2  ij)  (3)  being i and j the indices of the neurons from the previous layer and the current layer respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Applying  the reparametrization trick we arrive at wij = µij + σij ∗ ϵij, where ϵij is drawn from a standard normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Furthermore, if the prior is also a product of independent Gaussians, the KL divergence  between the prior and the variational posterior be computed analytically, which makes this approach  computationally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  Flipout  In case where sampling from q(w|θ) is not fully independently for different examples in a mini-batch, we  well obtain gradient estimates with high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Flipout method provides an alternative to decorrelate the  3  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  gradients within a mini batch by implicitly sampling pseudo-independent weights for each example (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  The method requires two assumptions about the properties of q(w|θ): symmetric with respect to zero,  and the weights of the network are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Under these assumptions, the distribution is invariant to  element wise multiplication by a random sign matrix ˆr, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=', ˆw = w◦ˆr, implies that w ∼ q(w) ≈ ˆw ∼ ˆq( ˆw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Therefore, the marginal distribution over gradients computed for individual examples will be identical to  the distribution computed using shared weights samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Hence, Flipout achieves much lower variance  updates when averaging over a mini batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We validate this approach experimentally by comparing against  Multiplicative normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  Uncertainty in BNNs  BNNs offer a groundwork to incorporate from the posterior distribution both, the uncertainty inherent to  the data (aleatoric uncertainty), and the uncertainty in the model parameters due to a limited amount of  training data (epistemic uncertainty) (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Following (16), assuming that the top of the BNNs consist of a  mean vector µ ∈ RN and a covariance matrix Σ ∈ RN(N+1)/21, and for a given fixed input x∗, T forward  passes of the network are computed, obtaining for each of their mean µt and covariance matrix Σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Then,  an estimator for approximate the predictive covariance can be written as  �  Cov(y∗, y∗|x∗) ≈ 1  T  T  �  t=1  Σt  �  ��  �  Aleatoric  + 1  T  T  �  t=1  (µt − µ)(µt − µ)T  �  ��  �  Epistemic  ,  (4)  with µ = 1  T  �T  t=1 µt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Notice that in case Σ is diagonal, and σ2 = diag(Σ), the last equation reduces to the  results obtained in (29, 30)  �  Var(y∗|x∗) ≈ 1  T  T  �  t=1  σ2  t  �  ��  �  Aleatoric  + 1  T  T  �  t=1  (µt − ¯µ)2  �  ��  �  Epistemic  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (5)  In this scenario, BNNs can be used to learn the correlations between the the targets and produce estimates  of their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Unfortunately, the uncertainty computed from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4, 5, tends to be miscalibrated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=',  the predicted uncertainty (taking into account both epistemic and aleatoric uncertainty) is underestimated  and does not allow robust detection of uncertain predictions at inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Therefore, calibration diagrams  along with methods to jointly calibrate aleatoric and epistemic uncertainties, must be employed before  inferring predictions from BNNs (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We come back to this point in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  Multiplicative normalizing flows  As mentioned previously, the most common family for the variational posterior used in BNNs is the  mean-field Gaussian distributions defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This simple distribution is unable to capture the  complexity of the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Therefore, we expect that increasing the complexity of the variational  posterior, BNNs achieve significant performance gains since we are now able to sample from a complicate  distribution that more closely resembles the true posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Certainly, transforming the variational posterior  must be followed with fast computations and still being numerically tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We now describe in detail  the Multiplicative Normalizing Flows (MNFs) method that provides flexible posterior distributions in an  1 Where the targets y ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  4  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  efficient way by employing auxiliary random variables and normalizing flows proposed by (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' MNFs  propose that the variational posterior can be expressed as an infinite mixture of distributions  q(w|θ) =  �  q(w|z, θ)q(z|θ)dz  (6)  where θ is the learnable posterior parameter, and z ∼ q(z|θ) ≡ q(z)2 is a vector with the same  dimension on the input layer, which plays the role of an auxiliary latent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Moreover, allowing local  reparametrizations, the variational posterior for fully connected layers become a modification of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 3  written as  w ∼ q(w|z) =  �  ij  N(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' ziµij, σ2  ij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (7)  Notice that by enhancing the complexity of q(z), we can increase the flexibility of the variational posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  This can be done using Normalizing Flows since the dimensionality of z is much lower compared to the  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Starting from samples z0 ∼ q(z0) from fully factorized Gaussian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 3, a rich distribution q(zK)  can be obtained by applying a successively invertible K-transformations fK on z0  zK = NF(z0) = fK ◦ · · · ◦ f1(z0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  log q(zK) = log q(z0) −  K  �  k=1  log  ����det ∂fk  ∂zk−1  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (8)  Unfortunately, the KL divergence in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2 becomes generally intractable as the posterior q(w) is an  infinite mixture as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This is addressed also in (33) by evoking Bayes law q(zK)q(w|zK) =  q(w)q(zK|w) and introducing an auxiliary distribution r(zK|w, φ) parameterized by φ, with the purpose  of approximating the posterior distribution of the original variational parameters q(zK|w) to further lower  bound the KL divergence term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' KL divergence term can be bounded as follows  − KL  �  q(w)  ��p(w)  �  = −Eq(w)  �  log  �q(w)  p(w)  ��  ≥ −Eq(w)  �  log  �q(w)  p(w)  �  + KL  �  q(zK|w)  ��r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ)  ��  = −Eq(w)  �  log  �q(w)  p(w)  �  + Eq(zK|w)  �  log  � q(zK|w)  r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ)  ���  = −Eq(w)  �  Eq(zK|w)  �  log  �q(w)  p(w)  ��  + Eq(zK|w)  �  log  � q(zK|w)  r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ)  ���  = −Eq(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='zK)  �  log  �q(w)  p(w)  �  + log  � q(zK|w)  r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ)  ��  = Eq(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='zK) [− log (q(w)q(zK|w)) + log r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ) + log p(w)] ⇒  − KL  �  q(w)  ��p(w)  �  ≥ Eq(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='zK)  �  − KL  �  q(w|zK)  ��p(w)  �  + log q(zK) + log r(zK|w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' φ)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (9)  where we have taken into account that KL[P∥Q] ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' and the equality is satisfied iff P = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In the last  line, the first term can be analytically computed since it will be the KL divergence between two Gaussian  distributions, while the second term is given by the Normalizing flow generated by fK as we observe in  2 The parameter θ will be omitted in this section for clarity of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  5  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Finally, the auxiliary posterior term is parameterized by inverse normalizing flows as follows (34)  z0 = NF−1(zK) = g−1  1  · · · ◦ g−1  K (zK);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  log r(zK|w, φ) = log r(z0|w, φ) +  K  �  k=1  log  �����det ∂g−1  k  ∂zk  ����� , (10)  where one can parameterize g−1  K as another normalizing flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In the paper (32), the authors also propose a  flexible parametrization of the auxiliary posterior as  z0 ∼ r(zK|w, φ) =  �  i  N(z0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' ˜µi(w, φ), ˜σ2  i (w, φ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (11)  We will use the parameterization of the mean ˜µ, and the variance ˜σ2 as in the original paper as well as the  masked RealNVP (35) as choice of Normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  3  N-BODY SIMULATIONS DATASET  In this work, we leverage 2000 hypercubes simulation taken from The Quijote project (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' They  have been run using the TreePM code Gadget-III (36), and their initial conditions were generated at  z = 127 using 2LPT (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The set chosen for this work is made of standard simulations with different  random seeds with the intention of emulating the cosmic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Each instance corresponds to a three-  dimensional distribution of the density field with size 643.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The cosmological parameters vary according  to Ωm ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5], Ωb ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='07], h ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9], ns ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2], σ8 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0], while neutrino mass  (Mν = 0eV) and the equation of state parameter (w = −1) are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The dataset was split into  training(70%), validation (10%), and test (20%), while hypercubes were logarithmic transformed and the  cosmological parameters normalized between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In this paper we will build BNNs with the ability to  predict three out of five aforementioned parameters, Ωm, σ8 and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  4  BNNS IMPLEMENTATION  We will consider three different BNNs architectures based on the discussion presented in Section 2: standard  BNNs (prior and variational posterior defined as a mean-field Normal distributions) [sBNNs];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' BNNs with  Flipout estimator [FlipoutBNNs];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' and Multiplicative normalizing flows [VBNNs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The experiments were  implemented using the TensorFlow v:2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9 and TensorFlow-probability v:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='19 (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' All BNNs designed in this  paper are comprised of three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' First, all experiments start with a 643-voxel input layer corresponding to  the normalised 3D density field followed by the fully-convolutional ResNet-18 backbone as it is presented  schematically in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' All the Resblock are fully pre-activated and their representation can be seen  in figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The repository Classification models 3D was used to build the backbone of BNNs (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Subsequently, the second part of BNNs represents the stochasticity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This is comprised  of just one layer and it depends on the type of BNN used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' For sBNNs, we employ the dense variational  layer which uses variational inference to fit an approximate posterior to the distribution over both the  kernel matrix and the bias terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Here, we use as posterior and prior(no-trainable) Normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Experiments with FlipoutBNNs for instance, are made via Flipout dense layer where the mean field normal  distribution are also utilized to parameterize the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' These two layers are already implemented in  the package TF-probability (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' On the other hand, for VBNNs we have adapted the class DenseMNF  implemented in the repositories TF-MNF, MNF-VBNN (32) to our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Here, we use 50 layers for the  masked RealNVP NF, and the maximum variance for layer weights is around the unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Finally, the last  6  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  ResNet-18 backbone  Layer Name  Input Shape  Output Shape  Batch Norm  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3)  3D Convolutional  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='70,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Max Pooling 3D  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='34,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Resblock 1  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 64)  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='64)  Resblock 2  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 64)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 128)  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='128)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='128 )  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='128)  Resblock 3  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 128)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 256)  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='256)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='256 )  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='256)  Resblock 4  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 256)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 512)  �  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='512)  Batch Norm+ReLU  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='512 )  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='512)  Global Avg Pooling  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='512)  (Nbatch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 512)  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Configuration of the backbone BNNs used for all experiments presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  part of all BNNs account for the output of the network, which is dependent on the aleatoric uncertainty  parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We use a 3D multivariate Gaussian distribution with nine parameters to be learnt (three  means µ for the cosmological parameters, and six elements for the covariance matrix Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  The loss function to be optimized during training is given by the ELBO 2 where the second term is  associated to the negative log-likelihood (NLL)  − NLL ∼ 1  2 log |s · Σ| + 1  2(y − µ)⊤ (s · Σ)−1 (y − µ),  (12)  averaged over the mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The scalar variable s is equal to one during the training process, and it  becomes a trainable variable during post-training to recalibrate the probability density function (16, 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  The algorithm used to minimize the objective function is the Adam optimizer with first and second moment  exponential decay rates of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='999, respectively (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The learning rate starts from 10−3 and it  will be reduced by a factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8 in case that any improvement has not been observed after 10 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Furthermore, we have applied warm-up period for which the model turns on progressively the KL term  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This is achieved by introducing a β variable in the ELBO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=', β · KL  �  q(w|θ)  ��p(w)  �  , so, this  parameter starts being equal to 0 and grows linearly to 1 during 10 epochs (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' BNNs were trained with  32 batches and early stopping callback for avoiding over-fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The infrastructure used was the Google  Cloud Platform (GCP) using a nvidia-tesla-t4 of 16 GB GDDR6 in a N1 machine series shared-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  7  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Illustration of the first skip connection in  a residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Illustration of the second skip connection  in the residual block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Each Resblock includes both skip connection configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' (A) The Resblock starts with this  configuration applied to the input tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' (B) The output of the previous configuration is fed into this  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  Metrics  We compare all BNN results in terms of performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=', the precision of their predictions for the  cosmological parameters quantified through Mean Square Error (MSE), ELBO, and plotting the true vs  predicted values with its coefficient of determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Also, it is important to quantify the quality of the  uncertainty estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' One of the ways to diagnostic the quality of the uncertainty estimates is through  reliability diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Following (31, 11), we can define perfect calibration of regression uncertainty as  Eˆσ2  ����  E[(y − µ)2]  �� ˆσ2 = α2�  − α2���  ∀  �  α2 ∈ R  �� α2 ≥ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (13)  Hence, the predicted uncertainty ˆσ2 is partitioned into K bins with equal width, and the variance per bin is  defined as  var(Bk) :=  1  ��Bk  ��  �  i∈Bm  1  N  N  �  n=1  (µi,n − yi)2,  (14)  with N stochastic forward passes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' On the other hand, the uncertainty per bin is defined as  uncert(Bk) :=  1  |Bk|  �  i∈Bk  ˆσ2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (15)  With these two quantities, we can generate reliability diagrams to assess the quality of the estimated  uncertainty via plotting var(Bk) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' uncert(Bk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In addition, we can compute the expected uncertainty  8  skip connection  (identity)  F() +  Conv3D  BN+ReLU  Conv3D  + (。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=')  add  output  F()skip connection  (identity)  F() +  BN+ReLU  Conv3D  BN+ReLU  Conv3D  + (,)  ppe  output  F()Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Metrics  FlipoutBNNs  VBNNs  sBNNs  Ωm  σ8  h  Ωmh2  σ8Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  m  Ωm  σ8  h  Ωmh2  σ8Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  m  Ωm  σ8  h  Ωmh2  σ8Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  m  MSE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='190  ELBO  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='85  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='71  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='57  r2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='80  UCE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='109  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='010  >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Metrics test set results for all BNNs architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' High UCE values indicate miscalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' MSE  and ELBO are computed only over the cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  calibration error (UCE) in order to quantify the miscalibration  UCE :=  K  �  k=1  |Bk|  m  ��var(Bk) − uncert(Bk)  ��,  (16)  with number of inputs m and set of indices Bk of inputs, for which the uncertainty falls into the bin k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' A  more general approach proposed in (16) consists in computing the expected coverage probabilities defined  as the x% of samples for which the true value of the parameters falls in the x%-confidence region defined  by the joint posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Clearly, this option is more precise since it captures higher-order statistics through  the full posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' However, for simplicity, we will follow the UCE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  5  ANALYSIS AND RESULTS OF PARAMETER INFERENCE WITH BNNS  In this section we discuss the results obtained by comparing three different versions of BNNs, the one  with MNFs, the standard BNN, and the third one using Flipout as estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The results reported in this  section were computed on the Test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' shows the metrics obtained for each BNN approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  As mentioned, MSE, ELBO and r2 provide well estimates for determining the precision of the model,  while UCE measures the miscalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Here, we can observe that VBNNs outperform all experiments,  not only taking into account the average error, but also the precision for each cosmological parameter along  with a good calibration in its uncertainty predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Followed by VBBNs, we have the FlipoutBNNs,  however, although this approach yields good cosmological parameter estimation, it understimates their  uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Therefore, VBNNs avoids indeed the application of an extra post training step in the Machine  Learning pipeline related to calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Notice that in all experiments, h becomes hardly predicted for all  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Figure 2 displays the predicted against true values for Ωm, ωm (instead of h), σ8 and the degeneracy  direction defined as σ8Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Error bars report the epistemic plus aleatoric uncertainties predicted by BNNs,  which illustrates the advantages of these probabilistic models where the certainty prediction of the model is  captured instead of traditional DNNs where only point estimates are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This uncertainty was taken  from the diagonal part of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  Calibration metrics  In figure 3, we analyze the quality of our uncertainty measurement using calibration diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We show  the predicted uncertainty vs observed uncertainty from our model on the Test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Better performing  uncertainty estimates should correlate more accurately with the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We can see that estimating  uncertainty from VBNNs reflect better the real uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Furthermore, the scale for VBNNs is two  orders of magnitude lower than FlipoutBNN, which also implies how reliable is this models according  to their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Notice that the even if we partitioned the variance into K = 10 bins with equal  width, FlipoutBNNs and sBNNs yield underestimate uncertainties (many examples concentrates in lower  bin values), for this reason we see that while VBNNs supply all ten samples in the calibration plots, for  9  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Plots of True vs Predicted values provided by the best experiment VBNNs, for Ωm, σ8, and  some derivative parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Points are the mean of the predicted distributions, and error bars stand for the  heteroscedastic uncertainty associated to epistemic plus aleatoric uncertainty at 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  the others we have just 3-4 of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Next, we employed the σ-scaling methodology for calibrating the  FlipoutBNNs predictions (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' For doing so, we optimize uniquely the loss function described in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 12  where all parameters related to the BNNs where frozen, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=', the only trainable parameter was s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' After  training, we got s ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='723, reducing UCE only up to 10%, and the number of samples in the calibration  diagrams enlarged to 4-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This minor performance enhancement means that σ-scaling is not suitable to  calibrate all BNNs, and alternative re-calibration techniques must be taken into account in order to build  reliable intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' At this point, we have noticed the advantages of working with methods that leading with  networks already well-calibrated after the training step (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  Joint analysis for Cosmological parameters  In order to show the parameter intervals and contours from the N-body simulations, we choose randomly  an example from the test set with true values shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The two-dimensional posterior distribution  of the cosmological parameters are shown in figure 4 and the parameter 95% intervals are reported in  table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We can observe that VBNNs provides considerably tighter and well constraints on all parameters  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  perfectmatch  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  perfectmatch  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9  Predicted  Predicted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9  True08  perfect match  perfect match  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  ywu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  Predicted  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  True Ωmh?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Calibration diagrams for the best experiments, VBNNs and FlipoutBNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The lower is the  UCE value, the higher is the calibration of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Dashes lines stand for the perfect calibration, so, the  discrepancy to this identity curve reveals miscalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  with respect to the sBNNs (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Most important, this technique offers also the correlation among parameters  and the measurement about how reliable the model in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  6  CONCLUSIONS  N-body simulations offer one of the most powerful ways to understand the initial conditions of the  Universe and improve our knowledge on fundamental physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' In this paper we used QUIJOTE dataset, in  order to show how convolutional DNNs capture non-Gaussian patters without requiring a specifying the  summary statistic (such as PS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Additionally, we have show how we can build probabilistic DNNs to obtain  uncertainties which account for the reliability in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' One of the main goals of this paper was  11  CalibrationforQm withVBNN  Calibrationforo:withVBNN  1e-3  1e-3  UCE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0008  UCE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0008  4  2  m  2  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  Expecteduncertainty  1e-3  Expecteduncertainty  1e-3  le-2 Calibration for h with VBNN  Calibration for Qm with FlipoutBNNs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  rtainty  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  UCE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0105  Observed uncertainty  UCE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1095  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  uncer  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  Observed  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='30  Expected uncertainty  1e-2  Expecteduncertainty  CalibrationforO: withFlipoutBNNs  CalibrationforhwithFlipoutBNNs  Observed uncertainty  20  UCE=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='099  UCE=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2595  15  4  3  10  2  5  1  0  5  10  15  20  0  1  2  3  4  5  Expected uncertainty  ExpecteduncertaintyHort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 68% and 95% parameter constraint contours from one example of Quijote test dataset using  VBNNs and FlipoutBNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The diagonal plots are the marginalized parameter constraints, the dashed lines  stand for the the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' This plot was made using Getdist (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  also reporting how improves these BNNs when we integrate them with techniques such as a Multiplicative  normalizing flows to enhance the variational posterior complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' We found that VBNNs not only provides  considerably tighter and well constraints on all cosmological parameters as we observed in figure 4, but  also yields with well-calibrated estimate uncertainties as it was shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Nevertheless, some  limitations in this research includes simple prior assumptions (mean-field approximations), lower resolution  in the simulations, and absence of additional calibration techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' These restrictions will be analysed in  detail in a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='65  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='6  Qm  h  0:Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  Qmh2  m  VBNNs  FlipoutBNNsHort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  Parameter  95% limits VBNNs  95% limits FlipoutBNNs  True Value  Ωm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='47+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='11  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='495  σ8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='697+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='038  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='699+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='059  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='060  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='699  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='17  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='20  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='800  σ8Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='25  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='577+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='051  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='573+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='063  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='587  Ωmh2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='19  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='573+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='063  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='317  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Parameter 95% intervals taken from the parameter constraint contours (figure 4) from one example  of Quijote test dataset using VBNN and FlipoutBNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This paper is based upon work supported by the Google Cloud Research Credits program with the award  GCP19980904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Leonardo Casta˜neda was supported by patrimonio aut´onomo fondo Nacional de financiamiento para  la ciencia y la tecnolog´ıa y la innovacion Francisco Jos´e de Caldas (Minciencias Colombia) grant No  110685269447 RC-80740-465-2020 projects 69723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Hort´ua acknowledges the support from cr´editos  educaci´on de doctorados nacionales y en el exterior- colciencias, and the grant provided by the Google  Cloud Research Credits program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  REFERENCES  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Stefano B, Kravtsov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Cosmological simulations of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Advanced Science Letters 4  (2011) 204–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1166/asl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Dodelson S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Modern Cosmology (Academic Press, Elsevier Science) (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Planck C, Aghanim N, Akrami Y, Ashdown M, Aumont J, Baccigalupi C, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Astronomy and Astrophysics 641 (2020) A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1051/0004-6361/201833910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Tinker JL, Sheldon ES, Wechsler RH, Becker MR, Rozo E, Zu Y, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  COSMOLOGICAL  CONSTRAINTS FROM GALAXY CLUSTERING AND THE MASS-TO-NUMBER RATIO OF  GALAXY CLUSTERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The Astrophysical Journal 745 (2011) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1088/0004-637x/745/1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Yusofi E, Ramzanpour MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Cosmological constant problem and h0 tension in void-dominated  cosmology (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='12180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Mesinger A, Furlanetto S, Cen R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 21cmfast: a fast, seminumerical simulation of the high-redshift  21-cm signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Monthly Notices of the Royal Astronomical Society 411 (2011) 955–972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='17731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  7 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Hamann J, Hannestad S, Lesgourgues J, Rampf C, Wong YY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Cosmological parameters from large  scale structure - geometric versus shape information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Journal of Cosmology and Astroparticle Physics  2010 (2010) 022–022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1088/1475-7516/2010/07/022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Abdalla E, Abell’an GF, Aboubrahim A, Agnello A, Akarsu O, Akrami Y, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Cosmology intertwined:  A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions  and anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Journal of High Energy Astrophysics (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Gillet N, Mesinger A, Greig B, Liu A, Ucci G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Deep learning from 21-cm tomography of the cosmic  dawn and reionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Monthly Notices of the Royal Astronomical Society 484 (2019) 282–293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1093/mnras/stz010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Dvorkin C, Mishra-Sharma S, Nord B, Villar VA, Avestruz C, Bechtol K, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Machine learning and  cosmology (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='08056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  13  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  11 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Guo C, Pleiss G, Sun Y, Weinberger KQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' On calibration of modern neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Proceedings of  the 34th International Conference on Machine Learning - Volume 70 (JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='org) (2017), ICML 17,  1321–1330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Chang DT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Bayesian neural networks: Essentials (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='13594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Ravanbakhsh S, Oliva J, Fromenteau S, Price LC, Ho S, Schneider J, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Estimating cosmological  parameters from the dark matter distribution (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='02033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Lazanu A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Extracting cosmological parameters from n-body simulations using machine learning  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Journal of Cosmology and Astroparticle Physics 2021 (2021) 039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1088/1475-7516/  2021/09/039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  15 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Wang BY, Pisani A, Villaescusa-Navarro F, Wandelt BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Machine learning cosmology from void  properties (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='06860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Hort´ua HJ, Volpi R, Marinelli D, Malag`o L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Parameter estimation for the cosmic microwave background  with bayesian neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Physical Review D 102 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1103/physrevd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='103509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  17 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Hort´ua HJ, Malag`o L, Volpi R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Constraining the reionization history using bayesian normalizing flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Machine Learning: Science and Technology 1 (2020) 035014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1088/2632-2153/aba6f1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Hortua HJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Constraining cosmological parameters from n-body simulations with bayesian neural  networks (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='11865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  19 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Mancarella M, Kennedy J, Bose B, Lombriser L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Seeking new physics in cosmology with bayesian  neural networks: Dark energy and modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' D 105 (2022) 023531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1103/  PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='023531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='List F, Rodd NL, Lewis GF, Bhat I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Galactic center excess in a new light: Disentangling the gamma-  ray sky with bayesian graph convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Physical Review Letters 125 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1103/physrevlett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='241102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Wagner-Carena S, Park JW, Birrer S, Marshall PJ, Roodman A, Wechsler RH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Hierarchical inference  with bayesian neural networks: An application to strong gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The Astrophysical  Journal 909 (2021) 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3847/1538-4357/abdf59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Graves A, editor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Practical Variational Inference for Neural Networks, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 24 (Curran Associates,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=') (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  23 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Charnock T, Perreault-Levasseur L, Lanusse F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Bayesian neural networks (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='01490.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Villaescusa-Navarro F, Hahn C, Massara E, Banerjee A, Delgado AM, Ramanah DK, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The quijote  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The Astrophysical Journal Supplement Series 250 (2020) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='3847/1538-4365/  ab9d82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Abdar M, Pourpanah F, Hussain S, Rezazadegan D, Liu L, Ghavamzadeh M, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' A review of  uncertainty quantification in deep learning: Techniques, applications and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Information  Fusion 76 (2021) 243–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='inffus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Gal Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Uncertainty in Deep Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' thesis, University of Cambridge (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  27 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Wen Y, Vicol P, Ba J, Tran D, Grosse R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Flipout: Efficient pseudo-independent weight perturbations on  mini-batches (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='04386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  28 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Kiureghian AD, Ditlevsen O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Aleatory or epistemic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' does it matter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Structural Safety 31 (2009) 105 –  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='strusafe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Risk Acceptance and Risk Communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  29 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Kendall A, Gal Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' What uncertainties do we need in bayesian deep learning for computer vision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  30 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Kwon Y, Won JH, Joon Kim B, Paik M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Ininternational conference on medical imaging with deep  learning (2018) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  14  Hort´ua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Parameter estimation via BNNs  31 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Laves MH, Ihler S, Fast JF, Kahrs LA, Ortmaier T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Well-calibrated regression uncertainty in medical  imaging with deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Medical Imaging with Deep Learning (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  32 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Louizos C, Welling M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Multiplicative normalizing flows for variational bayesian neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Proceedings of the 34th International Conference on Machine Learning - Volume 70 (JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='org)  (2017), ICML’17, 2218–2227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Ranganath R, Tran D, Blei DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Hierarchical variational models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Proceedings of the 33rd International  Conference on International Conference on Machine Learning - Volume 48 (JMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='org) (2016),  ICML’16, 2568–2577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  34 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Touati A, Satija H, Romoff J, Pineau J, Vincent P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Randomized value functions via multiplicative  normalizing flows (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='02315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  35 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Dinh L, Sohl-Dickstein J, Bengio S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Density estimation using real NVP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' International Conference on  Learning Representations (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  36 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Springel V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' The cosmological simulation code gadget-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Monthly Notices of the Royal Astronomical  Society 364 (2005) 1105–1134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='09655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  37 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Scoccimarro R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Transients from initial conditions: a perturbative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Monthly Notices of the  Royal Astronomical Society 299 (1998) 1097–1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1046/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1365-8711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='01845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  38 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Abadi M, Agarwal A, Barham P, Brevdo E, Chen Z, Citro C, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' TensorFlow: Large-scale machine  learning on heterogeneous systems (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  39 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Solovyev R, Kalinin AA, Gabruseva T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' 3d convolutional neural networks for stalled brain capillary  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Computers in Biology and Medicine 141 (2022) 105089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='compbiomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  105089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Kingma DP, Ba J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' Adam: A method for stochastic optimization (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  6980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  41 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Sønderby CK, Raiko T, Maaløe L, Sønderby SK, Winther O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Ladder variational autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  Proceedings of the 30th International Conference on Neural Information Processing Systems (Red  Hook, NY, USA: Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=') (2016), NIPS’16, 3745–3753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  42 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='Lewis A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content=' GetDist: a Python package for analysing Monte Carlo samples (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NE2T4oBgHgl3EQflwcz/content/2301.03991v1.pdf'}
diff --git a/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/2301.03552v1.pdf.txt b/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/2301.03552v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..894c5c245a9c31ea31ac86e19d516881f50c769e
--- /dev/null
+++ b/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/2301.03552v1.pdf.txt
@@ -0,0 +1,1395 @@
+Lieb lattices and pseudospin-1 dynamics under barrier- and
+well-like electrostatic interactions
+V. Jakubsk´y1 and K. Zelaya1
+1Nuclear Physics Institute, Czech Academy of Science, 250 68 ˇReˇz, Czech Republic
+Abstract
+This work considers the confining and scattering phenomena of electrons in a Lieb lattice
+subjected to the influence of a rectangular electrostatic barrier.
+In this setup, hopping
+amplitudes between nearest neighbors in orthogonal directions are considered different, and
+the next-nearest neighbor interaction describes spin-orbit coupling. This makes it possible to
+confine electrons and generate bound states, the exact number of which is exactly determined
+for null parallel momentum to the barrier. In such a case, it is proved that one even and one
+odd bound state is always generated, and the number of bound states increases for non-null
+and increasing values of the parallel momentum. That is, bound states carry current. In the
+scattering regime, the exact values of energy are determined where the resonant tunneling
+occurs. The existence of perfect tunneling energy in the form of super-Klein tunneling is
+proved to exist regardless of the bang gap opening. Finally, it is shown that perfect reflection
+appears when solutions are coupled to the intermediate flat-band solution.
+1
+Introduction
+The theoretical and experimental progress in the physics of graphene and other Dirac materials
+has become a trending topic in material science and theoretical physics [1,2]. Many remarkable
+properties of these materials follow from the fact that dynamics of low-energy quasi-particles is
+described by equations known in relativistic quantum mechanics. It makes it possible to test
+relativistic properties such as Klein tunneling [3,4], relativistic Landau levels, and the existence
+of pseudoparticles violating the Lorentz invariance [5, 6] (type-II Dirac fermions). Graphene
+mono- and multi-layer systems exhibit transport properties such as quantum Hall effect [7]
+and anomalous quantum Hall effect in graphene [8], and Josephson effect in twisted cuprate
+bilayers [9].
+Graphene has shown to be a helpful benchmark system to test the properties of relativistic
+pseudospin-1/2 particles in low-energy systems.
+Nevertheless, the family of Dirac materials
+contains also other, equally interesting, members.
+Their geometries can extend beyond the
+honeycomb lattice.
+For instance, there are Kagome [10], Dice or α − T3 [11, 12], and Lieb
+lattices [13, 14], which lead to effective pseudospin-1 Dirac equations. It was recently showed
+that the Kagome lattice can be obtained from a geometrical deformation of the Lieb lattice [15].
+For a recent survey of two-dimensional lattices and their physical properties and realization,
+see [16].
+1
+arXiv:2301.03552v1  [cond-mat.mes-hall]  9 Jan 2023
+
+Particularly, the Lieb lattice is a two-dimensional array with a periodicity of a square lattice.
+The sites are located in the corners of each square and at the midpoints on its sides. To our
+best knowledge, the Lieb lattice has not been found in nature. However, it has been prepared
+artificially in diverse ways [17, 18]. It was realized in experiments with optical fibers [19–23].
+Furthermore, it was formed by ultracold atoms trapped in optical lattices [24] or by electrons
+of Cu(111) atoms confined by an array of CO molecules [25]. It was also prepared in covalent-
+organic frameworks [26].
+The tight-binding model can well describe the band structure of the Lieb lattice. It reveals
+the existence of two bands with positive and negative energies and an additional so-called flat
+band. The latter is associated with the states that have fixed (zero) energy independent of the
+value of momentum. It is worth mentioning that the flat band solutions were prepared in the
+optical experiments, see [22, 23]. Similarly to graphene, the dynamics of the low-energy quasi-
+particles in the Lieb lattice is dictated by a relativistic Dirac-type equation. Nevertheless, these
+quasi-particles have pseudospin-1 due to three atoms per unit cell.
+In the current article, we investigate the scattering and confinement of the relativistic quasi-
+particles by a rectangular electric potential in the Lieb lattice with a gapped band structure.
+Gap-opening can be induced by on-site energy that differs on three sublattices or by the phase
+acquired by the electron when jumping between the neighboring sites [24], see also [27]. In
+the article, we adopt the second approach where a purely imaginary next nearest-neighbor
+interaction, attributed to spin-orbital coupling [13], is taken into account.
+Effects such as electron confinement and transmission are obtained with the aid of the proper
+boundary conditions, which enforce the continuity on two out of the three pseudospin-1 compo-
+nents. The third component can be discontinuous, which leads to a spatial discontinuity in the
+probability density. Nevertheless, it does not compromise the associated continuity equation.
+Electron dynamics for electrostatic interactions in graphene have been discussed in the litera-
+ture, such as the transmission properties in square barriers [28,29] and electron confinement with
+cylindrical quantum dots [30]. We thus focus on the related properties of the quasi-particle dy-
+namics in the Lieb lattice. We further analyze the influence of the flat-band solution in electron
+dynamics. As shown in the manuscript, solutions in this regime are described by degenerate
+Bloch-wave solutions whose linear combinations can compose wavepackets of arbitrary form.
+These are shown to be current-free solutions regardless of the nature of the wavepacket. As a
+result, one obtains perfectly reflected waves when they couple to flat-band solutions.
+The manuscript is structured as follows. In Sec. 2 we briefly introduce and discuss the main
+properties of the Lieb lattice with nearest next-nearest neighbor interactions, from which the
+effective low-energy Dirac equation is obtained.
+In Sec. 3, we present the general solutions
+and the transfer matrix associated with the rectangular electrostatic interaction. The latter
+is then exploited in Sec. 4 and Sec. 5 to discuss in full detail the localization of electrons and
+scattering dynamics, respectively. Finally, discussions and perspectives are provided in Sec. 7,
+and complementary details about the proof of the number of bound states are given in App. A.
+2
+
+(a)
+(b)
+Figure 1: (a) Lieb lattice, composed by the atoms A (blue-filled circle), B (green square), and C
+(red-filled square). The dashed arrows denote the direction of positive phase hopping parameter
+between next-nearest neighbors B − C. (b) Composition of a unit cell of the Lieb lattice. The
+unit displacement vectors ⃗δ1 = aˆx and ˆδ2 = aˆy connect the atom A with B and A with C,
+respectively. The corresponding nearest hopping parameters are t1, t2, whereas the next-nearest
+neighbor hopping parameter is +it3 and −it3 depending if it occurs in the direction denotes by
+the arrows.
+2
+Lieb lattice and pseudospin-1 Dirac equation
+Let us consider an electronic Lieb lattice 1 so that the separation between two nearest atoms is
+a, the length of each side of the square is ℓ = 2a. There are three sites in the elementary cell, see
+Fig. 1a. The primitive translation vectors are ⃗r1 = 2aˆx and ⃗r2 = 2aˆy. It is customary to denote
+the atoms at the corners of the square as A, whereas the atoms at the sides of the square are B
+(horizontal) and C (vertical). The lattice vectors ⃗δ1 = aˆx = ⃗r1/2 and ⃗δ2 = aˆy = ⃗r2/2 connect an
+atom on the site A to those on the sites B and C, respectively (see Fig. 1b). The atoms A, B and
+C form the three sublattices RA = n1⃗r1 +n2⃗r2, RB = ˜RA +⃗δ1, and RC = ⃗RA +⃗δ2, respectively,
+with n1, n2 ∈ Z. The reciprocal space is spanned by the translation vectors of the reciprocal
+space ˆrk1 and ˆrk2, ˆrp · ˆrkq = 2πδp,q, p, q = 1, 2. This leads to ˆrk1 = π
+a ˆx and ˆrk2 = π
+a ˆy. The
+first Brillouin zone, constructed from the Wigner-Seitz rule, restricts to the region composed by
+kx ∈ [− π
+2a, π
+2a] and ky ∈ [− π
+2a, π
+2a].
+The band structure of the electrons on the Lieb lattice can be analyzed with the use of the
+tight-binding model. There are considered the nearest neighbor (NN) interactions between the
+sites A − B and A − C, represented by the hopping parameters t1 and t2, respectively. We take
+into account also the next-nearest neighbor (NNN) transition B − C, which can be complex
+valued, with the sign of phase dependent on the orientation of the hopping. This emerges due
+to external time-dependent driven fields in photonic Lieb lattices [31], and magnon Lieb and
+Kagome lattices [32].
+In particular, we consider a purely imaginary NNN hopping parameter e±iπ/2t3, where the
+1The results here obtained apply to optical Lieb lattices as well.
+3
+
+C
+A
+BA
+B(a) t3 = 0
+(b) t3 ̸= 0
+Figure 2: Dispersion bands w+(⃗k) (yellow-upper), w−(⃗k) (green-lower), and w0(⃗k) (blue-middle)
+for the gapless (a) and gapped (b) configurations.
+hopping phase is positive (+) is the hopping occurs counter-clock-wise, and negative (−) oth-
+erwise. Such a hopping dynamics is depicted in Fig. 1a. This type of hopping was introduced
+by Haldane in [8] as a model for quantum anomalous Hall effect in graphene without strong
+external magnetic fields, which was latter found experimentally in [33]. See also [34] for a recent
+review.
+The spectral analysis of the tight-binding Hamiltonian reveals that there are three bands in
+its spectrum [13],
+w0(⃗k) = 0,
+w±(⃗k) = ±2
+�
+t2
+1 cos2(akx) + t2
+2 cos2(aky) + 4t2
+3 sin2(akx) sin2(aky).
+(1)
+The bands have linear dependence on the momentum in the four Dirac points that are
+situated in the first Brillouin zone. Their explicit position depends on the relative strength of
+t3. In this work, we focus on the most relevant situation where t3 < t1
+2 , t3 < t2
+2 . In that case,
+the Dirac point is ⃗K = ( π
+2a, π
+2a), see Fig.2b for illustration. A similar analysis holds for higher
+values of t3, where the Dirac points are displaced with respect to ⃗K. For a detailed discussion,
+see [13].
+Let us calculate the approximate form of the tight-binding Hamiltonian in the vicinity of the
+Dirac point ⃗K. We denote the effective operator as H(⃗k) ≡ H( ⃗K + ⃗k), where |⃗k| is considered
+small enough so that we can keep terms up to first-order in ⃗k. The proper expansion of H(⃗k)
+at the Dirac point ⃗K can be conveniently written as
+H(⃗k) = 2at1kxS1 + 2at2kyS2 + 4t3S3.
+(2)
+The matrices
+S1 =
+�
+�
+0
+1
+0
+1
+0
+0
+0
+0
+0
+�
+� ,
+S2 =
+�
+�
+0
+0
+1
+0
+0
+0
+1
+0
+0
+�
+� ,
+S3 =
+�
+�
+0
+0
+0
+0
+0
+−i
+0
+i
+0
+�
+� ,
+(3)
+form the three-dimensional representation of su(2) algebra, [Sp, Sq] = iεpqrSr, with εpqr the
+three-dimensional anti-symmetric tensor. Therefore, the quasi-particles described by the effec-
+tive Hamiltonian (2) have pseudospin 1.
+4
+
+元
+0
+2a
+a
+2
+w(k)
+0
+-2
+0
+元
+2a
+ky
+a元
+0
+2a
+a
+2
+w(k)
+-2
+0
+元
+2a
+ky
+aIt is worth noting that, for t3 = 0, the resulting Dirac Hamiltonians in (2) becomes linear
+combinations of the spin-1 matrices S1 and S2. In such a case, the matrix �S,
+�S =
+�
+�
+−1
+0
+0
+0
+1
+0
+0
+0
+1
+�
+� ,
+(4)
+satisfies {�S, Sj} = 0, with j = 1, 2, and represents the chiral symmetry of H as there holds
+{�S, H|t3=0} = 0. The later relation implies that the eigenvalues E of H|t3=0 are symmetric
+with respect to E = 0. When an eigenstate ΨE of H has energy E, then there is an eigenstate
+Ψ−E = �SΨE with the energy of the opposite sign.
+2.1
+External electrostatic interaction
+Throughout this manuscript, we consider a piece-wise continuous external electric field dis-
+tributed in the ˆx direction, while we discard any magnetic interaction. The corresponding effec-
+tive Hamiltonian is obtained from (2) through the Peierls transformation [35,36], ⃗k → −iℏ⃗∇ and
+iℏ∂t → iℏ∂t − U(⃗x)I, with I the 3 × 3 identity matrix. Since the Hamiltonian becomes invariant
+on the ˆy direction, the eigenstates can be cast in the form Ψ(x, y) → e±ik2yΨ(x), where Ψ(x)
+solve the following stationary equation:
+H(x)Ψ(x) = (−iℏv1S1∂x + ℏv2kyS2 + mS3 + Ua I)Ψ(x) = EΨ(x),
+(5)
+with Ψ(x) = (ψA(x), ψB(x), ψC(x))T .
+In (5), we have used v1 = 2at1, v2 = 2at2 and m = 4t3 to simplify the notation. This allows
+us relating v1 and v2 to the Fermi velocities along the ˆx and ˆy directions, respectively, whereas
+m plays the role of the mass term in the Dirac equation. Furthermore, we have considered
+a constant electrostatic potential, which is valid for our purposes since we are dealing with
+piece-wise continuous interactions.
+From the previous considerations, we may decouple the eigensolution components ψA,B,C as
+follows:
+− ℏ2v2
+1ψ′′
+A + ℏ2v2
+2k2
+yψA = ((E − Ua)2 − m2)ψA,
+(6)
+ψB = −iℏv1(E − Ua)ψ′
+A + ℏmv2kyψA
+(E − Ua)2 − m2
+,
+ψC = ℏmv1ψ′
+A + ℏv2ky(E − Ua)ψA
+(E − Ua)2 − m2
+,
+(7)
+where the hopping parameters tj, for j = 1, 2, 3.
+The probability current associated with Ψ can be calculated in standard manner from the
+continuity equation ∂tρ + ⃗∇ · j = 0. Here, ρ = Ψ†Ψ stands for the probability density, and the
+probability current takes the form
+j = (2v1 Re ψ∗
+AψB, 2v2 Re ψ∗
+AψC) .
+(8)
+Let us consider briefly the situation when the potential has a finite discontinuity at x = x0.
+It is necessary to specify the behavior of the wave functions at this point. It can be done by
+integrating (5) in the vicinity of x0. Alternatively, one can require the component of the density
+5
+
+current perpendicular to the barrier to be continuous. The second approach is more general and
+covers the boundary conditions provided by the integration as the special case that read as
+ψA(x−
+0 ) = ψA(x+
+0 ),
+ψB(x−
+0 ) = ψB(x+
+0 ).
+(9)
+It is worth noting that only two of the three eigensolution components are required to be
+continuous in x0, and the third component ψC can have a discontinuity at this point.
+The
+corresponding probability density is not necessarily continuous. This observation was made in
+pseudospin-1 photonic lattices [37]. The boundary conditions obtained in (9) keep the current
+of probability density in the ˆx direction continuous, which is the component perpendicular to
+the discontinuity. As the component ΨC(x) can be discontinuous at x0, the tangent current and
+the probability densities are not necessarily continuous.
+3
+Rectangular electrostatic barrier
+Let us consider an external electrostatic electric potential homogeneous along the ˆy direction
+and piece-wise continuous across the ˆx direction, with
+U(x) =
+�
+0
+|x| > L
+2
+U0
+|x| ≤ L
+2
+.
+(10)
+We consider, without loss of generality, U0 > 2m. Solutions of the stationary equation are split
+into three regions, namely, the region I (x < L/2), region II (−L/2 ≤ x ≤ L/2), and region III
+(x > L/2). The latter are written as
+Ξka = eikyyeikax
+�
+�
+�
+1
+−iℏmv2ky+ℏv1ka(E−Ua)
+(E−Ua)2−m2
+iℏmv1ka+ℏv2ky(E−Ua)
+(E−Ua)2−m2
+�
+�
+� ,
+a = I, II, III,
+(11)
+where UI = UIII = 0, UII = U0, and consequently kI = kIII.
+We consider ky is fixed as a
+real quantity to obtain plane-wave solutions propagating parallel to the barrier. In turn, ka is
+considered a complex parameter so that we can distinguish two different regimes (see discussion
+below).
+The solution (11) satisfies the eigenvalue equation (5) with the eigenvalue
+E = Ua ±
+�
+m2 + ℏ2(v2
+1k2a + v2
+2k2y) = Ua ±
+�
+˜m2 + ℏ2v2
+1k2a,
+(12)
+where we have introduced the effective mass term
+�m =
+�
+m2 + ℏ2v2
+2k2y.
+(13)
+From (11), we distinguish two behaviors, namely, plane-wave solutions for ka ∈ R and
+evanescent-wave solutions for ka = −ipa, with pa ∈ R.
+In both cases, the wave functions
+are associated with real eigenvalues. They are classified as
+ka ∈ R,
+E(ka, ky) = Ua ±
+�
+�m + v2
+1ℏ2k2a,
+E(pa, ky) ∈ (−∞, U0 − �m) ∪ (Ua + �m, ∞),
+(14)
+ka = −i pa,
+E(pa, ky) = Ua ±
+�
+�m − v2
+1ℏ2p2a,
+E(pa, ky) ∈ (Ua − �m, U0 + �m).
+(15)
+6
+
+(a)
+(b) |E| > m
+(c) |E| < m
+Figure 3: (a) Sketch of the energy surfaces spanned by the dispersion relations (14) (orange)
+and (15) (blue), together with two energy planes located at arbitrary energies |E| > m and
+|E| < m. Panel (b) and panel (c) depict the contour plot generated by the interception between
+the dispersion relations and the energy planes |E| > m and |E| < m, respectively. In panel (b),
+ξ = arctan(ky/kI) denotes the incidence angle of the plane wave and ky;c =
+√
+E2 − m2/ℏv2 the
+critical value of ky separating the evanescent-wave and plane-wave regimes.
+These dispersion relations span paraboloid and hyperboloid surfaces for plane-wave and evanescent-
+wave solutions, respectively. This behavior is depicted in Fig. 3a for UI = 0 (case a = I).
+For |E| > m, the behavior of the solutions is classified according to the values of ky, with
+ky;c =
+√
+E2 − m2/ℏv2 being the critical value. That is, for |ky| < ky;c, the solutions are plane-
+wave-like and the momenta kI and ky span an elliptic curve for a fixed energy. For |ky| > ky;c,
+the solutions become evanescent waves and pI and ky span a hyperbolic curve for the same fixed
+energy. This is sketched in Fig. 3b. For |E| < m, no plane-wave solutions exist for ky ∈ R,
+and only evanescent-wave solutions are generated. Here, pI, ky span a rotated hyperbola with
+respect to the case |E| > m, as depicted in Fig. 3c.
+The partial solutions Ξka at the regions I, II, III have to be combined in order to comply
+with the boundary conditions (9) at x0 = ±L and |x| → ∞. The wave function takes the general
+form
+Ξa(x, y) = αaΞ±
+ka(x, y) + βaΞ±
+−ka(x, y),
+a = I, II, III .
+(16)
+The boundary conditions (9) impose the continuity of the two upper components of the wave
+function, from which we find the set of relations between the coefficients αI, βI and αIII, βIII.
+That is,
+M
+�αI
+βI
+�
+=
+�αIII
+βIII
+�
+,
+M =
+�m11
+m12
+m21
+m22
+�
+,
+(17)
+with M being the transfer matrix, whose elements mij are functions of E, ky, m and U0. The
+7
+
+El>m
+E(k,k2)
+Ek<m?
+Plane-wave
+<region
+K
+Evanescentwave
+K=ki
+region
+K=iP1(p1,k2)
+k2;c
+(k1,k2)
+-K2:c
+K(p1,k2)
+Klatter are explicitly given by
+m11 = eiLkI
+�
+cos(LkII) − i sin(LkII)2v2
+1ℏ2k2
+I k2
+II + ˜m2(k2
+I + k2
+II)
+2E(E − V )kIkII
+�
+,
+(18)
+m22 = e−iLkI
+�
+cos(LkII) + i sin(LkII)2v2
+1ℏ2k2
+I k2
+II + ˜m2(k2
+I + k2
+II)
+2E(E − V )kIkII
+�
+,
+(19)
+m12 = isin(LkII)(k2
+yv2
+2(m2 + ℏ2k2
+yv2
+2) − v2
+1(m2 − ℏ2k2
+yv2
+2)k2
+1 − 2iv1v2kIkyE)( U0
+2 − E)
+ℏ2v2
+1kIkII(E2 − m2)(E − U0)
+,
+(20)
+where kI =
+√
+E2 − ˜m2/ℏv1 and kII =
+�
+(E − V )2 − ˜m2/ℏv1.
+The determinant of the transfer matrix is equal to one, in coherence with conservation of
+the probability current at the boundary. When kI and kII are real, there also holds m11 = m∗
+22
+and m12 = m∗
+21. In the next section, we shall use the transfer matrix for determinantion of the
+bound state energies as well as of the scattering characteristics of the plane-wave solutions.
+Additionally to (11), the Lieb lattice supports an additional solution in the form of a flat
+band, which is depicted in Fig. 2. This appears whenever E = Ua, and the eigensolutions cannot
+be determined from Eq. (6)-(7). Instead, one shall solve the Dirac Hamiltonian for E = Ua,
+which leads to the eigensolution Ξfb = (mχ, iℏv2kyχ, −ℏv1χ′)T , where χ = χ(x) is an arbitrary
+complex-valued function. Such an indeterminacy is better understood if one chooses χ such that
+Ξfb(ν, x) = eikyyeiνx
+�
+�
+m
+iℏv2ky
+−iℏv1ν
+�
+� ,
+(21)
+which is a flat band eigensolution for any ν ∈ C. Particularly, for ν ∈ R, Eq. (21) form a set of
+degenerate plane-wave solutions, usually known as degenerate Bloch waves [38] (see also Sec. 2.1
+in [18]). These degenerate waves form a continuous basis that can be used to construct arbitrary
+wavepackets through Fourier transforms. The latter has been exploited to construct the so-called
+compact localizes states [39], which are specific linear combinations of degenerate waves localized
+in each unitary cell of a finite-dimensional lattice. See [18, 40] for a more extensive discussion
+on the matter.
+It is clear that degenerate Bloch waves do not carry current on the x-direction, as jx =
+2v1 Re ψ∗
+AψB vanishes for any ν ∈ C.
+This also holds for any linear combination (finite or
+infinite) of degenerate Bloch state. Thus, the current states belonging to the flat band energy
+are current-free states.
+Notice that the dispersion and flat bands have a touching point only for m = 0 (See Fig. 2a),
+and thus one can explore the behavior of the solutions on the dispersion band when they approach
+the flat band interception. It is straightforward to realize that Ξka,ky,m→0 leads to the null vector,
+which is only one of the infinitely many solutions inside the flat band. For this reason, we shall
+discuss the flat band and the dispersion bands separately.
+4
+Electron confinement
+Let us explore the possibility of bound states trapped by the electrostatic potential (10). Here,
+we look for eigenvalues E so that the corresponding eigensolutions have finite norm in L2 ⊗ C3,
+8
+
+(a) ky = 0
+(b)
+(c)
+Figure 4: Sketch for the energy configuration associated with (10) for ky = 0 (a) and increasing
+values of ky (b)-(c). The diagonal-pattern and color-shaded regions denote the area covered by
+mass term m and effective mass term �m, respectively. In the panel (b), the energy (red-dashed
+line) inside the region II lies out of the effective mass term (plane-wave solution), whereas in
+the panel (c) they lie inside the effective mass term (evanescent-wave solution).
+which implies that eigensolutions must decay asymptotically to zero in the regions I and III for
+x → −∞ and x → ∞, respectively.
+Following (11), we thus use evanescent-wave solutions for the regions I and III. By fixing
+kI = kIII = ipI,
+pI > 0,
+(22)
+one restricts the energies into the interval E ∈ (− �m, �m), as depicted in all the cases of Fig. 4.
+The wave function composed from (16) has an exponentially vanishing behavior for |x| → ∞.
+This implies that we fix αI = 0, βI = 1 and βIII = 0, and the relation (17) turns into
+m12 = αIII,
+m22 = 0.
+(23)
+The first relation determines the amplitude of the wave function in the region III, whereas
+the second relation fixes the energies for the bound states. This can be written, after some
+simplifications, in the following form:
+tanh
+��
+�m2 − (E − U0)2
+ℏv1
+L
+�
+= −E(E − U0)
+�
+�m2 − (E − U0)2√
+�m2 − E2
+(E − U0)2( �m2 − E2) + �m2U0
+�
+E − U0
+2
+� .
+(24)
+The wave function ΞII in the intermediate region II can be either oscillatory for (E − U0)2 >
+˜m2 (we can set kII > 0 without loss of generality) or evanescent for (E − U0)2 < ˜m2 (kII = i pII,
+pII > 0), see Fig. 4b and Fig. 4c, respectively.
+The transcendental equation (24) allows us
+determining the bound state energies as a function of ky for both cases.
+Although the explicit solution E = E(ky) of (24) has to be found numerically, some pre-
+liminary information can be extracted by considering large values ℏv2ky ≫ U0, m in the tran-
+scendental equation (24). Here, �m ≈ ℏv2ky and the dispersion relation reduces to E2 ≈ E2
+∞ =
+ℏ2(−v2
+1p2
+I + v2
+2k2
+y). Since pI should be a real quantity in order to remain in the evanescent-wave
+regime in the regions I and III, we find that ℏv2|ky| ≥ E(ky) holds for asymptotic values of
+ℏv2ky. The behavior of E(ky) is thus bounded for ℏv2ky → ∞ and can be classified into the
+following in three asymptotic cases:
+9
+
+Uo+m
+Uo+m
+Uo
+m
+Uo-m
+10%m/7/7
+0
+-m
+1
+II
+III
+-mJo+m
+Uo
+Uo-m
+m 
+E
+0
+-m
+II
+IIIUo+m
+Uo
+Uo-m
+Uo-m
+m
+m
+E
+0
+-m
+-m
+II
+III(a)
+(b)
+Figure 5: (In units of ℏ=1) (a) Bound state energies E(ky), computed from (24), as a function
+of the transverse momentum ky for v1 = v2 = L = 1, m = 0.5, and U0 = 1.5. The blue-solid and
+red-dashed curves indicate bound state energies for arbitrary ky, whereas green-dot-dashed and
+black-dotted curves are energies emerging from a specific ky ̸= 0. The shaded area marks the
+scattering-state energy region. (b) Current parallel to the barrier Jy = ∂E(ky)/∂ky associated
+with the dispersion relations in (a).
+• First, a valid asymptotic behavior may be of the form E(ky → ∞) → C < ∞. Substituting
+the latter into (24) leads to a unique solution of the form E(ky → ∞) → C = U0/2.
+• Another possible asymptotic behavior is |E(ky)| = ℏv2|ky|, which vanishes both sides
+of (24). That is, |E(ky)| = ℏv2|ky| is a valid asymptotic behavior.
+• The last possible asymptotic behavior is |E(ky)| < ℏv2|ky|, which leads to a contradiction
+once substituted into (24). That is, such an asymptotic behavior is do not generate bound
+state solutions.
+We thus conclude that the eigenvalues associated with bound states, if they exist, either
+converge asymptotically to U0/2 or ℏv2|ky|. Since the current density on the direction parallel
+to the barrier is Jy = ∂E(ky)/∂ky ≡
+�
+R j(x, y)dx (see [41] or Appendix E in [42]), it converges
+either to zero or ±ℏv2 for ky → ∞.
+As an illustrative example, let us consider numerical values such that we have homogeneous
+Fermi velocities v1 = v2 = 1, a mass term m = 0.5, together with a rectangular potential well
+with L/ℏ = 1 and U0 = 1.5. Numerical solutions of (24) reveal the existence of two bound
+states for ky = 02, and new bound states appear for increasing values of ky. This is depicted in
+Fig. 5a, where one may see that energies indeed converge to either U0
+2 = 0.75 or become linear
+in ℏv2ky for large enough ky. Likewise, we depict in Fig. 5b the corresponding current density
+parallel to the barrier (Jy), which becomes finite or null for asymptotic ky, as predicted from
+our former analysis.
+• Further information is available for direct incidence, that is, ky = 0, �m = m. Here, the
+effective Hamiltonian possesses the additional symmetry represented [H, Px �S] = 0, with Px is
+the parity operator and �S defined in (4). This allows establishing a parity-symmetric criteria for
+the wave function �Ξ with respect to Px �S, namely, we classify the solutions fulfilling the condition
+Px �SΞ = ±Ξ as even (Ξ(e) for +) and odd (Ξ(e) for −). In this form, the coefficients of ΞII
+2This result agrees with the analytic formula presented in (26).
+10
+
+2
+0
+-4
+-2
+2
+4
+k2(k2
+-8
+-4
+4
+8
+K(a) U0 = 1.5
+(b)
+(c)
+Figure 6: (In units of ℏ=1) (a) Number of even (dotted) and odd (blue-thick) bound states
+as a function of L for v1 = 1, m = 0.5 and U0 = 1.5. (b) Eigensolution component ψC and
+(E − U(x))ψC for L = v1 = 1 and U0 = 1.5 and the even bound state energy E ≈ 0.281398.
+(c) Probability distribution associated with the eigenvalues E ≈ 0.281398 (blue-solid) and E ≈
+−0.32653 (red-dashing) and the same parameters as in (b).
+in (16) are αII = ±βII for even (+) and odd (−) functions, so that after evaluating the boundary
+condition at x = L one obtains relations to determine the energies of even and odd states as
+tan
+�kIIL
+2
+�
+= F(E),
+− cot
+�kIIL
+2
+�
+= F(E),
+F(E) =
+E
+U0 − E
+�
+(E − U0)2 − m2
+m2 − E2
+,
+(25)
+respectively, with kII =
+�
+(E − U0)2 − m2/ℏv1.
+Although the exact values of E cannot be analytically determined for arbitrary L, one can
+still determine the exact number of even (N(e)) and odd (N(o)) bound states. The thorough
+analysis (see App. A for a detailed proof) leads to
+N(e) =
+�
+L
+πℏv1
+�
+U0
+2
+�
+m + U0
+2
+�
++ 1
+2
+�
+−
+�
+L
+πℏv1
+�
+U0
+2
+�
+−m + U0
+2
+�
++ 1
+2
+�
++ 1,
+N(o) =
+�
+L
+πℏv1
+�
+U0
+2
+�
+m + U0
+2
+��
+−
+�
+L
+πℏv1
+�
+U0
+2
+�
+−m + U0
+2
+��
++ 1,
+(26)
+with ⌊·⌋ the floor function.
+From the latter, it is clear that at least one even and one odd bound state always exist,
+regardless of the potential width and strength.
+Particularly, for small enough L → 0, one
+obtains the E → 0 and odd E → −m as the even and odd bound state energies, respectively.
+Since the floor function is discontinuous, the number of bound states does not necessarily
+grow continuously for increasing values of L. That is, for L = L0 with N(e,o) bound states, there
+might be a L = L1 > L0 such that (N(e,o) − 1) are generated. This is indeed depicted in Fig. 6a
+for fixed potential depth and different potential length L.
+As discussed in Sec. 2.1, the component ψC might not be continuous, which can lead to
+discontinuous probability densities. Still, one may verify the validity of the bound state eigen-
+values E obtained from (25) by substituting it into (E −U(x))ψC, which should be a continuous
+function3. Particularly, from Fig. 6a, one notices that L = 1 and U0 = 1.5 lead to one even
+3It follows from kyΨB + (U(x) − E)ΨC = 0, which the third of the coupled equations represented by (5).
+11
+
+4
+N(e)
+N(o)
+3
+2
+1
+5
+13
+17
+L0.4
+c
+(E-U(x)Wc
+0
+-0.4
+-2
+-1
+1
+2
+X0.4
+0.2
+0
+-2
+-1
+1
+2
+X(E(e)
+0
+≈ 0.281398) and one odd (E(o)
+0
+≈ −0.32653) bound state energy eigenvalue. The compo-
+nent ψC and (E −U(x))ψC are depicted in Fig. 6b for E ≈ 0.281398, which verifies the required
+continuity condition for the latter function. The same conclusion is drawn for E ≈ −0.32653.
+Furthermore, the corresponding probability distributions associated with �Ψ
+(e) and �Ψ
+(o) are de-
+picted in Fig. 6c in blue-solid and red-dashed, respectively, which are discontinuous.
+5
+Scattering states and transmission amplitudes
+Let us now focus on the scattering of the plane waves on the barrier and the related phenomena.
+This is obtained when plane-wave-like solutions are present in the regions I and III, which
+corresponds to the eigenvalues E ∈ (−∞, − �m)∪( �m, ∞). Without loss of generality, we consider
+only outgoing waves in region III and outgoing together with incoming waves in region I. The
+coefficients of the wave function (16) are then fixed in the following manner,
+αI = 1,
+βI = r,
+αIII = t,
+βIII = 0.
+(27)
+The complex constants t and r can be calculated from (17) as
+t =
+1
+m22
+,
+r = −m21
+m22
+.
+(28)
+The coefficients r and t define the reflection and transmission coefficients R = |r|2 and T = |t|2
+that satisfy R + T = 1. The later expression can be directly verified by substituting from (28)
+when taking into account that there holds m11 = m∗
+22 and m12 = m∗
+21. After some calculations,
+one obtains,
+r = sin(kIIL)
+−2A(B − B′) + i
+�
+A′2 − A2 + (B − B′)2�
+2AA′ cos(kIIL) − i sin(kIIL) ((B − B′)2 + A2 + A′2),
+(29)
+where
+A
+v1
+=
+EkI
+E2 − m2 ,
+A′
+v1
+=
+(E − U0)kII
+(E − U0)2 − m2 ,
+B
+v2
+=
+mky
+E2 − m2 ,
+B′
+v2
+=
+mky
+(E − U0)2 − m2 ,
+(30)
+and kI =
+√
+E2− �m2
+ℏv1
+, kII =
+√
+(E−U0)2− �m2
+ℏv1
+. This expression also holds in cases where solutions in
+the region II are evanescent waves.
+Eq. (29) is a handy expression to understand the transmission of incoming waves from the
+region I and traveling to the region III. Particular interest is paid to cases in which perfect
+tunneling exists, T = 1. Such a tunneling is obtained whenever r = 0, which ensures that t
+is a unimodular complex number. In this case, the incident and transmitted waves share their
+amplitude, but the later carries a relative phase shift t as a leftover of its interaction with the
+barrier. For the sake of clarity, we split our discussion in two cases.
+Normal incidence (ky = 0)
+In this case, the reflection coefficient becomes simpler since B = B′ = 0, kI =
+√
+E2 − m2/ℏv1,
+and kII =
+�
+(E − U0)2 − m2/ℏv1. The numerator in r becomes proportional to m sin(kIIL).
+12
+
+Therefore, for the gapless lattice setup (m = 0), perfect tunneling occurs for any arbitrary ener-
+gies in E ∈ (−∞, −m)∪(m, ∞). This effect was reported in graphene [3,4,43] and pseudospin-1
+lattices [11,44].
+For m ̸= 0, perfect tunneling does exist for specific energies so that kIIL = nπ, with n = 1, . . ..
+The exact resonant energies are straightforward to compute and are presented in a much general
+case below. However, it is worth to analyze the behavior of T = 1 − |r|2 when the barrier is
+large enough, U0 ≫ m, E, for fixed and finite E. The straightforward calculations show that
+T ≈
+1
+1 +
+m4
+4E2(E2−m2) sin2 �
+U0L
+ℏv1
+�.
+(31)
+It reveals that despite the lack of the perfect tunneling, the transmission converges to a non-null
+value as the electrostatic barrier increases indefinitely. This is known as Klein paradox [45], and
+it is in sharp contrast with the non-relativistic case, where transmission becomes smaller for
+larger barrier heights.
+Oblique incidence (ky ̸= 0)
+• Super-Klein tunneling When B = B′ and A = ±A′ in from (29), the reflection coefficient
+vanishes and the transmission becomes perfect (T = 1). This is achieved when E = U0/2. One
+thus has perfect tunneling regardless of the incidence angle for E = U0/2. This phenomenon is
+called the super-Klein tunneling, already reported for pseudospin-1 lattice models with gapless
+dispersion and flat bands [37,44,46], as well as in pseudospin-1/2 graphene lattices [47]. Here,
+we note that the presence of the mass term (m ̸= 0) does not break the super-Klein tunneling as
+long as U0 > 2m. However, super-Klein tunneling is altogether lost by tuning the electrostatic
+barrier such that 0 < U0 < 2m, as no plane-wave solutions exist for E = U0/2. This highlights
+the effects of the mass term (band-gap) on the transmission properties.
+• Generalized Snell-Descartes law It is convenient to define the two-dimensional momentum
+vectors ⃗k = (kI, ky) and ⃗k′ = (kII, ky) that characterize the incident wave and the wave traveling
+through the electric barrier, respectively. The incident and transmitted angles are defined as
+ξ = arctan(ky/kI) and ξ′ = arctan(ky/kII), respectively, see Fig. 7a. Contrary to the bound
+state case of Sec. 4, plane-wave solutions only exist in the region I for bounded values of ky, i.e.,
+|ky| < ky;c =
+√
+E2 − m2/ℏv2. This alternatively implies that scattering phenomenon is available
+for restricted values of the effective-mass term �m. This is depicted in Fig. 7b, from which it is
+also clear that, for |ky| > ky;c, the shaded are covered by the effective-mass region overlaps with
+the energy E, leading to evanescent-wave solutions in the region I.
+From the dispersion relations in the regions I and II, together with the fact that ky is constant
+across all regions, one can establish a relation between the incident and transmitted angles ξ
+and ξ′ of Fig. 7a,
+tan ξ′
+tan ξ
+�
+v2
+1 + v2
+2 tan2 ξ
+v2
+1 + v2
+2 tan2 ξ′ =
+�
+E2 − m2
+(E − U0)2 − m2 .
+(32)
+For v1 = v2, one recovers the same Snell-Descartes law previously reported for graphene [4], and
+to the Snell’s law obtained for pseudospin-1 lattices with m = 0 reported in [46].
+Since we are considering U0 > 2m, we get the following information about the transmitted
+angle:
+13
+
+(a)
+(b)
+(c)
+Figure 7: (b) Scattering configuration (upper-view) for an incident wave ⃗k (region I), with inci-
+dent angle ξ, traveling through an electrostatic barrier (green-shaded area). The wave refracts
+into region II as a wave with vector ⃗k′ and transmitted angle ξ′. (b) Energy configuration of
+the panel (a) with an incident wave with energy E > �m (red-dashed line). (c) Energy curves
+spanned by κ, ky (region I and III) and κ′, ky (region II) for E > �m fixed as in panel (b).
+• For E ∈
+�
+m, U0
+2
+�
+, there exists a transmitted angle ξ′ for every incident angle ξ ∈ (−π/2, π/2).
+• For E = U0
+2 , the transmitted and incident angles are equal, ξ′ = ξ.
+• For E ∈
+� U0
+2 , U0 − m
+�
+∪ (U0 + m, ∞), there are transmitted angles ξ′ ∈ (−π/2, π/2) only
+for ξ ∈ (−ξc, ξc), with the critical angle tan2 ξc = v2
+1
+v2
+2
+(E−U0)2−m2
+2U0
+�
+E− U0
+2
+� . For other values of ξ, the
+solutions in the region II are evanescent waves.
+• For E ∈ (U0 − m, U0 + m), there are only evanescent waves in the region II.
+• Fabry-P´erot resonances Perfect transmission occurs for other energies as well, nevertheless,
+it gets angle dependent. The reflection coefficient (29) vanishes for kIIL = nπ, with n ∈ Z+.
+Since kII is in turn a function of the incidence angle ξ, once may conclude that perfect reflection
+appears only for some specific incidence angles. These are usually known as tunneling resonances
+or Fabry-P´erot resonances [4], and are given as a function of the incident angles ξ as
+E(res)
+±;n =
+�
+1 + v2
+2
+v2
+1
+tan2 ξ
+�
+�
+�
+�U0 ±
+�
+�
+�
+�
+�U 2
+0 −
+1
+1 + v2
+2
+v2
+1 tan2 ξ
+�
+�U 2
+0 − π2v2
+1(n + 1)2
+L2
+−
+m2
+1 + v2
+2
+v2
+1 tan2 ξ
+�
+�
+�
+�
+� ,
+(33)
+with n = 0, 1, . . ..
+These resonant energies behave asymptotically as limξ→±π/2 E(res)
++;n → ∞ and limξ→±π/2 E(res)
+−;n →
+(2U0)−1 �
+U 2
+0 − π2 v2
+1(n+1)2
+L2
+�
+. Thus, for almost perpendicular incident waves (ξ ∼ ±π/2), one re-
+quires larger and larger energies in order to recover the resonances at E(res)
++;n , whereas finite and
+well-defined energy values are required for the resonances E(res)
+−;n . This behavior is depicted in
+Fig. 8a.
+14
+
+I
+I1
+III
+k
+Ki
+k
+3
+1KUo+m
+Uo+m
+Uo
+Uo-m
+Uo-m
+E
+m
+m
+0
+-m
+-m
+II
+IIIRegion
+Region
+1
+II
+K(a)
+Figure 8: (Units of ℏ = 1) (a) Tunneling resonance energies E+;n (blue-solid) and E−;n (orange-
+dashed) as a function of ξ(−π/2, π/2). The inset depicts the transmission amplitude T as a
+function of E for ξ = 0. The shaded area denotes the region where the duple (ξ, E) produces
+evanescent waves in the region I. The parameters have been fixed as v1 = v2 = 1, m = 0.5 and
+U0 = 1.5.
+6
+Remarks on the flat-band solutions
+The piece-wise continuous nature of the electrostatic interaction (10) allows the generation of
+two flat band energies, one located at E = U0 for the region II, and another one at E = 0 for
+the regions I and III. Although the boundary conditions are the same in both cases, the allowed
+matching solutions have a different behavior.
+Let us first consider E = U0 and ky so that plane-wave solutions exist for the regions I and
+III. For generality, we consider incoming and outgoing plane waves in regions I and III, and a
+general flat-band solution in II. Here, the waves entering the interaction zone from the left and
+right have an amplitude I1 and I2, respectively, with I1,2 ∈ R. Additionally, we fix I2
+1 + I2
+2 = 1.
+Under these considerations, we have the general solutions
+Ξ =
+�
+�
+�
+�
+�
+I1ΞkI + A1Ξ−kI
+x < −L/2
+Ξfb
+|x| < L/2
+I2ΞkI + A2Ξ−kI
+x > L/2
+(34)
+where A1,2 ∈ C, Ξ = (mχ, iℏv2kyχ, −iℏv1χ′)T , with χ a complex-valued function, and Ξ±kI the
+solutions (11) evaluated at E = U0. By imposing the boundary conditions (9), one obtains the
+relations A1 = I1e−2iφeikIL and A2 = I2e2iφe−ikIL, with φ = arctan(v2kyU0/mv1kI), whereas the
+arbitrary function χ is restricted to fulfill the following relations at the boundaries,
+χ
+�
+− L
+2
+�
+=
+2v1kII1e−i(φ−kI L)
+�
+m2v1k2
+I + v2k2yU 2
+0
+,
+χ
+� L
+2
+�
+= −
+2v1kII2ei(φ−kI L)
+�
+m2v1k2
+I + v2k2yU 2
+0
+.
+(35)
+Given the arbitrary nature of χ, one may alternatively rewrite it as χ = 2v1kIei(φ−kI L)2x/L
+√
+m2v1k2
+I +v2k2yU2
+0 �χ, where
+�χ(−L/2) = I1 and �χ(L/2) = −I2.
+15
+
+20
+10
+-20
+E
+20
+E(res)
+n
+-10
+-20
+爪
+3
+2
+4
+4
+2Thus, the coupling of incident waves to the flat-band solution leads to a scattering problem
+in which the waves entering the interaction region are completely reflected inside their respective
+regions. Still, the flat band solutions allowed during such a process must fulfill the boundary
+conditions (35). Note that one also has the conservation property |A1|2 + |A2|2 = I2
+1 + I2
+2 = 1.
+The latter results hold whenever waves enter from only one region, say I1 = 1 and I2 = 0. In
+such a case, we have a perfect reflection in region I, up to a phase in the reflected wave.
+Flat-band solutions also occur for E = 0 in regions I and III. The arbitrary nature of the
+solutions in those flat bands can be tuned so that finite-norm solutions appear. The correspond-
+ing wave function in region II can be found using the boundary conditions, and the calculations
+are as straightforward as the scattering case presented above.
+7
+Concluding remarks
+In this manuscript, it was shown that the existence of a rectangular electrostatic barrier always
+produces at least two bound states for ky = 0, and generates more bound states at different
+energies for increasing values of ky. Interestingly, it was found that even for the asymptotic
+values ℏv2ky → ∞, the associated current density parallel to the barrier is bounded by ±ℏv2,
+where v2 = 2at2. Thus, the current is linear on the hopping amplitude across the ˆy-direction,
+as expected.
+It is worth remarking that dispersion relations obtained from (24) identify the energies for
+which electrons localize in the x-direction, and propagation is allowed in the ˆy-direction is still
+possible. However, by exploiting the separability of free-particle solutions, one can always con-
+struct linear combinations so that electrons localize in the ˆy-direction as well. Such a procedure
+has been discussed in [48] for graphene. For instance, in the example provided in Fig. 5a, one
+can take the energies associated with blue-solid and red-dashed curves as they exist for any
+ky ∈ R. From the relations (26), one can ensure that at least two of such dispersion relations
+always exist. Additional caution must be taken for the other dispersion relations, as they only
+exist for intervals ky ∈ S ⊆ R, and the linear combination must be constructed accordingly to
+that interval. Devising such packages is a task beyond the scope of the current work and will
+be discussed elsewhere, as it deserves attention by itself.
+On the one hand, for the scattering-wave regime, we have proved that even in the gapped
+case (m ̸= 0), the Lieb lattice supports super-Klein tunneling for an energy equal to half of
+the electric barrier, E = U0/2, provided that U0 > 2m. For the gapless case, we recover the
+same results previously reported for gapless T3 lattices [44] and ultra-cold atoms trapped in
+optical lattices [24]. On the other hand, we identified a new modified Snell-like law valid for
+anisotropic Fermi velocities v1 ̸= v2. The latter allows us to identify the Fabry-P´erot resonant
+transmission, which defines a relation between the incident energy and the incident-wave angle
+required to produce perfect tunneling up to a phase factor. Interestingly, for negative energies,
+perfect transmission is achievable for finite energies at incident waves almost perpendicular to
+the barrier. This is not the case for positive energies, as it is shown that the required energies
+diverge.
+The existence of flat-band solutions poses an additional case not available in graphene lattices.
+The latter allows the coupling of the solutions and determining the transmission properties,
+which in this case, leads to perfectly reflected waves. Since the flat-band solutions are defined
+16
+
+in terms of degenerate Bloch waves, there is an infinite family of solutions that allows such a
+reflection, as long as they fulfill the boundary condition (35).
+Acknowledgments
+K.Z. acknowledges the support from the project “Physicists on the move II” (KINE´O II)
+funded by the Ministry of Education, Youth, and Sports of the Czech Republic, Grant No.
+CZ.02.2.69/0.0/0.0/18 053/0017163.
+A
+Determining the number of even and odd bound states
+In this appendix, we present the derivation of the number of bound states for even bound
+states presented in (26). The procedure applies straightforwardly to the odd case as well. It is
+convenient to define the intervals
+I0 =
+�
+0, π
+2
+�
+,
+In =
+�π
+2 + (n − 1)π, π
+2 + nπ
+�
+,
+n = 1, 2, . . . ,
+(A-1)
+so that tan(x) is nonsingular for x ∈ In. Furthermore, if x ∈ ∪k=p2
+k=p1Ik, then tan(x) has (p2 − p1)
+singularities.
+To determine the number of even bound states, one must find the number of interceptions
+of F(E) in (25) and the periodic function tan(kIIL
+2 ) in the interval E ∈ (−m, m). To this end,
+one may notice that F(E) is a monotonously increasing function of E ∈ (−m, m) that tends to
+∓∞ for E → ∓m, and vanishes for E = 0. The latter means that F(E) defines the bijection
+F(E) : (−m, m) �→ R.
+On the other hand, ∂kII/∂E < 0 for E ∈ (−m, m), and one thus
+concludes that tan(kIIL/2) (and also − cot(kIIL/2)) is a monotonously decreasing function of E
+in each of the intervals kIIL
+2
+∈ In, with n = 0, 1, . . .. This property, combined with the fact that
+F(E) : (−m, m) �→ R is a monotonously increasing function, one concludes that interception of
+both functions always exist. One must determine the exact number of interceptions.
+By exploiting the fact that tan(kII L/2) is a periodic function, one just needs to count the
+number of periods inside the interval E ∈ (−m, m) for arbitrary U0 and L, which is equal to the
+number of singularities plus one. m is a lattice parameter, so it is assumed to be a fixed value.
+Be σ± := kIIL
+2 |E=∓m =
+L
+ℏv1
+�
+U0
+2
+�
+±m + U0
+2
+�
+so that the domain of tan
+�
+kIIL
+2
+�
+lies in the interval
+(σ−, σ+) for E ∈ (−m, m).
+Now, if σ− ∈ Ir1 and σ+ ∈ Ir2, with r2 > r1 and r1,2 = 0, 1, . . ., then, tan(kIIL/2) has
+r2 − r1 singularities for E ∈ (−m, m) and intercepts F(E) exactly (r2 − r1 + 1)-times. That is,
+N(e) = r2 − r1 + 1. The values of r1,2 are found by exploiting the fact that ⌊ x
+π + 1
+2⌋ = r for
+x ∈ Ir. One thus has ⌊ σ−
+π + 1
+2⌋ = r1 and ⌊ σ+
+π + 1
+2⌋ = r2, which leads to the expression presented
+in (26).
+The same procedure applies to the odd solutions, where we define the intervals �In = (nπ, (n+
+1)π), with n = 0, 1, . . ., so that cot(x) is nonsingular for x ∈ �In.
+Since − cot(kIIL/2) is
+monotonously decreasing for kIIL/2 ∈ �In, the same same reasoning used in the even case applies
+to the odd case, and one obtains N(o) in (26).
+17
+
+References
+[1] T.O. Wehling, A.M. Black-Shaffer, and A.V. Balatsky, “Dirac materials,” Adv. Phys. 63, 1
+(2014).
+[2] H. Aoki, M.S. Dresselhaus (eds.), Physics of Graphene (Springer International Publishing,
+Switzerland, 2014).
+[3] M.I. Katsnelson, K.S. Novoselov, and A.K, Geim, “Chiral tunnelling and the Klein paradox
+in graphene,” Nat. Phys. 2, 620–625 (2006).
+[4] P.E. Allain and J.N. Fuchs, “Klein tunneling in graphene: optics with massless electrons,”
+Eur. Phys. J. B 83, 301 (2011).
+[5] M. Yang et al., “Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide
+PtTe2,” Nat. Commun. 8, 257 (2017).
+[6] H. Zhang, Y. Xie , C. Zhong, Z. Zhang, and Y. Chen, “Tunable type-I and type-II Dirac
+Fermions in graphene with nitrogen line defects,” J. Phys. Chem. C 121, 12476 (2017).
+[7] E. McCann, V.I. Falko, “Landau-level degeneracy and quantum hall effect in a graphite
+bilayer,” Phys. Rev. Lett. 96, 086805 (2006).
+[8] F.D.M. Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-
+Matter Realization of the ‘Parity Anomaly’,” Phys. Rev. Lett. 61, 2015 (1988).
+[9] T. Tummuru, S. Plugge, and M. Franz, “Josephson effects in twisted cuprate bilayers,”
+Phys. Rev. B 105, 064501 (2022).
+[10] M. Mekata, “Kagome: The Story of the Basketweave Lattice,” Phys. Today 56, 12 (2003)
+[11] E. Illes, “Properties of the α − T3 model,” Ph.D. thesis.
+[12] B. Dey and T.K. Ghosh, “Floquet topological phase transition in the α−T3 lattice,” Phys.
+Rev. B 99, 205429 (2019).
+[13] V. Jakubsk´y and K. Zelaya, “Landau levels and snake states of pseudo-spin-1 Dirac-like
+electrons in gapped Lieb lattices,” J. Phys.: Condense Matt. 51 (2023) 025302.
+[14] N. Goldman, D.F. Urban, and D. Bercioux, “Topological phases for fermionic cold atoms
+on the Lieb lattice,” Phys. Rev. A 83, 063601 (2011).
+[15] W. Jiang et al., “Topological band evolution between Lieb and kagome lattices,” Physical
+Review B, 99(12) (2019). doi:10.1103/physrevb.99.125131
+[16] R. Fan, L. Sun, X. Shao, Y. Li, and M. Zhao, “Two-dimensional Dirac materials:
+tight-binding lattice models and material candidates,” ChemPhysMater (2022) accepted
+(https://doi.org/10.1016/j.chphma.2022.04.009).
+[17] L. Yan and P. Liljeroth, “Engineered electronic states in atomically precise artificial lattices
+and graphene nanoribbons,” Advances in Physics: X 4, 1651672 (2019). doi:
+18
+
+[18] D. Leykam, A. Andreanov and S. Flach, “Artificial flat band systems: from lattice models
+to experiments,” Advances in Physics: X 3, 1473052 (2018).
+[19] D. Guzm´an-Silva et al., “Experimental observation of bulk and edge transport in photonic
+Lieb lattices,” New J. Phys. 16, 063061 (2014).
+[20] R. A. Vicencio et al., “Observation of Localized States in Lieb Photonic Lattices,” Phys.
+Rev. Lett. 114, 245503 (2015).
+[21] F. Diebel et al., “Conical Diffraction and Composite Lieb Bosons in Photonic Lattices,”
+Phys. Rev. Lett. 116, 183902 (2016).
+[22] S. Mukherjee, A. Spracklen, D. Choudhury, N. Goldman, P. ¨Ohberg, E. Andersson, R.R.
+Thomson, “Observation of a Localized Flat-Band State in a Photonic Lieb Lattice,” Phys.
+Rev. Lett. 114 (24), 245504 (2015).
+[23] R. A. Vicencio, C. Cantillano, L. Morales-Inostroza, B. Real, C. Mej´ıa-Cort´es, S. Weimann,
+A. Szameit, and M.I. Molina, “Observation of Localized States in Lieb Photonic Lattices,”
+Phys. Rev. Lett. 114, 245503 (2015).
+[24] R. Shen, L.B. Shao, B. Wang, and D.Y. Xing, “Single Dirac cone with a flat band touching
+on line-centered-square optical lattices,” Phys. Rev. B 81, 041410(R) (2010).
+[25] M.R. Slot et al., “Experimental realization and characterization of an electronic Lieb lat-
+tice,” Nat. Phys. 13, 672 (2017).
+[26] B. Cui, X. Zheng, J. Wang, D. Liu, S. Xie, B. Huang, “Realization of Lieb lattice in covalent-
+organic frameworks with tunable topology and magnetism,” Nature Communications 11,
+66 (2020).
+[27] D. Green, L. Santos, and C. Chamon, “Isolated Flat Bands and Spin-1 Conical Bands in
+Two-Dimensional Lattices,” arXiv:1004.0708
+[28] Y. Klymenko, L. Malysheva, and A. Onipko, “Electron transmission through step- and
+barrier-like potentials in graphene ribbons,” Phys. Status Solidi B 245, 2181-2184 (2008).
+[29] L. Wei-Tao, L. Wen, and Y. Cheng-Zhi, “Enlarged band gap and electron switch in
+graphene-based step-barrier structure,” Appl. Phys. Lett. 103, 192102 (2013).
+[30] J.H. Bardarson, M. Titov, and P.W. Brouwer, “Electrostatic Confinement of Electrons in
+an Integrable Graphene Quantum Dot,” Phys. Rev. Lett. 102, 226803 (2009).
+[31] Y. Long and J. Ren, “Topological Landau-Zener Bloch Oscillations in Photonic Floquet
+Lieb Lattices,” arXiv:1706.01107 (2017).
+[32] S.A. Owerre, “Photoinduced Topological Phase Transitions in Topological Magnon Insula-
+tors,” Sci Rep 8, 4431 (2018).
+[33] C.-Z. Chang et al., “Experimental Observation of the Quantum Anomalous Hall Effect in
+a Magnetic Topological Insulator,” Science 340, 167 (2013).
+[34] S. Xing et al., “Theory, properties and engineering of 2D magnetic materials” Progress in
+Material Science 132, 101036 (2023).
+19
+
+[35] R. Peierls, “Zur Theorie der galvanomagnetischen Effekte,” Zeits. fur Physik 53, 255 (1929).
+[36] F. Bloch, “¨Uber die Quantenmechanik der Elektronen in Kristallgittern,” Zeits. fur Physik
+52, 555 (1928).
+[37] H. Xu and Y.-C. Lai, “Superscattering of a pseudospin-1 wave in a photonic lattice,” Phys.
+Rev. A 95, 012119 (2017).
+[38] H. Aoki, H. Ando, and H. Hatsumura, “Hofstadter butterflies for flat bands,” Phys. Rev.
+B. 54, 17296 (1996).
+[39] D.L. Bergman, C. Wu, and L. Balents, “Band touching from real-space topology in frus-
+trated hopping models,” Phys. Rev. B 78, 125104 (2008).
+[40] J.-W. Rhim and B.-Jung Yang, “Singular flat bands,” Adv. Phys. X 6, 1901606 (2021).
+[41] T. K. Ghosh, A. De Martino, W. H¨ausler, L. Dell’Anna, and R. Egger, “Conductance
+quantization and snake states in graphene magnetic waveguides,” Phys. Rev. B 77, 081404
+(2008).
+[42] W. Ashcroft and N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).
+[43] V. Jakubsk´y, L. M. Nieto and M. S. Plyushchay, “Klein tunneling in carbon nanostructures:
+A Free particle dynamics in disguise,” Phys. Rev. D 83, 047702 (2011).
+[44] D.F. Urban, D. Bercioux, M. Wimmer, and W. H¨ausler, Barrier transmission of Dirac-like
+pseudospin-one particles, Phys. Rev. B 84, 115136 (2011).
+[45] N. Dombey and A. Calogeracos, “Seventy years of the Klein paradox,” Phys. Rep. 315, 41
+(1999).
+[46] Y. Betancur-Ocampo, G. Cordourier-Maruri, V. Gupta, and R. de Coss, “Super-Klein
+tunneling of massive pseudospin-one particles,” Phys. Rev. B 96, 023404 (2017).
+[47] A. Contreras-Astorga, F. Correa, and V. Jakubsk´y, Super-Klein tunneling of Dirac fermions
+through electrostatic gratings in graphene, Phys. Rev. B 102, 115429 (2020).
+[48] V. Jakubsk´y and M. Tuˇsek, “Dispersionless wave packets in Dirac materials,” Annals of
+Physics 378, 171 (2017).
+20
+
diff --git a/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/load_file.txt b/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..4870422c1d1c3ae84bde569da9e4f28efe4f3ad7
--- /dev/null
+++ b/AdE1T4oBgHgl3EQf9Aai/content/tmp_files/load_file.txt
@@ -0,0 +1,727 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf,len=726
+page_content='Lieb lattices and pseudospin-1 dynamics under barrier- and  well-like electrostatic interactions  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jakubsk´y1 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zelaya1  1Nuclear Physics Institute, Czech Academy of Science, 250 68 ˇReˇz, Czech Republic  Abstract  This work considers the confining and scattering phenomena of electrons in a Lieb lattice  subjected to the influence of a rectangular electrostatic barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In this setup, hopping  amplitudes between nearest neighbors in orthogonal directions are considered different, and  the next-nearest neighbor interaction describes spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This makes it possible to  confine electrons and generate bound states, the exact number of which is exactly determined  for null parallel momentum to the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In such a case, it is proved that one even and one  odd bound state is always generated, and the number of bound states increases for non-null  and increasing values of the parallel momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is, bound states carry current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In the  scattering regime, the exact values of energy are determined where the resonant tunneling  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The existence of perfect tunneling energy in the form of super-Klein tunneling is  proved to exist regardless of the bang gap opening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Finally, it is shown that perfect reflection  appears when solutions are coupled to the intermediate flat-band solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  1  Introduction  The theoretical and experimental progress in the physics of graphene and other Dirac materials  has become a trending topic in material science and theoretical physics [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Many remarkable  properties of these materials follow from the fact that dynamics of low-energy quasi-particles is  described by equations known in relativistic quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It makes it possible to test  relativistic properties such as Klein tunneling [3,4], relativistic Landau levels, and the existence  of pseudoparticles violating the Lorentz invariance [5, 6] (type-II Dirac fermions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Graphene  mono- and multi-layer systems exhibit transport properties such as quantum Hall effect [7]  and anomalous quantum Hall effect in graphene [8], and Josephson effect in twisted cuprate  bilayers [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Graphene has shown to be a helpful benchmark system to test the properties of relativistic  pseudospin-1/2 particles in low-energy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Nevertheless, the family of Dirac materials  contains also other, equally interesting, members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Their geometries can extend beyond the  honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For instance, there are Kagome [10], Dice or α − T3 [11, 12], and Lieb  lattices [13, 14], which lead to effective pseudospin-1 Dirac equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It was recently showed  that the Kagome lattice can be obtained from a geometrical deformation of the Lieb lattice [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For a recent survey of two-dimensional lattices and their physical properties and realization,  see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='03552v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='mes-hall]  9 Jan 2023  Particularly, the Lieb lattice is a two-dimensional array with a periodicity of a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The sites are located in the corners of each square and at the midpoints on its sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' To our  best knowledge, the Lieb lattice has not been found in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' However, it has been prepared  artificially in diverse ways [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It was realized in experiments with optical fibers [19–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Furthermore, it was formed by ultracold atoms trapped in optical lattices [24] or by electrons  of Cu(111) atoms confined by an array of CO molecules [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It was also prepared in covalent-  organic frameworks [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The tight-binding model can well describe the band structure of the Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It reveals  the existence of two bands with positive and negative energies and an additional so-called flat  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter is associated with the states that have fixed (zero) energy independent of the  value of momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It is worth mentioning that the flat band solutions were prepared in the  optical experiments, see [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Similarly to graphene, the dynamics of the low-energy quasi-  particles in the Lieb lattice is dictated by a relativistic Dirac-type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Nevertheless, these  quasi-particles have pseudospin-1 due to three atoms per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In the current article, we investigate the scattering and confinement of the relativistic quasi-  particles by a rectangular electric potential in the Lieb lattice with a gapped band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Gap-opening can be induced by on-site energy that differs on three sublattices or by the phase  acquired by the electron when jumping between the neighboring sites [24], see also [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In  the article, we adopt the second approach where a purely imaginary next nearest-neighbor  interaction, attributed to spin-orbital coupling [13], is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Effects such as electron confinement and transmission are obtained with the aid of the proper  boundary conditions, which enforce the continuity on two out of the three pseudospin-1 compo-  nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The third component can be discontinuous, which leads to a spatial discontinuity in the  probability density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Nevertheless, it does not compromise the associated continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Electron dynamics for electrostatic interactions in graphene have been discussed in the litera-  ture, such as the transmission properties in square barriers [28,29] and electron confinement with  cylindrical quantum dots [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' We thus focus on the related properties of the quasi-particle dy-  namics in the Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' We further analyze the influence of the flat-band solution in electron  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' As shown in the manuscript, solutions in this regime are described by degenerate  Bloch-wave solutions whose linear combinations can compose wavepackets of arbitrary form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  These are shown to be current-free solutions regardless of the nature of the wavepacket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' As a  result, one obtains perfectly reflected waves when they couple to flat-band solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The manuscript is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2 we briefly introduce and discuss the main  properties of the Lieb lattice with nearest next-nearest neighbor interactions, from which the  effective low-energy Dirac equation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 3, we present the general solutions  and the transfer matrix associated with the rectangular electrostatic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter  is then exploited in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 4 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 5 to discuss in full detail the localization of electrons and  scattering dynamics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Finally, discussions and perspectives are provided in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 7,  and complementary details about the proof of the number of bound states are given in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  2  (a)  (b)  Figure 1: (a) Lieb lattice, composed by the atoms A (blue-filled circle), B (green square), and C  (red-filled square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The dashed arrows denote the direction of positive phase hopping parameter  between next-nearest neighbors B − C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (b) Composition of a unit cell of the Lieb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The  unit displacement vectors ⃗δ1 = aˆx and ˆδ2 = aˆy connect the atom A with B and A with C,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The corresponding nearest hopping parameters are t1, t2, whereas the next-nearest  neighbor hopping parameter is +it3 and −it3 depending if it occurs in the direction denotes by  the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  2  Lieb lattice and pseudospin-1 Dirac equation  Let us consider an electronic Lieb lattice 1 so that the separation between two nearest atoms is  a, the length of each side of the square is ℓ = 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' There are three sites in the elementary cell, see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The primitive translation vectors are ⃗r1 = 2aˆx and ⃗r2 = 2aˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It is customary to denote  the atoms at the corners of the square as A, whereas the atoms at the sides of the square are B  (horizontal) and C (vertical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The lattice vectors ⃗δ1 = aˆx = ⃗r1/2 and ⃗δ2 = aˆy = ⃗r2/2 connect an  atom on the site A to those on the sites B and C, respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The atoms A, B and  C form the three sublattices RA = n1⃗r1 +n2⃗r2, RB = ˜RA +⃗δ1, and RC = ⃗RA +⃗δ2, respectively,  with n1, n2 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The reciprocal space is spanned by the translation vectors of the reciprocal  space ˆrk1 and ˆrk2, ˆrp · ˆrkq = 2πδp,q, p, q = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This leads to ˆrk1 = π  a ˆx and ˆrk2 = π  a ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The  first Brillouin zone, constructed from the Wigner-Seitz rule, restricts to the region composed by  kx ∈ [− π  2a, π  2a] and ky ∈ [− π  2a, π  2a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The band structure of the electrons on the Lieb lattice can be analyzed with the use of the  tight-binding model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' There are considered the nearest neighbor (NN) interactions between the  sites A − B and A − C, represented by the hopping parameters t1 and t2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' We take  into account also the next-nearest neighbor (NNN) transition B − C, which can be complex  valued, with the sign of phase dependent on the orientation of the hopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This emerges due  to external time-dependent driven fields in photonic Lieb lattices [31], and magnon Lieb and  Kagome lattices [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In particular, we consider a purely imaginary NNN hopping parameter e±iπ/2t3, where the  1The results here obtained apply to optical Lieb lattices as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  3  C  A  BA  B(a) t3 = 0  (b) t3 ̸= 0  Figure 2: Dispersion bands w+(⃗k) (yellow-upper), w−(⃗k) (green-lower), and w0(⃗k) (blue-middle)  for the gapless (a) and gapped (b) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  hopping phase is positive (+) is the hopping occurs counter-clock-wise, and negative (−) oth-  erwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Such a hopping dynamics is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This type of hopping was introduced  by Haldane in [8] as a model for quantum anomalous Hall effect in graphene without strong  external magnetic fields, which was latter found experimentally in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' See also [34] for a recent  review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The spectral analysis of the tight-binding Hamiltonian reveals that there are three bands in  its spectrum [13],  w0(⃗k) = 0,  w±(⃗k) = ±2  �  t2  1 cos2(akx) + t2  2 cos2(aky) + 4t2  3 sin2(akx) sin2(aky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (1)  The bands have linear dependence on the momentum in the four Dirac points that are  situated in the first Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Their explicit position depends on the relative strength of  t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In this work, we focus on the most relevant situation where t3 < t1  2 , t3 < t2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In that case,  the Dirac point is ⃗K = ( π  2a, π  2a), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='2b for illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A similar analysis holds for higher  values of t3, where the Dirac points are displaced with respect to ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For a detailed discussion,  see [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Let us calculate the approximate form of the tight-binding Hamiltonian in the vicinity of the  Dirac point ⃗K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' We denote the effective operator as H(⃗k) ≡ H( ⃗K + ⃗k), where |⃗k| is considered  small enough so that we can keep terms up to first-order in ⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The proper expansion of H(⃗k)  at the Dirac point ⃗K can be conveniently written as  H(⃗k) = 2at1kxS1 + 2at2kyS2 + 4t3S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (2)  The matrices  S1 =  �  �  0  1  0  1  0  0  0  0  0  �  � ,  S2 =  �  �  0  0  1  0  0  0  1  0  0  �  � ,  S3 =  �  �  0  0  0  0  0  −i  0  i  0  �  � ,  (3)  form the three-dimensional representation of su(2) algebra, [Sp, Sq] = iεpqrSr, with εpqr the  three-dimensional anti-symmetric tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Therefore, the quasi-particles described by the effec-  tive Hamiltonian (2) have pseudospin 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  4  元  0  2a  a  2  w(k)  0  2  0  元  2a  ky  a元  0  2a  a  2  w(k)  2  0  元  2a  ky  aIt is worth noting that, for t3 = 0, the resulting Dirac Hamiltonians in (2) becomes linear  combinations of the spin-1 matrices S1 and S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In such a case, the matrix �S,  �S =  �  �  −1  0  0  0  1  0  0  0  1  �  � ,  (4)  satisfies {�S, Sj} = 0, with j = 1, 2, and represents the chiral symmetry of H as there holds  {�S, H|t3=0} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The later relation implies that the eigenvalues E of H|t3=0 are symmetric  with respect to E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' When an eigenstate ΨE of H has energy E, then there is an eigenstate  Ψ−E = �SΨE with the energy of the opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='1  External electrostatic interaction  Throughout this manuscript, we consider a piece-wise continuous external electric field dis-  tributed in the ˆx direction, while we discard any magnetic interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The corresponding effec-  tive Hamiltonian is obtained from (2) through the Peierls transformation [35,36], ⃗k → −iℏ⃗∇ and  iℏ∂t → iℏ∂t − U(⃗x)I, with I the 3 × 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Since the Hamiltonian becomes invariant  on the ˆy direction, the eigenstates can be cast in the form Ψ(x, y) → e±ik2yΨ(x), where Ψ(x)  solve the following stationary equation:  H(x)Ψ(x) = (−iℏv1S1∂x + ℏv2kyS2 + mS3 + Ua I)Ψ(x) = EΨ(x),  (5)  with Ψ(x) = (ψA(x), ψB(x), ψC(x))T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In (5), we have used v1 = 2at1, v2 = 2at2 and m = 4t3 to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This allows  us relating v1 and v2 to the Fermi velocities along the ˆx and ˆy directions, respectively, whereas  m plays the role of the mass term in the Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Furthermore, we have considered  a constant electrostatic potential, which is valid for our purposes since we are dealing with  piece-wise continuous interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  From the previous considerations, we may decouple the eigensolution components ψA,B,C as  follows:  − ℏ2v2  1ψ′′  A + ℏ2v2  2k2  yψA = ((E − Ua)2 − m2)ψA,  (6)  ψB = −iℏv1(E − Ua)ψ′  A + ℏmv2kyψA  (E − Ua)2 − m2  ,  ψC = ℏmv1ψ′  A + ℏv2ky(E − Ua)ψA  (E − Ua)2 − m2  ,  (7)  where the hopping parameters tj, for j = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The probability current associated with Ψ can be calculated in standard manner from the  continuity equation ∂tρ + ⃗∇ · j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here, ρ = Ψ†Ψ stands for the probability density, and the  probability current takes the form  j = (2v1 Re ψ∗  AψB, 2v2 Re ψ∗  AψC) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (8)  Let us consider briefly the situation when the potential has a finite discontinuity at x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  It is necessary to specify the behavior of the wave functions at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It can be done by  integrating (5) in the vicinity of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Alternatively, one can require the component of the density  5  current perpendicular to the barrier to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The second approach is more general and  covers the boundary conditions provided by the integration as the special case that read as  ψA(x−  0 ) = ψA(x+  0 ),  ψB(x−  0 ) = ψB(x+  0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (9)  It is worth noting that only two of the three eigensolution components are required to be  continuous in x0, and the third component ψC can have a discontinuity at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The  corresponding probability density is not necessarily continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This observation was made in  pseudospin-1 photonic lattices [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The boundary conditions obtained in (9) keep the current  of probability density in the ˆx direction continuous, which is the component perpendicular to  the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' As the component ΨC(x) can be discontinuous at x0, the tangent current and  the probability densities are not necessarily continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  3  Rectangular electrostatic barrier  Let us consider an external electrostatic electric potential homogeneous along the ˆy direction  and piece-wise continuous across the ˆx direction, with  U(x) =  �  0  |x| > L  2  U0  |x| ≤ L  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (10)  We consider, without loss of generality, U0 > 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Solutions of the stationary equation are split  into three regions, namely, the region I (x < L/2), region II (−L/2 ≤ x ≤ L/2), and region III  (x > L/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter are written as  Ξka = eikyyeikax  �  �  �  1  −iℏmv2ky+ℏv1ka(E−Ua)  (E−Ua)2−m2  iℏmv1ka+ℏv2ky(E−Ua)  (E−Ua)2−m2  �  �  � ,  a = I, II, III,  (11)  where UI = UIII = 0, UII = U0, and consequently kI = kIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  We consider ky is fixed as a  real quantity to obtain plane-wave solutions propagating parallel to the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In turn, ka is  considered a complex parameter so that we can distinguish two different regimes (see discussion  below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The solution (11) satisfies the eigenvalue equation (5) with the eigenvalue  E = Ua ±  �  m2 + ℏ2(v2  1k2a + v2  2k2y) = Ua ±  �  ˜m2 + ℏ2v2  1k2a,  (12)  where we have introduced the effective mass term  �m =  �  m2 + ℏ2v2  2k2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (13)  From (11), we distinguish two behaviors, namely, plane-wave solutions for ka ∈ R and  evanescent-wave solutions for ka = −ipa, with pa ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  In both cases, the wave functions  are associated with real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' They are classified as  ka ∈ R,  E(ka, ky) = Ua ±  �  �m + v2  1ℏ2k2a,  E(pa, ky) ∈ (−∞, U0 − �m) ∪ (Ua + �m, ∞),  (14)  ka = −i pa,  E(pa, ky) = Ua ±  �  �m − v2  1ℏ2p2a,  E(pa, ky) ∈ (Ua − �m, U0 + �m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (15)  6  (a)  (b) |E| > m  (c) |E| < m  Figure 3: (a) Sketch of the energy surfaces spanned by the dispersion relations (14) (orange)  and (15) (blue), together with two energy planes located at arbitrary energies |E| > m and  |E| < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Panel (b) and panel (c) depict the contour plot generated by the interception between  the dispersion relations and the energy planes |E| > m and |E| < m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In panel (b),  ξ = arctan(ky/kI) denotes the incidence angle of the plane wave and ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c =  √  E2 − m2/ℏv2 the  critical value of ky separating the evanescent-wave and plane-wave regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  These dispersion relations span paraboloid and hyperboloid surfaces for plane-wave and evanescent-  wave solutions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This behavior is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 3a for UI = 0 (case a = I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For |E| > m, the behavior of the solutions is classified according to the values of ky, with  ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c =  √  E2 − m2/ℏv2 being the critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is, for |ky| < ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c, the solutions are plane-  wave-like and the momenta kI and ky span an elliptic curve for a fixed energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For |ky| > ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c,  the solutions become evanescent waves and pI and ky span a hyperbolic curve for the same fixed  energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is sketched in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For |E| < m, no plane-wave solutions exist for ky ∈ R,  and only evanescent-wave solutions are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here, pI, ky span a rotated hyperbola with  respect to the case |E| > m, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The partial solutions Ξka at the regions I, II, III have to be combined in order to comply  with the boundary conditions (9) at x0 = ±L and |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The wave function takes the general  form  Ξa(x, y) = αaΞ±  ka(x, y) + βaΞ±  −ka(x, y),  a = I, II, III .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (16)  The boundary conditions (9) impose the continuity of the two upper components of the wave  function, from which we find the set of relations between the coefficients αI, βI and αIII, βIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  That is,  M  �αI  βI  �  =  �αIII  βIII  �  ,  M =  �m11  m12  m21  m22  �  ,  (17)  with M being the transfer matrix, whose elements mij are functions of E, ky, m and U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The  7  El>m  E(k,k2)  Ek<m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Plane-wave  <region  K  Evanescentwave  K=ki  region  K=iP1(p1,k2)  k2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c  (k1,k2)  K2:c  K(p1,k2)  Klatter are explicitly given by  m11 = eiLkI  �  cos(LkII) − i sin(LkII)2v2  1ℏ2k2  I k2  II + ˜m2(k2  I + k2  II)  2E(E − V )kIkII  �  ,  (18)  m22 = e−iLkI  �  cos(LkII) + i sin(LkII)2v2  1ℏ2k2  I k2  II + ˜m2(k2  I + k2  II)  2E(E − V )kIkII  �  ,  (19)  m12 = isin(LkII)(k2  yv2  2(m2 + ℏ2k2  yv2  2) − v2  1(m2 − ℏ2k2  yv2  2)k2  1 − 2iv1v2kIkyE)( U0  2 − E)  ℏ2v2  1kIkII(E2 − m2)(E − U0)  ,  (20)  where kI =  √  E2 − ˜m2/ℏv1 and kII =  �  (E − V )2 − ˜m2/ℏv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The determinant of the transfer matrix is equal to one, in coherence with conservation of  the probability current at the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' When kI and kII are real, there also holds m11 = m∗  22  and m12 = m∗  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In the next section, we shall use the transfer matrix for determinantion of the  bound state energies as well as of the scattering characteristics of the plane-wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Additionally to (11), the Lieb lattice supports an additional solution in the form of a flat  band, which is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This appears whenever E = Ua, and the eigensolutions cannot  be determined from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (6)-(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Instead, one shall solve the Dirac Hamiltonian for E = Ua,  which leads to the eigensolution Ξfb = (mχ, iℏv2kyχ, −ℏv1χ′)T , where χ = χ(x) is an arbitrary  complex-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Such an indeterminacy is better understood if one chooses χ such that  Ξfb(ν, x) = eikyyeiνx  �  �  m  iℏv2ky  −iℏv1ν  �  � ,  (21)  which is a flat band eigensolution for any ν ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Particularly, for ν ∈ R, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (21) form a set of  degenerate plane-wave solutions, usually known as degenerate Bloch waves [38] (see also Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='1  in [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' These degenerate waves form a continuous basis that can be used to construct arbitrary  wavepackets through Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter has been exploited to construct the so-called  compact localizes states [39], which are specific linear combinations of degenerate waves localized  in each unitary cell of a finite-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' See [18, 40] for a more extensive discussion  on the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  It is clear that degenerate Bloch waves do not carry current on the x-direction, as jx =  2v1 Re ψ∗  AψB vanishes for any ν ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  This also holds for any linear combination (finite or  infinite) of degenerate Bloch state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Thus, the current states belonging to the flat band energy  are current-free states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Notice that the dispersion and flat bands have a touching point only for m = 0 (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2a),  and thus one can explore the behavior of the solutions on the dispersion band when they approach  the flat band interception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It is straightforward to realize that Ξka,ky,m→0 leads to the null vector,  which is only one of the infinitely many solutions inside the flat band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For this reason, we shall  discuss the flat band and the dispersion bands separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  4  Electron confinement  Let us explore the possibility of bound states trapped by the electrostatic potential (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here,  we look for eigenvalues E so that the corresponding eigensolutions have finite norm in L2 ⊗ C3,  8  (a) ky = 0  (b)  (c)  Figure 4: Sketch for the energy configuration associated with (10) for ky = 0 (a) and increasing  values of ky (b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The diagonal-pattern and color-shaded regions denote the area covered by  mass term m and effective mass term �m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In the panel (b), the energy (red-dashed  line) inside the region II lies out of the effective mass term (plane-wave solution), whereas in  the panel (c) they lie inside the effective mass term (evanescent-wave solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  which implies that eigensolutions must decay asymptotically to zero in the regions I and III for  x → −∞ and x → ∞, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Following (11), we thus use evanescent-wave solutions for the regions I and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' By fixing  kI = kIII = ipI,  pI > 0,  (22)  one restricts the energies into the interval E ∈ (− �m, �m), as depicted in all the cases of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The wave function composed from (16) has an exponentially vanishing behavior for |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  This implies that we fix αI = 0, βI = 1 and βIII = 0, and the relation (17) turns into  m12 = αIII,  m22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (23)  The first relation determines the amplitude of the wave function in the region III, whereas  the second relation fixes the energies for the bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This can be written, after some  simplifications, in the following form:  tanh  ��  �m2 − (E − U0)2  ℏv1  L  �  = −E(E − U0)  �  �m2 − (E − U0)2√  �m2 − E2  (E − U0)2( �m2 − E2) + �m2U0  �  E − U0  2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (24)  The wave function ΞII in the intermediate region II can be either oscillatory for (E − U0)2 >  ˜m2 (we can set kII > 0 without loss of generality) or evanescent for (E − U0)2 < ˜m2 (kII = i pII,  pII > 0), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 4b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 4c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The transcendental equation (24) allows us  determining the bound state energies as a function of ky for both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Although the explicit solution E = E(ky) of (24) has to be found numerically, some pre-  liminary information can be extracted by considering large values ℏv2ky ≫ U0, m in the tran-  scendental equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here, �m ≈ ℏv2ky and the dispersion relation reduces to E2 ≈ E2  ∞ =  ℏ2(−v2  1p2  I + v2  2k2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Since pI should be a real quantity in order to remain in the evanescent-wave  regime in the regions I and III, we find that ℏv2|ky| ≥ E(ky) holds for asymptotic values of  ℏv2ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The behavior of E(ky) is thus bounded for ℏv2ky → ∞ and can be classified into the  following in three asymptotic cases:  9  Uo+m  Uo+m  Uo  m  Uo-m  10%m/7/7  0  m  1  II  III  mJo+m  Uo  Uo-m  m  E  0  m  II  IIIUo+m  Uo  Uo-m  Uo-m  m  m  E  0  m  m  II  III(a)  (b)  Figure 5: (In units of ℏ=1) (a) Bound state energies E(ky), computed from (24), as a function  of the transverse momentum ky for v1 = v2 = L = 1, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5, and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The blue-solid and  red-dashed curves indicate bound state energies for arbitrary ky, whereas green-dot-dashed and  black-dotted curves are energies emerging from a specific ky ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The shaded area marks the  scattering-state energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (b) Current parallel to the barrier Jy = ∂E(ky)/∂ky associated  with the dispersion relations in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  First, a valid asymptotic behavior may be of the form E(ky → ∞) → C < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Substituting  the latter into (24) leads to a unique solution of the form E(ky → ∞) → C = U0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Another possible asymptotic behavior is |E(ky)| = ℏv2|ky|, which vanishes both sides  of (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is, |E(ky)| = ℏv2|ky| is a valid asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The last possible asymptotic behavior is |E(ky)| < ℏv2|ky|, which leads to a contradiction  once substituted into (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is, such an asymptotic behavior is do not generate bound  state solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  We thus conclude that the eigenvalues associated with bound states, if they exist, either  converge asymptotically to U0/2 or ℏv2|ky|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Since the current density on the direction parallel  to the barrier is Jy = ∂E(ky)/∂ky ≡  �  R j(x, y)dx (see [41] or Appendix E in [42]), it converges  either to zero or ±ℏv2 for ky → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  As an illustrative example, let us consider numerical values such that we have homogeneous  Fermi velocities v1 = v2 = 1, a mass term m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5, together with a rectangular potential well  with L/ℏ = 1 and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Numerical solutions of (24) reveal the existence of two bound  states for ky = 02, and new bound states appear for increasing values of ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is depicted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 5a, where one may see that energies indeed converge to either U0  2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='75 or become linear  in ℏv2ky for large enough ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Likewise, we depict in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 5b the corresponding current density  parallel to the barrier (Jy), which becomes finite or null for asymptotic ky, as predicted from  our former analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Further information is available for direct incidence, that is, ky = 0, �m = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here, the  effective Hamiltonian possesses the additional symmetry represented [H, Px �S] = 0, with Px is  the parity operator and �S defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This allows establishing a parity-symmetric criteria for  the wave function �Ξ with respect to Px �S, namely, we classify the solutions fulfilling the condition  Px �SΞ = ±Ξ as even (Ξ(e) for +) and odd (Ξ(e) for −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In this form, the coefficients of ΞII  2This result agrees with the analytic formula presented in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  10  2  0  4  2  2  4  k2(k2  8  4  4  8  K(a) U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5  (b)  (c)  Figure 6: (In units of ℏ=1) (a) Number of even (dotted) and odd (blue-thick) bound states  as a function of L for v1 = 1, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5 and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (b) Eigensolution component ψC and  (E − U(x))ψC for L = v1 = 1 and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5 and the even bound state energy E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='281398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (c) Probability distribution associated with the eigenvalues E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='281398 (blue-solid) and E ≈  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='32653 (red-dashing) and the same parameters as in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  in (16) are αII = ±βII for even (+) and odd (−) functions, so that after evaluating the boundary  condition at x = L one obtains relations to determine the energies of even and odd states as  tan  �kIIL  2  �  = F(E),  − cot  �kIIL  2  �  = F(E),  F(E) =  E  U0 − E  �  (E − U0)2 − m2  m2 − E2  ,  (25)  respectively, with kII =  �  (E − U0)2 − m2/ℏv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Although the exact values of E cannot be analytically determined for arbitrary L, one can  still determine the exact number of even (N(e)) and odd (N(o)) bound states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The thorough  analysis (see App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A for a detailed proof) leads to  N(e) =  �  L  πℏv1  �  U0  2  �  m + U0  2  �  + 1  2  �  −  �  L  πℏv1  �  U0  2  �  −m + U0  2  �  + 1  2  �  + 1,  N(o) =  �  L  πℏv1  �  U0  2  �  m + U0  2  ��  −  �  L  πℏv1  �  U0  2  �  −m + U0  2  ��  + 1,  (26)  with ⌊·⌋ the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  From the latter, it is clear that at least one even and one odd bound state always exist,  regardless of the potential width and strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Particularly, for small enough L → 0, one  obtains the E → 0 and odd E → −m as the even and odd bound state energies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Since the floor function is discontinuous, the number of bound states does not necessarily  grow continuously for increasing values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is, for L = L0 with N(e,o) bound states, there  might be a L = L1 > L0 such that (N(e,o) − 1) are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is indeed depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 6a  for fixed potential depth and different potential length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='1, the component ψC might not be continuous, which can lead to  discontinuous probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Still, one may verify the validity of the bound state eigen-  values E obtained from (25) by substituting it into (E −U(x))ψC, which should be a continuous  function3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Particularly, from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 6a, one notices that L = 1 and U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5 lead to one even  3It follows from kyΨB + (U(x) − E)ΨC = 0, which the third of the coupled equations represented by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  11  4  N(e)  N(o)  3  2  1  5  13  17  L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='4  c  (E-U(x)Wc  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='4  2  1  1  2  X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='2  0  2  1  1  2  X(E(e)  0  ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='281398) and one odd (E(o)  0  ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='32653) bound state energy eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The compo-  nent ψC and (E −U(x))ψC are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 6b for E ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='281398, which verifies the required  continuity condition for the latter function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The same conclusion is drawn for E ≈ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='32653.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Furthermore, the corresponding probability distributions associated with �Ψ  (e) and �Ψ  (o) are de-  picted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 6c in blue-solid and red-dashed, respectively, which are discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  5  Scattering states and transmission amplitudes  Let us now focus on the scattering of the plane waves on the barrier and the related phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  This is obtained when plane-wave-like solutions are present in the regions I and III, which  corresponds to the eigenvalues E ∈ (−∞, − �m)∪( �m, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Without loss of generality, we consider  only outgoing waves in region III and outgoing together with incoming waves in region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The  coefficients of the wave function (16) are then fixed in the following manner,  αI = 1,  βI = r,  αIII = t,  βIII = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (27)  The complex constants t and r can be calculated from (17) as  t =  1  m22  ,  r = −m21  m22  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (28)  The coefficients r and t define the reflection and transmission coefficients R = |r|2 and T = |t|2  that satisfy R + T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The later expression can be directly verified by substituting from (28)  when taking into account that there holds m11 = m∗  22 and m12 = m∗  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' After some calculations,  one obtains,  r = sin(kIIL)  −2A(B − B′) + i  �  A′2 − A2 + (B − B′)2�  2AA′ cos(kIIL) − i sin(kIIL) ((B − B′)2 + A2 + A′2),  (29)  where  A  v1  =  EkI  E2 − m2 ,  A′  v1  =  (E − U0)kII  (E − U0)2 − m2 ,  B  v2  =  mky  E2 − m2 ,  B′  v2  =  mky  (E − U0)2 − m2 ,  (30)  and kI =  √  E2− �m2  ℏv1  , kII =  √  (E−U0)2− �m2  ℏv1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This expression also holds in cases where solutions in  the region II are evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (29) is a handy expression to understand the transmission of incoming waves from the  region I and traveling to the region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Particular interest is paid to cases in which perfect  tunneling exists, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Such a tunneling is obtained whenever r = 0, which ensures that t  is a unimodular complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In this case, the incident and transmitted waves share their  amplitude, but the later carries a relative phase shift t as a leftover of its interaction with the  barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For the sake of clarity, we split our discussion in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Normal incidence (ky = 0)  In this case, the reflection coefficient becomes simpler since B = B′ = 0, kI =  √  E2 − m2/ℏv1,  and kII =  �  (E − U0)2 − m2/ℏv1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The numerator in r becomes proportional to m sin(kIIL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  12  Therefore, for the gapless lattice setup (m = 0), perfect tunneling occurs for any arbitrary ener-  gies in E ∈ (−∞, −m)∪(m, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This effect was reported in graphene [3,4,43] and pseudospin-1  lattices [11,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For m ̸= 0, perfect tunneling does exist for specific energies so that kIIL = nπ, with n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='.  The exact resonant energies are straightforward to compute and are presented in a much general  case below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' However, it is worth to analyze the behavior of T = 1 − |r|2 when the barrier is  large enough, U0 ≫ m, E, for fixed and finite E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The straightforward calculations show that  T ≈  1  1 +  m4  4E2(E2−m2) sin2 �  U0L  ℏv1  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (31)  It reveals that despite the lack of the perfect tunneling, the transmission converges to a non-null  value as the electrostatic barrier increases indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is known as Klein paradox [45], and  it is in sharp contrast with the non-relativistic case, where transmission becomes smaller for  larger barrier heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Oblique incidence (ky ̸= 0)  Super-Klein tunneling When B = B′ and A = ±A′ in from (29), the reflection coefficient  vanishes and the transmission becomes perfect (T = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is achieved when E = U0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' One  thus has perfect tunneling regardless of the incidence angle for E = U0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This phenomenon is  called the super-Klein tunneling, already reported for pseudospin-1 lattice models with gapless  dispersion and flat bands [37,44,46], as well as in pseudospin-1/2 graphene lattices [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here,  we note that the presence of the mass term (m ̸= 0) does not break the super-Klein tunneling as  long as U0 > 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' However, super-Klein tunneling is altogether lost by tuning the electrostatic  barrier such that 0 < U0 < 2m, as no plane-wave solutions exist for E = U0/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This highlights  the effects of the mass term (band-gap) on the transmission properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Generalized Snell-Descartes law It is convenient to define the two-dimensional momentum  vectors ⃗k = (kI, ky) and ⃗k′ = (kII, ky) that characterize the incident wave and the wave traveling  through the electric barrier, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The incident and transmitted angles are defined as  ξ = arctan(ky/kI) and ξ′ = arctan(ky/kII), respectively, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Contrary to the bound  state case of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 4, plane-wave solutions only exist in the region I for bounded values of ky, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=',  |ky| < ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c =  √  E2 − m2/ℏv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This alternatively implies that scattering phenomenon is available  for restricted values of the effective-mass term �m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 7b, from which it is  also clear that, for |ky| > ky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='c, the shaded are covered by the effective-mass region overlaps with  the energy E, leading to evanescent-wave solutions in the region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  From the dispersion relations in the regions I and II, together with the fact that ky is constant  across all regions, one can establish a relation between the incident and transmitted angles ξ  and ξ′ of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 7a,  tan ξ′  tan ξ  �  v2  1 + v2  2 tan2 ξ  v2  1 + v2  2 tan2 ξ′ =  �  E2 − m2  (E − U0)2 − m2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (32)  For v1 = v2, one recovers the same Snell-Descartes law previously reported for graphene [4], and  to the Snell’s law obtained for pseudospin-1 lattices with m = 0 reported in [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Since we are considering U0 > 2m, we get the following information about the transmitted  angle:  13  (a)  (b)  (c)  Figure 7: (b) Scattering configuration (upper-view) for an incident wave ⃗k (region I), with inci-  dent angle ξ, traveling through an electrostatic barrier (green-shaded area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The wave refracts  into region II as a wave with vector ⃗k′ and transmitted angle ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (b) Energy configuration of  the panel (a) with an incident wave with energy E > �m (red-dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' (c) Energy curves  spanned by κ, ky (region I and III) and κ′, ky (region II) for E > �m fixed as in panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For E ∈  �  m, U0  2  �  , there exists a transmitted angle ξ′ for every incident angle ξ ∈ (−π/2, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For E = U0  2 , the transmitted and incident angles are equal, ξ′ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For E ∈  � U0  2 , U0 − m  �  ∪ (U0 + m, ∞), there are transmitted angles ξ′ ∈ (−π/2, π/2) only  for ξ ∈ (−ξc, ξc), with the critical angle tan2 ξc = v2  1  v2  2  (E−U0)2−m2  2U0  �  E− U0  2  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For other values of ξ, the  solutions in the region II are evanescent waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  For E ∈ (U0 − m, U0 + m), there are only evanescent waves in the region II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Fabry-P´erot resonances Perfect transmission occurs for other energies as well, nevertheless,  it gets angle dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The reflection coefficient (29) vanishes for kIIL = nπ, with n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Since kII is in turn a function of the incidence angle ξ, once may conclude that perfect reflection  appears only for some specific incidence angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' These are usually known as tunneling resonances  or Fabry-P´erot resonances [4], and are given as a function of the incident angles ξ as  E(res)  ±;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n =  �  1 + v2  2  v2  1  tan2 ξ  �  �  �  �U0 ±  �  �  �  �  �U 2  0 −  1  1 + v2  2  v2  1 tan2 ξ  �  �U 2  0 − π2v2  1(n + 1)2  L2  −  m2  1 + v2  2  v2  1 tan2 ξ  �  �  �  �  � ,  (33)  with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='.  These resonant energies behave asymptotically as limξ→±π/2 E(res)  +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n → ∞ and limξ→±π/2 E(res)  −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n →  (2U0)−1 �  U 2  0 − π2 v2  1(n+1)2  L2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Thus, for almost perpendicular incident waves (ξ ∼ ±π/2), one re-  quires larger and larger energies in order to recover the resonances at E(res)  +;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n , whereas finite and  well-defined energy values are required for the resonances E(res)  −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This behavior is depicted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  14  I  I1  III  k  Ki  k  3  1KUo+m  Uo+m  Uo  Uo-m  Uo-m  E  m  m  0  m  m  II  IIIRegion  Region  1  II  K(a)  Figure 8: (Units of ℏ = 1) (a) Tunneling resonance energies E+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n (blue-solid) and E−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='n (orange-  dashed) as a function of ξ(−π/2, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The inset depicts the transmission amplitude T as a  function of E for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The shaded area denotes the region where the duple (ξ, E) produces  evanescent waves in the region I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The parameters have been fixed as v1 = v2 = 1, m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5 and  U0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  6  Remarks on the flat-band solutions  The piece-wise continuous nature of the electrostatic interaction (10) allows the generation of  two flat band energies, one located at E = U0 for the region II, and another one at E = 0 for  the regions I and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Although the boundary conditions are the same in both cases, the allowed  matching solutions have a different behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Let us first consider E = U0 and ky so that plane-wave solutions exist for the regions I and  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For generality, we consider incoming and outgoing plane waves in regions I and III, and a  general flat-band solution in II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Here, the waves entering the interaction zone from the left and  right have an amplitude I1 and I2, respectively, with I1,2 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Additionally, we fix I2  1 + I2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Under these considerations, we have the general solutions  Ξ =  �  �  �  �  �  I1ΞkI + A1Ξ−kI  x < −L/2  Ξfb  |x| < L/2  I2ΞkI + A2Ξ−kI  x > L/2  (34)  where A1,2 ∈ C, Ξ = (mχ, iℏv2kyχ, −iℏv1χ′)T , with χ a complex-valued function, and Ξ±kI the  solutions (11) evaluated at E = U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' By imposing the boundary conditions (9), one obtains the  relations A1 = I1e−2iφeikIL and A2 = I2e2iφe−ikIL, with φ = arctan(v2kyU0/mv1kI), whereas the  arbitrary function χ is restricted to fulfill the following relations at the boundaries,  χ  �  − L  2  �  =  2v1kII1e−i(φ−kI L)  �  m2v1k2  I + v2k2yU 2  0  ,  χ  � L  2  �  = −  2v1kII2ei(φ−kI L)  �  m2v1k2  I + v2k2yU 2  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  (35)  Given the arbitrary nature of χ, one may alternatively rewrite it as χ = 2v1kIei(φ−kI L)2x/L  √  m2v1k2  I +v2k2yU2  0 �χ, where  �χ(−L/2) = I1 and �χ(L/2) = −I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  15  20  10  20  E  20  E(res)  n  10  20  爪  3  2  4  4  2Thus, the coupling of incident waves to the flat-band solution leads to a scattering problem  in which the waves entering the interaction region are completely reflected inside their respective  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Still, the flat band solutions allowed during such a process must fulfill the boundary  conditions (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Note that one also has the conservation property |A1|2 + |A2|2 = I2  1 + I2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The latter results hold whenever waves enter from only one region, say I1 = 1 and I2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' In  such a case, we have a perfect reflection in region I, up to a phase in the reflected wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Flat-band solutions also occur for E = 0 in regions I and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The arbitrary nature of the  solutions in those flat bands can be tuned so that finite-norm solutions appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The correspond-  ing wave function in region II can be found using the boundary conditions, and the calculations  are as straightforward as the scattering case presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  7  Concluding remarks  In this manuscript, it was shown that the existence of a rectangular electrostatic barrier always  produces at least two bound states for ky = 0, and generates more bound states at different  energies for increasing values of ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Interestingly, it was found that even for the asymptotic  values ℏv2ky → ∞, the associated current density parallel to the barrier is bounded by ±ℏv2,  where v2 = 2at2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Thus, the current is linear on the hopping amplitude across the ˆy-direction,  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  It is worth remarking that dispersion relations obtained from (24) identify the energies for  which electrons localize in the x-direction, and propagation is allowed in the ˆy-direction is still  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' However, by exploiting the separability of free-particle solutions, one can always con-  struct linear combinations so that electrons localize in the ˆy-direction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Such a procedure  has been discussed in [48] for graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For instance, in the example provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 5a, one  can take the energies associated with blue-solid and red-dashed curves as they exist for any  ky ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' From the relations (26), one can ensure that at least two of such dispersion relations  always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Additional caution must be taken for the other dispersion relations, as they only  exist for intervals ky ∈ S ⊆ R, and the linear combination must be constructed accordingly to  that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Devising such packages is a task beyond the scope of the current work and will  be discussed elsewhere, as it deserves attention by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  On the one hand, for the scattering-wave regime, we have proved that even in the gapped  case (m ̸= 0), the Lieb lattice supports super-Klein tunneling for an energy equal to half of  the electric barrier, E = U0/2, provided that U0 > 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' For the gapless case, we recover the  same results previously reported for gapless T3 lattices [44] and ultra-cold atoms trapped in  optical lattices [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' On the other hand, we identified a new modified Snell-like law valid for  anisotropic Fermi velocities v1 ̸= v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter allows us to identify the Fabry-P´erot resonant  transmission, which defines a relation between the incident energy and the incident-wave angle  required to produce perfect tunneling up to a phase factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Interestingly, for negative energies,  perfect transmission is achievable for finite energies at incident waves almost perpendicular to  the barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' This is not the case for positive energies, as it is shown that the required energies  diverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The existence of flat-band solutions poses an additional case not available in graphene lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The latter allows the coupling of the solutions and determining the transmission properties,  which in this case, leads to perfectly reflected waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Since the flat-band solutions are defined  16  in terms of degenerate Bloch waves, there is an infinite family of solutions that allows such a  reflection, as long as they fulfill the boundary condition (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Acknowledgments  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' acknowledges the support from the project “Physicists on the move II” (KINE´O II)  funded by the Ministry of Education, Youth, and Sports of the Czech Republic, Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='69/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='0/18 053/0017163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  A  Determining the number of even and odd bound states  In this appendix, we present the derivation of the number of bound states for even bound  states presented in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The procedure applies straightforwardly to the odd case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' It is  convenient to define the intervals  I0 =  �  0, π  2  �  ,  In =  �π  2 + (n − 1)π, π  2 + nπ  �  ,  n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' ,  (A-1)  so that tan(x) is nonsingular for x ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Furthermore, if x ∈ ∪k=p2  k=p1Ik, then tan(x) has (p2 − p1)  singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  To determine the number of even bound states, one must find the number of interceptions  of F(E) in (25) and the periodic function tan(kIIL  2 ) in the interval E ∈ (−m, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' To this end,  one may notice that F(E) is a monotonously increasing function of E ∈ (−m, m) that tends to  ∓∞ for E → ∓m, and vanishes for E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The latter means that F(E) defines the bijection  F(E) : (−m, m) �→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  On the other hand, ∂kII/∂E < 0 for E ∈ (−m, m), and one thus  concludes that tan(kIIL/2) (and also − cot(kIIL/2)) is a monotonously decreasing function of E  in each of the intervals kIIL  2  ∈ In, with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='. This property, combined with the fact that  F(E) : (−m, m) �→ R is a monotonously increasing function, one concludes that interception of  both functions always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' One must determine the exact number of interceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  By exploiting the fact that tan(kII L/2) is a periodic function, one just needs to count the  number of periods inside the interval E ∈ (−m, m) for arbitrary U0 and L, which is equal to the  number of singularities plus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' m is a lattice parameter, so it is assumed to be a fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Be σ± := kIIL  2 |E=∓m =  L  ℏv1  �  U0  2  �  ±m + U0  2  �  so that the domain of tan  �  kIIL  2  �  lies in the interval  (σ−, σ+) for E ∈ (−m, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Now, if σ− ∈ Ir1 and σ+ ∈ Ir2, with r2 > r1 and r1,2 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', then, tan(kIIL/2) has  r2 − r1 singularities for E ∈ (−m, m) and intercepts F(E) exactly (r2 − r1 + 1)-times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' That is,  N(e) = r2 − r1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' The values of r1,2 are found by exploiting the fact that ⌊ x  π + 1  2⌋ = r for  x ∈ Ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' One thus has ⌊ σ−  π + 1  2⌋ = r1 and ⌊ σ+  π + 1  2⌋ = r2, which leads to the expression presented  in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  The same procedure applies to the odd solutions, where we define the intervals �In = (nπ, (n+  1)π), with n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', so that cot(x) is nonsingular for x ∈ �In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Since − cot(kIIL/2) is  monotonously decreasing for kIIL/2 ∈ �In, the same same reasoning used in the even case applies  to the odd case, and one obtains N(o) in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  17  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wehling, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Black-Shaffer, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Balatsky, “Dirac materials,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 63, 1  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Aoki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Dresselhaus (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' ), Physics of Graphene (Springer International Publishing,  Switzerland, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [3] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Katsnelson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Novoselov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='K, Geim, “Chiral tunnelling and the Klein paradox  in graphene,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 2, 620–625 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Allain and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Fuchs, “Klein tunneling in graphene: optics with massless electrons,”  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 83, 301 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Lorentz-violating type-II Dirac fermions in transition metal dichalcogenide  PtTe2,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 8, 257 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Xie , C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Chen, “Tunable type-I and type-II Dirac  Fermions in graphene with nitrogen line defects,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' C 121, 12476 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' McCann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Falko, “Landau-level degeneracy and quantum hall effect in a graphite  bilayer,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 96, 086805 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Haldane, “Model for a Quantum Hall Effect without Landau Levels: Condensed-  Matter Realization of the ‘Parity Anomaly’,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 61, 2015 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Tummuru, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Plugge, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Franz, “Josephson effects in twisted cuprate bilayers,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 105, 064501 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Mekata, “Kagome: The Story of the Basketweave Lattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Today 56, 12 (2003)  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Illes, “Properties of the α − T3 model,” Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [12] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Dey and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Ghosh, “Floquet topological phase transition in the α−T3 lattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 99, 205429 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jakubsk´y and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zelaya, “Landau levels and snake states of pseudo-spin-1 Dirac-like  electrons in gapped Lieb lattices,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' : Condense Matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 51 (2023) 025302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Goldman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Urban, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Bercioux, “Topological phases for fermionic cold atoms  on the Lieb lattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A 83, 063601 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [15] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Topological band evolution between Lieb and kagome lattices,” Physical  Review B, 99(12) (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='1103/physrevb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='125131  [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Fan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Shao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zhao, “Two-dimensional Dirac materials:  tight-binding lattice models and material candidates,” ChemPhysMater (2022) accepted  (https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='chphma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Yan and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Liljeroth, “Engineered electronic states in atomically precise artificial lattices  and graphene nanoribbons,” Advances in Physics: X 4, 1651672 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' doi:  18  [18] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Leykam, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Andreanov and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Flach, “Artificial flat band systems: from lattice models  to experiments,” Advances in Physics: X 3, 1473052 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Guzm´an-Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Experimental observation of bulk and edge transport in photonic  Lieb lattices,” New J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 16, 063061 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Vicencio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Observation of Localized States in Lieb Photonic Lattices,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 114, 245503 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [21] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Diebel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Conical Diffraction and Composite Lieb Bosons in Photonic Lattices,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 116, 183902 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Mukherjee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Spracklen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Choudhury, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Goldman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' ¨Ohberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Andersson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Thomson, “Observation of a Localized Flat-Band State in a Photonic Lieb Lattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 114 (24), 245504 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Vicencio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Cantillano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Morales-Inostroza, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Real, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Mej´ıa-Cort´es, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Weimann,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Szameit, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Molina, “Observation of Localized States in Lieb Photonic Lattices,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 114, 245503 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [24] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Shen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Shao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Xing, “Single Dirac cone with a flat band touching  on line-centered-square optical lattices,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 81, 041410(R) (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Slot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Experimental realization and characterization of an electronic Lieb lat-  tice,” Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 13, 672 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [26] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Cui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Xie, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Huang, “Realization of Lieb lattice in covalent-  organic frameworks with tunable topology and magnetism,” Nature Communications 11,  66 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [27] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Green, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Santos, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Chamon, “Isolated Flat Bands and Spin-1 Conical Bands in  Two-Dimensional Lattices,” arXiv:1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='0708  [28] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Klymenko, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Malysheva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Onipko, “Electron transmission through step- and  barrier-like potentials in graphene ribbons,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Status Solidi B 245, 2181-2184 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [29] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wei-Tao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Cheng-Zhi, “Enlarged band gap and electron switch in  graphene-based step-barrier structure,” Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 103, 192102 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Bardarson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Titov, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Brouwer, “Electrostatic Confinement of Electrons in  an Integrable Graphene Quantum Dot,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 102, 226803 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Long and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Ren, “Topological Landau-Zener Bloch Oscillations in Photonic Floquet  Lieb Lattices,” arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='01107 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Owerre, “Photoinduced Topological Phase Transitions in Topological Magnon Insula-  tors,” Sci Rep 8, 4431 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='-Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Experimental Observation of the Quantum Anomalous Hall Effect in  a Magnetic Topological Insulator,” Science 340, 167 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Xing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=', “Theory, properties and engineering of 2D magnetic materials” Progress in  Material Science 132, 101036 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  19  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Peierls, “Zur Theorie der galvanomagnetischen Effekte,” Zeits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' fur Physik 53, 255 (1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [36] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Bloch, “¨Uber die Quantenmechanik der Elektronen in Kristallgittern,” Zeits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' fur Physik  52, 555 (1928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Xu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Lai, “Superscattering of a pseudospin-1 wave in a photonic lattice,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' A 95, 012119 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [38] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Aoki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Ando, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Hatsumura, “Hofstadter butterflies for flat bands,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 54, 17296 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Bergman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Balents, “Band touching from real-space topology in frus-  trated hopping models,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 78, 125104 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [40] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rhim and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='-Jung Yang, “Singular flat bands,” Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' X 6, 1901606 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [41] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Ghosh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' De Martino, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' H¨ausler, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Dell’Anna, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Egger, “Conductance  quantization and snake states in graphene magnetic waveguides,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 77, 081404  (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [42] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Ashcroft and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Mermin, Solid State Physics (Saunders College, Philadelphia, 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [43] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jakubsk´y, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Nieto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Plyushchay, “Klein tunneling in carbon nanostructures:  A Free particle dynamics in disguise,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' D 83, 047702 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [44] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Urban, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Bercioux, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Wimmer, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' H¨ausler, Barrier transmission of Dirac-like  pseudospin-one particles, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 84, 115136 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [45] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Dombey and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Calogeracos, “Seventy years of the Klein paradox,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' 315, 41  (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [46] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Betancur-Ocampo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Cordourier-Maruri, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Gupta, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' de Coss, “Super-Klein  tunneling of massive pseudospin-one particles,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 96, 023404 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Contreras-Astorga, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Correa, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jakubsk´y, Super-Klein tunneling of Dirac fermions  through electrostatic gratings in graphene, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' B 102, 115429 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  [48] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Jakubsk´y and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content=' Tuˇsek, “Dispersionless wave packets in Dirac materials,” Annals of  Physics 378, 171 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
+page_content='  20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AdE1T4oBgHgl3EQf9Aai/content/2301.03552v1.pdf'}
diff --git a/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf b/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..09fc09acbe13222e20ad22606210c6e569cb1919
--- /dev/null
+++ b/BtFKT4oBgHgl3EQfXS78/content/2301.11794v1.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:25a98f12b951e4eb1ba486615982763c875f354a03b252ae1a128126503ba235
+size 2910540
diff --git a/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/2301.05552v1.pdf.txt b/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/2301.05552v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f8b4f5441cc5d3435a28c0994bbc8740c63686b4
--- /dev/null
+++ b/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/2301.05552v1.pdf.txt
@@ -0,0 +1,1587 @@
+Application of the partial Dirichlet-Neumann contact algorithm to simulate
+low-velocity impact events on composite structures
+G. Guillameta,∗, A. Quintanas-Corominasa,b,c, M. Riveroa, G. Houzeauxa, M. V´azqueza, A. Turonb
+aBarcelona Supercomputing Center (BSC), Pla¸ca Eusebi G¨uell, 1-3, Barcelona, 08034, Catalonia, Spain
+bAMADE, Universitat de Girona, Av. Universitat de Girona 4, Girona, 17003, Catalonia, Spain
+cDepartment of Civil and Environmental Engineering, Imperial College London, London, SW7 2AZ, UK
+Abstract
+Impact simulations for damage resistance analysis are computationally intensive due to contact algo-
+rithms and advanced damage models. Both methods, which are the main ingredients in an impact event, re-
+quire refined meshes at the contact zone to obtain accurate predictions of the contact force and damage onset
+and propagation through the material. This work presents the application of the partial Dirichlet-Neumann
+contact algorithm to simulate low-velocity impact problems on composite structures using High-Performance
+Computing. This algorithm is devised for parallel finite element codes running on supercomputers, and it
+is extended to explicit time integration schemes to solve impact problems including damage. The proposed
+framework is validated with a standard test for damage resistance on fiber-reinforced polymer matrix com-
+posites. Moreover, the parallel performance of the proposed algorithm has been evaluated in a mesh of 74M
+of elements running with 2400 processors.
+Keywords:
+Contact mechanics, Damage modeling, Finite element analysis, High-Performance Computing
+1. Introduction
+Impacts by foreign objects against any part of the aircraft are a major concern for the aerospace industry
+because they may compromise the structural integrity of the aircraft. Impact events can be classified into
+three main categories: low, high (including ballistics), and hyper-high velocity impacts [1].
+During the
+impact, the energy by the foreign object (projectile) is transferred to the target (structure), and consequently,
+the material can be damaged. Concretely, low-velocity impact events on composite materials (e.g., tool drops
+during maintenance or manufacturing) can drastically reduce the residual strength of the part even for case
+scenarios of Barely Visible Impact Damage (BVID). Therefore, designing composite structures with damage
+resistance cannot be avoided.
+Moreover, extensive experimental campaigns particularly focused on the
+∗Corresponding author
+Email address: gerard.guillamet@bsc.es (G. Guillamet)
+Preprint submitted to Composites Part A: Applied Science and Manufacturing
+January 16, 2023
+arXiv:2301.05552v1  [cs.CE]  11 Jan 2023
+
+investigation and evaluation of damage resistance of a specific material may be prohibitive by the industry
+in terms of costs.
+Thus, virtual testing of impact events is of great interest as mathematical models and technology advance.
+However, solving the physics behind this problem, particularly from the material point of view, is still one
+of the most complex and challenging problems today. A review of existing software for composite impact
+modeling focused on low-velocity events is conducted by Nguyen et al. [38]. In this review, the constitutive
+damage models play an essential role apart from the methods such as the contact algorithm or temporal
+integration scheme. Most of them can capture the trends and peak forces reasonably well. The research
+community has put and continues to put a lot of effort into developing reliable constitutive damage models
+for composites. Remarkable progress has been made on specific methodologies and constitutive damage
+models for predicting the damage resistance and damage tolerance of composite structures [28, 10, 17, 53,
+29, 52, 51, 15]. However, in terms of computational performance, the resolution of an impact problem,
+including different sources of damage, is still computationally demanding. The use of sophisticated contact
+algorithms and advanced damage models require refined element meshes to accurately predict the onset and
+propagation of the damage in such materials.
+The most commonly used contact algorithms for the resolution of an impact problem are the Penalty
+methods [21], Classical Lagrange multipliers [4, 16] or the Augmented Lagrange multipliers. The latter is
+often chosen to solve the contact inequality constraints, see [58]. However, the parallel aspects of these
+traditional contact algorithms are not trivial, and to the authors’ knowledge, little effort has been invested
+in the parallel aspects of such algorithms and their scalability in supercomputers. Some research works
+dealing with the parallel aspects of contact algorithms are [34, 35, 21].
+A completely different approach to the previous algorithms is the method of partial Dirichlet-Neumann
+(PDN) conditions. The contact is tackled as a coupled problem, in which the contacting bodies are treated
+separately, in a staggered way. The coupling is performed through the exchange of boundary conditions at the
+contact interface following a Gauss-Seidel strategy. The pioneering works using this approach are conducted
+by Krause and Wohlmuth [24] and Yastrebov [59], showing the capabilities of solving nonlinear contact
+problems. To the authors’ knowledge, one of the first applications of this method for explicit dynamics is
+made by Lapeer et al. [25], where the PDN method was used to simulate natural childbirth using explicit
+dynamics and executed in a hybrid system with Central Processing Units (CPU) and Graphics Processing
+Unit (GPU) architectures. However, little attention is dedicated to the computational performance and
+the parallel aspects of dealing with large-scale models. More recently, this method has been adapted and
+implemented in parallel in the Alya multiphysics code [57] by Rivero [46] and published by the authors in
+[19]. The mathematical and the parallel aspects are described in detail in these works, demonstrating the
+benefits of the PDN contact algorithm in High-Performance Computing (HPC) systems.
+In this paper, we present the application of the aforementioned method proposed by the authors in [46, 19]
+2
+
+for the resolution of low-velocity impact problems for composite materials. Existing time integration schemes
+and constitutive models from the literature have also been adapted and implemented within a parallel
+framework. So the main contribution of the present paper is focused on the extension of the PDN contact
+algorithm for explicit time integration schemes and its use in HPC systems involving impact events in the
+field of composite materials. Additionally, a new mesh multiplication algorithm is presented to deal with
+cohesive elements and element technologies such as continuum shell elements.
+The content of this paper is structured as follows. Firstly, the methods for the resolution of low-velocity
+impact events on composite materials including damage are explained with a strong emphasis on the contact
+algorithm and its implementation in parallel codes based on the finite element method. Then, the algorithm
+is validated through three benchmark tests.
+The first one consists of a quasi-static indentation test to
+verify that the contact pressure is well captured by using implicit and explicit time integration schemes.
+The second and the third examples correspond to a low-velocity impact on a composite plate following the
+ASTM International standard to measure the damage resistance of fiber-reinforced polymers. These last
+examples, use different material systems which are quite used by the aerospace industry, the T800S/M21 and
+the AS4/8552 carbon/epoxy systems. The numerical predictions obtained are correlated with experiments,
+and the computational performance is analyzed and discussed for the coupon made of AS4/8552 material.
+Finally, the conclusions of this work are commented on together with future work to improve the simulation
+of impact events including damage.
+2. Modeling framework for low velocity impact events using High-performance systems
+This section describes the modeling framework and the application of the partial Dirichlet-Neumann
+(PDN) contact algorithm for the simulation of impact events. Particular emphasis is put on the extension of
+such contact algorithm for explicit dynamic analysis. All the methods presented here are implemented in the
+Alya multiphysics code [57] based on the Finite Element Method (FEM). This parallel code is based on high-
+performance programming techniques for distributed and shared memory supercomputers. Moreover, the
+methods are programmed using the total Langrangian formulation, where stresses and strains are measured
+with respect to the original configuration. The Green strain measure and the 2nd Piola-Kirchoff stress are
+used, and we follow the notation from Belytschko et al. [6] throughout the paper. As an impact event is
+a complex and computationally demanding engineering problem is very attractive to be solved using High-
+Performance Computing. It is worth highlighting that all the methods described here can be implemented
+in other parallel FEM codes.
+2.1. Partial Dirichlet-Neumann contact algorithm
+The low-velocity impact event proposed in this paper can be assumed as a non-linear contact problem,
+where the striker is considered as a rigid body and the plate as a deformable body. Let’s assume that both
+3
+
+body instances are of arbitrary shape, and we do not consider friction. Therefore, this contact problem can
+be written as a boundary value problem, see Yastrebov [60], which includes the Hertz-Signorini-Moreau law
+for normal contact. So the balance of momentum and the contact conditions can be written as follows:
+∇ · σ + f v = 0
+in Ω
+σ · n = σ0
+on ΓN
+u = u0
+on ΓD
+g ≥ 0, σn ≤ 0, σn g = 0, σt = 0
+on ΓC
+(1)
+being σ the Cauchy stress tensor, f v a vector of volumetric forces, σ0 a set of prescribed tractions on the
+Neumann boundary, ΓN; u0 a set of prescribed displacements on the Dirichlet boundary, ΓD. Over the
+contact boundary, ΓC, we have imposed the following conditions: g represents the gap between contacting
+bodies, σn is the normal contact pressure, and σt is the tangential stress. The tangential stress equal to
+zero (σt = 0) in Eq. (1) characterizes a frictionless contact case.
+In order to satisfy the conditions in Eq. (1), the present paper uses the partial Dirichlet-Neumann
+contact algorithm proposed by Rivero [46, 19] which is based on the work from Yastrebov [60]. In the works
+mentioned above, the method was applied for implicit time integration schemes, while in the present work,
+the algorithm is extended to explicit schemes. The main benefits of the PDN contact algorithm to typical
+Penalty or Lagrange Multipliers methods are the following: (i) the size of the problem does not increase due
+to the Lagrange multipliers methods as unknowns (ii) no restriction with respect to the mesh partitioner
+due to the use of contact elements (iii) absence of contact tangent matrices (implicit schemes) and residual
+contact force vectors and (iv) easy to be parallelized as it can be treated as a solid-to-solid coupling using
+existing methods for multiphysics applications such as the Gauss-Seidel scheme.
+The iterative process of the PDN contact algorithm in a frictionless problem is shown in Fig. 1. Let’s
+assume that the time of the simulation is 0 < t < tE and it is subdivided into nT S time steps ranging
+from n = 1...nT S and tE is the time at the end of the simulation. At time step n there is no interaction
+between both code instances, so no contact is detected (Fig. 1a). Then, at time tn+1, contact is detected
+as we have overlapping between both bodies. At this step, the non-penetration boundary conditions are
+treated kinematically, i.e., as Mulitple Point Constraints (MPC) by the projection of the nodes belonging
+to the slave surface (deformable’s body) to the master surface (rigid’s body) using a Dirichlet condition.
+Then, a local coordinate system with normal-tangent basis vectors nj and tj is created for each detected
+node j. The contact node is restricted to only move to the tangent line defined by the vector tj, see Fig.
+1e. In a hypothetical frictional contact problem, the friction force would be imposed in this direction as a
+Neumann boundary condition. In this work, friction is not considered for the low-velocity impact as the
+relative velocities at the contact zone are sufficiently small. After that, the contact algorithm checks the
+4
+
+2
+(a)
+(c)
+(b)
+(d)
+Rigid 
+Deformable
+Contact node
+Released (free) node
+(e) Kinematic constraint for node 1 and 2
+n2
+t2
+n1
+t1
+1
+2
+f c
+f c
+f c
+f c
+f c
+f c
+Figure 1: Iterative process of the parallel PDN contact algorithm. (a) Interaction; (b) Overlapping; (c) Dirichlet boundary
+conditions (projections) (d) Released nodes and equilibrium. (e) Kinematic constraint for node 1 and 2. The reader is referred
+to the web version of this paper for the color representation of this figure.
+presence of adhesion or artificial contact nodes, i.e., nodes in traction (Fig. 1c). The reaction contact force
+f c
+j has to satisfy the following condition:
+f c
+j · nj ≥ 0
+(2)
+Those adhesion nodes have to be released, so the current time step tn+1 has to be repeated; the i index
+shown in Fig. 1c and 1d represents the sub-iterations for node release. The whole kinematic constraint
+process is depicted for two of the contacting nodes in Fig. 1e. The nodes release algorithm for explicit time
+schemes is described in Algo. 2 in Appendix
+A. The condition to distinguish a true contact node or an
+adhesion (artificial) contact node is by means of the contact force (reaction due to the Dirichlet condition).
+The vector of contact forces using total Lagrangian formulation can be expressed as:
+(f c
+j)T =
+�
+Ω0
+BT
+0jP dΩ0
+(3)
+where BT
+0j is the matrix containing the derivatives of the shape functions with respect to the reference
+system and P is the nominal stress tensor, see [6].
+An exciting aspect of the PDN method is that the computational cost of the projections is very small
+compared to Penalty or Langrange approaches [25]. One of the most consuming parts and a vital issue for
+further research is the contact searching and communication between the subdomains (belonging to different
+code instances), as stated in the previous work from the authors [19]. In our case, we use the PLE++ library
+[61], which is an adaptation of the Parallel Location and Exchange PLE library [14]. The main algorithm
+of the PDN contact method is described in Algo. 1 in Appendix
+A and the nodes release algorithm for
+5
+
+explicit schemes is summarized in Algo. 2. The reader is referred to Rivero [46] and Guillamet et al. [19]
+works for more details on the implementation aspects.
+2.2. Time integration schemes
+2.2.1. Deformable body
+Spurious oscillations may appear when using explicit time schemes for dynamic and wave propagation
+problems such as impact events. These oscillations occur due to the mismatch of two different types of wave
+components. Thus, dissipative explicit time schemes are often used to reduce the numerical instabilities
+induced by the spatial and time discretization procedures. Among the many dissipative methods available,
+the Tchamwa–Wielgosz (TW) explicit scheme [31] is beneficial because it damps out the spurious oscillations
+occurring in the highest frequency domain. This is the time integration scheme selected in this work, but
+any other explicit time scheme such as the Central Difference (CD) [6] including bulk viscosity could also
+be used.
+The motion described by the TW scheme is the following:
+˙d
+n+1 = ˙d
+n + ∆t¨d
+n
+(4)
+dn+1 = dn + ∆t ˙d
+n + ϕ(∆t)2¨d
+n
+(5)
+where d, ˙d, ¨d are the displacement, velocity and acceleration nodal vectors, respectively; ∆t is the time
+increment or step size; and ϕ is a numerical viscous parameter, which in the current work is set to 1.033
+[31]. The key to the computational efficiency of explicit time integration schemes is the use of the lumped
+mass matrix for the resolution of the linear system of equations, which is simplified as an easy inversion of
+the diagonal mass matrix [6]. The global stiffness matrix is not required to be assembled as it is needed for
+implicit time integration schemes. The explicit time integration scheme solves accelerations, so their values
+at the beginning of the increment are computed by making use of the equation of motion of the system:
+m ¨d
+n = f n(dn, tn) = f e(dn, tn) − f i(dn, tn) − f c(dn, tn)
+(6)
+where m is the vector representation of the lumped mass matrix, f e is the global vector of external forces, f i
+is the global vector of the internal forces, and f c is the global vector of the contact forces from the Dirichlet
+condition imposed on that nodes. Thanks to m, the acceleration can be computed without invoking any
+solver as:
+¨d
+n = m−1(f e − f i − f c)
+(7)
+6
+
+2.2.2. Rigid body
+The striker in the present work is considered as a rigid body. The resolution of the equations of motion
+for the rigid bodies we use a 4th order Runge Kutta scheme. Let’s consider the following differential equation
+where the right hand side is a function of both time and another function dependent on time.
+dy
+dt = f(t, y(t))
+(8)
+From this equation, the Runge-Kutta method estimates the solution at n + 1 taking into account four
+evaluations of the right hand side step dt as follows,
+k1 = dt · f(t, y(t))
+k2 = dt · f(t + dt
+2 , y(t) + k1
+2 )
+k3 = dt · f(t + dt
+2 , y(t) + k2
+2 )
+k4 = dt · f(t + dt, y(t) + k3)
+yn+1 = y(t + dt) = y(t) + k1
+6 + k2
+3 + k3
+4 + k4
+6
+(9)
+In the present paper, the motion of the striker is solved by making use of the following differential
+equation:
+m · ¨d = m · g − f e
+(10)
+where m is a scalar value of the mass of the rigid body, g is the gravity force vector at the center of mass,
+¨d is the linear acceleration, and f e is the external force also at the center of mass from the rigid body. It is
+worth mentioning that the rigid body is represented by a point (center or mass), so the above vectors have
+a dimension of 2 for 2-d problems and 3 for 3-d problems. When contact occurs the external force from the
+rigid body is calculated by f e = �nc
+j=1 f c
+j, where j denotes a contact node and nc is the total contact nodes
+belonging to the deformable body.
+2.3. Mesoscale damage modeling for fiber-reinforced composites
+The mesoscopic length scale is the most suitable for virtual testing of low-velocity impacts on structures
+made of composite materials. At this scale, the numerical predictions have a good trade-off between infor-
+mation about the damage mechanisms driving the failure process and the structural response without the
+complexity of dealing with intricate microstructures. It is worth emphasizing that the mesoscopic length
+scale is not only appropriate for the bottom levels of the building block approach (coupon and elements)
+[28, 53, 15, 50] but also for the top levels (sub-components and components) [43, 44, 13].
+7
+
+From the constitutive modelling viewpoint, the mesoscopic length scale simplifies the intricate microstruc-
+ture of long fibre composite laminates by homogenising the properties and mechanisms at the lamina level.
+The outcome is a layered material with two well-defined regions: intralaminar and interlaminar. The former
+is modelled as a transversally isotropic material, which can fail due to fibre breaking and matrix cracking
+according to the loading scenario. The latter is modelled as a very thin region, usually tending to zero
+thickness, where delamination can onset and propagate.
+Regarding the modelling architecture, several strategies exist in the literature suitable for modelling
+composite at a mesoscopic length scale using FEM [37]. We adopt a continuous approach for the intralam-
+inar region (CDM with linear elements) and a discontinuous one for the interlaminar (CZM with interface
+elements). The straightforward implementation of this strategy in a standard FEM code aids in preserv-
+ing the scalability of Alya multiphysics [41]. Thus, the mesoscale damage modeling strategy exploits the
+computational resources to maximize the accuracy of the impacts thanks to very thin meshes.
+2.3.1. Intralaminar damage model
+The intralaminar damage model for predicting ply failure is based on the continuum damage mechanics
+framework.
+Fiber and matrix cracks are smeared in the continuum and represented by state variables.
+Accordingly, the crack’s kinematics is not explicitly represented, but their effects on the degradation of the
+capacities of sustaining loads. In turn, the onset and growth of the damage failure mechanisms are governed
+by the failure surfaces and evolution laws. In this work, we employ a local damage model based on the
+constitutive modeling framework for long fiber composite materials proposed by Maim´ı et al. [32, 33]. This
+framework has been used widely in the literature, demonstrating outstanding accuracy and performance not
+only for static scenarios [11, 7, 41] but also for impact [17, 51, 48] and fatigue [26, 27].
+The main ingredients of the intralaminar damage model are: i) transversally isotropic elasto-plastic re-
+sponse, ii) damage activation functions related to the different ply failure mechanisms through the maximum
+strain criterion for the fiber breaking and the LaRC criteria for the matrix cracking, iii) the damage evolution
+laws are defined to dissipate the fracture energy associated to the opening mode ensuring mesh objectivity
+by the crack-band theory [5], and iv) the thermodynamic consistency is ensured by imposing irreversibility
+of the damage variables.
+Fig. 2 illustrates the intralaminar failure modes schematically modelled, while Algo. 4 in Appendix A
+summarises the material model workflow. Note that a plastic response under shear loads is considered,
+and five damage mechanisms are modelled: fibre breaking, fibre kinking, tensile and compressive matrix
+cracking, and shear matrix cracking. The details of the expressions employed and their justification from a
+physical standpoint can be found in [32, 33, 51].
+Besides the constitutive response, the intralaminar damage model also encloses the computation of the
+critical time step, which is required by the explicit time integration scheme. For the sake of simplicity, we
+8
+
+Fibre breaking
+Fibre kinking
+Matrix cracking
+Matrix cracking
+Matrix cracking
+Figure 2: Schematic representation of the intralaminar damage mechanisms. Adapted from [26].
+utilise the same formula of a transversally isotropic material:
+∆t = vsound
+ℓc
+=
+�
+max Cij
+ρ
+(11)
+where Cij are the components of the effective stiffness matrix, ρ is the density of the material, and ℓc is
+the characteristic element length. Considering a structured hexahedral mesh employed, we approximate the
+characteristic element length with the element volume Ve [33]:
+ℓc ≈
+3�
+Ve
+(12)
+2.3.2. Interlaminar damage model
+The interlaminar damage model for predicting the onset and propagation of delamination is based on
+the cohesive zone approach and formulated in the context of damage mechanics. Accordingly, a damage
+state variable is employed to account for the gradual loss of the bearing capacities of the material in the
+cohesive zone due to the separation of crack surfaces. In turn, the separation or opening of the crack is
+represented by a kinematic quantity noted as displacement jump, which is approximated by employing the
+interface element technology [2, 39, 9]. Thus, the interlaminar damage model is a constitutive model that
+computes the cohesive reactions as a function of displacement jumps. More details of the mesoscale modeling
+of delamination using cohesive zone models can be found in Carreras et al. [12].
+In this work, we use the cohesive zone model proposed by Turon et al. [54, 56], which has been employed
+extensively in the literature [20, 47, 40, 42]. The main characteristics of this CZ model are: i) linear response
+9
+
+125 μm125 μm125 μm125 μm125 μmbefore initiation of the softening, ii) linear relation between the cohesive tractions and crack openings,
+iii) onset and propagation of the damage in compliance with the Benzeggagh-Kenane criterion, and iv)
+thermodynamic consistency despite the loading scenario, even when the mix-mode ratio varies. Algo. 5
+summarizes the cohesive zone model workflow.
+Regarding the element technology, we employ zero-thickness interface elements for capturing the delam-
+ination, implemented using the formulation presented in [45]. As standard interface elements are used, the
+integrals are computed using a Newton-Cotes integration scheme to mitigate the spurious oscillations in the
+traction profile along the interface [49]. The stable time increment, which is necessary for the explicit time
+integration scheme is obtained through [51]:
+∆tcoh =
+�
+¯ρ
+Kcoh
+(13)
+where ¯ρ and Kcoh are numerical parameters known as cohesive surface density and penalty stiffness, re-
+spectively. The cohesive surface density for zero-thickness elements is approximated by the expression in
+[52]. In turn, the cohesive penalty stiffness is defined to avoid affecting the compliance of the system as
+Kcoh ≥ 50ET /tlam, where ET is the transverse elastic modulus and tlam the adjacent laminate thickness
+[55].
+2.4. Mesh refinement algorithm for interface elements with a cohesive law
+The mesh multiplication algorithm proposed by Houzeaux et al. [22] has been extended to deal with
+the presence of interface elements or even continuum shell element formulations.
+Focusing on interface
+elements, they are zero-thickness elements with a cohesive material law that are inserted between plies in
+a laminated composite material in order to predict the delamination damage mechanism. It is well known
+that an accurate prediction of the onset and propagation of delamination in composite materials requires
+very refined meshes, as stated in [54, 55]. However, depending on the number of interface layers or the
+geometry size, it can be challenging to place these elements between plies and computationally demanding
+to solve the problem.
+On the other hand, mesh generation of large meshes is often a bottleneck in engineering applications
+to deal with thousands of millions of elements. Thus, integrated tools for mesh refinement within parallel
+codes devised for High-Performance systems allow a parallel and fast refinement of the coarse mesh without
+the need to create the mesh again.
+Therefore, this paper also introduces a new capability of the mesh multiplication algorithm from [22],
+which enables the refinement of large-scale problems, including interface elements. Let’s assume a configu-
+ration of two bulk elements together with an interface element between them, as shown in Fig. 3.
+10
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+Interface
+element 
+Solid
+element 
+2
+3
+4
+1
+6
+7
+8
+5
+Original node
+New edge node
+New face node
+New center node
+2
+3
+4
+6
+8
+5
+(ne
+ELINT= 4) 
+Bulk element 
+Interface element 
+(ne
+BULK = 8) 
+Global numbering
+Local numbering
+1
+7
+Figure 3: Mesh multiplication between bulk and interface (cohesive) elements.
+The 8-node interface element can only be divided into four elements to avoid the duplication of the
+element at the interface mid-plane between the bulk elements. The criterion used for the correct division
+is by making use of the element normal, also known as stacking direction, which is required for the proper
+behaviour of the element due to its kinematics. Thus, those parallel planes to the element normal are used
+to divide the element. The dimensions of the new mesh can be calculated as follows:
+ne = 8 · n0
+e,BULK − n0
+e,ELINT · 4
+nn = n0
+n + nedges + nfaces + n0
+e,BULK − n0
+e,ELINT − n0
+edges,ELINT − n0
+faces,ELINT
+nb = 4 · n0
+b − 2 · n0
+b,ELINT
+(14)
+where ne, nn and nb are the total number of elements, nodes and boundaries for the new mesh. In order
+to refine the hybrid mesh is important to know the total number of n0
+e, n0
+n and n0
+b from the original mesh
+and also information about the edges and faces that have to be divide or not. Algo. 3 summarizes the
+different steps and functions for the mesh division and reconstruction of the interface domains in a parallel
+framework.
+3. Benchmark tests
+Three benchmark tests are conducted to validate the application of the parallel partial Dirichlet-Neumann
+contact algorithm using an explicit time integration scheme. The first example consists of a quasi-static
+indentation test, which has already been solved using an implicit time integration scheme in [19].
+The
+solution using explicit analysis is compared with the numerical solution obtained for implicit analysis. The
+second and the third examples consist of a low-velocity impact event on two coupons manufactured with two
+11
+
+well-known material systems for the damage prediction: T800S/M21 and AS4/8552 respectively. Thanks
+to the proposed algorithm’s flexibility and generality, we use a multi-code approach, where the motion of
+each body (rigid and deformable) is solved using different instances of Alya. Regarding the partitioning
+of the mesh, we use the Space-Filling Curve (SFC) based partitioner described in [8], which performs the
+partitioning in parallel and maximizes the load balance. It is worth highlighting that all the executions
+here are in parallel (pre-process, solution, and post-process steps). In all the examples, the contact bodies
+are discretized with a refined finite element mesh to assess the geometrical localization between both code
+instances and to obtain an accurate prediction of the contact force. All the simulations are conducted in
+MareNostrum4 supercomputer. This cluster has 3456 nodes, each of them with 48 processors Intel Xeon
+Platinum @ 2.1 [GHz], giving a total processor count of 165 888 processors.
+3.1. Quasi-static indentation test
+This example has already been solved using an implicit time solution scheme in [19, 46]. This case is
+now solved as a quasi-static problem using explicit dynamics. The example consists of a rigid rounded head
+(indenter) and a deformable beam, see Fig. 4. The geometrical dimensions of the indenter (rigid body) are
+ri = 1 m and wi = 0.5 m, while the beam (deformable body) are hb = 0.25 m, lb = 1.5 m and wb = 0.3 m. The
+relative position of the indenter with respect to the beam is given by the parameters ax = 0.25 m, az = 0.1 m
+and ay = 0.01 m (gap). The beam is modelled with an hyperelastic Neo-Hookean formulation [3] and finite
+strains, with material properties Eb = 6.896 × 108 Pa (Young modulus), νb = 0.32 (Poisson ratio) and density
+ρ = 1000 kg m−3. The beam is fully clamped at the bottom face, and a prescribed vertical displacement of
+δ = 0.11 m is applied at the top surface belonging to the indenter. Both bodies are discretized with finite
+elements using full integration: 8-node linear solid elements for the beam and 4-node linear tetrahedrons
+for the indenter. The beam has a base mesh of 3510 elements, while the indenter has 15 960 elements. A
+non-linear dynamic analysis is performed with a total time of the simulation of 0.05 s and a fixed time step of
+1 × 10−5. The selected time step value is smaller than the stable time increment, which is 2.796 × 10−5, and
+no mass scaling is used. In order to perform a quasi-static event and minimize the kinetic energy, a smooth
+step function (fifth-order polynomial) is applied. This function has the form A0+(AE −A0)ξ3(10−15ξ+6ξ2)
+for t0 ≤ t < tE, where A0 and A1 are the initial and final amplitude, t0 and tE are the initial and final time
+of the simulation and ξ =
+t−t0
+tE−t0 . This smooth load rate ensures that the first and second time derivatives
+are zero at the beginning and the end of the transition.
+12
+
+Beam (deformable)
+hb
+x
+y
+z
+y
+lb
+wi
+wb
+az
+ax
+Indenter (rigid)
+ri
+ux = uy = uz = 0
+ux = uz = 0,
+ay
+uy =
+Figure 4: Setup for the quasi-static indentation test. Adapted from [19].
+Displacements and forces obtained at the contact zone are shown in Fig. 5 for two different paths. Line
+path a is centered and goes from one side to the other in the length direction of the beam, while line path b
+is also centered in the width direction. The numerical prediction using the explicit time integration scheme
+is compared with the implicit solution obtained in [19]. We can observe an excellent agreement between
+both numerical predictions in terms of the displacements and contact force.
+(a)
+(b)
+(c)
+(e)
+(d)
+Path b
+Path a
+[18]
+[18]
+[18]
+[18]
+[18]
+Figure 5: Displacements and contact forces at straight lines path a and b. (a) Tangential displacement in x-direction for path
+a. (b) Normal displacement in y-direction for path a. (c) Tangential displacement in z-direction for path b. (d) Contact force
+at line path a. (e) Contact force at line path b.
+13
+
+3.2. Low velocity impact on a composite plate
+The proposed benchmark consists of a drop-weight of a rigid hemispherical striker on a rectangular plate
+made of composite material, see Fig. 6. Two impact scenarios using different material systems, layups, and
+impact energies are considered for the validation of the proposed framework. The materials selected are
+the unidirectional prepreg M21/194/34%/T800S (T800S/M21) and the unidirectional prepreg AS4/8552,
+both carbon-epoxy systems. On the one hand, the coupon made of T800S/M21 is manufactured by Hellenic
+Aerospace Industry and tested at Element Materials Technology Seville facilities within the framework of the
+CleanSky2 SHERLOC project. Most of the material properties from the T800S/M21 are also characterized
+by Hellenic Aerospace Industry and Element Materials Technology Seville. On the other hand, the coupon
+made of AS4/8552 is chosen from literature through the works conducted by Gonz´alez et al.
+[17] and
+Soto et al. [51]. All the material properties from the aforementioned materials, including damage model
+parameters, are summarized in Tab. 1. The intralaminar damage model is fed by the in-situ strengths which
+are calculated following the works by Furtado et al. [15] and Soto et al. [51].
+Rubber clamp
+(ux = uy = uz = 0)
+Striker
+rs = 8 mm
+(ux = uy = 0)
+ms = 5 kg
+Top view
+ 
+75 mm
+125 mm
+R7
+Refined region
+(75 mm x 75 mm)
+ 
+(uz = 0)
+Window cut
+ 
+Stacking sequence
+ [454/04/-454/904]s
+Cohesive elements between clusters
+tply = 0.18125 mm
+tcoh = 1.0 x 10-4 mm
+Figure 6: Numerical setup for the low velocity impact test. The mesh and layup correspond to the coupon made of AS4/8552
+material used for the parallel performance analysis.
+Both impact case scenarios follow the standard ASTM D7136/D7136M-20 [23] for damage resistance
+evaluation of fiber-reinforced polymers. Each plate has the same dimensions: 150 mm×100 mm and each
+of them are supported on a metallic frame with a cut-out of 125 mm×75 mm.
+Rubber-tipped clamps
+clamp the plate instance at the four corners. We consider equivalent boundary conditions to represent this
+experiment. As we can see in Fig. 6, the metallic frame and the rubber clamps from the experiment do not
+exist as physical entities, so we only consider the contact surface from the rubber cylinder-shaped clamps
+and the contact edges of the cut-out window of the metallic frame, where we apply the boundary conditions.
+14
+
+T800S/M21
+AS4/8552
+Property
+Value
+CV(%)
+Ref.
+Value
+Ref.
+Density (t/mm3)
+1.59 × 10−9
+-
+1.59 × 10−9
+[51, 17]
+Elastic
+E11 (MPa)
+138.4 × 103
+1.95
+128.0 × 103
+[51, 17]
+E22 = E33 (MPa)
+8.54 × 103
+3
+7.63 × 103
+[51, 17]
+ν12 = ν13 (-)
+0.311
+16
+0.35
+[51, 17]
+ν23 (-)
+0.45
+-
+0.45
+[51, 17]
+G12 = G13 (MPa)
+4.29 × 103
+3
+4.358 × 103
+[51, 17]
+G23 (MPa)
+2.945 × 103
+-
+2.631 × 103
+[51, 17]
+Strength
+XT (MPa)
+2854.0
+4
+2300.0
+[51, 17]
+XC (MPa)
+1109.0
+13
+1531.0
+[51, 17]
+YT (MPa)
+56.6
+5.8
+74.2
+YC (MPa)
+250.0
+[15]
+199.8
+[51, 17]
+SL (MPa)
+93.7
+0.6
+94.36a
+[36]
+αo (◦)
+53
+[32]
+53
+[32]
+In-situ strengthsc
+Y is
+T,int (MPa)
+132.5 (1tply)
+-
+117.5 (4tply)
+Y is
+T,int (MPa)
+93.7 (2tply)
+-
+117.5 (8tply)
+Y is
+T,out (MPa)
+83.8 (1tply)
+-
+74.2 (4tply)
+Y is
+C,int (MPa)
+250.0 (1tply)
+-
+199.8 (4tply)
+Y is
+C,int (MPa)
+250.0 (2tply)
+-
+199.8 (8tply)
+Y is
+C,out (MPa)
+250.0 (1tply)
+-
+199.8 (4tply)
+Sis
+L,int (MPa)
+116.0 (1tply)
+-
+120.8 (4tply)
+Sis
+L,int (MPa)
+116.0 (2tply)
+-
+120.8 (8tply)
+Sis
+L,out (MPa)
+93.7 (1tply)
+-
+94.4 (4tply)
+Fracture toughness
+GXT (N/mm)
+340
+[15]
+81.5
+[51, 17]
+GXC (N/mm)
+60.0
+[15]
+106.3
+[51, 17]
+GY T (N/mm)
+GIc
+7.3
+GIc
+[51, 17]
+GY C (N/mm)
+1.38b
+20
+1.313b
+[51, 17]
+GSL (N/mm)
+GIIc
+20
+GIIc
+[51, 17]
+Traction separation law
+fXT (-)
+0.1
+-
+0.1
+[51, 17]
+fGT (-)
+0.6
+-
+0.6
+[51, 17]
+fXC (-)
+0.1
+-
+0.1
+[51, 17]
+fGC (-)
+0.9
+-
+0.9
+[51, 17]
+Matrix plasticity
+Sp (N/mm)
+66.9
+-
+[15]
+62.0a
+Kp (N/mm)
+0.09
+-
+[15]
+0.1936a
+Interface properties
+GIc (N/mm)
+0.308
+7.3
+0.28
+[51, 17]
+GIIc (N/mm)
+0.828
+20
+0.79
+[51, 17]
+τI (MPa)
+49.2d
+5.8
+YT
+[51, 17]
+τII (MPa)
+80.7d
+0.6
+SL
+[51, 17]
+η (-)
+1.75
+-
+1.45
+[51, 17]
+Kcoh (M/mm3)
+1.1 × 106
+-
+2.5 × 104
+[51]
+a Best fitted based on properties from [36]
+b GY C = GSL/cos(αo) [32]
+c Calculated considering plasticity using equations from [51]
+d Engineering solution by Turon et al. 2007 [55] using Ne=5
+Table 1: Material properties for the M21/194/34%/T800S (T800S/M21) and Hexply AS4/8552 including damage models
+parameters.
+15
+
+The velocity of the striker is given as an initial condition set in the impact direction, while the remaining
+degrees of freedom are constrained. The initial velocity of the striker is calculated based on the impact
+energy of each case study. The initial position of the striker has a gap of 0.01 mm between the striker
+tip and the top surface of the plate in order to avoid overlapping between bodies at the beginning of the
+simulation. Moreover, gravity forces are included in both body instances, considering a gravity value of
+9.81 m/s2.
+We employ 8-node full integration hexahedron elements for the plate using the inter- and intra-laminar
+damage models described in Sec. 2.3.2 and Sec. 2.3.1 respectively. Cohesive elements are inserted at each
+interface between different ply angles.
+It is worth highlighting that other constitutive material models
+and element technologies would also be feasible in combination with the proposed contact algorithm. With
+regards to the strikers used for each impact case scenario, they are discretized with 4-node linear tetrahedron
+elements with a biased mesh of 0.1 mm at the center of the half-sphere and 1 mm at the end of the edge.
+The total number of elements for the striker used for the T800S/M21 and AS4/8552 materials are 32 685
+and 79 934, respectively. Regarding the plates, they both have a refined centered region of 75 mm×75 mm
+with an in-plane element size equal or multiple to the ply thickness, depending on the material system in
+order to guarantee an aspect ratio close or equal to 1.
+3.2.1. Coupon made of T800S/M21 material
+This impact coupon has a stacking sequence of [45/ − 45/02/90/0]S and is made of T800S/M21. The
+nominal ply thickness is 0.192 mm. This case study is submitted to an impact energy of 10 J, which falls into
+the Barely Visible Impact Damage (BVID) analysis. The striker has a diameter of 25 mm and a mass of 2 kg,
+which is modeled as a rigid body. The global element size for the plate is 1 mm, and each lamina and the
+clusters of two plies have one element through the thickness. The in-plane element size is 0.192 mm which
+is equal to the ply thickness resulting in an aspect ratio of 1 for those elements at plies without clustering
+and located at the refined region. The mesh of the plate has a total of 1 042 525 hexahedron elements (≈
+3.3 million of Degrees Of Freedom (DOF)). The total time for this simulation is set to 5.0 ms. The initial
+velocity of the striker considering the gap previously mentioned is 3.16 m/s.
+The numerical predictions for this impact case scenario and their comparison with experimental data
+are shown in Fig. 7 and summarized in Tab. 2. The experimental test campaign consisted of testing a
+batch of five coupons to ensure proper repeatability of the results. The force-time for each impact was
+recorded with a limited number of points (52 points on average for each impact test). The reduced number
+of points only allows for validation of the global behavior of the impact case scenario. Energy-time and the
+force-displacement curves are calculated by integrating once and twice the experimental force history curve.
+16
+
+(a)
+(b)
+(c)
+Figure 7: Experimental and numerical curves for the 10J impact on the coupon made of T800S/M21 material. (a) Impact
+force-time. (b) Impact force-displacement. (c) Energy-time.
+As we can see either in Fig. 7 and Tab. 2 a good agreement is obtained between experiments and
+numerical predictions.
+On the one hand, the proposed contact algorithm combined with the proposed
+damage models is able to capture the maximum impact force very well and the maximum displacement
+pretty well with errors below 10%, respectively. On the contrary, the different dissipated energies obtained by
+the experiments show a high dispersity between them, resulting in difficulty in conducting a fair comparison
+between the predicted value 1.1 J and the experimental mean value.
+Experiment
+Prediction
+Difference (%)
+Mean
+Std.
+Maximum impact force, f c
+max (kN)
+5.3
+0.2
+5.2
+-1.0
+Maximum displacement, dmax (mm)
+5.4
+0.1
+4.9
+-9.3
+Dissipated energy, Edis (J)
+0.2
+0.3
+1.1
+>10
+Table 2: Comparison of numerical results with experimental data for the impact case scenario of the plate made of T800S/M21
+material.
+3.2.2. Coupon made of AS4/8552 material
+This second case consists of a coupon made with the AS4/8552 material.
+The plate has a stacking
+sequence of [454/04/ − 454/904]S with a nominal ply thickness of 0.181 mm resulting a plate thickness of
+5.8 mm. This case study has higher energy (19.3 J) than the previous one, and it also includes clusters of four
+and eight plies which are potential for extensive matrix cracks and delaminations. The energy of 19.3 J also
+falls into BVID analysis. The in-plane element sizes used in [17] and [51] are 0.3 mm and 0.5 mm respectively.
+In the present work, two element sizes are studied using the mesh refinement algorithm described in Sec. 2.4,
+see Tab. 3. The base mesh for the plate has a total of 335 622 hexahedron elements (≈ 1 million of Degrees
+Of Freedom (DOF)). The total time for the simulation is set to 5.0 ms. In this case, the striker has a mass
+17
+
+of 5 kg, and its radius is 8 mm. The initial velocity of the striker considering the gap previously mentioned
+is 2.78 m/s.
+Refinement
+Element
+No. element
+No. elem.
+No. nodes
+Initial stable
+level, ndivi
+size (mm)
+through ply cluster
+plate
+plate
+time increment (s)
+0
+0.7250
+1
+335 622
+364 320
+7.378 × 10−8
+1
+0.3625
+2
+2 109 624
+2 219 983
+3.515 × 10−8
+Table 3: Element sizes used on the coupon made of AS4/8552 material system and initial stable time increment for each case
+study.
+The numerical predictions of the impact force-displacement and energy - time curves are shown in Fig.
+8 and Fig. 9 respectively, using different element sizes. The most important physics variables for a proper
+validation are summarized in Tab. 4. This table compares the experimental results from [51] with the
+numerical predictions.
+Case
+f c
+del (kN)
+f c
+max (kN)
+dmax (mm)
+Edis (J)
+Aproj
+del
+(mm2)
+Experiment [17]
+4.41
+7.74
+3.72
+12.03
+3898.3
+Numerical (le = 0.7250 mm)
+4.20
+8.70
+3.60
+7.70
+4723.1
+Numerical (le = 0.3625 mm)
+4.30
+8.30
+3.70
+7.90
+5249.20
+Table 4: Comparison of the numerical results obtained with the proposed framework with experimental data from [51]. fc
+del is
+the delamination threshold force, fc
+max is the maximum contact force, dmax is the maximum indentation, Edis is the dissipated
+energy and Aproj
+del
+is the projected delamination area.
+The initial elastic deflection of the plate is very well captured for all the meshes (Fig. 8), meaning that
+the stiffness of the plate is accurately predicted by the PDN contact algorithm. After that, delamination
+onset occurs at the top of the elastic part, around 4.5 kN. This point is also very well captured by the
+interlaminar damage model using cohesive elements between each of the ply clustering. Then, a combination
+of interlaminar and intralaminar damage occurs until the striker reaches both the maximum load and
+displacement, resulting with a pretty good prediction as also shown in Tab. 4. Since damage appears, the
+continuum damage models and the characterization of the material properties play a fundamental role in the
+simulation of this benchmark case. Despite the delamination threshold, maximum force and displacement
+are very well captured; the dissipated energy and the projected delamination area are overpredicted, see
+Tab. 4. According to Soto et al. [51], the projected delamination and the corresponding energy dissipated
+could be considerably improved when using solid elements with one integration point for the bulk material
+and cohesive contact surfaces instead of cohesive elements to be able to better predict the delamination
+shapes at each interface of the layup.
+18
+
+Figure 8: Numerical prediction of the force-displacement curved using two element sizes and correlation with the experiment
+from Gonz´alez et al. [17].
+Figure 9: Numerical prediction of the impact energy vs. time using two element sizes and correlation with the experiment from
+Gonz´alez et al. [17].
+Fig. 10 depicts and aims to quantify the most important failure mechanisms that appear on the plate.
+Fiber damage is represented by damage variable D1, which includes both fiber breakage and fiber kinking,
+see Fig. 10a. As we can see, this source of damage is not the most predominant and mostly appears at the
+bottom of the striker. Matrix cracking is represented with the damage variable D2, which includes matrix
+tension and compression (Fig. 10b). Finally, the last source of damage is delamination (Fig. 10b). Its
+prediction is compared with the shape obtained from the experiment, which is represented in dashed lines.
+As we discussed previously, this source of damage is overpredicted for all the element sizes studied, see Tab.
+4 and further research would be required in that direction as the values of the material properties, and
+19
+
+the damage models play a fundamental role. Furthermore, the extensive matrix cracks and delamination
+predicted for this impact case scenario corroborate the experimental observations by Gonz´alez et al. [18] on
+the effect of ply clustering to originate extensive matrix cracks and large delaminations.
+3898.3 mm2
+(a)
+(b)
+(c)
+Experiment
+Numerical
+5248.2 mm2
+10 mm
+Dcoh
+D1
+D2
+Figure 10: Numerical prediction of the damage occurred in the coupon. (a) Fiber damage, D1. (b) Matrix cracking, D2. (c)
+Projected delamination, Dcoh. The numerical result correspond to the most refined mesh.
+3.3. Parallel performance
+The speedup and the parallel efficiency of the proposed contact algorithm for solving low-velocity impact
+events are evaluated in this section. All the executions are conducted in MareNostrum4 supercomputer. A
+strong scalability analysis has been conducted using a larger mesh than the ones studied in Sec. 3.2. The
+model corresponds to the AS4/8552 impact case scenario. The new mesh has a total of 74M elements with
+228M of DOF, which results from a base mesh of 1 472 328 elements using two levels of the mesh refinement
+algorithm. Strong scalability consists of fixing the mesh and solving the problem with a different number
+of processors, Central Processing Unit (CPU). The strong speedup is calculated as
+t0
+tN while the parallel
+efficiency is calculated as
+t0N0
+tNN , where N is the number of processors and t0 is the reference simulation
+time for N0 processors. The number of processors used for this analysis ranges from 192 to 2400. Due to
+the resolution of the problem following a multibody/multicode approach, the number of processors for the
+striker is fixed to 16 (sufficiently for its mesh) while the number of processors for the plate is changed. It
+is worth mentioning that the strong computational effort falls in the resolution (deformation) of the plate
+and the localization and exchange of information phases, as explained in [19]. Due to the small time step
+in this simulation, 9.261 × 10−9 s, the simulations for the scalability curve are limited to the first 7460 time
+20
+
+DAM01
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.7
+0.80.9 1.0e+00
+Y
+Y
+L.
+DAM02
+DCOHE
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9 1.0e+00
+0.0e+00 0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9 1.0e+00steps. The end of the execution (last time step) corresponds to an impact force of approximately 1 kN, which
+falls into the linear elastic regime of the force-displacement curve shown in Fig. 8. The strong speedup and
+parallel efficiency are shown in Fig. 11. The ideal scalability and efficiency are represented with a dashed
+line.
+Average No. of elements per core
+Strong speedup
+Parallel Efficiency
+Total No. of processors
+Total No. of processors
+Figure 11: Strong scalability of the low velocity impact test with a plate mesh of 74M hexahedron elements.
+The model
+corresponds to the benchmark case using AS4/8552 material system.
+The results obtained in Fig. 11 show that the scalability of the problem in explicit analysis is really
+good up to 2400 processors using a mesh of 74M elements. The parallel efficiency is maintained above 90%,
+which demonstrates the good scalability of the proposed framework to deal with large-scale problems. This
+linear behavior is also shown in Tab. 5 where we summarize the total CPU time for each execution using a
+different number of processors while maintaing fixed the size of the problem.
+21
+
+No. of CPUs
+192
+384
+768
+1536
+1824
+2064
+2400
+17:20
+08:33
+4:15
+02:11
+01:51
+01:39
+01:27
+Table 5: Total CPU time expressed in hh:mm for different executions of the low-velocity impact simulation considering a
+fixed mesh of 74M of elements (228M of DOF) with a total of 7460 time steps. This CPU time includes the preprocess part,
+where two mesh refinement levels are performed and the solution of the contact problem within the elastic regime of the
+force-displacement curve.
+It is also worth mentioning that the application of the proposed contact algorithm in explicit dynam-
+ics improves both the speedup and the parallel efficiency in comparison to an implicit resolution for the
+deformable body (plate), as already studied by the authors in [19]. This improvement in computational per-
+formance is mainly attributed to the time integration scheme for the deformable body. In explicit dynamics,
+it is not required to invert the global matrix of the system. In this case, the unknown is the acceleration, and
+the system is solved directly using the lumped mass matrix and the global force vector on the right-hand
+side. The reader is referred to [6] for more details.
+4. Conclusions
+In this paper, we apply the parallel PDN contact algorithm to simulate low-velocity impact events on
+fiber-reinforced polymer composites using a High-Performance Computing environment. Existing damage
+models from the literature have been implemented in our multiphysics finite element code Alya to simulate
+the material damage. Moreover, we introduce a new capability in the in-house mesh refinement algorithm to
+deal with cohesive elements and other element types, such as continuum shell elements. This is really attrac-
+tive as we can refine the finite element mesh at the beginning of the simulation with a meager computational
+cost.
+We validate the whole framework with several benchmark tests. The last example corresponds to a
+well-known low-velocity impact test following the ASTM standard for damage resistance analysis. In this
+case, we study two impact case scenarios with two different material systems: the T800S/M21 and the
+AS4/8552, obtaining excellent predictions for impact behavior and pretty good damage occurrence compared
+to experimental data from the literature. Additionally, the mesh refinement algorithm’s capabilities have
+been demonstrated for the plate made of AS4/8552 material.
+Finally, we evaluate the parallel performance of the impact simulation. Despite not using ”very” large
+meshes for the physics validation cases, we have generated a new larger mesh using the mesh refinement
+algorithm. The reason behind this is the stable time increment, which becomes smaller as the element size
+decrease. The new mesh has 74M hexahedron elements (228M of DOF) using full integration. An excellent
+computational efficiency (above 90%) has been obtained up to 2400 CPUs, demonstrating its applicability
+to solve large mesh models ranging from micro-scale to macro-scale.
+22
+
+A further conclusion of this work is that we demonstrate the potential application of the parallel PDN
+contact algorithm for low-velocity impact events and its parallel efficiency for large models compared to
+traditional Penalty or Lagrange contact-based methods. As we commented previously, we use full integration
+elements for all the examples; the use of reduced integration elements, which are more appropriate for
+explicit schemes and overcome the well-known locking pathologies from solid brick elements, can considerably
+increase the speedup of the simulations. Moreover, the localization of contact nodes and the communication
+between subdomains created by the domain decomposition method is a crucial issue for further research as
+it is the main bottleneck regarding the computational efficiency of contact algorithms.
+Acknowledgements
+This work has received funding from the Clean Sky 2 Joint Undertaking (JU) under grant agreements
+No. 807083 and No. 945521 (SHERLOC project). The JU receives support from the European Union’s
+Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union.
+The authors gratefully acknowledge Hellenic Aerospace Industry for manufacturing of the coupons made
+of T800S/M21 material and Kirsa Mu˜noz and Miguel ´Angel Jim´enez from Element Materials Technology
+Seville for conducting the experimental impact tests and providing all the experimental data. A. Quintanas-
+Corominas acknowledges financial support from the European Union-NextGenerationEU and the Ministry
+of Universities and Recovery, Transformation and Resilience Plan of the Spanish Government through a call
+of the University of Girona (grant REQ2021-A-30). G. Guillamet thankfully acknowledges the computer
+resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center (FI-
+2019-2-0010). Last but not least, the authors would also like to thank the late Claudio Lopes for all the
+interesting discussions and contributions to the simulation of impact events and damage on composites.
+Appendix A. Algorithms
+Here we summarize the main algorithms of the whole modeling framework to solve low-velocity impact
+events for damage resistance of fiber-reinforced polymer composites by making use of High-Performance
+Computing.
+23
+
+Algorithm 1 Main code for the partial Dirichlet-Neumann (PDN) contact algorithm.
+This PDN contact algorithm is treated as a coupling problem between two or more body instances.
+In the present algorithm,
+we describe the contact algorithm between two code instances: a rigid body represented by the domain Ωa and the deformable
+body represented by the domain Ωb.
+The coupling is performed through the exchange of boundary conditions at the contact
+interface following a Gauss-Seidel strategy. At each time step, contact detection is done for both instances, and synchronization
+and localization is executed. When contact is detected (at least one boundary node belonging to the deformable body is penetrated
+inside the rigid body), the rigid one computes and sends to the deformable body all the information required for the enforcement
+of the kinematic boundary conditions.
+The reader is referred to the Ph.D. from Rivero [46], or [19] for more details on the
+implementation aspects of the proposed contact algorithm.
+Require: Ωa, Ωb
+1: loop time
+2:
+Compute time step, tn+1
+3:
+loop reset
+4:
+if Rigid body, Ωa then
+5:
+Contact detection (localization)
+▷ Contact detection & localization, Algo. 1 in [19]
+6:
+Exchange data: receive f cont from Ωb
+▷ Exchange & communication data, Algo. 2 in [19]
+7:
+call calculateProjections()
+▷ Projections & local coordinate system, Algo. 3 in [19]
+8:
+call RK4Scheme()
+▷ Solve system
+9:
+Exchange data: send projection data to Ωb
+▷ Exchange & communication data, Algo. 2 in [19]
+10:
+end if
+11:
+if Deformable body, Ωb then
+12:
+Contact detection (localization)
+▷ Contact detection & localization, Algo. 1 in [19]
+13:
+Exchange data: receive data (projections) from Ωa
+▷ Algo. 2 in [19]
+14:
+call EssentialBoundaryCondition()
+▷ Contact nodes & Dirichlet condition, Algo. 4 in [19]
+15:
+call ExplicitScheme()
+▷ Solve system
+16:
+call ReleaseNodes()
+▷ Algo. 2
+17:
+Exchange: send f cont to Ωa
+▷ Exchange & communication data, Algo. 2 in [19]
+18:
+end if
+19:
+if kfl reset = 0 then
+20:
+exit loop reset
+21:
+end if
+22:
+end loop
+23: end loop
+Algorithm 2 ReleaseNodes() algorithm for explicit time integration schemes
+This algorithm is executed concurrently and for each subdomain at the end of the time step tn+1. The kfl reset is the key flag for
+the repetition of the current time step tn+1 when exists adhesion contact nodes. The sign of the contact force is checked according
+to Eq. 2. The key flag to release the adhesion nodes is called kfl nodes to release. Then the adhesion contact nodes are released
+(as free non-contacting nodes), and the time step is repeated, activating the reset key flag. As all the subdomains require to know
+if the time step has to be repeated or not, the MPI_MAX is in charge to collect the value of the reset for all the subdomains of the
+mesh.
+1: kfl reset ← 0
+2: Get contact force f c and mark adhesion nodes
+3: if kfl nodes to release then
+4:
+Adhesion nodes are set to free nodes
+5:
+kfl reset ← 1
+6: end if
+7: call MPI_MAX(kfl reset)
+24
+
+Algorithm 3 Recursive mesh multiplication algorithm
+The level of mesh refinement is set with the parameter ndivi. ne, nn and nb are the total number of elements, nodes and boundaries
+of the new mesh. The same parameters with the superscript 0 indicate the initial dimension of the mesh. In order to define the
+dimensions of the new mesh is necessary to know the total number of edges (nedgg) and faces nfacg of the initial mesh. Then,
+once the dimensions are known, the DivideMesh() subroutine is in charge of doing the following actions: i) divide each edge and
+face from the initial mesh, ii) define the new element connectivities, iv) define the new element boundary connectivities and v)
+assign the material codes and the corresponding fields such as material coordinate systems. The last step of the mesh division
+algorithm is to reconstruct the interface domains through the ReconstructInterfaceDomains() subroutine. The reader is referred
+to Houzeuax et al. [22] for more details on the implementation and parallel aspects of the proposed algorithm.
+Input n0
+e, n0
+n, n0
+b , ndivi
+Output ne, nn, nb
+1: for idivi = 1, ndivi do
+2:
+nedgg = GetEdges()
+3:
+nfacg = GetFaces()
+4:
+ne, nn, nb = GetDimensions()
+▷ Eq. 14
+5:
+call DivideMesh()
+▷ Sec. 3.2 [22]
+6:
+call ReconstructInterfaceDomains()
+▷ Sec. 3.2 [22]
+7: end for
+Algorithm 4 Workflow of the intralaminar damage model.
+The strain and stress tensors, ε and σ, are defined in the material coordinate system using compact notation [6]. The superscripts
+n and n + 1 define the past and current time steps, respectively. The subscripts N indicate the four damage mechanisms associated
+with the loading function φN and internal threshold variables rN (fibre breaking, fibre kinking, tensile matrix cracking, and
+compressive matrix cracking).
+In turn, the subscript M indicates the five uniaxial damage states DM, represented in 2.
+The
+required material properties are i) elastic properties (E11, E22, ν12, ν23, G12 ), ii) ply strengths (XT , XC, YT , YC, SL), iii) fracture
+toughness (GXT , GXC, GY T , GY C, GSL) associated with the damage mechanism, and iv) yield strength and hardening (Sp, Kp);
+all these properties can be obtained through standardised tests or computational micromechanics simulations [30]. The required
+parameters are: characteristic element length ℓc [33] and state variables at the past time step, i.e. εn
+p , and rt
+M. At the initial time
+step, the state variables are initialised as εn
+p = 0 and rn
+M = 1.
+Input εn+1, εn
+p , rn
+M, ℓc, material properties
+Output σn+1, εn+1
+p
+, rn+1
+M
+1: εn+1
+p
+(εn
+p )
+▷ Plastic strains, yield function in [51]
+2: εn+1
+e
+← εn+1 − εn+1
+p
+▷ Effective elastic strains
+3: σn+1
+e
+← H−1 · εn+1
+e
+▷ Effective compliance matrix H in [30]g
+4: φn+1
+M
+(σn+1
+e
+)
+▷ Loading functions (failure criteria), Eqs. 8, 13, 20, 21 in [32]
+5: rn+1
+N
+(φn+1
+N
+, rn
+N)
+▷ Damage thresholds, Eqs. 24, 26 in [32]
+6: Dn+1
+M
+(rn+1
+N
+)
+▷ Damage state variables according [51] and Eq 6 in [32]
+7: σn+1 ← H−1(Dn+1
+M
+) · εn+1
+e
+▷ Nominal compliance matrix H(Dn+1
+M
+) in [30]
+25
+
+Algorithm 5 Workflow of the cohesive zone model.
+The displacement jumps and interface tractions, ∆ = {∆1, ∆2, ∆3}T and τ = {τ1, τ2, τ3}T , are defined at the mid-plane being
+1 and 2 tangential and 3 normal directions. The superscripts t and t + 1 define the past and current time steps, respectively. In
+turn, the subscript M indicates the pure-mode I and II openings associated with the opening directions, I ↔ {3} and II ↔ {1, 2}.
+The latter is also referred with the subscript sh in [56]. The required input parameters are i) onset displacement jumps (∆Mo),
+ii) critical displacement jumps (∆Mc), iii) penalty stiffness (KM), and iv) Benzeggagh-Kenane exponent for the mixed-mode ratio
+(η). The onset and critical jumps can be obtained from the cohesive strengths (τM) and fracture toughness material properties by
+∆Mo = τM/KM and ∆Mc = 2GM/τM, respectively. The damage threshold state variable at the past time rn
+D, which is initialised
+at the initial time step as rn
+D = 0, is also required to evaluate the model.
+Input ∆n+1, rn
+D, material properties
+Output τ n+1, rn+1
+D
+1: Kn+1
+B
+(∆n+1)
+▷ Local mixed-mode penatly stiffness, Eq. 13 in [56]
+2: Bn+1(∆n+1)
+▷ Local mixed-mode ratio, Eq. 17 in [56]
+3: λn+1
+o
+(Bn+1, Kn+1
+B
+)
+▷ Local mixed-mode onset jump, Eq. 26 in [56]
+4: λn+1
+c
+(Bn+1, Kn+1
+B
+, λn+1
+o
+)
+▷ Local mixed-mode propagation jump, Eq. 24 in [56]
+5: λt+1(∆t+1)
+▷ Local mixed-mode equivalent jump, Eq. 12 in [56]
+6: Hn+1(λn+1, λn+1
+o
+, λn+1
+c
+)
+▷ Loading function (failure criteria), Eq. 20 in [56]
+7: rn+1
+D
+(Hn+1, rn
+D)
+▷ Damage threshold, Eq. 21 in [56]
+8: Dn+1(rn+1
+D
+, λn+1
+o
+, λn+1
+c
+)
+▷ Damage state, Eq. 20 in [56]
+9: τ n+1
+coh (Dn+1, ∆t+1)
+▷ Cohesive tractions, Eq. 7 in [56]
+10: τ n+1
+con (∆n+1)
+▷ Contact tractions, Eq. 8 in [56]
+11: τ n+1 ← τ n+1
+coh + τ n+1
+con
+References
+[1] Abrate, S., 1994. Impact on Laminated Composites: Recent Advances. Applied Mechanics Reviews 47, 517–544. doi:10.
+1115/1.3111065.
+[2] Allix, O., Corigliano, A., 1996.
+Modeling and simulation of crack propagation in mixed-modes interlaminar fracture
+specimens. International Journal of Fracture 77, 111–140. doi:10.1007/BF00037233.
+[3] Ansys Mechanical APDL, . ANSYS Inc., Help system, Theory Reference Chapter 4: Structures with Material Nonlinear-
+ities.
+[4] Bathe, K., Chaudhary, A., 1985. A solution method for planar and axisymmetric contact problems. International Journal
+for Numerical Methods in Engineering 21, 65–85. doi:10.1016/b978-1-85617-802-0.00007-4.
+[5] Baˇzant, Z.P., Oh, B.H., 1983.
+Crack band theory for fracture of concrete.
+Mat´eriaux et Construction 16, 155–177.
+doi:10.1007/BF02486267.
+[6] Belytschko, T., Kam Liu, W., Moran, B., Elkhodary, K.I., 2014. Nonlinear Finite Elements for Continua and Structures.
+1. second edi ed., Wiley. doi:10.1088/1751-8113/44/8/085201.
+[7] Bisagni, C., Vescovini, R., D´avila, C.G., 2011.
+Single-Stringer Compression Specimen for the Assessment of Damage
+Tolerance of Postbuckled Structures. Journal of Aircraft 48, 495–502. doi:10.2514/1.c031106.
+[8] Borrell, R., Cajas, J.C., Mira, D., Taha, A., Koric, S., V´azquez, M., Houzeaux, G., 2018. Parallel mesh partitioning based
+on space filling curves. Computers and Fluids 173, 264–272. doi:10.1016/j.compfluid.2018.01.040.
+[9] de Borst, R., Remmers, J.J.C., 2006. Computational modelling of delamination. Composites Science and Technology 66,
+713–722. doi:10.1016/j.compscitech.2004.12.025. advances in statics and dynamics of delamination.
+[10] Bouvet, C., Castani´e, B., Bizeul, M., Barrau, J.J., 2009. Low velocity impact modelling in laminate composite panels
+with discrete interface elements. International Journal of Solids and Structures 46, 2809–2821. doi:10.1016/j.ijsolstr.
+2009.03.010.
+26
+
+[11] Camanho, P.P., Maim´ı, P., D´avila, C.G., 2007. Prediction of size effects in notched laminates using continuum damage
+mechanics. Composites Science and Technology 67, 2715–2727. doi:10.1016/j.compscitech.2007.02.005.
+[12] Carreras, L., Guillamet, G., Quintanas-Corominas, A., Renart, J., Turon, A., 2021. Mesoscale modelling of delamination
+using the cohesive zone model approach, in: Van Paepegem, W. (Ed.), Multi-Scale Continuum Mechanics Modelling of
+Fibre-Reinforced Polymer Composites. Woodhead Publishing. Woodhead Publishing Series in Composites Science and
+Engineering, pp. 555–577. doi:10.1016/b978-0-12-818984-9.00018-4.
+[13] Cheng, Z.Q., Tan, W., Xiong, J.J., 2022. Modelling pre-fatigue, low-velocity impact and post-impact fatigue behaviours of
+composite helicopter tail structures under multipoint coordinated loading spectrum. Thin-Walled Structures 176, 109349.
+doi:10.1016/j.tws.2022.109349.
+[14] Fournier, Y., 2014. Parallel location and exchange. Technical Report. ´Electricite de France (EDF).
+[15] Furtado, C., Catalanotti, G., Arteiro, A., Gray, P.J., Wardle, B.L., Camanho, P.P., 2019. Simulation of failure in laminated
+polymer composites: Building-block validation.
+Composite Structures 226, 111168.
+doi:10.1016/j.compstruct.2019.
+111168.
+[16] Gallego, F.J., Anza, J.J., 1989. A mixed finite element model for the elastic contact problem. International Journal for
+Numerical Methods in Engineering 28, 1249–1264. doi:10.1002/nme.1620280603.
+[17] Gonz´alez, E.V., Maim´ı, P., Camanho, P.P., Turon, A., Mayugo, J.A., 2012.
+Simulation of drop-weight impact and
+compression after impact tests on composite laminates. Composite Structures 94, 3364–3378. doi:10.1016/j.compstruct.
+2012.05.015.
+[18] Gonz´alez, E.V., Maim´ı, P., Camanho, P.P., Lopes, C.S., Blanco, N., 2011. Effects of ply clustering in laminated composite
+plates under low-velocity impact loading. Composites Science and Technology 71, 805–817. doi:10.1016/j.compscitech.
+2010.12.018.
+[19] Guillamet, G., Rivero, M., Zavala-Ak´e, M., V´azquez, M., Houzeaux, G., Oller, S., 2022. A parallel algorithm for unilateral
+contact problems. Computers and Structures 271, 106862. doi:10.1016/j.compstruc.2022.106862.
+[20] Guillamet, G., Turon, A., Costa, J., Linde, P., 2016. A quick procedure to predict free-edge delamination in thin-ply
+laminates under tension. Engineering Fracture Mechanics 168, 28–39. doi:10.1016/j.engfracmech.2016.01.019. modeling
+of fracture and damage in composite materials.
+[21] Har, J., Fulton, R.E., 2003. A parallel finite element procedure for contact-impact problems. Engineering with Computers
+19, 67–84. doi:10.1007/s00366-003-0252-4.
+[22] Houzeaux, G., de la Cruz, R., Owen, H., V´azquez, M., 2013.
+Parallel Uniform Mesh Multiplication Applied To A
+Navier–stokes Solver. Computers and Fluids 80, 142–151. doi:10.1016/j.compfluid.2012.04.017. selected contributions
+of the 23rd International Conference on Parallel Fluid Dynamics ParCFD2011.
+[23] International, A., 2020. ASTM D7136/D7136M-20, Standard Test Method for Measuring the Damage Resistance of a
+Fiber-Reinforced Polymer Matrix Composite to a Drop-Weight Impact Event.
+[24] Krause, R.H., Wohlmuth, B.I., 2002. A Dirichlet-Neumann type algorithm for contact problems with friction. Computing
+and Visualization in Science doi:10.1007/s00791-002-0096-2.
+[25] Lapeer, R., Gerikhanov, Z., Sadulaev, S.M., Audinis, V., Rowland, R., Crozier, K., Morris, E., 2019. A computer-based
+simulation of childbirth using the partial Dirichlet–Neumann contact method with total Lagrangian explicit dynamics on
+the GPU. Biomechanics and Modeling in Mechanobiology 18, 681–700. doi:10.1007/s10237-018-01109-x.
+[26] Llobet, J., Maim´ı, P., Essa, Y., de la Escalera, F.M., 2021a. A continuum damage model for composite laminates: Part
+III - Fatigue. Mechanics of Materials 153, 103659. doi:10.1016/j.mechmat.2020.103659.
+[27] Llobet, J., Maim´ı, P., Turon, A., Bak, B.L.V., Lindgaard, E., Carreras, L., Essa, Y., de la Escalera, F.M., 2021b. A
+continuum damage model for composite laminates: Part IV- Experimental and numerical tests. Mechanics of Materials
+154, 103686. doi:10.1016/j.mechmat.2020.103686.
+27
+
+[28] Lopes, C.S., Camanho, P.P., G¨urdal, Z., Maim´ı, P., Gonz´alez, E.V., 2009. Low-velocity impact damage on dispersed
+stacking sequence laminates. Part II: Numerical simulations. Composites Science and Technology 69, 937–947. doi:10.
+1016/j.compscitech.2009.02.015.
+[29] Lopes, C.S., Gonz´alez, C., Falc´o, O., Naya, F., LLorca, J., Tijs, B., 2016a. Multiscale virtual testing: the roadmap to
+efficient design of composites for damage resistance and tolerance. CEAS Aeronautical Journal 7, 607–619. doi:10.1007/
+s13272-016-0210-7.
+[30] Lopes, C.S., S´adaba, S., Gonz´alez, C., Llorca, J., Camanho, P.P., 2016b. Physically-sound simulation of low-velocity impact
+on fiber reinforced laminates. International Journal of Impact Engineering 92, 3–17. doi:10.1016/j.ijimpeng.2015.05.014.
+impact Loading on Lightweight Structures.
+[31] Maheo, L., Grolleau, V., Rio, G., 2009. Damping efficiency of the Tchamwa-Wielgosz explicit dissipative scheme under
+instantaneous loading conditions. Comptes Rendus - Mecanique 337, 722–732. doi:10.1016/j.crme.2009.10.005.
+[32] Maim´ı, P., Camanho, P.P., Mayugo, J.A., D´avila, C.G., 2007a. A continuum damage model for composite laminates: Part
+I - Constitutive model. Mechanics of Materials 39, 897–908. doi:10.1016/j.mechmat.2007.03.005.
+[33] Maim´ı, P., Camanho, P.P., Mayugo, J.A., D´avila, C.G., 2007b. A continuum damage model for composite laminates: Part
+II - Computational implementation and validation. Mechanics of Materials 39, 909–919. doi:10.1016/j.mechmat.2007.
+03.006.
+[34] Malone, J.G., Johnsok, K.L., 1994. A parallel finite element contact/impact algorithm for non-linear explicit transient
+analysis: Part I - The search algorithm and contact mechanics. International Journal for Numerical Methods in Engineering
+37, 559–590. doi:10.1002/nme.1620370403.
+[35] Malone, J.G., Johnson, N.L., 1994. A parallel finite element contact/impact algorithm for non-linear explicit transient
+analysis: Part II-Parallel implementation. International Journal for Numerical Methods in Engineering doi:10.1002/nme.
+1620370404.
+[36] Marlett, K., 2011. Hexcel 8552 AS4 Unidirectional Prepreg at 190gsm and 35% RC Qualification Material Property Data
+Report. Technical Report. National institute for aviation research.
+[37] van der Meer, F.P., 2012. Mesolevel Modeling of Failure in Composite Laminates: Constitutive, Kinematic and Algorithmic
+Aspects. Archives of Computational Methods in Engineering 19, 381–425. doi:10.1007/s11831-012-9076-y.
+[38] Nguyen, M.Q., Elder, D.J., Bayandor, J., Thomson, R.S., Scott, M.L., 2005. A review of explicit finite element software
+for composite impact analysis. Journal of Composite Materials 39, 375–386. doi:10.1177/0021998305046739.
+[39] Ortiz, M., Pandolfi, A., 1999. Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation
+analysis.
+Numerical Methods in Engineering 44, 1267–1282.
+doi:10.1002/(SICI)1097-0207(19990330)44:9<1267::
+AID-NME486>3.0.CO;2-7.
+[40] Plagianakos, T., Mu˜noz, K., Guillamet, G., Prentzias, V., Quintanas-Corominas, A., Jimenez, M., Karachalios, E., 2020.
+Assessment of CNT-doping and hot-wet storage aging effects on Mode I, II and I/II interlaminar fracture toughness of a UD
+Graphite/Epoxy material system. Engineering Fracture Mechanics 224, 106761. doi:10.1016/j.engfracmech.2019.106761.
+[41] Quintanas-Corominas, A., Maim´ı, P., Casoni, E., Turon, A., Mayugo, J.A., Guillamet, G., V´azquez, M., 2018. A 3D
+transversally isotropic constitutive model for advanced composites implemented in a high performance computing code.
+European Journal of Mechanics - A/Solids 71, 278–291. doi:10.1016/j.euromechsol.2018.03.021.
+[42] Quintanas-Corominas, A., Turon, A., Reinoso, J., Casoni, E., Paggi, M., Mayugo, J.A., 2020. A phase field approach
+enhanced with a cohesive zone model for modeling delamination induced by matrix cracking.
+Computer Methods in
+Applied Mechanics and Engineering 358, 112618. doi:10.1016/j.cma.2019.112618.
+[43] Reinoso, J., Bl´azquez, A., Estefani, A., Par´ıs, F., Ca˜nas, J., 2013.
+A composite runout specimen subjected to ten-
+sion–compression loading conditions: Experimental and global–local finite element analysis. Composite Structures 101,
+274–289. doi:10.1016/j.compstruct.2012.12.056.
+28
+
+[44] Reinoso, J., Bl´azquez, A., T´avara, L., Par´ıs, F., Arellano, C., 2016. Damage tolerance of composite runout panels under
+tensile loading. Composites Part B: Engineering 96, 79–93. doi:10.1016/j.compositesb.2016.03.083.
+[45] Reinoso, J., Paggi, M., 2014.
+A consistent interface element formulation for geometrical and material nonlinearities.
+Comput Mech 54, 1569–1581. doi:10.1007/s00466-014-1077-2.
+[46] Rivero, M., 2018. A Parallel Algorithm for Deformable Contact Problems. Phd thesis. Universitat Polit`ecnica de Catalunya,
+Escola T`ecnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona.
+[47] Sarrado, C., Leone, F.A., Turon, A., 2016. Finite-thickness cohesive elements for modeling thick adhesives. Engineer-
+ing Fracture Mechanics 168, 105–113.
+doi:10.1016/j.engfracmech.2016.03.020. modeling of fracture and damage in
+composite materials.
+[48] Sasikumar, A., Costa, J., Trias, D., Llobet, J., C´ozar, I.R., Turon, A., Linde, P., 2020. A virtual testing based search
+for optimum compression after impact strength in thin laminates using ply-thickness hybridization and unsymmetrical
+designs. Composites Science and Technology 196, 108188. doi:10.1016/j.compscitech.2020.108188.
+[49] Schellekens, J.C.J., De Borst, R., 1993. On the numerical integration of interface elements. International Journal for
+Numerical Methods in Engineering 36, 43–66. doi:10.1002/nme.1620360104.
+[50] Sommer, D.E., Thomson, D., Falc´o, O., Quino, G., Cui, H., Petrinic, N., 2022.
+Damage modelling of carbon fibre
+composite crush tubes: Numerical simulation and experimental validation of drop weight impact. Composites Part A:
+Applied Science and Manufacturing 160, 107033. doi:10.1016/j.compositesa.2022.107033.
+[51] Soto, A., Gonz´alez, E.V., Maim´ı, P., Mayugo, J.A., Pasquali, P.R., Camanho, P.P., 2018a. A methodology to simulate
+low velocity impact and compression after impact in large composite stiffened panels. Composite Structures 204, 223–238.
+doi:10.1016/j.compstruct.2018.07.081.
+[52] Soto, A., Gonz´alez, E.V., Maim´ı, P., Mart´ın de la Escalera, F., Sainz de Aja, J.R., Alvarez, E., 2018b. Low velocity impact
+and compression after impact simulation of thin ply laminates. Composites Part A: Applied Science and Manufacturing
+109, 413–427. doi:10.1016/j.compositesa.2018.03.017.
+[53] Tan, W., Falzon, B.G., Chiu, L.N.S., Price, M., 2015. Predicting low velocity impact damage and Compression-After-
+Impact (CAI) behaviour of composite laminates. Composites Part A: Applied Science and Manufacturing 71, 212–226.
+doi:10.1016/j.compositesa.2015.01.025.
+[54] Turon, A., Camanho, P.P., Costa, J., D´avila, C.G., 2006. A damage model for the simulation of delamination in advanced
+composites under variable-mode loading. Mechanics of Materials 38, 1072–1089. doi:10.1016/j.mechmat.2005.10.003.
+[55] Turon, A., D´avila, C.G., Camanho, P.P., Costa, J., 2007. An engineering solution for mesh size effects in the simulation of
+delamination using cohesive zone models. Engineering Fracture Mechanics 74, 1665–1682. doi:10.1016/j.engfracmech.
+2006.08.025.
+[56] Turon, A., Gonz´alez, E.V., Sarrado, C., Guillamet, G., Maim´ı, P., 2018.
+Accurate simulation of delamination under
+mixed-mode loading using a cohesive model with a mode-dependent penalty stiffness. Composite Structures 184, 506–511.
+doi:10.1016/j.compstruct.2017.10.017.
+[57] V´azquez, M., Houzeaux, G., Koric, S., Artigues, A., Aguado-Sierra, J., Ar´ıs, R., Mira, D., Calmet, H., Cucchietti, F.,
+Owen, H., Taha, A., Burness, E.D., Cela, J.M., Valero, M., 2016.
+Alya: Multiphysics engineering simulation toward
+exascale. Journal of Computational Science 14, 15–27. doi:10.1016/j.jocs.2015.12.007.
+[58] Weyler, R., Oliver, J., Sain, T., Cante, J.C., 2012. On the contact domain method: A comparison of penalty and Lagrange
+multiplier implementations. Computer Methods in Applied Mechanics and Engineering 205-208, 68–82. doi:10.1016/j.
+cma.2011.01.011. special Issue on Advances in Computational Methods in Contact Mechanics.
+[59] Yastrebov, V.A., 2011. Computational Contact Mechanics. Phd thesis. MINES ParisTech. doi:10.1017/CBO9781107415324.
+004.
+[60] Yastrebov, V.A., Breitkopf, P., 2013. Numerical Methods in Contact Mechanics. doi:10.1002/9781118647974.
+29
+
+[61] Zavala-Ak´e, J.M., 2018. A high-performance computing coupling tool for partitioned multi-physics applications. Phd
+thesis. Universitat Polit`ecnica de Catalunya, Departament de F´ısica.
+30
+
diff --git a/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/load_file.txt b/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1c11ae52035195b31bf01480aff47bf5a4f4e789
--- /dev/null
+++ b/F9E5T4oBgHgl3EQfVg_E/content/tmp_files/load_file.txt
@@ -0,0 +1,1512 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf,len=1511
+page_content='Application of the partial Dirichlet-Neumann contact algorithm to simulate  low-velocity impact events on composite structures  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Guillameta,∗, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Quintanas-Corominasa,b,c, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Riveroa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Houzeauxa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' V´azqueza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Turonb  aBarcelona Supercomputing Center (BSC), Pla¸ca Eusebi G¨uell, 1-3, Barcelona, 08034, Catalonia, Spain  bAMADE, Universitat de Girona, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Universitat de Girona 4, Girona, 17003, Catalonia, Spain  cDepartment of Civil and Environmental Engineering, Imperial College London, London, SW7 2AZ, UK  Abstract  Impact simulations for damage resistance analysis are computationally intensive due to contact algo-  rithms and advanced damage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Both methods, which are the main ingredients in an impact event, re-  quire refined meshes at the contact zone to obtain accurate predictions of the contact force and damage onset  and propagation through the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This work presents the application of the partial Dirichlet-Neumann  contact algorithm to simulate low-velocity impact problems on composite structures using High-Performance  Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This algorithm is devised for parallel finite element codes running on supercomputers, and it  is extended to explicit time integration schemes to solve impact problems including damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The proposed  framework is validated with a standard test for damage resistance on fiber-reinforced polymer matrix com-  posites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Moreover, the parallel performance of the proposed algorithm has been evaluated in a mesh of 74M  of elements running with 2400 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Keywords:  Contact mechanics, Damage modeling, Finite element analysis, High-Performance Computing  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Introduction  Impacts by foreign objects against any part of the aircraft are a major concern for the aerospace industry  because they may compromise the structural integrity of the aircraft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Impact events can be classified into  three main categories: low, high (including ballistics), and hyper-high velocity impacts [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  During the  impact, the energy by the foreign object (projectile) is transferred to the target (structure), and consequently,  the material can be damaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Concretely, low-velocity impact events on composite materials (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', tool drops  during maintenance or manufacturing) can drastically reduce the residual strength of the part even for case  scenarios of Barely Visible Impact Damage (BVID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Therefore, designing composite structures with damage  resistance cannot be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Moreover, extensive experimental campaigns particularly focused on the  ∗Corresponding author  Email address: gerard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='guillamet@bsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='es (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Guillamet)  Preprint submitted to Composites Part A: Applied Science and Manufacturing  January 16, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='05552v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='CE]  11 Jan 2023  investigation and evaluation of damage resistance of a specific material may be prohibitive by the industry  in terms of costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Thus, virtual testing of impact events is of great interest as mathematical models and technology advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  However, solving the physics behind this problem, particularly from the material point of view, is still one  of the most complex and challenging problems today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A review of existing software for composite impact  modeling focused on low-velocity events is conducted by Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this review, the constitutive  damage models play an essential role apart from the methods such as the contact algorithm or temporal  integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Most of them can capture the trends and peak forces reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The research  community has put and continues to put a lot of effort into developing reliable constitutive damage models  for composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Remarkable progress has been made on specific methodologies and constitutive damage  models for predicting the damage resistance and damage tolerance of composite structures [28, 10, 17, 53,  29, 52, 51, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' However, in terms of computational performance, the resolution of an impact problem,  including different sources of damage, is still computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The use of sophisticated contact  algorithms and advanced damage models require refined element meshes to accurately predict the onset and  propagation of the damage in such materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The most commonly used contact algorithms for the resolution of an impact problem are the Penalty  methods [21], Classical Lagrange multipliers [4, 16] or the Augmented Lagrange multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The latter is  often chosen to solve the contact inequality constraints, see [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' However, the parallel aspects of these  traditional contact algorithms are not trivial, and to the authors’ knowledge, little effort has been invested  in the parallel aspects of such algorithms and their scalability in supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Some research works  dealing with the parallel aspects of contact algorithms are [34, 35, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  A completely different approach to the previous algorithms is the method of partial Dirichlet-Neumann  (PDN) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The contact is tackled as a coupled problem, in which the contacting bodies are treated  separately, in a staggered way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The coupling is performed through the exchange of boundary conditions at the  contact interface following a Gauss-Seidel strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The pioneering works using this approach are conducted  by Krause and Wohlmuth [24] and Yastrebov [59], showing the capabilities of solving nonlinear contact  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' To the authors’ knowledge, one of the first applications of this method for explicit dynamics is  made by Lapeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [25], where the PDN method was used to simulate natural childbirth using explicit  dynamics and executed in a hybrid system with Central Processing Units (CPU) and Graphics Processing  Unit (GPU) architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' However, little attention is dedicated to the computational performance and  the parallel aspects of dealing with large-scale models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' More recently, this method has been adapted and  implemented in parallel in the Alya multiphysics code [57] by Rivero [46] and published by the authors in  [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The mathematical and the parallel aspects are described in detail in these works, demonstrating the  benefits of the PDN contact algorithm in High-Performance Computing (HPC) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In this paper, we present the application of the aforementioned method proposed by the authors in [46, 19]  2  for the resolution of low-velocity impact problems for composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Existing time integration schemes  and constitutive models from the literature have also been adapted and implemented within a parallel  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' So the main contribution of the present paper is focused on the extension of the PDN contact  algorithm for explicit time integration schemes and its use in HPC systems involving impact events in the  field of composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Additionally, a new mesh multiplication algorithm is presented to deal with  cohesive elements and element technologies such as continuum shell elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The content of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Firstly, the methods for the resolution of low-velocity  impact events on composite materials including damage are explained with a strong emphasis on the contact  algorithm and its implementation in parallel codes based on the finite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Then, the algorithm  is validated through three benchmark tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The first one consists of a quasi-static indentation test to  verify that the contact pressure is well captured by using implicit and explicit time integration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The second and the third examples correspond to a low-velocity impact on a composite plate following the  ASTM International standard to measure the damage resistance of fiber-reinforced polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' These last  examples, use different material systems which are quite used by the aerospace industry, the T800S/M21 and  the AS4/8552 carbon/epoxy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The numerical predictions obtained are correlated with experiments,  and the computational performance is analyzed and discussed for the coupon made of AS4/8552 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Finally, the conclusions of this work are commented on together with future work to improve the simulation  of impact events including damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Modeling framework for low velocity impact events using High-performance systems  This section describes the modeling framework and the application of the partial Dirichlet-Neumann  (PDN) contact algorithm for the simulation of impact events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Particular emphasis is put on the extension of  such contact algorithm for explicit dynamic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' All the methods presented here are implemented in the  Alya multiphysics code [57] based on the Finite Element Method (FEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This parallel code is based on high-  performance programming techniques for distributed and shared memory supercomputers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Moreover, the  methods are programmed using the total Langrangian formulation, where stresses and strains are measured  with respect to the original configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The Green strain measure and the 2nd Piola-Kirchoff stress are  used, and we follow the notation from Belytschko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [6] throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As an impact event is  a complex and computationally demanding engineering problem is very attractive to be solved using High-  Performance Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It is worth highlighting that all the methods described here can be implemented  in other parallel FEM codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Partial Dirichlet-Neumann contact algorithm  The low-velocity impact event proposed in this paper can be assumed as a non-linear contact problem,  where the striker is considered as a rigid body and the plate as a deformable body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Let’s assume that both  3  body instances are of arbitrary shape, and we do not consider friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Therefore, this contact problem can  be written as a boundary value problem, see Yastrebov [60], which includes the Hertz-Signorini-Moreau law  for normal contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' So the balance of momentum and the contact conditions can be written as follows:  ∇ · σ + f v = 0  in Ω  σ · n = σ0  on ΓN  u = u0  on ΓD  g ≥ 0, σn ≤ 0, σn g = 0, σt = 0  on ΓC  (1)  being σ the Cauchy stress tensor, f v a vector of volumetric forces, σ0 a set of prescribed tractions on the  Neumann boundary, ΓN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' u0 a set of prescribed displacements on the Dirichlet boundary, ΓD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Over the  contact boundary, ΓC, we have imposed the following conditions: g represents the gap between contacting  bodies, σn is the normal contact pressure, and σt is the tangential stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The tangential stress equal to  zero (σt = 0) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (1) characterizes a frictionless contact case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In order to satisfy the conditions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (1), the present paper uses the partial Dirichlet-Neumann  contact algorithm proposed by Rivero [46, 19] which is based on the work from Yastrebov [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In the works  mentioned above, the method was applied for implicit time integration schemes, while in the present work,  the algorithm is extended to explicit schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The main benefits of the PDN contact algorithm to typical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='Penalty or Lagrange Multipliers methods are the following: (i) the size of the problem does not increase due  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='to the Lagrange multipliers methods as unknowns (ii) no restriction with respect to the mesh partitioner  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='due to the use of contact elements (iii) absence of contact tangent matrices (implicit schemes) and residual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='contact force vectors and (iv) easy to be parallelized as it can be treated as a solid-to-solid coupling using  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='existing methods for multiphysics applications such as the Gauss-Seidel scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The iterative process of the PDN contact algorithm in a frictionless problem is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Let’s  assume that the time of the simulation is 0 < t < tE and it is subdivided into nT S time steps ranging  from n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='nT S and tE is the time at the end of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' At time step n there is no interaction  between both code instances, so no contact is detected (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Then, at time tn+1, contact is detected  as we have overlapping between both bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' At this step, the non-penetration boundary conditions are  treated kinematically, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', as Mulitple Point Constraints (MPC) by the projection of the nodes belonging  to the slave surface (deformable’s body) to the master surface (rigid’s body) using a Dirichlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Then, a local coordinate system with normal-tangent basis vectors nj and tj is created for each detected  node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The contact node is restricted to only move to the tangent line defined by the vector tj, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In a hypothetical frictional contact problem, the friction force would be imposed in this direction as a  Neumann boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this work, friction is not considered for the low-velocity impact as the  relative velocities at the contact zone are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' After that, the contact algorithm checks the  4  2  (a)  (c)  (b)  (d)  Rigid  Deformable  Contact node  Released (free) node  (e) Kinematic constraint for node 1 and 2  n2  t2  n1  t1  1  2  f c  f c  f c  f c  f c  f c  Figure 1: Iterative process of the parallel PDN contact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (a) Interaction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (b) Overlapping;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (c) Dirichlet boundary  conditions (projections) (d) Released nodes and equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (e) Kinematic constraint for node 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reader is referred  to the web version of this paper for the color representation of this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  presence of adhesion or artificial contact nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', nodes in traction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reaction contact force  f c  j has to satisfy the following condition:  f c  j · nj ≥ 0  (2)  Those adhesion nodes have to be released, so the current time step tn+1 has to be repeated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' the i index  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1c and 1d represents the sub-iterations for node release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The whole kinematic constraint  process is depicted for two of the contacting nodes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The nodes release algorithm for explicit time  schemes is described in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 in Appendix  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The condition to distinguish a true contact node or an  adhesion (artificial) contact node is by means of the contact force (reaction due to the Dirichlet condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The vector of contact forces using total Lagrangian formulation can be expressed as:  (f c  j)T =  �  Ω0  BT  0jP dΩ0  (3)  where BT  0j is the matrix containing the derivatives of the shape functions with respect to the reference  system and P is the nominal stress tensor, see [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  An exciting aspect of the PDN method is that the computational cost of the projections is very small  compared to Penalty or Langrange approaches [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' One of the most consuming parts and a vital issue for  further research is the contact searching and communication between the subdomains (belonging to different  code instances), as stated in the previous work from the authors [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In our case, we use the PLE++ library  [61], which is an adaptation of the Parallel Location and Exchange PLE library [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The main algorithm  of the PDN contact method is described in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1 in Appendix  A and the nodes release algorithm for  5  explicit schemes is summarized in Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reader is referred to Rivero [46] and Guillamet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [19]  works for more details on the implementation aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Time integration schemes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Deformable body  Spurious oscillations may appear when using explicit time schemes for dynamic and wave propagation  problems such as impact events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' These oscillations occur due to the mismatch of two different types of wave  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thus, dissipative explicit time schemes are often used to reduce the numerical instabilities  induced by the spatial and time discretization procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Among the many dissipative methods available,  the Tchamwa–Wielgosz (TW) explicit scheme [31] is beneficial because it damps out the spurious oscillations  occurring in the highest frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This is the time integration scheme selected in this work, but  any other explicit time scheme such as the Central Difference (CD) [6] including bulk viscosity could also  be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The motion described by the TW scheme is the following:  ˙d  n+1 = ˙d  n + ∆t¨d  n  (4)  dn+1 = dn + ∆t ˙d  n + ϕ(∆t)2¨d  n  (5)  where d, ˙d, ¨d are the displacement, velocity and acceleration nodal vectors, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ∆t is the time  increment or step size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' and ϕ is a numerical viscous parameter, which in the current work is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='033  [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The key to the computational efficiency of explicit time integration schemes is the use of the lumped  mass matrix for the resolution of the linear system of equations, which is simplified as an easy inversion of  the diagonal mass matrix [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The global stiffness matrix is not required to be assembled as it is needed for  implicit time integration schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The explicit time integration scheme solves accelerations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' so their values  at the beginning of the increment are computed by making use of the equation of motion of the system:  m ¨d  n = f n(dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' tn) = f e(dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' tn) − f i(dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' tn) − f c(dn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' tn)  (6)  where m is the vector representation of the lumped mass matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' f e is the global vector of external forces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' f i  is the global vector of the internal forces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' and f c is the global vector of the contact forces from the Dirichlet  condition imposed on that nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thanks to m, the acceleration can be computed without invoking any  solver as:  ¨d  n = m−1(f e − f i − f c)  (7)  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Rigid body  The striker in the present work is considered as a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The resolution of the equations of motion  for the rigid bodies we use a 4th order Runge Kutta scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Let’s consider the following differential equation  where the right hand side is a function of both time and another function dependent on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  dy  dt = f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' y(t))  (8)  From this equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' the Runge-Kutta method estimates the solution at n + 1 taking into account four  evaluations of the right hand side step dt as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  k1 = dt · f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' y(t))  k2 = dt · f(t + dt  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' y(t) + k1  2 )  k3 = dt · f(t + dt  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' y(t) + k2  2 )  k4 = dt · f(t + dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' y(t) + k3)  yn+1 = y(t + dt) = y(t) + k1  6 + k2  3 + k3  4 + k4  6  (9)  In the present paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' the motion of the striker is solved by making use of the following differential  equation:  m · ¨d = m · g − f e  (10)  where m is a scalar value of the mass of the rigid body,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' g is the gravity force vector at the center of mass,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  ¨d is the linear acceleration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' and f e is the external force also at the center of mass from the rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It is  worth mentioning that the rigid body is represented by a point (center or mass), so the above vectors have  a dimension of 2 for 2-d problems and 3 for 3-d problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' When contact occurs the external force from the  rigid body is calculated by f e = �nc  j=1 f c  j, where j denotes a contact node and nc is the total contact nodes  belonging to the deformable body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mesoscale damage modeling for fiber-reinforced composites  The mesoscopic length scale is the most suitable for virtual testing of low-velocity impacts on structures  made of composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' At this scale, the numerical predictions have a good trade-off between infor-  mation about the damage mechanisms driving the failure process and the structural response without the  complexity of dealing with intricate microstructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It is worth emphasizing that the mesoscopic length  scale is not only appropriate for the bottom levels of the building block approach (coupon and elements)  [28, 53, 15, 50] but also for the top levels (sub-components and components) [43, 44, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  7  From the constitutive modelling viewpoint, the mesoscopic length scale simplifies the intricate microstruc-  ture of long fibre composite laminates by homogenising the properties and mechanisms at the lamina level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The outcome is a layered material with two well-defined regions: intralaminar and interlaminar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The former  is modelled as a transversally isotropic material, which can fail due to fibre breaking and matrix cracking  according to the loading scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The latter is modelled as a very thin region, usually tending to zero  thickness, where delamination can onset and propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Regarding the modelling architecture, several strategies exist in the literature suitable for modelling  composite at a mesoscopic length scale using FEM [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' We adopt a continuous approach for the intralam-  inar region (CDM with linear elements) and a discontinuous one for the interlaminar (CZM with interface  elements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The straightforward implementation of this strategy in a standard FEM code aids in preserv-  ing the scalability of Alya multiphysics [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thus, the mesoscale damage modeling strategy exploits the  computational resources to maximize the accuracy of the impacts thanks to very thin meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Intralaminar damage model  The intralaminar damage model for predicting ply failure is based on the continuum damage mechanics  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Fiber and matrix cracks are smeared in the continuum and represented by state variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Accordingly, the crack’s kinematics is not explicitly represented, but their effects on the degradation of the  capacities of sustaining loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In turn, the onset and growth of the damage failure mechanisms are governed  by the failure surfaces and evolution laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this work, we employ a local damage model based on the  constitutive modeling framework for long fiber composite materials proposed by Maim´ı et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This  framework has been used widely in the literature, demonstrating outstanding accuracy and performance not  only for static scenarios [11, 7, 41] but also for impact [17, 51, 48] and fatigue [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The main ingredients of the intralaminar damage model are: i) transversally isotropic elasto-plastic re-  sponse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ii) damage activation functions related to the different ply failure mechanisms through the maximum  strain criterion for the fiber breaking and the LaRC criteria for the matrix cracking,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' iii) the damage evolution  laws are defined to dissipate the fracture energy associated to the opening mode ensuring mesh objectivity  by the crack-band theory [5],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' and iv) the thermodynamic consistency is ensured by imposing irreversibility  of the damage variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 illustrates the intralaminar failure modes schematically modelled, while Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4 in Appendix A  summarises the material model workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Note that a plastic response under shear loads is considered,  and five damage mechanisms are modelled: fibre breaking, fibre kinking, tensile and compressive matrix  cracking, and shear matrix cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The details of the expressions employed and their justification from a  physical standpoint can be found in [32, 33, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Besides the constitutive response, the intralaminar damage model also encloses the computation of the  critical time step, which is required by the explicit time integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' For the sake of simplicity, we  8  Fibre breaking  Fibre kinking  Matrix cracking  Matrix cracking  Matrix cracking  Figure 2: Schematic representation of the intralaminar damage mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Adapted from [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  utilise the same formula of a transversally isotropic material:  ∆t = vsound  ℓc  =  �  max Cij  ρ  (11)  where Cij are the components of the effective stiffness matrix, ρ is the density of the material, and ℓc is  the characteristic element length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Considering a structured hexahedral mesh employed, we approximate the  characteristic element length with the element volume Ve [33]:  ℓc ≈  3�  Ve  (12)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Interlaminar damage model  The interlaminar damage model for predicting the onset and propagation of delamination is based on  the cohesive zone approach and formulated in the context of damage mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Accordingly, a damage  state variable is employed to account for the gradual loss of the bearing capacities of the material in the  cohesive zone due to the separation of crack surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In turn, the separation or opening of the crack is  represented by a kinematic quantity noted as displacement jump, which is approximated by employing the  interface element technology [2, 39, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thus, the interlaminar damage model is a constitutive model that  computes the cohesive reactions as a function of displacement jumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' More details of the mesoscale modeling  of delamination using cohesive zone models can be found in Carreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In this work, we use the cohesive zone model proposed by Turon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [54, 56], which has been employed  extensively in the literature [20, 47, 40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The main characteristics of this CZ model are: i) linear response  9  125 μm125 μm125 μm125 μm125 μmbefore initiation of the softening, ii) linear relation between the cohesive tractions and crack openings,  iii) onset and propagation of the damage in compliance with the Benzeggagh-Kenane criterion, and iv)  thermodynamic consistency despite the loading scenario, even when the mix-mode ratio varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 5  summarizes the cohesive zone model workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Regarding the element technology, we employ zero-thickness interface elements for capturing the delam-  ination, implemented using the formulation presented in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As standard interface elements are used, the  integrals are computed using a Newton-Cotes integration scheme to mitigate the spurious oscillations in the  traction profile along the interface [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The stable time increment, which is necessary for the explicit time  integration scheme is obtained through [51]:  ∆tcoh =  �  ¯ρ  Kcoh  (13)  where ¯ρ and Kcoh are numerical parameters known as cohesive surface density and penalty stiffness, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The cohesive surface density for zero-thickness elements is approximated by the expression in  [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In turn, the cohesive penalty stiffness is defined to avoid affecting the compliance of the system as  Kcoh ≥ 50ET /tlam, where ET is the transverse elastic modulus and tlam the adjacent laminate thickness  [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mesh refinement algorithm for interface elements with a cohesive law  The mesh multiplication algorithm proposed by Houzeaux et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [22] has been extended to deal with  the presence of interface elements or even continuum shell element formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Focusing on interface  elements, they are zero-thickness elements with a cohesive material law that are inserted between plies in  a laminated composite material in order to predict the delamination damage mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It is well known  that an accurate prediction of the onset and propagation of delamination in composite materials requires  very refined meshes, as stated in [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' However, depending on the number of interface layers or the  geometry size, it can be challenging to place these elements between plies and computationally demanding  to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  On the other hand, mesh generation of large meshes is often a bottleneck in engineering applications  to deal with thousands of millions of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thus, integrated tools for mesh refinement within parallel  codes devised for High-Performance systems allow a parallel and fast refinement of the coarse mesh without  the need to create the mesh again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Therefore, this paper also introduces a new capability of the mesh multiplication algorithm from [22],  which enables the refinement of large-scale problems, including interface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Let’s assume a configu-  ration of two bulk elements together with an interface element between them, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  10  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  Interface  element  Solid  element  2  3  4  1  6  7  8  5  Original node  New edge node  New face node  New center node  2  3  4  6  8  5  (ne  ELINT= 4)  Bulk element  Interface element  (ne  BULK = 8)  Global numbering  Local numbering  1  7  Figure 3: Mesh multiplication between bulk and interface (cohesive) elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The 8-node interface element can only be divided into four elements to avoid the duplication of the  element at the interface mid-plane between the bulk elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The criterion used for the correct division  is by making use of the element normal, also known as stacking direction, which is required for the proper  behaviour of the element due to its kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thus, those parallel planes to the element normal are used  to divide the element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The dimensions of the new mesh can be calculated as follows:  ne = 8 · n0  e,BULK − n0  e,ELINT · 4  nn = n0  n + nedges + nfaces + n0  e,BULK − n0  e,ELINT − n0  edges,ELINT − n0  faces,ELINT  nb = 4 · n0  b − 2 · n0  b,ELINT  (14)  where ne, nn and nb are the total number of elements, nodes and boundaries for the new mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In order  to refine the hybrid mesh is important to know the total number of n0  e, n0  n and n0  b from the original mesh  and also information about the edges and faces that have to be divide or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3 summarizes the  different steps and functions for the mesh division and reconstruction of the interface domains in a parallel  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Benchmark tests  Three benchmark tests are conducted to validate the application of the parallel partial Dirichlet-Neumann  contact algorithm using an explicit time integration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The first example consists of a quasi-static  indentation test, which has already been solved using an implicit time integration scheme in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The  solution using explicit analysis is compared with the numerical solution obtained for implicit analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The  second and the third examples consist of a low-velocity impact event on two coupons manufactured with two  11  well-known material systems for the damage prediction: T800S/M21 and AS4/8552 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thanks  to the proposed algorithm’s flexibility and generality, we use a multi-code approach, where the motion of  each body (rigid and deformable) is solved using different instances of Alya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Regarding the partitioning  of the mesh, we use the Space-Filling Curve (SFC) based partitioner described in [8], which performs the  partitioning in parallel and maximizes the load balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It is worth highlighting that all the executions  here are in parallel (pre-process, solution, and post-process steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In all the examples, the contact bodies  are discretized with a refined finite element mesh to assess the geometrical localization between both code  instances and to obtain an accurate prediction of the contact force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' All the simulations are conducted in  MareNostrum4 supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This cluster has 3456 nodes, each of them with 48 processors Intel Xeon  Platinum @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 [GHz], giving a total processor count of 165 888 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Quasi-static indentation test  This example has already been solved using an implicit time solution scheme in [19, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This case is  now solved as a quasi-static problem using explicit dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The example consists of a rigid rounded head  (indenter) and a deformable beam, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The geometrical dimensions of the indenter (rigid body) are  ri = 1 m and wi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 m, while the beam (deformable body) are hb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='25 m, lb = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 m and wb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The  relative position of the indenter with respect to the beam is given by the parameters ax = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='25 m, az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 m  and ay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01 m (gap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The beam is modelled with an hyperelastic Neo-Hookean formulation [3] and finite  strains, with material properties Eb = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='896 × 108 Pa (Young modulus), νb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='32 (Poisson ratio) and density  ρ = 1000 kg m−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The beam is fully clamped at the bottom face, and a prescribed vertical displacement of  δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='11 m is applied at the top surface belonging to the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Both bodies are discretized with finite  elements using full integration: 8-node linear solid elements for the beam and 4-node linear tetrahedrons  for the indenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The beam has a base mesh of 3510 elements, while the indenter has 15 960 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A  non-linear dynamic analysis is performed with a total time of the simulation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='05 s and a fixed time step of  1 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The selected time step value is smaller than the stable time increment, which is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='796 × 10−5, and  no mass scaling is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In order to perform a quasi-static event and minimize the kinetic energy, a smooth  step function (fifth-order polynomial) is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This function has the form A0+(AE −A0)ξ3(10−15ξ+6ξ2)  for t0 ≤ t < tE, where A0 and A1 are the initial and final amplitude, t0 and tE are the initial and final time  of the simulation and ξ =  t−t0  tE−t0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This smooth load rate ensures that the first and second time derivatives  are zero at the beginning and the end of the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  12  Beam (deformable)  hb  x  y  z  y  lb  wi  wb  az  ax  Indenter (rigid)  ri  ux = uy = uz = 0  ux = uz = 0,  ay  uy =  Figure 4: Setup for the quasi-static indentation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Adapted from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Displacements and forces obtained at the contact zone are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 5 for two different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Line  path a is centered and goes from one side to the other in the length direction of the beam, while line path b  is also centered in the width direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The numerical prediction using the explicit time integration scheme  is compared with the implicit solution obtained in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' We can observe an excellent agreement between  both numerical predictions in terms of the displacements and contact force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  (a)  (b)  (c)  (e)  (d)  Path b  Path a  [18]  [18]  [18]  [18]  [18]  Figure 5: Displacements and contact forces at straight lines path a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (a) Tangential displacement in x-direction for path  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (b) Normal displacement in y-direction for path a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (c) Tangential displacement in z-direction for path b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (d) Contact force  at line path a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (e) Contact force at line path b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Low velocity impact on a composite plate  The proposed benchmark consists of a drop-weight of a rigid hemispherical striker on a rectangular plate  made of composite material, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Two impact scenarios using different material systems, layups, and  impact energies are considered for the validation of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The materials selected are  the unidirectional prepreg M21/194/34%/T800S (T800S/M21) and the unidirectional prepreg AS4/8552,  both carbon-epoxy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' On the one hand, the coupon made of T800S/M21 is manufactured by Hellenic  Aerospace Industry and tested at Element Materials Technology Seville facilities within the framework of the  CleanSky2 SHERLOC project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Most of the material properties from the T800S/M21 are also characterized  by Hellenic Aerospace Industry and Element Materials Technology Seville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' On the other hand, the coupon  made of AS4/8552 is chosen from literature through the works conducted by Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [17] and  Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' All the material properties from the aforementioned materials, including damage model  parameters, are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The intralaminar damage model is fed by the in-situ strengths which  are calculated following the works by Furtado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [15] and Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Rubber clamp  (ux = uy = uz = 0)  Striker  rs = 8 mm  (ux = uy = 0)  ms = 5 kg  Top view  75 mm  125 mm  R7  Refined region  (75 mm x 75 mm)  (uz = 0)  Window cut  Stacking sequence  [454/04/-454/904]s  Cohesive elements between clusters  tply = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='18125 mm  tcoh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 x 10-4 mm  Figure 6: Numerical setup for the low velocity impact test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The mesh and layup correspond to the coupon made of AS4/8552  material used for the parallel performance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Both impact case scenarios follow the standard ASTM D7136/D7136M-20 [23] for damage resistance  evaluation of fiber-reinforced polymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Each plate has the same dimensions: 150 mm×100 mm and each  of them are supported on a metallic frame with a cut-out of 125 mm×75 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Rubber-tipped clamps  clamp the plate instance at the four corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' We consider equivalent boundary conditions to represent this  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As we can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 6, the metallic frame and the rubber clamps from the experiment do not  exist as physical entities, so we only consider the contact surface from the rubber cylinder-shaped clamps  and the contact edges of the cut-out window of the metallic frame, where we apply the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  14  T800S/M21  AS4/8552  Property  Value  CV(%)  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Value  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Density (t/mm3)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='59 × 10−9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='59 × 10−9  [51, 17]  Elastic  E11 (MPa)  138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4 × 103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='95  128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 × 103  [51, 17]  E22 = E33 (MPa)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='54 × 103  3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='63 × 103  [51, 17]  ν12 = ν13 (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='311  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='35  [51, 17]  ν23 (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='45  [51, 17]  G12 = G13 (MPa)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='29 × 103  3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='358 × 103  [51, 17]  G23 (MPa)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='945 × 103 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='631 × 103  [51, 17]  Strength  XT (MPa)  2854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  4  2300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  [51, 17]  XC (MPa)  1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  13  1531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  [51, 17]  YT (MPa)  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  YC (MPa)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  [15]  199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8  [51, 17]  SL (MPa)  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='36a  [36]  αo (◦)  53  [32]  53  [32]  In-situ strengthsc  Y is  T,int (MPa)  132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 (1tply) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 (4tply)  Y is  T,int (MPa)  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7 (2tply) 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 (8tply)  Y is  T,out (MPa)  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (1tply) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2 (4tply)  Y is  C,int (MPa)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 (1tply) 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (4tply)  Y is  C,int (MPa)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 (2tply) 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (8tply)  Y is  C,out (MPa)  250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 (1tply) 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (4tply)  Sis  L,int (MPa)  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 (1tply) 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (4tply)  Sis  L,int (MPa)  116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 (2tply) 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 (8tply)  Sis  L,out (MPa)  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7 (1tply) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4 (4tply)  Fracture toughness  GXT (N/mm)  340  [15]  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5  [51, 17]  GXC (N/mm)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  [15]  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  [51, 17]  GY T (N/mm)  GIc  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  GIc  [51, 17]  GY C (N/mm)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='38b  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='313b  [51, 17]  GSL (N/mm)  GIIc  20  GIIc  [51, 17]  Traction separation law  fXT (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  [51, 17]  fGT (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  [51, 17]  fXC (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  [51, 17]  fGC (-)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9  [51, 17]  Matrix plasticity  Sp (N/mm)  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9 [15]  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0a  Kp (N/mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='09 [15]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1936a  Interface properties  GIc (N/mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='308  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='28  [51, 17]  GIIc (N/mm)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='828  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='79  [51, 17]  τI (MPa)  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2d  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8  YT  [51, 17]  τII (MPa)  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7d  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  SL  [51, 17]  η (-)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='45  [51, 17]  Kcoh (M/mm3)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 × 106 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 × 104  [51]  a Best fitted based on properties from [36]  b GY C = GSL/cos(αo) [32]  c Calculated considering plasticity using equations from [51]  d Engineering solution by Turon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2007 [55] using Ne=5  Table 1: Material properties for the M21/194/34%/T800S (T800S/M21) and Hexply AS4/8552 including damage models  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  15  The velocity of the striker is given as an initial condition set in the impact direction, while the remaining  degrees of freedom are constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The initial velocity of the striker is calculated based on the impact  energy of each case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The initial position of the striker has a gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01 mm between the striker  tip and the top surface of the plate in order to avoid overlapping between bodies at the beginning of the  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Moreover, gravity forces are included in both body instances, considering a gravity value of  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='81 m/s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  We employ 8-node full integration hexahedron elements for the plate using the inter- and intra-laminar  damage models described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Cohesive elements are inserted at each  interface between different ply angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  It is worth highlighting that other constitutive material models  and element technologies would also be feasible in combination with the proposed contact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' With  regards to the strikers used for each impact case scenario, they are discretized with 4-node linear tetrahedron  elements with a biased mesh of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 mm at the center of the half-sphere and 1 mm at the end of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The total number of elements for the striker used for the T800S/M21 and AS4/8552 materials are 32 685  and 79 934, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Regarding the plates, they both have a refined centered region of 75 mm×75 mm  with an in-plane element size equal or multiple to the ply thickness, depending on the material system in  order to guarantee an aspect ratio close or equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Coupon made of T800S/M21 material  This impact coupon has a stacking sequence of [45/ − 45/02/90/0]S and is made of T800S/M21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The  nominal ply thickness is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='192 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This case study is submitted to an impact energy of 10 J, which falls into  the Barely Visible Impact Damage (BVID) analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The striker has a diameter of 25 mm and a mass of 2 kg,  which is modeled as a rigid body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The global element size for the plate is 1 mm, and each lamina and the  clusters of two plies have one element through the thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The in-plane element size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='192 mm which  is equal to the ply thickness resulting in an aspect ratio of 1 for those elements at plies without clustering  and located at the refined region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The mesh of the plate has a total of 1 042 525 hexahedron elements (≈  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 million of Degrees Of Freedom (DOF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The total time for this simulation is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The initial  velocity of the striker considering the gap previously mentioned is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='16 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The numerical predictions for this impact case scenario and their comparison with experimental data  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 7 and summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The experimental test campaign consisted of testing a  batch of five coupons to ensure proper repeatability of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The force-time for each impact was  recorded with a limited number of points (52 points on average for each impact test).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reduced number  of points only allows for validation of the global behavior of the impact case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Energy-time and the  force-displacement curves are calculated by integrating once and twice the experimental force history curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  16  (a)  (b)  (c)  Figure 7: Experimental and numerical curves for the 10J impact on the coupon made of T800S/M21 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (a) Impact  force-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (b) Impact force-displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (c) Energy-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  As we can see either in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 7 and Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 a good agreement is obtained between experiments and  numerical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  On the one hand, the proposed contact algorithm combined with the proposed  damage models is able to capture the maximum impact force very well and the maximum displacement  pretty well with errors below 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' On the contrary, the different dissipated energies obtained by  the experiments show a high dispersity between them, resulting in difficulty in conducting a fair comparison  between the predicted value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1 J and the experimental mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Experiment  Prediction  Difference (%)  Mean  Std.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Maximum impact force, f c  max (kN)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0  Maximum displacement, dmax (mm)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  Dissipated energy, Edis (J)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  >10  Table 2: Comparison of numerical results with experimental data for the impact case scenario of the plate made of T800S/M21  material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Coupon made of AS4/8552 material  This second case consists of a coupon made with the AS4/8552 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The plate has a stacking  sequence of [454/04/ − 454/904]S with a nominal ply thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='181 mm resulting a plate thickness of  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This case study has higher energy (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 J) than the previous one, and it also includes clusters of four  and eight plies which are potential for extensive matrix cracks and delaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The energy of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 J also  falls into BVID analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The in-plane element sizes used in [17] and [51] are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 mm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 mm respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In the present work, two element sizes are studied using the mesh refinement algorithm described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4,  see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The base mesh for the plate has a total of 335 622 hexahedron elements (≈ 1 million of Degrees  Of Freedom (DOF)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The total time for the simulation is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this case, the striker has a mass  17  of 5 kg, and its radius is 8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The initial velocity of the striker considering the gap previously mentioned  is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='78 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Refinement  Element  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' element  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' elem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' nodes  Initial stable  level, ndivi  size (mm)  through ply cluster  plate  plate  time increment (s)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7250  1  335 622  364 320  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='378 × 10−8  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3625  2  2 109 624  2 219 983  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='515 × 10−8  Table 3: Element sizes used on the coupon made of AS4/8552 material system and initial stable time increment for each case  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The numerical predictions of the impact force-displacement and energy - time curves are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  8 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 9 respectively, using different element sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The most important physics variables for a proper  validation are summarized in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This table compares the experimental results from [51] with the  numerical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Case  f c  del (kN)  f c  max (kN)  dmax (mm)  Edis (J)  Aproj  del  (mm2)  Experiment [17]  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='74  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='72  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03  3898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  Numerical (le = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7250 mm)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='20  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='70  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='60  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='70  4723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  Numerical (le = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3625 mm)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='30  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='70  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='90  5249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='20  Table 4: Comparison of the numerical results obtained with the proposed framework with experimental data from [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' fc  del is  the delamination threshold force, fc  max is the maximum contact force, dmax is the maximum indentation, Edis is the dissipated  energy and Aproj  del  is the projected delamination area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The initial elastic deflection of the plate is very well captured for all the meshes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 8), meaning that  the stiffness of the plate is accurately predicted by the PDN contact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' After that, delamination  onset occurs at the top of the elastic part, around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5 kN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This point is also very well captured by the  interlaminar damage model using cohesive elements between each of the ply clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Then, a combination  of interlaminar and intralaminar damage occurs until the striker reaches both the maximum load and  displacement, resulting with a pretty good prediction as also shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Since damage appears, the  continuum damage models and the characterization of the material properties play a fundamental role in the  simulation of this benchmark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Despite the delamination threshold, maximum force and displacement  are very well captured;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' the dissipated energy and the projected delamination area are overpredicted, see  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' According to Soto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [51], the projected delamination and the corresponding energy dissipated  could be considerably improved when using solid elements with one integration point for the bulk material  and cohesive contact surfaces instead of cohesive elements to be able to better predict the delamination  shapes at each interface of the layup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  18  Figure 8: Numerical prediction of the force-displacement curved using two element sizes and correlation with the experiment  from Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Figure 9: Numerical prediction of the impact energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' time using two element sizes and correlation with the experiment from  Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 10 depicts and aims to quantify the most important failure mechanisms that appear on the plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Fiber damage is represented by damage variable D1, which includes both fiber breakage and fiber kinking,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As we can see, this source of damage is not the most predominant and mostly appears at the  bottom of the striker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Matrix cracking is represented with the damage variable D2, which includes matrix  tension and compression (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Finally, the last source of damage is delamination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 10b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Its  prediction is compared with the shape obtained from the experiment, which is represented in dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  As we discussed previously, this source of damage is overpredicted for all the element sizes studied, see Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  4 and further research would be required in that direction as the values of the material properties, and  19  the damage models play a fundamental role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Furthermore, the extensive matrix cracks and delamination  predicted for this impact case scenario corroborate the experimental observations by Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [18] on  the effect of ply clustering to originate extensive matrix cracks and large delaminations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3 mm2  (a)  (b)  (c)  Experiment  Numerical  5248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2 mm2  10 mm  Dcoh  D1  D2  Figure 10: Numerical prediction of the damage occurred in the coupon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (a) Fiber damage, D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (b) Matrix cracking, D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (c)  Projected delamination, Dcoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The numerical result correspond to the most refined mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Parallel performance  The speedup and the parallel efficiency of the proposed contact algorithm for solving low-velocity impact  events are evaluated in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' All the executions are conducted in MareNostrum4 supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A  strong scalability analysis has been conducted using a larger mesh than the ones studied in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The  model corresponds to the AS4/8552 impact case scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The new mesh has a total of 74M elements with  228M of DOF, which results from a base mesh of 1 472 328 elements using two levels of the mesh refinement  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Strong scalability consists of fixing the mesh and solving the problem with a different number  of processors, Central Processing Unit (CPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The strong speedup is calculated as  t0  tN while the parallel  efficiency is calculated as  t0N0  tNN , where N is the number of processors and t0 is the reference simulation  time for N0 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The number of processors used for this analysis ranges from 192 to 2400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Due to  the resolution of the problem following a multibody/multicode approach, the number of processors for the  striker is fixed to 16 (sufficiently for its mesh) while the number of processors for the plate is changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' It  is worth mentioning that the strong computational effort falls in the resolution (deformation) of the plate  and the localization and exchange of information phases, as explained in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Due to the small time step  in this simulation, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='261 × 10−9 s, the simulations for the scalability curve are limited to the first 7460 time  20  DAM01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00  Y  Y  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  DAM02  DCOHE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0e+00steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The end of the execution (last time step) corresponds to an impact force of approximately 1 kN, which  falls into the linear elastic regime of the force-displacement curve shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The strong speedup and  parallel efficiency are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The ideal scalability and efficiency are represented with a dashed  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Average No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' of elements per core  Strong speedup  Parallel Efficiency  Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' of processors  Total No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' of processors  Figure 11: Strong scalability of the low velocity impact test with a plate mesh of 74M hexahedron elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The model  corresponds to the benchmark case using AS4/8552 material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The results obtained in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 11 show that the scalability of the problem in explicit analysis is really  good up to 2400 processors using a mesh of 74M elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The parallel efficiency is maintained above 90%,  which demonstrates the good scalability of the proposed framework to deal with large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This  linear behavior is also shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 5 where we summarize the total CPU time for each execution using a  different number of processors while maintaing fixed the size of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  21  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' of CPUs  192  384  768  1536  1824  2064  2400  17:20  08:33  4:15  02:11  01:51  01:39  01:27  Table 5: Total CPU time expressed in hh:mm for different executions of the low-velocity impact simulation considering a  fixed mesh of 74M of elements (228M of DOF) with a total of 7460 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This CPU time includes the preprocess part,  where two mesh refinement levels are performed and the solution of the contact problem within the elastic regime of the  force-displacement curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  It is also worth mentioning that the application of the proposed contact algorithm in explicit dynam-  ics improves both the speedup and the parallel efficiency in comparison to an implicit resolution for the  deformable body (plate), as already studied by the authors in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This improvement in computational per-  formance is mainly attributed to the time integration scheme for the deformable body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In explicit dynamics,  it is not required to invert the global matrix of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this case, the unknown is the acceleration, and  the system is solved directly using the lumped mass matrix and the global force vector on the right-hand  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reader is referred to [6] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Conclusions  In this paper, we apply the parallel PDN contact algorithm to simulate low-velocity impact events on  fiber-reinforced polymer composites using a High-Performance Computing environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Existing damage  models from the literature have been implemented in our multiphysics finite element code Alya to simulate  the material damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Moreover, we introduce a new capability in the in-house mesh refinement algorithm to  deal with cohesive elements and other element types, such as continuum shell elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' This is really attrac-  tive as we can refine the finite element mesh at the beginning of the simulation with a meager computational  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  We validate the whole framework with several benchmark tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The last example corresponds to a  well-known low-velocity impact test following the ASTM standard for damage resistance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In this  case, we study two impact case scenarios with two different material systems: the T800S/M21 and the  AS4/8552, obtaining excellent predictions for impact behavior and pretty good damage occurrence compared  to experimental data from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Additionally, the mesh refinement algorithm’s capabilities have  been demonstrated for the plate made of AS4/8552 material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Finally, we evaluate the parallel performance of the impact simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Despite not using ”very” large  meshes for the physics validation cases, we have generated a new larger mesh using the mesh refinement  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reason behind this is the stable time increment, which becomes smaller as the element size  decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The new mesh has 74M hexahedron elements (228M of DOF) using full integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' An excellent  computational efficiency (above 90%) has been obtained up to 2400 CPUs, demonstrating its applicability  to solve large mesh models ranging from micro-scale to macro-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  22  A further conclusion of this work is that we demonstrate the potential application of the parallel PDN  contact algorithm for low-velocity impact events and its parallel efficiency for large models compared to  traditional Penalty or Lagrange contact-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As we commented previously, we use full integration  elements for all the examples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' the use of reduced integration elements, which are more appropriate for  explicit schemes and overcome the well-known locking pathologies from solid brick elements, can considerably  increase the speedup of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Moreover, the localization of contact nodes and the communication  between subdomains created by the domain decomposition method is a crucial issue for further research as  it is the main bottleneck regarding the computational efficiency of contact algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Acknowledgements  This work has received funding from the Clean Sky 2 Joint Undertaking (JU) under grant agreements  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 807083 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 945521 (SHERLOC project).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The JU receives support from the European Union’s  Horizon 2020 research and innovation program and the Clean Sky 2 JU members other than the Union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The authors gratefully acknowledge Hellenic Aerospace Industry for manufacturing of the coupons made  of T800S/M21 material and Kirsa Mu˜noz and Miguel ´Angel Jim´enez from Element Materials Technology  Seville for conducting the experimental impact tests and providing all the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Quintanas-  Corominas acknowledges financial support from the European Union-NextGenerationEU and the Ministry  of Universities and Recovery, Transformation and Resilience Plan of the Spanish Government through a call  of the University of Girona (grant REQ2021-A-30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Guillamet thankfully acknowledges the computer  resources at MareNostrum and the technical support provided by Barcelona Supercomputing Center (FI-  2019-2-0010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Last but not least, the authors would also like to thank the late Claudio Lopes for all the  interesting discussions and contributions to the simulation of impact events and damage on composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Algorithms  Here we summarize the main algorithms of the whole modeling framework to solve low-velocity impact  events for damage resistance of fiber-reinforced polymer composites by making use of High-Performance  Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  23  Algorithm 1 Main code for the partial Dirichlet-Neumann (PDN) contact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  This PDN contact algorithm is treated as a coupling problem between two or more body instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In the present algorithm,  we describe the contact algorithm between two code instances: a rigid body represented by the domain Ωa and the deformable  body represented by the domain Ωb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The coupling is performed through the exchange of boundary conditions at the contact  interface following a Gauss-Seidel strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' At each time step, contact detection is done for both instances, and synchronization  and localization is executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' When contact is detected (at least one boundary node belonging to the deformable body is penetrated  inside the rigid body), the rigid one computes and sends to the deformable body all the information required for the enforcement  of the kinematic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The reader is referred to the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' from Rivero [46], or [19] for more details on the  implementation aspects of the proposed contact algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Require: Ωa, Ωb  1: loop time  2:  Compute time step, tn+1  3:  loop reset  4:  if Rigid body, Ωa then  5:  Contact detection (localization)  ▷ Contact detection & localization, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1 in [19]  6:  Exchange data: receive f cont from Ωb  ▷ Exchange & communication data, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 in [19]  7:  call calculateProjections()  ▷ Projections & local coordinate system, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3 in [19]  8:  call RK4Scheme()  ▷ Solve system  9:  Exchange data: send projection data to Ωb  ▷ Exchange & communication data, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 in [19]  10:  end if  11:  if Deformable body, Ωb then  12:  Contact detection (localization)  ▷ Contact detection & localization, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 1 in [19]  13:  Exchange data: receive data (projections) from Ωa  ▷ Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 in [19]  14:  call EssentialBoundaryCondition()  ▷ Contact nodes & Dirichlet condition, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 4 in [19]  15:  call ExplicitScheme()  ▷ Solve system  16:  call ReleaseNodes()  ▷ Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2  17:  Exchange: send f cont to Ωa  ▷ Exchange & communication data, Algo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2 in [19]  18:  end if  19:  if kfl reset = 0 then  20:  exit loop reset  21:  end if  22:  end loop  23: end loop  Algorithm 2 ReleaseNodes() algorithm for explicit time integration schemes  This algorithm is executed concurrently and for each subdomain at the end of the time step tn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The kfl reset is the key flag for  the repetition of the current time step tn+1 when exists adhesion contact nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The sign of the contact force is checked according  to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The key flag to release the adhesion nodes is called kfl nodes to release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Then the adhesion contact nodes are released  (as free non-contacting nodes), and the time step is repeated, activating the reset key flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' As all the subdomains require to know  if the time step has to be repeated or not, the MPI_MAX is in charge to collect the value of the reset for all the subdomains of the  mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1: kfl reset ← 0  2: Get contact force f c and mark adhesion nodes  3: if kfl nodes to release then  4:  Adhesion nodes are set to free nodes  5:  kfl reset ← 1  6: end if  7: call MPI_MAX(kfl reset)  24  Algorithm 3 Recursive mesh multiplication algorithm  The level of mesh refinement is set with the parameter ndivi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ne, nn and nb are the total number of elements, nodes and boundaries  of the new mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The same parameters with the superscript 0 indicate the initial dimension of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In order to define the  dimensions of the new mesh is necessary to know the total number of edges (nedgg) and faces nfacg of the initial mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Then,  once the dimensions are known, the DivideMesh() subroutine is in charge of doing the following actions: i) divide each edge and  face from the initial mesh, ii) define the new element connectivities, iv) define the new element boundary connectivities and v)  assign the material codes and the corresponding fields such as material coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The last step of the mesh division  algorithm is to reconstruct the interface domains through the ReconstructInterfaceDomains() subroutine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The reader is referred  to Houzeuax et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' [22] for more details on the implementation and parallel aspects of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Input n0  e, n0  n, n0  b , ndivi  Output ne, nn, nb  1: for idivi = 1, ndivi do  2:  nedgg = GetEdges()  3:  nfacg = GetFaces()  4:  ne, nn, nb = GetDimensions()  ▷ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 14  5:  call DivideMesh()  ▷ Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2 [22]  6:  call ReconstructInterfaceDomains()  ▷ Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2 [22]  7: end for  Algorithm 4 Workflow of the intralaminar damage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The strain and stress tensors, ε and σ, are defined in the material coordinate system using compact notation [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The superscripts  n and n + 1 define the past and current time steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The subscripts N indicate the four damage mechanisms associated  with the loading function φN and internal threshold variables rN (fibre breaking, fibre kinking, tensile matrix cracking, and  compressive matrix cracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  In turn, the subscript M indicates the five uniaxial damage states DM, represented in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The  required material properties are i) elastic properties (E11, E22, ν12, ν23, G12 ), ii) ply strengths (XT , XC, YT , YC, SL), iii) fracture  toughness (GXT , GXC, GY T , GY C, GSL) associated with the damage mechanism, and iv) yield strength and hardening (Sp, Kp);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  all these properties can be obtained through standardised tests or computational micromechanics simulations [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The required  parameters are: characteristic element length ℓc [33] and state variables at the past time step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' εn  p , and rt  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' At the initial time  step, the state variables are initialised as εn  p = 0 and rn  M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Input εn+1, εn  p , rn  M, ℓc, material properties  Output σn+1, εn+1  p  , rn+1  M  1: εn+1  p  (εn  p )  ▷ Plastic strains, yield function in [51]  2: εn+1  e  ← εn+1 − εn+1  p  ▷ Effective elastic strains  3: σn+1  e  ← H−1 · εn+1  e  ▷ Effective compliance matrix H in [30]g  4: φn+1  M  (σn+1  e  )  ▷ Loading functions (failure criteria), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 8, 13, 20, 21 in [32]  5: rn+1  N  (φn+1  N  , rn  N)  ▷ Damage thresholds, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 24, 26 in [32]  6: Dn+1  M  (rn+1  N  )  ▷ Damage state variables according [51] and Eq 6 in [32]  7: σn+1 ← H−1(Dn+1  M  ) · εn+1  e  ▷ Nominal compliance matrix H(Dn+1  M  ) in [30]  25  Algorithm 5 Workflow of the cohesive zone model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The displacement jumps and interface tractions, ∆ = {∆1, ∆2, ∆3}T and τ = {τ1, τ2, τ3}T , are defined at the mid-plane being  1 and 2 tangential and 3 normal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The superscripts t and t + 1 define the past and current time steps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' In  turn, the subscript M indicates the pure-mode I and II openings associated with the opening directions, I ↔ {3} and II ↔ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  The latter is also referred with the subscript sh in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The required input parameters are i) onset displacement jumps (∆Mo),  ii) critical displacement jumps (∆Mc), iii) penalty stiffness (KM), and iv) Benzeggagh-Kenane exponent for the mixed-mode ratio  (η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The onset and critical jumps can be obtained from the cohesive strengths (τM) and fracture toughness material properties by  ∆Mo = τM/KM and ∆Mc = 2GM/τM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' The damage threshold state variable at the past time rn  D, which is initialised  at the initial time step as rn  D = 0, is also required to evaluate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Input ∆n+1, rn  D, material properties  Output τ n+1, rn+1  D  1: Kn+1  B  (∆n+1)  ▷ Local mixed-mode penatly stiffness, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 13 in [56]  2: Bn+1(∆n+1)  ▷ Local mixed-mode ratio, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 17 in [56]  3: λn+1  o  (Bn+1, Kn+1  B  )  ▷ Local mixed-mode onset jump, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 26 in [56]  4: λn+1  c  (Bn+1, Kn+1  B  , λn+1  o  )  ▷ Local mixed-mode propagation jump, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 24 in [56]  5: λt+1(∆t+1)  ▷ Local mixed-mode equivalent jump, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 12 in [56]  6: Hn+1(λn+1, λn+1  o  , λn+1  c  )  ▷ Loading function (failure criteria), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 20 in [56]  7: rn+1  D  (Hn+1, rn  D)  ▷ Damage threshold, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 21 in [56]  8: Dn+1(rn+1  D  , λn+1  o  , λn+1  c  )  ▷ Damage state, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 20 in [56]  9: τ n+1  coh (Dn+1, ∆t+1)  ▷ Cohesive tractions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 7 in [56]  10: τ n+1  con (∆n+1)  ▷ Contact tractions, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 8 in [56]  11: τ n+1 ← τ n+1  coh + τ n+1  con  References  [1] Abrate, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Impact on Laminated Composites: Recent Advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Applied Mechanics Reviews 47, 517–544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='3111065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [2] Allix, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Corigliano, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Modeling and simulation of crack propagation in mixed-modes interlaminar fracture  specimens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal of Fracture 77, 111–140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/BF00037233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [3] Ansys Mechanical APDL, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ANSYS Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Help system, Theory Reference Chapter 4: Structures with Material Nonlinear-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [4] Bathe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Chaudhary, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A solution method for planar and axisymmetric contact problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal  for Numerical Methods in Engineering 21, 65–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/b978-1-85617-802-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='00007-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [5] Baˇzant, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Oh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Crack band theory for fracture of concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Mat´eriaux et Construction 16, 155–177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/BF02486267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [6] Belytschko, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Kam Liu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Moran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Elkhodary, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Nonlinear Finite Elements for Continua and Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' second edi ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Wiley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1088/1751-8113/44/8/085201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [7] Bisagni, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Vescovini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Single-Stringer Compression Specimen for the Assessment of Damage  Tolerance of Postbuckled Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Journal of Aircraft 48, 495–502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2514/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='c031106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [8] Borrell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Cajas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mira, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Taha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Koric, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Houzeaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Parallel mesh partitioning based  on space filling curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computers and Fluids 173, 264–272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [9] de Borst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Remmers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computational modelling of delamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Science and Technology 66,  713–722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compscitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' advances in statics and dynamics of delamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [10] Bouvet, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Castani´e, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Bizeul, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Barrau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Low velocity impact modelling in laminate composite panels  with discrete interface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal of Solids and Structures 46, 2809–2821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='ijsolstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  26  [11] Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Prediction of size effects in notched laminates using continuum damage  mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Science and Technology 67, 2715–2727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compscitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [12] Carreras, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Quintanas-Corominas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Renart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mesoscale modelling of delamination  using the cohesive zone model approach, in: Van Paepegem, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' (Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ), Multi-Scale Continuum Mechanics Modelling of  Fibre-Reinforced Polymer Composites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Woodhead Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Woodhead Publishing Series in Composites Science and  Engineering, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' 555–577.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/b978-0-12-818984-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='00018-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [13] Cheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Tan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Xiong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Modelling pre-fatigue, low-velocity impact and post-impact fatigue behaviours of  composite helicopter tail structures under multipoint coordinated loading spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Thin-Walled Structures 176, 109349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='tws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='109349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [14] Fournier, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Parallel location and exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ´Electricite de France (EDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [15] Furtado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Catalanotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Arteiro, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gray, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Wardle, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Simulation of failure in laminated  polymer composites: Building-block validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Composite Structures 226, 111168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  111168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [16] Gallego, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Anza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A mixed finite element model for the elastic contact problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal for  Numerical Methods in Engineering 28, 1249–1264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/nme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1620280603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [17] Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Simulation of drop-weight impact and  compression after impact tests on composite laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composite Structures 94, 3364–3378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [18] Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Lopes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Blanco, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Effects of ply clustering in laminated composite  plates under low-velocity impact loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Science and Technology 71, 805–817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compscitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [19] Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Rivero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Zavala-Ak´e, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Houzeaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Oller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A parallel algorithm for unilateral  contact problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computers and Structures 271, 106862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='106862.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [20] Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Costa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Linde, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A quick procedure to predict free-edge delamination in thin-ply  laminates under tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Engineering Fracture Mechanics 168, 28–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='engfracmech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' modeling  of fracture and damage in composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [21] Har, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Fulton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A parallel finite element procedure for contact-impact problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Engineering with Computers  19, 67–84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/s00366-003-0252-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [22] Houzeaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', de la Cruz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Owen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Parallel Uniform Mesh Multiplication Applied To A  Navier–stokes Solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computers and Fluids 80, 142–151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' selected contributions  of the 23rd International Conference on Parallel Fluid Dynamics ParCFD2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [23] International, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' ASTM D7136/D7136M-20, Standard Test Method for Measuring the Damage Resistance of a  Fiber-Reinforced Polymer Matrix Composite to a Drop-Weight Impact Event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [24] Krause, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Wohlmuth, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A Dirichlet-Neumann type algorithm for contact problems with friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computing  and Visualization in Science doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/s00791-002-0096-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [25] Lapeer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gerikhanov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Sadulaev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Audinis, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Rowland, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Crozier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Morris, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A computer-based  simulation of childbirth using the partial Dirichlet–Neumann contact method with total Lagrangian explicit dynamics on  the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Biomechanics and Modeling in Mechanobiology 18, 681–700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/s10237-018-01109-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [26] Llobet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Essa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', de la Escalera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A continuum damage model for composite laminates: Part  III - Fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mechanics of Materials 153, 103659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='mechmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='103659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [27] Llobet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Bak, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Lindgaard, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Carreras, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Essa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', de la Escalera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A  continuum damage model for composite laminates: Part IV- Experimental and numerical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mechanics of Materials  154, 103686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='mechmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='103686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  27  [28] Lopes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', G¨urdal, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Low-velocity impact damage on dispersed  stacking sequence laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Part II: Numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Science and Technology 69, 937–947.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compscitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [29] Lopes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Falc´o, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Naya, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', LLorca, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Tijs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Multiscale virtual testing: the roadmap to  efficient design of composites for damage resistance and tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' CEAS Aeronautical Journal 7, 607–619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/  s13272-016-0210-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [30] Lopes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', S´adaba, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Llorca, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Physically-sound simulation of low-velocity impact  on fiber reinforced laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal of Impact Engineering 92, 3–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='ijimpeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  impact Loading on Lightweight Structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [31] Maheo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Grolleau, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Rio, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Damping efficiency of the Tchamwa-Wielgosz explicit dissipative scheme under  instantaneous loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Comptes Rendus - Mecanique 337, 722–732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='crme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [32] Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2007a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A continuum damage model for composite laminates: Part  I - Constitutive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mechanics of Materials 39, 897–908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='mechmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [33] Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2007b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A continuum damage model for composite laminates: Part  II - Computational implementation and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mechanics of Materials 39, 909–919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='mechmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [34] Malone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Johnsok, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A parallel finite element contact/impact algorithm for non-linear explicit transient  analysis: Part I - The search algorithm and contact mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal for Numerical Methods in Engineering  37, 559–590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/nme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1620370403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [35] Malone, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Johnson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A parallel finite element contact/impact algorithm for non-linear explicit transient  analysis: Part II-Parallel implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal for Numerical Methods in Engineering doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/nme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  1620370404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [36] Marlett, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Hexcel 8552 AS4 Unidirectional Prepreg at 190gsm and 35% RC Qualification Material Property Data  Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Technical Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' National institute for aviation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [37] van der Meer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mesolevel Modeling of Failure in Composite Laminates: Constitutive, Kinematic and Algorithmic  Aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Archives of Computational Methods in Engineering 19, 381–425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/s11831-012-9076-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [38] Nguyen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Elder, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Bayandor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Thomson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Scott, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A review of explicit finite element software  for composite impact analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Journal of Composite Materials 39, 375–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1177/0021998305046739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [39] Ortiz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Pandolfi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Finite-deformation irreversible cohesive elements for three-dimensional crack-propagation  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Numerical Methods in Engineering 44, 1267–1282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/(SICI)1097-0207(19990330)44:9<1267::  AID-NME486>3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='CO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [40] Plagianakos, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mu˜noz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Prentzias, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Quintanas-Corominas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Jimenez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Karachalios, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Assessment of CNT-doping and hot-wet storage aging effects on Mode I, II and I/II interlaminar fracture toughness of a UD  Graphite/Epoxy material system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Engineering Fracture Mechanics 224, 106761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='engfracmech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='106761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [41] Quintanas-Corominas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Casoni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A 3D  transversally isotropic constitutive model for advanced composites implemented in a high performance computing code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  European Journal of Mechanics - A/Solids 71, 278–291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='euromechsol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [42] Quintanas-Corominas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Reinoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Casoni, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Paggi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A phase field approach  enhanced with a cohesive zone model for modeling delamination induced by matrix cracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Computer Methods in  Applied Mechanics and Engineering 358, 112618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='112618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [43] Reinoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Bl´azquez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Estefani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Par´ıs, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Ca˜nas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  A composite runout specimen subjected to ten-  sion–compression loading conditions: Experimental and global–local finite element analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composite Structures 101,  274–289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  28  [44] Reinoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Bl´azquez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', T´avara, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Par´ıs, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Arellano, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Damage tolerance of composite runout panels under  tensile loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Part B: Engineering 96, 79–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compositesb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [45] Reinoso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Paggi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  A consistent interface element formulation for geometrical and material nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Comput Mech 54, 1569–1581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1007/s00466-014-1077-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [46] Rivero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A Parallel Algorithm for Deformable Contact Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Phd thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Universitat Polit`ecnica de Catalunya,  Escola T`ecnica Superior d’Enginyers de Camins, Canals i Ports de Barcelona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [47] Sarrado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Leone, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Finite-thickness cohesive elements for modeling thick adhesives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Engineer-  ing Fracture Mechanics 168, 105–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='engfracmech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' modeling of fracture and damage in  composite materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [48] Sasikumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Costa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Trias, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Llobet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', C´ozar, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Linde, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A virtual testing based search  for optimum compression after impact strength in thin laminates using ply-thickness hybridization and unsymmetrical  designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Science and Technology 196, 108188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compscitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='108188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [49] Schellekens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', De Borst, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' On the numerical integration of interface elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' International Journal for  Numerical Methods in Engineering 36, 43–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/nme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1620360104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [50] Sommer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Thomson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Falc´o, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Quino, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Cui, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Petrinic, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Damage modelling of carbon fibre  composite crush tubes: Numerical simulation and experimental validation of drop weight impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Part A:  Applied Science and Manufacturing 160, 107033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compositesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='107033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [51] Soto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mayugo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Pasquali, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A methodology to simulate  low velocity impact and compression after impact in large composite stiffened panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composite Structures 204, 223–238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='081.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [52] Soto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mart´ın de la Escalera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Sainz de Aja, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Alvarez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Low velocity impact  and compression after impact simulation of thin ply laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Part A: Applied Science and Manufacturing  109, 413–427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compositesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [53] Tan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Falzon, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Chiu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Price, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Predicting low velocity impact damage and Compression-After-  Impact (CAI) behaviour of composite laminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composites Part A: Applied Science and Manufacturing 71, 212–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compositesa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [54] Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Costa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A damage model for the simulation of delamination in advanced  composites under variable-mode loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Mechanics of Materials 38, 1072–1089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='mechmat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [55] Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', D´avila, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Camanho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Costa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' An engineering solution for mesh size effects in the simulation of  delamination using cohesive zone models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Engineering Fracture Mechanics 74, 1665–1682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='engfracmech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [56] Turon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Gonz´alez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Sarrado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Guillamet, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Maim´ı, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Accurate simulation of delamination under  mixed-mode loading using a cohesive model with a mode-dependent penalty stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Composite Structures 184, 506–511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='compstruct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [57] V´azquez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Houzeaux, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Koric, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Artigues, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Aguado-Sierra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Ar´ıs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Mira, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Calmet, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Cucchietti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=',  Owen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Taha, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Burness, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Cela, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Valero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  Alya: Multiphysics engineering simulation toward  exascale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Journal of Computational Science 14, 15–27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='jocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [58] Weyler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Oliver, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Sain, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Cante, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' On the contact domain method: A comparison of penalty and Lagrange  multiplier implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computer Methods in Applied Mechanics and Engineering 205-208, 68–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  cma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' special Issue on Advances in Computational Methods in Contact Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [59] Yastrebov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Computational Contact Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Phd thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' MINES ParisTech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1017/CBO9781107415324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  [60] Yastrebov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', Breitkopf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Numerical Methods in Contact Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='1002/9781118647974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  29  [61] Zavala-Ak´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' A high-performance computing coupling tool for partitioned multi-physics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Phd  thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content=' Universitat Polit`ecnica de Catalunya, Departament de F´ısica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
+page_content='  30' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E5T4oBgHgl3EQfVg_E/content/2301.05552v1.pdf'}
diff --git a/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/2301.02199v1.pdf.txt b/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/2301.02199v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..861c797948b34e185ae0227d86f96413599cc9c6
--- /dev/null
+++ b/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/2301.02199v1.pdf.txt
@@ -0,0 +1,684 @@
+arXiv:2301.02199v1  [math.GR]  5 Jan 2023
+On the Generalized Fitting Height and Nonsoluble
+Length of the Mutually Permutable Products of Finite
+Groups∗
+Viachaslau I. Murashka1,2 and Alexander F. Vasil’ev3,2
+Abstract
+The generalized Fitting height h∗(G) of a finite group G is the least number h such
+that F∗
+h(G) = G, where F∗
+(0)(G) = 1, and F∗
+(i+1)(G) is the inverse image of the generalized
+Fitting subgroup F∗(G/F∗
+(i)(G)). Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G
+be the shortest normal series in which for i odd the factor Gi+1/Gi is p-soluble (possibly
+trivial), and for i even the factor Gi+1/Gi is a (non-empty) direct product of nonabelian
+simple groups. Then h = λp(G) is called the non-p-soluble length of a group G. We proved
+that if a finite group G is a mutually permutable product of of subgroups A and B then
+max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1 and max{λp(A), λp(B)} = λp(G).
+Also we introduced and studied the non-Frattini length.
+Keywords: Finite group; generalized Fitting subgroup; mutually permutable product
+of groups; generalized Fitting height; non-p-soluble length; Plotkin radical.
+1
+Introduction and the Main Results
+All groups considered here are finite. E.I. Khukhro and P. Shumyatsky introduced and
+studied interesting invariants of a group: the generalized Fitting height and the nonsoluble
+length [11–13]. The first one is the extension of the well known Fitting height to the class of
+all groups and the second one implicitly appeared in [8,20].
+Definition 1.1 (Khukhro, Shumyatsky). (1) The generalized Fitting height h∗(G) of a finite
+group G is the least number h such that F∗
+h(G) = G, where F∗
+(0)(G) = 1, and F∗
+(i+1)(G) is the
+inverse image of the generalized Fitting subgroup F∗(G/F∗
+(i)(G)).
+(2) Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series
+in which for i odd the factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor
+Gi+1/Gi is a (non-empty) direct product of nonabelian simple groups. Then h = λp(G) is called
+the non-p-soluble length of a group G.
+(3) Recall that λ2(G) = λ(G) is the nonsoluble length of a group G.
+In [12] E.I. Khukhro and P. Shumyatsky showed that in the general case the generalized
+Fitting height of a factorized group is not bounded in terms of the generalized Fitting heights
+of factors. The same situation is also for the nonsoluble length.
+Recall [1, Definition 4.1.1] that a group G is called a mutually permutable product of its
+subgroups A and B if G = AB, A permutes with every subgroup of B and B permutes with
+1email: mvimath@yandex.ru
+2Francisk Skorina Gomel State University, Gomel, Belarus
+3email: formation56@mail.ru
+∗Supported by BFFR Φ23PHΦ-237
+1
+
+every subgroup of A. The products of mutually permutable subgroups is the very interesting
+topic of the theory of groups (for example, see [1, Chapter 4]).
+The main result of our paper is
+Theorem 1.1. Let a group G be the product of the mutually permutable subgroups A and B.
+Then
+(1) max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.
+(2) max{λp(A), λp(B)} = λp(G) for any prime p. In particular, max{λ(A), λ(B)} = λ(G).
+If a group G is soluble, then h∗(G) = h(G) is the Fitting height of a group G.
+Corollary 1.2 ([10]). If a soluble group G is the product of the mutually permutable subgroups
+A and B, then max{h(A), h(B)} ≤ h(G) ≤ max{h(A), h(B)} + 1.
+Example 1.1. Note that the symmetric group S3 of degree 3 is the mutually permutable product
+of the cyclic groups Z2 and Z3 of orders 2 and 3 respectively. Hence h∗(S3) = max{h∗(Z2), h∗(Z3)}+
+1 = max{h(Z2), h(Z3)} + 1.
+2
+The Functorial Method
+According to B.I. Plotkin [15] a functorial is a function γ which assigns to each group G its
+characteristic subgroup γ(G) satisfying f(γ(G)) = γ(f(G)) for any isomorphism f : G → G∗.
+We are interested in functorials with some properties:
+(F1) f(γ(G)) ⊆ γ(f(G)) for every epimorphism f : G → G∗.
+(F2) γ(N) ⊆ γ(G) for every N ⊴ G.
+(F3) γ(G) ∩ N ⊆ γ(N) for every N ⊴ G.
+Remark 2.1. (0) Functions F∗ and Rp that assign to every group respectively its the generalized
+Fitting subgroup and the p-soluble radical are examples of functorials. It is well known that they
+satisfy (F1), (F2), (F3).
+(1) Recall that a functorial γ is called a Plotkin radical if it satisfies (F1), idempotent (i.e.
+γ(γ(G)) = γ(G)) and N ⊆ γ(G) for every γ(N) = N ⊴ G [5, p. 28].
+(2) A functorial that satisfies (F3) is often called hereditary (nevertheless, the same word
+means different in the theory of classes of groups).
+(3) A functorial γ is a hereditary Plotkin radical if and only if it satisfies (F1), (F2), (F3).
+Let prove it. Assume that γ is a hereditary Plotkin radical. We need only to prove that it satisfies
+(F2). If N ⊴ G, then γ(N) char N ⊴ G. So γ(N) ⊴ G. Now γ(N) = γ(γ(N)) ⊆ γ(G). Thus
+a hereditary Plotkin radical satisfies (F1), (F2), (F3). Assume that γ satisfies (F1), (F2), (F3).
+We need only to prove that it is idempotent. By (F3) we have γ(G) = γ(G) ∩ G ⊆ γ(γ(G)) ⊆
+γ(G). Thus γ(γ(G)) = γ(G).
+(4) The functorial Φ which assigns to every group G its Frattini subgroup Φ(G) satisfies
+(F1) and (F2) but not (F3).
+(5) If γ satisfies (F2) and (F3), then γ(G) ∩ N = γ(N) for every group G and N ⊴ G.
+Lemma 2.1. If γ satisfies (F1) and (F2), then γ(G1 × G2) = γ(G1) × γ(G2) for any groups
+G1 and G2.
+Proof. From Gi ⊴ G1 × G2 it follows that γ(Gi) ⊆ γ(G1 × G2) by (F2) for i ∈ {1, 2}. Note
+that γ(G1 × G2)Gi/Gi ⊆ γ((G1 × G2)/Gi) = (γ(G¯i) × Gi)/Gi by (F1) for i ∈ {1, 2}. Now
+γ(G1 × G2) ⊆ (γ(G1 × G2)G2) ∩ (γ(G1 × G2)G1) ⊆
+(γ(G1) × G2) ∩ (G1 × γ(G2)) = γ(G1) × γ(G2).
+Thus γ(G1 × G2) = γ(G1) × γ(G2).
+2
+
+Recall [15] that for functorials γ1 and γ2 the upper product γ2 ⋆ γ1 is defined by
+(γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)).
+Proposition 2.2. Let γ1 and γ2 be functorials. If γ1 and γ2 satisfy (F1) and (F2), then γ2 ⋆γ1
+satisfies (F1) and (F2). Moreover if γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).
+Proof. (1) γ2 ⋆ γ1 satisfies (F1).
+Let f : G → f(G) be an epimorphism. From f(γ2(G)) ⊆ γ2(f(G)) it follows that the
+following diagram is commutative.
+G
+f
+�
+f4
+�P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+P
+f1
+�
+f(G)
+f3
+�
+G/γ2(G)
+f2� f(G)/γ2(f(G))
+Let X = γ1(G/γ2(G)) and Y = γ1(f(G)/γ2(f(G))). Note that (γ2 ⋆ γ1)(G) = f −1
+1 (X) and
+(γ2 ⋆ γ1)(f(G)) = f −1
+3 (Y ) by the definition of γ2 ⋆ γ1. Since γ1 satisfies (F1), we see that
+f2(X) ⊆ Y . Hence X ⊆ f −1
+2 (Y ). Now (γ2 ⋆ γ1)(G) ⊆ f −1
+1 (f −1
+2 (Y )) = f −1
+4 (Y ). So
+f((γ2 ⋆ γ1)(G)) ⊆ f(f −1
+4 (Y )) = f −1
+3 (Y ) = (γ2 ⋆ γ1)(f(G)).
+Thus γ2 ⋆ γ1 satisfies (F1).
+(2) γ2 ⋆ γ1 satisfies (F2).
+Let N ⊴ G. From γ2(N) char N ⊴ G it follows that γ2(N) ⊴ G. Since γ2 satisfies (F2),
+we see that γ2(N) ⊆ γ2(G). So the following diagram is commutative.
+G
+f1�
+f3
+�❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+G/γ2(N)
+f2
+�
+G/γ2(G)
+Let X = γ1(G/γ2(N)), Y = γ1(N/γ2(N)) and Z = γ1(G/γ2(G)). Note that (γ2 ⋆ γ1)(G) =
+f −1
+3 (Z) and (γ1 ⋆ γ2)(N) ⊆ f −1
+1 (Y ). Since γ1 satisfies (F1) and (F2), we see that f2(X) ⊆ Z
+and Y ⊆ X. Now
+(γ2 ⋆ γ1)(N) ⊆ f −1
+1 (Y ) ⊆ f −1
+1 (X) ⊆ f −1
+1 (f −1
+2 (Z)) = f −1
+3 (Z) = (γ2 ⋆ γ1)(G).
+Hence γ2 ⋆ γ1 satisfies (F2).
+(3) If γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).
+Assume that γ1 and γ2 satisfy (F2) and (F3). Let N ⊴ G.
+Since Nγ2(G)/γ2(G) ∩ (γ2 ⋆ γ1)(G)/γ2(G) ⊴ (γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)), we see by
+(5) of Remark 2.1 that
+γ1((Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G)) = (Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G).
+Note that
+(Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G) =
+(N ∩ (γ2 ⋆ γ1)(G))γ2(G)/γ2(G) ≃ (N ∩ (γ2 ⋆ γ1)(G))/(N ∩ γ2(G))
+= (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊴ N/γ2(N).
+It means that (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊆ γ1(N/γ2(N)). Thus N ∩ (γ2 ⋆ γ1)(G) ⊆ (γ2 ⋆ γ1)(N),
+i.e γ2 ⋆ γ1 satisfies (F3).
+3
+
+Here we introduce the height hγ(G) of a group G which corresponds to a given functorial γ.
+Definition 2.1. Let γ be a functorial. Then the γ-series of G is defined starting from γ(0)(G) =
+1, and then by induction γ(i+1)(G) = (γ(i) ⋆ γ)(G) is the inverse image of γ(G/γ(i)(G)). The
+least number h such that γ(h)(G) = G is defined to be γ-height hγ(G) of G. If there is no such
+number, then hγ(G) = ∞.
+The following Lemma directly follows from Proposition 2.2.
+Lemma 2.3. Let γ be a functorial. If γ satisfies (F1) and (F2), then γ(n) satisfies (F1) and
+(F2) for all natural n. Moreover if γ satisfies (F3), then γ(n) satisfies (F3) for all natural n.
+Lemma 2.4. Let γ be a functorial. If γ satisfies (F1) and (F2), then hγ(G/N) ≤ hγ(G) ≤
+hγ(N) + hγ(G/N) for every N ⊴ G. Moreover, if γ also satisfies (F3), then hγ(N) ≤ hγ(G).
+Proof. Note that γ(n) satisfies (F1) and (F2) for every n by Lemma 2.3.
+Since γ(n) satisfies (F1), G/N = γhγ(G)(G)/N ≤ γ(hγ(G))(G/N) ≤ G/N. So γ(hγ(G))(G/N) =
+G/N. Thus hγ(G/N) ≤ hγ(G).
+Since γ(n) satisfies (F2), we see that N = γ(hγ(N))(N) ⊆ γ(hγ(N))(G). Note that hγ(G/γ(hγ(N))(G)) ≤
+hγ(G/N). Thus hγ(G) ≤ hγ(N) + hγ(G/N).
+Assume that γ also satisfies (F3).
+Then γ(n) satisfies (F3) by Lemma 2.3.
+Now N =
+G ∩ N = γ(hγ(G))(G) ∩ N ⊆ γ(hγ(G))(N) ≤ N. So γ(hγ(G))(N) = N. Thus hγ(N) ≤ hγ(G).
+If γ = F∗, then hγ(G) = h∗(G) for every group G. The non-p-soluble length can also be
+defined with the help of functorials. Here by Rp(G) we denote the p-soluble radical of a group
+G.
+Lemma 2.5. Let Fp = Rp ⋆F∗ ⋆Rp and G be a non-p-soluble group. Then λp(G) is the smallest
+natural i with Fp(i)(G) = G.
+Proof. Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series in which for i odd the
+factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor Gi+1/Gi is a (non-empty)
+direct product of nonabelian simple groups.
+Note that G1 ≤ Rp(G) and G2/G1 is quasinilpotent. Hence G2Rp(G)/Rp(G) is quasinilpo-
+tent. It means that G2Rp(G)/Rp(G) ≤ F∗(G/Rp(G)). Hence G2 ≤ (Rp ⋆ F∗)(G). Since G3/G2
+is p-soluble, we see that G3(Rp ⋆ F∗)(G)/(Rp ⋆ F∗)(G) is p-soluble. Hence G3(Rp ⋆ F∗)(G)/(Rp ⋆
+F∗)(G) ≤ Rp(G/(Rp ⋆ F∗)(G)). It means that G3 ≤ Fp(G) = Fp(1)(G).
+Assume that we proved G2i+1 ≤ Fp(i)(G). Let prove that G2(i+1)+1 ≤ Fp(i+1)(G).
+From G2i+1 ≤ Fp(i)(G) it follows that G2i+1 ≤ (Fp(i) ⋆ Rp)(G).
+Note that G2i+2/G2i+1
+is quasinilpotent. It means that G2i+2(Fp(i) ⋆ Rp)(G)/(Fp(i) ⋆ Rp)(G) is quasinilpotent. Hence
+G2i+2 ≤ ((Fp(i)⋆Rp)⋆F∗)(G). Since G2(i+1)+1/G2i+2 is p-soluble, we see that G2(i+1)+1(Fp(i)⋆Rp⋆
+F∗)(G)/(Fp(i)⋆Rp⋆F∗)(G) is p-soluble. Hence G2(i+1)+1(Fp(i)⋆Rp⋆F∗)(G)/((Fp(i)⋆Rp⋆F∗)(G) ≤
+Rp(G/(Fp(i) ⋆ Rp ⋆ F∗)(G)). It means that G2(i+1)+1 ≤ (Fp(i) ⋆ Rp ⋆ F∗ ⋆ Rp)(G) = Fp(i+1)(G).
+Therefore λp(G) ≥ n where n is the smallest integer with Fp(n)(G) = n. Since Rp ⋆Rp = Rp,
+we see that Fp(n)(G) presents a normal series 1 ≤ F1 ≤ F2 ≤ · · · ≤ F2n+1 in which for i odd the
+factor Fi+1/Fi is p-soluble (possibly trivial), and for i even the factor Fi+1/Fi is a (non-empty)
+direct product of nonabelian simple groups. So λp(G) ≤ n. Thus λp(G) = n.
+Now we are able to estimate the γ-height of the direct product subgroups and of the join
+of subnormal subgroups:
+4
+
+Theorem 2.6. Let γ be a functorial with γ(H) > 1 for every group H that satisfies (F1)
+and (F2).
+(1) If G = ×n
+i=1Ai is the direct product of its normal subgroups Ai, then hγ(G) = max{hγ(Ai) |
+1 ≤ i ≤ n}.
+(2) Let G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai. Then hγ(G) ≤
+max{hγ(Ai) | 1 ≤ i ≤ n}. If γ satisfies (F3), then hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.
+Proof. Note that γ(n) satisfies (F1) and (F2) for every n by Proposition 2.2.
+(1) From Lemma 2.1 it follows that if G = ×n
+i=1Ai, then γ(n)(G) = ×n
+i=1γ(n)(Ai). It means
+that hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.
+(2) Assume that G = ⟨Ai | 1 ≤ i ≤ n⟩ is the join of its subnormal subgroups Ai, h1 =
+max{hγ(Ai) | 1 ≤ i ≤ n} and h2 = hγ(G). Since γ(n) satisfies (F2), we see that γ(n)(N) ⊆
+γ(n)(G) for every subnormal subgroup N of G and every n. Now
+G = ⟨Ai | 1 ≤ i ≤ n⟩ = ⟨γ(h1)(Ai) | 1 ≤ i ≤ n⟩ ⊆ γ(h1)(G) ⊆ G.
+Hence γ(h1)(G) = G. It means that h2 ≤ h1.
+Suppose that γ satisfies (F3). Now γ(n) satisfies (F3) for every n by Proposition 2.2. From
+(5) of Remark 2.1 it follows that γ(n)(G) ∩ N = γ(n)(N) for every subnormal subgroup N of G.
+Now Ai = Ai ∩ G = Ai ∩ γ(h2)(G) = γ(h2)(Ai). It means that hγ(Ai) ≤ h2 for every i. Hence
+h1 ≤ h2. Thus h1 = h2.
+Corollary 2.7. Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.
+Then h∗(G) = max{h∗(Ai) | 1 ≤ i ≤ n} and λp(G) = max{λp(Ai) | 1 ≤ i ≤ n}.
+3
+The Classes of Groups Method
+Recall that a formation is a class F of groups with the following properties: (a) every
+homomorphic image of an F-group is an F-group, and (b) if G/M and G/N are F-groups, then
+also G/(M ∩ N) ∈ F. Recall that the F-residual of a group G is the smallest normal subgroup
+GF of G with G/GF ∈ F.
+A formation is called Fitting if (a) from N ⊴ G ∈ F it follows that N ∈ F and (b) a group
+G ∈ F whenever it is a product of normal F-subgroups. Recall that the F-radical GF of a group
+G is the greatest normal F-subgroup.
+The classes N∗ of all quasinilpotent groups and Sp of all p-soluble groups are Fitting for-
+mations.
+From [3, IX, Remarks 1.11 and Theorem 1.12] and [3, IV, Theorem 1.8] follows
+Lemma 3.1. Let F and H be non-empty Fitting formations. Then
+FH = (G | GF ∈ H) = (G | G/GH ∈ F)
+is a Fitting formation.
+Corollary 3.2. The class Hp = (G | Fp(G) = G) is a Fitting formation.
+It is straightforward to check that for a Fitting formation F, the F-radical can be considered
+as a functorial γ which satisfies (F1), (F2) and (F3). For convenience in this case denote hγ by
+hF. Now h∗(G) = hF∗(G) = hN∗(G) and for a non-p-soluble group λp(G) = hFp(G) = hHp(G).
+Lemma 3.3. Let F be a Fitting formation. If H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H)−1.
+Proof. Let prove that if H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H) − 1. Let hF(H) = n
+and hF(HF) = k. Then HF(n−1)(H) < H and H/HF(n−1) ∈ F. It means that HF ≤ HF(n−1).
+Since HF(n−1) satisfies (F3), we see that (HF)F(n−1) = HF. Hence k ≤ n − 1.
+Note that HF = (HF)F(k) ≤ HF(k). It means that H/HF(k) ∈ F. Hence k ≥ n − 1. Thus
+k = n − 1.
+5
+
+If F, H, K ̸= ∅ are formations, then (FH)K = F(HK) by [3, IV, Theorem 1.8]. That is why
+the class Fn = F . . . F
+� �� �
+n
+is a well defined formation.
+Lemma 3.4. For a natural number n and a Fitting formation F holds Fn = (G | hF(G) ≤ n).
+Proof. From Lemma 3.3 it follows that if G ∈ (G | hF(G) ≤ n), then GFn = 1. It means that
+(G | hF(G) ≤ n) ⊆ Fn. Assume that there is a group G ∈ Fn with hF(G) > n. Note that
+G
+F ̸= G for every quotient group G ̸≃ 1 of G. Then hF(GFn) > 0 by Lemma 3.3. It means that
+GFn ̸= 1, a contradiction. Therefore Fn ⊆ (G | hF(G) ≤ n). Thus Fn = (G | h(G) ≤ n).
+In the next lemma we recall the key properties of mutually permutable products
+Lemma 3.5. Let a group G = AB be a mutually permutable product of subgroups A and B.
+Then
+(1) [1, Lemma 4.1.10] G/N = (AN/N)(BN/N) is a mutually permutable product of sub-
+groups AN/N and BN/N for every normal subgroup N of G.
+(2) [1, Lemma 4.3.3(4)] If N is a minimal normal subgroup of a group G, then {N ∩A, N ∩
+B} ⊆ {1, N}.
+(3) [1, Lemma 4.3.3(5)] If N is a minimal normal subgroup of G contained in A and B∩N =
+1, then N ≤ CG(A) or N ≤ CG(B). If furthermore N is not cyclic, then N ≤ CG(B).
+(4) [1, Theorem 4.3.11] AGBG ̸= 1.
+(5) [1, Corollary 4.1.26] A′ and B′ are subnormal in G.
+Recall that π(G) is the set of all prime divisors of |G|, π(F) = ∪
+G∈Fπ(G) and Nπ denote the
+class of all nilpotent π-groups.
+Lemma 3.6. Let F be a Fitting formation. Assume that hF(G) ≤ h + 1 for every mutually
+permutable product G of two F-subgroups. Then
+max{hF(A), hF(B)} − 1 ≤ hF(G) ≤ max{hF(A), hF(B)} + h
+for every mutually permutable product G of two subgroups A and B with hF(A), hF(B) < ∞.
+Proof. If A = 1 or B = 1, then there is nothing to prove. Assume that A, B ̸= 1. Let a group
+G = AB be the product of mutually permutable subgroups A and B. From hF(A), hF(B) < ∞
+it follows that π(G) ⊆ π(F). According to [3, IX, Lemma 1.8] Nπ(F) ⊆ F. Note that A′ and B′
+are subnormal in G by (5) of Lemma 3.5. Since HF ⊴ HNπ(F) ⊴ H′ holds for every π(F)-group
+H, subgroups AF and BF are subnormal in G. Let C = ⟨AF, BF⟩G = ⟨{(AF)x | x ∈ G}∪{(BF)x |
+x ∈ G}⟩. Then by (2) of Theorem 2.6 and by Lemma 3.3
+hF(C) = max
+�
+{(hF(AF)x) | x ∈ G} ∪ {(hF(BF)x) | x ∈ G}
+�
+= max{hF(AF), hF(BF)} = max{hF(A), hF(B)} − 1.
+Now G/C = (AC/C)(BC/C) is a mutually permutable product of F-subgroups AC/C and
+BC/C by (1) of Lemma 3.5. It means that hF(G/C) ≤ h + 1 by our assumption. With the
+help of Lemma 2.4 we see that
+hF(G) ≤ hF(C) + hF(G/C) ≤ max{hF(A), hF(B)} − 1 + 1 + h = max{hF(A), hF(B)} + h.
+From the other hand, hF(G) ≥ hF(C) = max{hF(A), hF(B)} − 1 by (2) of Theorem 2.6.
+Lemma 3.7. Let F be a Fitting formation. Assume that a group G is the least order group
+with
+(1) G is a mutually permutable product of two subgroups A and B with hF(A) ≥ hF(B);
+(2) hF(G) = hF(A) − 1.
+Then G has the unique minimal normal subgroup N, N ≤ A and hF(A/N) = hF(A) − 1.
+6
+
+Proof. Let N be a minimal normal subgroup of G. Then N ∩ A ∈ {N, 1} by (2) of Lemma 3.5.
+Assume that N ∩ A = 1. Now G/N = (AN/N)(BN/N) is a mutually permutable product
+of groups AN/N and BN/N by (1) of Lemma 3.5. By our assumption and hF(G) ≥ hF(G/N) ≥
+hF(AN/N) = hF(A), a contradiction. Hence N ∩ A = N for every minimal normal subgroup N
+of G.
+Now hF(G) + 1 = hF(A) > hF(G) ≥ hF(G/N) ≥ hF(A/N) ≥ hF(A) − 1. It means that
+hF(G) = hF(A/N) = hF(A) − 1.
+If G has two minimal normal subgroups N1 and N2, then hF(A/N1) = hF(A/N2) = hF(A)−1.
+It means hF(A) < hF(A) − 1 by Lemma 3.4, a contradiction. Hence G has a unique minimal
+normal subgroup N.
+4
+Proof of Theorem 1.1(1)
+Our proof relies on the notion of the X-hypercenter. A chief factor H/K of G is called
+X-central in G provided
+(H/K) ⋊ (G/CG(H/K)) ∈ X
+(see [18, p. 127–128] or [7, 1, Definition 2.2]). A normal subgroup N of G is said to be X-
+hypercentral in G if N = 1 or N ̸= 1 and every chief factor of G below N is X-central. The
+symbol ZX(G) denotes the X-hypercenter of G, that is, the product of all normal X-hypercentral
+in G subgroups. According to [18, Lemma 14.1] or [7, 1, Theorem 2.6] ZX(G) is the largest
+normal X-hypercentral subgroup of G. If X = N is the class of all nilpotent groups, then
+ZN(G) = Z∞(G) is the hypercenter of G.
+Lemma 4.1. Let n be a natural number.
+Then (N∗)n = (G | h∗(G) ≤ n) = (G | G =
+Z(N∗)n(G)).
+Proof. First part follows from Lemma 3.4. It is well known that the class of all quasinilpotent
+groups is a composition (or Baer-local, or solubly saturated) formation (see [2, Example 2.2.17]).
+According to [18, Theorem 7.9] (N∗)n is a composition formation. Now (N∗)n = (G | G =
+Z(N∗)n(G)) by [7, 1, Theorem 2.6].
+For a normal section H/K of G the subgroup C∗
+G(H/K) = HCG(H/K) is called an inneriser
+(see [2, Definition 1.2.2]). It is the set of all elements of G that induce inner automorphisms
+on H/K.
+From the definition of the generalized Fitting subgroup it follows that it is the
+intersection of innerisers of all chief factors.
+Lemma 4.2. Let N be a normal subgroup of a group G. If N is a direct product of isomorphic
+simple groups and h∗(G/C∗
+G(N)) ≤ k − 1, then F∗
+(k)(G/N) = F∗
+(k)(G)/N.
+Proof. Assume that h∗(G/C∗
+G(N)) ≤ k − 1.
+Let F/N = F∗
+(k)(G/N).
+Then F∗
+(k)(G) ⊆ F.
+Now F/C∗
+F(N) ≃ FC∗
+G(N)/C∗
+G(N) ⊴ G/C∗
+G(N). Therefore h∗(F/C∗
+F(N)) ≤ k − 1. It means
+that h∗(F/C∗
+F(H/K)) ≤ k − 1 for every chief factor H/K of F below N. Hence (H/K) ⋊
+(F/CF(H/K)) ∈ (N∗)k for every chief factor H/K of F below N. It means that N ≤ Z(N∗)k(F).
+Thus F ∈ (N∗)k by Lemma 4.1. So F ⊆ F∗
+(k)(G). Thus F∗
+(k)(G) = F.
+Lemma 4.3. If a group G = AB is a product of mutually permutable quasinilpotent subgroups
+A and B, then h∗(G) ≤ 2.
+Proof. To prove this lemma we need only to prove that if a group G = AB is a product
+of mutually permutable quasinilpotent subgroups A and B, then G ∈ (N∗)2 by Lemma 4.1.
+Assume the contrary. Let G be a minimal order counterexample.
+(1) G has a unique minimal normal subgroup N and G/N ∈ (N∗)2.
+7
+
+Note that G/N is a mutually permutable product of quasinilpotent subgroups (AN/N) and
+(BN/N) by (1) of Lemma 3.5. Hence G/N ∈ (N∗)2 by our assumption. Since (N∗)2 is a
+formation, we see that G has a unique minimal normal subgroup. According to (4) of Lemma
+3.5 AGBG ̸= 1. WLOG we may assume that G has a minimal normal subgroup N ≤ A.
+(2) N ≤ A ∩ B.
+Suppose that N ∩ B = 1. Then A ≤ CG(N) or B ≤ CG(N) by (3) of Lemma 3.5. If A ≤
+CG(N), then N ⋊ G/CG(N) ≃ N ⋊ B/CB(N) ∈ (N∗)2. If B ≤ CG(N), then N ⋊ G/CG(N) ≃
+N ⋊ A/CA(N) ∈ (N∗) ⊆ (N∗)2 by [2, Corollary 2.2.5]. In both cases N ≤ Z(N∗)2(G). It means
+that G ∈ (N∗)2, a contradiction. Now N ∩ B ̸= 1. Hence N ≤ A ∩ B by (2) of Lemma 3.5.
+(3) N is non-abelian.
+Assume that N is abelian. Since A is quasinilpotent, we see that A/CA(N) is a p-group.
+By analogy B/CB(N) is a p-group. Note that A/CA(N) ≃ ACG(N)/CG(N) and B/CB(N) ≃
+BCG(N)/CG(N). From G = AB it follows that G/CG(N) is a p-group. Since N is a chief
+factor of G, we see that G/CG(N) ≃ 1. So N ≤ Z∞(G) ≤ Z(N∗)2(G). Thus G ∈ (N∗)2, a
+contradiction. It means that N is non-abelian.
+(4) The final contradiction.
+Now N is a direct product of minimal normal subgroups of A. Since A is quasinilpotent, we
+see that every element of A induces an inner automorphism on every minimal normal subgroup
+of A. Hence every element of A induces an inner automorphism on N.
+By analogy every
+element of B induces an inner automorphism on N.
+From G = AB it follows that every
+element of G induces an inner automorphism on N. So NCG(N) = G or G/CG(N) ≃ N. Now
+N ⋊ (G/CG(N)) ∈ (N∗)2. It means that N ≤ Z(N∗)2(G). Thus G ∈ (N∗)2 and h∗(G) ≤ 2, the
+final contradiction.
+Proof of Theorem 1.1(1). Let a group G be a mutually permutable product of subgroups A
+and B. From Theorem 2.6 and Lemma 4.3 it follows that
+max{h∗(A), h∗(B)} − 1 ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.
+Assume that max{h∗(A), h∗(B)} − 1 = h∗(G). WLOG let h∗(A) = h∗(G) − 1. We may
+assume that a group G is the least order group with such properties. Then G has the unique
+minimal normal subgroup N, N ≤ A and h∗(A/N) = h∗(A) − 1 by Lemma 3.7.
+Assume that h∗(A/C∗
+A(N)) < h∗(A) − 1. Then
+F∗
+(h∗(A)−1)(A/N) = F∗
+(h∗(A)−1)(A)/N < A/N
+by Lemma 4.2. It means that h∗(A) = h∗(A/N), a contradiction. Hence h∗(A/C∗
+A(N)) =
+h∗(A) − 1.
+Since G/C∗
+G(N) = (AC∗
+G(N)/C∗
+G(N))(BC∗
+G(N)/C∗
+G(N)) is a mutually permutable products
+of subgroups AC∗
+G(N)/C∗
+G(N) and BC∗
+G(N)/C∗
+G(N) by (1) of Lemma 3.5 and A/C∗
+A(N) ≃
+AC∗
+G(N)/C∗
+A(N), we see that h∗(G/C∗
+G(N)) ≥ h∗(A/C∗
+A(N)) = h∗(A) − 1 by our assumptions.
+Note that F∗(G) ≤ C∗
+G(N). Now h∗(G)−1 = h∗(G/F∗(G)) ≥ h∗(G/C∗
+G(N)) ≥ h∗(A/C∗
+A(N)) =
+h∗(A) − 1. It means that h∗(G) ≥ h∗(A), the final contradiction.
+5
+Proof of Theorem 1.1(2)
+Lemma 5.1. Let p be a prime and H = Hp. If a group G = AB is a product of mutually
+permutable H-subgroups A and B, then G ∈ H.
+Proof. Assume the contrary. Let G be a minimal order counterexample.
+(1) G has a unique minimal normal subgroup N, G/N ∈ H and N is not p-soluble.
+Note that G/N is a mutually permutable product of H-subgroups (AN/N) and (BN/N) by
+(1) of Lemma 3.5. Hence G/N ∈ H by our assumption. Since H is a formation, we see that G
+8
+
+has a unique minimal normal subgroup. According to (4) of Lemma 3.5 AGBG ̸= 1. WLOG
+we may assume that G has a minimal normal subgroup N ≤ A.
+If N is p-soluble, then Fp(G)/N = Fp(G/N) = G, i.e. So Fp(G) = G. Thus G ∈ H, a
+contradiction.
+(2) N ≤ A ∩ B.
+Suppose that N ∩ B = 1. Note that N is not cyclic by (1). Then B ≤ CG(N) by (3) of
+Lemma 3.5. Hence N ⋊ G/CG(N) ≃ N ⋊ A/CA(N) ∈ H by [2, Corollary 2.2.5]. It means that
+N ≤ ZH(G). Therefore G ∈ H, a contradiction. Now N ∩ B ̸= 1. Hence N ≤ A ∩ B by (2) of
+Lemma 3.5.
+(4) The final contradiction.
+Since N is the unique minimal normal subgroup of G and non-abelian, we see that CG(N) =
+1. So CA(N) = CB(N) = 1. Hence Rp(A) = Rp(B) = 1. In particular F(A) = F(B) = 1.
+Note that all minimal normal subgroups of A are in N. For B is the same situation. Thus
+N = F∗(A) = F∗(B). So G/N is a mutually permutable product of p-soluble groups. Since the
+class of all p-soluble groups is closed by extensions by p-soluble groups, G/N is p-soluble by (1)
+and (4) of Lemma 3.5. From N ≤ F∗(G) it follows that G ∈ H, the contradiction.
+Proof of Theorem 1.1(2). Let H = Hp and a group G be a mutually permutable product of
+subgroups A and B. First we a going to prove that max{hH(A), hH(B)} = hH(G).
+By Lemmas 3.6 and 4.3 we have
+max{hH(A), hH(B)} − 1 ≤ hH(G) ≤ max{hH(A), hH(B)}.
+Assume that max{hH(A), hH(B)} − 1 = hH(G) for some mutually permutable product G of
+A and B. Assume that G is a minimal order group with this property. WLOG let hH(A) =
+hH(G) − 1. Then G has the unique minimal normal subgroup N, N ≤ A and hH(A/N) =
+hH(A) − 1 by Lemma 3.7.
+If N is p-soluble, then Rp(A/N) = Rp(A)/N. It means that Fp(A/N) = Fp(A)/N. Thus
+hH(A/N) = hH(A), a contradiction.
+It means that Rp(G) = 1. Note that now N is a simple non-abelian group. Since N is a
+unique minimal normal subgroup of G, we see that N = F∗(G). Now hH(G/N) = hH(G) − 1.
+Therefore
+hH(G) − 1 = hH(G/N) ≥ hH(A/N) = hH(A) − 1.
+Thus hH(G) ≥ hH(A), the contradiction.
+We proved that max{hH(A), hH(B)} = hH(G).
+Let G be a mutually permutable product of groups A and B. If A, B are p-soluble, then
+G is p-soluble by (1) and (4) of Lemma 3.5.
+Hence λp(G) = λp(A) = λp(B) = 0.
+Now
+assume that at least one of subgroups A, B is not p-soluble. Then G is not p-soluble by (1)
+and (4) of Lemma 3.5. WLOG let hH(A) ≥ hH(B). Hence A is not p-soluble. We proved
+that hH(A) = hH(G). Note that hH(G) = λp(G), hH(A) = λp(A), hH(B) = λp(B) if B is not
+p-soluble by Lemma 2.5 and 0 = λp(B) < 1 = hH(B) ≤ hH(A) = λp(A) otherwise. Thus
+max{λp(A), λp(B)} = λp(G).
+6
+Non-Frattini length
+The Frattini subgroup Φ(G) play an important role in the theory of classes of groups. One
+of the useful properties of the Fitting subgroup of a soluble group is that it is strictly greater
+than the Frattini subgroup of the same group. Note that the generalized Fitting subgroup is
+non-trivial in every group but there are groups in which it coincides with the Frattini subgroup.
+That is why the following length seems interesting.
+9
+
+Definition 6.1. Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h = G be a shortest normal series in which for i
+even Gi+1/Gi ≤ Φ(G/Gi), and for i odd the factor Gi+1/Gi is a (non-empty) direct product of
+simple groups. Then h = ˜h(G) will be called the non-Frattini length of a group G.
+Note that if G is a soluble group, then ˜h(G) = h(G). Another reason that leads us to this
+length is the generalization of the Fitting subgroup ˜F(G) introduced by P. Schmid [16] and
+L.A. Shemetkov [17, Definition 7.5] and defined by
+Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = Soc(G/Φ(G)).
+P. F¨orster [4] showed that ˜F(G) can be defined by
+Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = F∗(G/Φ(G)).
+Let Φ and ˜F be functorials that assign Φ(G) and ˜F(G) to every group G. Then ˜F = Φ ⋆ F∗. It
+is well known that Φ satisfies (F1) and (F2). Hence ˜F satisfies (F1) and (F2) by Proposition
+2.2.
+Note that Φ(G/Φ(G)) ≃ 1. By analogy with the proof of Lemma 2.5 one can show that the
+non-Frattini length ˜h(G) of a group G and h˜F(G) coincide for every group G. The following
+theorem shows connections between the non-Frattini length and the generalized Fitting height.
+Theorem 6.1. For any group G holds ˜h(G) ≤ h∗(G) ≤ 2˜h(G). There exists a group H with
+˜h(H) = n and h∗(H) = 2n for any natural n.
+Proof. Since Φ(G) and Soc(G/Φ(G)) are quasinilpotent, we see that F∗(G) ≤ ˜F(G) ≤ F∗
+(2)(G).
+Now F∗
+(n)(G) ≤ ˜F(n)(G) ≤ F∗
+(2n)(G). Hence if ˜F(n)(G) = G, then F∗
+(n)(G) ≤ G and F∗
+(2n)(G) = G.
+It means ˜h(G) ≤ h∗(G) ≤ 2˜h(G).
+Let K be a group, K1 be isomorphic to the regular wreath product of A5 and K. Note
+that the base B of it is the unique minimal normal subgroup of K1 and non-abelian. According
+to [6], there is a Frattini F3K1-module A which is faithful for K1 and a Frattini extension
+A ֌ K2 ։ K1 such that A
+K1
+≃ Φ(K2) and K2/Φ(K2) ≃ K1.
+Let denote K2 by f(K). Now f(K)/˜F(f(K)) ≃ K. From the definition of h˜F = ˜h it follows
+that ˜h(f(K)) = ˜h(K) + 1.
+Note that Φ(f(K)) ⊆ F∗(f(K)).
+Assume that Φ(f(K)) ̸= F∗(f(K)).
+It means that
+F∗(f(K)) = ˜F(f(K)) is quasinilpotent. By [9, X, Theorem 13.8] it follows that Φ(f(K)) ⊆
+Z(F∗(f(K))). It means that 1 < B ≤ CK1(A). Thus A is not faithful, a contradiction.
+Thus Φ(f(K)) = F∗(f(K)) and f(K)/F∗(f(K)) ≃ K1.
+Since K1 has a unique mini-
+mal normal subgroup B and it is non-abelian, we see that F∗(K1) = B.
+It means that
+f(K)/F∗
+(2)(f(K)) ≃ K. From the definition of h∗ it follows that h∗(f(K)) = h∗(K) + 2.
+As usual, let f(1)(K) = f(K) and f(i+1)(K) = f(f(i)(K)).
+Then ˜h(f(n)(1)) = n and
+h∗(f(n)(1)) = 2n for any natural n.
+The following proposition directly follows from Theorem 2.6.
+Proposition 6.2. Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.
+Then ˜h(G) ≤ max{˜h(Ai) | 1 ≤ i ≤ n}.
+One of the main differences between the non-Frattini length and the generalized Fitting
+height is that the non-Frattini length of a normal subgroup can be greater than the non-Frattini
+length of a group.
+Example 6.1. Let E ≃ A5. There is an F5E-module V such that R = Rad(V ) is a faithful
+irreducible F5E-module and V/R is an irreducible trivial F5E-module (how to construct such
+module, for example, see [14]). Let G = V ⋋ E. Now Φ(G) = R by [3, B, Lemma 3.14]. Note
+10
+
+that G/Φ(G) = G/R ≃ Z5 × E. So ˜F(G) = G and ˜h(G) = 1. Note that G = V (RE) where V
+and RE are normal subgroups of G. Since V is abelian, we see that ˜h(V ) = 1. Note that R
+is a unique minimal normal subgroup of RE and Φ(RE) = 1. It means that ˜F(RE) = R and
+˜h(RE) = 2. Thus ˜h(G) < max{˜h(V ), ˜h(RE)} and ˜F does not satisfy (F3).
+Recall [1, Definition 4.1.1] that a group G is called a totally permutable product of its
+subgroups A and B if G = AB and every subgroup of A permutes with every subgroup of B.
+Theorem 6.3. Let a group G = AB be a totally permutable product of subgroups A and B.
+Then
+max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.
+Proof. If A = 1 or B = 1, then max{˜h(A), ˜h(B)} = ˜h(G). Assume that A, B ̸= 1.
+According to [1, Proposition 4.1.16] A ∩ B ≤ F(G). Hence A ∩ B ≤ F∗(G). Now G =
+G/F∗(G) is a totally permutable product of A = AF∗(G)/F∗(G) and B = BF∗(G)/F∗(G)
+by [1, Corollary 4.1.11]. Note that A ∩ B ≃ 1. According to [1, Lemma 4.2.2] [A, B] ≤ F(G).
+So [A, B] ≤ F∗(G). It means that
+G/F∗(G) = (AF∗(G)/F∗(G)) × (BF∗(G)/F∗(G)).
+Note that for the formation U of all supersoluble groups we have U ⊂ N2 ⊂ (N∗)2. Hence
+if H = H1H2 is a product of totally permutable (N∗)2-subgroups H1 and H2, then H ∈ (N∗)2
+by [1, Theorem 5.2.1]. Analyzing the proof of [1, Theorem 5.2.2] we see that this theorem is
+true not only for saturated formation, but for formations F = (G | G = ZF(G)). In particular,
+it is true for (N∗)2. Thus if H = H1H2 ∈ (N∗)2 is a product of totally permutable subgroups
+H1 and H2, then H1, H2 ∈ (N∗)2. Now (N∗)2 satisfies conditions of [1, Proposition 5.3.9].
+Therefore A ∩ F∗
+(2)(G) = F∗
+(2)(A) and B ∩ F∗
+(2)(G) = F∗
+(2)(B). Note that
+AF∗(G)/F∗(G) ≃ AF∗
+(2)(G)/F∗
+(2)(G) ≃ A/F∗
+(2)(A).
+By analogy BF∗(G)/F∗(G) ≃ B/F∗
+(2)(B). Hence
+G/F∗
+(2)(G) ≃ (A/F∗
+(2)(A)) × (B/F∗
+(2)(B)).
+By Theorem 2.6 and ˜h = h˜F we have ˜h(G/F∗
+(2)(G)) = max{˜h(A/F∗
+(2)(A)), ˜h(B/F∗
+(2)(B))}.
+From ˜F(H) ≤ F∗
+(2)(H) ≤ ˜F(2)(H) and Lemma 2.4 it follows that for any group H ̸= 1 holds
+˜h(H) − 1 = ˜h(H/˜F(H)) ≥ ˜h(H/F∗
+(2)(H)) ≥ ˜h(H/˜F(2)(H)) ≥ ˜h(H) − 2.
+Therefore
+{˜h(G) − ˜h(G/F∗
+(2)(G)), ˜h(A) − ˜h(A/F∗
+(2)(A)), ˜h(B) − ˜h(B/F∗
+(2)(B))} ⊆ {1, 2}.
+Thus max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.
+While proving Theorem 6.3 we were not able to answer the following question:
+Question 6.1. Let a group G = AB be a totally permutable product of subgroups A and B. Is
+max{˜h(A), ˜h(B)} ≤ ˜h(G)?
+The following question seems interesting
+Question 6.2. Do there exists a constant h with | max{˜h(A), ˜h(B)} − ˜h(G)| ≤ h for any
+mutually permutable product G = AB of subgroups A and B?
+D.A. Towers [19] defined and studied analogues of F∗(G) and ˜F(G) for Lie algebras. Using
+these subgroups and the radical (of a Lie algebra) one can introduce the generalized Fitting
+height, the non-soluble length and the non-Frattini length of a (finite dimension) Lie algebra.
+Question 6.3. Estimate the generalized Fitting height, the non-soluble length and the non-
+Frattini length of a (finite dimension) Lie algebra that is the sum of its two subalgebras (ideals,
+subideals, mutually or totally permutable subalgebras).
+11
+
+References
+[1] A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad. Products of Finite Groups. De
+Gruyter, 2010.
+[2] A. Ballester-Bollinches and L. M. Ezquerro. Classes of Finite Groups, volume 584 of Math.
+Appl. Springer Netherlands, 2006.
+[3] K. Doerk and T. O. Hawkes. Finite Soluble Groups, volume 4 of De Gruyter Exp. Math.
+De Gruyter, Berlin, New York, 1992.
+[4] P. F¨orster. Projektive Klassen endlicher Gruppen: IIa. Ges¨attigte Formationen: Ein all-
+gemeiner Satz von Gasch¨utz-Lubeseder-Baer-Typ. Pub, Mat. UAB, 29(2/3):39–76, 1985.
+[5] B. J. Gardner and R. Wiegandt. Radical theory of rings. Marcel Dekker New York, 2003.
+[6] R. L. Griess and P. Schmid. The Frattini module. Arch. Math., 30(1):256–266, 1978.
+[7] W. Guo. Structure Theory for Canonical Classes of Finite Groups. Springer-Verlag, Berlin,
+Heidelberg, 2015.
+[8] P. Hall and G. Higman. On the p-Length of p-Soluble Groups and Reduction Theorems
+for Burnside’s Problem. Proc. London Math. Soc., s3-6(1):1–42, 1956.
+[9] B. Huppert and N. Blackburn. Finite groups III, volume 243 of Grundlehren Math. Wiss.
+Springer-Verlag, Berlin, Heidelberg, 1982.
+[10] E. Jabara. The Fitting length of a product of mutually permutable finite groups. Acta
+Math. Hung., 159(1):206–210, 2019.
+[11] E. I. Khukhro and P. Shumyatsky. Nonsoluble and non-p-soluble length of finite groups.
+Isr. J. Math., 207(2):507–525, 2015.
+[12] E. I. Khukhro and P. Shumyatsky. On the length of finite factorized groups. Ann. Mat.
+Pura Appl., 194(6):1775–1780, 2015.
+[13] E. I. Khukhro and P. Shumyatsky. On the length of finite groups and of fixed points. Proc.
+Amer. Math. Soc., 143(9):3781–3790, 2015.
+[14] V.I. Murashka.
+On one conjecture about supersoluble groups.
+Publ. Math. Debrecen,
+100(3-4):399–404, 2022.
+[15] B. I. Plotkin. Radicals in groups, operations on group classes and radical classes. In Selected
+questions of algebra and logic, pages 205–244. Nauka, Novosibirsk, 1973. In Russian.
+[16] P. Schmid. ¨Uber die Automorphismengruppen endlicher Gruppen. Arch. Math., 23(1):236–
+242, 1972.
+[17] L. A. Shemetkov. Formations of finite groups. Nauka, Moscow, 1978. In Russian.
+[18] L. A. Shemetkov and A. N. Skiba. Formations of algebraic systems. Nauka, Moscow, 1989.
+In Russian.
+[19] D. A. Towers. The generalised nilradical of a Lie algebra. J. Algebra, 470:197–218, 2017.
+[20] J. S. Wilson. On the structure of compact torsion groups. Monatsh. Math., 96(1):57–66,
+1983.
+12
+
diff --git a/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/load_file.txt b/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0c3db18f96475d0baf361407bd2975492c69a6b3
--- /dev/null
+++ b/FNE0T4oBgHgl3EQfQwDu/content/tmp_files/load_file.txt
@@ -0,0 +1,811 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf,len=810
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='02199v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='GR]  5 Jan 2023  On the Generalized Fitting Height and Nonsoluble  Length of the Mutually Permutable Products of Finite  Groups∗  Viachaslau I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Murashka1,2 and Alexander F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Vasil’ev3,2  Abstract  The generalized Fitting height h∗(G) of a finite group G is the least number h such  that F∗  h(G) = G, where F∗  (0)(G) = 1, and F∗  (i+1)(G) is the inverse image of the generalized  Fitting subgroup F∗(G/F∗  (i)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G  be the shortest normal series in which for i odd the factor Gi+1/Gi is p-soluble (possibly  trivial), and for i even the factor Gi+1/Gi is a (non-empty) direct product of nonabelian  simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then h = λp(G) is called the non-p-soluble length of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' We proved  that if a finite group G is a mutually permutable product of of subgroups A and B then  max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1 and max{λp(A), λp(B)} = λp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Also we introduced and studied the non-Frattini length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Keywords: Finite group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' generalized Fitting subgroup;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' mutually permutable product  of groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' generalized Fitting height;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' non-p-soluble length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Plotkin radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  1  Introduction and the Main Results  All groups considered here are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Khukhro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shumyatsky introduced and  studied interesting invariants of a group: the generalized Fitting height and the nonsoluble  length [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The first one is the extension of the well known Fitting height to the class of  all groups and the second one implicitly appeared in [8,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1 (Khukhro, Shumyatsky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' (1) The generalized Fitting height h∗(G) of a finite  group G is the least number h such that F∗  h(G) = G, where F∗  (0)(G) = 1, and F∗  (i+1)(G) is the  inverse image of the generalized Fitting subgroup F∗(G/F∗  (i)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) Let p be a prime, 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series  in which for i odd the factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor  Gi+1/Gi is a (non-empty) direct product of nonabelian simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then h = λp(G) is called  the non-p-soluble length of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (3) Recall that λ2(G) = λ(G) is the nonsoluble length of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  In [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Khukhro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shumyatsky showed that in the general case the generalized  Fitting height of a factorized group is not bounded in terms of the generalized Fitting heights  of factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The same situation is also for the nonsoluble length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Recall [1, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1] that a group G is called a mutually permutable product of its  subgroups A and B if G = AB, A permutes with every subgroup of B and B permutes with  1email: mvimath@yandex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='ru  2Francisk Skorina Gomel State University, Gomel, Belarus  3email: formation56@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='ru  ∗Supported by BFFR Φ23PHΦ-237  1  every subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The products of mutually permutable subgroups is the very interesting  topic of the theory of groups (for example, see [1, Chapter 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  The main result of our paper is  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G be the product of the mutually permutable subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then  (1) max{h∗(A), h∗(B)} ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) max{λp(A), λp(B)} = λp(G) for any prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In particular, max{λ(A), λ(B)} = λ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  If a group G is soluble, then h∗(G) = h(G) is the Fitting height of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2 ([10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If a soluble group G is the product of the mutually permutable subgroups  A and B, then max{h(A), h(B)} ≤ h(G) ≤ max{h(A), h(B)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that the symmetric group S3 of degree 3 is the mutually permutable product  of the cyclic groups Z2 and Z3 of orders 2 and 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence h∗(S3) = max{h∗(Z2), h∗(Z3)}+  1 = max{h(Z2), h(Z3)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  2  The Functorial Method  According to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Plotkin [15] a functorial is a function γ which assigns to each group G its  characteristic subgroup γ(G) satisfying f(γ(G)) = γ(f(G)) for any isomorphism f : G → G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  We are interested in functorials with some properties:  (F1) f(γ(G)) ⊆ γ(f(G)) for every epimorphism f : G → G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (F2) γ(N) ⊆ γ(G) for every N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (F3) γ(G) ∩ N ⊆ γ(N) for every N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' (0) Functions F∗ and Rp that assign to every group respectively its the generalized  Fitting subgroup and the p-soluble radical are examples of functorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It is well known that they  satisfy (F1), (F2), (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (1) Recall that a functorial γ is called a Plotkin radical if it satisfies (F1), idempotent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  γ(γ(G)) = γ(G)) and N ⊆ γ(G) for every γ(N) = N ⊴ G [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) A functorial that satisfies (F3) is often called hereditary (nevertheless, the same word  means different in the theory of classes of groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (3) A functorial γ is a hereditary Plotkin radical if and only if it satisfies (F1), (F2), (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let prove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that γ is a hereditary Plotkin radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' We need only to prove that it satisfies  (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If N ⊴ G, then γ(N) char N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So γ(N) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now γ(N) = γ(γ(N)) ⊆ γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus  a hereditary Plotkin radical satisfies (F1), (F2), (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that γ satisfies (F1), (F2), (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  We need only to prove that it is idempotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' By (F3) we have γ(G) = γ(G) ∩ G ⊆ γ(γ(G)) ⊆  γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus γ(γ(G)) = γ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (4) The functorial Φ which assigns to every group G its Frattini subgroup Φ(G) satisfies  (F1) and (F2) but not (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (5) If γ satisfies (F2) and (F3), then γ(G) ∩ N = γ(N) for every group G and N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If γ satisfies (F1) and (F2), then γ(G1 × G2) = γ(G1) × γ(G2) for any groups  G1 and G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From Gi ⊴ G1 × G2 it follows that γ(Gi) ⊆ γ(G1 × G2) by (F2) for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note  that γ(G1 × G2)Gi/Gi ⊆ γ((G1 × G2)/Gi) = (γ(G¯i) × Gi)/Gi by (F1) for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now  γ(G1 × G2) ⊆ (γ(G1 × G2)G2) ∩ (γ(G1 × G2)G1) ⊆  (γ(G1) × G2) ∩ (G1 × γ(G2)) = γ(G1) × γ(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus γ(G1 × G2) = γ(G1) × γ(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  2  Recall [15] that for functorials γ1 and γ2 the upper product γ2 ⋆ γ1 is defined by  (γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let γ1 and γ2 be functorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If γ1 and γ2 satisfy (F1) and (F2), then γ2 ⋆γ1  satisfies (F1) and (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Moreover if γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' (1) γ2 ⋆ γ1 satisfies (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let f : G → f(G) be an epimorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From f(γ2(G)) ⊆ γ2(f(G)) it follows that the  following diagram is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  G  f  �  f4  �P  P  P  P  P  P  P  P  P  P  P  P  P  P  P  f1  �  f(G)  f3  �  G/γ2(G)  f2� f(G)/γ2(f(G))  Let X = γ1(G/γ2(G)) and Y = γ1(f(G)/γ2(f(G))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that (γ2 ⋆ γ1)(G) = f −1  1 (X) and  (γ2 ⋆ γ1)(f(G)) = f −1  3 (Y ) by the definition of γ2 ⋆ γ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since γ1 satisfies (F1), we see that  f2(X) ⊆ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence X ⊆ f −1  2 (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now (γ2 ⋆ γ1)(G) ⊆ f −1  1 (f −1  2 (Y )) = f −1  4 (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So  f((γ2 ⋆ γ1)(G)) ⊆ f(f −1  4 (Y )) = f −1  3 (Y ) = (γ2 ⋆ γ1)(f(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus γ2 ⋆ γ1 satisfies (F1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) γ2 ⋆ γ1 satisfies (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From γ2(N) char N ⊴ G it follows that γ2(N) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since γ2 satisfies (F2),  we see that γ2(N) ⊆ γ2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So the following diagram is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  G  f1�  f3  �❍  ❍  ❍  ❍  ❍  ❍  ❍  ❍  ❍  ❍  G/γ2(N)  f2  �  G/γ2(G)  Let X = γ1(G/γ2(N)), Y = γ1(N/γ2(N)) and Z = γ1(G/γ2(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that (γ2 ⋆ γ1)(G) =  f −1  3 (Z) and (γ1 ⋆ γ2)(N) ⊆ f −1  1 (Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since γ1 satisfies (F1) and (F2), we see that f2(X) ⊆ Z  and Y ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now  (γ2 ⋆ γ1)(N) ⊆ f −1  1 (Y ) ⊆ f −1  1 (X) ⊆ f −1  1 (f −1  2 (Z)) = f −1  3 (Z) = (γ2 ⋆ γ1)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Hence γ2 ⋆ γ1 satisfies (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (3) If γ1 and γ2 also satisfy (F3), then γ2 ⋆ γ1 satisfies (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that γ1 and γ2 satisfy (F2) and (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since Nγ2(G)/γ2(G) ∩ (γ2 ⋆ γ1)(G)/γ2(G) ⊴ (γ2 ⋆ γ1)(G)/γ2(G) = γ1(G/γ2(G)), we see by  (5) of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1 that  γ1((Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G)) = (Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that  (Nγ2(G) ∩ (γ2 ⋆ γ1)(G))/γ2(G) =  (N ∩ (γ2 ⋆ γ1)(G))γ2(G)/γ2(G) ≃ (N ∩ (γ2 ⋆ γ1)(G))/(N ∩ γ2(G))  = (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊴ N/γ2(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means that (N ∩ (γ2 ⋆ γ1)(G))/γ2(N) ⊆ γ1(N/γ2(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus N ∩ (γ2 ⋆ γ1)(G) ⊆ (γ2 ⋆ γ1)(N),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='e γ2 ⋆ γ1 satisfies (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  3  Here we introduce the height hγ(G) of a group G which corresponds to a given functorial γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let γ be a functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then the γ-series of G is defined starting from γ(0)(G) =  1, and then by induction γ(i+1)(G) = (γ(i) ⋆ γ)(G) is the inverse image of γ(G/γ(i)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The  least number h such that γ(h)(G) = G is defined to be γ-height hγ(G) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If there is no such  number, then hγ(G) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  The following Lemma directly follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let γ be a functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If γ satisfies (F1) and (F2), then γ(n) satisfies (F1) and  (F2) for all natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Moreover if γ satisfies (F3), then γ(n) satisfies (F3) for all natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let γ be a functorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If γ satisfies (F1) and (F2), then hγ(G/N) ≤ hγ(G) ≤  hγ(N) + hγ(G/N) for every N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Moreover, if γ also satisfies (F3), then hγ(N) ≤ hγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that γ(n) satisfies (F1) and (F2) for every n by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since γ(n) satisfies (F1), G/N = γhγ(G)(G)/N ≤ γ(hγ(G))(G/N) ≤ G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So γ(hγ(G))(G/N) =  G/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus hγ(G/N) ≤ hγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since γ(n) satisfies (F2), we see that N = γ(hγ(N))(N) ⊆ γ(hγ(N))(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that hγ(G/γ(hγ(N))(G)) ≤  hγ(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus hγ(G) ≤ hγ(N) + hγ(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that γ also satisfies (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then γ(n) satisfies (F3) by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now N =  G ∩ N = γ(hγ(G))(G) ∩ N ⊆ γ(hγ(G))(N) ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So γ(hγ(G))(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus hγ(N) ≤ hγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  If γ = F∗, then hγ(G) = h∗(G) for every group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The non-p-soluble length can also be  defined with the help of functorials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Here by Rp(G) we denote the p-soluble radical of a group  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let Fp = Rp ⋆F∗ ⋆Rp and G be a non-p-soluble group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then λp(G) is the smallest  natural i with Fp(i)(G) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h+1 = G be the shortest normal series in which for i odd the  factor Gi+1/Gi is p-soluble (possibly trivial), and for i even the factor Gi+1/Gi is a (non-empty)  direct product of nonabelian simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that G1 ≤ Rp(G) and G2/G1 is quasinilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G2Rp(G)/Rp(G) is quasinilpo-  tent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that G2Rp(G)/Rp(G) ≤ F∗(G/Rp(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G2 ≤ (Rp ⋆ F∗)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since G3/G2  is p-soluble, we see that G3(Rp ⋆ F∗)(G)/(Rp ⋆ F∗)(G) is p-soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G3(Rp ⋆ F∗)(G)/(Rp ⋆  F∗)(G) ≤ Rp(G/(Rp ⋆ F∗)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that G3 ≤ Fp(G) = Fp(1)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that we proved G2i+1 ≤ Fp(i)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let prove that G2(i+1)+1 ≤ Fp(i+1)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From G2i+1 ≤ Fp(i)(G) it follows that G2i+1 ≤ (Fp(i) ⋆ Rp)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that G2i+2/G2i+1  is quasinilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that G2i+2(Fp(i) ⋆ Rp)(G)/(Fp(i) ⋆ Rp)(G) is quasinilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence  G2i+2 ≤ ((Fp(i)⋆Rp)⋆F∗)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since G2(i+1)+1/G2i+2 is p-soluble, we see that G2(i+1)+1(Fp(i)⋆Rp⋆  F∗)(G)/(Fp(i)⋆Rp⋆F∗)(G) is p-soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G2(i+1)+1(Fp(i)⋆Rp⋆F∗)(G)/((Fp(i)⋆Rp⋆F∗)(G) ≤  Rp(G/(Fp(i) ⋆ Rp ⋆ F∗)(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that G2(i+1)+1 ≤ (Fp(i) ⋆ Rp ⋆ F∗ ⋆ Rp)(G) = Fp(i+1)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Therefore λp(G) ≥ n where n is the smallest integer with Fp(n)(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since Rp ⋆Rp = Rp,  we see that Fp(n)(G) presents a normal series 1 ≤ F1 ≤ F2 ≤ · · · ≤ F2n+1 in which for i odd the  factor Fi+1/Fi is p-soluble (possibly trivial), and for i even the factor Fi+1/Fi is a (non-empty)  direct product of nonabelian simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So λp(G) ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus λp(G) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now we are able to estimate the γ-height of the direct product subgroups and of the join  of subnormal subgroups:  4  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let γ be a functorial with γ(H) > 1 for every group H that satisfies (F1)  and (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (1) If G = ×n  i=1Ai is the direct product of its normal subgroups Ai, then hγ(G) = max{hγ(Ai) |  1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) Let G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then hγ(G) ≤  max{hγ(Ai) | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If γ satisfies (F3), then hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that γ(n) satisfies (F1) and (F2) for every n by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (1) From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1 it follows that if G = ×n  i=1Ai, then γ(n)(G) = ×n  i=1γ(n)(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means  that hγ(G) = max{hγ(Ai) | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) Assume that G = ⟨Ai | 1 ≤ i ≤ n⟩ is the join of its subnormal subgroups Ai, h1 =  max{hγ(Ai) | 1 ≤ i ≤ n} and h2 = hγ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since γ(n) satisfies (F2), we see that γ(n)(N) ⊆  γ(n)(G) for every subnormal subgroup N of G and every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now  G = ⟨Ai | 1 ≤ i ≤ n⟩ = ⟨γ(h1)(Ai) | 1 ≤ i ≤ n⟩ ⊆ γ(h1)(G) ⊆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Hence γ(h1)(G) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that h2 ≤ h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Suppose that γ satisfies (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now γ(n) satisfies (F3) for every n by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From  (5) of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1 it follows that γ(n)(G) ∩ N = γ(n)(N) for every subnormal subgroup N of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now Ai = Ai ∩ G = Ai ∩ γ(h2)(G) = γ(h2)(Ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that hγ(Ai) ≤ h2 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence  h1 ≤ h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus h1 = h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then h∗(G) = max{h∗(Ai) | 1 ≤ i ≤ n} and λp(G) = max{λp(Ai) | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  3  The Classes of Groups Method  Recall that a formation is a class F of groups with the following properties: (a) every  homomorphic image of an F-group is an F-group, and (b) if G/M and G/N are F-groups, then  also G/(M ∩ N) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Recall that the F-residual of a group G is the smallest normal subgroup  GF of G with G/GF ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  A formation is called Fitting if (a) from N ⊴ G ∈ F it follows that N ∈ F and (b) a group  G ∈ F whenever it is a product of normal F-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Recall that the F-radical GF of a group  G is the greatest normal F-subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  The classes N∗ of all quasinilpotent groups and Sp of all p-soluble groups are Fitting for-  mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From [3, IX, Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='11 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='12] and [3, IV, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='8] follows  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let F and H be non-empty Fitting formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then  FH = (G | GF ∈ H) = (G | G/GH ∈ F)  is a Fitting formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The class Hp = (G | Fp(G) = G) is a Fitting formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It is straightforward to check that for a Fitting formation F, the F-radical can be considered  as a functorial γ which satisfies (F1), (F2) and (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' For convenience in this case denote hγ by  hF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now h∗(G) = hF∗(G) = hN∗(G) and for a non-p-soluble group λp(G) = hFp(G) = hHp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let F be a Fitting formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let prove that if H ̸= 1 and hF(H) < ∞, then hF(HF) = hF(H) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let hF(H) = n  and hF(HF) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then HF(n−1)(H) < H and H/HF(n−1) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that HF ≤ HF(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since HF(n−1) satisfies (F3), we see that (HF)F(n−1) = HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that HF = (HF)F(k) ≤ HF(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that H/HF(k) ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence k ≥ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus  k = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  5  If F, H, K ̸= ∅ are formations, then (FH)K = F(HK) by [3, IV, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' That is why  the class Fn = F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' F  � �� �  n  is a well defined formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' For a natural number n and a Fitting formation F holds Fn = (G | hF(G) ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3 it follows that if G ∈ (G | hF(G) ≤ n), then GFn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that  (G | hF(G) ≤ n) ⊆ Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that there is a group G ∈ Fn with hF(G) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that  G  F ̸= G for every quotient group G ̸≃ 1 of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then hF(GFn) > 0 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that  GFn ̸= 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Therefore Fn ⊆ (G | hF(G) ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus Fn = (G | h(G) ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  In the next lemma we recall the key properties of mutually permutable products  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G = AB be a mutually permutable product of subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then  (1) [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='10] G/N = (AN/N)(BN/N) is a mutually permutable product of sub-  groups AN/N and BN/N for every normal subgroup N of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3(4)] If N is a minimal normal subgroup of a group G, then {N ∩A, N ∩  B} ⊆ {1, N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (3) [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3(5)] If N is a minimal normal subgroup of G contained in A and B∩N =  1, then N ≤ CG(A) or N ≤ CG(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If furthermore N is not cyclic, then N ≤ CG(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (4) [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='11] AGBG ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (5) [1, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='26] A′ and B′ are subnormal in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Recall that π(G) is the set of all prime divisors of |G|, π(F) = ∪  G∈Fπ(G) and Nπ denote the  class of all nilpotent π-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let F be a Fitting formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that hF(G) ≤ h + 1 for every mutually  permutable product G of two F-subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then  max{hF(A), hF(B)} − 1 ≤ hF(G) ≤ max{hF(A), hF(B)} + h  for every mutually permutable product G of two subgroups A and B with hF(A), hF(B) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If A = 1 or B = 1, then there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that A, B ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group  G = AB be the product of mutually permutable subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From hF(A), hF(B) < ∞  it follows that π(G) ⊆ π(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According to [3, IX, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='8] Nπ(F) ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that A′ and B′  are subnormal in G by (5) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since HF ⊴ HNπ(F) ⊴ H′ holds for every π(F)-group  H, subgroups AF and BF are subnormal in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let C = ⟨AF, BF⟩G = ⟨{(AF)x | x ∈ G}∪{(BF)x |  x ∈ G}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then by (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6 and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3  hF(C) = max  �  {(hF(AF)x) | x ∈ G} ∪ {(hF(BF)x) | x ∈ G}  �  = max{hF(AF), hF(BF)} = max{hF(A), hF(B)} − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now G/C = (AC/C)(BC/C) is a mutually permutable product of F-subgroups AC/C and  BC/C by (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that hF(G/C) ≤ h + 1 by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' With the  help of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4 we see that  hF(G) ≤ hF(C) + hF(G/C) ≤ max{hF(A), hF(B)} − 1 + 1 + h = max{hF(A), hF(B)} + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From the other hand, hF(G) ≥ hF(C) = max{hF(A), hF(B)} − 1 by (2) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let F be a Fitting formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that a group G is the least order group  with  (1) G is a mutually permutable product of two subgroups A and B with hF(A) ≥ hF(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) hF(G) = hF(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then G has the unique minimal normal subgroup N, N ≤ A and hF(A/N) = hF(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  6  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let N be a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then N ∩ A ∈ {N, 1} by (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that N ∩ A = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now G/N = (AN/N)(BN/N) is a mutually permutable product  of groups AN/N and BN/N by (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' By our assumption and hF(G) ≥ hF(G/N) ≥  hF(AN/N) = hF(A), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence N ∩ A = N for every minimal normal subgroup N  of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now hF(G) + 1 = hF(A) > hF(G) ≥ hF(G/N) ≥ hF(A/N) ≥ hF(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that  hF(G) = hF(A/N) = hF(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  If G has two minimal normal subgroups N1 and N2, then hF(A/N1) = hF(A/N2) = hF(A)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means hF(A) < hF(A) − 1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G has a unique minimal  normal subgroup N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  4  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1(1)  Our proof relies on the notion of the X-hypercenter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' A chief factor H/K of G is called  X-central in G provided  (H/K) ⋊ (G/CG(H/K)) ∈ X  (see [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' 127–128] or [7, 1, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' A normal subgroup N of G is said to be X-  hypercentral in G if N = 1 or N ̸= 1 and every chief factor of G below N is X-central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The  symbol ZX(G) denotes the X-hypercenter of G, that is, the product of all normal X-hypercentral  in G subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According to [18, Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1] or [7, 1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6] ZX(G) is the largest  normal X-hypercentral subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If X = N is the class of all nilpotent groups, then  ZN(G) = Z∞(G) is the hypercenter of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let n be a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then (N∗)n = (G | h∗(G) ≤ n) = (G | G =  Z(N∗)n(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' First part follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It is well known that the class of all quasinilpotent  groups is a composition (or Baer-local, or solubly saturated) formation (see [2, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  According to [18, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='9] (N∗)n is a composition formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now (N∗)n = (G | G =  Z(N∗)n(G)) by [7, 1, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  For a normal section H/K of G the subgroup C∗  G(H/K) = HCG(H/K) is called an inneriser  (see [2, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It is the set of all elements of G that induce inner automorphisms  on H/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From the definition of the generalized Fitting subgroup it follows that it is the  intersection of innerisers of all chief factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let N be a normal subgroup of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If N is a direct product of isomorphic  simple groups and h∗(G/C∗  G(N)) ≤ k − 1, then F∗  (k)(G/N) = F∗  (k)(G)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that h∗(G/C∗  G(N)) ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let F/N = F∗  (k)(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then F∗  (k)(G) ⊆ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now F/C∗  F(N) ≃ FC∗  G(N)/C∗  G(N) ⊴ G/C∗  G(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Therefore h∗(F/C∗  F(N)) ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means  that h∗(F/C∗  F(H/K)) ≤ k − 1 for every chief factor H/K of F below N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence (H/K) ⋊  (F/CF(H/K)) ∈ (N∗)k for every chief factor H/K of F below N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that N ≤ Z(N∗)k(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus F ∈ (N∗)k by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So F ⊆ F∗  (k)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus F∗  (k)(G) = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If a group G = AB is a product of mutually permutable quasinilpotent subgroups  A and B, then h∗(G) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' To prove this lemma we need only to prove that if a group G = AB is a product  of mutually permutable quasinilpotent subgroups A and B, then G ∈ (N∗)2 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let G be a minimal order counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (1) G has a unique minimal normal subgroup N and G/N ∈ (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  7  Note that G/N is a mutually permutable product of quasinilpotent subgroups (AN/N) and  (BN/N) by (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G/N ∈ (N∗)2 by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since (N∗)2 is a  formation, we see that G has a unique minimal normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According to (4) of Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5 AGBG ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' WLOG we may assume that G has a minimal normal subgroup N ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) N ≤ A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Suppose that N ∩ B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then A ≤ CG(N) or B ≤ CG(N) by (3) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If A ≤  CG(N), then N ⋊ G/CG(N) ≃ N ⋊ B/CB(N) ∈ (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If B ≤ CG(N), then N ⋊ G/CG(N) ≃  N ⋊ A/CA(N) ∈ (N∗) ⊆ (N∗)2 by [2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In both cases N ≤ Z(N∗)2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means  that G ∈ (N∗)2, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now N ∩ B ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence N ≤ A ∩ B by (2) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (3) N is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that N is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since A is quasinilpotent, we see that A/CA(N) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  By analogy B/CB(N) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that A/CA(N) ≃ ACG(N)/CG(N) and B/CB(N) ≃  BCG(N)/CG(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From G = AB it follows that G/CG(N) is a p-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since N is a chief  factor of G, we see that G/CG(N) ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So N ≤ Z∞(G) ≤ Z(N∗)2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus G ∈ (N∗)2, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that N is non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (4) The final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now N is a direct product of minimal normal subgroups of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since A is quasinilpotent, we  see that every element of A induces an inner automorphism on every minimal normal subgroup  of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence every element of A induces an inner automorphism on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  By analogy every  element of B induces an inner automorphism on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From G = AB it follows that every  element of G induces an inner automorphism on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So NCG(N) = G or G/CG(N) ≃ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now  N ⋊ (G/CG(N)) ∈ (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that N ≤ Z(N∗)2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus G ∈ (N∗)2 and h∗(G) ≤ 2, the  final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G be a mutually permutable product of subgroups A  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3 it follows that  max{h∗(A), h∗(B)} − 1 ≤ h∗(G) ≤ max{h∗(A), h∗(B)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that max{h∗(A), h∗(B)} − 1 = h∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' WLOG let h∗(A) = h∗(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' We may  assume that a group G is the least order group with such properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then G has the unique  minimal normal subgroup N, N ≤ A and h∗(A/N) = h∗(A) − 1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that h∗(A/C∗  A(N)) < h∗(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then  F∗  (h∗(A)−1)(A/N) = F∗  (h∗(A)−1)(A)/N < A/N  by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that h∗(A) = h∗(A/N), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence h∗(A/C∗  A(N)) =  h∗(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since G/C∗  G(N) = (AC∗  G(N)/C∗  G(N))(BC∗  G(N)/C∗  G(N)) is a mutually permutable products  of subgroups AC∗  G(N)/C∗  G(N) and BC∗  G(N)/C∗  G(N) by (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5 and A/C∗  A(N) ≃  AC∗  G(N)/C∗  A(N), we see that h∗(G/C∗  G(N)) ≥ h∗(A/C∗  A(N)) = h∗(A) − 1 by our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that F∗(G) ≤ C∗  G(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now h∗(G)−1 = h∗(G/F∗(G)) ≥ h∗(G/C∗  G(N)) ≥ h∗(A/C∗  A(N)) =  h∗(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that h∗(G) ≥ h∗(A), the final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  5  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1(2)  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let p be a prime and H = Hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If a group G = AB is a product of mutually  permutable H-subgroups A and B, then G ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let G be a minimal order counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (1) G has a unique minimal normal subgroup N, G/N ∈ H and N is not p-soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that G/N is a mutually permutable product of H-subgroups (AN/N) and (BN/N) by  (1) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence G/N ∈ H by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since H is a formation, we see that G  8  has a unique minimal normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According to (4) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5 AGBG ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' WLOG  we may assume that G has a minimal normal subgroup N ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  If N is p-soluble, then Fp(G)/N = Fp(G/N) = G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So Fp(G) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus G ∈ H, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (2) N ≤ A ∩ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Suppose that N ∩ B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that N is not cyclic by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then B ≤ CG(N) by (3) of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence N ⋊ G/CG(N) ≃ N ⋊ A/CA(N) ∈ H by [2, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that  N ≤ ZH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Therefore G ∈ H, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now N ∩ B ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence N ≤ A ∩ B by (2) of  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  (4) The final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since N is the unique minimal normal subgroup of G and non-abelian, we see that CG(N) =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So CA(N) = CB(N) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence Rp(A) = Rp(B) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In particular F(A) = F(B) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that all minimal normal subgroups of A are in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' For B is the same situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus  N = F∗(A) = F∗(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So G/N is a mutually permutable product of p-soluble groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since the  class of all p-soluble groups is closed by extensions by p-soluble groups, G/N is p-soluble by (1)  and (4) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From N ≤ F∗(G) it follows that G ∈ H, the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let H = Hp and a group G be a mutually permutable product of  subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' First we a going to prove that max{hH(A), hH(B)} = hH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3 we have  max{hH(A), hH(B)} − 1 ≤ hH(G) ≤ max{hH(A), hH(B)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that max{hH(A), hH(B)} − 1 = hH(G) for some mutually permutable product G of  A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that G is a minimal order group with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' WLOG let hH(A) =  hH(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then G has the unique minimal normal subgroup N, N ≤ A and hH(A/N) =  hH(A) − 1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  If N is p-soluble, then Rp(A/N) = Rp(A)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that Fp(A/N) = Fp(A)/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus  hH(A/N) = hH(A), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means that Rp(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that now N is a simple non-abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since N is a  unique minimal normal subgroup of G, we see that N = F∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now hH(G/N) = hH(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Therefore  hH(G) − 1 = hH(G/N) ≥ hH(A/N) = hH(A) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus hH(G) ≥ hH(A), the contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  We proved that max{hH(A), hH(B)} = hH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let G be a mutually permutable product of groups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If A, B are p-soluble, then  G is p-soluble by (1) and (4) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Hence λp(G) = λp(A) = λp(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now  assume that at least one of subgroups A, B is not p-soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then G is not p-soluble by (1)  and (4) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' WLOG let hH(A) ≥ hH(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence A is not p-soluble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' We proved  that hH(A) = hH(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that hH(G) = λp(G), hH(A) = λp(A), hH(B) = λp(B) if B is not  p-soluble by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5 and 0 = λp(B) < 1 = hH(B) ≤ hH(A) = λp(A) otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus  max{λp(A), λp(B)} = λp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  6  Non-Frattini length  The Frattini subgroup Φ(G) play an important role in the theory of classes of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' One  of the useful properties of the Fitting subgroup of a soluble group is that it is strictly greater  than the Frattini subgroup of the same group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that the generalized Fitting subgroup is  non-trivial in every group but there are groups in which it coincides with the Frattini subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  That is why the following length seems interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  9  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let 1 = G0 ≤ G1 ≤ · · · ≤ G2h = G be a shortest normal series in which for i  even Gi+1/Gi ≤ Φ(G/Gi), and for i odd the factor Gi+1/Gi is a (non-empty) direct product of  simple groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then h = ˜h(G) will be called the non-Frattini length of a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that if G is a soluble group, then ˜h(G) = h(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Another reason that leads us to this  length is the generalization of the Fitting subgroup ˜F(G) introduced by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Schmid [16] and  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shemetkov [17, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5] and defined by  Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = Soc(G/Φ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' F¨orster [4] showed that ˜F(G) can be defined by  Φ(G) ⊆ ˜F(G) and ˜F(G)/Φ(G) = F∗(G/Φ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let Φ and ˜F be functorials that assign Φ(G) and ˜F(G) to every group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Then ˜F = Φ ⋆ F∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It  is well known that Φ satisfies (F1) and (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence ˜F satisfies (F1) and (F2) by Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that Φ(G/Φ(G)) ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' By analogy with the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='5 one can show that the  non-Frattini length ˜h(G) of a group G and h˜F(G) coincide for every group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The following  theorem shows connections between the non-Frattini length and the generalized Fitting height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' For any group G holds ˜h(G) ≤ h∗(G) ≤ 2˜h(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' There exists a group H with  ˜h(H) = n and h∗(H) = 2n for any natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since Φ(G) and Soc(G/Φ(G)) are quasinilpotent, we see that F∗(G) ≤ ˜F(G) ≤ F∗  (2)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Now F∗  (n)(G) ≤ ˜F(n)(G) ≤ F∗  (2n)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence if ˜F(n)(G) = G, then F∗  (n)(G) ≤ G and F∗  (2n)(G) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means ˜h(G) ≤ h∗(G) ≤ 2˜h(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let K be a group, K1 be isomorphic to the regular wreath product of A5 and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note  that the base B of it is the unique minimal normal subgroup of K1 and non-abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According  to [6], there is a Frattini F3K1-module A which is faithful for K1 and a Frattini extension  A \u058c K2 ։ K1 such that A  K1  ≃ Φ(K2) and K2/Φ(K2) ≃ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Let denote K2 by f(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now f(K)/˜F(f(K)) ≃ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From the definition of h˜F = ˜h it follows  that ˜h(f(K)) = ˜h(K) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that Φ(f(K)) ⊆ F∗(f(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Assume that Φ(f(K)) ̸= F∗(f(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means that  F∗(f(K)) = ˜F(f(K)) is quasinilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' By [9, X, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='8] it follows that Φ(f(K)) ⊆  Z(F∗(f(K))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that 1 < B ≤ CK1(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus A is not faithful, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus Φ(f(K)) = F∗(f(K)) and f(K)/F∗(f(K)) ≃ K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Since K1 has a unique mini-  mal normal subgroup B and it is non-abelian, we see that F∗(K1) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  It means that  f(K)/F∗  (2)(f(K)) ≃ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' From the definition of h∗ it follows that h∗(f(K)) = h∗(K) + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  As usual, let f(1)(K) = f(K) and f(i+1)(K) = f(f(i)(K)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then ˜h(f(n)(1)) = n and  h∗(f(n)(1)) = 2n for any natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  The following proposition directly follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G = ⟨Ai | 1 ≤ i ≤ n⟩ be the join of its subnormal subgroups Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then ˜h(G) ≤ max{˜h(Ai) | 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  One of the main differences between the non-Frattini length and the generalized Fitting  height is that the non-Frattini length of a normal subgroup can be greater than the non-Frattini  length of a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let E ≃ A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' There is an F5E-module V such that R = Rad(V ) is a faithful  irreducible F5E-module and V/R is an irreducible trivial F5E-module (how to construct such  module, for example, see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let G = V ⋋ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now Φ(G) = R by [3, B, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note  10  that G/Φ(G) = G/R ≃ Z5 × E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' So ˜F(G) = G and ˜h(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that G = V (RE) where V  and RE are normal subgroups of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Since V is abelian, we see that ˜h(V ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that R  is a unique minimal normal subgroup of RE and Φ(RE) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that ˜F(RE) = R and  ˜h(RE) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus ˜h(G) < max{˜h(V ), ˜h(RE)} and ˜F does not satisfy (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Recall [1, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1] that a group G is called a totally permutable product of its  subgroups A and B if G = AB and every subgroup of A permutes with every subgroup of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G = AB be a totally permutable product of subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Then  max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' If A = 1 or B = 1, then max{˜h(A), ˜h(B)} = ˜h(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Assume that A, B ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  According to [1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='16] A ∩ B ≤ F(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence A ∩ B ≤ F∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now G =  G/F∗(G) is a totally permutable product of A = AF∗(G)/F∗(G) and B = BF∗(G)/F∗(G)  by [1, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that A ∩ B ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' According to [1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2] [A, B] ≤ F(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  So [A, B] ≤ F∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' It means that  G/F∗(G) = (AF∗(G)/F∗(G)) × (BF∗(G)/F∗(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Note that for the formation U of all supersoluble groups we have U ⊂ N2 ⊂ (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence  if H = H1H2 is a product of totally permutable (N∗)2-subgroups H1 and H2, then H ∈ (N∗)2  by [1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Analyzing the proof of [1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2] we see that this theorem is  true not only for saturated formation, but for formations F = (G | G = ZF(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In particular,  it is true for (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Thus if H = H1H2 ∈ (N∗)2 is a product of totally permutable subgroups  H1 and H2, then H1, H2 ∈ (N∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Now (N∗)2 satisfies conditions of [1, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Therefore A ∩ F∗  (2)(G) = F∗  (2)(A) and B ∩ F∗  (2)(G) = F∗  (2)(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Note that  AF∗(G)/F∗(G) ≃ AF∗  (2)(G)/F∗  (2)(G) ≃ A/F∗  (2)(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  By analogy BF∗(G)/F∗(G) ≃ B/F∗  (2)(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hence  G/F∗  (2)(G) ≃ (A/F∗  (2)(A)) × (B/F∗  (2)(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='6 and ˜h = h˜F we have ˜h(G/F∗  (2)(G)) = max{˜h(A/F∗  (2)(A)), ˜h(B/F∗  (2)(B))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  From ˜F(H) ≤ F∗  (2)(H) ≤ ˜F(2)(H) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='4 it follows that for any group H ̸= 1 holds  ˜h(H) − 1 = ˜h(H/˜F(H)) ≥ ˜h(H/F∗  (2)(H)) ≥ ˜h(H/˜F(2)(H)) ≥ ˜h(H) − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Therefore  {˜h(G) − ˜h(G/F∗  (2)(G)), ˜h(A) − ˜h(A/F∗  (2)(A)), ˜h(B) − ˜h(B/F∗  (2)(B))} ⊆ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Thus max{˜h(A), ˜h(B)} − 1 ≤ ˜h(G) ≤ max{˜h(A), ˜h(B)} + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  While proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3 we were not able to answer the following question:  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Let a group G = AB be a totally permutable product of subgroups A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Is  max{˜h(A), ˜h(B)} ≤ ˜h(G)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  The following question seems interesting  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Do there exists a constant h with | max{˜h(A), ˜h(B)} − ˜h(G)| ≤ h for any  mutually permutable product G = AB of subgroups A and B?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Towers [19] defined and studied analogues of F∗(G) and ˜F(G) for Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Using  these subgroups and the radical (of a Lie algebra) one can introduce the generalized Fitting  height, the non-soluble length and the non-Frattini length of a (finite dimension) Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Estimate the generalized Fitting height, the non-soluble length and the non-  Frattini length of a (finite dimension) Lie algebra that is the sum of its two subalgebras (ideals,  subideals, mutually or totally permutable subalgebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  11  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Ballester-Bolinches, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Esteban-Romero, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Asaad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Products of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' De  Gruyter, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Ballester-Bollinches and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Ezquerro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Classes of Finite Groups, volume 584 of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Springer Netherlands, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Doerk and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hawkes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Finite Soluble Groups, volume 4 of De Gruyter Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  De Gruyter, Berlin, New York, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' F¨orster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Projektive Klassen endlicher Gruppen: IIa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Ges¨attigte Formationen: Ein all-  gemeiner Satz von Gasch¨utz-Lubeseder-Baer-Typ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Pub, Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' UAB, 29(2/3):39–76, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Gardner and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Wiegandt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Radical theory of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Marcel Dekker New York, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Griess and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The Frattini module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 30(1):256–266, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [7] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Structure Theory for Canonical Classes of Finite Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Springer-Verlag, Berlin,  Heidelberg, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hall and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Higman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' On the p-Length of p-Soluble Groups and Reduction Theorems  for Burnside’s Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', s3-6(1):1–42, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Huppert and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Blackburn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Finite groups III, volume 243 of Grundlehren Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Springer-Verlag, Berlin, Heidelberg, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Jabara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The Fitting length of a product of mutually permutable finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Acta  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Hung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 159(1):206–210, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Khukhro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shumyatsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Nonsoluble and non-p-soluble length of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Isr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 207(2):507–525, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [12] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Khukhro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shumyatsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' On the length of finite factorized groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Pura Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 194(6):1775–1780, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [13] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Khukhro and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shumyatsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' On the length of finite groups and of fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 143(9):3781–3790, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Murashka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  On one conjecture about supersoluble groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Debrecen,  100(3-4):399–404, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [15] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Plotkin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Radicals in groups, operations on group classes and radical classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In Selected  questions of algebra and logic, pages 205–244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Nauka, Novosibirsk, 1973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Schmid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' ¨Uber die Automorphismengruppen endlicher Gruppen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 23(1):236–  242, 1972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shemetkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Formations of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Nauka, Moscow, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' In Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Shemetkov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Skiba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Formations of algebraic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Nauka, Moscow, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  In Russian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [19] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' The generalised nilradical of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Algebra, 470:197–218, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' On the structure of compact torsion groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Monatsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content=', 96(1):57–66,  1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FNE0T4oBgHgl3EQfQwDu/content/2301.02199v1.pdf'}
diff --git a/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/2301.11440v1.pdf.txt b/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/2301.11440v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fa14827c4f9f6963fb76306821fee302158ed6d3
--- /dev/null
+++ b/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/2301.11440v1.pdf.txt
@@ -0,0 +1,809 @@
+Secure synchronization of artificial neural networks
+used to correct errors in quantum cryptography
+Marcin Niemiec∗, Tymoteusz Widlarz∗, Miralem Mehic†‡
+∗ AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Krakow, Poland
+† Department of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo,
+Zmaja od Bosne bb, 71000, Sarajevo, Bosnia and Herzegovina
+‡ VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czechia
+∗niemiec@agh.edu.pl †widlarztymoteusz@gmail.com ‡miralem.mehic@ieee.org
+Abstract—Quantum cryptography can provide a very high
+level of data security. However, a big challenge of this technique
+is errors in quantum channels. Therefore, error correction
+methods must be applied in real implementations. An example is
+error correction based on artificial neural networks. This paper
+considers the practical aspects of this recently proposed method
+and analyzes elements which influence security and efficiency.
+The synchronization process based on mutual learning processes
+is analyzed in detail. The results allowed us to determine the
+impact of various parameters. Additionally, the paper describes
+the recommended number of iterations for different structures of
+artificial neural networks and various error rates. All this aims
+to support users in choosing a suitable configuration of neural
+networks used to correct errors in a secure and efficient way.
+Index Terms—quantum cryptography, key reconciliation, error
+correction, artificial neural networks
+I. INTRODUCTION
+The emergence and intensive development of the field of
+quantum computing has put many cryptography algorithms at
+risk. However, quantum physics also allows to achieve multi-
+ple cryptography tasks. One of the most popular is quantum
+key distribution [1]. Unfortunately, quantum communication
+is not perfect and additional solutions are required to correct
+any errors after the key distribution in the quantum channel.
+Artificial neural networks can be utilized to correct these errors
+[2]. It is a recently proposed solution which provides high
+level of security and efficiency comparing to other existing
+error correction methods.
+This paper analyzes the impact of different neural networks’
+parameters on the synchronization process. These parameters
+influence the number of iterations required as well as the
+security and efficiency of quantum cryptography. Therefore,
+it is important to know which neural network scheme should
+be chosen and which should be avoided. Additionally, the syn-
+chronization requires the number of iterations to be specified.
+Therefore, a recommended number of iterations for a particular
+multiple neural network’s scheme is provided.
+The paper is structured as follows. Related work is re-
+viewed in Section 2. Section 3 presents the basics of quantum
+cryptography, the architecture of the tree parity machine,
+and error correction using this structure of artificial neural
+networks. Analysis of synchronization parameters including
+the recommended number of iterations for typical keys and
+error rates is described in Section 4. Section 5 concludes the
+paper.
+II. RELATED WORK
+The first quantum key distribution (QKD) protocol, intro-
+duced in 1984 by Bennet and Brassard, is BB84 [3]. This
+scheme uses the polarization state of a single photon to
+transmit information. Since then, several other protocols have
+been presented. One of them is the E91 protocol introduced
+in 1991 by Ekerd [4]. It utilizes entangled pairs of photons
+in the QKD process. However, some errors usually appear
+during data exchange in the quantum channel. After the initial
+QKD, there is a specific step: quantum bit error rate (QBER)
+estimation based on the acquired keys. The QBER value is
+usually low [5]. It must to be lower than the chosen threshold
+used to detect the eavesdropper.
+Several methods of correcting error incurred in the quan-
+tum key distribution process have been developed. The first
+described method – BBBSS – was proposed in 1992 [6].
+However, the most popular is the Cascade key reconciliation
+protocol [7]. It is based on multiple random permutations.
+The Winnow protocol, based on the exchange of parity and
+Hamming codes, is another method of error correction in the
+raw key [8]. Its main improvement is the reduction of the
+required communication between both parties. The third most
+popular error reconciliation scheme is the low density parity
+check approach. It offers a significant reduction of exchanged
+information; however, it introduces more computation and
+memory costs than the Cascade and Winnow protocols [7].
+In 2019, another method of error correction in quantum
+cryptography was proposed by Niemiec in [2]. The solution
+uses mutual synchronization of two artificial neural networks
+(ANN) to correct the errors. The tree parity machine (TPM)
+is proposed as a neural network used in this approach. It is
+a well-known structure in cryptography – the synchronization
+of two TPMs can be used as a key exchange protocol. TPMs
+arXiv:2301.11440v1  [cs.CR]  26 Jan 2023
+
+cannot be used as a general method to correct a selected error
+because it is not possible to predict the final string of bits after
+the synchronization process. However, it is a desirable feature
+for shared keys which should be random strings of bits.
+III. QUANTUM CRYPTOGRAPHY SUPPORTED BY
+ARTIFICIAL NEURAL NETWORKS
+Symmetric cryptography uses a single key to encrypt and
+decrypt secret messages. Let’s assume that Alice and Bob, the
+two characters used in describing cryptography protocols, are
+using symmetric encryption. The goal is to send information
+from Alice to Bob in a way that provides confidentiality. To
+achieve this, Alice and Bob need to agree on a shared secret
+key. Alice encrypts confidential data using the previously
+chosen key and Bob decrypts it using the same key. The same
+key is applied to encrypt and decrypt the information, hence
+the name: symmetric-key encryption. It is worth mentioning
+only the one-time-pad symmetric scheme has been proven
+secure but it requires a key not smaller than the message being
+sent.
+In general, symmetric-key encryption algorithms – for ex-
+ample the Advanced Encryption Standard (AES) [9] – per-
+form better than asymmetric-key algorithms [10]. However,
+symmetric-key algorithms have an important disadvantage
+compared to asymmetric-key schemes. In the symmetric key
+encryption scheme, the key needs to be safely distributed
+or established between Alice and Bob [11]. The symmetric
+key can be exchanged in a number of ways, including via
+a trusted third party or by direct exchange between involved
+parties. However, both methods introduce some vulnerabili-
+ties, including passive scanning of network traffic. A method
+where the eavesdropper can be easily detected uses quantum
+mechanics to establish keys between Alice and Bob. It is called
+the quantum key distribution protocol.
+A. Quantum key distribution
+Quantum mechanics allows for secure key distribution1
+among network users. Two main principles are the core of
+the security of QKD: an unknown quantum state cannot be
+copied [12], and the quantum state cannot be estimated without
+disturbing it. One of the most popular QKD protocols which
+uses those principles is the BB84 scheme [3].
+The BB84 protocol uses photons with two polarization
+bases: rectilinear or diagonal. Alice encodes a string of bits
+using photons on a randomly chosen basis. After that, all the
+photons are sent through a quantum channel. Bob randomly
+chooses a basis for each photon to decode the binary 0 or
+1. Alice and Bob’s bases are compared through a public
+communication channel. Each bit where both parties chose the
+same basis should be the same. However, when Bob measures
+the photon in a different basis than Alice, this bit is rejected.
+The remaining bits are the same for both parties and can be
+considered as a symmetric key. Next, the error estimation
+1In fact, a key is not distributed but negotiated. However, the term
+’distribution’ is consistently used in this paper to be consistent with the
+commonly accepted name of the technique.
+is performed. Randomly chosen parts of the keys between
+Alice and Bob are compared to compute the QBER value.
+If the comparison results in a high error rate, it means that
+the eavesdropper (Eve) is trying to gain information about
+the exchanged photons. However, the quantum channel is not
+perfect, and errors are usually detected due to disturbance,
+noise in the detectors or other elements. The number of errors
+introduced by the quantum channel’s imperfections must be
+considered while deciding the maximum acceptable error rate.
+The differences between Alice and Bob’s keys need to
+be corrected. Several error correction methods are known.
+BBBSS is the earliest scheme proposed in [6]. It is mainly
+based on parity checks. The most popular method is the
+Cascade protocol [13]. It is an improved version of BBBSS
+and requires less information to be sent between Alice and
+Bob through the public channel. The Cascade protocol and
+its predecessor are based on multiple parity checks. The basic
+idea is that the keys are divided into blocks of a fixed size.
+The number of bits in each block depends on the previously
+calculated QBER value. Alice and Bob compare the parities
+of each block to allow them to find an odd number of errors.
+If errors are detected in a given block, it is split into two.
+The process is repeated recursively for each block until all
+errors are corrected. It concludes a single iteration after which
+Alice and Bob have keys with an even number of errors or
+without any errors. Before performing the following iterations,
+the keys are scrambled, and the size of the block is increased.
+The number of iterations is predetermined. As a result of this
+process, Alice and Bob should have the same keys. However,
+it is not always the case. A number of iterations or block sizes
+can be chosen incorrectly and cause failure in error correction.
+Additionally, the algorithm performs multiple parity checks
+over the public channel, which can be intercepted by an
+eavesdropper (Eve). As a result, Eve can construct a partial
+key. Alice and Bob should discard parts of their keys to
+increase the lost security. This reduces the performance of
+this method since the confidential keys must be shortened in
+the process. Another error reconciliation method is based on
+mutual synchronization of artificial neural networks.
+B. Tree parity machine
+An artificial neural network (ANN) is a computing system
+inspired by biological neural networks [14]. ANNs are used
+to recognize patterns and in many other solutions in the fields
+of machine learning. ANNs consist of multiple connected
+nodes (artificial neurons), with each neuron representing a
+mathematical function [15]. These nodes are divided into three
+types of layers: the first (input) layer, at least one hidden layer,
+and the output layer. The connections between neurons in each
+layer can be characterized by weights.
+In cryptography, the most commonly used neural network is
+the tree parity machine (TPM) [16]. A scheme of this model
+is presented in Fig. 1. There are K ×N input neurons, divided
+into K groups. There is a single hidden layer with K nodes.
+Each of these nodes has N inputs. The TPM has a single
+output neuron. The connections between input neurons and
+
+hidden layer neurons are described by weights W – integers
+in the range [−L, L], thus L is the maximum and −L is
+the minimum weight value. The values of σ characterize the
+connections between the hidden layer neurons and an output
+neuron. The output value of the TPM is described by τ.
+The value of σ is calculated using the following formulas:
+σk = sgn(
+N
+�
+n=1
+xkn ∗ wkn)
+(1)
+sgn(z) =
+�
+−1
+z ≤ 0
+1
+z > 0
+(2)
+Due to the usage of the presented signum function, σ can take
+two values: 1 or −1. The output value of TPM is calculated
+as:
+τ =
+K
+�
+k=1
+σk
+(3)
+This neural network has two possible outcomes: 1 or −1.
+For the TPM structure, multiple learning algorithms are
+proposed. Most popular are Hebbian, anti-Hebbian, and ran-
+dom walk. The leading is the Hebbian rule [17]. The Hebbian
+algorithm updates ANN weights in the following manner:
+w∗
+kn = vL(wkn + xkn ∗ σk ∗ θ(σk, τ))
+(4)
+where θ limits the impact of hidden layer neurons whose value
+was different than τ:
+θ(σk, τ) =
+�
+0
+if σk ̸= τ
+1
+if σk = τ
+(5)
+The vL function makes sure that the new weights are kept
+within the [−L, L] range:
+vL(z) =
+�
+�
+�
+�
+�
+−L
+if z ≤ −L
+z
+if − L < z < L
+L
+if z ≥ L
+(6)
+The TPM structure allows for mutual learning of the two
+neural networks [18], primarily based on updating weights
+only when the outputs from both neural networks are the same.
+The input values are random and the same for both Alice and
+Bob’s TPMs. Inputs are updated in each iteration. The security
+of this process relies on the fact that cooperating TPMs can
+achieve convergence significantly faster than Eve’s machine,
+which can update weights less frequently. The TPM is most
+commonly used in cryptography to exchange a secret key. This
+usage is defined as neural cryptography [19]. Alice and Bob
+mutually synchronize their TPMs to achieve the same weights.
+After the synchronization process, these weights provide a
+secure symmetric key.
+C. Error correction based on TPMs
+TPMs can be utilized during the error correction process
+in quantum cryptography [2]. The neural network’s task is to
+correct all errors to achieve the same string of confidential bits
+at both endpoints. Firstly, Alice and Bob prepare their TPMs.
+The number of neurons in the hidden layer (K) and the number
+of input neurons (N) is determined by Alice and passed on
+to Bob. The value L must also be agreed between the users.
+The keys achieved using the QKD protocol are changed into
+integer values in the range [−L, L]. These values are used
+in the appropriate TPMs as weights between neurons in the
+input layer and the hidden layer. Since Alice’s string of bits
+is similar to Bob’s (QBER is usually not high), the weights
+in the created TPMs are almost synchronized. At this point,
+Alice and Bob have constructed TPMs with the same structure
+but with a few differences in the weight values.
+After establishing the TPM structure and changing bits to
+weights, the synchronization process starts. It consists of mul-
+tiple iterations, repeated until common weights are achieved
+between Alice and Bob. A single iteration starts from Alice
+choosing the input string and computing the result using the
+TPM. After that, the generated input string is passed on to Bob,
+who computes the output of his TPM using the received input.
+Then, the results are compared. If the outputs of both TPMs
+match, the weights can be updated. Otherwise, the process is
+repeated with a different input string.
+After an appropriate number of iterations, the TPMs are
+synchronized and Alice and Bob can change the weights back
+into a string of bits. The resulting bits are the same. However,
+the privacy amplification process after error correction is still
+recommended [20]. The reduction of the key protecting Alice
+and Bob from information leakage is defined as [2]:
+Z = log2L+12i
+(7)
+where i is the number of TPM iterations.
+This usage of TPM is safer than the neural cryptography
+solution, because weights are similar before the synchroniza-
+tion. Therefore, significantly fewer iterations are required to
+achieve convergence than the randomly initialized weights
+in key establishing algorithms. It is worth mentioning this
+method of error correction is characterized by high efficiency,
+e.g. requires approximately 30% less iterations than Cascade
+algorithm [2].
+IV. ANALYSIS OF THE SYNCHRONIZATION PROCESS
+The crucial decision regarding the error detection approach
+based on TPMs is the number of iterations during the syn-
+chronization process. This value should be as low as possible
+for security reasons. However, it cannot be too low, since
+neural networks will not be able to correct all errors in the
+key otherwise. It is the user’s responsibility to select the
+appropriate value for the error correction. The main objective
+of the analysis is to determine the impact of various neural
+network parameters on the synchronization process. Another
+goal is to provide a recommended number of iterations for
+users.
+
+X11
+X12
+X13
+X1N
+X21
+X22
+X23
+X2N
+XK1
+XK2
+XK3
+XKN
+∑
+∏
+∑
+∑
+W11
+W1N
+W21
+W2N
+WKN= {-L, … ,L}
+WK1
+σK= {-1, 1}
+σ2
+σ1
+τ={-1, 1}
+Fig. 1. Model of tree parity machine.
+A. Testbed
+The experiments require an application to simulate the error
+correction process based on artificial neural networks. The
+application for correcting errors arising in quantum key distri-
+bution was written in Python and uses the NumPy package – a
+library for scientific computing which provides fast operations
+on arrays required by the TPM. The functions provided by
+NumPy satisfy all necessary calculations to achieve neural
+network convergence. Synchronization of TPMs is performed
+over sockets to allow real-world usage of this tool. The
+Hebbian learning algorithm for updating weights is used.
+The developed application makes it possible to correct errors
+in the keys using quantum key distribution protocols. The users
+are also able to correct simulated keys with the chosen error
+rate. It helps if users do not have strings of bits created by a
+real QKD system. An important feature of the tool is its ability
+to select neural network parameters. The user can personalize
+the synchronization process, starting from the key length and
+error rate. The least sufficient number of bits was used for
+translation into a single integer (values of the weights must be
+in the range [−L, L]). It was demonstrated that the number of
+hidden neurons and the number of inputs depend on the chosen
+key length and L value. Therefore, users need to select these
+parameters taking into account the requirements and needs.
+During the experiments the minimum number of returned
+required iterations for a single TPM configuration was set
+to 200. The maximum number of iterations was limited to
+1000. Additionally, the maximum number of retries in a single
+iteration was limited to 10 to speed up the simulation process.
+Finally, 1880 different scenarios were analyzed. All possible
+TPM configurations for key lengths varying between 100 and
+700 with a 100 bit step are available. Moreover, the data is
+available for other keys with lengths varying between 128 and
+352 with an 8 bit step. Between 350 and 500 synchronizations
+were performed for each TPM. It was assumed that this
+number of iterations is sufficient to achieve convergence.
+B. Recommended number of iterations
+To obtain the recommended number of iterations of TPMs
+for successful error correction, the sum of means and standard
+deviations of the results was calculated. The median and
+variance values were calculated as well for comparison. The
+full results are available online2. The selected part – the neural
+network configurations where the key length equals 256 bits
+with the recommended number of iterations – is presented in
+Tab. I.
+Fig. 2. Histogram for number of iterations (TPM with a 256 bit key, N = 16,
+K = 4, L = 4, QBER = 3%).
+2Recommended numbers of iterations for 1880 different scenarios –
+TPM structures and QBER values – are available from: http://kt.agh.edu.pl/
+∼niemiec/ICC-2023 This is mainly based on possible key lengths which vary
+between 128 and 500 bits with 4 bit steps. Additionally, keys with lengths
+between 500 and 700 with 100 bit steps are included.
+
+180
+160
+140
+120
+Count
+100
+80
+60
+40
+20
+0
+[11, 69]
+(69, 127)
+(127, 185)
+(185, 243)
+(243, 301)
+(301, 359)
+(359, 400)
+> 400
+Number of iterationsTABLE I
+RECOMMENDED NUMBER OF ITERATIONS FOR TPMS GENERATED FOR
+256 BIT KEYS
+Weights
+range
+{−L, L}
+QBER
+[%]
+Number
+of
+inputs
+to a single
+hidden
+neuron
+[N]
+Number
+of
+hidden
+neurons
+[K]
+Recommended
+number
+of
+iterations
+2
+1
+2
+43
+154
+2
+1
+43
+2
+51
+2
+2
+2
+43
+179
+2
+2
+43
+2
+59
+2
+2
+86
+1
+24
+2
+3
+2
+43
+188
+2
+3
+43
+2
+64
+2
+3
+86
+1
+25
+3
+1
+2
+43
+218
+3
+1
+43
+2
+71
+3
+1
+86
+1
+33
+3
+2
+2
+43
+309
+3
+2
+43
+2
+94
+3
+2
+86
+1
+39
+3
+3
+2
+43
+325
+3
+3
+43
+2
+97
+3
+3
+86
+1
+40
+4
+1
+2
+32
+450
+4
+1
+4
+16
+496
+4
+1
+8
+8
+301
+4
+1
+16
+4
+176
+4
+1
+32
+2
+125
+4
+2
+2
+32
+554
+4
+2
+4
+16
+701
+4
+2
+8
+8
+483
+4
+2
+16
+4
+264
+4
+2
+32
+2
+152
+4
+3
+2
+32
+609
+4
+3
+4
+16
+772
+4
+3
+8
+8
+542
+4
+3
+16
+4
+302
+4
+3
+32
+2
+164
+Fig. 2 shows the histogram of data gathered for a sin-
+gle neural network configuration. The distribution is right-
+skewed. The mean value is greater than the median. It is a
+common characteristic for other tested TPM configurations. If
+the distribution is not positively skewed, it is symmetrical.
+The recommended number of iterations for the presented
+configuration, according to Tab. I, equals 302. It is based on
+the sum of the mean and standard deviation values. For all
+presented TPM configurations, this sum gives an 84% chance
+of successful synchronization, assuming a normal distribution
+of results. For the right-skewed distribution, similar to the one
+presented in Fig. 2, the probability of success is higher. The
+85-th percentile for the given set is equal to 276 – less than
+the proposed value. In this case, after choosing the suggested
+number of iterations the user has more than an 88% chance
+of success.
+Knowing the lowest required number of iterations is im-
+portant because it reduces the risk of a successful attack by
+Eve. The attacker could create independent TPMs and try
+to synchronize one of them with Alice or Bob’s machine.
+The recommended number of iterations increases the security
+of this solution because Alice and Bob require far fewer
+iterations to synchronize, compared to Alice (or Bob) and Eve
+synchronizing using random weights.
+C. Impact of TPM structures
+The results of simulations allow us to analyze how TPM
+structures affect the number of required iterations during the
+synchronization process. Fig. 3 shows the number of required
+iterations depending on the K and N parameters. It shows
+two different TPM configurations: one with a 144 bit key and
+another with a 216 bit key. These configurations were chosen
+due to having a similar number of possible K and N pairs.
+For a given key length, L value and error rate there is a limited
+number of possible N and K values. The K value changes
+in inverse proportion to the N value. As presented in Fig.
+3 the speed of the TPM synchronization process depends on
+the neural network structure (N and K values). The number
+of required iterations increases alongside the higher number
+of neurons in the hidden layer (K). The trend is similar for
+both presented TPMs. After achieving a certain threshold,
+the number of recommended iterations increases slowly. The
+results fit the logarithmic trend line. It means that after a
+certain K value, increasing this parameter further does not
+affect the synchronization speed as much as under a certain
+threshold.
+Fig. 3. Number of iterations for TPMs with 144 and 216 bit keys for different
+K value.
+Other configurations of the selected TPMs were studied
+based on the increasing error rate of the keys. Two configura-
+tions with 128 and 256 bit keys were tested. The average of
+every possible configuration of the recommended number of it-
+erations was calculated for different QBER values. The results
+are presented in Fig. 4. This confirms that a greater number
+of errors results in a higher average number of recommended
+iterations. It confirms the applicability of TPMs to correct
+errors emerging in quantum key distribution, where the error
+rate should not be higher than a few percent. Therefore, the
+eavesdropper needs more iterations to synchronize its TPM.
+Additionally, it was verified that value L has an exponential
+impact on the average recommended number of iterations. The
+data was gathered using a similar approach to the study with
+
+180
+160
+Recommended number of iterations
+140
+120
+100
+144 bits
+80
+216 bits
+60
+40
+20
+0
+6
+8
+9
+10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
+Number of hidden layer neurons (K)Fig. 4.
+Number of iterations for TPMs with 128 and 256 bit keys depended
+on the QBER.
+the impact of QBER. The average recommended number of
+iterations of each configuration for a given L was calculated.
+Fig. 5 shows the exponential trend line. It is worth mentioning
+that the impact of L value on the synchronization time is
+significant.
+Fig. 5.
+Number of iterations for TPMs with 128 and 256 bit keys dependent
+on the L value.
+It is the user’s responsibility to choose the best possible
+configuration for a given key length and QBER value. The
+analysis shows that the L value should be chosen carefully
+since it exponentially affects the required number of iterations.
+Additionally, the choice of the K value should be made
+with caution due to its logarithmic impact on the number of
+iterations.
+V. SUMMARY
+The analysis of the TPM synchronization process used for
+error correction purposes was presented in this paper. It shows
+that the parameters of the TPM structure have an impact on
+the synchronization time and security of this error correction
+method. However, different parameters of artificial neural
+networks have different effects. Therefore, users should be
+aware of how to choose the configuration of neural networks
+used to correct errors in a secure and efficient way. One of
+the deciding factors which need to be selected is the number
+of iterations. The paper describes the recommended number
+of iterations for different TPM structures and QBER values
+to assist users in this step. The numbers recommended by the
+authors are as low as possible but with a high probability of
+successful synchronization to ensure secure and efficient error
+correction based on artificial neural networks.
+ACKNOWLEDGMENT
+This work was supported by the ECHO project which has
+received funding from the European Union’s Horizon 2020
+research and innovation programme under the grant agreement
+no. 830943.
+REFERENCES
+[1] S. Abidin, A. Swami, E. Ramirez-As´ıs, J. Alvarado-Tolentino, R. K.
+Maurya, and N. Hussain, “Quantum cryptography technique: A way
+to improve security challenges in mobile cloud computing (mcc),”
+Materials Today: Proceedings, vol. 51, pp. 508–514, 2022.
+[2] M. Niemiec, “Error correction in quantum cryptography based on
+artificial neural networks,” Quantum Information Processing, 2019.
+[3] C. Bennett and G. Brassard, “Quantum cryptography: Public key dis-
+tribution and coin tossing,” Theoretical Computer Science - TCS, pp.
+175–179, 1984.
+[4] A. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev.
+Lett., pp. 661–663, 1991.
+[5] M. Khodr, “Evaluations of quantum bit error rate using the three stage
+multiphoton protocol,” 2017 International Conference on Electrical and
+Computing Technologies and Applications (ICECTA), pp. 1–4, 2017.
+[6] C. Bennett, F. Bessette, G. Brassard, L. Salvail, and J. Smolin, “Exper-
+imental quantum cryptography,” Journal of Cryptology, pp. 3–28, 1992.
+[7] M. Mehic, M. Niemiec, H. Siljak, and M. Voznak, “Error reconcilia-
+tion in quantum key distribution protocols,” Reversible Computation:
+Extending Horizons of Computing: Selected Results of the COST Action
+IC1405, pp. 222–236, 2020.
+[8] W. T. Buttler, S. K. Lamoreaux, J. R. Torgerson, G. H. Nickel, C. H.
+Donahue, and C. G. Peterson, “Fast, efficient error reconciliation for
+quantum cryptography,” Phys. Rev. A, 2003.
+[9] H. Delfs and H. Knebl, “Symmetric-key encryption,” Introduction to
+Cryptography: Principles and Applications, pp. 11–31, 2007.
+[10] M. Panda, “Performance analysis of encryption algorithms for security,”
+2016 International Conference on Signal Processing, Communication,
+Power and Embedded System (SCOPES), pp. 278–284, 2016.
+[11] M. Umaparvathi and D. K. Varughese, “Evaluation of symmetric en-
+cryption algorithms for manets,” 2010 IEEE International Conference
+on Computational Intelligence and Computing Research, pp. 1–3, 2010.
+[12] W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,”
+Nature, pp. 802–803, 1982.
+[13] G. Brassard and L. Salvail, “Secret-key reconciliation by public discus-
+sion,” Advances in Cryptology, pp. 410–423, 1994.
+[14] J. Hopfield, “Artificial neural networks,” IEEE Circuits and Devices
+Magazine, pp. 3–10, 1988.
+[15] P. P. Hadke and S. G. Kale, “Use of neural networks in cryptography:
+A review,” in 2016 World Conference on Futuristic Trends in Research
+and Innovation for Social Welfare (Startup Conclave), 2016, pp. 1–4.
+[16] A. Sarkar, “Secure exchange of information using artificial intelligence
+and chaotic system guided neural synchronization,” Multimedia Tools
+and Applications, vol. 80, pp. 1–31, 05 2021.
+[17] M. Aleksandrov and Y. Bashkov, “Factors affecting synchronization time
+of tree parity machines in cryptography,” 2020 IEEE 2nd International
+Conference on Advanced Trends in Information Theory (ATIT), pp. 108–
+112, 2020.
+[18] R. Metzler, W. Kinzel, and I. Kanter, “Interacting neural networks,”
+Phys. Rev. E, pp. 2555–2565, 2000.
+[19] W. Kinzel and I. Kanter, “Neural cryptography,” Proceedings of the 9th
+International Conference on Neural Information Processing, pp. 1351–
+1354, 2002.
+[20] C. Bennett, G. Brassard, and J. Robert, “Privacy amplification by public
+discussion,” SIAM J. Comput., p. 210–229, 1988.
+
+300
+ iterations
+250
+I number of i
+200
+150
+●128 bits
+●256 bits
+100
+50
+0
+1
+2
+3
+QBER [%]400
+300
+200
+·128 bits
+·256 bits
+100
+0
+2
+3
+4
+7
\ No newline at end of file
diff --git a/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/load_file.txt b/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..624446bf7ae332a300d5ca80db54787abbf7fadd
--- /dev/null
+++ b/HdFJT4oBgHgl3EQfFCyj/content/tmp_files/load_file.txt
@@ -0,0 +1,589 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf,len=588
+page_content='Secure synchronization of artificial neural networks  used to correct errors in quantum cryptography  Marcin Niemiec∗, Tymoteusz Widlarz∗, Miralem Mehic†‡  ∗ AGH University of Science and Technology, al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Mickiewicza 30, 30-059 Krakow, Poland  † Department of Telecommunications, Faculty of Electrical Engineering, University of Sarajevo,  Zmaja od Bosne bb, 71000, Sarajevo, Bosnia and Herzegovina  ‡ VSB – Technical University of Ostrava, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' listopadu 2172/15, 708 00 Ostrava, Czechia  ∗niemiec@agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='pl †widlarztymoteusz@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='com ‡miralem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='mehic@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='org  Abstract—Quantum cryptography can provide a very high  level of data security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, a big challenge of this technique  is errors in quantum channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore, error correction  methods must be applied in real implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' An example is  error correction based on artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This paper  considers the practical aspects of this recently proposed method  and analyzes elements which influence security and efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The synchronization process based on mutual learning processes  is analyzed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The results allowed us to determine the  impact of various parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Additionally, the paper describes  the recommended number of iterations for different structures of  artificial neural networks and various error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' All this aims  to support users in choosing a suitable configuration of neural  networks used to correct errors in a secure and efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Index Terms—quantum cryptography, key reconciliation, error  correction, artificial neural networks  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' INTRODUCTION  The emergence and intensive development of the field of  quantum computing has put many cryptography algorithms at  risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, quantum physics also allows to achieve multi-  ple cryptography tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' One of the most popular is quantum  key distribution [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Unfortunately, quantum communication  is not perfect and additional solutions are required to correct  any errors after the key distribution in the quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Artificial neural networks can be utilized to correct these errors  [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is a recently proposed solution which provides high  level of security and efficiency comparing to other existing  error correction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  This paper analyzes the impact of different neural networks’  parameters on the synchronization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' These parameters  influence the number of iterations required as well as the  security and efficiency of quantum cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore,  it is important to know which neural network scheme should  be chosen and which should be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Additionally, the syn-  chronization requires the number of iterations to be specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Therefore, a recommended number of iterations for a particular  multiple neural network’s scheme is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Related work is re-  viewed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Section 3 presents the basics of quantum  cryptography, the architecture of the tree parity machine,  and error correction using this structure of artificial neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Analysis of synchronization parameters including  the recommended number of iterations for typical keys and  error rates is described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Section 5 concludes the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' RELATED WORK  The first quantum key distribution (QKD) protocol, intro-  duced in 1984 by Bennet and Brassard, is BB84 [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This  scheme uses the polarization state of a single photon to  transmit information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Since then, several other protocols have  been presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' One of them is the E91 protocol introduced  in 1991 by Ekerd [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It utilizes entangled pairs of photons  in the QKD process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, some errors usually appear  during data exchange in the quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' After the initial  QKD, there is a specific step: quantum bit error rate (QBER)  estimation based on the acquired keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The QBER value is  usually low [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It must to be lower than the chosen threshold  used to detect the eavesdropper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Several methods of correcting error incurred in the quan-  tum key distribution process have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The first  described method – BBBSS – was proposed in 1992 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  However, the most popular is the Cascade key reconciliation  protocol [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is based on multiple random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The Winnow protocol, based on the exchange of parity and  Hamming codes, is another method of error correction in the  raw key [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Its main improvement is the reduction of the  required communication between both parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The third most  popular error reconciliation scheme is the low density parity  check approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It offers a significant reduction of exchanged  information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' however, it introduces more computation and  memory costs than the Cascade and Winnow protocols [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  In 2019, another method of error correction in quantum  cryptography was proposed by Niemiec in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The solution  uses mutual synchronization of two artificial neural networks  (ANN) to correct the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The tree parity machine (TPM)  is proposed as a neural network used in this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is  a well-known structure in cryptography – the synchronization  of two TPMs can be used as a key exchange protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' TPMs  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='11440v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='CR]  26 Jan 2023  cannot be used as a general method to correct a selected error  because it is not possible to predict the final string of bits after  the synchronization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, it is a desirable feature  for shared keys which should be random strings of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' QUANTUM CRYPTOGRAPHY SUPPORTED BY  ARTIFICIAL NEURAL NETWORKS  Symmetric cryptography uses a single key to encrypt and  decrypt secret messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Let’s assume that Alice and Bob, the  two characters used in describing cryptography protocols, are  using symmetric encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The goal is to send information  from Alice to Bob in a way that provides confidentiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' To  achieve this, Alice and Bob need to agree on a shared secret  key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice encrypts confidential data using the previously  chosen key and Bob decrypts it using the same key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The same  key is applied to encrypt and decrypt the information, hence  the name: symmetric-key encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is worth mentioning  only the one-time-pad symmetric scheme has been proven  secure but it requires a key not smaller than the message being  sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  In general, symmetric-key encryption algorithms – for ex-  ample the Advanced Encryption Standard (AES) [9] – per-  form better than asymmetric-key algorithms [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However,  symmetric-key algorithms have an important disadvantage  compared to asymmetric-key schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' In the symmetric key  encryption scheme, the key needs to be safely distributed  or established between Alice and Bob [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The symmetric  key can be exchanged in a number of ways, including via  a trusted third party or by direct exchange between involved  parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, both methods introduce some vulnerabili-  ties, including passive scanning of network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' A method  where the eavesdropper can be easily detected uses quantum  mechanics to establish keys between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is called  the quantum key distribution protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Quantum key distribution  Quantum mechanics allows for secure key distribution1  among network users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Two main principles are the core of  the security of QKD: an unknown quantum state cannot be  copied [12], and the quantum state cannot be estimated without  disturbing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' One of the most popular QKD protocols which  uses those principles is the BB84 scheme [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The BB84 protocol uses photons with two polarization  bases: rectilinear or diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice encodes a string of bits  using photons on a randomly chosen basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' After that, all the  photons are sent through a quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bob randomly  chooses a basis for each photon to decode the binary 0 or  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice and Bob’s bases are compared through a public  communication channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Each bit where both parties chose the  same basis should be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, when Bob measures  the photon in a different basis than Alice, this bit is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The remaining bits are the same for both parties and can be  considered as a symmetric key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Next, the error estimation  1In fact, a key is not distributed but negotiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, the term  ’distribution’ is consistently used in this paper to be consistent with the  commonly accepted name of the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Randomly chosen parts of the keys between  Alice and Bob are compared to compute the QBER value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  If the comparison results in a high error rate, it means that  the eavesdropper (Eve) is trying to gain information about  the exchanged photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, the quantum channel is not  perfect, and errors are usually detected due to disturbance,  noise in the detectors or other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The number of errors  introduced by the quantum channel’s imperfections must be  considered while deciding the maximum acceptable error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The differences between Alice and Bob’s keys need to  be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Several error correction methods are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  BBBSS is the earliest scheme proposed in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is mainly  based on parity checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The most popular method is the  Cascade protocol [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is an improved version of BBBSS  and requires less information to be sent between Alice and  Bob through the public channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The Cascade protocol and  its predecessor are based on multiple parity checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The basic  idea is that the keys are divided into blocks of a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The number of bits in each block depends on the previously  calculated QBER value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice and Bob compare the parities  of each block to allow them to find an odd number of errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  If errors are detected in a given block, it is split into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The process is repeated recursively for each block until all  errors are corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It concludes a single iteration after which  Alice and Bob have keys with an even number of errors or  without any errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Before performing the following iterations,  the keys are scrambled, and the size of the block is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The number of iterations is predetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' As a result of this  process, Alice and Bob should have the same keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However,  it is not always the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' A number of iterations or block sizes  can be chosen incorrectly and cause failure in error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Additionally, the algorithm performs multiple parity checks  over the public channel, which can be intercepted by an  eavesdropper (Eve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' As a result, Eve can construct a partial  key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice and Bob should discard parts of their keys to  increase the lost security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This reduces the performance of  this method since the confidential keys must be shortened in  the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Another error reconciliation method is based on  mutual synchronization of artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Tree parity machine  An artificial neural network (ANN) is a computing system  inspired by biological neural networks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' ANNs are used  to recognize patterns and in many other solutions in the fields  of machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' ANNs consist of multiple connected  nodes (artificial neurons), with each neuron representing a  mathematical function [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' These nodes are divided into three  types of layers: the first (input) layer, at least one hidden layer,  and the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The connections between neurons in each  layer can be characterized by weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  In cryptography, the most commonly used neural network is  the tree parity machine (TPM) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' A scheme of this model  is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' There are K ×N input neurons, divided  into K groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' There is a single hidden layer with K nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Each of these nodes has N inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The TPM has a single  output neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The connections between input neurons and  hidden layer neurons are described by weights W – integers  in the range [−L, L], thus L is the maximum and −L is  the minimum weight value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The values of σ characterize the  connections between the hidden layer neurons and an output  neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The output value of the TPM is described by τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The value of σ is calculated using the following formulas:  σk = sgn(  N  �  n=1  xkn ∗ wkn)  (1)  sgn(z) =  �  −1  z ≤ 0  1  z > 0  (2)  Due to the usage of the presented signum function, σ can take  two values: 1 or −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The output value of TPM is calculated  as:  τ =  K  �  k=1  σk  (3)  This neural network has two possible outcomes: 1 or −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  For the TPM structure, multiple learning algorithms are  proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Most popular are Hebbian, anti-Hebbian, and ran-  dom walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The leading is the Hebbian rule [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The Hebbian  algorithm updates ANN weights in the following manner:  w∗  kn = vL(wkn + xkn ∗ σk ∗ θ(σk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' τ))  (4)  where θ limits the impact of hidden layer neurons whose value  was different than τ:  θ(σk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' τ) =  �  0  if σk ̸= τ  1  if σk = τ  (5)  The vL function makes sure that the new weights are kept  within the [−L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' L] range:  vL(z) =  �  �  �  �  �  −L  if z ≤ −L  z  if − L < z < L  L  if z ≥ L  (6)  The TPM structure allows for mutual learning of the two  neural networks [18],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' primarily based on updating weights  only when the outputs from both neural networks are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The input values are random and the same for both Alice and  Bob’s TPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Inputs are updated in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The security  of this process relies on the fact that cooperating TPMs can  achieve convergence significantly faster than Eve’s machine,  which can update weights less frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The TPM is most  commonly used in cryptography to exchange a secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This  usage is defined as neural cryptography [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alice and Bob  mutually synchronize their TPMs to achieve the same weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  After the synchronization process, these weights provide a  secure symmetric key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Error correction based on TPMs  TPMs can be utilized during the error correction process  in quantum cryptography [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The neural network’s task is to  correct all errors to achieve the same string of confidential bits  at both endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Firstly, Alice and Bob prepare their TPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The number of neurons in the hidden layer (K) and the number  of input neurons (N) is determined by Alice and passed on  to Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The value L must also be agreed between the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The keys achieved using the QKD protocol are changed into  integer values in the range [−L, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' These values are used  in the appropriate TPMs as weights between neurons in the  input layer and the hidden layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Since Alice’s string of bits  is similar to Bob’s (QBER is usually not high), the weights  in the created TPMs are almost synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' At this point,  Alice and Bob have constructed TPMs with the same structure  but with a few differences in the weight values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  After establishing the TPM structure and changing bits to  weights, the synchronization process starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It consists of mul-  tiple iterations, repeated until common weights are achieved  between Alice and Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' A single iteration starts from Alice  choosing the input string and computing the result using the  TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' After that, the generated input string is passed on to Bob,  who computes the output of his TPM using the received input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Then, the results are compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' If the outputs of both TPMs  match, the weights can be updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Otherwise, the process is  repeated with a different input string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  After an appropriate number of iterations, the TPMs are  synchronized and Alice and Bob can change the weights back  into a string of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The resulting bits are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However,  the privacy amplification process after error correction is still  recommended [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The reduction of the key protecting Alice  and Bob from information leakage is defined as [2]:  Z = log2L+12i  (7)  where i is the number of TPM iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  This usage of TPM is safer than the neural cryptography  solution, because weights are similar before the synchroniza-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore, significantly fewer iterations are required to  achieve convergence than the randomly initialized weights  in key establishing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is worth mentioning this  method of error correction is characterized by high efficiency,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' requires approximately 30% less iterations than Cascade  algorithm [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' ANALYSIS OF THE SYNCHRONIZATION PROCESS  The crucial decision regarding the error detection approach  based on TPMs is the number of iterations during the syn-  chronization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This value should be as low as possible  for security reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, it cannot be too low, since  neural networks will not be able to correct all errors in the  key otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is the user’s responsibility to select the  appropriate value for the error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The main objective  of the analysis is to determine the impact of various neural  network parameters on the synchronization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Another  goal is to provide a recommended number of iterations for  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  X11  X12  X13  X1N  X21  X22  X23  X2N  XK1  XK2  XK3  XKN  ∑  ∏  ∑  ∑  W11  W1N  W21  W2N  WKN= {-L, … ,L}  WK1  σK= {-1, 1}  σ2  σ1  τ={-1, 1}  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Model of tree parity machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Testbed  The experiments require an application to simulate the error  correction process based on artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  application for correcting errors arising in quantum key distri-  bution was written in Python and uses the NumPy package – a  library for scientific computing which provides fast operations  on arrays required by the TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The functions provided by  NumPy satisfy all necessary calculations to achieve neural  network convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Synchronization of TPMs is performed  over sockets to allow real-world usage of this tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  Hebbian learning algorithm for updating weights is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The developed application makes it possible to correct errors  in the keys using quantum key distribution protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The users  are also able to correct simulated keys with the chosen error  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It helps if users do not have strings of bits created by a  real QKD system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' An important feature of the tool is its ability  to select neural network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The user can personalize  the synchronization process, starting from the key length and  error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The least sufficient number of bits was used for  translation into a single integer (values of the weights must be  in the range [−L, L]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It was demonstrated that the number of  hidden neurons and the number of inputs depend on the chosen  key length and L value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore, users need to select these  parameters taking into account the requirements and needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  During the experiments the minimum number of returned  required iterations for a single TPM configuration was set  to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The maximum number of iterations was limited to  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Additionally, the maximum number of retries in a single  iteration was limited to 10 to speed up the simulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Finally, 1880 different scenarios were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' All possible  TPM configurations for key lengths varying between 100 and  700 with a 100 bit step are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Moreover, the data is  available for other keys with lengths varying between 128 and  352 with an 8 bit step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Between 350 and 500 synchronizations  were performed for each TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It was assumed that this  number of iterations is sufficient to achieve convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Recommended number of iterations  To obtain the recommended number of iterations of TPMs  for successful error correction, the sum of means and standard  deviations of the results was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The median and  variance values were calculated as well for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  full results are available online2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The selected part – the neural  network configurations where the key length equals 256 bits  with the recommended number of iterations – is presented in  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Histogram for number of iterations (TPM with a 256 bit key, N = 16,  K = 4, L = 4, QBER = 3%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  2Recommended numbers of iterations for 1880 different scenarios –  TPM structures and QBER values – are available from: http://kt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='agh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='pl/  ∼niemiec/ICC-2023 This is mainly based on possible key lengths which vary  between 128 and 500 bits with 4 bit steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Additionally, keys with lengths  between 500 and 700 with 100 bit steps are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  180  160  140  120  Count  100  80  60  40  20  0  [11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 69]  (69,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 127)  (127,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 185)  (185,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 243)  (243,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 301)  (301,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 359)  (359,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 400)  > 400  Number of iterationsTABLE I  RECOMMENDED NUMBER OF ITERATIONS FOR TPMS GENERATED FOR  256 BIT KEYS  Weights  range  {−L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' L}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='QBER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='[%]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='inputs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='to a single  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='hidden  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='neuron  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='[N]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='hidden  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='neurons  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='[K]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='Recommended  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='iterations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='154  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='51  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='179  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='59  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='188  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='25  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='218  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='71  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='33  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='309  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='39  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='325  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='43  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='97  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='450  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='496  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='301  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='176  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='125  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='554  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='701  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='483  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='264  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='152  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='609  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='772  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='542  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='302  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='164  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 2 shows the histogram of data gathered for a sin-  gle neural network configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The distribution is right-  skewed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The mean value is greater than the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is a  common characteristic for other tested TPM configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' If  the distribution is not positively skewed, it is symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The recommended number of iterations for the presented  configuration, according to Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' I, equals 302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is based on  the sum of the mean and standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' For all  presented TPM configurations, this sum gives an 84% chance  of successful synchronization, assuming a normal distribution  of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' For the right-skewed distribution, similar to the one  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 2, the probability of success is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  85-th percentile for the given set is equal to 276 – less than  the proposed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' In this case, after choosing the suggested  number of iterations the user has more than an 88% chance  of success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Knowing the lowest required number of iterations is im-  portant because it reduces the risk of a successful attack by  Eve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The attacker could create independent TPMs and try  to synchronize one of them with Alice or Bob’s machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  The recommended number of iterations increases the security  of this solution because Alice and Bob require far fewer  iterations to synchronize, compared to Alice (or Bob) and Eve  synchronizing using random weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Impact of TPM structures  The results of simulations allow us to analyze how TPM  structures affect the number of required iterations during the  synchronization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 3 shows the number of required  iterations depending on the K and N parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It shows  two different TPM configurations: one with a 144 bit key and  another with a 216 bit key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' These configurations were chosen  due to having a similar number of possible K and N pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  For a given key length, L value and error rate there is a limited  number of possible N and K values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The K value changes  in inverse proportion to the N value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' As presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  3 the speed of the TPM synchronization process depends on  the neural network structure (N and K values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The number  of required iterations increases alongside the higher number  of neurons in the hidden layer (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The trend is similar for  both presented TPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' After achieving a certain threshold,  the number of recommended iterations increases slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  results fit the logarithmic trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It means that after a  certain K value, increasing this parameter further does not  affect the synchronization speed as much as under a certain  threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Number of iterations for TPMs with 144 and 216 bit keys for different  K value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Other configurations of the selected TPMs were studied  based on the increasing error rate of the keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Two configura-  tions with 128 and 256 bit keys were tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The average of  every possible configuration of the recommended number of it-  erations was calculated for different QBER values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The results  are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' This confirms that a greater number  of errors results in a higher average number of recommended  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It confirms the applicability of TPMs to correct  errors emerging in quantum key distribution, where the error  rate should not be higher than a few percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore, the  eavesdropper needs more iterations to synchronize its TPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Additionally, it was verified that value L has an exponential  impact on the average recommended number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  data was gathered using a similar approach to the study with  180  160  Recommended number of iterations  140  120  100  144 bits  80  216 bits  60  40  20  0  6  8  9  10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  Number of hidden layer neurons (K)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Number of iterations for TPMs with 128 and 256 bit keys depended  on the QBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  the impact of QBER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The average recommended number of  iterations of each configuration for a given L was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 5 shows the exponential trend line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It is worth mentioning  that the impact of L value on the synchronization time is  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Number of iterations for TPMs with 128 and 256 bit keys dependent  on the L value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  It is the user’s responsibility to choose the best possible  configuration for a given key length and QBER value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The  analysis shows that the L value should be chosen carefully  since it exponentially affects the required number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Additionally, the choice of the K value should be made  with caution due to its logarithmic impact on the number of  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' SUMMARY  The analysis of the TPM synchronization process used for  error correction purposes was presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' It shows  that the parameters of the TPM structure have an impact on  the synchronization time and security of this error correction  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' However, different parameters of artificial neural  networks have different effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Therefore, users should be  aware of how to choose the configuration of neural networks  used to correct errors in a secure and efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' One of  the deciding factors which need to be selected is the number  of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The paper describes the recommended number  of iterations for different TPM structures and QBER values  to assist users in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' The numbers recommended by the  authors are as low as possible but with a high probability of  successful synchronization to ensure secure and efficient error  correction based on artificial neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  ACKNOWLEDGMENT  This work was supported by the ECHO project which has  received funding from the European Union’s Horizon 2020  research and innovation programme under the grant agreement  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 830943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Abidin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Swami, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Ramirez-As´ıs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Alvarado-Tolentino, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Maurya, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Hussain, “Quantum cryptography technique: A way  to improve security challenges in mobile cloud computing (mcc),”  Materials Today: Proceedings, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 51, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 508–514, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Niemiec, “Error correction in quantum cryptography based on  artificial neural networks,” Quantum Information Processing, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bennett and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Brassard, “Quantum cryptography: Public key dis-  tribution and coin tossing,” Theoretical Computer Science - TCS, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  175–179, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Ekert, “Quantum cryptography based on Bell’s theorem,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 661–663, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Khodr, “Evaluations of quantum bit error rate using the three stage  multiphoton protocol,” 2017 International Conference on Electrical and  Computing Technologies and Applications (ICECTA), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1–4, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bennett, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bessette, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Brassard, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Salvail, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Smolin, “Exper-  imental quantum cryptography,” Journal of Cryptology, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 3–28, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Mehic, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Niemiec, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Siljak, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Voznak, “Error reconcilia-  tion in quantum key distribution protocols,” Reversible Computation:  Extending Horizons of Computing: Selected Results of the COST Action  IC1405, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 222–236, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Buttler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Lamoreaux, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Torgerson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Nickel, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  Donahue, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Peterson, “Fast, efficient error reconciliation for  quantum cryptography,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' A, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Delfs and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Knebl, “Symmetric-key encryption,” Introduction to  Cryptography: Principles and Applications, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 11–31, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Panda, “Performance analysis of encryption algorithms for security,”  2016 International Conference on Signal Processing, Communication,  Power and Embedded System (SCOPES), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 278–284, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Umaparvathi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Varughese, “Evaluation of symmetric en-  cryption algorithms for manets,” 2010 IEEE International Conference  on Computational Intelligence and Computing Research, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1–3, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Wootters and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Zurek, “A single quantum cannot be cloned,”  Nature, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 802–803, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Brassard and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Salvail, “Secret-key reconciliation by public discus-  sion,” Advances in Cryptology, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 410–423, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Hopfield, “Artificial neural networks,” IEEE Circuits and Devices  Magazine, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 3–10, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [15] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Hadke and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Kale, “Use of neural networks in cryptography:  A review,” in 2016 World Conference on Futuristic Trends in Research  and Innovation for Social Welfare (Startup Conclave), 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [16] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Sarkar, “Secure exchange of information using artificial intelligence  and chaotic system guided neural synchronization,” Multimedia Tools  and Applications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 80, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1–31, 05 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Aleksandrov and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bashkov, “Factors affecting synchronization time  of tree parity machines in cryptography,” 2020 IEEE 2nd International  Conference on Advanced Trends in Information Theory (ATIT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 108–  112, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Metzler, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Kinzel, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Kanter, “Interacting neural networks,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' E, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 2555–2565, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [19] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Kinzel and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Kanter, “Neural cryptography,” Proceedings of the 9th  International Conference on Neural Information Processing, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 1351–  1354, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Bennett, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Brassard, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Robert, “Privacy amplification by public  discussion,” SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=', p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content=' 210–229, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
+page_content='  300  iterations  250  I number of i  200  150  128 bits  256 bits  100  50  0  1  2  3  QBER [%]400  300  200  128 bits  256 bits  100  0  2  3  4  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFJT4oBgHgl3EQfFCyj/content/2301.11440v1.pdf'}
diff --git a/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf b/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..ba58546d1303c1ff5e39827f8dd2671ed3aa6ebf
--- /dev/null
+++ b/MtFJT4oBgHgl3EQfzS0r/content/2301.11642v1.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9b6e90f0ee6cae3b5f4c76db62894a3e01a673b792d933d9b8dc274f1fb769cd
+size 573711
diff --git a/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/2301.13305v1.pdf.txt b/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/2301.13305v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..edda5ad47718299d799393ea26f534b5d254035e
--- /dev/null
+++ b/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/2301.13305v1.pdf.txt
@@ -0,0 +1,371 @@
+arXiv:2301.13305v1  [math.CO]  30 Jan 2023
+Graph-Codes
+Noga Alon ∗
+Abstract
+The symmetric difference of two graphs G1, G2 on the same set of vertices [n] =
+{1, 2, . . . , n} is the graph on [n] whose set of edges are all edges that belong to exactly
+one of the two graphs G1, G2. Let H be a fixed graph with an even (positive) number
+of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs
+on [n] containing no two members whose symmetric difference is a copy of H. Is it
+true that DH(n) = o(2(n
+2)) for any such H? We discuss this problem, compute the
+value of DH(n) up to a constant factor for stars and matchings, and discuss several
+variants of the problem including ones that have been considered in earlier work.
+1
+Introduction
+1.1
+The problem
+The symmetric difference of two graph G1 = (V, E1) and G2 = (V, E2) on the same set of
+vertices V is the graph (V, E1 ⊕ E2) where E1 ⊕ E2 is the symmetric difference between
+E1 and E2, that is, the set of all edges that belong to exactly one of the two graphs. Put
+V = [n] = {1, 2, . . . , n} and let H be a family of graphs on the set of vertices [n] which is
+closed under isomorphism. A collection of graphs F on [n] is called an H-(graph)-code if
+it contains no two members whose symmetric difference is a graph in H. For the special
+case that H contains all copies of a single graph H on [n] this is called an H-code. Here
+we are interested in the maximum possible cardinality of such codes for various families
+H. Let DH(n) denote this maximum, and let
+dH(n) = DH(n)
+2(n
+2)
+denote the maximum possible fraction of the total number of graphs on [n] in an H-code.
+If H consists of all graphs isomorphic to one graph H, we denote dH(n) by dH(n). Note
+that if H consists of all graphs with less than d edges, then DH(n) is simply the maximum
+∗Princeton University, Princeton, NJ, USA and Tel Aviv University, Tel Aviv, Israel.
+Email:
+nalon@math.princeton.edu. Research supported in part by NSF grant DMS-2154082 and by USA-Israel
+BSF grant 2018267.
+1
+
+possible cardinality of a binary code of length
+�n
+2
+�
+and minimum distance at least d. This
+motivates the terminology “graph-codes” used here.
+The case H = K where K is the family of all cliques is of particular interest. This case
+is motivated by a conjecture of Gowers raised in his blog post [8] in 2009 and is discussed
+briefly in the comments of that blog. If H consists of all graphs with independence number
+at most 2, then dH(n) ≥ 1/8 for all n ≥ 3, as shown by the family of all graphs on [n]
+containing a triangle on the set of vertices {1, 2, 3}. An interesting result of Ellis, Filmus
+and Friedgut [5], settling a conjecture of Simonovits and S´os, asserts that this is tight
+for all n ≥ 3. The corresponding result, that dH′(n) = 1/26 for all n ≥ 4, where H′ is
+the family of all graphs with independence number at most 3, is proved in [3]. A more
+systematic study of the parameters DH(n) and dH(n) for various families of graphs H
+appears in the recent paper [1]. The families H considered in this work include the family
+of all disconnected graphs, the family of all graphs that are not 2-connected, the family
+of all non-Hamiltonian graphs and the family of all graphs that contain or do not contain
+a spanning star. Additional families studied are all graphs that contain an induced or
+non-induced copy of a fixed graph T, or all graphs that do not contain such a subgraph.
+In this note we focus on the case that H consists of a single graph H and the case that
+H is the family of all cliques, or all cliques up to a prescribed size. Note that trivially,
+if every member of H has an odd number of edges then dH(n) ≥ 1
+2 as the family of all
+graphs on [n] with an even number of edges forms an H-code.
+This suggests the following intriguing question.
+Question 1.1. Let H be a family of graphs closed under isomorphism. Is it true that
+dH(n) tends to 0 as n tends to infinity if and only if H contains a graph with an even
+number of edges ? Equivalently: is it true that for any fixed graph H with an even number
+of edges, dH(n) tends to 0 as n tends to infinity ?
+We also study the linear variant of these problems, where the H-codes considered are
+restricted to linear subspaces, that is, to families of graphs on [n] closed under symmetric
+difference.
+1.2
+Results
+Recall that K is the family of all cliques. Let K(r) denote the set of all cliques on at most
+r vertices. Let K1,t denote the star with t edges and let Mt denote the matching of t edges.
+Theorem 1.2. For every positive integer k,
+dK1,2k(n) = Θk(1/nk) and
+dM1,2k(n) = Θk(1/nk).
+Proposition 1.3. For every integer r ≥ 1,
+dK(4r+3)(n) ≥ Ω( 1
+nr ).
+2
+
+Proposition 1.4. For the family K of all cliques, dK(n) ≥
+1
+2[n/2] .
+Proposition 1.5. Let H be a fixed graph obtained from two copies of a graph H′ by
+identifying the vertices of an independent set of H′. Then
+dH(n) ≤ |V (H)|
+n
+for all n ≥ |V (H)|.
+In particular, dH(n) tends to 0 as n tends to infinity.
+Remark:
+all lower bounds are proved by exhibiting proper colorings of the relevant
+Cayley graphs, and in all cases the constructed family is an affine space over Z2. Using
+a simple Ramsey-theoretic argument it is not difficult to show that for an affine space
+the maximum possible cardinality obtained is at most a fraction O(log log n/ log n) of all
+graphs on n vertices whenever the defining family contains a fixed graph with an even
+number of edges.
+Since all lower bounds are obtained by what may be called linear graph-codes one can
+study this separately, as done for standard error correcting codes. For the family of all
+cliques K we get here an exact result (strengthening the assertion of Proposition 1.4).
+Theorem 1.6. For any n ≥ 2, the minimum possible co-dimension of a linear space of
+graphs on n vertices that contains no member of K is exactly [n/2].
+2
+Proofs
+2.1
+Upper bounds
+For a family of graphs H and an integer n, the Cayley graph C(n, H) is the graph whose
+vertices are all graphs on the n vertices [n], where two are adjacent iff their symmetric
+difference is a member of H. This is clearly a Cayley graph over the elementary abelian
+2-group ZN
+2 with N =
+�n
+2
+�
+. The function DH(n) is just the independence number of this
+graph, dH(n) is the so called independence ratio. Since the graph C(n, H) is vertex tran-
+sitive, its independence ratio is exactly the reciprocal of its fractional chromatic number.
+In order to prove an upper bound of α for its independence ratio it suffices to exhibit a
+set S of vertices that contains no independent set of size larger than α|S|. This applies
+also to weighted sets of vertices, but we will not use weights here.
+Proof of Proposition 1.5: Let a + b denote the number of vertices of H′ where b is the
+size of its independent set so that H is obtained from two copies of H′ by identifying the
+vertices in this independent set. Thus the number of vertices of H is 2a + b. Consider
+the following set of m = ⌊(n − b)/a⌋ copies of H′ on subsets of the vertex set [n]. All of
+3
+
+them contain the same independent set on the vertices {n − b + 1, n − b + 2, . . . , n}, and
+the additional vertices of copy number i are the vertices (i − 1)a + 1, (i − 1)a + 2, . . . , ia},
+where 1 ≤ i ≤ m. Each of these copies can be viewed as a vertex of the Cayley graph
+C = C(n, {H}). Since the symmetric difference of every pair of such copies forms a copy
+of H, this set forms a clique of size m in C, implying that dH(n) ≤ 1
+m ≤ |V (H)|/n.
+□
+The proofs of Theorem 1.2 for stars and for matchings are very similar. We describe the
+proof for stars and briefly mention the modification needed for matchings. The upper
+bound in Theorem 1.2 for the star K1,1 is a special case of the result above (with H′ being
+a single edge). The upper bound for any prime k can be proved using the following result
+of Frankl and Wilson.
+Theorem 2.1 ([7]). Let p be a prime, and let a0, a1, . . . , ar be distinct residue classes
+modulo p. Let F be a family of subsets of [n] and suppose that |F| ≡ a0 mod p for all
+F ∈ F and that for every two distinct F1, F2 ∈ F, |F1∩F2| ≡ ai mod p for some 1 ≤ i ≤ r.
+Then |F| ≤ �r
+i=0
+�n
+i
+�
+.
+Suppose k is a prime, n ≥ 2k and consider the family G of all stars K1,2k−1 with
+center 1 and 2k − 1 leaves among the vertices {2, 3, . . . , n}. Thus |G| =
+� n−1
+2k−1
+�
+. If two
+such stars share exactly k − 1 common leaves then their symmetric difference is a copy of
+K1,2k. A subset of G which is independent in the Cayley graph C(n, K1,2k) corresponds to
+a collection of subsets of the set {2, 3, . . . , n}, each of size 2k − 1, where the intersection of
+no two of these subsets is of cardinality k−1. Therefore, each of these sets is of cardinality
+−1 modulo k and no intersection is of cardinality −1 modulo k. By the Frankl-Wilson
+Theorem (Theorem 2.1) the cardinality of such a family is at most �k−1
+i=0
+�n−1
+i
+�
+. Therefore,
+for every prime k,
+dK1,2k(n) ≤
+�k−1
+i=0
+�n−1
+i
+�
+� n−1
+2k−1
+�
+≤ Ok( 1
+nk ).
+In order to prove the upper bound for all k we need the following result of Frankl and
+F¨uredi.
+Theorem 2.2 ([6]). For every fixed positive integers ℓ > ℓ1 +ℓ2 there exist n0 = n0(ℓ) and
+dℓ > 0 so that for all n > n0, if F is a family of ℓ-subsets of [n] in which the intersection
+of each pair of distinct members is of cardinality either at least ℓ − ℓ1 or strictly smaller
+than ℓ2, then
+|F| ≤ dℓ · nmax{ℓ1,ℓ2}.
+Proof of Theorem 1.2, upper bound: The proof for stars is essentially identical to
+the one described above for prime k, using Theorem 2.2 instead of Theorem 2.1.
+Let
+G be the family of all stars K1,2k−1 with center 1 and 2k − 1 leaves among the vertices
+4
+
+{2, 3, . . . , n}. Thus |G| =
+� n−1
+2k−1
+�
+. If two such stars share exactly k − 1 common leaves
+then their symmetric difference is a copy of K1,2k. Therefore, by Theorem 2.2 above with
+ℓ = 2k−1, ℓ1 = ℓ2 = k−1, the maximum cardinality of a subset of G which is independent
+in the Cayley graph C(n, K1,2k) is at most some ck(n − 1)k−1 for all sufficiently large n.
+This supplies the required upper bound
+ck(n − 1)k
+|G|
+≤ Ok( 1
+nk ),
+for dK1,2k(n). The proof for matchings is similar, starting with the family of all subsets
+of cardinality 2k − 1 of a fixed matching of cardinality ⌊n/2⌋. The symmetric difference
+of any two matchings that share exactly k − 1 common edges is a copy of M2k. Thus the
+proof can proceed exactly as in the case of stars.
+□
+2.2
+Lower bounds
+In order to lower bound the independence number of a Cayley graph C = C(n, H) it
+suffices to upper bound its chromatic number. One way to do so is to assign to each edge
+e of the complete graph on [n] a vector ve ∈ Zr
+2 for some r, so that for every H ∈ H,
+�
+e∈E(H) ve ̸= 0, where the sum is computed in Zr
+2. Given these vectors, we can assign
+to each graph G on [n] the color �
+e∈E(G) ve (computed, of course, in Zr
+2). This is clearly
+a proper coloring of C by at most 2r colors. Note that the matrix whose columns are
+the
+�n
+2
+�
+vectors ve is the analogue of the parity-check matrix of a linear error correcting
+code in the traditional theory of codes, and the color defined above is the analogue of the
+syndrome of a word, see, e.g., [9] for more information about these basic notions.
+Proof of Theorem 1.2, lower bound: For stars, it suffices to show that the chromatic
+number of the Cayley graph C = C(n, K1,2k) is at most O(nk). Let s be the smallest
+integer so that 2s − 1 ≥ n. As shown by the columns of the parity check matrix of a
+BCH-code with designed distance 2k + 1 there is a collection S of 2s − 1 binary vectors
+of length r = ks so that no sum of at most 2k of them (in Zks
+2 ) is the zero vector. Fix a
+proper edge coloring c of Kn by n colors. For each edge e let ve be the vector number c(e)
+in S. This gives the desired lower bound for stars. For matchings we use essentially the
+same construction, starting with a (non-proper) edge coloring of Kn by n colors in which
+each color class forms a star.
+□
+Proof of Proposition 1.3, lower bound:
+As in the previous proof, but the initial
+edge-coloring now is defined by c(ij) = i for all i < j and the binary vectors selected
+are taken from the columns of the parity check matrix of a code with designed distance
+2r + 2. Let U be the set of vertices of a clique of size at least 2 and at most 4r + 3. Then
+U contains at least 1 and at most 2r + 1 vertices i for which there is an odd number of
+5
+
+vertices of U with index strictly larger than i. Therefore the sum of vectors corresponding
+to the edges of the clique on U is equal to a sum of at most 2r + 1 column vectors of the
+parity check matrix, which is nonzero.
+□
+Proof of Proposition 1.4, lower bound: This follows from the construction in the
+proof of Theorem 1.6 described in the next section.
+3
+Linear graph-codes
+Proof of Theorem 1.6: The theorem is equivalent to the statement that for all n ≥ 2
+the minimum possible r = r(n) so that there are graphs G1, . . . , Gr on the vertex set [n]
+such that every clique on a subset of cardinality at least 2 of [n] contains an odd number
+of edges of at least one graph Gi, is r = [n/2]. It clearly suffices to prove the upper bound
+for odd n (that imply the result for n − 1) and the lower bound for even n (implying the
+result for n + 1). The upper bound is described in what follows. Let n ≥ 3 be odd. Split
+the numbers [n − 1] = {1, 2, . . . , n − 1} into the (n − 1)/2 blocks Bi = {2i − 1, 2i} for
+1 ≤ i ≤ (n − 1)/2. Let Gi be the graph consisting of all edges of the n − 2i triangles with
+a common base Bi on the vertices Bi ∪ {j} for 2i < j ≤ n. Our family of graphs is the
+set of these (n − 1)/2 graphs Gi. Let K be an arbitrary clique on a subset A of at least
+2 vertices in [n]. If A contains a full block Bi for some i, then it contains exactly 2x + 1
+edges of Gi, where x is the cardinality of the intersection of A with {2i + 1, 2i + 2, . . . , n}.
+As this is odd for all x ≥ 0 we may assume that A contains no block Bi. In this case,
+let j be the second largest element in A (recall that |A| ≥ 2). Clearly j ≤ n − 1, hence
+it is contained in one of the blocks Bi. But in this case Gi contains exactly one edge
+of the clique K, completing the proof of the upper bound. Note that it is simple to give
+additional constructions with the same properties as any set of graphs that spans the same
+subspace as the graphs above will do. In particular, we can replace one of the graphs Gi
+by the complete graph Kn, which is the sum of all graphs Gi.
+To prove the lower bound assume n is even and let G1, . . . Gn/2−1 be a family of n/2−1
+graphs on [n]. We have to show that there is a clique on at least 2 vertices containing an
+even number of edges of each Gi. We show that in fact there is such a clique on an even
+number of vertices. To do so we apply the classical theorem of Chevalley and Warning
+(cf., e.g., [2] or [12]). Recall that it asserts that any system of polynomials with n variables
+over a finite field in which the number of variables exceeds the sum of the degrees, which
+admits a solution, must admit another one (in fact, the number of solutions is divisible by
+the characteristics). Associate each vertex i with a variable xi over Z2 and consider the
+following homogeneous system of polynomial equations over Z2. For each graph Gs in our
+6
+
+family,
+�
+ij∈E(Gs)
+xixj = 0.
+In addition, add the linear equation �n
+i=1 xi = 0.
+The sum of the degrees of the polynomials here is 2(n/2 − 1) + 1 = n − 1, which
+is smaller than the number of variables.
+Since the system is homogeneous it admits
+the trivial solution xi = 0 for all i. Any other solution (which exists by the Chevalley
+Warning Theorem) gives a clique on the set of vertices {i : xi = 1} which is nonempty, of
+even cardinality, and contains an even number of edges (possibly zero) of each Gi. This
+establishes the lower bound and completes the proof of Theorem 1.6.
+□
+4
+Concluding remarks and open problems
+• Question 1.1, which is equivalent to the problem of deciding whether or not for any
+fixed nonempty graph H with an even number of edges dH(n) tends to 0 as n tends
+to infinity, remains wide open.
+An interesting special case is whether or not dK4(n) = o(1). It is also interesting
+to decide whether or not dK4(n) ≥
+1
+no(1) . It is not difficult to show that the latter
+would follow from the existence (if true) of an edge coloring of Kn by no(1) colors
+with no copy of K4 in which every color appears an even number of times. This may
+be related to the construction in [10], see also [4].
+• Gowers conjectured in [8] that any family of a constant fraction of all graphs on [n],
+where n is sufficiently large, contains two graphs G1, G2 such that G2 is a subgraph
+of G1 and the symmetric difference of the two graphs (that is, the set of all edges of
+G1 that are not in G2) forms a clique. This is clearly stronger than the conjecture
+that dK(n) tends to 0 as n tends to infinity, which is also open. As explained in
+[8] the question of Gowers can be viewed as the first unknown case of a polynomial
+version of the density Hales-Jewett Theorem.
+• As mentioned in the remark following the statement of Proposition 1.5, it is not
+difficult to show that for every graph H with an even number of eges the maximum
+possible cardinality of a linear family of graphs on [n] in which no symmetric differ-
+ence is a copy of H, is o(2(n
+2)). As the proof applies Ramsey’s Theorem, it provides
+very weak bounds. It will be interesting to establish tighter bounds for the linear
+case. Theorem 1.6 provides an example of a tight result of this form.
+• The problem considered above can be extended to hypergraphs. More generally, it
+can be extended to other versions of problems about binary codes, where the coordi-
+nates of each codeword are indexed by the elements of some combinatorial structure,
+7
+
+and the forbidden symmetric differences correspond to a prescribed family of sub-
+structures. Here is an example of a problem of this type. What is the maximum
+possible cardinality of a collection of binary vectors whose coordinates are indexed
+by the elements of the ordered set [n], where no symmetric difference of two dis-
+tinct members of the collection forms an interval of length which is a cube of an
+integer? The corresponding Cayley graph here has 2n vertices, and it is triangle-free
+by Fermat’s last Theorem for cubes. Its independece number, which is the answer
+to the question above, is o(2n). Indeed, this follows from the Furstenberg-S´ark¨ozy
+Theorem and its extensions [11], by considering the maximum possible cardinality
+of an independent set in the induced subgraph on the set of all vertices that are
+characteristic vectors of an interval [i] = {1, . . . , i} for 0 ≤ i ≤ n.
+References
+[1] N. Alon, A. Gujgiczer, J. K¨orner, A. Milojevi´c and G. Simonyi, Structured codes of
+graphs, SIAM J. Discrete Math., to appear.
+[2] Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, New York,
+1966.
+[3] A. Berger and Y. Zhao, K4-intersecting families of graphs, arXiv:2103.12671, 2021.
+[4] D. Conlon, J. Fox, C. Lee and B. Sudakov, The Erd˝os-Gy´arf´as problem on generalized
+Ramsey numbers, Proc. London Math. Soc. 110 (2015), 1–18.
+[5] D. Ellis, Y. Filmus and E. Friedgut, Triangle-intersecting families of graphs, Journal
+of the European Mathematical Society 14 (2012), No. 3, 841–885.
+[6] P. Frankl and Z. F¨uredi, Forbidding just one intersection, J. Combin. Theory Ser. A
+39 (1985), no. 2, 160–176.
+[7] P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences,
+Combinatorica 1 (1981), 357–368.
+[8] W. T. Gowers, https://gowers.wordpress.com/2009/11/14/the-first-unknown-case-of-
+polynomial-dhj/
+[9] F. MacWilliams and N. Sloane, The Theory of Error-Correcting Codes, I. North-
+Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-
+New York-Oxford (1977).
+[10] D. Mubayi, Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18
+(1998), no. 2, 293–296.
+8
+
+[11] A. S´ark¨ozy, On difference sets of sequences of integers. III. Acta Math. Acad. Sci.
+Hungar. 31 (1978), no. 3-4, 355–386.
+[12] W. M. Schmidt, Equations over Finite Fields, an Elementary Approach, Springer
+Verlag Lecture Notes in Math., 1976.
+9
+
diff --git a/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/load_file.txt b/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8b9222293ef724e9f91cb9c1035a903c13ba1dab
--- /dev/null
+++ b/NtFQT4oBgHgl3EQfWjbh/content/tmp_files/load_file.txt
@@ -0,0 +1,301 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf,len=300
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='13305v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='CO]  30 Jan 2023  Graph-Codes  Noga Alon ∗  Abstract  The symmetric difference of two graphs G1, G2 on the same set of vertices [n] =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n} is the graph on [n] whose set of edges are all edges that belong to exactly  one of the two graphs G1, G2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a fixed graph with an even (positive) number  of edges, and let DH(n) denote the maximum possible cardinality of a family of graphs  on [n] containing no two members whose symmetric difference is a copy of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Is it  true that DH(n) = o(2(n  2)) for any such H?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We discuss this problem, compute the  value of DH(n) up to a constant factor for stars and matchings, and discuss several  variants of the problem including ones that have been considered in earlier work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1  The problem  The symmetric difference of two graph G1 = (V, E1) and G2 = (V, E2) on the same set of  vertices V is the graph (V, E1 ⊕ E2) where E1 ⊕ E2 is the symmetric difference between  E1 and E2, that is, the set of all edges that belong to exactly one of the two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Put  V = [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n} and let H be a family of graphs on the set of vertices [n] which is  closed under isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A collection of graphs F on [n] is called an H-(graph)-code if  it contains no two members whose symmetric difference is a graph in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the special  case that H contains all copies of a single graph H on [n] this is called an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Here  we are interested in the maximum possible cardinality of such codes for various families  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let DH(n) denote this maximum, and let  dH(n) = DH(n)  2(n  2)  denote the maximum possible fraction of the total number of graphs on [n] in an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  If H consists of all graphs isomorphic to one graph H, we denote dH(n) by dH(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note  that if H consists of all graphs with less than d edges, then DH(n) is simply the maximum  ∗Princeton University, Princeton, NJ, USA and Tel Aviv University, Tel Aviv, Israel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Email:  nalon@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='princeton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Research supported in part by NSF grant DMS-2154082 and by USA-Israel  BSF grant 2018267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1  possible cardinality of a binary code of length  �n  2  �  and minimum distance at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This  motivates the terminology “graph-codes” used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The case H = K where K is the family of all cliques is of particular interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This case  is motivated by a conjecture of Gowers raised in his blog post [8] in 2009 and is discussed  briefly in the comments of that blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If H consists of all graphs with independence number  at most 2, then dH(n) ≥ 1/8 for all n ≥ 3, as shown by the family of all graphs on [n]  containing a triangle on the set of vertices {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' An interesting result of Ellis, Filmus  and Friedgut [5], settling a conjecture of Simonovits and S´os, asserts that this is tight  for all n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The corresponding result, that dH′(n) = 1/26 for all n ≥ 4, where H′ is  the family of all graphs with independence number at most 3, is proved in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A more  systematic study of the parameters DH(n) and dH(n) for various families of graphs H  appears in the recent paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The families H considered in this work include the family  of all disconnected graphs, the family of all graphs that are not 2-connected, the family  of all non-Hamiltonian graphs and the family of all graphs that contain or do not contain  a spanning star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Additional families studied are all graphs that contain an induced or  non-induced copy of a fixed graph T, or all graphs that do not contain such a subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In this note we focus on the case that H consists of a single graph H and the case that  H is the family of all cliques, or all cliques up to a prescribed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that trivially,  if every member of H has an odd number of edges then dH(n) ≥ 1  2 as the family of all  graphs on [n] with an even number of edges forms an H-code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  This suggests the following intriguing question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a family of graphs closed under isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Is it true that  dH(n) tends to 0 as n tends to infinity if and only if H contains a graph with an even  number of edges ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Equivalently: is it true that for any fixed graph H with an even number  of edges, dH(n) tends to 0 as n tends to infinity ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  We also study the linear variant of these problems, where the H-codes considered are  restricted to linear subspaces, that is, to families of graphs on [n] closed under symmetric  difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2  Results  Recall that K is the family of all cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K(r) denote the set of all cliques on at most  r vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K1,t denote the star with t edges and let Mt denote the matching of t edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every positive integer k,  dK1,2k(n) = Θk(1/nk) and  dM1,2k(n) = Θk(1/nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every integer r ≥ 1,  dK(4r+3)(n) ≥ Ω( 1  nr ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  2  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the family K of all cliques, dK(n) ≥  1  2[n/2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let H be a fixed graph obtained from two copies of a graph H′ by  identifying the vertices of an independent set of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Then  dH(n) ≤ |V (H)|  n  for all n ≥ |V (H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In particular, dH(n) tends to 0 as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Remark:  all lower bounds are proved by exhibiting proper colorings of the relevant  Cayley graphs, and in all cases the constructed family is an affine space over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Using  a simple Ramsey-theoretic argument it is not difficult to show that for an affine space  the maximum possible cardinality obtained is at most a fraction O(log log n/ log n) of all  graphs on n vertices whenever the defining family contains a fixed graph with an even  number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Since all lower bounds are obtained by what may be called linear graph-codes one can  study this separately, as done for standard error correcting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For the family of all  cliques K we get here an exact result (strengthening the assertion of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For any n ≥ 2, the minimum possible co-dimension of a linear space of  graphs on n vertices that contains no member of K is exactly [n/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  2  Proofs  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1  Upper bounds  For a family of graphs H and an integer n, the Cayley graph C(n, H) is the graph whose  vertices are all graphs on the n vertices [n], where two are adjacent iff their symmetric  difference is a member of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly a Cayley graph over the elementary abelian  2-group ZN  2 with N =  �n  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The function DH(n) is just the independence number of this  graph, dH(n) is the so called independence ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Since the graph C(n, H) is vertex tran-  sitive, its independence ratio is exactly the reciprocal of its fractional chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In order to prove an upper bound of α for its independence ratio it suffices to exhibit a  set S of vertices that contains no independent set of size larger than α|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This applies  also to weighted sets of vertices, but we will not use weights here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5: Let a + b denote the number of vertices of H′ where b is the  size of its independent set so that H is obtained from two copies of H′ by identifying the  vertices in this independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus the number of vertices of H is 2a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Consider  the following set of m = ⌊(n − b)/a⌋ copies of H′ on subsets of the vertex set [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' All of  3  them contain the same independent set on the vertices {n − b + 1, n − b + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}, and  the additional vertices of copy number i are the vertices (i − 1)a + 1, (i − 1)a + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , ia},  where 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Each of these copies can be viewed as a vertex of the Cayley graph  C = C(n, {H}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Since the symmetric difference of every pair of such copies forms a copy  of H, this set forms a clique of size m in C, implying that dH(n) ≤ 1  m ≤ |V (H)|/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  The proofs of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 for stars and for matchings are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We describe the  proof for stars and briefly mention the modification needed for matchings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper  bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 for the star K1,1 is a special case of the result above (with H′ being  a single edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper bound for any prime k can be proved using the following result  of Frankl and Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let p be a prime, and let a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , ar be distinct residue classes  modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let F be a family of subsets of [n] and suppose that |F| ≡ a0 mod p for all  F ∈ F and that for every two distinct F1, F2 ∈ F, |F1∩F2| ≡ ai mod p for some 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Then |F| ≤ �r  i=0  �n  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Suppose k is a prime, n ≥ 2k and consider the family G of all stars K1,2k−1 with  center 1 and 2k − 1 leaves among the vertices {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus |G| =  � n−1  2k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If two  such stars share exactly k − 1 common leaves then their symmetric difference is a copy of  K1,2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A subset of G which is independent in the Cayley graph C(n, K1,2k) corresponds to  a collection of subsets of the set {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}, each of size 2k − 1, where the intersection of  no two of these subsets is of cardinality k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore, each of these sets is of cardinality  −1 modulo k and no intersection is of cardinality −1 modulo k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' By the Frankl-Wilson  Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1) the cardinality of such a family is at most �k−1  i=0  �n−1  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore,  for every prime k,  dK1,2k(n) ≤  �k−1  i=0  �n−1  i  �  � n−1  2k−1  �  ≤ Ok( 1  nk ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In order to prove the upper bound for all k we need the following result of Frankl and  F¨uredi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For every fixed positive integers ℓ > ℓ1 +ℓ2 there exist n0 = n0(ℓ) and  dℓ > 0 so that for all n > n0, if F is a family of ℓ-subsets of [n] in which the intersection  of each pair of distinct members is of cardinality either at least ℓ − ℓ1 or strictly smaller  than ℓ2, then  |F| ≤ dℓ · nmax{ℓ1,ℓ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2, upper bound: The proof for stars is essentially identical to  the one described above for prime k, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 instead of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Let  G be the family of all stars K1,2k−1 with center 1 and 2k − 1 leaves among the vertices  4  {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus |G| =  � n−1  2k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If two such stars share exactly k − 1 common leaves  then their symmetric difference is a copy of K1,2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2 above with  ℓ = 2k−1, ℓ1 = ℓ2 = k−1, the maximum cardinality of a subset of G which is independent  in the Cayley graph C(n, K1,2k) is at most some ck(n − 1)k−1 for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  This supplies the required upper bound  ck(n − 1)k  |G|  ≤ Ok( 1  nk ),  for dK1,2k(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The proof for matchings is similar, starting with the family of all subsets  of cardinality 2k − 1 of a fixed matching of cardinality ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The symmetric difference  of any two matchings that share exactly k − 1 common edges is a copy of M2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Thus the  proof can proceed exactly as in the case of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2  Lower bounds  In order to lower bound the independence number of a Cayley graph C = C(n, H) it  suffices to upper bound its chromatic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' One way to do so is to assign to each edge  e of the complete graph on [n] a vector ve ∈ Zr  2 for some r, so that for every H ∈ H,  �  e∈E(H) ve ̸= 0, where the sum is computed in Zr  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Given these vectors, we can assign  to each graph G on [n] the color �  e∈E(G) ve (computed, of course, in Zr  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly  a proper coloring of C by at most 2r colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that the matrix whose columns are  the  �n  2  �  vectors ve is the analogue of the parity-check matrix of a linear error correcting  code in the traditional theory of codes, and the color defined above is the analogue of the  syndrome of a word, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', [9] for more information about these basic notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='2, lower bound: For stars, it suffices to show that the chromatic  number of the Cayley graph C = C(n, K1,2k) is at most O(nk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let s be the smallest  integer so that 2s − 1 ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As shown by the columns of the parity check matrix of a  BCH-code with designed distance 2k + 1 there is a collection S of 2s − 1 binary vectors  of length r = ks so that no sum of at most 2k of them (in Zks  2 ) is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Fix a  proper edge coloring c of Kn by n colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For each edge e let ve be the vector number c(e)  in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This gives the desired lower bound for stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For matchings we use essentially the  same construction, starting with a (non-proper) edge coloring of Kn by n colors in which  each color class forms a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='3, lower bound:  As in the previous proof, but the initial  edge-coloring now is defined by c(ij) = i for all i < j and the binary vectors selected  are taken from the columns of the parity check matrix of a code with designed distance  2r + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let U be the set of vertices of a clique of size at least 2 and at most 4r + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Then  U contains at least 1 and at most 2r + 1 vertices i for which there is an odd number of  5  vertices of U with index strictly larger than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Therefore the sum of vectors corresponding  to the edges of the clique on U is equal to a sum of at most 2r + 1 column vectors of the  parity check matrix, which is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='4, lower bound: This follows from the construction in the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6 described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  3  Linear graph-codes  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6: The theorem is equivalent to the statement that for all n ≥ 2  the minimum possible r = r(n) so that there are graphs G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , Gr on the vertex set [n]  such that every clique on a subset of cardinality at least 2 of [n] contains an odd number  of edges of at least one graph Gi, is r = [n/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It clearly suffices to prove the upper bound  for odd n (that imply the result for n − 1) and the lower bound for even n (implying the  result for n + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The upper bound is described in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let n ≥ 3 be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Split  the numbers [n − 1] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n − 1} into the (n − 1)/2 blocks Bi = {2i − 1, 2i} for  1 ≤ i ≤ (n − 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let Gi be the graph consisting of all edges of the n − 2i triangles with  a common base Bi on the vertices Bi ∪ {j} for 2i < j ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Our family of graphs is the  set of these (n − 1)/2 graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Let K be an arbitrary clique on a subset A of at least  2 vertices in [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' If A contains a full block Bi for some i, then it contains exactly 2x + 1  edges of Gi, where x is the cardinality of the intersection of A with {2i + 1, 2i + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  As this is odd for all x ≥ 0 we may assume that A contains no block Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' In this case,  let j be the second largest element in A (recall that |A| ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Clearly j ≤ n − 1, hence  it is contained in one of the blocks Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' But in this case Gi contains exactly one edge  of the clique K, completing the proof of the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Note that it is simple to give  additional constructions with the same properties as any set of graphs that spans the same  subspace as the graphs above will do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' In particular, we can replace one of the graphs Gi  by the complete graph Kn, which is the sum of all graphs Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  To prove the lower bound assume n is even and let G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Gn/2−1 be a family of n/2−1  graphs on [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We have to show that there is a clique on at least 2 vertices containing an  even number of edges of each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' We show that in fact there is such a clique on an even  number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' To do so we apply the classical theorem of Chevalley and Warning  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', [2] or [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Recall that it asserts that any system of polynomials with n variables  over a finite field in which the number of variables exceeds the sum of the degrees, which  admits a solution, must admit another one (in fact, the number of solutions is divisible by  the characteristics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Associate each vertex i with a variable xi over Z2 and consider the  following homogeneous system of polynomial equations over Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' For each graph Gs in our  6  family,  �  ij∈E(Gs)  xixj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  In addition, add the linear equation �n  i=1 xi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The sum of the degrees of the polynomials here is 2(n/2 − 1) + 1 = n − 1, which  is smaller than the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Since the system is homogeneous it admits  the trivial solution xi = 0 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Any other solution (which exists by the Chevalley  Warning Theorem) gives a clique on the set of vertices {i : xi = 1} which is nonempty, of  even cardinality, and contains an even number of edges (possibly zero) of each Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This  establishes the lower bound and completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  □  4  Concluding remarks and open problems  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='1, which is equivalent to the problem of deciding whether or not for any  fixed nonempty graph H with an even number of edges dH(n) tends to 0 as n tends  to infinity, remains wide open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  An interesting special case is whether or not dK4(n) = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It is also interesting  to decide whether or not dK4(n) ≥  1  no(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It is not difficult to show that the latter  would follow from the existence (if true) of an edge coloring of Kn by no(1) colors  with no copy of K4 in which every color appears an even number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This may  be related to the construction in [10], see also [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Gowers conjectured in [8] that any family of a constant fraction of all graphs on [n],  where n is sufficiently large, contains two graphs G1, G2 such that G2 is a subgraph  of G1 and the symmetric difference of the two graphs (that is, the set of all edges of  G1 that are not in G2) forms a clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' This is clearly stronger than the conjecture  that dK(n) tends to 0 as n tends to infinity, which is also open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As explained in  [8] the question of Gowers can be viewed as the first unknown case of a polynomial  version of the density Hales-Jewett Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  As mentioned in the remark following the statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='5, it is not  difficult to show that for every graph H with an even number of eges the maximum  possible cardinality of a linear family of graphs on [n] in which no symmetric differ-  ence is a copy of H, is o(2(n  2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' As the proof applies Ramsey’s Theorem, it provides  very weak bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' It will be interesting to establish tighter bounds for the linear  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='6 provides an example of a tight result of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  The problem considered above can be extended to hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' More generally, it  can be extended to other versions of problems about binary codes, where the coordi-  nates of each codeword are indexed by the elements of some combinatorial structure,  7  and the forbidden symmetric differences correspond to a prescribed family of sub-  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Here is an example of a problem of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' What is the maximum  possible cardinality of a collection of binary vectors whose coordinates are indexed  by the elements of the ordered set [n], where no symmetric difference of two dis-  tinct members of the collection forms an interval of length which is a cube of an  integer?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' The corresponding Cayley graph here has 2n vertices, and it is triangle-free  by Fermat’s last Theorem for cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Its independece number, which is the answer  to the question above, is o(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Indeed, this follows from the Furstenberg-S´ark¨ozy  Theorem and its extensions [11], by considering the maximum possible cardinality  of an independent set in the induced subgraph on the set of all vertices that are  characteristic vectors of an interval [i] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' , i} for 0 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Alon, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Gujgiczer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' K¨orner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Milojevi´c and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Simonyi, Structured codes of  graphs, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Discrete Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [2] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Borevich and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Shafarevich, Number Theory, Academic Press, New York,  1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Berger and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Zhao, K4-intersecting families of graphs, arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='12671, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Conlon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Fox, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Lee and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Sudakov, The Erd˝os-Gy´arf´as problem on generalized  Ramsey numbers, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 110 (2015), 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Ellis, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Filmus and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Friedgut, Triangle-intersecting families of graphs, Journal  of the European Mathematical Society 14 (2012), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 3, 841–885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [6] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Frankl and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' F¨uredi, Forbidding just one intersection, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Theory Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' A  39 (1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 2, 160–176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Frankl and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Wilson, Intersection theorems with geometric consequences,  Combinatorica 1 (1981), 357–368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Gowers, https://gowers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='com/2009/11/14/the-first-unknown-case-of-  polynomial-dhj/  [9] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' MacWilliams and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Sloane, The Theory of Error-Correcting Codes, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' North-  Holland Mathematical Library, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' North-Holland Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', Amsterdam-  New York-Oxford (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Mubayi, Edge-coloring cliques with three colors on all 4-cliques, Combinatorica 18  (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 2, 293–296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  8  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' S´ark¨ozy, On difference sets of sequences of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 31 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' 3-4, 355–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=' Schmidt, Equations over Finite Fields, an Elementary Approach, Springer  Verlag Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content=', 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
+page_content='  9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NtFQT4oBgHgl3EQfWjbh/content/2301.13305v1.pdf'}
diff --git a/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/2301.00835v1.pdf.txt b/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/2301.00835v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..59514955ae785c5c1d9030ce8c64638fb082cd96
--- /dev/null
+++ b/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/2301.00835v1.pdf.txt
@@ -0,0 +1,3589 @@
+Timed Model-Based Mutation Operators for Simulink Models
+Jian Chen* 1 Manar H. Alalfi 2 Thomas R. Dean 3
+1 Department of Electrical and Computer Engineering, Queen’s University
+Kingston, ON, Canada
+E-mail: jian.chen@queensu.ca
+2 Department of Computer Science, Ryerson University
+Toronto, ON, Canada
+E-mail: manar.alalfi@cs.ryerson.ca
+3 Department of Electrical and Computer Engineering, Queen’s University
+Kingston, ON, Canada
+E-mail: tom.dean@queensu.ca
+Abstract
+Model-based mutation analysis is a recent research area, and real-time system testing can benefit from
+using model mutants. Model-based mutation testing (MBMT) is a particular branch of model-based test-
+ing. It generates faulty versions of a model using mutation operators to evaluate and improve test cases.
+Mutation testing is an effective way to ensure software correctness and has been applied to various appli-
+cation areas. Simulink is a vital modeling language for real-time systems. This paper introduces Simulink
+model mutation analysis to improve Model-in-the-loop (MIL) testing. We propose a set of Simulink mu-
+tation operators based on AUTOSAR, which reflects the temporal correctness when a Simulink model
+is mapped to Operating System tasks. We implement a mutation framework that generates mutants for
+implicit clock Simulink models. Finally, we demonstrate how this framework generates mutants to reveal
+task interference issues in the simulation. Our work integrates the Simulink model with the timed systems
+to better support mutation testing automation.
+Keywords: Mutation Testing, Model-Based Testing, Model-Based Mutation Testing, Mutation Operator,
+Simulink, Real-Time System, Scheduling, AUTOSAR
+1.
+Introduction
+Today, cars come equipped with advanced technolo-
+gies that did not exist before, such as Automatic
+Emergency Braking (AEB), Adaptive Cruise Con-
+trol (ACC), Lane Departure Warning/Lane Keeping,
+and Autonomous driving. All of these features rely
+on software to realize sophisticated control algo-
+rithms. Generally, such software is developed within
+the timed system context, in which the system cor-
+rectness not only relies on the software implemented
+functions correctness but also depends on the sys-
+tem to meet time constraints. Many factors can con-
+tribute to the execution time of a system running on
+a target platform. Issues such as task interference
+may cause delays during task execution. Software
+quality plays a crucial role in such safety-critical ap-
+plications.
+Model-Based Testing (MBT) is a promising
+technique for the automated testing of timed sys-
+arXiv:2301.00835v1  [cs.SE]  2 Jan 2023
+
+J. Chen et al. / Mutation Operators for Simulink Models
+tems. A model represents the behavior of software,
+and the model is usually abstracted from real-time
+specifications. However, some modeling environ-
+ments support this feature in the Hardware-in-the-
+loop (HIL) simulation testing instead of the MIL.
+For example, Matlab/Simulink (ML/SL) simulations
+assume block behaviors are completed in nearly zero
+execution time, while real execution requires a finite
+execution time, which may cause a failure. ML/SL
+models are based on the Synchronous Reactive (SR)
+model 23 that may assume the task execution times
+are zero. Errors in the model may not be apparent
+without an explicit real-time execution in the MIL
+phase. Usually, a Simulink model can be well simu-
+lated in the MIL, but it may have errors in the real-
+time context.
+Hence, MBT needs an extension to accommo-
+date the real-time context, which includes modeling
+the system through a timed formalism, and check-
+ing the implementation conforms to its specification.
+Traditionally, this is done via conformance checks
+35. Recently, several tools have been proposed to
+simulate the real-time execution effects for ML/SL
+models in MIL, such as TrueTime 21, TRES 12,
+Timing-aware blocks 27, and SimSched 9. SimSched
+uses a model transformation to integrate scheduling
+into the model to validate the real-time context dur-
+ing simulation. To evaluate SimSched, we turn to
+mutation testing using mutation analysis to assist the
+evaluation of the SimSched tool.
+In this paper, we propose a set of mutation op-
+erators with a timed task model, which is based
+on the AUTomotive Open System ARchitecture
+(AUTOSAR), that reflects the temporal correctness
+when a Simulink model is mapped to Real-Time Op-
+erating System (RTOS) tasks in a real-time context.
+This paper is organized as follows:
+Section
+2 introduces background information.
+Section 3
+presents the set of proposed timed mutation oper-
+ators for Simulink models. Section 4 explains the
+usage of the timed mutation operators. Section 5
+presents validation experiments and results. Section
+6 summarizes related studies in MBT. Finally, Sec-
+tion 7 presents the conclusions of our work and out-
+lines future work.
+2.
+Background
+This section gives an overview of the background
+information on the material needed to explain our
+work. We begin with a basic introduction to mu-
+tation testing, Simulink, and AUTOSAR; then, we
+present our timed task model.
+2.1.
+Mutation testing
+Mutation testing was introduced in the 1970s 17,13,22
+and proved to be an effective way to reveal software
+faults 32. It is a fault-based software testing tech-
+nique, which has been extensively studied and used
+for decades. It contributes a range of methods, tools,
+and reliable results for software testing. Mutation
+testing is designed to find valid test cases and dis-
+cover real errors in the program.
+Model-Based Mutation Testing (MBMT) takes
+the advantages of both model-based testing and mu-
+tation testing and has been widely applied to mul-
+tiple types of models such as feature models 20,
+statechart-based models 41,1, timed automata 3,2, and
+Simulink 8,26,38. However, in real-time system de-
+velopment, both logical and temporal correctness is
+crucial to the correct system functionality. The tem-
+poral correctness depends on timing assumptions for
+each task. Timed Automata (TA) 4 is a common for-
+malism to model and verify real-time systems to see
+whether designs meet temporal requirements. Aich-
+ernig et al.3 propose an MBMT technique for timed
+automata that applies to input/output timed automata
+(TAIO) model. Nilsson et al.
+28 add an extension
+to the TA formalism with a task model, and their
+mutation operators focus on timeliness. Simulink∗is
+widely used for model-driven development of soft-
+ware within the automotive sector. Most of the mu-
+tation operators proposed for Simulink models are
+from a property point of view either run-time or
+design-time such as signal modification, arithmetic
+alternation, or block change 18,38,36,43. Some of the
+proposed mutation testings are targeted at test case
+generation for Simulink models 8,19. However, there
+is no mutation operator with an explicit clock model
+for Simulink.
+* https://www.mathworks.com/products/simulink.html
+
+J. Chen et al. / Mutation Operators for Simulink Models
+2.2.
+Simulink
+Simulink is one of the most popular modeling lan-
+guages for modeling dynamical systems, and MAT-
+LAB provides a graphical programming environ-
+ment to perform system simulations. Simulink mod-
+els are graphical blocks and lines, and they are con-
+nected by signals between input and output ports.
+The Simulink simulation engine determines the ex-
+ecution order of blocks based on the data depen-
+dencies among the blocks before a simulation exe-
+cution. Simulink defines two types of blocks, di-
+rect feedthrough, and non-direct feedthrough, to as-
+sure the correct data dependencies in the simula-
+tion. Simulink uses the following two basic rules
+25 to determine the sorted execution order: A block
+must be executed before any of the blocks whose
+direct-feedthrough ports it drives; Blocks without
+direct feedthrough inputs can execute in arbitrary or-
+der as long as they precede any block whose direct-
+feedthrough inputs they drive. All blocks are sched-
+uled in sorted order and executed in sequential exe-
+cution order. The Simulink engine maintains a vir-
+tual clock to execute each ordered block at each vir-
+tual time.
+Simulink Coder†supports code generation and of-
+fers a framework to execute the generated code in
+a real-time environment. Simulink Coder can gen-
+erate code for the periodic task, either using a sin-
+gle task or a multi-task. Single-task implementa-
+tions can preserve the semantics during the simula-
+tion because the generated code is invoked by a sim-
+ple scheduler in a single thread without preemptions.
+For multi-task implementations, the generated code
+is invoked by a rate monotonic (RM) 24 scheduler
+in a multithreaded RTOS environment, where each
+task is assigned a priority and preemptions occur be-
+tween tasks. As a consequence of preemption and
+scheduling, the implementation semantic can con-
+flict with the model semantic in a multi-rate system.
+2.3.
+AUTOSAR
+AUTOSAR is an open industry standard to meet the
+needs of future car development. AUTOSAR de-
+fines three main layers: the application, the runtime
+environment (RTE), and the basic software (BSW)
+layer 40. The functions in the application layer are
+implemented by SW-Cs, which encapsulate part or
+all of the automotive electronic functions, as shown
+in Figure 1. The components communicate via a
+Virtual Functional Bus (VFB), which is an abstrac-
+tion of all the communication mechanisms of AU-
+TOSAR. Engineers abstract the communication de-
+tails of software components employing VFBs. A
+set of runnables represents the SW-Cs internal be-
+haviors, and a runnable is the smallest executable
+code that can be individually scheduled, either by
+a timer or an event. Lastly, runnables are required
+to map to a set of tasks for a target platform, and
+the mapping has to preserve ordering relations and
+causal dependencies. Simulink has supported AU-
+TOSAR compliant code generation since version
+R2006a‡.
+All AUTOSAR concepts can be repre-
+sented by Simulink blocks and the existing Simulink
+blocks can be easily used in the AUTOSAR devel-
+opment process. Some of AUTOSAR concepts and
+Simulink concepts mapping relation is shown in Ta-
+ble 1 39.
+Fig. 1. AUTOSAR components, interfaces, and runnables.
+(Adapted from 5)
+† https://www.mathworks.com/products/simulink-coder.html
+‡ https://www.mathworks.com/products/simulink.html
+
+SW-C 1
+SW-C 2
+SW-C 3
+SW-C n
+Runnable 2a
+Runnable 3a
+Runnable na
+Runnable 1a 
+！
+Runnable 2b
+Runnable nb
+NTOIA
+V
+OIDID
+Virtual Function Bus(VFB)
+Tool supporting deployment
+System
+ECU
+of sW components
+Constraints
+Descriptions
+个
+ECU I
+ECU II
+ECU M
+SW-C 1
+SW-C 3
+SW-C 2
+SW-C n
+RTE
+RTE
+RTE
+Basic Software
+Basic Software
+Basic SoftwareJ. Chen et al. / Mutation Operators for Simulink Models
+Table 1. Examples of ML/SL and AUTOSAR Concepts Map-
+ping.
+ML/SL
+AUTOSAR
+Subsystem
+Atomic Software
+Component
+Function-call subsystem
+Runnable
+Function calls
+RTEEvents
+2.4.
+Task model
+In automotive software, Simulink models are often
+drawn from real-time specifications and are realized
+as a set of tasks running on an RTOS. In order to
+better test this kind of software in the MIL phase,
+model-based testing needs to be scaled to the real-
+time context, which includes a timed formalism to
+model the system under test conforming with the
+real-time requirements. We define a task model to
+model the timing properties of tasks in the Simulink
+environment and the application is modeled as a set
+of periodic tasks.
+A task model, T, is represented by a tuple
+{φ,ρ,c,γ, prect, precr, prio, jitter}, where φ is an
+offset of the task, ρ is the period of the task, c is the
+Worst Case Execution Time (WCET) of the task, γ
+is a list of runnables that belong to the task, prect is
+the precedence constraint of the task, precr is the
+precedence constraint of the runnables within the
+task, prio is the priority associated with the task,
+and jitter is the deviation of a task from the periodic
+release times. Every task has an implicit deadline
+which means the deadline of a task is equal to ρ. An
+offset φ refers to the time delay between the arrival
+of the first instance of a periodic task and its release
+time. A WCET is the summation of each runnable
+execution time. A precedence constraint prect is a
+list of tasks that specifies the task execution order,
+and precr is a list of runnables within a task.
+Fig. 2. Task states and transitions of task model.
+Figure 2 shows the task-state and transition dia-
+grams of the task model that is based on OSEK’s ba-
+sic task-state model. The task model includes three
+states: suspended, ready, and running, and four tran-
+sitions: Active, Stare, Preempt, and Terminate. The
+transitions represent the actions to activate, start,
+preempt, or terminate a task.
+Fig. 3. Task timing parameters shown in Gantt chart (all
+related to Task2).
+Figure 3 shows the timing parameters of a task
+model and different timing parameters can alter the
+application’s real-time behavior within a system.
+3.
+Mutation Operators for Simulink Model
+This section introduces a mutation analysis ap-
+proach to validate real-time context during the sim-
+ulation. Mutation operators are the key elements of
+mutation testing. Model-based mutation testing is
+a method of injecting faults into models to check
+whether the tests are passed or failed, thus validat-
+ing the software. The injecting faults are the muta-
+tion operators. Before we apply mutation operators
+to the model, we need to identify them which is to
+
+Start
+Ready
+Running
+Preempt
+Active
+Terminate
+SuspendedPriority
+Active
+Start
+Preempt
+Terminate
+Task1
+Task2
+offseti
+C1
+C2
+jitter
+period
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+understand what kind of errors can cause failure. We
+have proposed the following task-related mutation
+operators.
+3.1.
+Offset mutation operators
+The task release offset is one of the factors that affect
+the computation result in terms of task interference.
+In order to take the offset into account for analy-
+sis, we introduced an offset mutation operator. For a
+known offset φ, a task can now execute after φ time
+units with respect to the start of its period. The exe-
+cution time of the task is unchanged at c time units
+before the next period starts.
+3.1.1. mITO
+This operator adds δ time to the current offset. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the offset φi to
+φi +δ.
+3.1.2. mDTO
+This operator subtracts δ time to the current offset.
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the offset φi to
+φi −δ.
+3.2.
+Period mutation operators
+An RTOS usually applies a preemptive multitask-
+ing scheduling algorithm to determine the execu-
+tion order of tasks, and the most picked algorithm is
+fixed-priority scheduling (FPS). The algorithm as-
+signs each task a static priority level. The RTOS
+scheduler executes the highest priority task from the
+ready task queue. Simulink Coder supports an RM
+scheduler, where the priority of a task is associated
+with its period, if a task has a smaller period, then it
+has a higher priority. Furthermore, a lower-priority
+task can be preempted by a more top-priority task
+during the execution.
+3.2.1. mITPER
+This operator increases the period of a task, which
+changes the task to a slower rate. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the period of the task i
+to ρi +δ.
+3.2.2. mDTPER
+This operator decreases the period of a task, which
+changes the task to a faster rate. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the period of the task i
+to ρi −δ.
+3.3.
+Execution time mutation operators
+The correctness of a real-time system is determined
+on one hand by the computation results of the log-
+ical program, and on the other hand, is strictly re-
+lated to the time at which the results are produced.
+Hence, execution time analysis is essential during
+the process of designing and verifying embedded
+systems. For this reason, we propose execution time
+operators, which can adjust the execution time of
+each task at the runnable level to simulate the run
+time execution on different processor speeds. The
+longer execution time of a task may lead to a sce-
+nario where a lower-rate task blocks a higher-rate
+task so that it misses its deadline.
+3.3.1. mITET
+This operator adds δ time to the current exe-
+cution time of each runnable, which increases
+the total execution time.
+For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the execution time ci to
+ci +δ.
+3.3.2.
+mDTET
+This operator subtracts δ
+time from the cur-
+rent execution time of each runnable, which de-
+creases the total execution time. For a given task
+τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this
+mutation operator changes the execution time ci to
+ci −δ.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+3.4.
+Execution precedence mutation operators
+The RTOS scheduler selects tasks to execute accord-
+ing to the priority level of the task. However, the
+spawn order determines the execution order of tasks
+with the same priority. Whichever task is spawned
+first is realized and gets the CPU first to run. This re-
+sults in situations where a pair of tasks have a prece-
+dence relation in the implementation that does not
+exist in the design phase lost an existing precedence
+relation in the implementation. The incorrect prece-
+dence can cause a wrong execution order of tasks.
+Hence, we proposed the precedence mutation oper-
+ators which can specify a precedence relation be-
+tween a pair of tasks and runnables. This operator
+creates mutants by assigning a specific execution or-
+der to a set of tasks or runnable to reflect the prece-
+dence relation.
+3.4.1. mATPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each task τ j ∈ T (j ̸= i), if τj /∈
+precti, this mutation operator adds τj to precti.
+3.4.2. mRTPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each task τj ∈ T (j ̸= i), if
+τj ∈ precti, this mutation operator removes τ j from
+precti.
+3.4.3. mARPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each runnable γim ∈ τi, if γim /∈
+precri, this mutation operator adds γim to precri.
+3.4.4. mRRPREC
+For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,
+jitteri} ∈ T, for each runnable γim ∈ τi, if γim ∈
+precri, this mutation operator removes γim from
+precri.
+3.5.
+Priority mutation operators
+In an RTOS, each task is assigned a relative priority,
+which is a static integer to identify the degree of im-
+portance of tasks. The highest priority task always
+gets the CPU when it becomes ready to run. The
+most common RTOS scheduling algorithm is pre-
+emptive scheduling.
+3.5.1. mITPRI
+This operator increases the priority of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the priority of the
+task prioi to proii +δ.
+3.5.2. mDTPRI
+This operator decreases the priority of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the period of the
+task i to proii −δ.
+3.6.
+Jitter mutation operators
+Timing jitter exists in the RTOS, and it is the delay
+between subsequent periods of time for a given task.
+3.6.1. mITJ
+This operator increases the jitter time of a task. For a
+given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the priority of the
+task jitteri to jitteri +δ.
+3.6.2.
+mDTJ
+This operator decreases the jitter time of a task. For
+a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈
+T, this mutation operator changes the period of the
+task jitteri to jitteri −δ.
+3.7.
+Shared memory mutation operators
+It is common that RTOS tasks exchange data or
+information via shared memory(e.g., global vari-
+able, memory buffer, hardware register). The shared
+memory can easily cause access conflict if the logi-
+cal software design is neglected in any corner case.
+Here we introduce a set of variable mutation opera-
+tors.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+3.7.1. mDSM
+This operator defines a new value to a global vari-
+able in a task. If a task reads this global variable,
+then we define a new value right before the reads
+occurred.
+3.7.2. mUDSM
+This operator un-defines a global variable in a task.
+If a task writes this global variable, then ignore this
+writes operation.
+3.7.3. mRDSM
+This operator removes the definition of a global vari-
+able. If a global variable is initialized in a task then
+do not initialize it.
+3.7.4. mRSM
+This operator adds a reference to a global variable.
+3.7.5. mRMSMR
+This operator removes reference to a global variable.
+3.7.6. mRSMR
+This operator replaces a reference of a global vari-
+able with a different global variable.
+Table 2. Simuilnk Mutation Operators
+Mutation Key
+Title
+mITO
+Increase Task Offset
+mDTO
+Decrease Task Offset
+mITPER
+Increase Task Period
+mDTPER
+Decrease Task Period
+mITET
+Increase Task Execution Time
+mDTET
+Decrease Task Execution Time
+mATPREC
+Add Task Precedence
+mRTPREC
+Remove Task Precedence
+mARPREC
+Add Runnable Precedence
+mRRPREC
+Remove Runnable Precedence
+mITPRI
+Increase Task Priority
+mDTPRI
+Decrease Task Priority
+mITJ
+Increase Task Jitter
+mDTJ
+Decrease Task Jitter
+mDSM
+Define Shared Memory
+mUDSM
+Un-define Shared Memory
+mRDSM
+Remove Definition Shared Memory
+mRSM
+Reference a Shared Memory
+mRMSMR
+Remove a Shared Memory Reference
+mRSMR
+Replace a Shared Memory Reference
+4.
+Mutation operators demonstration
+We have introduced twenty mutation operators cate-
+gorized into seven classes and explained each mu-
+tation class.
+The mutation operators are summa-
+rized in Table 2. We use simple examples to demon-
+strate the use of each mutation operator. To demon-
+strate our mutation operators, we use the tool Sim-
+Sched to alter the properties of software applications
+realized as Simulink models.
+From Table 1, we
+know that each function-call subsystem represents
+an AUTOSAR runnable. The function-call subsys-
+tem can be executed conditionally when a function-
+call event signal arrives. Both an S-function block
+and a Stateflow block can provide such a function-
+call event. SimSched applies the function-call in-
+vocation mechanism to use an S-function to gener-
+ate a function-call event to schedule each runnable
+(function-call subsystem). Figure 4 shows the Sim-
+Sched parameters dialogue that we can utilize it to
+adjust the timing properties to implement the muta-
+tion operator for Simulink models.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 4. SimSched Parameter setting dialogue.
+In this section, we use several simple examples
+to exhibit the mutants generated by our mutation op-
+erators. Figure 5 illustrates the use of SimSched to
+schedule a Simulink model. In this example, Sim-
+Sched is at the top left corner, which schedules three
+runnables (R1, R2, R3), and they are mapped to two
+tasks (T1, T2). Runnable R1 is mapped to T1, the pe-
+riod of T1 is 10ms, priority is 2, and the execution
+time of R1 is 3ms. R2 and R3 are mapped to T2. The
+period of T2 is 20ms, priority is 1, and the execu-
+tion time of R2 and R3 are 3ms and 3ms accordingly.
+The detailed parameter settings are listed in Table 3.
+There is a Data Store Memory block in this exam-
+ple, named A, which defines a shared data store that
+is a memory area used by Data Store Read and Data
+Store Write block with the same data store name. R1
+writes a constant value to a global variable A. R2
+reads A first then writes the summation of A and its
+delay value to A. R3 reads A then subtracts its delay
+value from A, and outputs the result.
+Table 3. The simple example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+3
+2
+R1
+T2
+20
+3
+1
+R2
+T2
+20
+3
+1
+R3
+Fig. 5. A simple example of using SimSched to schedule
+AUTOSAR SW-Cs.
+Fig. 6. Task executions Gantt chart of the running example.
+Fig. 7. A simple example of using Stateflow to schedule
+AUTOSAR SW-Cs.
+Fig. 8. Simple example output of Stateflow scheduler
+simulation.
+
+BlockParameters:SimSched
+S-Function (mask)
+Parameters
+Task:
+[1, 2]
+Priority:
+[2,1]
+Period:
+[10,20]
+Runnable:
+"R1','R2',"'R3'
+[1, 2, 2]
+Task Mapping:
+Execution Time:
+[3, 3, 3]
+[0,0,0]
+Offset
+[0,0,0]
+Jitter
+OK
+Cancel
+Help
+ApplyScheduler
+Runnable(period, execution time, priority) Task()
+R1(10ms, 3ms, 2)
+ Task(1)
+R2( 20ms, 3ms,
+ Task(2)
+R3 20ms, 3ms,
+Task(2)
+function()
+function()
+function(
+SimSched
+function
+Runnable1 subsystem
+Runnable2 subsystem
+Runnable3 subsystem1
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.060:0
+call()
+0:1
+R1()
+1 ms Clock
+R2()
+0:1
+R3()
+call(
+Temporal Logic
+0:6
+Scheduler
+0:1
+0:1
+0:1
+R1()
+R2()
+R3()
+R1
+0:7
+A
+subrater
+Runnable1 subsystem1
+Runnable2 subsystem1
+Runnable3 subsystem'11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+40
+R2
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+We use a Stateflow scheduler version of the sim-
+ple example shown in Figure 7 to show the typical
+Stateflow scheduler simulation result, then compare
+it with the SimSched scheduler simulation result.
+The task parameters are all the same shown in Table
+3. We apply the same task configurations for both
+the Stateflow scheduler and SimSched models for
+simulation. Figure 8 shows the Stateflow scheduler
+simulation result, and Figure 9 shows the SimSched
+simulation result. The result figures show the output
+value of each runnable. From the figure, we can see
+that R1, R2 and R3 are all executed at time 0 in Fig-
+ure 8; R1, is executed time 0, R2 is executed at 3ms,
+and R3 is executed at 6ms in Figure 9. R2 and R3 are
+executed later than the Stateflow scheduler simula-
+tion in Figure 8. This is because that SimSched takes
+into account execution time, and each task must be
+executed until the previous task is completed on a
+single core platform.
+Fig. 9. Simple example output of SimSched simulation without
+applying any mutation operator.
+4.1.
+Offset mutation operators
+We first apply the mITO mutation operator to the
+running example, let’s say that increase δ1 = 3ms
+for T1 then we have the task execution timeline in
+Figure 10. We can see T2 is executed first at time
+0, and T1 preempts T2 at 3ms in the first period due
+to the offset effect. After the first period, there is
+no preemption between T1 and T2. Then, we apply
+the mDTO mutation operator based on the previous
+settings. We set δ1 = −1ms to T1 then the offset
+for T1 is 2ms. Figure 11 shows the task execution
+timeline. T2 is preempted by T1 during the execution
+of the first period. Compared to the task execution
+Gantt chart of our running example shown in Figure
+6 with no offset, we can clearly see the preemption
+effect.
+Fig. 10. Task executions Gantt chart of the running example
+after increase offset mutation operator is applied.
+Fig. 11. Task executions Gantt chart of the running example
+after decrease offset mutation operator is applied.
+The running example’s output after applying the
+mITO mutation operator is shown in Figure 12
+which is different from Figure 9. Because T1 pre-
+empts T2 at the first instance execution, the output
+of R3 is from zero to ten then goes back to zero
+then goes up instead of always increasing value. The
+running example’s output after applying the mDTO
+mutation operator is the same as Figure 9 because
+the offset operator only affects the initial execution
+of each task, and the preemption occurs before the
+first execution of R2 instance completion. Inside our
+model scheduler program, we trigger each subsys-
+tem at the end of each execution time slot. Techni-
+cally, the execution order of this example is still R1,
+R2, and R3 so the output of the simulation keeps the
+same.
+Fig. 12. Simple example output of SimSched simulation after
+applying mITO mutation operator.
+
+T1
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05
+0.06
+Time(sec.)
+T2
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)T1
+06
+0
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+T2
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)Runnable1 subsystem
+11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+Runnable2_subsystem
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+Runnable3_subsystem
+20
+10
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)11
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+30
+20
+oR2 
+0
+0.003
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+20
+10
+0
+0
+0.006
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+TimeJ. Chen et al. / Mutation Operators for Simulink Models
+4.2.
+Period mutation operators
+We apply the mITPER operator to this example to
+increase the period of a task. We set δ1 = 1ms to T1
+so the period of the task T1 is 11ms now. Figure 13
+shows that T2 is preempted at the time of 22ms, and
+the simulation yields a wrong result due to this pre-
+emption shown in Figure 14. The output of R3 is an
+alternating value instead of an increasing value.
+Fig. 13. Task executions Gantt chart of the running example
+after increase period mutation operator is applied.
+Fig. 14. Simple example output of SimSched simulation after
+applying mITPER mutation operator.
+We apply the mDTPER operator to this example
+to decrease the period of a task. We set δ1 = −4ms
+to T1 so the period of the task T1 is 6ms now. Then,
+we run the simulation, T2 is preempted by T1 shown
+in Figure 15 and it yields a wrong simulation result
+shown in Figure 16. The output of R3 is either zero
+or ten instead of an increasing value.
+Fig. 15. Task executions Gantt chart of the running example
+after decreasing period mutation operator is applied.
+Fig. 16. Simple example output of SimSched simulation
+after applying mDTPER mutation operator.
+4.3.
+Execution time mutation operators
+We apply the mITET operator to this example to in-
+crease the execution time of a task. We can specify
+any runnable to increase its execution time within a
+task. For example, we set δ1 = 4ms to R2 in T2 so
+the execution of R2 is 7ms and T2 takes 10ms to ex-
+ecute now. Figure 17 shows that T2 is preempted at
+the time of 10ms, and the simulation yields a wrong
+result due to this preemption. The wrong result is
+the same as the example of applying decreasing task
+period. We apply the mDTET operator to this exam-
+ple to decrease the execution time of a task. We set
+δ1 = −1ms to T1 so the execution time of the task T1
+is 2ms now. Then, we run the simulation, there is no
+preemption that occurs between these two tasks and
+the output is as expected as the original model.
+Fig. 17. Task executions Gantt chart of the running example
+after increase execution time mutation operator is applied.
+4.4.
+Execution precedence mutation operators
+We introduce the second example as Table 4 to ex-
+plain the mATPREC and mRTPREC operators. Fig-
+ure 18 shows the task execution Gantt chart of this
+example. From the task execution chart, we can see
+the execution order of the tasks is T1,T2,T1,T3.
+
+1
+0.5
+0E
+0
+0.01
+0.03
+0.04
+0.05
+0.06
+Time(sec.)
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Time(sec.)Runnable1_sybsystem
+11
+10
+9 
+0.1
+0.2
+0.3
+0.4
+0.5 
+0.6
+0.7 
+0
+0.8
+0.9
+Runnable2_subsystem
+400
+200
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Runnable3_subsystem
+400 
+200
+0
+0
+0.1
+0.2
+0.3
+9:0
+0.0
+0.8
+0.9T1
+1
+0.5 
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+T2
+1
+0.5 
+0
+0
+0.02
+0.03
+0.04
+0.05
+0.06Runnable1_sybsystem
+11
+10
+9.
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+Runnable2_subsystem
+40 
+20 
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+Runnable3_subsystem
+10 
+5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.11
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0.06
+1
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06J. Chen et al. / Mutation Operators for Simulink Models
+Table 4. The simple example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+5
+1
+3
+R1
+T2
+10
+4
+2
+R2
+T3
+10
+3
+1
+R3
+Fig. 18. Task executions Gantt chart of example 2.
+First, we assume there is no precedence rela-
+tion among tasks so we use the mATPREC mutation
+operator to add a precedence relation τ3 to prect2,
+which specifies that a new instance of T2 cannot start
+unless T3 has executed after the last instance of T2.
+Hence, we set the execution order that T3 is executed
+before T2 in the setting dialogue. Figure 19 shows
+the execution result that T2 is preempted by T1. If T2
+is not a re-entrant function then this preemption may
+cause potential failure execution.
+Fig. 19. Task executions Gantt chart of example 2 after task
+precedence mutation operator is applied.
+Then, we assume there is a precedence relation
+between T1 and T3 and the task execution diagram
+is the same in Figure 18. We apply the mRTPREC
+mutation operator to remove the precedence relation
+prect3 from τ1. The result is the same as shown in
+Figure 19.
+We add one runnable R4 to the first example and
+assign it to T1. This new task configuration is shown
+in Table 5. R4 writes a different constant value from
+R1 to the global variable A. We apply mARPREC
+mutation operator to this new example, which adds
+γ1 to precr4. R4 requires R1 execute first so R4 over-
+writes the value written by R1. The operator changes
+the execution order of runnables.
+Table 5. Task configuration settings for runnable precedence
+mutation operators.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+2
+2
+R1
+T2
+20
+2
+1
+R2
+T3
+20
+2
+1
+R3
+T1
+10
+2
+1
+R4
+In example one, T2 has two runnables R2 and R3
+with a precedence relation between them. We apply
+mRRPREC runnable remove precedence mutation
+operator to remove the precedence γ2 from precr3.
+We schedule R3 runs before R2 since no precedence
+constraint that turns out different than the original
+simulation. The original output of R3 is an increas-
+ing value along with the execution instead of a value
+of either zero or a fixed value. The reason is that
+R3 executes first and it reads A before R2 writes any
+new value to A. The bottom output line in Figure 20
+shows the execution result.
+Fig. 20. The outputs of example one three runnables.
+4.5.
+Priority mutation operators
+Table 6. Priority mutation operator example settings
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+10
+1
+4
+R1
+T2
+10
+2
+3
+R2
+T3
+10
+3
+2
+R3
+We apply mITPRI operator to the example in Ta-
+ble 6 to increase the priority of T3. This mutation
+operator changes the priority of prio3 to proi3 + 3
+so the T3 has the highest priority 5 in this example,
+which results in T3 being executed at first. Figure 21
+shows T3 is triggered first in the task execution Gantt
+chart. This mutation alters the task execution order.
+
+1
+口
+口
+口
+口
+口
+口
+0.5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05 
+0.06
+1
+0
+0
+0. b1
+0.02 
+0.03
+0.04 
+0.05
+0.06
+1
+1
+1
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.061 
+0.5
+0
+4.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+0.5
+0
+0
+0. 01
+0.02
+0.03
+0.04 
+0.05
+0.06
+L
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.0611
+10
+9
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+40
+20 
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1
+10 F
+5
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+0.08
+0.09
+0.1J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 21. Task executions Gantt chart after applying increas-
+ing task priority mutation operator.
+We apply mDTPRI operator to decrease the pri-
+ority of T1. This mutation operator changes the pri-
+ority of prioi to proi−3 so the T1 has the lowest pri-
+ority 1 in this example, which results in T1 being
+executed at last. The task execution Gantt chart is
+shown in Figure 22.
+Fig. 22. Task executions Gantt chart after applying decreas-
+ing task priority mutation operator.
+4.6.
+Jitter mutation operators
+We apply mITJ operator to increase a jitter time of
+a task. For example, let δ = 2, this mutation op-
+erator changes the real release time of the task to
+jitter1 = 0+2. Figure 23 shows the execution of T2
+is preempted by T1 caused by the jitter.
+Fig. 23. Task executions Gantt chart after applying increas-
+ing jitter mutation operator.
+Then mDTJ mutation operator decreases the jit-
+ter time of a task. We apply this operator to the
+above example and let δ = −1 so the task jitter1 =
+2 − 1. T2 is preempted by T1 during the simulation
+phase.
+4.7.
+Shared memory mutation operators
+In this shared memory category, we introduce five
+mutation operators.
+The first one is mDSM, and
+this operator assigns a new value to the memory
+store before a read.
+For our example, we add a
+Data Store Write block right before the Data Store
+Read execution so that the Data Store Write block
+defines a new value to the variable, and we chose
+the initial value of this variable as the default new
+value. The mutant using mUDSM operator is shown
+in Figure 24, which only shows the changes of
+Runnable2 subsystem. We add a constant block and
+a Data Store Write block at the top left corner.
+Fig. 24. A simple example of DSM mutant.
+The second mutant operator is mUDSM, and this
+operator disregards a write to a Data Store block.
+For our example, we remove the Data Store Write
+block. Figure 25 shows the mUDSM mutant that the
+Data Store Write has been removed.
+Fig. 25. A simple example of UDSM mutant.
+The third mutant operator is mRDSM, and this
+operator removes an initialization value to a Data
+Store memory. In many programs, variables require
+
+1F
+1
+0.5 
+0
+/o
+0.01
+ 0.02 
+0.03
+ 0.04 
+0.05 
+0.06
+1
+1
+0.5 
+0
+0.01
+0.02 
+0.03 
+ 0.04 
+0.05
+0.06 
+0.5
+0.01
+0.02 
+0.03 
+0.04 
+0.05
+0.061 F
+0.5
+0
+/
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1 
+0.01
+0
+0.02
+0.03
+0.04 
+0.05
+0.06
+0.5
+0
+0.01
+0.02
+0.03
+0.04 
+0.05
+0.061
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05
+0.06
+0.5
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06f()
+2:0
+2:1
+function
+A
+0
+2:4
+2:2
+A
+2:5
+A
+2:3
+Zf()
+function
+A
+1
+Z.J. Chen et al. / Mutation Operators for Simulink Models
+an initial value before they can use properly. For
+our example, Runnable1 subsystem is such a pro-
+cess of initializing Data Store A; then, we remove
+the Data Store Write block in Runnable1 subsystem.
+The Simulink model can still run simulations with-
+out any issues however the output of the simulation
+only yields a single value.
+Mutant operator mRSM adds a new reference to
+shared memory. Figure 26 shows the block diagrams
+of a Simulink model with three subsystems and they
+are mapped to two tasks. The model has a DSM
+block A in the root-level system. There is a Data
+Store Write block inside subsystems Task B1 and a
+Data Store Read block in Task B2. The period of
+Task A is 5ms and the period of Task B is 10ms.
+To implement the mRSM, we add a Data Store Read
+block to the TaskA subsystem which shows in Figure
+27. In the original example, Task A executes first
+then Task B1 writes A and Task B2 reads A. The
+mutant program has the same execution order as the
+original model. However, when the Data Store Read
+block in Task A executes, the block reads data from
+an uninitialized data store or a previous instant of
+Task B1 as Task B has not executed yet or has been
+executed previously.
+Fig. 26. A simple Simulink model.
+Fig. 27. An example of mRSM mutant operator. Adding a
+Data Store Read block to Task A block.
+Mutant operator mRMSMR deletes a reference to
+shared memory. In Figure 26, Task B2 has a refer-
+ence to a DSM block A in the root-level system. To
+implement the mRMSMR, we delete the Data Store
+Read block in the TaskB2 subsystem. In the mu-
+tant program, Task B2 has a constant output value
+of zero since there is no reference.
+5.
+Evaluation Phase
+In the previous section, we describe how a model
+scheduler SimSched can validate the real-time con-
+text during a simulation, and we utilize mutation
+testing to evaluate SimSched. In this section, we
+perform experiments to demonstrate the use of our
+mutation testing framework to evaluate the quality
+of SimSched and Stateflow schedulers in scheduling
+tasks in real-time systems.
+5.1.
+Evaluation Process
+To validate the proposed mutation operators, we ap-
+ply them to ML/SL models. We separate the evalu-
+ation process into two parts base and extension, ac-
+cording to the ability of ML/SL. We apply the first-
+order mutants (FOMs) 22 to ML/SL models to gen-
+erate a mutant, which means we generate a mutant
+by using a mutation operator only once.
+5.1.1. Base Case
+In the base case, we examine the simulation results
+of the original models and the SimSched models and
+their mutants. An original model M is an ML/SL
+model scheduled by Stateflow scheduler; A Sim-
+Sched model M ′ is an original model scheduled
+by SimSched; The mutants (Mµ or M ′
+µ) are ei-
+ther original model or SimSched models mutated by
+one of our mutation operators. Figure 28 shows the
+schematic diagram of our mutants generation pro-
+cess. We use the simulation result of M as a com-
+parison baseline, and then we compare the baseline
+with every other simulation result of Mµ, and M ′
+µ.
+We examine the comparison result to see if the re-
+sult reaches a verdict failure during model simula-
+tion. We say a mutant is killed if a verdict of failure
+is reached.
+
+In2
+Out1
+A
+Task A
+Out1
+In2
+In1
+Qut3
+Task B1
+Task_B2A
+2
+DSRA
+2J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 28. Schematic diagram of the model mutants genera-
+tion process.
+Fig. 29. Simple evaluation example scheduled by Stateflow
+scheduler.
+We use a simple example shown in Figure 29 to
+explain the base case evaluation process. This ex-
+ample is an original model. We replace the State-
+flow scheduler with a SimSched scheduler to form
+a SimSched model. We generate mutants for both
+the original and SimSched models by a specific mu-
+tation operator, e.g., mDTPER, to decrease the task
+period. Then we run the simulation for both mutants
+and analyzed the results to see if there is any errors.
+If the simulation result of Mµ or M ′
+µ is different
+from the original model and shows a verdict failure,
+then we say the mutant is killed.
+In this example,
+we have three runnables
+R1,R2,R3 and they are mapped to two tasks T1,T2.
+R1 is mapped to T1 and R2,R3 are mapped to T2. The
+period of T1 is 3ms and The period of T2 is 6ms.
+The execution time of each runnable is 1ms. The
+simulation result of the M is shown in Figure 30
+and it shows each runnable output is a rising non-
+interlaced polyline. We apply the mDTPER muta-
+tion operator as decreasing 1ms to both the origi-
+nal model and SimSched model to generate mutants.
+The task T1 in the mutants has a period of 2ms. The
+simulation result of these simulations is shown in
+Figure 31 and Figure 32. The simulation result of
+M ′
+µ is different from the result of M , and it shows
+the output of R2 and R3 are two rising interlaced
+polylines because SimSched can simulate the exe-
+cution time and preemption. T1 preempts T2 in the
+SimSched mutant model to yield an alternative ex-
+ecution trace, and we say a verdict fail is reached.
+However, the simulation result of Mµ is similar to
+the result of M . Thus, the mDTPER mutant is killed
+to the M ′
+µ and is alive to the Mµ. We can not apply
+this means to all mutation operators due to the nature
+of ML/SL. We combine this method and the follow-
+ing method to evaluate the mutation operators.
+Fig. 30. M simulation result.
+Fig. 31. Mµ simulation result.
+Fig. 32. M ′µ simulation result
+5.1.2. Extension
+To evaluate the rest of the mutation operators, we
+implement a mutation generator with additional
+functionalities to assist the validation process. One
+feature is to check the mutant model’s schedulabil-
+ity at the given set of tasks configuration to decide if
+all task deadlines are met. The other function is to
+
+M'
+M
+SimSched
+Mutate
+Mutate
+M
+M'
+μ
+n1 ms Clock
+A
+R1()
+R3()
+R2()
+Sall(
+R2
+R1
+call()
+R3
+R10
+R2()
+R30
+Scheduler90
+R1
+R2
+R3
+80
+70
+60
+50
+40
+30
+20
+10
+0
+-10
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06R1
+R2
+120
+R3
+100
+80
+60
+40
+20
+0
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06R1
+60
+R2
+R3
+50
+40
+30
+20
+10
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Offset=0J. Chen et al. / Mutation Operators for Simulink Models
+check the data access sequence. If there is a DataS-
+tore block in the mutated model, every read or write
+to this DataStore block is recorded. Then we use
+this mutated model data access sequence to com-
+pare with the original model data access sequence.
+The mutation generator is implemented as a Matlab
+script written in m-file.
+The validation process takes a Stateflow sched-
+uled ML/SL model and a test specification as input.
+The test specification specifies which mutation op-
+erator to use. A mutant generator applies the speci-
+fied mutation operator to the ML/SL model via Sim-
+Sched and generates a mutant. The mutant genera-
+tor then executes the simulation both for the original
+model and the mutated model using the additional
+functionalities to analyze the simulation. If the anal-
+ysis shows at least one task misses its deadline in a
+mutated model, then we say a mutant is killed. Or
+at least one variable comparison result of the DataS-
+tore access sequence is unmatching, and then we say
+a mutant is killed; otherwise, we report the mutant
+is benign.
+Fig. 33. A simple example of using Model Scheduler to
+schedule AUTOSAR SW-Cs.
+We use an example shown in Figure 33 to ex-
+plain the validation process. It has three runnables
+and is mapped to two tasks, R1 map to T1, R2, and
+R3 map to T2. The period of task T1 is 10ms, and
+T2 is 20ms, every runnable’s execution time is 3ms.
+There is a DataStore block named A as a shared vari-
+able in this example model. If we apply the period
+mutation operator mDTPER ρi − δ where i − 1 and
+δ = 6 to this model to decrease the period of T1 and
+generate a mutant, run it. The analysis result shows
+the T2 missed deadline, then we say this mutant is
+killed. If we apply the execution time mutation op-
+erator mITET ci + δ where i = 1 and δ = 3 to this
+model to increase the execution time for T1 and gen-
+erate a mutant. The DataStore access sequence of
+the original model is a pattern of WRWR where W
+represents a write to the shared variable, and R rep-
+resents a read to the shared variable. The mutant
+generates a different sequence, which is WRWWR.
+It is because the T1 has a longer execution time than
+the original model, and it preempts T2 during the ex-
+ecution of T2. Hence, there is one more W in the
+DataStore access sequence.
+5.2.
+Experiments
+We employ two examples to demonstrate the use of
+our mutation testing framework. We first explain the
+two examples in detail. We then apply the mutation
+operators to the two models scheduled by both the
+Stateflow scheduler and SimSched.
+Fig. 34.
+The three-servo example adapted from 12 with
+Stateflow scheduler.
+5.2.1. Three Servos Model
+We adapt an example from the TrueTime 21 exam-
+ple library, which shows a possible implementation
+of a three-servo PID control system. The example
+is shown in Figure 34 with a Stateflow scheduler.
+In this example, three DC servos are modeled by a
+continuous-time system, and three PID controllers
+are implemented as three subsystems. We map three
+controller subsystems to three runnables R1, R2, and
+R3 then they are mapped to tasks T1, T2, and T3. The
+task periods are T1=4 , T2 = 5 and T3 = 6 ms re-
+spectively. Each task has the same execution time as
+
+Scheduler
+Runnable(period, execution time, priority) Task()
+function()
+function()
+function()
+R1(10ms,3ms,2)j
+Task(1)
+R2( 20ms, 3ms, 1
+Task(2)
+R3( 20ms, 5ms, 1
+Task(2
+function
+irv2
+irv1
+irv2
+SimSched
+irv3 
+Aader
+Runnable1_subsystem
+Runnable2_subsystem
+Runnable3_subsystem1 ms Clock
+callo
+R10
+R2()
+R30
+call(
+Temporal Logic
+Scheduler
+functionO
+1000
+u
+s2+s
+DCServo1
+PID1
+functionO
+1000
+u
+2+s
+DCServo2
+PID2
+function()
+1000
+u
+s?+s
+DCServo3
+PID3J. Chen et al. / Mutation Operators for Simulink Models
+1ms. Task settings are shown in Table 7. The simula-
+tion result is shown in Figure 35 based on the above
+task settings. The three graphs show the output of
+the motors using the three PID controllers when the
+corresponding task parameters are assigned accord-
+ingly. In the graph, the square wave is the reference
+input signal for the motors, where the computation
+delays are not taken into account. Three PID con-
+trollers are all smooth output signals as expected.
+Table 7. Three Servo example settings.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T1
+4
+1
+3
+R1
+T2
+5
+1
+2
+R2
+T3
+6
+1
+1
+R3
+Fig. 35.
+The three servos example output with Stateflow
+scheduler.
+We replace the Stateflow scheduler with the Sim-
+Sched scheduler, and the updated example is shown
+in Figure 36. In this example, three DC servos have
+the same task setting as the Stateflow scheduler ex-
+ample. Each runnable has the same execution time
+as 1ms. The simulation result is the same as the
+Stateflow scheduler example based on the above task
+settings. There is no deadline missing for any task,
+so the simulation result shows every task has smooth
+control. Figure 37 shows the task active chart gen-
+erated by SimSched. Every task has been executed
+within its own deadline.
+Fig. 36. The three servos example adapted from 12.
+Fig. 37. The three servos example task active chart gener-
+ated by SimSched.
+Fig. 38. The adjusted Stateflow scheduler for mITO muta-
+tion operator to increase o f fset as 1ms for DCServo1.
+
+2
+DCServo1
+1
+SignalGenerator
+0
+-1
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo2
+1
+SignalGenerator
+0
+-1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9Scheduler
+Runnable(period, execution time, priority) Task()
+E
+R1(4ms, 2ms, 3) Task(1)
+R2( 5ms, 2ms, 2
+Task(2)
+R3( 6ms, 2ms, 1
+ Task(3
+SimSched
+function(
+○○
+1000
+u
+s2+s
+DCServo1
+PID1
+function()
+1000
+u
+2+s
+y
+DCServo2
+PID2
+function()
+1000
+u
+32+s
+DCServo3
+PID30.5 
+o
+0
+0. 01
+0.02 
+0.03 
+to0
+0. 05
+0.06 
+0.5
+0
+0.01
+0.02 
+0.03 
+to0
+0.05 
+0.06 
+1F
+0.5 
+0
+0.01
+0.02 
+0.03 
+to0
+0.05 
+0.06 Scheduler
+on at(1,tick): Period_4_ms;
+du: on every(5,tick) : Rate5ms;
+du: on every(6,tick) : Rate6ms;
+'Sched_4_MS
+Periodicl
+on every(4,tick) : Period_4_ms;
+Sched 5 MS
+Periodicl
+en: Period_5_ms;
+Rate5ms
+Sched 6 MS
+Periodic/
+en: Period_6_ms;
+Rate6msJ. Chen et al. / Mutation Operators for Simulink Models
+Next step, we apply mITO, mDTO, mITPER,
+mDTPER, mARPREC, mRRPREC, mITJ, mDTJ
+mutation operators to both Stateflow scheduler and
+SimShced examples to generate two versions of mu-
+tants with the same mutation operators.
+To ap-
+ply some of the mutation operators to evaluate the
+Stateflow scheduler, we need to adjust the State-
+flow scheduler so that it can be used on the gen-
+erated mutants. Figure 38 shows an example that
+is adjusted for the Offset mutation operator. This
+example uses a temporal logic operator at in the
+state to set the Offset parameter to generate a mu-
+tant for PID1 which runs at the period of 4ms in
+this example. This mutant increases Offset as 1ms
+for DCServo1 controlled by PID1. The mutant of
+the SimSched version can be easily generated by our
+model scheduler SimSiched.
+Fig. 39. The Stateflow scheduled three-servo example task
+active chart after applying mITO mutation operator to in-
+crease o f fset as 1ms for DCServo1.
+We run simulations for both versions of the mu-
+tants generated by Offset mutation operator. Both
+mutant versions’ output of three servos is the same
+as shown in Figure 35. The only difference occurs
+at the beginning of the simulation but it does not af-
+fect the smooth control of DCServos. We can see
+the difference from the following comparison. Fig-
+ure 39 shows the Stateflow scheduled task active
+chart after applying mITO mutation operator to in-
+crease offset as 1ms for DCServo1. Before apply-
+ing the mutation operator, every task is released at
+time 0. After applying the offset mutation opera-
+tor, Task 1 is delayed by 1ms shown on the top of
+the figure. Task 2 and Task 3 are both released at
+time 0. Figure 40 shows the SimSched scheduled
+three servos example task active chart after applying
+mITO mutation operator to increase offset as 1ms for
+DCServo1. There are three output signals represent-
+ing three tasks from top to bottom T1, T2, and T3.
+As T1 has a 1ms offset, Task 2 is executed first as
+shown in the figure the second line starts at time 0.
+Because the SimSched scheduler has the execution
+time parameter, Task 2 is executed at time 1 and Task
+3 at time 2 respectively.
+Fig. 40. The SimSched scheduled three servos example task
+active chart after applying mITO mutation operator to in-
+crease Offset as 1ms for DCServo1.
+We use a similar approach to apply Period muta-
+tion operator to the three-servo example and gener-
+ate mutants for both the Stateflow scheduler model
+and SimSched model. We use two mutation con-
+figurations to show the similarities and differences
+between the two schedulers. The first configuration
+is [1,5,6]. It means Task 1 has a period of 1ms and
+the period of Task 2 and Task 3 keep the same as
+5ms and 6ms, respectively.
+Figure 41 shows the
+Stateflow scheduler mutant simulation result. Be-
+cause Task 1 has a 1ms period, it has too many times
+of calculations, and the output value is out of the
+chart. Task 2 and Task 3 keep the same output as be-
+fore. Figure 42 shows the SimSched mutant simula-
+tion result. The output of Task 1 is the same as the
+Stateflow scheduler mutant. However, Task 2 and
+Task 3 are different from the Stateflow one. Because
+the SimSched takes the execution time into account,
+Task 1 has the highest priority and has an execution
+time of 1ms, and Task 1 always runs during the sim-
+ulation. Task 2 and Task 3 always are preempted by
+Task 1 because they have a lower priority than Task
+1.
+Fig. 41. The Stateflow scheduled three servos example out-
+put after applying mDTPER mutation operator to decrease
+period as 1ms for DCServo1.
+
+PID1  4 ms 1ms offset
+0.01
+0.02
+0.03
+0.04 
+0.05 
+0.06
+PID2 5 ms no offset
+1
+0.5 
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0
+0.06
+PID3 6 ms no offset
+1
+0.5
+0
+0
+0.01
+0.02 
+0.03
+0.04 
+0.05 
+0.061
+0.5,
+0 
+0
+0.01
+0.02 
+0.03
+0.04
+0.05
+0.06
+1 
+0.5 
+0
+0.01 
+0.02 
+0.03
+to'0
+0.05
+0.06
+1 
+0.5 
+0 
+0
+0.01
+0.02 
+0.03 
+to'0
+0.05
+0.062
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3
+0.4 
+0.5 
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8
+0.9J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 42. The SimSched scheduled three servos example out-
+put after applying mDTPER mutation operator to decrease
+period as 1ms for DCServo1.
+The second configuration is [13,5,6]. It means
+Task 1 has a period of 13ms and the period of Task
+2 and Task 3 keep the same as 5ms and 6ms, re-
+spectively. Figure 43 shows the Stasflow scheduler
+mutant simulation result and the SimSched mutant
+has the same output as shown in the figure. Because
+Task 1 has a 13ms period, it has fewer computations
+than the original model. Although the output behav-
+ior looks like Task 1 misses its deadline in the figure,
+every execution of Task 1 meets its deadline and it
+is executed as scheduled.
+Fig. 43. The Stateflow scheduler three servos example out-
+put after applying mITPER mutation operator to increase
+period as 13ms for DCServo1.
+We
+generate
+mutants
+for
+mARPREC
+and
+mRRPREC operators by setting each parallel state’s
+execution order in the Stateflow scheduler model
+and configuring the parameters and connections for
+the SimSched model. The two mutants’ simulation
+results are the same as the original model, except
+that each task’s execution order is different from the
+original model.
+We generate mutants for mITJ and mDTJ op-
+erators by adapting the Stateflow scheduler in the
+model and configuring the parameters in the Sim-
+Sched model. We set the configuration as [1,0,0].
+It means only Task 1 has a jitter as 1ms. Figure 44
+shows the SimSched scheduler three servos example
+task active chart after applying mITJ mutation oper-
+ator to increase jitter as 1ms for DCServo1. Be-
+cause Task 1 has 1ms jitter, Task 2 is executed first
+then Task1 and Task3.
+Fig. 44. The SimSched scheduler three servos example task
+active chart after applying mITJ mutation operator to in-
+crease jitter as 1ms for DCServo1.
+We only apply the execution time operator to the
+SimSched model due to the lack of support for the
+Stateflow scheduler. Figure 45 shows the effect out-
+put of mITET mutation operator. We set c1 = 3ms
+using the mITET mutation operator for Task 1 to
+generate a mutant. The output of Task 3, shown as
+DCservo 3 at the bottom of the figure, is a curly
+wave. It is an unstable control due to the preemp-
+tion by T1 and T3 missing its deadline. Figure 46
+shows the task preemption effect. Task 1 takes 3ms
+to execute and Task 2 takes 1ms to execute. After
+the execution of Task 1 and Task 2, Task 3 should be
+executed; however, it is the time that Task 1 is sched-
+uled to run. Task 1 has a higher priority, so Task 3 is
+preempted by Task 1. The first instance of Task 3 is
+executed at 19ms so Task 3 does not have a smooth
+control signal output as the other tasks. Although
+some task preemptions occur in Task 2, it does not
+miss enough deadlines to significantly affect the out-
+put. Task 2 still has a smooth output signal as shown
+in Figure 45 as the second chart.
+Fig. 45. The SimSched scheduler three servos example sig-
+nal output after applying mITET mutation operator to in-
+crease executiontime to 3ms for DCServo1.
+
+2
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3 
+0.4
+0.5
+0.6
+0.7 
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.92
+DCServo1
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+2
+DCServo2
+SignalGenerator
+0
+-2 
+0
+0.1
+0.2 
+0.3
+0.4 
+0.5 
+0.6
+0.7
+0.8
+0.9
+2
+DCServo3
+SignalGenerator
+0
+-2
+0
+0.1
+0.2
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8
+0.9DCServo1
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+DCServo2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+DCServo3
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05DCServo1
+2
+0
+-1
+2
+0.1
+0.2 
+0.3 
+0.4 
+0.5 
+0.6 
+0.7 
+0.8
+0.9 
+DCServo2
+2
+-1
+。
+0.1
+0.2
+0.3 
+0.4 
+0.5 
+0.6 
+0.7 
+0.8
+0.9 
+DCServo3
+2 
+1
+0
+-1
+-2
+0.1 
+0.2 
+0.3
+0.4 
+0.5
+0.6
+0.7
+0.8 
+0.9J. Chen et al. / Mutation Operators for Simulink Models
+Fig. 46. The SimSched scheduler three servos example task
+active chart after applying mITET mutation operator to in-
+crease executiontime to 3ms for DCServo1.
+5.2.2. Throttle Position Control Model
+We adopt an AUTOSAR software component
+Simulink model from Mathworks shown in Figure
+47. It implements a throttle position control sys-
+tem for an automobile and contains three sensors,
+one monitor, one controller, and one actuator. They
+are implemented as six subsystems and mapped to
+six runnables TPSSecondary, Monitor, Controller,
+Actuator, APPSnsr and TPSPrimary then they are
+mapped to tasks T1, and T2. The task periods are
+T1=5ms and T2 = 10 ms respectively. Each runnable
+has the same execution time of 1ms. Task settings
+are shown in Table 8. This example uses seven Data-
+StoreMemory blocks to access the shared resources.
+Fig. 47. Throttle position control Simulink model contains
+six runnables.
+Table 8. Throttle control example settings.
+Task
+Period
+Execution
+Priority
+Runnable
+(ms)
+Time(ms)
+T2
+10
+1
+1
+TPSPrimary
+T1
+5
+1
+2
+TPSSecondary
+T1
+5
+1
+2
+Monitor
+T1
+5
+1
+2
+Controller
+T1
+5
+1
+2
+Actuator
+T2
+10
+1
+1
+APPSnsr
+Figure 48 shows the simulation result, which is
+generated by a Stateflow scheduler. The square wave
+in the figure is the simulated pedal input, and the
+curly wave is the output of the throttle body, repre-
+senting the current throttle position. The Stateflow
+scheduler simulates the throttle control controller’s
+process well and simulates the entire control pro-
+cess.
+Fig. 48. The simulated throttle position of the throttle posi-
+tion control model scheduled by the Stateflow scheduler.
+Figure 49 shows the runnable active chart sched-
+uled by the Stateflow scheduler. All runnables are
+scheduled and executed at time 0. Runnable TP-
+SPrimary and APPSensor are mapped to T2 and they
+are scheduled and executed every 10ms and the top
+and bottom chars shown in the figure are their active
+charts. The active charts of runnable TPPSSendary,
+Monitor, Controller, and Actuator are the four charts
+in the middle of the figure. They are scheduled and
+executed every 5ms.
+Fig. 49. The runnable active chart of the throttle position
+control model scheduled by the Stateflow scheduler.
+We apply SimSched to the Stateflow scheduler
+model, and we can get a similar simulation result as
+the Stateflow scheduler example based on the above
+task settings. Figure 50 shows the task active chart
+generated by SimSched. Every task has been exe-
+cuted within its own deadline. Both task T1 and T2
+are scheduled at time 0 but only T1 is executed at
+
+DCServo1
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05 
+DCServo2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+DCServo3
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.054
+TPSSecondaryRun5ms
+MonitorRun5ms
+ControllerRun5ms
+function()
+function()
+function(
+汇
+ThrottlePositionSensorSecondary
+ThrottlePositionMonitor
+Controller
+6
+APPSnsrRunl
+ActuatorRun5ms
+TPSPrimaryRuniOms
+function()
+function()
+function()
+口
+APPHwlOValueread
+ThrCmdHwlOValuewrite1
+ThrottlePositionSensorPrimary
+AccelerationPedalPositionSensor
+ThrottlePositionActuator
+APP_HwlO_Value
+ThrCmd HwlO ValueThrottle Pos
+0.7
+Simulated Pedal Input
+0.6
+Throttle Pos
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5TPSPrimary
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+TPSSecondary
+1
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03 
+0.035
+0.04 
+0.045
+0.05
+Monitor
+1
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045 
+0.05
+Controller
+1
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Actuator
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04
+0
+0.045
+0.05
+APPSensor
+3
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+s0'0
+0J. Chen et al. / Mutation Operators for Simulink Models
+time 0 due to its higher priority. T1 takes up 4ms to
+run. After the first instance of T1 is finished, T2 is
+executed. The second instance of T1 arrives at 5ms,
+which is during the middle of the execution of T2. T2
+is preempted by T1 and resumes at the completion of
+the second instance of T1.
+Fig. 50. The task level active chart of the throttle position
+control model scheduled by the SimSched scheduler.
+Figure 51 shows the runnable level active chart
+of the throttle position control model scheduled by
+the SimSched scheduler. From this figure, we can
+clearly see the activity of each runnable.
+It ex-
+actly shows the execution order of each runnable.
+Runnable TPPSSendary, Monitor, Controller, and
+Actuator are executed one after another followed by
+TPSPrimary. Runnable APPSensor is executed af-
+ter the second instance of T1 and it is the preemption
+point of T2.
+Fig. 51. The runnable active chart of the throttle position
+control model scheduled by the SimSched scheduler.
+We use the same means applied to three servos
+example to apply it to the throttle position control
+model. We adopt the Stateflow schedule and replace
+it with SimSched to generate mutants for both the
+Stateflow scheduler and SimSched for the experi-
+ments.
+Figure 52 shows the runnable active chart of the
+Throttle Position Control model scheduled by Sim-
+Sched after applying mTIO mutation operator as in-
+creasing 2ms offset for T1. The runnable Actuator
+active chart shown in the second bottom chart in the
+figure is missing its first execution. T1 takes 4ms to
+run, and it also has 2ms offset, so the total execution
+time of T1 exceeds its period 5ms. On the other hand,
+the mutant generated by the Stateflow scheduler can
+not simulate this overrun situation due to the lack of
+execution time simulation support.
+Fig. 52. The runnable active chart of Throttle Position Con-
+trol model scheduled by SimSched after applying mTIO
+mutation operator as increasing 2ms offset for T1.
+Figure 53 shows the simulated throttle position
+of the throttle position control model scheduled by
+SimSched after applying mITPER mutation opera-
+tor for T1 at 100ms. The Stateflow scheduler also
+can output the same figure. The mITPER mutation
+operator can reduce the computation times of a task
+at the same amount of time, which results in unsta-
+ble control as shown in the figure.
+Fig. 53. The simulated throttle position of the throttle po-
+sition control model scheduled by the Stateflow scheduler
+after applying mITPER mutation operator.
+Figure 54 shows the simulated throttle position
+of the throttle position control model scheduled by
+SimSched after applying mDTPER mutation opera-
+tor for T1 at 4ms. The mDTPER can result in no
+output signal for T2 because a higher rate task in-
+creases the computation times, and the lower rate
+task does not get an execution. In this example, we
+set ρ1 = 4ms using the mDTPER mutation operator,
+
+Task1
+1
+0.5
+oE
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03 
+0.035
+0.04 
+0.045
+0.05
+Task2
+1
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05TPsPrimary
+0.5 
+10
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+TPSSecondary
+0.5
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Monitor
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Controller
+0.5 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Actuator
+0.5
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0
+0.05
+APPSnsr
+0.5 /
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05TPsPrimary
+1
+0.5
+10
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+TPSSecondary
+1 
+0.5
+0
+0
+0.005
+0.01
+0.015
+0.02 
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05
+Monitor
+1
+0.5
+0 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Controller
+1 
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+Actuator
+1 F
+0.5 
+0
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+0.05
+APPSnsr
+0. E
+0
+0.005
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04 
+0.045
+0.05Throttle Pos
+Simulated Pedal Input
+Throttle Pos
+0.8
+0.6
+0.4
+0.2
+0
+0.5
+1.5J. Chen et al. / Mutation Operators for Simulink Models
+then the T2 is always preempted by T1 and does not
+have a chance to execute.
+Fig. 54. The simulated throttle position of the throttle po-
+sition control model scheduled by SimSched after applying
+mDTPER mutation operator.
+We only apply the execution time operator to
+SimSched models. We set c1 = 5ms using mITET
+mutation operator for T1 to generate a mutant. This
+mutant just outputs the same throttle position as
+shown in Figure 54. Since T1 has increased its exe-
+cution time by 1ms, it just takes up all the time slots
+in its period. T2 is preempted by T1 during the simu-
+lation process.
+We apply mARPREC and mRRPREC mutation
+operators to this throttle position control exam-
+ple. First, there is no precedence between runnable
+APPSnsr and TPSPrimary, and we use a mARPREC
+mutation operator to add precedence γAPPSnsr to
+precrTPSPrimary to generate mutants for both sched-
+ulers.
+Runnable Controller consumes the values
+produced by APPSnsr and TPSPrimary to calcu-
+late the throttle percent value for the throttle actu-
+ator. The changes in the simulation results of both
+mutants are trivial. Figure 55 shows the simulation
+result comparison between the original model and
+the SimSched mutant.
+Second, there is precedence between Controller
+and Actuator, and Controller is executed before
+Actuator. We remove the precedence from the pair
+of runnables, so the Actuator (destination) runs be-
+fore Controller (source), which changes the data de-
+pendency and delays the data. The difference in sim-
+ulation is similar to Figure 55.
+Fig. 55. The difference of simulation result between the
+original model and the mutant with mARPREC mutation
+operator scheduled by SimSichde.
+We apply mDSM, mUDSM, mRDSM, mRSM,
+mRMSMR, and mRSMR mutation operators to this
+example and generate accordingly mutants for both
+the Stateflow scheduler and SimSched.
+Inter-
+estingly, since the task time properties have not
+changed for the shared memory mutation operators,
+the simulation results are consistent between the two
+types of mutants. For example, we apply the mDSM
+mutation operator to variable ThrCmdPercentValus
+and set the variable to a new constant value. The
+simulation result of both types of mutants is the
+same shown in Figure 56. Since this variable is an
+input to a Lookup table, there is always an output
+that matches the input value and yields a correspond-
+ing value to the output. The input value is constant,
+so the throttle position’s output is a smooth curve.
+Fig. 56. The throttle position output after applying mDSM
+mutation operator to variable ThrCmdPercentValus sched-
+uled.
+5.3.
+Evaluation Result
+We apply the evaluation process to the example
+models to investigate the mutation operator’s effec-
+tiveness to kill mutants.
+Table 9 and 10 summa-
+rize the results of the efficacy of mutation operators,
+
+Throttle Pos
+Simulated Pedal Input
+0.6
+Throttle Pos
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5ThrottlePos(Run2:TPCWithTaskAndDSM_CaseStudy_Stateflow)
+ThrottlePos (Run5:TPCWithTaskAndDSM_CaseStudy_SimSched)
+Tolerance
+0.6 
+0.5
+0.4 
+0.3 
+0.2
+0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+1.2
+1.3
+1.4
+1.5Throttle Pos
+0.7
+Simulated Pedal Input
+Throttle Pos
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+0
+0.5
+1.5J. Chen et al. / Mutation Operators for Simulink Models
+where each row provides the number of mutants and
+the mutation score of our mutation operators. The
+mutation score is a measure that gives the percent-
+age of killed mutants with the total number of muta-
+tions.
+Table 9. Mutation analysis of mutation operators for three ser-
+vos example.
+Stateflow Scheduler
+SimSched
+Operator
+Mutants
+Kills
+Mutants
+Kills
+Offset
+36
+24
+36
+27
+Period
+39
+25
+39
+25
+Execution
+N/A
+N/A
+36
+31
+Time
+Precedence
+5
+0
+5
+0
+Priority
+18
+0
+18
+0
+Jitter
+36
+24
+36
+27
+Mutation
+54.48%
+64.71%
+Score
+For the three servo example, we generate 134
+mutants for the Stateflow scheduler models and 170
+mutants for the SimSched models. We achieve a
+mutation score of 54.48% for the Stateflow sched-
+uler model and 64.71% for the SimSched models.
+Evaluation results show that the time-related muta-
+tion operators have the most effect on the mutation
+testing, such as the Offset, Period, Execution Time,
+and Jitter mutation operator. We observe that the
+Precedence and Priority mutation operators kill zero
+mutant because, in this example, each controller in-
+dividually controls a motor. There is no connection
+between them, so the change of precedence and pri-
+ority does not cause any simulation changes. How-
+ever, the three controllers run on a single CPU, and
+one task execution time’s length affects other tasks.
+We also observe the Offset mutation operator only
+affects each task’s initial execution, and tasks miss
+the deadline. Still, each mutant’s simulation results
+show each controller can have stable control of each
+servo.
+Table 10. Mutation analysis of mutation operators for throttle
+position control example.
+Stateflow Scheduler
+SimSched
+Operator
+Mutants
+Kills
+Mutants
+Kills
+Offset
+19
+7
+19
+15
+Period
+30
+6
+30
+19
+Execution
+N/A
+N/A
+34
+34
+Time
+Precedence
+23
+10
+23
+10
+Priority
+11
+0
+11
+0
+Jitter
+19
+12
+19
+15
+Shared
+72
+46
+72
+46
+Memory
+Mutation
+49.39%
+70.2%
+Score
+For the throttle position control example, we
+generate 164 mutants for the Stateflow scheduler
+models and 198 mutants for the SimSched models.
+We achieve a mutation score of 49.39% for the State-
+flow scheduler model and 70.2% for the SimSched
+models. Evaluation results are similar to the previ-
+ous example. The time-related mutation operators
+have the most effective for mutation testing.
+We
+observe that the Shared Memory mutation operators
+have the same kills for both Mµ and M ′
+µ. In this
+example, task T2 has two Runnable TPSPrimary and
+APPSnsr each has a shared variable to update at each
+execution, and there is no direct relation between
+them. Though SimSched can simulate the preemp-
+tion of T2 to interrupt its execution, shared memory
+mutants’ model behaviors are the same for both Mµ
+and M ′
+µ.
+From the above two examples, we can see the
+mutation operators are application-dependent, and
+SimSched can achieve a higher mutation score at
+the time-related mutation operators. For example,
+the Precedence mutation operator kills zero mutants
+for the three servo example, but it kills ten mutants
+for the throttle position control example for both
+Stateflow Scheduler and SimSched. The three-servo
+example does not require any precedence at all;
+each task only controls itself. However, the throt-
+tle position control example requires precedence. A
+runnable consumes data from a previous runnable
+execution. If we alter the precedence, the execu-
+tion order is different from the original model ex-
+
+J. Chen et al. / Mutation Operators for Simulink Models
+ecution; it produces a different data flow. For exam-
+ple, in the Period mutation operator both Stateflow
+scheduler and SimSched kill the same mutants for
+the three serve example, but SimSched kills more
+mutants than the Stateflow Scheduler in the throttle
+position control example. Because the throttle po-
+sition control example has four runnables in T1 and
+each runnable has 1ms execution time. We use a
+mDTPER to T1 and generate a mutant that the pe-
+riod of T1 is 4ms instead of 5ms, then T1 will occupy
+all the execution time slots, and T2 will not be ex-
+ecuted for the SimSched. However, the Stateflow
+Scheduler does not consider the execution time, so
+both T1 and T2 are executed as scheduled.
+6.
+Related Work
+Our work aligns with the MBMT, and it has been
+applied to various models.
+Trakhtenbrot 41 pro-
+poses mutation testing on statechart-based models
+for reactive systems. This approach mainly deals
+with the conformance between specific semantics
+of statechart models and the model’s implementa-
+tion.
+Mutation testing has been applied to fea-
+ture models to test software product lines 20. The
+feature models represent the variability of software
+product lines and configurable systems.
+El-Fakih
+et al.14 develop a technique of mutation-based test
+case generation toward extended finite state ma-
+chines (EFSM) that examines the EFSM under test
+agrees to user-defined faults. Belli et al.6 have sur-
+veyed MBMT approach a great deal and detailed the
+approach applied to graph-based models, including
+directed graphs, event sequence graphs, finite-state
+machines, and statecharts. A recent mutation testing
+survey31 presents up-to-date advanced approaches
+that use mutants to support software engineering ac-
+tivities to model artifacts.
+Our work also fits in the timed system test-
+ing, which requires a real-time environment. Re-
+searchers 3,28,29 have utilized the most studied for-
+malisms TA to inject faults to the timed system and
+reveal that the time-related errors are unable to find
+by using randomly generated test suites.
+Nilsson
+et al.28 first proposed a set of extended TA muta-
+tion operators based on TA with Tasks (TAT) 30 to
+test real-time systems which depend on the execu-
+tion time and execution order of individual tasks.
+The mutation operators are interested in the time-
+liness of a task to meet its deadlines. Aichernig et
+al.3 propose a mutation testing framework for timed
+systems, where they define eight mutation operators
+to mutate the model and its mutants are expressed
+as a variant of TA in Uppaal specification format.
+The authors also develop a mutation-based test case
+generation framework for real-time, where they use
+symbolic bounded model checking techniques and
+incremental solving. Cornaglia et al.11 presents an
+automated framework MODELTime that facilitates
+the study of target platform-dependent timing dur-
+ing the model-based development of embedded ap-
+plications using MATLAB/Simulink simulations.
+ML/SL is one of the most popular formalisms to
+model and simulate embedded systems, and many
+researchers have explored the various type of mu-
+tation operators applied to ML/SL models. Hanh
+et al.18 propose a set of mutation operators based
+on investigating common faults in ML/SL models
+to validate test suites and present a process of muta-
+tion testing for ML/SL models. They provide twelve
+mutation operators and divide them into five cate-
+gories. Stephan et al.38 utilize the mutation testing
+technique to compare model-clone detection tools
+for ML/SL models.
+They present a taxonomy of
+ML/SL and prose a set of structural mutation op-
+erators based on three clone types. The mutation
+operators are used to evaluate the model-clone de-
+tectors. Using the mutation-based technique to gen-
+erate test cases for ML/SL models automatically has
+been studied 8,19.
+They can effectively generate
+small sets of test cases that achieve high coverage
+on a collection of Simulink models from the auto-
+motive domain. A recent work SLEMI 10 has ap-
+plied mutation techniques to the Simulink compiler
+and uses tools to generate mutants of the seed model
+and found 9 confirmed bugs in Simulink models.
+Our work intends to exploit mutation analysis
+to identify potential time-related errors in ML/SL
+models.
+Roy and Cordy 33,34 first propose using
+mutation analysis to assist the evaluation of soft-
+ware clone detection tools. They develop a frame-
+work for testing code-clone detectors based on mu-
+
+J. Chen et al. / Mutation Operators for Simulink Models
+tation. Stephan et al.37,36 proposed a framework that
+can objectively and quantitatively evaluate and com-
+pare model-clone detectors using mutation analysis.
+Their work is based on a structural mutation method
+for ML/SL model mutation. Our mutation operators
+are based on a timed system task model, whereas,
+there are no relevant existing studies that directly
+integrated the ML/SL models in the timed systems
+in the MIL phase; thus, we carry out the work pre-
+sented in this paper.
+Co-simulation16 is a widely used technique in
+model-based testing to verify as much of the system
+functionality, among subsystems, as possible. Com-
+posing the simulations of sub-simulators can achieve
+a joint simulation of a coupled system. Many differ-
+ent languages and tools are used for other purposes
+in the model-based design domain, either designing
+continuous plants or discrete controllers.
+A rela-
+tively recent open standard functional mock-up in-
+terface (FMI) is developed for exchange simulation
+models in a standardized format, including support
+for co-simulation. Gomes et al.15 propose an ap-
+proach to facilitate the implementation of the Func-
+tional Mock-up Interface standard.
+They use the
+MBT methodology to evaluate the tools that export
+Functional Mock-up Units (FMUs). Hence, they can
+root out the ambiguities and improve conformance
+to the FMI standard. Garro et al.7 employs FMI to
+perform co-simulation to verify the system require-
+ments based on the FOrmal Requirements Model-
+ing Language and the Modelica language.
+Zafar
+et al.42 present a systematic tool-supported MBT
+workflow to facilitate the simulation-based testing
+process of an embedded system. The workflow ex-
+pends from the requirements phase, and generation
+of executable test scripts, to the execution of gener-
+ated test scripts on simulation levels.
+7.
+Conclusion and future work
+In this paper, we proposed a set of timed muta-
+tion operators for the ML/SL model that is primar-
+ily intended to integrate the timed task model in the
+ML/SL model to better support MIL simulation us-
+ing mutation analysis. Moreover, testing at an ear-
+lier stage during the development process reduces
+development costs since earlier changes and fixing
+errors are much more manageable. We introduce a
+timed task model and present a set of mutation op-
+erators for the ML/SL based on this task model. We
+implement a mutation analysis framework that can
+apply mutation operators to the simple ML/SL mod-
+els. We demonstrate the approach on several ML/SL
+models. The results validate that mutation analysis
+can reveal time-related faults. We intend to automate
+the mutation testing process for the ML/SL environ-
+ment and improve the mutation operators to expose
+defects in the future. We will further validate our
+mutation analysis method to more industrial com-
+plex ML/SL model sets.
+Acknowledgments
+This work was supported in part by the Natural Sci-
+ences and Engineering Research Council of Canada
+(NSERC), as part of the NECSIS Automotive Part-
+nership with General Motors, IBM Canada, and Ma-
+lina Software Corp.
+1. Bernhard K. Aichernig, Harald Brandl, Elisabeth
+J¨obstl, Willibald Krenn, Rupert Schlick, and Stefan
+Tiran.
+Killing strategies for model-based mutation
+testing. Software Testing, Verification and Reliability,
+25(8):716–748, dec 2015.
+2. Bernhard K. Aichernig and Florian Lorber. Towards
+generation of adaptive test cases from partial models
+of determinized timed automata. In 2015 IEEE Eighth
+International Conference on Software Testing, Verifi-
+cation and Validation Workshops (ICSTW), pages 1–6.
+IEEE, apr 2015.
+3. Bernhard K. Aichernig, Florian Lorber, and Dejan
+Niˇckovi´c. Time for mutants - Model-based mutation
+testing with timed automata. In Lecture Notes in Com-
+puter Science (including subseries Lecture Notes in
+Artificial Intelligence and Lecture Notes in Bioinfor-
+matics), volume 7942 LNCS, pages 20–38. Springer,
+Berlin, Heidelberg, 2013.
+4. Rajeev Alur and David L. Dill. A theory of timed au-
+tomata. Theoretical Computer Science, 126(2):183–
+235, apr 1994.
+5. AUTOSAR. Autosar development partnership. http:
+//www.autosar.org, 2018.
+6. Fevzi Belli, Christof J. Budnik, Axel Hollmann,
+Tugkan Tuglular, and W. Eric Wong.
+Model-based
+mutation testing—Approach and case studies. Science
+of Computer Programming, 120:25–48, may 2016.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+7. Daniel Bouskela, Alberto Falcone, Alfredo Garro, Au-
+drey Jardin, Martin Otter, Nguyen Thuy, and Andrea
+Tundis.
+Formal requirements modeling for cyber-
+physical systems engineering: an integrated solution
+based on form-l and modelica. Requirements Engi-
+neering, pages 1–30, 2021.
+8. Angelo Brillout, Nannan He, Michele Mazzucchi,
+Daniel Kroening, Mitra Purandare, Philipp R¨ummer,
+and Georg Weissenbacher. Mutation-based test case
+generation for Simulink models.
+In Lecture Notes
+in Computer Science (including subseries Lecture
+Notes in Artificial Intelligence and Lecture Notes in
+Bioinformatics), volume 6286 LNCS, pages 208–227,
+2010.
+9. Jian Chen, Manar H Alalfi, Thomas R Dean, and
+S Ramesh.
+Modeling autosar implementations
+in simulink.
+In European Conference on Mod-
+elling Foundations and Applications, pages 279–292.
+Springer, 2018.
+10. Shafiul Azam Chowdhury, Sohil Lal Shrestha, Tay-
+lor T. Johnson, and Christoph Csallner. Slemi: Equiva-
+lence modulo input (emi) based mutation of cps mod-
+els for finding compiler bugs in simulink.
+In 2020
+IEEE/ACM 42nd International Conference on Soft-
+ware Engineering (ICSE), pages 335–346, 2020.
+11. Alessandro Cornaglia, Shakib Hasan, Alexander
+Viehl, Oliver Bringmann, and Wolfgang Rosenstiel.
+Modeltime: Fully automated timing exploration of
+simulink models for embedded processors. In AmE
+2019 - Automotive meets Electronics; 10th GMM-
+Symposium, pages 1–6, 2019.
+12. Fabio Cremona, Matteo Morelli, and Marco Di Na-
+tale.
+TRES: A Modular Representation of Sched-
+ulers, Tasks, and Messages to Control Simulations
+in Simulink. Proceedings of the 30th Annual ACM
+Symposium on Applied Computing, pages 1940–1947,
+2015.
+13. R.A. DeMillo, R.J. Lipton, and F.G. Sayward. Hints
+on Test Data Selection: Help for the Practicing Pro-
+grammer. Computer, 11(4):34–41, apr 1978.
+14. Khaled
+El-Fakih,
+Anton
+Kolomeez,
+Svetlana
+Prokopenko, and Nina Yevtushenko.
+Extended
+Finite State Machine Based Test Derivation Driven
+by User Defined Faults.
+In 2008 International
+Conference on Software Testing, Verification, and
+Validation, pages 308–317. IEEE, apr 2008.
+15. Cl´audio Gomes, Romain Franceschini, Nick Battle,
+Casper Thule, Kenneth Lausdahl, Hans Vangheluwe,
+and Peter Gorm Larsen. Application of model-based
+testing to dynamic evaluation of functional mockup
+units. 2020.
+16. Cl´audio Gomes, Casper Thule, David Broman, Pe-
+ter Gorm Larsen, and Hans Vangheluwe.
+Co-
+simulation: a survey.
+ACM Computing Surveys
+(CSUR), 51(3):1–33, 2018.
+17. R.G. Hamlet.
+Testing Programs with the Aid of a
+Compiler. IEEE Transactions on Software Engineer-
+ing, SE-3(4):279–290, jul 1977.
+18. Le Thi My Hanh and Nguyen Thanh Binh.
+Muta-
+tion operators for simulink models. In Proceedings -
+4th International Conference on Knowledge and Sys-
+tems Engineering, KSE 2012, pages 54–59. IEEE, aug
+2012.
+19. N. He, P. R¨ummer, and D. Kroening. Test-case gener-
+ation for embedded simulink via formal concept anal-
+ysis. In 2011 48th ACM/EDAC/IEEE Design Automa-
+tion Conference (DAC), pages 224–229, 2011.
+20. Christopher Henard, Mike Papadakis, Gilles Perrouin,
+Jacques Klein, and Yves Le Traon. Assessing Soft-
+ware Product Line Testing Via Model-Based Mu-
+tation: An Application to Similarity Testing.
+In
+2013 IEEE Sixth International Conference on Soft-
+ware Testing, Verification and Validation Workshops,
+pages 188–197. IEEE, mar 2013.
+21. Dan Henriksson, Anton Cervin, and Karl-Erik ˚Arz´en.
+TrueTime : Real-time Control System Simulation with
+MATLAB / Simulink.
+Proceedings of the Nordic
+MATLAB Conference, 2003.
+22. Yue Jia and Mark Harman. An Analysis and Survey
+of the Development of Mutation Testing. IEEE Trans-
+actions on Software Engineering, 37(5):649–678, sep
+2011.
+23. Edward A. Lee, Stephen Neuendorffer, and Gang
+Zhou.
+Synchronous Reactive Models.
+In Claudius
+Ptolemaeus, editor, System Design, Modeling, and
+Simulation using Ptolemy II. Ptolemy.org, 2014.
+24. J. Lehoczky, L. Sha, and Y. Ding. The rate monotonic
+scheduling algorithm: exact characterization and av-
+erage case behavior. [1989] Proceedings. Real-Time
+Systems Symposium, pages 0–5, 1989.
+25. MathWorks. Simulink User’s Guide, r2017b. http:
+//www.mathworks.com, 2017.
+26. Reza Matinnejad, Shiva Nejati, Lionel C. Briand, and
+Thomas Bruckmann. Automated test suite generation
+for time-continuous simulink models. In Proceedings
+of the 38th International Conference on Software En-
+gineering - ICSE ’16, pages 595–606, New York, New
+York, USA, 2016. ACM Press.
+27. Andreas Naderlinger. Simulating preemptive schedul-
+ing with timing-aware blocks in Simulink. In Design,
+Automation & Test in Europe Conference & Exhibi-
+tion (DATE), 2017, pages 758–763. IEEE, mar 2017.
+28. R. Nilsson, J. Offutt, and S.F. Andler.
+Mutation-
+based testing criteria for timeliness.
+In Proceed-
+ings of the 28th Annual International Computer Soft-
+ware and Applications Conference, 2004. COMPSAC
+2004., pages 306–311. IEEE, 2004.
+29. Robert Nilsson and Jeff Offutt. Automated testing of
+timeliness: A case study. In Proceedings - Interna-
+tional Conference on Software Engineering, 2007.
+
+J. Chen et al. / Mutation Operators for Simulink Models
+30. C. Norstrom, A. Wall, and Wang Yi. Timed automata
+as task models for event-driven systems.
+In Pro-
+ceedings Sixth International Conference on Real-Time
+Computing Systems and Applications. RTCSA’99
+(Cat. No.PR00306), pages 182–189, 1999.
+31. Mike Papadakis, Marinos Kintis, Jie Zhang, Yue Jia,
+Yves Le Traon, and Mark Harman.
+Mutation test-
+ing advances: an analysis and survey.
+In Advances
+in Computers, volume 112, pages 275–378. Elsevier,
+2019.
+32. Mike Papadakis, Marinos Kintis, Jie Zhang, Yue Jia,
+Yves Le Traon, and Mark Harman. Mutation Test-
+ing Advances: An Analysis and Survey. Advances in
+Computers, 112:275–378, jan 2019.
+33. Chanchal K. Roy and James R. Cordy.
+A muta-
+tion / injection-based automatic framework for eval-
+uating code clone detection tools. In IEEE Interna-
+tional Conference on Software Testing, Verification,
+and Validation Workshops, ICSTW 2009, pages 157–
+166, 2009.
+34. Chanchal K. Roy, James R. Cordy, and Rainer
+Koschke.
+Comparison and evaluation of code
+clone detection techniques and tools: A qualita-
+tive approach.
+Science of Computer Programming,
+74(7):470–495, may 2009.
+35. Julien Schmaltz and Jan Tretmans.
+On confor-
+mance testing for timed systems.
+In Lecture Notes
+in Computer Science (including subseries Lecture
+Notes in Artificial Intelligence and Lecture Notes in
+Bioinformatics), volume 5215 LNCS, pages 250–264.
+Springer, Berlin, Heidelberg, 2008.
+36. Matthew Stephan.
+Model clone detector evaluation
+using mutation analysis.
+In Proceedings - 30th In-
+ternational Conference on Software Maintenance and
+Evolution, ICSME 2014, pages 633–638. Institute of
+Electrical and Electronics Engineers Inc., dec 2014.
+37. Matthew Stephan, Manar H. Alafi, Andrew Steven-
+son, and James R. Cordy.
+Using mutation analysis
+for a model-clone detector comparison framework. In
+Proceedings - International Conference on Software
+Engineering, pages 1261–1264, 2013.
+38. Matthew Stephan, Manar H Alalfi, and James R
+Cordy. Towards a taxonomy for Simulink model mu-
+tations. In Proceedings - IEEE 7th International Con-
+ference on Software Testing, Verification and Valida-
+tion Workshops, ICSTW 2014, pages 206–215, 2014.
+39. The AUTOSAR Consortium. Applying simulink to
+autosar, r3.1., 2006.
+40. The AUTOSAR Consortium. The AUTOSAR Stan-
+dard, r4.3., 2018.
+41. Mark Trakhtenbrot. Implementation-Oriented Muta-
+tion Testing of Statechart Models. In 2010 Third Inter-
+national Conference on Software Testing, Verification,
+and Validation Workshops, pages 120–125. IEEE, apr
+2010.
+42. Muhammad Nouman Zafar, Wasif Afzal, and Eduard
+Enoiu. Towards a workflow for model-based testing of
+embedded systems. In Proceedings of the 12th Inter-
+national Workshop on Automating TEST Case Design,
+Selection, and Evaluation, pages 33–40, 2021.
+43. Yuan Zhan and John A. Clark. Search-based muta-
+tion testing for simulink models.
+In GECCO 2005
+- Genetic and Evolutionary Computation Conference,
+pages 1061–1068, New York, New York, USA, 2005.
+ACM Press.
+
diff --git a/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/load_file.txt b/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..44250dba1a4d65a601bc64dc0f934388a29d70e1
--- /dev/null
+++ b/OdAyT4oBgHgl3EQf7PpY/content/tmp_files/load_file.txt
@@ -0,0 +1,2135 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf,len=2134
+page_content='Timed Model-Based Mutation Operators for Simulink Models  Jian Chen* 1 Manar H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Alalfi 2 Thomas R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Dean 3  1 Department of Electrical and Computer Engineering, Queen’s University  Kingston, ON, Canada  E-mail: jian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='chen@queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='ca  2 Department of Computer Science, Ryerson University  Toronto, ON, Canada  E-mail: manar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='alalfi@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='ryerson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='ca  3 Department of Electrical and Computer Engineering, Queen’s University  Kingston, ON, Canada  E-mail: tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='dean@queensu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='ca  Abstract  Model-based mutation analysis is a recent research area, and real-time system testing can benefit from  using model mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Model-based mutation testing (MBMT) is a particular branch of model-based test-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It generates faulty versions of a model using mutation operators to evaluate and improve test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation testing is an effective way to ensure software correctness and has been applied to various appli-  cation areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink is a vital modeling language for real-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This paper introduces Simulink  model mutation analysis to improve Model-in-the-loop (MIL) testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We propose a set of Simulink mu-  tation operators based on AUTOSAR, which reflects the temporal correctness when a Simulink model  is mapped to Operating System tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We implement a mutation framework that generates mutants for  implicit clock Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Finally, we demonstrate how this framework generates mutants to reveal  task interference issues in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Our work integrates the Simulink model with the timed systems  to better support mutation testing automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Keywords: Mutation Testing, Model-Based Testing, Model-Based Mutation Testing, Mutation Operator,  Simulink, Real-Time System, Scheduling, AUTOSAR  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Introduction  Today, cars come equipped with advanced technolo-  gies that did not exist before, such as Automatic  Emergency Braking (AEB), Adaptive Cruise Con-  trol (ACC), Lane Departure Warning/Lane Keeping,  and Autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' All of these features rely  on software to realize sophisticated control algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Generally, such software is developed within  the timed system context, in which the system cor-  rectness not only relies on the software implemented  functions correctness but also depends on the sys-  tem to meet time constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Many factors can con-  tribute to the execution time of a system running on  a target platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Issues such as task interference  may cause delays during task execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Software  quality plays a crucial role in such safety-critical ap-  plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Model-Based Testing (MBT) is a promising  technique for the automated testing of timed sys-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='00835v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='SE]  2 Jan 2023  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A model represents the behavior of software,  and the model is usually abstracted from real-time  specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, some modeling environ-  ments support this feature in the Hardware-in-the-  loop (HIL) simulation testing instead of the MIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For example, Matlab/Simulink (ML/SL) simulations  assume block behaviors are completed in nearly zero  execution time, while real execution requires a finite  execution time, which may cause a failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' ML/SL  models are based on the Synchronous Reactive (SR)  model 23 that may assume the task execution times  are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Errors in the model may not be apparent  without an explicit real-time execution in the MIL  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Usually, a Simulink model can be well simu-  lated in the MIL, but it may have errors in the real-  time context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Hence, MBT needs an extension to accommo-  date the real-time context, which includes modeling  the system through a timed formalism, and check-  ing the implementation conforms to its specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Traditionally, this is done via conformance checks  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Recently, several tools have been proposed to  simulate the real-time execution effects for ML/SL  models in MIL, such as TrueTime 21, TRES 12,  Timing-aware blocks 27, and SimSched 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' SimSched  uses a model transformation to integrate scheduling  into the model to validate the real-time context dur-  ing simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' To evaluate SimSched, we turn to  mutation testing using mutation analysis to assist the  evaluation of the SimSched tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In this paper, we propose a set of mutation op-  erators with a timed task model, which is based  on the AUTomotive Open System ARchitecture  (AUTOSAR), that reflects the temporal correctness  when a Simulink model is mapped to Real-Time Op-  erating System (RTOS) tasks in a real-time context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  This paper is organized as follows:  Section  2 introduces background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Section 3  presents the set of proposed timed mutation oper-  ators for Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Section 4 explains the  usage of the timed mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Section 5  presents validation experiments and results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Section  6 summarizes related studies in MBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Finally, Sec-  tion 7 presents the conclusions of our work and out-  lines future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Background  This section gives an overview of the background  information on the material needed to explain our  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We begin with a basic introduction to mu-  tation testing, Simulink, and AUTOSAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' then, we  present our timed task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation testing  Mutation testing was introduced in the 1970s 17,13,22  and proved to be an effective way to reveal software  faults 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It is a fault-based software testing tech-  nique, which has been extensively studied and used  for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It contributes a range of methods, tools,  and reliable results for software testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation  testing is designed to find valid test cases and dis-  cover real errors in the program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Model-Based Mutation Testing (MBMT) takes  the advantages of both model-based testing and mu-  tation testing and has been widely applied to mul-  tiple types of models such as feature models 20,  statechart-based models 41,1, timed automata 3,2, and  Simulink 8,26,38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, in real-time system de-  velopment, both logical and temporal correctness is  crucial to the correct system functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The tem-  poral correctness depends on timing assumptions for  each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Timed Automata (TA) 4 is a common for-  malism to model and verify real-time systems to see  whether designs meet temporal requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Aich-  ernig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3 propose an MBMT technique for timed  automata that applies to input/output timed automata  (TAIO) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Nilsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  28 add an extension  to the TA formalism with a task model, and their  mutation operators focus on timeliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink∗is  widely used for model-driven development of soft-  ware within the automotive sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Most of the mu-  tation operators proposed for Simulink models are  from a property point of view either run-time or  design-time such as signal modification, arithmetic  alternation, or block change 18,38,36,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Some of the  proposed mutation testings are targeted at test case  generation for Simulink models 8,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, there  is no mutation operator with an explicit clock model  for Simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='com/products/simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='html  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Simulink  Simulink is one of the most popular modeling lan-  guages for modeling dynamical systems, and MAT-  LAB provides a graphical programming environ-  ment to perform system simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink mod-  els are graphical blocks and lines, and they are con-  nected by signals between input and output ports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The Simulink simulation engine determines the ex-  ecution order of blocks based on the data depen-  dencies among the blocks before a simulation exe-  cution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink defines two types of blocks, di-  rect feedthrough, and non-direct feedthrough, to as-  sure the correct data dependencies in the simula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink uses the following two basic rules  25 to determine the sorted execution order: A block  must be executed before any of the blocks whose  direct-feedthrough ports it drives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Blocks without  direct feedthrough inputs can execute in arbitrary or-  der as long as they precede any block whose direct-  feedthrough inputs they drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' All blocks are sched-  uled in sorted order and executed in sequential exe-  cution order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Simulink engine maintains a vir-  tual clock to execute each ordered block at each vir-  tual time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Simulink Coder†supports code generation and of-  fers a framework to execute the generated code in  a real-time environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink Coder can gen-  erate code for the periodic task, either using a sin-  gle task or a multi-task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Single-task implementa-  tions can preserve the semantics during the simula-  tion because the generated code is invoked by a sim-  ple scheduler in a single thread without preemptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For multi-task implementations, the generated code  is invoked by a rate monotonic (RM) 24 scheduler  in a multithreaded RTOS environment, where each  task is assigned a priority and preemptions occur be-  tween tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' As a consequence of preemption and  scheduling, the implementation semantic can con-  flict with the model semantic in a multi-rate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  AUTOSAR  AUTOSAR is an open industry standard to meet the  needs of future car development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' AUTOSAR de-  fines three main layers: the application, the runtime  environment (RTE), and the basic software (BSW)  layer 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The functions in the application layer are  implemented by SW-Cs, which encapsulate part or  all of the automotive electronic functions, as shown  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The components communicate via a  Virtual Functional Bus (VFB), which is an abstrac-  tion of all the communication mechanisms of AU-  TOSAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Engineers abstract the communication de-  tails of software components employing VFBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A  set of runnables represents the SW-Cs internal be-  haviors, and a runnable is the smallest executable  code that can be individually scheduled, either by  a timer or an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Lastly, runnables are required  to map to a set of tasks for a target platform, and  the mapping has to preserve ordering relations and  causal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink has supported AU-  TOSAR compliant code generation since version  R2006a‡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  All AUTOSAR concepts can be repre-  sented by Simulink blocks and the existing Simulink  blocks can be easily used in the AUTOSAR devel-  opment process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Some of AUTOSAR concepts and  Simulink concepts mapping relation is shown in Ta-  ble 1 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' AUTOSAR components, interfaces, and runnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  (Adapted from 5)  † https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='com/products/simulink-coder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='html  ‡ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='com/products/simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='html  SW-C 1  SW-C 2  SW-C 3  SW-C n  Runnable 2a  Runnable 3a  Runnable na  Runnable 1a  ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Runnable 2b  Runnable nb  NTOIA  V  OIDID  Virtual Function Bus(VFB)  Tool supporting deployment  System  ECU  of sW components  Constraints  Descriptions  个  ECU I  ECU II  ECU M  SW-C 1  SW-C 3  SW-C 2  SW-C n  RTE  RTE  RTE  Basic Software  Basic Software  Basic SoftwareJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Examples of ML/SL and AUTOSAR Concepts Map-  ping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  ML/SL  AUTOSAR  Subsystem  Atomic Software  Component  Function-call subsystem  Runnable  Function calls  RTEEvents  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task model  In automotive software, Simulink models are often  drawn from real-time specifications and are realized  as a set of tasks running on an RTOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In order to  better test this kind of software in the MIL phase,  model-based testing needs to be scaled to the real-  time context, which includes a timed formalism to  model the system under test conforming with the  real-time requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We define a task model to  model the timing properties of tasks in the Simulink  environment and the application is modeled as a set  of periodic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  A task model, T, is represented by a tuple  {φ,ρ,c,γ, prect, precr, prio, jitter}, where φ is an  offset of the task, ρ is the period of the task, c is the  Worst Case Execution Time (WCET) of the task, γ  is a list of runnables that belong to the task, prect is  the precedence constraint of the task, precr is the  precedence constraint of the runnables within the  task, prio is the priority associated with the task,  and jitter is the deviation of a task from the periodic  release times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Every task has an implicit deadline  which means the deadline of a task is equal to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' An  offset φ refers to the time delay between the arrival  of the first instance of a periodic task and its release  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A WCET is the summation of each runnable  execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A precedence constraint prect is a  list of tasks that specifies the task execution order,  and precr is a list of runnables within a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task states and transitions of task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 2 shows the task-state and transition dia-  grams of the task model that is based on OSEK’s ba-  sic task-state model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The task model includes three  states: suspended, ready, and running, and four tran-  sitions: Active, Stare, Preempt, and Terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  transitions represent the actions to activate, start,  preempt, or terminate a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task timing parameters shown in Gantt chart (all  related to Task2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 3 shows the timing parameters of a task  model and different timing parameters can alter the  application’s real-time behavior within a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation Operators for Simulink Model  This section introduces a mutation analysis ap-  proach to validate real-time context during the sim-  ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation operators are the key elements of  mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Model-based mutation testing is  a method of injecting faults into models to check  whether the tests are passed or failed, thus validat-  ing the software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The injecting faults are the muta-  tion operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Before we apply mutation operators  to the model, we need to identify them which is to  Start  Ready  Running  Preempt  Active  Terminate  SuspendedPriority  Active  Start  Preempt  Terminate  Task1  Task2  offseti  C1  C2  jitter  period  TimeJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  understand what kind of errors can cause failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We  have proposed the following task-related mutation  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Offset mutation operators  The task release offset is one of the factors that affect  the computation result in terms of task interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In order to take the offset into account for analy-  sis, we introduced an offset mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a  known offset φ, a task can now execute after φ time  units with respect to the start of its period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The exe-  cution time of the task is unchanged at c time units  before the next period starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mITO  This operator adds δ time to the current offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a  given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the offset φi to  φi +δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mDTO  This operator subtracts δ time to the current offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the offset φi to  φi −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Period mutation operators  An RTOS usually applies a preemptive multitask-  ing scheduling algorithm to determine the execu-  tion order of tasks, and the most picked algorithm is  fixed-priority scheduling (FPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The algorithm as-  signs each task a static priority level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The RTOS  scheduler executes the highest priority task from the  ready task queue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink Coder supports an RM  scheduler, where the priority of a task is associated  with its period, if a task has a smaller period, then it  has a higher priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Furthermore, a lower-priority  task can be preempted by a more top-priority task  during the execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mITPER  This operator increases the period of a task, which  changes the task to a slower rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a given task  τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this  mutation operator changes the period of the task i  to ρi +δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mDTPER  This operator decreases the period of a task, which  changes the task to a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a given task  τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this  mutation operator changes the period of the task i  to ρi −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Execution time mutation operators  The correctness of a real-time system is determined  on one hand by the computation results of the log-  ical program, and on the other hand, is strictly re-  lated to the time at which the results are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Hence, execution time analysis is essential during  the process of designing and verifying embedded  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For this reason, we propose execution time  operators, which can adjust the execution time of  each task at the runnable level to simulate the run  time execution on different processor speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  longer execution time of a task may lead to a sce-  nario where a lower-rate task blocks a higher-rate  task so that it misses its deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mITET  This operator adds δ time to the current exe-  cution time of each runnable, which increases  the total execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For a given task  τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this  mutation operator changes the execution time ci to  ci +δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  mDTET  This operator subtracts δ  time from the cur-  rent execution time of each runnable, which de-  creases the total execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a given task  τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈ T, this  mutation operator changes the execution time ci to  ci −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Execution precedence mutation operators  The RTOS scheduler selects tasks to execute accord-  ing to the priority level of the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, the  spawn order determines the execution order of tasks  with the same priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Whichever task is spawned  first is realized and gets the CPU first to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This re-  sults in situations where a pair of tasks have a prece-  dence relation in the implementation that does not  exist in the design phase lost an existing precedence  relation in the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The incorrect prece-  dence can cause a wrong execution order of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Hence, we proposed the precedence mutation oper-  ators which can specify a precedence relation be-  tween a pair of tasks and runnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This operator  creates mutants by assigning a specific execution or-  der to a set of tasks or runnable to reflect the prece-  dence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mATPREC  For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,  jitteri} ∈ T, for each task τ j ∈ T (j ̸= i), if τj /∈  precti, this mutation operator adds τj to precti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRTPREC  For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,  jitteri} ∈ T, for each task τj ∈ T (j ̸= i), if  τj ∈ precti, this mutation operator removes τ j from  precti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mARPREC  For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,  jitteri} ∈ T, for each runnable γim ∈ τi, if γim /∈  precri, this mutation operator adds γim to precri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRRPREC  For a given task τi{φi,ρi,ci,γi, precti, precri, prioi,  jitteri} ∈ T, for each runnable γim ∈ τi, if γim ∈  precri, this mutation operator removes γim from  precri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Priority mutation operators  In an RTOS, each task is assigned a relative priority,  which is a static integer to identify the degree of im-  portance of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The highest priority task always  gets the CPU when it becomes ready to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  most common RTOS scheduling algorithm is pre-  emptive scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mITPRI  This operator increases the priority of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a  given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the priority of the  task prioi to proii +δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mDTPRI  This operator decreases the priority of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a  given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the period of the  task i to proii −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Jitter mutation operators  Timing jitter exists in the RTOS, and it is the delay  between subsequent periods of time for a given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mITJ  This operator increases the jitter time of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For a  given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the priority of the  task jitteri to jitteri +δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  mDTJ  This operator decreases the jitter time of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For  a given task τi{φi,ρi,ci,γi, precti, precri, prioi, jitteri} ∈  T, this mutation operator changes the period of the  task jitteri to jitteri −δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Shared memory mutation operators  It is common that RTOS tasks exchange data or  information via shared memory(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', global vari-  able, memory buffer, hardware register).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The shared  memory can easily cause access conflict if the logi-  cal software design is neglected in any corner case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Here we introduce a set of variable mutation opera-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mDSM  This operator defines a new value to a global vari-  able in a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If a task reads this global variable,  then we define a new value right before the reads  occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mUDSM  This operator un-defines a global variable in a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  If a task writes this global variable, then ignore this  writes operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRDSM  This operator removes the definition of a global vari-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If a global variable is initialized in a task then  do not initialize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRSM  This operator adds a reference to a global variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRMSMR  This operator removes reference to a global variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' mRSMR  This operator replaces a reference of a global vari-  able with a different global variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simuilnk Mutation Operators  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Mutation Key  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Title  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mITO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Increase Task Offset  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDTO  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Decrease Task Offset  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mITPER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Increase Task Period  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDTPER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Decrease Task Period  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mITET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Increase Task Execution Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDTET  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Decrease Task Execution Time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mATPREC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Add Task Precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRTPREC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Remove Task Precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mARPREC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Add Runnable Precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRRPREC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Remove Runnable Precedence  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mITPRI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Increase Task Priority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDTPRI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Decrease Task Priority  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mITJ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Increase Task Jitter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDTJ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Decrease Task Jitter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mDSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Define Shared Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mUDSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Un-define Shared Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRDSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Remove Definition Shared Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Reference a Shared Memory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRMSMR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Remove a Shared Memory Reference  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mRSMR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='Replace a Shared Memory Reference  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation operators demonstration  We have introduced twenty mutation operators cate-  gorized into seven classes and explained each mu-  tation class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The mutation operators are summa-  rized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We use simple examples to demon-  strate the use of each mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' To demon-  strate our mutation operators, we use the tool Sim-  Sched to alter the properties of software applications  realized as Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  From Table 1, we  know that each function-call subsystem represents  an AUTOSAR runnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The function-call subsys-  tem can be executed conditionally when a function-  call event signal arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Both an S-function block  and a Stateflow block can provide such a function-  call event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' SimSched applies the function-call in-  vocation mechanism to use an S-function to gener-  ate a function-call event to schedule each runnable  (function-call subsystem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 4 shows the Sim-  Sched parameters dialogue that we can utilize it to  adjust the timing properties to implement the muta-  tion operator for Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' SimSched Parameter setting dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In this section, we use several simple examples  to exhibit the mutants generated by our mutation op-  erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 5 illustrates the use of SimSched to  schedule a Simulink model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In this example, Sim-  Sched is at the top left corner, which schedules three  runnables (R1, R2, R3), and they are mapped to two  tasks (T1, T2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Runnable R1 is mapped to T1, the pe-  riod of T1 is 10ms, priority is 2, and the execution  time of R1 is 3ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R2 and R3 are mapped to T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  period of T2 is 20ms, priority is 1, and the execu-  tion time of R2 and R3 are 3ms and 3ms accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The detailed parameter settings are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  There is a Data Store Memory block in this exam-  ple, named A, which defines a shared data store that  is a memory area used by Data Store Read and Data  Store Write block with the same data store name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R1  writes a constant value to a global variable A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R2  reads A first then writes the summation of A and its  delay value to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R3 reads A then subtracts its delay  value from A, and outputs the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simple example settings  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T1  10  3  2  R1  T2  20  3  1  R2  T2  20  3  1  R3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple example of using SimSched to schedule  AUTOSAR SW-Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple example of using Stateflow to schedule  AUTOSAR SW-Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple example output of Stateflow scheduler  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  BlockParameters:SimSched  S-Function (mask)  Parameters  Task:  [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2]  Priority:  [2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1]  Period:  [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='20]  Runnable:  "R1\',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='\'R2\',"' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="'R3'  [1," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2]  Task Mapping:  Execution Time:  [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3]  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0]  Offset  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0]  Jitter  OK  Cancel  Help  ApplyScheduler  Runnable(period,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' execution time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' priority) Task()  R1(10ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2)  Task(1)  R2( 20ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task(2)  R3 20ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task(2)  function()  function()  function(  SimSched  function  Runnable1 subsystem  Runnable2 subsystem  Runnable3 subsystem1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="060:0  call()  0:1  R1()  1 ms Clock  R2()  0:1  R3()  call(  Temporal Logic  0:6  Scheduler  0:1  0:1  0:1  R1()  R2()  R3()  R1  0:7  A  subrater  Runnable1 subsystem1  Runnable2 subsystem1  Runnable3 subsystem'11  10  9  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  40  R2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  20  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  TimeJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  We use a Stateflow scheduler version of the sim-  ple example shown in Figure 7 to show the typical  Stateflow scheduler simulation result, then compare  it with the SimSched scheduler simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The task parameters are all the same shown in Table  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply the same task configurations for both  the Stateflow scheduler and SimSched models for  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 8 shows the Stateflow scheduler  simulation result, and Figure 9 shows the SimSched  simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The result figures show the output  value of each runnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' From the figure, we can see  that R1, R2 and R3 are all executed at time 0 in Fig-  ure 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R1, is executed time 0, R2 is executed at 3ms,  and R3 is executed at 6ms in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R2 and R3 are  executed later than the Stateflow scheduler simula-  tion in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This is because that SimSched takes  into account execution time, and each task must be  executed until the previous task is completed on a  single core platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple example output of SimSched simulation without  applying any mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Offset mutation operators  We first apply the mITO mutation operator to the  running example, let’s say that increase δ1 = 3ms  for T1 then we have the task execution timeline in  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We can see T2 is executed first at time  0, and T1 preempts T2 at 3ms in the first period due  to the offset effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' After the first period, there is  no preemption between T1 and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Then, we apply  the mDTO mutation operator based on the previous  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set δ1 = −1ms to T1 then the offset  for T1 is 2ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 11 shows the task execution  timeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T2 is preempted by T1 during the execution  of the first period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Compared to the task execution  Gantt chart of our running example shown in Figure  6 with no offset, we can clearly see the preemption  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example  after increase offset mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example  after decrease offset mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The running example’s output after applying the  mITO mutation operator is shown in Figure 12  which is different from Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Because T1 pre-  empts T2 at the first instance execution, the output  of R3 is from zero to ten then goes back to zero  then goes up instead of always increasing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  running example’s output after applying the mDTO  mutation operator is the same as Figure 9 because  the offset operator only affects the initial execution  of each task, and the preemption occurs before the  first execution of R2 instance completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Inside our  model scheduler program, we trigger each subsys-  tem at the end of each execution time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Techni-  cally, the execution order of this example is still R1,  R2, and R3 so the output of the simulation keeps the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple example output of SimSched simulation after  applying mITO mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  T1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=')  T2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' )T1  06  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=')  T2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' )Runnable1 subsystem  11  10  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=')  Runnable2_subsystem  20  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=')  Runnable3_subsystem  20  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' )11  10  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  30  20  oR2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  20  10  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  TimeJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Period mutation operators  We apply the mITPER operator to this example to  increase the period of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set δ1 = 1ms to T1  so the period of the task T1 is 11ms now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 13  shows that T2 is preempted at the time of 22ms, and  the simulation yields a wrong result due to this pre-  emption shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The output of R3 is an  alternating value instead of an increasing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example  after increase period mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple example output of SimSched simulation after  applying mITPER mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We apply the mDTPER operator to this example  to decrease the period of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set δ1 = −4ms  to T1 so the period of the task T1 is 6ms now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Then,  we run the simulation, T2 is preempted by T1 shown  in Figure 15 and it yields a wrong simulation result  shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The output of R3 is either zero  or ten instead of an increasing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example  after decreasing period mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple example output of SimSched simulation  after applying mDTPER mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Execution time mutation operators  We apply the mITET operator to this example to in-  crease the execution time of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We can specify  any runnable to increase its execution time within a  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For example, we set δ1 = 4ms to R2 in T2 so  the execution of R2 is 7ms and T2 takes 10ms to ex-  ecute now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 17 shows that T2 is preempted at  the time of 10ms, and the simulation yields a wrong  result due to this preemption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The wrong result is  the same as the example of applying decreasing task  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply the mDTET operator to this exam-  ple to decrease the execution time of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set  δ1 = −1ms to T1 so the execution time of the task T1  is 2ms now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Then, we run the simulation, there is no  preemption that occurs between these two tasks and  the output is as expected as the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of the running example  after increase execution time mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Execution precedence mutation operators  We introduce the second example as Table 4 to ex-  plain the mATPREC and mRTPREC operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Fig-  ure 18 shows the task execution Gantt chart of this  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' From the task execution chart, we can see  the execution order of the tasks is T1,T2,T1,T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0E  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Time(sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' )Runnable1_sybsystem  11  10  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  Runnable2_subsystem  400  200  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  Runnable3_subsystem  400  200  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  9:0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9T1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  T2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06Runnable1_sybsystem  11  10  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  Runnable2_subsystem  40  20  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  Runnable3_subsystem  10  5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simple example settings  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T1  5  1  3  R1  T2  10  4  2  R2  T3  10  3  1  R3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  First, we assume there is no precedence rela-  tion among tasks so we use the mATPREC mutation  operator to add a precedence relation τ3 to prect2,  which specifies that a new instance of T2 cannot start  unless T3 has executed after the last instance of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Hence, we set the execution order that T3 is executed  before T2 in the setting dialogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 19 shows  the execution result that T2 is preempted by T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If T2  is not a re-entrant function then this preemption may  cause potential failure execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart of example 2 after task  precedence mutation operator is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Then, we assume there is a precedence relation  between T1 and T3 and the task execution diagram  is the same in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply the mRTPREC  mutation operator to remove the precedence relation  prect3 from τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The result is the same as shown in  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We add one runnable R4 to the first example and  assign it to T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This new task configuration is shown  in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R4 writes a different constant value from  R1 to the global variable A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply mARPREC  mutation operator to this new example, which adds  γ1 to precr4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R4 requires R1 execute first so R4 over-  writes the value written by R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The operator changes  the execution order of runnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task configuration settings for runnable precedence  mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T1  10  2  2  R1  T2  20  2  1  R2  T3  20  2  1  R3  T1  10  2  1  R4  In example one, T2 has two runnables R2 and R3  with a precedence relation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply  mRRPREC runnable remove precedence mutation  operator to remove the precedence γ2 from precr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We schedule R3 runs before R2 since no precedence  constraint that turns out different than the original  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The original output of R3 is an increas-  ing value along with the execution instead of a value  of either zero or a fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The reason is that  R3 executes first and it reads A before R2 writes any  new value to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The bottom output line in Figure 20  shows the execution result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The outputs of example one three runnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Priority mutation operators  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Priority mutation operator example settings  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T1  10  1  4  R1  T2  10  2  3  R2  T3  10  3  2  R3  We apply mITPRI operator to the example in Ta-  ble 6 to increase the priority of T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This mutation  operator changes the priority of prio3 to proi3 + 3  so the T3 has the highest priority 5 in this example,  which results in T3 being executed at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 21  shows T3 is triggered first in the task execution Gantt  chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This mutation alters the task execution order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  1  口  口  口  口  口  口  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' b1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  1  1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  L  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0611  10  9  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  40  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  10 F  5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart after applying increas-  ing task priority mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We apply mDTPRI operator to decrease the pri-  ority of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This mutation operator changes the pri-  ority of prioi to proi−3 so the T1 has the lowest pri-  ority 1 in this example, which results in T1 being  executed at last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The task execution Gantt chart is  shown in Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart after applying decreas-  ing task priority mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Jitter mutation operators  We apply mITJ operator to increase a jitter time of  a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For example, let δ = 2, this mutation op-  erator changes the real release time of the task to  jitter1 = 0+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 23 shows the execution of T2  is preempted by T1 caused by the jitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task executions Gantt chart after applying increas-  ing jitter mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Then mDTJ mutation operator decreases the jit-  ter time of a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply this operator to the  above example and let δ = −1 so the task jitter1 =  2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T2 is preempted by T1 during the simulation  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Shared memory mutation operators  In this shared memory category, we introduce five  mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The first one is mDSM, and  this operator assigns a new value to the memory  store before a read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For our example, we add a  Data Store Write block right before the Data Store  Read execution so that the Data Store Write block  defines a new value to the variable, and we chose  the initial value of this variable as the default new  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutant using mUDSM operator is shown  in Figure 24, which only shows the changes of  Runnable2 subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We add a constant block and  a Data Store Write block at the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple example of DSM mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The second mutant operator is mUDSM, and this  operator disregards a write to a Data Store block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  For our example, we remove the Data Store Write  block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 25 shows the mUDSM mutant that the  Data Store Write has been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple example of UDSM mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The third mutant operator is mRDSM, and this  operator removes an initialization value to a Data  Store memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In many programs, variables require  1F  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  /o  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='061 F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06f()  2:0  2:1  function  A  0  2:4  2:2  A  2:5  A  2:3  Zf()  function  A  1  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  an initial value before they can use properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For  our example, Runnable1 subsystem is such a pro-  cess of initializing Data Store A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' then, we remove  the Data Store Write block in Runnable1 subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The Simulink model can still run simulations with-  out any issues however the output of the simulation  only yields a single value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutant operator mRSM adds a new reference to  shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 26 shows the block diagrams  of a Simulink model with three subsystems and they  are mapped to two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The model has a DSM  block A in the root-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' There is a Data  Store Write block inside subsystems Task B1 and a  Data Store Read block in Task B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The period of  Task A is 5ms and the period of Task B is 10ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  To implement the mRSM, we add a Data Store Read  block to the TaskA subsystem which shows in Figure  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In the original example, Task A executes first  then Task B1 writes A and Task B2 reads A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  mutant program has the same execution order as the  original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, when the Data Store Read  block in Task A executes, the block reads data from  an uninitialized data store or a previous instant of  Task B1 as Task B has not executed yet or has been  executed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple Simulink model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' An example of mRSM mutant operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Adding a  Data Store Read block to Task A block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutant operator mRMSMR deletes a reference to  shared memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Figure 26, Task B2 has a refer-  ence to a DSM block A in the root-level system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' To  implement the mRMSMR, we delete the Data Store  Read block in the TaskB2 subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In the mu-  tant program, Task B2 has a constant output value  of zero since there is no reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Evaluation Phase  In the previous section, we describe how a model  scheduler SimSched can validate the real-time con-  text during a simulation, and we utilize mutation  testing to evaluate SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In this section, we  perform experiments to demonstrate the use of our  mutation testing framework to evaluate the quality  of SimSched and Stateflow schedulers in scheduling  tasks in real-time systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Evaluation Process  To validate the proposed mutation operators, we ap-  ply them to ML/SL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We separate the evalu-  ation process into two parts base and extension, ac-  cording to the ability of ML/SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply the first-  order mutants (FOMs) 22 to ML/SL models to gen-  erate a mutant, which means we generate a mutant  by using a mutation operator only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Base Case  In the base case, we examine the simulation results  of the original models and the SimSched models and  their mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' An original model M is an ML/SL  model scheduled by Stateflow scheduler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A Sim-  Sched model M ′ is an original model scheduled  by SimSched;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutants (Mµ or M ′  µ) are ei-  ther original model or SimSched models mutated by  one of our mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 28 shows the  schematic diagram of our mutants generation pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We use the simulation result of M as a com-  parison baseline, and then we compare the baseline  with every other simulation result of Mµ, and M ′  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We examine the comparison result to see if the re-  sult reaches a verdict failure during model simula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We say a mutant is killed if a verdict of failure  is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In2  Out1  A  Task A  Out1  In2  In1  Qut3  Task B1  Task_B2A  2  DSRA  2J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Schematic diagram of the model mutants genera-  tion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simple evaluation example scheduled by Stateflow  scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We use a simple example shown in Figure 29 to  explain the base case evaluation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This ex-  ample is an original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We replace the State-  flow scheduler with a SimSched scheduler to form  a SimSched model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We generate mutants for both  the original and SimSched models by a specific mu-  tation operator, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', mDTPER, to decrease the task  period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Then we run the simulation for both mutants  and analyzed the results to see if there is any errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  If the simulation result of Mµ or M ′  µ is different  from the original model and shows a verdict failure,  then we say the mutant is killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In this example,  we have three runnables  R1,R2,R3 and they are mapped to two tasks T1,T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  R1 is mapped to T1 and R2,R3 are mapped to T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  period of T1 is 3ms and The period of T2 is 6ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The execution time of each runnable is 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  simulation result of the M is shown in Figure 30  and it shows each runnable output is a rising non-  interlaced polyline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We apply the mDTPER muta-  tion operator as decreasing 1ms to both the origi-  nal model and SimSched model to generate mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The task T1 in the mutants has a period of 2ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  simulation result of these simulations is shown in  Figure 31 and Figure 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simulation result of  M ′  µ is different from the result of M , and it shows  the output of R2 and R3 are two rising interlaced  polylines because SimSched can simulate the exe-  cution time and preemption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T1 preempts T2 in the  SimSched mutant model to yield an alternative ex-  ecution trace, and we say a verdict fail is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  However, the simulation result of Mµ is similar to  the result of M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Thus, the mDTPER mutant is killed  to the M ′  µ and is alive to the Mµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We can not apply  this means to all mutation operators due to the nature  of ML/SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We combine this method and the follow-  ing method to evaluate the mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' M simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mµ simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' M ′µ simulation result  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Extension  To evaluate the rest of the mutation operators, we  implement a mutation generator with additional  functionalities to assist the validation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' One  feature is to check the mutant model’s schedulabil-  ity at the given set of tasks configuration to decide if  all task deadlines are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=" The other function is to  M'  M  SimSched  Mutate  Mutate  M  M'  μ  n1 ms Clock  A  R1()  R3()  R2()  Sall(  R2  R1  call()  R3  R10  R2()  R30  Scheduler90  R1  R2  R3  80  70  60  50  40  30  20  10  0  10  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06R1  R2  120  R3  100  80  60  40  20  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06R1  60  R2  R3  50  40  30  20  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  Offset=0J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  check the data access sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If there is a DataS-  tore block in the mutated model, every read or write  to this DataStore block is recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Then we use  this mutated model data access sequence to com-  pare with the original model data access sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The mutation generator is implemented as a Matlab  script written in m-file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The validation process takes a Stateflow sched-  uled ML/SL model and a test specification as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The test specification specifies which mutation op-  erator to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A mutant generator applies the speci-  fied mutation operator to the ML/SL model via Sim-  Sched and generates a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutant genera-  tor then executes the simulation both for the original  model and the mutated model using the additional  functionalities to analyze the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If the anal-  ysis shows at least one task misses its deadline in a  mutated model, then we say a mutant is killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Or  at least one variable comparison result of the DataS-  tore access sequence is unmatching, and then we say  a mutant is killed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' otherwise, we report the mutant  is benign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A simple example of using Model Scheduler to  schedule AUTOSAR SW-Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We use an example shown in Figure 33 to ex-  plain the validation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It has three runnables  and is mapped to two tasks, R1 map to T1, R2, and  R3 map to T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The period of task T1 is 10ms, and  T2 is 20ms, every runnable’s execution time is 3ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  There is a DataStore block named A as a shared vari-  able in this example model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If we apply the period  mutation operator mDTPER ρi − δ where i − 1 and  δ = 6 to this model to decrease the period of T1 and  generate a mutant, run it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The analysis result shows  the T2 missed deadline, then we say this mutant is  killed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If we apply the execution time mutation op-  erator mITET ci + δ where i = 1 and δ = 3 to this  model to increase the execution time for T1 and gen-  erate a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The DataStore access sequence of  the original model is a pattern of WRWR where W  represents a write to the shared variable, and R rep-  resents a read to the shared variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutant  generates a different sequence, which is WRWWR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  It is because the T1 has a longer execution time than  the original model, and it preempts T2 during the ex-  ecution of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Hence, there is one more W in the  DataStore access sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Experiments  We employ two examples to demonstrate the use of  our mutation testing framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We first explain the  two examples in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We then apply the mutation  operators to the two models scheduled by both the  Stateflow scheduler and SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The three-servo example adapted from 12 with  Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Three Servos Model  We adapt an example from the TrueTime 21 exam-  ple library, which shows a possible implementation  of a three-servo PID control system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The example  is shown in Figure 34 with a Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In this example, three DC servos are modeled by a  continuous-time system, and three PID controllers  are implemented as three subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We map three  controller subsystems to three runnables R1, R2, and  R3 then they are mapped to tasks T1, T2, and T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  task periods are T1=4 , T2 = 5 and T3 = 6 ms re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Each task has the same execution time as  Scheduler  Runnable(period,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' execution time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' priority) Task()  function()  function()  function()  R1(10ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2)j  Task(1)  R2( 20ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 3ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 1  Task(2)  R3( 20ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 5ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 1  Task(2  function  irv2  irv1  irv2  SimSched  irv3  Aader  Runnable1_subsystem  Runnable2_subsystem  Runnable3_subsystem1 ms Clock  callo  R10  R2()  R30  call(  Temporal Logic  Scheduler  functionO  1000  u  s2+s  DCServo1  PID1  functionO  1000  u  2+s  DCServo2  PID2  function()  1000  u  s?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='+s  DCServo3  PID3J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task settings are shown in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simula-  tion result is shown in Figure 35 based on the above  task settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The three graphs show the output of  the motors using the three PID controllers when the  corresponding task parameters are assigned accord-  ingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In the graph, the square wave is the reference  input signal for the motors, where the computation  delays are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Three PID con-  trollers are all smooth output signals as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Three Servo example settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T1  4  1  3  R1  T2  5  1  2  R2  T3  6  1  1  R3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The three servos example output with Stateflow  scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We replace the Stateflow scheduler with the Sim-  Sched scheduler, and the updated example is shown  in Figure 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In this example, three DC servos have  the same task setting as the Stateflow scheduler ex-  ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Each runnable has the same execution time  as 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simulation result is the same as the  Stateflow scheduler example based on the above task  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' There is no deadline missing for any task,  so the simulation result shows every task has smooth  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 37 shows the task active chart gen-  erated by SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Every task has been executed  within its own deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The three servos example adapted from 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The three servos example task active chart gener-  ated by SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The adjusted Stateflow scheduler for mITO muta-  tion operator to increase o f fset as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2  DCServo1  1  SignalGenerator  0  1  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo2  1  SignalGenerator  0  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo3  SignalGenerator  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9Scheduler  Runnable(period, execution time, priority) Task()  E  R1(4ms, 2ms, 3) Task(1)  R2( 5ms, 2ms, 2  Task(2)  R3( 6ms, 2ms, 1  Task(3  SimSched  function(  ○○  1000  u  s2+s  DCServo1  PID1  function()  1000  u  2+s  y  DCServo2  PID2  function()  1000  u  32+s  DCServo3  PID30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  o  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  to0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  to0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  to0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06 Scheduler  on at(1,tick): Period_4_ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  du: on every(5,tick) : Rate5ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  du: on every(6,tick) : Rate6ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="  'Sched_4_MS  Periodicl  on every(4,tick) : Period_4_ms;" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Sched 5 MS  Periodicl  en: Period_5_ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Rate5ms  Sched 6 MS  Periodic/  en: Period_6_ms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Rate6msJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Next step, we apply mITO, mDTO, mITPER,  mDTPER, mARPREC, mRRPREC, mITJ, mDTJ  mutation operators to both Stateflow scheduler and  SimShced examples to generate two versions of mu-  tants with the same mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  To ap-  ply some of the mutation operators to evaluate the  Stateflow scheduler, we need to adjust the State-  flow scheduler so that it can be used on the gen-  erated mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 38 shows an example that  is adjusted for the Offset mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This  example uses a temporal logic operator at in the  state to set the Offset parameter to generate a mu-  tant for PID1 which runs at the period of 4ms in  this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This mutant increases Offset as 1ms  for DCServo1 controlled by PID1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutant of  the SimSched version can be easily generated by our  model scheduler SimSiched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Stateflow scheduled three-servo example task  active chart after applying mITO mutation operator to in-  crease o f fset as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We run simulations for both versions of the mu-  tants generated by Offset mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Both  mutant versions’ output of three servos is the same  as shown in Figure 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The only difference occurs  at the beginning of the simulation but it does not af-  fect the smooth control of DCServos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We can see  the difference from the following comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Fig-  ure 39 shows the Stateflow scheduled task active  chart after applying mITO mutation operator to in-  crease offset as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Before apply-  ing the mutation operator, every task is released at  time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' After applying the offset mutation opera-  tor, Task 1 is delayed by 1ms shown on the top of  the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 2 and Task 3 are both released at  time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 40 shows the SimSched scheduled  three servos example task active chart after applying  mITO mutation operator to increase offset as 1ms for  DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' There are three output signals represent-  ing three tasks from top to bottom T1, T2, and T3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  As T1 has a 1ms offset, Task 2 is executed first as  shown in the figure the second line starts at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Because the SimSched scheduler has the execution  time parameter, Task 2 is executed at time 1 and Task  3 at time 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The SimSched scheduled three servos example task  active chart after applying mITO mutation operator to in-  crease Offset as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We use a similar approach to apply Period muta-  tion operator to the three-servo example and gener-  ate mutants for both the Stateflow scheduler model  and SimSched model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We use two mutation con-  figurations to show the similarities and differences  between the two schedulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The first configuration  is [1,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It means Task 1 has a period of 1ms and  the period of Task 2 and Task 3 keep the same as  5ms and 6ms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 41 shows the  Stateflow scheduler mutant simulation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Be-  cause Task 1 has a 1ms period, it has too many times  of calculations, and the output value is out of the  chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 2 and Task 3 keep the same output as be-  fore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 42 shows the SimSched mutant simula-  tion result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The output of Task 1 is the same as the  Stateflow scheduler mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, Task 2 and  Task 3 are different from the Stateflow one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Because  the SimSched takes the execution time into account,  Task 1 has the highest priority and has an execution  time of 1ms, and Task 1 always runs during the sim-  ulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 2 and Task 3 always are preempted by  Task 1 because they have a lower priority than Task  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Stateflow scheduled three servos example out-  put after applying mDTPER mutation operator to decrease  period as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  PID1  4 ms 1ms offset  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  PID2 5 ms no offset  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  PID3 6 ms no offset  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5,  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="03  to'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='06  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="03  to'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='062  DCServo1  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo2  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo3  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The SimSched scheduled three servos example out-  put after applying mDTPER mutation operator to decrease  period as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The second configuration is [13,5,6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It means  Task 1 has a period of 13ms and the period of Task  2 and Task 3 keep the same as 5ms and 6ms, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 43 shows the Stasflow scheduler  mutant simulation result and the SimSched mutant  has the same output as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Because  Task 1 has a 13ms period, it has fewer computations  than the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Although the output behav-  ior looks like Task 1 misses its deadline in the figure,  every execution of Task 1 meets its deadline and it  is executed as scheduled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Stateflow scheduler three servos example out-  put after applying mITPER mutation operator to increase  period as 13ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We  generate  mutants  for  mARPREC  and  mRRPREC operators by setting each parallel state’s  execution order in the Stateflow scheduler model  and configuring the parameters and connections for  the SimSched model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The two mutants’ simulation  results are the same as the original model, except  that each task’s execution order is different from the  original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We generate mutants for mITJ and mDTJ op-  erators by adapting the Stateflow scheduler in the  model and configuring the parameters in the Sim-  Sched model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set the configuration as [1,0,0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  It means only Task 1 has a jitter as 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 44  shows the SimSched scheduler three servos example  task active chart after applying mITJ mutation oper-  ator to increase jitter as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Be-  cause Task 1 has 1ms jitter, Task 2 is executed first  then Task1 and Task3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The SimSched scheduler three servos example task  active chart after applying mITJ mutation operator to in-  crease jitter as 1ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We only apply the execution time operator to the  SimSched model due to the lack of support for the  Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 45 shows the effect out-  put of mITET mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set c1 = 3ms  using the mITET mutation operator for Task 1 to  generate a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The output of Task 3, shown as  DCservo 3 at the bottom of the figure, is a curly  wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It is an unstable control due to the preemp-  tion by T1 and T3 missing its deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 46  shows the task preemption effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 1 takes 3ms  to execute and Task 2 takes 1ms to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' After  the execution of Task 1 and Task 2, Task 3 should be  executed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' however, it is the time that Task 1 is sched-  uled to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 1 has a higher priority, so Task 3 is  preempted by Task 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The first instance of Task 3 is  executed at 19ms so Task 3 does not have a smooth  control signal output as the other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Although  some task preemptions occur in Task 2, it does not  miss enough deadlines to significantly affect the out-  put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task 2 still has a smooth output signal as shown  in Figure 45 as the second chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The SimSched scheduler three servos example sig-  nal output after applying mITET mutation operator to in-  crease executiontime to 3ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2  DCServo1  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo2  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo3  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='92  DCServo1  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo2  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  2  DCServo3  SignalGenerator  0  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9DCServo1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  DCServo2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  DCServo3  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05DCServo1  2  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  DCServo2  2  1  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  DCServo3  2  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The SimSched scheduler three servos example task  active chart after applying mITET mutation operator to in-  crease executiontime to 3ms for DCServo1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Throttle Position Control Model  We adopt an AUTOSAR software component  Simulink model from Mathworks shown in Figure  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' It implements a throttle position control sys-  tem for an automobile and contains three sensors,  one monitor, one controller, and one actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' They  are implemented as six subsystems and mapped to  six runnables TPSSecondary, Monitor, Controller,  Actuator, APPSnsr and TPSPrimary then they are  mapped to tasks T1, and T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The task periods are  T1=5ms and T2 = 10 ms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Each runnable  has the same execution time of 1ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Task settings  are shown in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This example uses seven Data-  StoreMemory blocks to access the shared resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Throttle position control Simulink model contains  six runnables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Throttle control example settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Task  Period  Execution  Priority  Runnable  (ms)  Time(ms)  T2  10  1  1  TPSPrimary  T1  5  1  2  TPSSecondary  T1  5  1  2  Monitor  T1  5  1  2  Controller  T1  5  1  2  Actuator  T2  10  1  1  APPSnsr  Figure 48 shows the simulation result, which is  generated by a Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The square wave  in the figure is the simulated pedal input, and the  curly wave is the output of the throttle body, repre-  senting the current throttle position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Stateflow  scheduler simulates the throttle control controller’s  process well and simulates the entire control pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simulated throttle position of the throttle posi-  tion control model scheduled by the Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 49 shows the runnable active chart sched-  uled by the Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' All runnables are  scheduled and executed at time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Runnable TP-  SPrimary and APPSensor are mapped to T2 and they  are scheduled and executed every 10ms and the top  and bottom chars shown in the figure are their active  charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The active charts of runnable TPPSSendary,  Monitor, Controller, and Actuator are the four charts  in the middle of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' They are scheduled and  executed every 5ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The runnable active chart of the throttle position  control model scheduled by the Stateflow scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We apply SimSched to the Stateflow scheduler  model, and we can get a similar simulation result as  the Stateflow scheduler example based on the above  task settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 50 shows the task active chart  generated by SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Every task has been exe-  cuted within its own deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Both task T1 and T2  are scheduled at time 0 but only T1 is executed at  DCServo1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  DCServo2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  DCServo3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='054  TPSSecondaryRun5ms  MonitorRun5ms  ControllerRun5ms  function()  function()  function(  汇  ThrottlePositionSensorSecondary  ThrottlePositionMonitor  Controller  6  APPSnsrRunl  ActuatorRun5ms  TPSPrimaryRuniOms  function()  function()  function()  口  APPHwlOValueread  ThrCmdHwlOValuewrite1  ThrottlePositionSensorPrimary  AccelerationPedalPositionSensor  ThrottlePositionActuator  APP_HwlO_Value  ThrCmd HwlO ValueThrottle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  Simulated Pedal Input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  Throttle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5TPSPrimary  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  TPSSecondary  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Monitor  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Controller  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Actuator  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  APPSensor  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content="045  s0'0  0J." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  time 0 due to its higher priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T1 takes up 4ms to  run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' After the first instance of T1 is finished, T2 is  executed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The second instance of T1 arrives at 5ms,  which is during the middle of the execution of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T2  is preempted by T1 and resumes at the completion of  the second instance of T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The task level active chart of the throttle position  control model scheduled by the SimSched scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 51 shows the runnable level active chart  of the throttle position control model scheduled by  the SimSched scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' From this figure, we can  clearly see the activity of each runnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  It ex-  actly shows the execution order of each runnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Runnable TPPSSendary, Monitor, Controller, and  Actuator are executed one after another followed by  TPSPrimary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Runnable APPSensor is executed af-  ter the second instance of T1 and it is the preemption  point of T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The runnable active chart of the throttle position  control model scheduled by the SimSched scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We use the same means applied to three servos  example to apply it to the throttle position control  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We adopt the Stateflow schedule and replace  it with SimSched to generate mutants for both the  Stateflow scheduler and SimSched for the experi-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 52 shows the runnable active chart of the  Throttle Position Control model scheduled by Sim-  Sched after applying mTIO mutation operator as in-  creasing 2ms offset for T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The runnable Actuator  active chart shown in the second bottom chart in the  figure is missing its first execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T1 takes 4ms to  run, and it also has 2ms offset, so the total execution  time of T1 exceeds its period 5ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' On the other hand,  the mutant generated by the Stateflow scheduler can  not simulate this overrun situation due to the lack of  execution time simulation support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The runnable active chart of Throttle Position Con-  trol model scheduled by SimSched after applying mTIO  mutation operator as increasing 2ms offset for T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 53 shows the simulated throttle position  of the throttle position control model scheduled by  SimSched after applying mITPER mutation opera-  tor for T1 at 100ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The Stateflow scheduler also  can output the same figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mITPER mutation  operator can reduce the computation times of a task  at the same amount of time, which results in unsta-  ble control as shown in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simulated throttle position of the throttle po-  sition control model scheduled by the Stateflow scheduler  after applying mITPER mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Figure 54 shows the simulated throttle position  of the throttle position control model scheduled by  SimSched after applying mDTPER mutation opera-  tor for T1 at 4ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mDTPER can result in no  output signal for T2 because a higher rate task in-  creases the computation times, and the lower rate  task does not get an execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In this example, we  set ρ1 = 4ms using the mDTPER mutation operator,  Task1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  oE  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Task2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05TPsPrimary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  TPSSecondary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Monitor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Controller  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Actuator  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  APPSnsr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5 /  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05TPsPrimary  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  10  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  TPSSecondary  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Monitor  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Controller  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  Actuator  1 F  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05  APPSnsr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' E  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='045  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='05Throttle Pos  Simulated Pedal Input  Throttle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  then the T2 is always preempted by T1 and does not  have a chance to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The simulated throttle position of the throttle po-  sition control model scheduled by SimSched after applying  mDTPER mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We only apply the execution time operator to  SimSched models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We set c1 = 5ms using mITET  mutation operator for T1 to generate a mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This  mutant just outputs the same throttle position as  shown in Figure 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Since T1 has increased its exe-  cution time by 1ms, it just takes up all the time slots  in its period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' T2 is preempted by T1 during the simu-  lation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We apply mARPREC and mRRPREC mutation  operators to this throttle position control exam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' First, there is no precedence between runnable  APPSnsr and TPSPrimary, and we use a mARPREC  mutation operator to add precedence γAPPSnsr to  precrTPSPrimary to generate mutants for both sched-  ulers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Runnable Controller consumes the values  produced by APPSnsr and TPSPrimary to calcu-  late the throttle percent value for the throttle actu-  ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The changes in the simulation results of both  mutants are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Figure 55 shows the simulation  result comparison between the original model and  the SimSched mutant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Second, there is precedence between Controller  and Actuator, and Controller is executed before  Actuator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We remove the precedence from the pair  of runnables, so the Actuator (destination) runs be-  fore Controller (source), which changes the data de-  pendency and delays the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The difference in sim-  ulation is similar to Figure 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The difference of simulation result between the  original model and the mutant with mARPREC mutation  operator scheduled by SimSichde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We apply mDSM, mUDSM, mRDSM, mRSM,  mRMSMR, and mRSMR mutation operators to this  example and generate accordingly mutants for both  the Stateflow scheduler and SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Inter-  estingly, since the task time properties have not  changed for the shared memory mutation operators,  the simulation results are consistent between the two  types of mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For example, we apply the mDSM  mutation operator to variable ThrCmdPercentValus  and set the variable to a new constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  simulation result of both types of mutants is the  same shown in Figure 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Since this variable is an  input to a Lookup table, there is always an output  that matches the input value and yields a correspond-  ing value to the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The input value is constant,  so the throttle position’s output is a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The throttle position output after applying mDSM  mutation operator to variable ThrCmdPercentValus sched-  uled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Evaluation Result  We apply the evaluation process to the example  models to investigate the mutation operator’s effec-  tiveness to kill mutants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 9 and 10 summa-  rize the results of the efficacy of mutation operators,  Throttle Pos  Simulated Pedal Input  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  Throttle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5ThrottlePos(Run2:TPCWithTaskAndDSM_CaseStudy_Stateflow)  ThrottlePos (Run5:TPCWithTaskAndDSM_CaseStudy_SimSched)  Tolerance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5Throttle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7  Simulated Pedal Input  Throttle Pos  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='5J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  where each row provides the number of mutants and  the mutation score of our mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  mutation score is a measure that gives the percent-  age of killed mutants with the total number of muta-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation analysis of mutation operators for three ser-  vos example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Stateflow Scheduler  SimSched  Operator  Mutants  Kills  Mutants  Kills  Offset  36  24  36  27  Period  39  25  39  25  Execution  N/A  N/A  36  31  Time  Precedence  5  0  5  0  Priority  18  0  18  0  Jitter  36  24  36  27  Mutation  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='48%  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='71%  Score  For the three servo example, we generate 134  mutants for the Stateflow scheduler models and 170  mutants for the SimSched models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We achieve a  mutation score of 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='48% for the Stateflow sched-  uler model and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='71% for the SimSched models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Evaluation results show that the time-related muta-  tion operators have the most effect on the mutation  testing, such as the Offset, Period, Execution Time,  and Jitter mutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We observe that the  Precedence and Priority mutation operators kill zero  mutant because, in this example, each controller in-  dividually controls a motor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' There is no connection  between them, so the change of precedence and pri-  ority does not cause any simulation changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' How-  ever, the three controllers run on a single CPU, and  one task execution time’s length affects other tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We also observe the Offset mutation operator only  affects each task’s initial execution, and tasks miss  the deadline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Still, each mutant’s simulation results  show each controller can have stable control of each  servo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation analysis of mutation operators for throttle  position control example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Stateflow Scheduler  SimSched  Operator  Mutants  Kills  Mutants  Kills  Offset  19  7  19  15  Period  30  6  30  19  Execution  N/A  N/A  34  34  Time  Precedence  23  10  23  10  Priority  11  0  11  0  Jitter  19  12  19  15  Shared  72  46  72  46  Memory  Mutation  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='39%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2%  Score  For the throttle position control example, we  generate 164 mutants for the Stateflow scheduler  models and 198 mutants for the SimSched models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We achieve a mutation score of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='39% for the State-  flow scheduler model and 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='2% for the SimSched  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Evaluation results are similar to the previ-  ous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The time-related mutation operators  have the most effective for mutation testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  We  observe that the Shared Memory mutation operators  have the same kills for both Mµ and M ′  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In this  example, task T2 has two Runnable TPSPrimary and  APPSnsr each has a shared variable to update at each  execution, and there is no direct relation between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Though SimSched can simulate the preemp-  tion of T2 to interrupt its execution, shared memory  mutants’ model behaviors are the same for both Mµ  and M ′  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  From the above two examples, we can see the  mutation operators are application-dependent, and  SimSched can achieve a higher mutation score at  the time-related mutation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For example,  the Precedence mutation operator kills zero mutants  for the three servo example, but it kills ten mutants  for the throttle position control example for both  Stateflow Scheduler and SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The three-servo  example does not require any precedence at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  each task only controls itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, the throt-  tle position control example requires precedence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A  runnable consumes data from a previous runnable  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' If we alter the precedence, the execu-  tion order is different from the original model ex-  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  ecution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' it produces a different data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' For exam-  ple, in the Period mutation operator both Stateflow  scheduler and SimSched kill the same mutants for  the three serve example, but SimSched kills more  mutants than the Stateflow Scheduler in the throttle  position control example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Because the throttle po-  sition control example has four runnables in T1 and  each runnable has 1ms execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We use a  mDTPER to T1 and generate a mutant that the pe-  riod of T1 is 4ms instead of 5ms, then T1 will occupy  all the execution time slots, and T2 will not be ex-  ecuted for the SimSched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' However, the Stateflow  Scheduler does not consider the execution time, so  both T1 and T2 are executed as scheduled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Related Work  Our work aligns with the MBMT, and it has been  applied to various models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Trakhtenbrot 41 pro-  poses mutation testing on statechart-based models  for reactive systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' This approach mainly deals  with the conformance between specific semantics  of statechart models and the model’s implementa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation testing has been applied to fea-  ture models to test software product lines 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The  feature models represent the variability of software  product lines and configurable systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  El-Fakih  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='14 develop a technique of mutation-based test  case generation toward extended finite state ma-  chines (EFSM) that examines the EFSM under test  agrees to user-defined faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Belli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='6 have sur-  veyed MBMT approach a great deal and detailed the  approach applied to graph-based models, including  directed graphs, event sequence graphs, finite-state  machines, and statecharts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A recent mutation testing  survey31 presents up-to-date advanced approaches  that use mutants to support software engineering ac-  tivities to model artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Our work also fits in the timed system test-  ing, which requires a real-time environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Re-  searchers 3,28,29 have utilized the most studied for-  malisms TA to inject faults to the timed system and  reveal that the time-related errors are unable to find  by using randomly generated test suites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Nilsson  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='28 first proposed a set of extended TA muta-  tion operators based on TA with Tasks (TAT) 30 to  test real-time systems which depend on the execu-  tion time and execution order of individual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The mutation operators are interested in the time-  liness of a task to meet its deadlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Aichernig et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3 propose a mutation testing framework for timed  systems, where they define eight mutation operators  to mutate the model and its mutants are expressed  as a variant of TA in Uppaal specification format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  The authors also develop a mutation-based test case  generation framework for real-time, where they use  symbolic bounded model checking techniques and  incremental solving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cornaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='11 presents an  automated framework MODELTime that facilitates  the study of target platform-dependent timing dur-  ing the model-based development of embedded ap-  plications using MATLAB/Simulink simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  ML/SL is one of the most popular formalisms to  model and simulate embedded systems, and many  researchers have explored the various type of mu-  tation operators applied to ML/SL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Hanh  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='18 propose a set of mutation operators based  on investigating common faults in ML/SL models  to validate test suites and present a process of muta-  tion testing for ML/SL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' They provide twelve  mutation operators and divide them into five cate-  gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Stephan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='38 utilize the mutation testing  technique to compare model-clone detection tools  for ML/SL models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  They present a taxonomy of  ML/SL and prose a set of structural mutation op-  erators based on three clone types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The mutation  operators are used to evaluate the model-clone de-  tectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Using the mutation-based technique to gen-  erate test cases for ML/SL models automatically has  been studied 8,19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  They can effectively generate  small sets of test cases that achieve high coverage  on a collection of Simulink models from the auto-  motive domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A recent work SLEMI 10 has ap-  plied mutation techniques to the Simulink compiler  and uses tools to generate mutants of the seed model  and found 9 confirmed bugs in Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Our work intends to exploit mutation analysis  to identify potential time-related errors in ML/SL  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Roy and Cordy 33,34 first propose using  mutation analysis to assist the evaluation of soft-  ware clone detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' They develop a frame-  work for testing code-clone detectors based on mu-  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Stephan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='37,36 proposed a framework that  can objectively and quantitatively evaluate and com-  pare model-clone detectors using mutation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Their work is based on a structural mutation method  for ML/SL model mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Our mutation operators  are based on a timed system task model, whereas,  there are no relevant existing studies that directly  integrated the ML/SL models in the timed systems  in the MIL phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' thus, we carry out the work pre-  sented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Co-simulation16 is a widely used technique in  model-based testing to verify as much of the system  functionality, among subsystems, as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Com-  posing the simulations of sub-simulators can achieve  a joint simulation of a coupled system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Many differ-  ent languages and tools are used for other purposes  in the model-based design domain, either designing  continuous plants or discrete controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  A rela-  tively recent open standard functional mock-up in-  terface (FMI) is developed for exchange simulation  models in a standardized format, including support  for co-simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Gomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='15 propose an ap-  proach to facilitate the implementation of the Func-  tional Mock-up Interface standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  They use the  MBT methodology to evaluate the tools that export  Functional Mock-up Units (FMUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Hence, they can  root out the ambiguities and improve conformance  to the FMI standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Garro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='7 employs FMI to  perform co-simulation to verify the system require-  ments based on the FOrmal Requirements Model-  ing Language and the Modelica language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Zafar  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='42 present a systematic tool-supported MBT  workflow to facilitate the simulation-based testing  process of an embedded system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The workflow ex-  pends from the requirements phase, and generation  of executable test scripts, to the execution of gener-  ated test scripts on simulation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Conclusion and future work  In this paper, we proposed a set of timed muta-  tion operators for the ML/SL model that is primar-  ily intended to integrate the timed task model in the  ML/SL model to better support MIL simulation us-  ing mutation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Moreover, testing at an ear-  lier stage during the development process reduces  development costs since earlier changes and fixing  errors are much more manageable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We introduce a  timed task model and present a set of mutation op-  erators for the ML/SL based on this task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We  implement a mutation analysis framework that can  apply mutation operators to the simple ML/SL mod-  els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We demonstrate the approach on several ML/SL  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The results validate that mutation analysis  can reveal time-related faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We intend to automate  the mutation testing process for the ML/SL environ-  ment and improve the mutation operators to expose  defects in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' We will further validate our  mutation analysis method to more industrial com-  plex ML/SL model sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Acknowledgments  This work was supported in part by the Natural Sci-  ences and Engineering Research Council of Canada  (NSERC), as part of the NECSIS Automotive Part-  nership with General Motors, IBM Canada, and Ma-  lina Software Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Bernhard K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Aichernig, Harald Brandl, Elisabeth  J¨obstl, Willibald Krenn, Rupert Schlick, and Stefan  Tiran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Killing strategies for model-based mutation  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Software Testing, Verification and Reliability,  25(8):716–748, dec 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Bernhard K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Aichernig and Florian Lorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Towards  generation of adaptive test cases from partial models  of determinized timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In 2015 IEEE Eighth  International Conference on Software Testing, Verifi-  cation and Validation Workshops (ICSTW), pages 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  IEEE, apr 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Bernhard K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Aichernig, Florian Lorber, and Dejan  Niˇckovi´c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Time for mutants - Model-based mutation  testing with timed automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Lecture Notes in Com-  puter Science (including subseries Lecture Notes in  Artificial Intelligence and Lecture Notes in Bioinfor-  matics), volume 7942 LNCS, pages 20–38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Springer,  Berlin, Heidelberg, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Rajeev Alur and David L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Dill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' A theory of timed au-  tomata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Theoretical Computer Science, 126(2):183–  235, apr 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' AUTOSAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Autosar development partnership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' http:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='autosar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='org, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Fevzi Belli, Christof J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Budnik, Axel Hollmann,  Tugkan Tuglular, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Eric Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Model-based  mutation testing—Approach and case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Science  of Computer Programming, 120:25–48, may 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Daniel Bouskela, Alberto Falcone, Alfredo Garro, Au-  drey Jardin, Martin Otter, Nguyen Thuy, and Andrea  Tundis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Formal requirements modeling for cyber-  physical systems engineering: an integrated solution  based on form-l and modelica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Requirements Engi-  neering, pages 1–30, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Angelo Brillout, Nannan He, Michele Mazzucchi,  Daniel Kroening, Mitra Purandare, Philipp R¨ummer,  and Georg Weissenbacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation-based test case  generation for Simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Lecture Notes  in Computer Science (including subseries Lecture  Notes in Artificial Intelligence and Lecture Notes in  Bioinformatics), volume 6286 LNCS, pages 208–227,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Jian Chen, Manar H Alalfi, Thomas R Dean, and  S Ramesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Modeling autosar implementations  in simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In European Conference on Mod-  elling Foundations and Applications, pages 279–292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Shafiul Azam Chowdhury, Sohil Lal Shrestha, Tay-  lor T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Johnson, and Christoph Csallner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Slemi: Equiva-  lence modulo input (emi) based mutation of cps mod-  els for finding compiler bugs in simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In 2020  IEEE/ACM 42nd International Conference on Soft-  ware Engineering (ICSE), pages 335–346, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Alessandro Cornaglia, Shakib Hasan, Alexander  Viehl, Oliver Bringmann, and Wolfgang Rosenstiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Modeltime: Fully automated timing exploration of  simulink models for embedded processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In AmE  2019 - Automotive meets Electronics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 10th GMM-  Symposium, pages 1–6, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Fabio Cremona, Matteo Morelli, and Marco Di Na-  tale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  TRES: A Modular Representation of Sched-  ulers, Tasks, and Messages to Control Simulations  in Simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Proceedings of the 30th Annual ACM  Symposium on Applied Computing, pages 1940–1947,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' DeMillo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Lipton, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Sayward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Hints  on Test Data Selection: Help for the Practicing Pro-  grammer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Computer, 11(4):34–41, apr 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Khaled  El-Fakih,  Anton  Kolomeez,  Svetlana  Prokopenko, and Nina Yevtushenko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Extended  Finite State Machine Based Test Derivation Driven  by User Defined Faults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In 2008 International  Conference on Software Testing, Verification, and  Validation, pages 308–317.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, apr 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cl´audio Gomes, Romain Franceschini, Nick Battle,  Casper Thule, Kenneth Lausdahl, Hans Vangheluwe,  and Peter Gorm Larsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Application of model-based  testing to dynamic evaluation of functional mockup  units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cl´audio Gomes, Casper Thule, David Broman, Pe-  ter Gorm Larsen, and Hans Vangheluwe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Co-  simulation: a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  ACM Computing Surveys  (CSUR), 51(3):1–33, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Hamlet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Testing Programs with the Aid of a  Compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE Transactions on Software Engineer-  ing, SE-3(4):279–290, jul 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Le Thi My Hanh and Nguyen Thanh Binh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Muta-  tion operators for simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Proceedings -  4th International Conference on Knowledge and Sys-  tems Engineering, KSE 2012, pages 54–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, aug  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' He, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R¨ummer, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Kroening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Test-case gener-  ation for embedded simulink via formal concept anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In 2011 48th ACM/EDAC/IEEE Design Automa-  tion Conference (DAC), pages 224–229, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Christopher Henard, Mike Papadakis, Gilles Perrouin,  Jacques Klein, and Yves Le Traon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Assessing Soft-  ware Product Line Testing Via Model-Based Mu-  tation: An Application to Similarity Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In  2013 IEEE Sixth International Conference on Soft-  ware Testing, Verification and Validation Workshops,  pages 188–197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, mar 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Dan Henriksson, Anton Cervin, and Karl-Erik ˚Arz´en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  TrueTime : Real-time Control System Simulation with  MATLAB / Simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Proceedings of the Nordic  MATLAB Conference, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Yue Jia and Mark Harman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' An Analysis and Survey  of the Development of Mutation Testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE Trans-  actions on Software Engineering, 37(5):649–678, sep  2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Edward A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Lee, Stephen Neuendorffer, and Gang  Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Synchronous Reactive Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Claudius  Ptolemaeus, editor, System Design, Modeling, and  Simulation using Ptolemy II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Ptolemy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='org, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Lehoczky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Sha, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The rate monotonic  scheduling algorithm: exact characterization and av-  erage case behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' [1989] Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Real-Time  Systems Symposium, pages 0–5, 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' MathWorks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulink User’s Guide, r2017b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' http:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='mathworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='com, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Reza Matinnejad, Shiva Nejati, Lionel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Briand, and  Thomas Bruckmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Automated test suite generation  for time-continuous simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Proceedings  of the 38th International Conference on Software En-  gineering - ICSE ’16, pages 595–606, New York, New  York, USA, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' ACM Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Andreas Naderlinger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Simulating preemptive schedul-  ing with timing-aware blocks in Simulink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Design,  Automation & Test in Europe Conference & Exhibi-  tion (DATE), 2017, pages 758–763.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, mar 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Nilsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Offutt, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Andler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation-  based testing criteria for timeliness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Proceed-  ings of the 28th Annual International Computer Soft-  ware and Applications Conference, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' COMPSAC  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', pages 306–311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Robert Nilsson and Jeff Offutt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Automated testing of  timeliness: A case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Proceedings - Interna-  tional Conference on Software Engineering, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' / Mutation Operators for Simulink Models  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Norstrom, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Wall, and Wang Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Timed automata  as task models for event-driven systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Pro-  ceedings Sixth International Conference on Real-Time  Computing Systems and Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' RTCSA’99  (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='PR00306), pages 182–189, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mike Papadakis, Marinos Kintis, Jie Zhang, Yue Jia,  Yves Le Traon, and Mark Harman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Mutation test-  ing advances: an analysis and survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Advances  in Computers, volume 112, pages 275–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Elsevier,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mike Papadakis, Marinos Kintis, Jie Zhang, Yue Jia,  Yves Le Traon, and Mark Harman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mutation Test-  ing Advances: An Analysis and Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Advances in  Computers, 112:275–378, jan 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chanchal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Roy and James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cordy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  A muta-  tion / injection-based automatic framework for eval-  uating code clone detection tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In IEEE Interna-  tional Conference on Software Testing, Verification,  and Validation Workshops, ICSTW 2009, pages 157–  166, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Chanchal K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Roy, James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cordy, and Rainer  Koschke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Comparison and evaluation of code  clone detection techniques and tools: A qualita-  tive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Science of Computer Programming,  74(7):470–495, may 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Julien Schmaltz and Jan Tretmans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  On confor-  mance testing for timed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Lecture Notes  in Computer Science (including subseries Lecture  Notes in Artificial Intelligence and Lecture Notes in  Bioinformatics), volume 5215 LNCS, pages 250–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Springer, Berlin, Heidelberg, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Matthew Stephan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Model clone detector evaluation  using mutation analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In Proceedings - 30th In-  ternational Conference on Software Maintenance and  Evolution, ICSME 2014, pages 633–638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Institute of  Electrical and Electronics Engineers Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', dec 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Matthew Stephan, Manar H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Alafi, Andrew Steven-  son, and James R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Cordy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  Using mutation analysis  for a model-clone detector comparison framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In  Proceedings - International Conference on Software  Engineering, pages 1261–1264, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Matthew Stephan, Manar H Alalfi, and James R  Cordy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Towards a taxonomy for Simulink model mu-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Proceedings - IEEE 7th International Con-  ference on Software Testing, Verification and Valida-  tion Workshops, ICSTW 2014, pages 206–215, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The AUTOSAR Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Applying simulink to  autosar, r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The AUTOSAR Consortium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' The AUTOSAR Stan-  dard, r4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Mark Trakhtenbrot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Implementation-Oriented Muta-  tion Testing of Statechart Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In 2010 Third Inter-  national Conference on Software Testing, Verification,  and Validation Workshops, pages 120–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' IEEE, apr  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Muhammad Nouman Zafar, Wasif Afzal, and Eduard  Enoiu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Towards a workflow for model-based testing of  embedded systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' In Proceedings of the 12th Inter-  national Workshop on Automating TEST Case Design,  Selection, and Evaluation, pages 33–40, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Yuan Zhan and John A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content=' Search-based muta-  tion testing for simulink models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  In GECCO 2005  Genetic and Evolutionary Computation Conference,  pages 1061–1068, New York, New York, USA, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
+page_content='  ACM Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OdAyT4oBgHgl3EQf7PpY/content/2301.00835v1.pdf'}
diff --git a/OtAzT4oBgHgl3EQfIftM/content/tmp_files/2301.01062v1.pdf.txt b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/2301.01062v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..67b9383d302264afb6d7c74372f3f913e55aaec8
--- /dev/null
+++ b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/2301.01062v1.pdf.txt
@@ -0,0 +1,2037 @@
+arXiv:2301.01062v1  [math.AT]  3 Jan 2023
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+OSCAR RANDAL-WILLIAMS
+Abstract. We describe the ring structure of the rational cohomology of the
+Torelli groups of the manifolds #gSn × Sn in a stable range, for 2n ≥ 6.
+Some of our results are also valid for 2n = 2, where they are closely related to
+unpublished results of Kawazumi and Morita.
+Contents
+1.
+Introduction
+1
+2.
+Characteristic classes
+3
+3.
+Twisted cohomology of diffeomorphism groups
+9
+4.
+Cohomology of Torelli groups
+19
+5.
+The case 2n = 2
+22
+References
+28
+1. Introduction
+This paper can be considered as a (somewhat extensive) addendum to our earlier
+work with Kupers [KRW20b]. We shall be concerned with the manifold Wg :=
+#gSn × Sn generalising to higher dimensions the orientable surface of genus g, its
+topological group Diff+(Wg) of orientation-preserving diffeomorphisms, and various
+subgroups of it. The first kind of subgroups are Diff(Wg, D2n) ≤ Diff+(Wg, ∗) ≤
+Diff+(Wg), the diffeomorphisms which fix a disc and a point respectively.
+The
+second kind are their Torelli subgroups
+Tor(Wg, D2n),
+Tor+(Wg, ∗),
+Tor+(Wg),
+consisting of those diffeomorphisms which in addition act trivially on Hn(Wg; Z).
+The intersection form on this middle cohomology group is nondegenerate and (−1)n-
+symmetric, giving a homomorphism
+αg : Diff+(Wg) −→ Gg :=
+�
+Sp2g(Z)
+if n is odd,
+Og,g(Z)
+if n is even.
+This map is not always surjective, but its image is a certain finite-index subgroup
+G′
+g ≤ Gg, even when restricted to Diff(Wg, D2n), so there is an outer G′
+g-action on
+each of the Torelli subgroups. This makes the rational cohomology of each of the
+Torelli groups into G′
+g-representations.
+In [KRW20b], for 2n ≥ 6 we determined H∗(BTor(Wg, D2n); Q) as a ring and
+as a G′
+g-representation in a range of degrees tending to infinity with g.
+Using
+the Serre spectral sequence associated to various simple fibrations relating the dif-
+ferent Torelli groups we were able to also determine H∗(BTor+(Wg, ∗); Q) and
+H∗(BTor+(Wg); Q) as G′
+g-representations. This kind of argument was not able to
+2010 Mathematics Subject Classification. 55R40, 11F75, 57S05, 18D10, 20G05.
+Key words and phrases. Cohomology of diffeomorphism groups, Torelli groups, cohomology of
+arithmetic groups, Miller-Morita-Mumford classes.
+1
+
+2
+OSCAR RANDAL-WILLIAMS
+determine the ring structure, however, as multiplicative information gets lost when
+passing to the associated graded of the Serre filtration. Here we shall determine
+H∗(BTor+(Wg, ∗); Q) and H∗(BTor+(Wg); Q) as Q-algebras too: this is achieved in
+Theorem 4.1. The statement given there is more powerful, but just as in [KRW20b,
+Section 5] one can extract from it the following presentation for H∗(BTor+(Wg); Q),
+which is easier to parse. (A presentation for H∗(BTor+(Wg, ∗); Q) can be extracted
+in a similar way.)
+Let us write H(g) := Hn(Wg; Q), on which G′
+g operates in the evident way. Let
+λ : H(g) ⊗ H(g) → Q denote the intersection form, and {ai}2g
+i=1 be a basis of H(g)
+with dual basis {a#
+i }2g
+i=1 in the sense that λ(a#
+i , aj) = δij, so that the form dual
+to the pairing λ is ω = �2g
+i=1 ai ⊗ a#
+i . In Section 2.2 we will construct certain
+“modified twisted Miller–Morita–Mumford classes”, which when restricted to the
+Torelli group yield G′
+g-equivariant maps
+¯κc : H(g)⊗r −→ Hn(r−2)+|c|(BTor+(Wg); Q)
+for each c ∈ Q[e, p1, p2, . . . , pn−1] = H∗(BSO(2n); Q) and each s ≥ 0. When r = 0
+we write ¯κc = ¯κc(1); these agree with the usual Miller–Morita–Mumford classes κc.
+Theorem A. If 2n ≥ 6 then, in a range of degrees tending to infinity with g,
+H∗(BTor+(Wg); Q) is generated as a Q-algebra by the classes ¯κc(v1 ⊗ · · · ⊗ vr) for
+c a monomial in e, p1, . . . , pn−1, and r ≥ 0, such that n(r −2)+|c| > 0. A complete
+set of relations in this range is given by
+(i) ¯κc(vσ(1) ⊗ · · · ⊗ vσ(r)) = sign(σ)n · ¯κc(v1 ⊗ · · · ⊗ vr),
+(ii) ¯κe(v1) = 0,
+(iii)
+�
+i
+¯κx(v ⊗ ai) · ¯κy(a#
+i ⊗ w) = ¯κx·y(v ⊗ w) +
+1
+χ2 ¯κe2 · ¯κx(v) · ¯κy(w)
+− 1
+χ
+�
+¯κe·x(v) · ¯κy(w) + ¯κx(v) · ¯κe·y(w)
+�
+,
+(iv)
+�
+i
+¯κx(v ⊗ ai ⊗ a#
+i ) = χ−2
+χ ¯κe·x(v) +
+1
+χ2 ¯κe2 · ¯κx(v),
+(v) ¯κLi = 0,
+for v ∈ H(g)⊗r and w ∈ H(g)⊗s.
+For 2n = 4 or 2n = 2 there is still a map from the Q-algebra given by this
+presentation to H∗(BTor+(Wg); Q). If 2n = 2 then (in a stable range) this map
+is an isomorphism onto the maximal algebraic subrepresentation in degrees ≤ N,
+assuming that H∗(BTor+(Wg); Q) is finite-dimensional in degrees < N for all large
+enough g. This is known to hold for N = 2 by work of Johnson [Joh85].
+1.1. Outline. The overall strategy is parallel to [KRW20b]. There we defined cer-
+tain twisted Miller–Morita–Mumford classes and used them to describe the twisted
+cohomology groups H∗(BDiff+(Wg, D2n); H⊗s) in a stable range of degrees, where
+H is the local coefficient system corresponding to Hn(Wg; Q) with the action by dif-
+feomorphisms of Wg. This calculation was valid for 2n = 2 as well. Using that for
+2n ≥ 6 the G′
+g-representations H∗(BTor+(Wg, D2n); Q) extend to representations
+of the ambient algebraic group (namely Sp2g or Og,g) by [KRW20a]1, the argument
+was completed by establishing the degeneration of the Serre spectral sequence
+Ep,q
+2
+= Hp(G′
+g; Hq(BTor+(Wg, D2n); Q)⊗H⊗s) =⇒ Hp+q(BDiff+(Wg, D2n); H⊗s)
+using work of Borel, and then using a categorical form of Schur–Weyl duality to ex-
+tract the structure of H∗(BTor+(Wg, D2n); Q) from the H∗(BDiff+(Wg, D2n); H⊗s)
+for all s’s and various structure maps between them.
+1In fact we did something more complicated in [KRW20b] because this algebraicity result was
+not known at the time, but please allow some narrative leeway.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+3
+The twisted Miller–Morita–Mumford classes may be defined on BDiff+(Wg, ∗)
+too, but not on BDiff+(Wg).
+Our first task will be to define so-called “modi-
+fied twisted Miller–Morita–Mumford classes” in H∗(BDiff+(Wg); H⊗s) and analyse
+their behaviour: it turns out that their behaviour is significantly more complicated
+than the unmodified version, though still understandable. We will then use them
+to describe the twisted cohomology groups H∗(BDiff+(Wg); H⊗s) in a stable range
+of degrees. This description will be in terms of a certain vector space of graphs
+with vertices labelled by monomials in Euler and Pontrjagin classes, which play the
+role here of the vector spaces of labelled partitions from [KRW20b]. The passage
+from this calculation to H∗(BTor+(Wg); Q) is as above.
+The case of dimension 2n = 2 is somewhat special, in precisely the same way as
+it was in [KRW20b]: the calculation of H∗(BDiff+(Wg); H⊗s) is valid in this case,
+but as the cohomology of BTor+(Wg) is not even known to be finite-dimensional
+in a stable range, we cannot make a conclusion about it. (Instead one can make
+a conclusion about the continuous cohomology of the Torelli group, i.e. the Lie
+algebra cohomology of its Mal’cev Lie algebra: see [KRW21], [FNW21], [Hai20].)
+In addition, in this case our modified twisted Miller–Morita–Mumford classes are
+essentially the same as those that have been defined by Kawazumi and Morita
+[Mor96, KM96, KM01], and the graphical calculus that we employ is similar to
+theirs. In Section 5 we fully explain this connection, and also relate it to work of
+Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].
+To avoid a great deal of repetition we have refrained from spelling out a lot of the
+background that was given in [KRW20b], and from giving in detail arguments that
+are very similar to those given there. As such this paper should not be considered
+as attempting to be self-contained: given that its interest will be to readers of
+[KRW20b] this should not present a problem.
+1.2. Acknowledgements. I am grateful to Alexander Kupers for feedback on an
+earlier draft. I was supported by the ERC under the European Union’s Horizon
+2020 research and innovation programme (grant agreement No. 756444) and by a
+Philip Leverhulme Prize from the Leverhulme Trust.
+2. Characteristic classes
+2.1. Recollection on twisted Miller–Morita–Mumford classes. If π′ : E′ →
+X′ is an oriented smooth W 2n
+g -bundle equipped with a section s : X′ → E′, and
+H denotes the local coefficient system x �→ Hn((π′)−1(x); Q) on X′, then it is
+explained in [KRW20b, Section 3.2] that there is a unique class ε = εs ∈ Hn(E′; H)
+characterised by
+(i) for each x ∈ X′ the element ε|(π′)−1(x) ∈ Hn((π′)−1(x); Q)⊗Hn((π′)−1(x); Q)
+is coevaluation, and
+(ii) s∗ε = 0.
+The proof is as follows. The Serre spectral sequence yields an exact sequence
+0 → Hn(X′; H)
+(π′)∗
+→ Hn(E′; H) → H0(X′; H∨⊗H)
+dn+1
+→ Hn+1(X′; H)
+(π′)∗
+→ Hn+1(E′; H)
+and the section s shows that the right-hand map (π′)∗ is injective, so that the map
+dn+1 is zero, and splits the left-hand map (π′)∗. The class coev ∈ H0(X′; H∨ ⊗ H)
+then gives rise to a unique ε satisfying the given properties.
+We then defined the twisted Miller–Morita–Mumford class
+(2.1)
+κεac = κεac(π′, s) :=
+�
+π′ εa · c(Tπ′E′) ∈ H(a−2)n+|c|(X′; H⊗a).
+
+4
+OSCAR RANDAL-WILLIAMS
+2.2. Modified twisted Miller–Morita–Mumford classes. If π : E → X is
+an oriented smooth W 2n
+g -bundle but is not equipped with a section then, as long
+as χ := χ(Wg) = 2 + (−1)n2g ̸= 0 (i.e. (n, g) ̸= (odd, 1), cf. Remark 2.1), the
+cohomological role of the section can instead be played by the transfer map
+1
+χπ!(e · −) : H∗(E; H) −→ H∗(X; H),
+where e := e(TπE) ∈ H2n(E; Q) denotes the Euler class of the vertical tangent
+bundle. The projection formula
+1
+χπ!(e · π∗(x)) = 1
+χπ!(e) · x = χ
+χx = x
+shows that this map splits π∗. Thus in this situation there is a unique class ¯ε ∈
+Hn(E; H) characterised by
+(i) for each x ∈ X the element ¯ε|π−1(x) ∈ Hn(π−1(x); Q) ⊗ Hn(π−1(x); Q) is
+coevaluation, and
+(ii)
+1
+χπ!(e · ¯ε) = 0.
+Remark 2.1. If (n, g) = (odd, 1) then there is no class ¯ε ∈ Hn(E; H) satisfying (i)
+and natural under pullback. To see this it suffices to give one example of a smooth
+oriented W1-bundle for which ¯ε does not exist. Consider the Borel construction for
+the evident action of S1 × S1 on W1 = Sn × Sn given by considering Sn as the unit
+sphere in C(n+1)/2. This gives a smoth oriented W1-bundle over B(S1 × S1) with
+total space E ≃ CP(n−1)/2 × CP(n−1)/2. Thus Hn(E; H) = 0 as n is odd but E has
+a cell structure with only even-dimensional cells.
+By analogy with (2.1) we may then define the modified twisted Miller–Morita–
+Mumford class
+(2.2)
+κ¯εac = κ¯εac(π) :=
+�
+π
+¯εa · c(TπE) ∈ H(a−2)n+|c|(X; H⊗a).
+If π : E → X does have a section s : X → E then the class ε ∈ Hn(E; H) is also
+defined, and we may compare it with ¯ε as follows:
+Lemma 2.2. If π : E → X has a section then ¯ε = ε − 1
+χπ∗κεe.
+Proof. The classes ε, ¯ε ∈ Hn(E; H) are both defined, and agree when restricted to
+the fibres of the map π, so by considering the Serre spectral sequence for π we must
+have ¯ε − ε = π∗(x) for some class x ∈ Hn(X; H). Applying 1
+χπ!(e · −) we see that
+x = 1
+χπ!(e · (¯ε − ε)) = 0 − 1
+χπ!(e · ε) = − 1
+χκεe. (Here we have used, as we often will,
+the fact that e has even degree to commute it past ε.)
+□
+Remark 2.3 (Splitting principle). The pullback
+(2.3)
+E1 ×X E2
+E2
+E1
+X,
+pr1
+pr2
+π2
+π1
+where πi : Ei → X are copies of the map π, is equipped with a section given by the
+diagonal map ∆ : E1 → E1 ×X E2. As the maps π∗
+1 and pr∗
+2 are injective (they are
+split by their corresponding transfer maps), for the purpose of establishing identities
+between the characteristic classes we have discussed it suffices to do so for bundles
+which do have a section.
+There is another description of ¯ε which is sometimes useful. Let pr1 : E ×X E →
+E be as in (2.3), which is an oriented Wg-bundle with section given by the diagonal
+map ∆, and so has the class κεe(pr1, ∆) defined.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+5
+Lemma 2.4. We have ¯ε = − 1
+χκεe(pr1, ∆) ∈ Hn(E; H).
+Proof. By Remark 2.3 we may suppose without loss of generality that π : E → X
+has a section s : X → E, defining a class ε = εs ∈ Hn(E; H).
+Consider the
+pullback square (2.3); let ei = (pri)∗(e) ∈ H2n(E1 ×X E2; Q) be the Euler class
+of the vertical tangent bundle on the ith factor. Considering pr1 as a Wg-bundle
+with section given by the diagonal map ∆, there is a class ε∆ ∈ Hn(E1 ×X E2; H)
+defined. As both ε∆ and pr∗
+2(εs) restrict to coevaluation on the fibres of pr1, we
+have ε∆ − pr∗
+2(εs) = pr∗
+1(x) for some class x ∈ Hn(E1; H). Pulling this equation
+back along ∆ shows that x = −εs, so ε∆ = pr∗
+2(εs) − pr∗
+1(εs). Then we have
+κεe(pr1, ∆) = (pr1)!(ε∆ · e2)
+= (pr1)!((pr∗
+2(εs) − pr∗
+1(εs)) · e2)
+= (pr1)!(pr∗
+2(εs · e)) − (pr1)!(pr∗
+1(εs) · e2)
+= π∗
+1(π2)!(εs · e) − χεs
+= π∗
+1κεe(π, s) − χεs
+= −χ¯ε
+as required.
+□
+The intersection form of the fibres of π : E → X provides a map of local coeffi-
+cient systems λ : H⊗H → Q; as we will often be concerned with applying it to two
+factors of a tensor power H⊗k and will have to specify which factors we apply it to,
+we will denote λ by λ1,2 and more generally write λi,j : H⊗k → H⊗k−2 for the map
+that applies λ to the ith and jth factors. We call such operations contraction.
+If p : E1 ×X E2 → X denotes the fibre product of two copies of π : E → X, and
+if this has a section s : X → E, then in [KRW20b, Lemma 3.9] we have established
+the formula
+(2.4)
+λ1,2(ε × ε) = ∆!(1) − 1 × v − v × 1 + p∗s∗e ∈ H2n(E1 ×X E2; Q),
+where v = s!(1) ∈ H2n(E; Q) is the fibrewise Poincar´e dual to the section s, cf.
+[KRW20b, Lemma 3.1]. The analogue of this formula for ¯ε is as follows.
+Lemma 2.5. We have
+λ1,2(¯ε × ¯ε) = ∆!(1) +
+1
+χ2 p∗κe2 − 1
+χ(e × 1 + 1 × e) ∈ H2n(E1 ×X E2; Q).
+Proof. As in Remark 2.3 we may suppose without loss of generality that π : E → X
+has a section s : X → E, so that ε ∈ Hn(E; H) is defined.
+By Lemma 2.2 we have ¯ε = ε − 1
+χπ∗κεe ∈ Hn(E; H), and so
+λ1,2(¯ε × ¯ε) = λ1,2((ε − 1
+χπ∗κe·ε) × (ε − 1
+χπ∗κe·ε))
+= λ1,2(ε × ε) − λ1,2( 1
+χπ∗κεe × ε)
+− λ1,2(ε × 1
+χπ∗κεe) + λ1,2( 1
+χπ∗κεe × 1
+χπ∗κεe).
+The first term is given by (2.4), and using [KRW20b, Proposition 3.10] the last
+term is given by
+λ1,2( 1
+χπ∗κεe × 1
+χπ∗κεe) =
+1
+χ2 p∗λ1,2(κεe · κεe) =
+1
+χ2 p∗(κe2 + (χ2 − 2χ)s∗e).
+For the middle two terms, note that
+ε × 1
+χπ∗κεe = 1
+χ(ε × 1) · p∗(κe·ε) = 1
+χ(ε · π∗κεe) × 1
+
+6
+OSCAR RANDAL-WILLIAMS
+so we need to calculate λ1,2(ε · π∗κεe) ∈ H2n(E; Q). The class ε · κεe is the fibre
+integral along pr1 : E1 ×X E2 → E1 of ε × (ε · e) = (ε × ε) · (1 × e), so
+λ1,2(ε · κεe) = (pr1)!(λ1,2(ε × ε) · (1 × e))
+= (pr1)!((∆!(1) − 1 × v − v × 1 + p∗s∗e) · (1 × e))
+= e − π∗s∗e − χv + χπ∗s∗e
+and hence
+λ1,2(ε × 1
+χκεe) = 1
+χ(e − π∗s∗e − χv + χπ∗s∗e) × 1
+= 1
+χe × 1 + χ−1
+χ p∗s∗e − v × 1
+and similarly
+λ1,2( 1
+χκεe × ε) = 1
+χ1 × e + χ−1
+χ p∗s∗e − 1 × v.
+Combining these gives
+λ1,2(¯ε × ¯ε) = ∆!(1) − 1 × v − v × 1 + p∗s∗e
++
+1
+χ2 p∗κe2 + χ−2
+χ p∗s∗e
+− ( 1
+χe × 1 + χ−1
+χ p∗s∗e − v × 1)
+− ( 1
+χ1 × e + χ−1
+χ p∗s∗e − 1 × v)
+= ∆!(1) +
+1
+χ2 p∗κe2 − 1
+χ(e × 1 + 1 × e)
+as required.
+□
+If in addition we have a lift ℓ : E → B of the fibrewise Gauss map along some
+fibration θ : B → BSO(2n) then for any c ∈ H∗(B; Q) we can define modified
+twisted Miller–Morita–Mumford classes by the formula
+κ¯εac := π!(¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X; H⊗a).
+Under the action of a permutation σ ∈ Sa of the tensor factors these classes
+transform as sign(σ)n, as ¯ε has degree n.
+Thus for any finite set S there is a
+well-defined element
+(2.5)
+κ¯εSc := π!(¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X; H⊗S) ⊗ (det QS)⊗n.
+To keep track of signs, for an ordered set S = {s1 < s2 < · · · < sa} we will often
+write κ¯εs1,...,sac ∈ Hn(a−2)+|c|(X; H⊗S) for the corresponding element, understand-
+ing that if σ is a reordering of S then κ¯εσ(s1),...,σ(sa)c = sign(σ)nκ¯εs1,...,sa c.
+Using Lemma 2.5 we immediately see that these characteristic classes satisfy the
+following analogue of the contraction formula from [KRW20b, Proposition 3.10].
+Proposition 2.6 (Modified contraction formula). In H∗(X; H⊗−) we have the
+identities
+λ1,2(π!(¯ε1,2,...,a · ℓ∗c)) = ( χ−2
+χ )π!(¯ε3,4,...,a · ℓ∗(e · c)) +
+1
+χ2 κe2 · π!(¯ε3,4,...,a · ℓ∗c)
+and
+λa,a+1(π!(¯ε1,2,...,a · ℓ∗c) · π!(¯εa+1,...,a+b · ℓ∗c′)) = π!(¯ε1,...,a−1,a+2,...,a+b · ℓ∗(c · c′))
++
+1
+χ2 κe2 · π!(¯ε1,...,a−1 · ℓ∗c) · π!(¯εa+2,...,a+b · ℓ∗c′)
+− 1
+χπ!(¯ε1,...,a−1 · ℓ∗(e · c)) · π!(¯εa+2,...,a+b · ℓ∗c′)
+− 1
+χπ!(¯ε1,...,a−1 · ℓ∗c) · π!(¯εa+2,...,a+b · ℓ∗(e · c′)).
+Similarly, from Lemma 2.2 we immediately obtain the following:
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+7
+Proposition 2.7. If the bundle π : E → X has a section, so that the class ε and
+hence κεSc is defined, then
+κ¯εSc =
+�
+I⊆S
+κεIc(− 1
+χκεe)S\I ∈ H∗(X; H⊗S) ⊗ (det QS)⊗n.
+Let us give an example of using the modified contraction formula to evaluate an
+expression.
+Example 2.8. Consider the class λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6). Then
+λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = λ1,5λ2,6
+�
+κ¯ε1,2,5,6 +
+1
+χ2 κe2κ¯ε1,2κ¯ε5,6
+− 1
+χ(κ¯ε1,2eκ¯ε5,6 + κ¯ε1,2κ¯ε5,6e)
+�
+.
+The first term is
+λ1,5λ2,6(κ¯ε1,2,5,6) = (−1)nλ1,5λ2,6(κ¯ε1,5,2,6)
+= (−1)nλ2,6( χ−2
+χ κ¯ε2,6e +
+1
+χ2 κe2κ¯ε2,6)
+= (−1)n χ−2
+χ ( χ−2
+χ κe2 +
+1
+χ2 κe2χ) + (−1)n 1
+χ2 κe2( χ−2
+χ χ)
+= (−1)n( (χ−2)2
+χ2
++ 2 χ−2
+χ2 )κe2.
+The second term is
+1
+χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε5,6) = (−1)n 1
+χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε6,5)
+= (−1)n 1
+χ2 κe2λ1,5(κ¯ε1,5)
+= (−1)n χ−2
+χ2 κe2.
+The third term is
+− 1
+χλ1,5λ2,6(κ¯ε1,2eκ¯ε5,6) = (−1)n+1 1
+χλ1,5λ2,6(κ¯ε1,2eκ¯ε6,5)
+= (−1)n+1 1
+χλ1,5(κ¯ε1,5e − 1
+χκ¯ε1eκ¯ε5e)
+= (−1)n+1 1
+χ
+�
+( χ−2
+χ κe2 +
+1
+χ2 κe2χ)
+− 1
+χ(κe2 +
+1
+χ2 κe2χ2 − 1
+χ(2χκe2))
+�
+= (−1)n+1( χ−2
+χ2 +
+1
+χ2 −
+1
+χ2 −
+1
+χ2 +
+2
+χ2 )κe2
+= (−1)n+1 χ−1
+χ2 κe2
+and the fourth term is the same as the third by the evident symmetry.
+In total we have
+λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = (−1)n( (χ−2)2
+χ2
++ 2 χ−2
+χ2 + χ−2
+χ2 − 2 χ−1
+χ2 )κe2
+= (−1)n χ−3
+χ κe2.
+□
+2.3. Graphical interpretation. In [KRW20b, Section 5] it was found to be very
+convenient to adopt a graphical formalism where κεac corresponds to a vertex with
+a half-edges incident to it and a formal label c, a product of κεac’s corresponds to
+a disjoint union of such vertices, and applying the contraction λi,j corresponds to
+pairing up the half-edges labelled i and j.
+It will be convenient to adopt a similar formalism here. Let S be a finite set, and
+V be a graded Q-algebra with a distinguished element e ∈ V2n. Slightly modifying2
+2The difference is that we allow labelled vertices whose contribution to the degree is 0.
+
+8
+OSCAR RANDAL-WILLIAMS
+the definition from [KRW20b, Proof of Theorem 5.1], a marked oriented graph with
+legs S and labelled by V consists of the following data:
+(i) a totally ordered finite set ⃗V (of vertices), a totally ordered finite set ⃗H (of
+half-edges), and a monotone function a: ⃗H → ⃗V (encoding that a half-edge
+h is incident to the vertex a(h)),
+(ii) an ordered matching m = {(ai, bi)}i∈I of the set H ⊔ S (encoding the
+oriented edges of the graph),
+(iii) a function c: V → V with homogeneous values, such that |c(v)|+n(|a−1(v)|−
+2) ≥ 0.
+Marked oriented graphs Γ = (⃗V , ⃗H, a, m, c) and Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) with the
+same set of legs S are isomorphic if there are order-preserving bijections ⃗V
+∼
+→ ⃗V ′
+and ⃗H
+∼
+→ ⃗H′ which intertwine a and a′, intertwine c and c′, and send m to m′. An
+oriented graph is an isomorphism class [Γ] of marked oriented graph. We assign to
+a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) the degree
+deg(Γ) :=
+�
+v∈V
+�
+|c(v)| + n(|a−1(v)| − 2)
+�
+= n(|H| − 2|V |) +
+�
+v∈V
+|c(v)|.
+Let π : E → X be an oriented Wg-bundle with a lift ℓ : E → B of the map clas-
+sifying the vertical tangent bundle along θ : B → BSO(2n), and let V := H∗(B; Q)
+and e := θ∗e ∈ V2n. Then given a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) with
+legs S we form a class
+¯κ(Γ) ∈ Hdeg(Γ)(X; H⊗S)
+by the following recipe. Firstly, we may form
+(2.6)
+�
+v∈V
+κ¯εa−1(v)c(v) ∈ H∗(X; H⊗H),
+where we have used the ordering on ⃗V to order the product, and the ordering on
+⃗H to trivialise the factor of (det QH)⊗n = (�
+v∈V det Qa−1(v))⊗n that arises from
+(2.5). Secondly, taking two copies S1 and S2 of the set S and writing si ∈ Si for
+the element corresponding to s ∈ S we can form
+(2.7)
+�
+s∈S
+κ¯εs1,s2 ∈ H∗(X, H⊗(S1⊔S2)).
+As each κ¯εs1,s2 has degree 0, the product does not depend on how the factors are
+ordered. Taking the product of (2.6) and (2.7) and then applying λx,y for each
+ordered pair (x, y) in the matching m on H ⊔ S = H ⊔ S1 gives the required class
+¯κ(Γ) ∈ H∗(X; H⊗S2) = H∗(X; H⊗S).
+Example 2.9. In this graphical interpretation we recognise the class evaluated
+in Example 2.8 as that associated to the theta-graph with a certain ordering and
+orientation.
+Clearly ¯κ(Γ) only depends on the underlying oriented graph [Γ]. We now describe
+how it transforms when the orderings on V , H, and the pairs m are changed, without
+changing the underlying labelled graph. If Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) is another marked
+oriented graph and there are bijections f : H → H′ and g : V → V ′ intertwining
+a and a′ and c and c′ and such that under these bijections the matching m′ differs
+from m by reversing the order of k pairs, then
+¯κ(Γ′) = (−1)nksign(f)sign(g) · ¯κ(Γ)
+for certain signs described on [KRW20b, pp. 55-56].
+Graphs considered as representing ¯κ’s behave differently to those representing κ’s
+described in [KRW20b, Section 5]. To distinguish them we will depict the graphs
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+9
+representing κ’s in red, as we did in that paper, and the graphs representing ¯κ’s in
+blue. The contraction formula of [KRW20b, Proposition 3.10] was interpreted in
+[KRW20b, Section 5] as giving relations among red graphs which yield equivalent
+κ-classes. In the generality of a smooth oriented Wg-bundle π : E → X with section
+s : X → E these may be depicted as follows:
+•c
+=
+•ce
++s∗e
+•c
+−2s∗c
+•c
+•c′
+=
+•cc′
++s∗e
+•c
+•c′
+−s∗c
+•c′
+−s∗c′
+•c
+Figure 1. The contraction formula, displayed graphically.
+Here the negative terms only arise when they make sense, i.e. when the vertex
+has valence 2 in the first case, when the vertex labelled c has valence 1 in the second
+case, and when the vertex labelled c′ has valence 1 in the third case.
+Convention. In these and the following figures, to avoid clutter we have adopted the
+following ordering conventions: vertices are numbered starting from 1 from left to
+right, half-edges around each vertex are ordered clockwise starting from the marked
+half-edge, and edges are oriented from the smaller half-edge to the larger one.
+Similarly, the modified contraction formula of Proposition 2.6 can be interpreted
+as giving the following relations among blue graphs which yield equivalent ¯κ-classes:
+•c
+=
+χ−2
+χ
+•ce
++ 1
+χ2
+•c
+•e2
+•c
+•c′
+=
+•cc′
++ 1
+χ2
+•c
+•c′
+•e2
+− 1
+χ(
+•ce
+•c′
++
+•c
+•
+c′e
+)
+Figure 2. The modified contraction formula, displayed graphically.
+3. Twisted cohomology of diffeomorphism groups
+The main goal of this section is to describe the twisted cohomology groups
+H∗(BDiff+(Wg); H⊗S) and H∗(BDiff+(Wg, ∗); H⊗S)
+in a stable range of degrees, of the classifying space BDiff+(Wg) of the group of
+orientation-preserving diffeomorphisms of Wg (which classifies oriented Wg-bundles),
+and the classifying space BDiff+(Wg, ∗) of the group of orientation-preserving dif-
+feomorphisms of Wg which fix a point ∗ ∈ Wg (which classifies oriented Wg-bundles
+with section). In [KRW20b, Theorem 3.15] the analogous calculation was given for
+
+10
+OSCAR RANDAL-WILLIAMS
+the classifying space BDiff(Wg, D2n) of the group of diffeomorphisms of Wg which
+fix a disc D2n ⊂ Wg.
+In order to do this we will also discuss the manifolds Wg equipped with θ-
+structures for the tangential structure θ : BSO(2n)⟨n⟩ → BO(2n), i.e. the n-
+connected cover of BO(2n). In this case we will consider the homotopy quotients
+BDiffθ(Wg) := Bun(T Wg, θ∗γ2n)//Diff(Wg)
+BDiffθ(Wg, ∗) := Bun(T Wg, θ∗γ2n)//Diff(Wg, ∗)
+where Bun(T Wg, θ∗γ2n) denotes the space of vector bundle maps T Wg → θ∗γ2n
+from the tangent bundle of Wg to the bundle classified by θ. The group Diff(Wg)
+acts on the space of bundle maps by precomposing with the derivative.
+There is a factorisation θ : BSO(2n)⟨n⟩
+θor
+→ BSO(2n)
+σ→ BO(2n), and by ob-
+struction theory one sees that the space Bun(T Wg, σ∗γ2n) has two contractible
+path components corresponding to the two orientations of Wg. In particular there
+are equivalences
+Bun(T Wg, σ∗γ2n)//Diff(Wg) ≃ BDiff+(Wg)
+Bun(T Wg, σ∗γ2n)//Diff(Wg, ∗) ≃ BDiff+(Wg, ∗)
+and so θor induces maps
+BDiffθ(Wg) −→ BDiff+(Wg)
+and
+BDiffθ(Wg, ∗) −→ BDiff+(Wg, ∗).
+It is shown in [GRW19, Section 5.2] that these are principal SO[0, n − 1]-fibrations.
+In particular the spaces BDiffθ(Wg) and BDiffθ(Wg, ∗) are path-connected.
+3.1. Spaces of graphs. Our description of the twisted cohomology groups of
+BDiff+(Wg), BDiff+(Wg, ∗), BDiff(Wg, D2n), BDiffθ(Wg) and BDiffθ(Wg, ∗) in a
+stable range will be—via the graphical interpretation given in Section 2.3—in terms
+of graded vector spaces of labelled graphs, modulo certain relations. (Readers of
+[KRW20b] may have been expecting vector spaces of labelled partitions instead:
+here we have found spaces of graphs more convenient for formulating results, cf.
+Remark 3.2, though spaces of labelled partitions will still play a role in the proofs.)
+To describe these spaces of graphs we will use the graded Q-algebras
+V := H∗(BSO(2n)⟨n⟩; Q) = Q[p⌈ n+1
+4
+⌉, . . . , pn−1, e]
+W := H∗(BSO(2n); Q) = Q[p1, . . . , pn−1, e]
+with distinguished elements e of degree 2n given by the Euler class. In order to work
+in a way which is agnostic about the genus g of the manifold Wg under consideration,
+we will work over the ring Q[χ±1] instead of Q, where χ is an invertible formal
+parameter which will—later—be set to the Euler characteristic 2 + (−1)n2g of Wg.
+Definition 3.1.
+(i) Let
+Graph1(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼
+where ∼
+(a) imposes the sign rule for changing orderings of vertices and half-edges and
+for reversing orientations of edges;
+(b) imposes linearity in the labels, and sets a graph containing an a-valent
+vertex labelled by c with |c| + n(a − 1) < 0 to zero;
+(c) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, and if 2n ≡ 0
+mod 4 sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0;
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+11
+(d) imposes the contraction relations3
+•c
+=
+•ce
+−
+2c
+•c
+•c′
+=
+•cc′
+−
+c
+•c′
+−
+c′
+•c
+where the negative terms only arises when they makes sense, i.e. in the
+first case when the vertex has valence 2 and its label c is a scalar multiple
+of 1 ∈ V0, in the second case when c is a scalar multiple of 1 ∈ V0 and has
+valence 1, and similarly in the third case.
+(ii) Let
+Graphθ
+∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V} ⊗ V/ ∼
+where ∼ imposes (a)–(c) as well as
+(d′) imposes the contraction relations of Figure 1.
+(iii) Let
+Graph∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W} ⊗ W/ ∼
+where ∼ imposes (a) and (b), as well as
+(c′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent
+vertex labelled by any degree 2n monomial in Pontrjagin classes equal
+to 0, and for any 1 ≤ i ≤ ⌊n/4⌋ sets
+•
+cpi
+= 1
+χ
+•c
+⊗pi
+and (d′).
+(iv) Let
+Graphθ(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼
+where ∼ imposes (a) and (b), as well as
+(c′′′) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, if 2n ≡ 0 mod 4
+sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0, and sets the
+1-valent vertex labelled by e ∈ V2n equal to 0,
+(d′′′) imposes the contraction relations of Figure 2.
+(v) Let
+Graph(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W}/ ∼
+where ∼ imposes (a), (b), as well as
+(c′′′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent
+vertex labelled by any degree 2n monomial in Pontrjagin classes equal
+to 0, sets the 1-valent vertex labelled by e ∈ W2n equal to 0, and for
+any 1 ≤ i ≤ ⌊n/4⌋ sets
+3These are the relations from Figure 1 when s∗ kills all positive-degree classes.
+
+12
+OSCAR RANDAL-WILLIAMS
+•
+cpi
+= 1
+χ
+•c
+•
+epi
+and (d′′′).
+Remark 3.2 (Graphs and partitions). In all cases one can apply the (modified)
+contraction formula to pass from a graph to a sum of graphs with strictly fewer
+edges, and so by iterating to a sum of graphs with no edges. These are disjoint
+unions of labelled corollas, and so correspond to partitions of S with labels in V or
+W, plus additional external labels in cases (ii) and (iii). There are two issues with
+this. The first is that in cases (iv) and (v) it is not clear that the resulting sum of
+disjoint unions of labelled corollas is unique, as one has to choose an order in which
+to eliminate edges. The second is that even if it is, then the functoriality on the
+Brauer category which we describe below would involve gluing in edges and then
+eliminating them, leading to a complicated formula. This is why we have found it
+convenient to work with spaces of graphs.
+We wish to consider each of the above as defining functors on the (signed) Brauer
+category as in [KRW20b, Section 2.3], but to take into account the parameter χ we
+must slightly generalise to a Q[χ]-linear version of the (signed) Brauer category.
+Definition 3.3. For finite sets S and T let preBrχ(S, T ) be the free Q[χ]-module
+on tuples (f, mS, mT ) of a bijection f from a subset S◦ ⊂ S to a subset T ◦ ⊂ T ,
+an ordered matching mS of S \ S◦, and an ordered matching mT of T \ T ◦.
+Let Brχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −
+(f, m′
+S, m′
+T ) whenever mS agrees with m′
+S after reversing some pairs, and mT agrees
+with m′
+T after reversing some pairs.
+Let sBrχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −
+(−1)kl(f, m′
+S, m′
+T ) whenever mS agrees with m′
+S after reversing k pairs, and mT
+agrees with m′
+T after reversing l pairs.
+Let (s)Brχ be the Q[χ]-linear category whose objects are finite sets, and whose
+morphisms are the Q[χ]-modules (s)Brχ(S, T ) defined above. In the case of Brχ
+we think of [f, mS, mT ] as representing 1-dimensional cobordisms with no closed
+components: then the composition law is given by composing cobordisms and then
+replacing each closed 1-manifold by a factor of χ − 2. In the case of sBrχ we think
+of (f, mS, mT ) as representing oriented 1-dimensional cobordisms with no closed
+components: then the composition law is given by composing cobordisms and then
+replacing each compatibly oriented closed 1-manifold by a factor of −(χ − 2).
+Let d(s)Brχ denote the subcategories having all objects and morphisms spanned
+by [f, mS, mT ] with T ◦ = ∅.
+For a central charge d ∈ Q let (d)(s)Brd denote the Q-linear category obtained
+by specialising the Q[χ]-linear category (d)(s)Brχ to χ = 2+d for (d)Br or χ = 2−d
+for (d)sBr. (This notation then agrees with [KRW20b, Definition 2.14, 2.19].)
+We consider the spaces of graphs above as defining Q[χ]-linear functors
+Graph1(−), Graphθ
+∗(−), Graph∗(−), Graphθ(−), Graph(−) : (s)Brχ → Gr(Q[χ±1]-mod)
+in the evident way, by gluing of oriented graphs (after orientations have been ar-
+ranged to be compatible). We endow them with a lax symmetric monoidality by
+disjoint union of graphs. We write Graph1(−)g : (s)Br2g → Gr(Q-mod) and so on
+for their specialisations at χ = 2 + (−1)n2g (defined for (n, g) ̸= (odd, 1)).
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+13
+3.2. The isomorphism theorem. Theorem 3.4 below extends [KRW20b, Theo-
+rem 3.15] to BDiffθ(Wg, ∗), BDiff+(Wg, ∗), BDiffθ(Wg), and BDiff+(Wg).
+To formulate it we first observe that when π : E → X is a smooth oriented
+Wg-bundle and H is the local coefficient system over X given by the fibrewise nth
+homology of this bundle, the fibrewise intersection form λ : H⊗H → Q and its dual
+ω : Q → H ⊗ H are (−1)n-symmetric and satisfy λ ◦ ω = (−1)n2g · Id, so provide a
+Q-linear functor S �→ H⊗S from (s)Br2g to the category of local coefficient systems
+of Q-modules over X. (Strictly speaking our definitions require χ = 2 + (−1)n2g to
+be invertible, so we omit the case (n, g) = (odd, 1).) Composing this with taking
+cohomology gives a functor
+H∗(X; H⊗−) : (s)Br2g −→ Gr(Q-mod)
+S �−→ H∗(X; H⊗S).
+The relations in the various spaces of graphs defined in Section 3.1 were chosen
+precisely to match the contraction formula of [KRW20b, Proposition 3.10] (in the
+case of Graph1) and the modified contraction formula of Proposition 2.6 (in the
+other cases), so that assigning to a graph its associated κ- or ¯κ-class provides
+natural transformations
+(i) κ : Graph1(−)g → H∗(BDiff(Wg, D2n); H⊗−),
+(ii) κ : Graphθ
+∗(−)g → H∗(BDiffθ(Wg, ∗); H⊗−),
+(iii) κ : Graph∗(−)g → H∗(BDiff+(Wg, ∗); H⊗−),
+(iv) ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg); H⊗−),
+(v) ¯κ : Graph(−)g → H∗(BDiff+(Wg); H⊗−),
+of functors (s)Br2g → Gr(Q-mod).
+Theorem 3.4. For 2n = 2 or 2n ≥ 6 the maps (i)–(v) are isomorphisms in a
+range of cohomological degrees tending to infinity with g.
+We will first give the proof in cases (i), (ii), (iii), and in case (v) assuming case
+(iv); the much more involved case (iv) will be treated afterwards.
+Proof of Theorem 3.4 (i), (ii), (iii), (v). For case (i) observe that Graph1(−)g is
+naturally isomorphic to the functor G(−, V) from [KRW20b, Proof of Theorem 5.1],
+which is shown there to be isomorphic to the functor P(−, V)≥0 ⊗ det⊗n. This case
+then follows from [KRW20b, Theorem 3.15].
+For case (ii) we first construct the homotopy fibre sequence
+(3.1)
+BDiff(Wg, D2n) −→ BDiffθ(Wg, ∗) −→ BSO(2n)⟨n⟩.
+The left-hand term may be written as the homotopy quotient of Diff(Wg, ∗) acting
+on the Stiefel manifold Fr(T∗Wg) given by the space of frames in the tangent space to
+Wg at the point ∗ ∈ Wg, as this action is transitive and its stabiliser is the subgroup
+which fixes a point and its tangent space, which is homotopy equivalent to fixing a
+disc. The middle term was defined as the homotopy quotient of Diff(Wg, ∗) acting
+on Bun(T Wg, θ∗γ2n). Evaluation at ∗ ∈ Wg defines a Diff(Wg, ∗)-invariant map
+ev : Bun(T Wg, θ∗γ2n) −→ BSO(2n)⟨n⟩,
+which is a fibration.
+If we choose a point x ∈ BSO(2n)⟨n⟩ and a framing ξ :
+(θ∗γ2n)x
+∼
+→ R2n, then there is a map ξ∗ : ev−1(x) → Fr(T∗Wg) given by sending a
+bundle map ˆℓ : T Wg → θ∗γ2n whose underlying map sends ∗ to x to the framing
+ξ ◦ ˆℓx : T∗Wg → (θ∗γ2n)x → R2n. One verifies by obstruction theory that ξ∗ :
+ev−1(x) → Fr(T∗Wg) is a weak equivalence. Taking homotopy orbits for Diff(Wg, ∗)
+then gives the required homotopy fibre sequence.
+
+14
+OSCAR RANDAL-WILLIAMS
+As H∗(BDiff(Wg, D2n); H⊗S) is spanned by products of twisted Miller–Morita–
+Mumford classes κεac with c ∈ V in a stable range by (i), and these classes may be
+defined on BDiffθ(Wg, ∗), the Serre spectral sequence
+H∗(BSO(2n)⟨n⟩; Q) ⊗ H∗(BDiff(Wg, D2n); H⊗S) ⇒ H∗(BDiffθ(Wg, ∗); H⊗S)
+for the homotopy fibre sequence (3.1) collapses in a stable range. The result then
+follows by observing that the analogue of the Serre filtration of Graphθ
+∗(−)g, induced
+by the descending filtration by degrees of H∗(BSO(2n)⟨n⟩; Q) = V, has
+gr(Graphθ
+∗(−)g) ∼= V ⊗ Graph1(−)g,
+because modulo V>0 the formula of (d′) specialises to that of (d). The induced map
+gr(κ) : gr(Graphθ
+∗(−)g) −→ gr(H∗(BDiffθ(Wg, ∗); H⊗−))
+therefore has the form V ⊗ {the map κ in case (i)} so is an isomorphism in a stable
+range by case (i). Case (ii) follows.
+Case (iii) is just like the above, using the homotopy fibre sequence
+BDiff(Wg, D2n) −→ BDiff+(Wg, ∗) −→ BSO(2n)
+instead, which is established in the analogous way, and W in place of V.
+Case (v) can be deduced from case (iv) by applying the same method to the
+homotopy fibre sequence
+BDiffθ(Wg) −→ BDiff+(Wg)
+ξ
+−→ BSO[0, n]
+established in [GRW19, Section 5.2]. The filtration step is a little different, so we
+give some details. It follows from (iv) that H∗(BDiffθ(Wg); H⊗S) is spanned by
+products of twisted Miller–Morita–Mumford classes κ¯εac with c ∈ V in a stable
+range, and these may be defined on BDiff+(Wg) (in fact they may be defined even
+for c ∈ W) so the corresponding Serre spectral sequence
+H∗(BSO[0, n]; Q) ⊗ H∗(BDiffθ(Wg); H⊗S) ⇒ H∗(BDiff+(Wg); H⊗S)
+degenerates in a stable range. In this case the analogue of the Serre filtration on
+Graph(−)g is induced by giving the graph Υi := ({0}, ∅, ∅ → {0}, ∅, c(0) = epi)
+filtration 4i for 1 ≤ i ≤ ⌊n/4⌋, giving all other connected graphs filtration 0, and
+extending multiplicatively. The associated graded of this filtration has the form
+gr(Graph(−)g) ∼= Q[Υ1, Υ2, . . . , Υ⌊n/4⌋] ⊗ Graphθ(−)g,
+because the relation in (c′′′′) shows that any graph with a vertex labelled cpi for
+1 ≤ i ≤ ⌊n/4⌋ is equivalent to a graph of strictly larger filtration, unless the vertex
+is 0-valent and the label is epi. As ¯κ(Υi) = κepi = χ · ξ∗(pi) by [GRW19, Remark
+5.5] it follows that the induced map
+gr(¯κ) : gr(Graph(−)g) −→ gr(H∗(BDiff+(Wg); H⊗S))
+has the form {an isomorphism} ⊗ {the map ¯κ in case (iv)} so is an isomorphism in
+a stable range by case (iv).
+□
+3.3. Proof of Theorem 3.4 (iv). The proof of Theorem 3.4 (iv) is of a less formal
+nature. It will be parallel to that of [KRW20b, Theorem 3.15], but algebraically
+more complicated.
+An important tool will be the following lemma, inspired by
+[Qui71, p. 566].
+Lemma 3.5. Let G be a topological group and p : P → X be a principal G-bundle
+with action a : G×P → P, which satisfies the Leray–Hirsch property in cohomology
+over a field F. Then
+H∗(X; F)
+H∗(P; F)
+H∗(G; F) ⊗F H∗(P; F)
+p∗
+a∗
+1⊗Id
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+15
+is an equaliser diagram.
+Proof. Let us leave F implicit.
+By the Leray–Hirsch property H∗(P) is a free
+H∗(X)-module and hence is faithfully flat. Thus it suffices to prove that the dia-
+gram is an equaliser diagram after applying −⊗H∗(X) H∗(P). By the Leray–Hirsch
+property we also have H∗(P) ⊗H∗(X) H∗(P)
+∼
+→ H∗(P ×X P). Thus it suffices to
+show that
+H∗(P)
+H∗(P ×X P)
+H∗(G) ⊗ H∗(P ×X P)
+pr∗
+2
+a∗
+1⊗Id
+is an equaliser diagram, which is the same question for the principal G-bundle
+pr2 : P ×X P → P. But this principal G-bundle has a section given by the diagonal
+map, which trivialises it: this trivialisation identifies the diagram with
+H∗(P)
+H∗(G) ⊗ H∗(P)
+H∗(G) ⊗ H∗(G) ⊗ H∗(P)
+1⊗Id
+µ∗⊗Id
+1⊗Id
+which is indeed an equaliser diagram as it has a contraction induced by a∗.
+□
+We adapt the proof of [KRW20b, Theorem 3.15], supposing for concreteness that
+n is odd. Consider the tangential structure θ × Y : BSO(2n)⟨n⟩ × Y → BSO(2n)
+with Y = K(W ∨, n + 1) and W a generic rational vector space. Then we have
+H∗(Y ; Q) ∼= Sym∗(W[n + 1]), the symmetric algebra on the vector space W places
+in (even) degree n + 1. If n is even then like at the end of the proof of [KRW20b,
+Theorem 3.15] we would take Y = K(W ∨, n + 2) instead, so H∗(Y ; Q) would still
+be a symmetric algebra. Apart from this there is no essential difference, and we
+will not comment further on the differences in the case n even.
+There are associated universal Wg-bundles
+π : Eθ −→ BDiffθ(Wg)
+πY : Eθ×Y −→ BDiffθ×Y (Wg)
+and an evaluation map ℓ : Eθ×Y → Y . Neglecting the “maps to Y ” part of the
+tangential structure gives a homotopy fibre sequence
+map(Wg, Y ) −→ BDiffθ×Y (Wg) −→ BDiffθ(Wg).
+We can take Y to be a topological abelian group, which then acts fibrewise on the
+map θ × Y and hence acts on compatibly Eθ×Y and BDiffθ×Y (Wg). Using this we
+can form the homotopy fibre sequence
+map(Wg, Y )//Y −→ BDiffθ×Y (Wg)//Y −→ BDiffθ(Wg).
+The space map(Wg, Y )//Y is a K(Hn(Wg; Q)⊗W ∨, 1), so there is an identification
+of graded local coefficient systems
+H∗(map(Wg, Y )//Y ; Q) = Λ∗(H ⊗ W[1]).
+This is natural in the vector space W, and scaling by u ∈ Q× acts on Λk(H⊗ W[1])
+by uk. It follows that it acts this way on the kth row of the Serre spectral sequence
+Ep,q
+2
+= Hp(BDiffθ(Wg); Λq(H ⊗ W[1])) ⇒ Hp+q(BDiffθ×Y (Wg)//Y ; Q).
+As the differentials in this spectral sequence must be equivariant for this Q×-action,
+it follows that they must all be trivial.
+Furthermore this action gives a weight
+decomposition of both sides, which identifies
+H∗(BDiffθ(Wg); Λk(H ⊗ W)) ∼= H∗+k(BDiffθ×Y (Wg)//Y ; Q)(k),
+the weight k-subspace.
+To access the latter groups, we use that there is a map
+α : BDiffθ×Y (Wg) −→ Ω∞
+0 (MTθ ∧ Y+)
+
+16
+OSCAR RANDAL-WILLIAMS
+which by the main theorems of [Bol12, RW16, GTMW09] (for 2n = 2) and [GRW18,
+GRW14, GRW17] for (2n ≥ 6) is an isomorphism on cohomology in a stable
+range of degrees. Here MTθ is the Thom spectrum of −θ∗γ2n, so writing u−2n ∈
+H−2n(MTθ; Q) for its Thom class, by the Thom isomorphism we have
+H∗(MTθ; Q) ∼= u−2n · H∗(BSO(2n)⟨n⟩; Q) = u−2n · Q[p⌈n+1
+4
+⌉, . . . , pn−1, e].
+The rational cohomology of Ω∞
+0 (MTθ ∧ Y+) is then given by
+Sym∗([H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0),
+which can be considered as the free (graded-)commutative algebra on the even-
+degree classes κc,w1···wr with c ∈ Q[p⌈ n+1
+4
+⌉, . . . , pn−1, e] and wi ∈ W, modulo lin-
+earity in c and in the wi, and modulo commutativity of the wi. The pullbacks
+of these classes along α we again denote κc,w1···wr, and they may be described
+intrinsically as the fibre integrals πY
+! (c(TπY Eθ×Y ) · ℓ∗(w1 · · · wr)).
+Lemma 3.6. There are unique classes ¯κc,w1···wr ∈ H∗(BDiffθ×Y (Wg)//Y ; Q) which
+pull back to
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj) ∈ H∗(BDiffθ×Y (Wg); Q),
+and in a stable range of degrees H∗(BDiffθ×Y (Wg)//Y ; Q) is the free graded-commutative
+algebra on the classes ¯κc,w1···wr, modulo linearity in c and in the wi, commutativity
+of the wi, and modulo ¯κe,w1 = 0.
+Proof. We wish to apply Lemma 3.5 to the principal Y -bundle
+(3.2)
+BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg)//Y.
+First observe that the fibre inclusion j : Y → BDiffθ×Y (Wg) classifies the Wg-
+bundle pr1 : Y × Wg → Y equipped with the product θ-structure and the map
+ℓ = pr1 : Y × Wg → Y . Thus for any w ∈ W we have
+j∗κe,w = χw ∈ Hn+1(Y ; Q),
+and so (3.2) satisfies the Leray–Hirsch property.
+Lemma 3.5 then describes H∗(BDiffθ×Y (Wg)//Y ; Q) as the equaliser of
+(3.3)
+H∗(BDiffθ×Y (Wg); Q)
+H∗(Y ; Q) ⊗ H∗(BDiffθ×Y (Wg); Q).
+a∗
+1⊗Id
+In a stable range H∗(BDiffθ×Y (Wg); Q) is described in terms of the classes κc,w1···wr,
+so to make use of this equaliser description we must determine how these classes
+pull back along the action map
+a : Y × BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg).
+This map classifies the Wg-bundle Y × πY : Y × Eθ×Y → Y × BDiffθ×Y (Wg)
+equipped with the structure map Y × Eθ×Y
+Y ×ℓ
+→
+Y × Y
+·→ Y .
+As the wi ∈
+W = Hn+1(Y ; Q) are primitive with respect to the coproduct induced by the
+multiplication on Y , we have
+a∗(κc,w1···wr) = (Y × πY )!((1 × c(TπY Eθ×Y )) ·
+r
+�
+i=1
+(wi × 1 + 1 × ℓ∗(wi)))
+=
+�
+I⊔J={1,2,...,r}
+wI × κc,wJ.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+17
+Our goal now is to show that the classes defined by
+¯κc,w1···wr :=
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj) ∈ H∗(BDiffθ×Y (Wg); Q)
+are equalised by the maps (3.3), so by Lemma 3.5Lemma 3.5 descend to unique
+classes of the same name in H∗(BDiffθ×Y (Wg)//Y ; Q). To see this, we calculate
+using the formula above that
+a∗(¯κc,w1···wr) =
+�
+I⊔J={1,2,...,r}
+� �
+S⊔T =I
+wS × κc,wT
+�
+· (−1)|J| �
+j∈J
+(wj × 1 + 1
+χ1 × κe,wj)
+=
+�
+S⊔T ⊔U⊔V ={1,2,...,r}
+(−1)|U|wS⊔U ×
+�
+κc,wT ·
+�
+v∈V
+(− 1
+χκe,wv)
+�
+.
+For each A ⊆ {1, 2, . . ., r} the coefficient of wA is
+
+ �
+U⊆A
+(−1)|U|
+
+
+
+
+�
+T ⊔V ={1,...,r}\A
+κc,wT ·
+�
+v∈V
+(− 1
+χκe,wv)
+
+
+and �
+U⊆A(−1)|U| vanishes if A ̸= ∅, and is 1 if A = ∅ (it is the binomial expansion
+of (1 − 1)|A|), which shows that a∗(¯κc,w1···wr) = 1 × ¯κc,w1···wr as required.
+Finally, that these classes (except ¯κe,w1 = 0) freely generate the Q-algebra
+H∗(BDiffθ×Y (Wg)//Y ; Q) in a stable range follows from the fact that the κc,w1···wr
+freely generate H∗(BDiffθ×Y (Wg); Q) in a stable range, together with the observa-
+tion that ¯κc,w1···wr ≡ κc,w1···wr modulo the ideal generated by classes κe,w and the
+Leray–Hirsch property again.
+□
+Let us provide a “fibre-integral” interpretation of the classes we have just con-
+structed. Consider the map of principal Y -bundles
+Y
+Eθ×Y
+Eθ×Y //Y
+Y
+BDiffθ×Y (Wg)
+BDiffθ×Y (Wg)//Y.
+i
+πY
+πY //Y
+j
+The composition ℓ ◦ i : Y → Y is the identity, so i∗ℓ∗(w) = w ∈ Hn+1(Y ; Q). We
+showed in the proof above that j∗κe,w = χw ∈ Hn+1(Y ; Q), so in particular both
+these principal Y -bundles satisfy the Leray–Hirsch property. Together these give
+that
+i∗(ℓ∗(w) − 1
+χ(πY )∗κe,w) = 0.
+As Y is n-connected it follows from the Serre spectral sequence that there exists a
+unique class ¯ℓ∗(w) ∈ Hn+1(Eθ×Y //Y ; Q) which pulls back to ℓ∗(w) − 1
+χ(πY )∗κe,w.
+Lemma 3.7. We have
+¯κc,w1···wr = (πY //Y )!(c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) ∈ H∗(BDiffθ×Y (Wg)//Y ; Q).
+Proof. As the lower of the above principal Y -bundles satisfies the Leray–Hirsch
+property, this identity may be verified after pulling back to BDiffθ×Y (Wg).
+In
+H∗(Eθ×Y ; Q) we have ¯ℓ∗(w) = ℓ∗(w) − 1
+χ(πY )∗κe,w, so expanding out gives
+(πY )!(c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) = (πY )!(c ·
+r
+�
+i=1
+(ℓ∗(wi) − 1
+χ(πY )∗κe,wi))
+=
+�
+I⊔J={1,2,...,r}
+κc,wI ·
+�
+j∈J
+(− 1
+χκe,wj)
+
+18
+OSCAR RANDAL-WILLIAMS
+as required.
+□
+The classes ¯κc,w1···wr provide an isomorphism
+Sym∗
+�[H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0
+u−2n · e ⊗ W[n + 1]
+�
+−→ H∗(BDiffθ×Y (Wg)//Y ; Q)
+in a stable range, natural in W, which with the discussion above gives an identifi-
+cation of graded vector spaces
+H∗(BDiffθ(Wg); Λ∗(H ⊗ W[1])) ∼= Sym∗
+�[H∗(MTθ; Q) ⊗ Sym∗(W[n + 1])]>0
+u−2n · e ⊗ W[n + 1]
+�
+natural in W.
+Just as in the proof of [KRW20b, Theorem 3.15], and using its notation, this
+implies that there is a natural transformation
+(3.4)
+Pbis(−, V)≥0 ⊗ det⊗n −→ H∗(BDiffθ(Wg); H⊗−)
+of lax symmetric monoidal functors FB → Gr(Q-mod) which is an isomorphism in a
+stable range, where P(−, V)≥0 → Pbis(−, V)≥0 is the quotient by those partitions
+containing a part of size 1 labelled by e ∈ V2n. Assigning to a labelled part the
+corolla with that label gives a natural transformation
+(3.5)
+Pbis(−, V)≥0 ⊗ det⊗n −→ Graphθ(−)g,
+of lax symmetric monoidal functors FB → Gr(Q-mod), and we claim that using this
+(3.4) factors through the map ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg); H⊗−). Assuming
+this claim for now, observe that using the contraction relations in Definition 3.1 (iv)
+(d′′′) to contract all edges shows that (3.5) is surjective, which with the fact that
+(3.4) is an isomorphism in a stable range will show that the map ¯κ is an isomorphism
+in a stable range too (as well as the map (3.5)).
+It remains to show the factorisation, i.e. that the map (3.4) sends a part of size
+a labelled by c ∈ V to the class κ¯εac. We again proceed as in the relevant step of
+the proof of [KRW20b, Theorem 3.15]. There is a fibration sequence
+map(Wg, Y ) −→ Eθ×Y −→ Eθ
+and so, taking homotopy orbits for the fibrewise Y -action, a fibration sequence
+map(Wg, Y )//Y −→ Eθ×Y //Y −→ Eθ.
+Again by functoriality in W the associated Serre spectral sequence collapses to
+identify the weight decomposition as
+H∗(Eθ; Λk(H ⊗ W)) ∼= H∗+k(Eθ×Y //Y ; Q)(k).
+Given the description in Lemma 3.7 we must show that the map
+¯ℓ(−) : W −→ Hn+1(Eθ×Y //Y ; Q)(1) ∼= Hn(Eθ; H) ⊗ W
+is given by w �→ ¯ε ⊗ w, which is the analogue of [KRW20b, Claim 3.16]. As it is
+natural in the vector space W it must certainly be given by ¯ℓ(w) = x ⊗ w for some
+x ∈ Hn(Eθ; H), and we must show that x = ¯ε. That the restriction of x to the
+fibre Wg of π : Eθ → BDiffθ(Wg) is given by coevaluation may be done precisely
+as in [KRW20b, Claim 3.16]. By the characterisation of ¯ε it remains to check that
+1
+χ(πY )!(e · ¯ℓ∗(w)) = 0 ∈ Hn+1(BDiffθ×Y (Wg)//Y ; Q).
+By the Leray–Hirsch property this may be checked after pulling back to BDiffθ×Y (Wg),
+but as ¯ℓ∗(w) = ℓ∗(w) − 1
+χ(πY )∗κe,w ∈ Hn+1(Eθ×Y ; Q) by definition, the vanishing
+is immediate.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+19
+3.4. Comparisons. There are natural maps
+BDiff(Wg, D2n)
+BDiffθ(Wg, ∗)
+BDiffθ(Wg)
+BDiff(Wg, D2n)
+BDiff+(Wg, ∗)
+BDiff+(Wg)
+a
+b
+c
+d
+e
+f
+which each induce maps on H∗(−; H⊗S). There are corresponding maps of spaces
+of graphs
+Graph1(−)g
+Graphθ
+∗(−)g
+Graphθ(−)g
+Graph1(−)g
+Graph∗(−)g
+Graph(−)g
+a∗
+b∗
+e∗
+c∗
+d∗
+f ∗
+given as follows. The maps c∗ and d∗ are induced by the projections W → V. The
+maps a∗ and e∗ are induced by applying the augmentations V → Q and W → Q
+to the second tensor factor. The maps b∗ and f ∗ are more subtle, as they involve
+converting between blue graphs and red graphs, via the formula of Proposition 2.7.
+Graphically it is given by
+•
+�→
+•
+− 1
+χ(
+•
+•
+e
+•
+•
+e
+•
+•
+e
++
++
+)
+with certain orderings.
+The maps b and f are also oriented Wg-bundles, so they also induce fibre-
+integration maps b! and f! on cohomology. These are b∗- and f ∗-linear respectively,
+so are determined by the maps (of degree −2n)
+b! : V −→ Graphθ(−)g
+f! : W −→ Graph(−)g
+which each send a monomial c in pi’s and e to the graph given by a single vertex
+labelled by c.
+4. Cohomology of Torelli groups
+The isomorphisms provided by Theorem 3.4 can be converted into information
+about the spaces
+BTor(Wg, D2n), BTorθ(Wg, ∗), BTor+(Wg, ∗), BTorθ(Wg), BTor+(Wg)
+just as [KRW20a, Theorem 4.1] is deduced from [KRW20a, Theorem 3.15]. Let us
+give the definition of these spaces and formulate the result: the following is largely
+a reminder of some points from [KRW20a], and we do not spell out all details again.
+The group Diff+(Wg) acts on Hn(Wg; Z) preserving the nondegenerate (−1)n-
+symmetric intersection form λ : Hn(Wg; Z) ⊗ Hn(Wg; Z) → Z. This provides a
+homomorphism
+αg : Diff+(Wg) −→ Gg :=
+�
+Sp2g(Z)
+if n is odd,
+Og,g(Z)
+if n is even.
+This map is not always surjective, but its image is a certain finite-index subgroup
+G′
+g ≤ Gg, an arithmetic group associated to the algebraic group Sp2g or Og,g. This
+subgroup has been determined by Kreck [Kre79]: it is the whole of Gg if n is even
+or n = 1, 3, 7, and otherwise is the subgroup Spq
+2g(Z) ≤ Sp2g(Z) of those matrices
+which preserve the standard quadratic refinement (of Arf invariant 0).
+
+20
+OSCAR RANDAL-WILLIAMS
+We define Tor+(Wg) to be the kernel of this homomorphism, and Tor+(Wg, ∗)
+and Tor(Wg, D2n) to be the kernel of its restriction to the subgroups Diff+(Wg, ∗)
+and Diff(Wg, D2n) respectively (these restrictions still have image G′
+g). Further-
+more, we define
+BTorθ(Wg) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg)
+BTorθ(Wg, ∗) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg, ∗),
+where Bun+(T Wg, θ∗γ2n) ⊂ Bun(T Wg, θ∗γ2n) consists of the orientation-preserving
+bundle maps (for some choice of orientation of θ∗γ2n that we make once and for
+all). By the discussion at the beginning of Section 3 the spaces Bun+(T Wg, θ∗γ2n)
+are path-connected, so each of the BTor’s we have defined are principal G′
+g-bundles
+over the corresponding BDiff’s. In particular, their rational cohomologies are both
+Q-algebras and G′
+g-representations, and we will describe them as such in a stable
+range. Before doing so, we recall that the work of Borel identifies
+H∗(G′
+g; Q) =
+�
+Q[σ2, σ6, σ10, . . .]
+if n is odd,
+Q[σ4, σ8, σ12, . . .]
+if n is even.
+in a stable range of degrees, where σ4i−2n may be chosen so that it pulls back to the
+Miller–Morita–Mumford class κLi ∈ H4i−2n(BDiff+(Wg; Q) associated to the ith
+Hirzebruch L-class. In particular the κLi vanish in the cohomology of BTor+(Wg).
+Let us write H(g) := Hn(Wg; Q), which is the standard representation of G′
+g.
+Pulled back from BDiff+(Wg) to BTor+(Wg) the coefficient system H is canonically
+trivialised, but has an action of G′
+g: it can be identified with the dual H(g)∨. The
+edge homomorphism of the Serre spectral sequence
+(4.1)
+H∗(BDiff+(Wg); H⊗S) −→
+�
+H∗(BTor+(Wg); Q) ⊗ (H(g)∨)⊗S�G′
+g
+allows us to consider the modified twisted Miller–Morita–Mumford classes ¯κεSc as
+providing G′
+g-equivariant homomorphisms
+¯κc : H(g)⊗S −→ Hn(|S|−2)+|c|(BTor+(Wg); Q).
+The identities from the modified contraction formula correspond to identities
+among these maps: this will give relations analogous to [KRW20b, Section 5.2],
+which we will spell out after the proof of Theorem 4.1 below. First we explain how
+these relations can be organised in a categorical way, as follows.
+Considering (4.1) as a natural transformation of functors on (s)Br2g, we may
+precompose it with the map
+¯κ : Graphg(−) −→ H∗(BDiff+(Wg); H⊗−)
+(which is an isomorphism in a stable range for n ̸= 2 by Theorem 3.4). This gives
+G′
+g-equivariant maps H(g)⊗S ⊗ Graphg(S) → H∗(BTor+(Wg); Q) which assemble
+to a map
+K∨ ⊗(s)Br2g Graphg(−) −→ H∗(BTor+(Wg); Q)
+out of the coend, where K : (s)Br2g → Rep(G′
+g) sends S to H(g)⊗S. The domain
+obtains a graded-commutative Q-algebra structure coming from the lax symmetric
+monoidality of Graphg(−) and strong symmetric monoidality of K(−). Theorem
+4.1 below will say that this is surjective in a stable range, with kernel the ideal
+generated by the κLi, but before stating it we explain a simplification.
+Let us write i : d(s)Br → (s)Br2g for the inclusion of the downward (signed)
+Brauer category. Thus subcategory is independent if g, as no circles can be created
+by composing morphisms in the downward Brauer category. Write Graph1(−)′ ⊂
+i∗Graph1(−)g for the subfunctor where we forbid bivalent vertices labelled by 1 ∈ V
+both of whose half-edges are legs; similarly, this functor is independent of g. Like
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+21
+just after [KRW20b, Proposition 3.11], Graph1(−)g is then the left Kan extension
+i∗Graph1(−)′ of Graph1(−)′ along i. We similarly define Graphθ
+∗(−)′, Graph∗(−)′,
+Graphθ(−)′, and Graph(−)′, whose left Kan extensions again recover the original
+functors. The following is the analogue of [KRW20b, Theorem 4.1].
+Theorem 4.1. There are G′
+g-equivariant ring homomorphisms
+i∗(K∨) ⊗d(s)Br Graph1(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor(Wg, D2n); Q)
+(i)
+i∗(K∨) ⊗d(s)Br Graphθ
+∗(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTorθ(Wg, ∗); Q)
+(ii)
+i∗(K∨) ⊗d(s)Br Graph∗(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor+(Wg, ∗); Q)
+(iii)
+i∗(K∨) ⊗d(s)Br Graphθ(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTorθ(Wg); Q)
+(iv)
+i∗(K∨) ⊗d(s)Br Graph(−)′
+(κLi | 4i − 2n > 0)
+−→ H∗(BTor+(Wg); Q)
+(v)
+which for 2n ≥ 6 are isomorphisms in a stable range of degrees.
+If 2n = 2 then, in a stable range of degrees and assuming that the target is
+finite-dimensional in degrees ∗ < N for all large enough g, these maps are iso-
+morphisms onto the maximal algebraic subrepresentations in degrees ∗ ≤ N, and
+monomorphisms in degrees ∗ ≤ N + 1.
+Proof. By the main theorem of [KRW20a], as long as 2n ≥ 6 the G′
+g-representations
+Hi(BTor(Wg, D2n); Q) are algebraic. Using the inheritance properties for algebraic
+representations from [KRW20a, Theorem 2.2], the Serre spectral sequences for the
+homotopy fibre sequences
+BTor(Wg, D2n) −→BTor+(Wg, ∗) −→ BSO(2n)
+BTor(Wg, D2n) −→BTorθ(Wg, ∗) −→ BSO(2n)⟨n⟩
+show that the cohomology groups of BTor+(Wg, ∗) and BTorθ(Wg, ∗) are also al-
+gebraic G′
+g-representations, and the same for the homotopy fibre sequences
+Wg −→BTor+(Wg, ∗) −→ BTor+(Wg)
+Wg −→BTorθ(Wg, ∗) −→ BTorθ(Wg)
+show that the cohomology groups of BTor+(Wg) and BTorθ(Wg) are algebraic
+G′
+g-representations too.
+Using this algebraicity property, case (i) is precisely [KRW20b, Theorem 4.1],
+using that by [KRW20b, Proof of Theorem 5.1] Graph1(−)g is isomorphic to the
+functor P(−, V)≥0⊗det⊗n. The other cases follow in the same way, using [KRW20b,
+Proposition 2.16], from Theorem 3.4, with one elaboration which we describe below.
+The addendum in the case 2n = 2 is precisely as in [KRW20b, Theorem 4.1].
+The elaboration comes when verifying the first hypothesis of [KRW20b, Lemma
+4.3], which in case (v) for example requires us to know that H∗(BDiff+(Wg); H⊗S)
+is a free H∗(G′
+g; Q)-module in a stable range. But by transfer H∗(BDiff+(Wg); H⊗S)
+is a summand of H∗(BDiff+(Wg, ∗); H⊗S) (as H∗(G′
+g; Q)-modules), and similarly
+with θ-structures, so cases (ii) and (iii) imply cases (iv) and (v).
+In the other
+hand in case (iii) for example we have discussed in the proof of Theorem 3.4 the
+degeneration of the Serre spectral sequence in a stable range, giving
+gr(H∗(BDiff+(Wg, ∗); H⊗S)) ∼= H∗(BSO(2n); Q) ⊗ H∗(BDiff(Wg, D2n); H⊗S).
+
+22
+OSCAR RANDAL-WILLIAMS
+The Serre filtration is one of H∗(G′
+g; Q)-modules, so as the associated graded is a
+free H∗(G′
+g; Q)-module in a stable range (because H∗(BDiff(Wg, D2n); H⊗S) is the
+case treated in [KRW20b, Theorem 4.1]), it follows that H∗(BDiff+(Wg, ∗); H⊗S)
+is too. The same argument applies in case (ii).
+□
+This quite categorical description can be used to get a more down-to-earth pre-
+sentation for these cohomology rings: in case (v) this is the presentation we have
+recorded in Theorem A. This is deduced just as in [KRW20b, Section 5], though
+most of the work has been done as we have already expressed things in terms of
+graphs. As in [KRW20b, Section 5.4] this is not the smallest possible presentation:
+it can be simplified by manipulating graphs; we leave the details to the interested
+reader.
+5. The case 2n = 2
+Although Theorem 4.1 is only known to hold in a limited range of degrees in the
+case 2n = 2 (N = 2 is currently the best known constant for g ≥ 3, using the work
+of Johnson [Joh85]), Theorem 3.4 does hold in a range of cohomological degrees
+tending to infinity with g. In this case our discussion is closely related to the work
+of Kawazumi and Morita [Mor96, KM96, KM01], and in this section we we take
+the opportunity to revisit that work from our perspective. Throughout this section
+we assume that g ≥ 2, so that χ(Wg) = 2 − 2g ̸= 0.
+In terms of Kawazumi and Morita’s notation we have
+Mg := π0(Diff+(Wg))
+Mg,∗ := π0(Diff+(Wg, ∗))
+Mg,1 := π0(Diff+(Wg, D2)).
+Under our assumption g ≥ 2 the groups Diff+(Wg), Diff+(Wg, ∗), and Diff+(Wg, D2)
+all have contractible path-components, so the group cohomology of Mg is the co-
+homology of BDiff+(Wg), and so on. Theorem 3.4 gives a natural transformation
+¯κ : Graph(−)g −→ H∗(Mg; H⊗−)
+of functors sBr2g → Gr(Q-mod), which is an isomorphism in a stable range of
+degrees. Note that in this case H∗(BSO(2); Q) = Q[e] so V = W = Q[e] and there
+is no difference between the tangential structure θ and an orientation. In particular
+if we denote by Γi ∈ Graph(∅) the graph with a single vertex, no edges, and labelled
+by ei+1, then ¯κ(Γi) = κi ∈ H2i(Mg; Q) is the usual Miller–Morita–Mumford class4.
+Our goal in Sections 5.1–5.4 is to analyse Graph(−) in several ways, making
+contact with the work of Kawazumi and Morita mentioned above as well as work
+of Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].
+5.1. Reduction to corollas. The possible labels for the vertices of graphs in
+Graph(S) are powers of the Euler class e. Given any graph we may iteratedly apply
+the modified contraction formula to write it as a linear combination of graphs with
+fewer edges, and hence any graph is equivalent to a linear combination of graphs
+with no edges: these are disjoint unions of corollas. Of these, by definition of Graph:
+the 0-valent corolla labelled by e is equal to the scalar χ, the 1-valent corolla labelled
+by 1 ∈ V is trivial, and the 1-valent corolla labelled by e ∈ V is trivial. Define a
+labelled partition of a finite set S to be a partition {Sα}α∈I of S into (possibly
+empty) subsets and a label enα for each part, such that
+(i) If |Sα| = 0 then nα ≥ 2,
+(ii) If |Sα| = 1 then nα ≥ 1.
+4Our κi is denoted ei in the work of Kawazumi and Morita.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+23
+We give a part (Sα, nα) degree 2nα + |Sα| − 2, and a labelled partition the degree
+given by the sums of the degrees of its parts. Similarly to the proof of Theorem 3.4
+(iv) (particularly around equation (3.5)), let Pbis(S, V)≥0 denote the free Q[χ±1]-
+module with basis the set of labelled partitions of S. Assigning to a labelled part
+(Sα, enα) the corolla with legs Sα and label enα defines a map
+(5.1)
+Pbis(S, V)≥0 ⊗ det QS −→ Graph(S),
+natural in S with respect to bijections.
+Lemma 5.1. The map (5.1) is an isomorphism.
+Proof. It is surjective, as explained above, by repeatedly applying the modified
+contraction formula to express a graph in terms of graphs without edges.
+If it were not injective then it would have some nontrivial Q[χ±1]-linear com-
+bination of labelled partitions in its kernel, of a given degree d, and this would
+remain a nontrivial Q-linear combination of labelled partitions when specialised to
+χ = 2 − 2g for all g ≫ 0 (as a Laurent polynomial in χ has finitely-many roots).
+But in the proof of Theorem 3.4 (iv), in the discussion after equation (3.5), it is
+explained that when specialised to χ = 2 − 2g this map is an isomorphism in a
+range of degrees tending to infinity with g; for large enough g the degree d will be
+in this stable range, a contradiction.
+□
+In particular, for the graphs Γi described above there is an isomorphism
+(5.2)
+Q[χ±1][Γ1, Γ2, . . .] ∼= Graph(∅).
+5.2. Reduction to trivalent graphs without labels. In this section we will
+prove the following.
+Theorem 5.2. Using the modified contraction formula any marked oriented graph
+is equivalent to a Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1]-linear combination of
+trivalent graphs with all vertices labelled by 1 ∈ V0.
+Let Graphtri(S) ≤ Graph(S) denote the sub-Q[χ±1]-module spanned by those
+marked oriented graphs which are trivalent and all of whose labels are 1 ∈ V.
+Corollary 5.3. The monomorphism i : Graphtri(−) → Graph(−) becomes an iso-
+morphism upon inverting χ − 2, χ − 3, and χ − 4. In particular Graphtri(−)g =
+Graph(−)g.
+Remark 5.4 (2-valent vertices labelled by 1). Using the relation
+λ2,3(κ¯ε1,2κ¯ε3,...,nc) = κ¯ε1,3,...,nc
+we can always remove 2-valent vertices labelled by 1. It is sometimes convenient
+when writing formulas for 3-valent graphs to also allow 2-valent vertices labelled
+by 1: we allow ourselves to do so, noting that the above can always be used to
+eliminate the 2-valent vertices.
+Proof of Theorem 5.2. As a matter of notation we will formally manipulate modi-
+fied twisted Miller–Morita–Mumford classes, but this is equivalent to manipulating
+marked oriented graphs. Rearranging the first contraction formula gives
+(5.3)
+κ¯εaeb =
+χ
+χ−2
+�
+λ1,2κ¯ε2+aeb−1 −
+1
+χ2 κe2κ¯εaeb−1
+�
+.
+Rearranging the second contraction formula gives
+κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −
+1
+χ2 (κe2 · κ¯εa · κ¯εb) + 1
+χ(κ¯εae · κ¯εb + κ¯εa · κ¯εbe)
+
+24
+OSCAR RANDAL-WILLIAMS
+and using (5.3) to eliminate the Euler classes from the last two terms gives
+κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −
+1
+χ2 (κe2 · κ¯εa · κ¯εb)
++
+1
+χ−2((λ1,2(κ¯ε2+a) −
+1
+χ2 κe2κ¯εa) · κ¯εb + κ¯εa · (λ1,2(κ¯ε2+b) −
+1
+χ2 κe2κ¯εb))
+= λa+1,a+2(κ¯εa+1 · κ¯ε1+b) +
+1
+χ−2 ((λ1,2(κ¯ε2+a) · κ¯εb + κ¯εa · λ1,2(κ¯ε2+b))
+−
+1
+χ(χ−2)κe2 · κ¯εa · κ¯εb.
+It suffices to show that each corolla κ¯εaeb may be represented by a linear combi-
+nation of trivalent graphs. By Example 2.8 the class κe2 may be represented by a
+trivalent graph (after inverting χ − 3) so by iteratedly applying (5.3) it suffices to
+show that each κ¯εn can too. By Remark 5.4 we may as well show that classes can
+be represented by 2- and 3-valent graphs. To get started we have κ¯ε = 0 as it has
+negative degree.
+Consider the class λ2,5λ3,4(κ¯ε1,2,3 ·κ¯ε4,5,6). Using the form of the relations above,
+which avoid creating Euler classes, this is
+λ2,5(κ¯ε1,2,5,6 −
+1
+χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6 + κ¯ε1,2λu,v(κ¯εu,v,5,6)) +
+1
+χ(χ−2)(κe2κ¯ε1,2κ¯ε5,6))
+= λ2,5(κ¯ε1,2,5,6) −
+2
+χ−2λu,v(κ¯εu,v,1,6) +
+1
+χ(χ−2)κe2κ¯ε1,6
+= χ−4
+χ−2λ2,5(κ¯ε1,2,5,6) +
+1
+χ(χ−2)κe2κ¯ε1,6
+Renumbering legs and rearranging, this shows that λ1,2(κ¯ε4) may be represented
+by 2- and 3-valent graphs.
+Applied with (a, b) = (2, 2) the second relation gives
+κ¯ε4 = λ3,4(κ¯ε3 · κ¯ε3) +
+1
+χ−2 ((λ1,2(κ¯ε4) · κ¯ε2 + κ¯ε2 · λ1,2(κ¯ε4)) −
+1
+χ(χ−2)κe2 · κ¯ε2 · κ¯ε2,
+which with the above shows that κ¯ε4 may be represented by 2- and 3-valent graphs.
+Similarly to the above, consider λ2,5λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6,7), which is
+λ2,5(κ¯ε1,2,5,6,7 +
+1
+χ(χ−2)κe2κ¯ε1,2κ¯ε5,6,7 −
+1
+χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6,7 + κ¯ε1,2λu,v(κ¯εu,v,5,6,7)))
+= λ2,5(κ¯ε1,2,5,6,7) +
+1
+χ(χ−2)κe2κ¯ε1,6,7 −
+1
+χ−2(λ2,5λu,v(κ¯εu,v,1,2κ¯ε5,6,7) + λu,v(κ¯εu,v,1,6,7))
+= χ−3
+χ−2λ2,5(κ¯ε1,2,5,6,7) +
+1
+χ(χ−2)κe2κ¯ε1,6,7 −
+1
+χ−2λ2,5λu,v(κ¯εu,v,1,2)κ¯ε5,6,7.
+Renumbering legs and rearranging, this shows that λ1,2(κ¯ε5) may be represented by
+2-, 3-, and 4-valent graphs; with the above it follows that it can also be represented
+by 2- and 3-valent graphs.
+Applied with (a, b) = (2, 3) the second relation gives
+κ¯ε5 = λ3,4(κ¯ε3 · κ¯ε4) +
+1
+χ−2 ((λ1,2(κ¯ε4) · κ¯ε3 + κ¯ε2 · λ1,2(κ¯ε5)) −
+1
+χ(χ−2)κe2 · κ¯ε2 · κ¯ε3,
+so it follows that κ¯ε5 may be represented by 2- and 3-valent graphs.
+If n ≥ 6 then we can write n = a + b with a, b ≥ 3, so a + 2, b + 2 < n and so the
+second relation expresses κ¯εn in terms of κ¯εm’s with m < n. Thus all κ¯εn’s may be
+represented by 2- and 3-valent graphs as required.
+□
+It is worth observing that we have the relation
+(5.4)
+λ1,2(κ¯ε3) = χ−2
+χ κ¯εe +
+1
+χ2 κe2κ¯ε = 0,
+using that κ¯εe = 0 (by definition) and that κ¯ε = 0 (as it has negative degree). This
+means that any graph having a trivalent vertex with a loop is trivial in Graph(−).
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+25
+5.3. A remark on orderings. A curious normalisation is possible when consider-
+ing trivalent graphs, allowing one to neglect the orderings of vertices, of half-edges,
+and the orientations of edges. In [Mor96, KM96, KM01] this is implemented ab
+initio and (marked) oriented graphs play no role. Let us explain this normalisation,
+extended to trivalent graphs with legs.
+A trivalent graph ˜Γ with legs S consists of a set V of vertices, a set H of half-
+edges, a 3-to-1 map a : H → V recording to which vertex each half-edge is incident,
+and an unordered matching µ on H ⊔ S recording which half-edges span an edge,
+and which half-edges are connected to which legs.
+Given a trivalent graph ˜Γ = (V, H, a : H → V, µ) with legs S, we may choose an
+ordering of V and choose an ordering of H such that a : H → V is weakly monotone
+(equivalently, choose an ordering of the half-edges incident at each vertex). We also
+choose an ordering of S.
+There is an induced ordering of H ⊔ S by putting ⃗S
+after ⃗H, and we form an ordered matching m of H ⊔ S by taking those pairs
+(a, b) with a < b and {a, b} ∈ µ. Using this we form an oriented trivalent graph
+Γchoice = (⃗V , ⃗H, a : H → V, m), depending on these choices. The normalisation is
+as follows. Let x1 < x2 < x3 < x4 < . . . < x2k ∈ H ⊔ S be the total order on
+H ⊔ S, and let a1 < b1, . . . , ak < bk be the ordered pairs which span an edge, with
+a1 < a2 < . . . < ak ∈ H ⊔ S. Then there is a bijection given by
+ρ :=
+� a1 b1 a2 b2 a3 b3 ···
+ak
+bk
+x1 x2 x3 x4 x5 x6 ··· x2k−1 x2k
+�
+and we define Γ := sign(ρ) · Γchoice.
+Claim. As long as ˜Γ has no vertices with loops, the element Γ does not depend
+on the choice of ordering of V or H, and depends on the ordering of S precisely as
+the sign representation.
+In particular if we set5
+Graphundec(S) := Q[χ±1][˜Γ trivalent graph with legs S]/(graphs with loops)
+then the Claim together with the relation (5.4) provides an epimorphism
+Φ : Graphundec(S) ⊗ det QS −→ Graphtri(S)
+of Q[χ±1]-modules, natural with respect to bijections in S. This can be extended to
+a natural transformation of functors on sBrχ by letting an ordered matching (a, b)
+of elements of S act by adding an edge to the trivalent graph connecting a and b,
+and contracting the determinant by a ∧ b. Doing so might create a circle with no
+vertices, which should be replaced by the scalar χ − 2.
+Proof of Claim. If (h1, h2, h3) are the half-edges incident at a vertex v and we
+change their ordering to (hσ(1), hσ(2), hσ(3)) giving Γ′
+choice, then (under the assump-
+tion that Γ does not have loops) the relative ordering of half-edges forming an
+edge has not changed, so m′ = m. Thus Γ′
+choice = sign(σ) · Γchoice. On the other
+hand ρ′ is obtained from ρ by postcomposing with σ, and precomposing with a
+permutation which permutes some (ai < bi)’s, which is an even permutation. Thus
+sign(ρ′) = sign(σ) · sign(ρ), so Γ′ = Γ.
+Suppose a vertex v1 has half edges (h1
+1, h1
+2, h1
+3) and v2 has half edges (h2
+1, h2
+2, h2
+3),
+and v1 < v2 ∈ ⃗V are adjacent in the ordering on V , and consider transposing the
+ordering of these vertices. For edges between a u < v1 and a vi or between a vi and
+a u > v2 the relative ordering of their half-edges does not change. Edges between
+v1 and v2 have the relative ordering of their half-edges reversed. Thus if there are
+N such edges we have Γ′
+choice = (−1)1+N ·Γchoice. But the permutation ρ is changed
+5In [Mor96, KM96, KM01] they restrict to “trivalent graphs without loops”, however we find
+it more natural to allow loops but set graphs with a loop to zero.
+
+26
+OSCAR RANDAL-WILLIAMS
+by permuting (h2
+1, h2
+2, h2
+3) past (h1
+1, h1
+2, h1
+3), which has sign −1, and N transpositions
+(aibi), which has sign (−1)N. Thus again Γ′ = Γ.
+Finally, changing the order of S by a permutation τ changes ρ by postcomposition
+with τ, so acts as sign(τ).
+□
+Example 5.5. For the ordering of vertices and half-edges corresponding to the
+theta-graph in Example 2.8 the associated permutation is ρ = (1)(235)(46) which
+is odd, so the undecorated theta-graph yields χ−3
+χ κe2. This is precisely minus the
+evaluation of βΓ2 on [KM01, p. 39] (unfortunately the theta-graph is denoted Γ2 in
+that paper). This minus comes from the use of a different sign convention, see the
+discussion at [KRW20b, top of p. 33].
+5.4. Relations among trivalent graphs. The modified contraction formula de-
+scribes relations among graphs involving contracting an edge, but this necessarily
+involves graphs with vertices of different valencies. In Theorem 5.2 we have ex-
+plained that, in the case of surfaces, all graphs may be expressed purely in terms of
+trivalent graphs: one may ask what relations among trivalent graphs Γ are imposed
+by the contraction formula.
+For the unmodified contraction formula discussed in [KRW20b], the answer is
+that it imposes the “I = H” relation among trivalent graphs: this is because
+both the I- and H-graphs admit contractions to the X-graph. Furthermore, as all
+connected trivalent graphs with the same number of legs and of the same genus are
+equivalent under the “I = H” relation, and the contraction formula never changes
+the genus or number of legs, there are no further relations.
+In the setting of the modified contraction formula discussed here it is more
+complicated. It is best given in the setting of undecorated trivalent graphs.
+Theorem 5.6. After inverting χ−2, χ−3, and χ−4, undecorated trivalent graphs
+which differ locally by
+(IHmod)
+=
++
+1
+(χ−4)(3−χ)(
+−
+)
++
+1
+χ−4(
++
+−
+−
+)
+give the same elements in Graphtri[(χ − 2)−1, (χ − 3)−1, (χ − 4)−1].
+Proof. We establish this relation in Graphtri({a, b, c, d})⊗detQ{a,b,c,d}, and it then
+follows in general using functoriality on the signed Brauer category. We order the
+legs as a < b < c < d.
+(i)
+a
+b
+c
+d
+1
+3
+5
+6 2
+4
+(ii)
+a
+b
+c
+d
+1 2
+6
+3 4
+5
+Figure 3. Some marked graphs.
+Consider first the H-shaped graph shown in Figure 3 (i), with the depicted names
+of half edges, ordered as 3 < 1 < 5 < 6 < 2 < 4. Its corresponding permutation is
+� 3 c 1 a 5 6 2 b 4 d
+3 1 5 6 2 4 a b c d
+�
+which is even. Thus this ordering data represents the underlying
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+27
+undecorated H-shaped trivalent graph. Ignoring for now the matchings to the legs
+(which are given by matching 1 with a, 2 with b, and so on), it corresponds to
+λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4). Using the form of the relations which avoid creating Euler
+classes from the proof of Theorem 5.2 we have
+λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4) = κ¯ε3,1,2,4 +
+1
+χ(χ−2)κe2κ¯ε3,1κ¯ε2,4
+−
+1
+χ−2(λu,v(κ¯εu,v,3,1)κ¯ε2,4 + κ¯ε3,1λu,v(κ¯εu,v,2,4)).
+Consider now the I-shaped graph shown in Figure 3 (ii), with the depicted names
+of the half-edges, ordered as 4 < 3 < 5 < 6 < 1 < 2. Its corresponding permutation
+is
+� 4 d 3 c 5 6 1 a 2 b
+4 3 5 6 1 2 a b c d
+�
+which is odd. Thus this ordering data represents minus the
+underlying undecorated I-shaped trivalent graph. Ignoring again the matchings to
+the legs, it corresponds to
+λ5,6(κ¯ε4,3,5 · κ¯ε6,1,2) = κ¯ε4,3,1,2 +
+1
+χ(χ−2)κe2κ¯ε4,3κ¯ε1,2
+−
+1
+χ−2(λu,v(κ¯εu,v,4,3)κ¯ε1,2 + κ¯ε4,3λu,v(κ¯εu,v,1,2)).
+The sum of these two expressions therefore represents the image under Φ of
+the difference H − I of the underlying undecorated trivalent graphs. Furthermore,
+κ¯ε4,3,1,2 = −κ¯ε3,1,2,4 so these terms cancel.
+From the proof of Theorem 5.2 we have the identity
+λu,v(κ¯εu,v,s,t) = χ−2
+χ−4λi,jλk,l(κ¯εs,i,k · κ¯εl,j,t) −
+1
+χ(χ−4)κe2κ¯εs,t,
+expressing terms of the form λu,v(κ¯εu,v,s,t) in terms of (2- and) 3-valent vertices.
+Applying it to the sum of the two expressions above, and collecting terms, therefore
+gives
+Φ(H − I) =
+1
+χ(χ−4)κe2�
+κ¯ε3,1κ¯ε2,4 + κ¯ε4,3κ¯ε2,1�
+−
+1
+χ−4
+�
+λi,jλk,l(κ¯ε3,i,k · κ¯εl,j,1)κ¯ε2,4 + κ¯ε3,1λi,jλk,l(κ¯ε2,i,k · κ¯εl,j,4)
+λi,jλk,l(κ¯ε4,i,k · κ¯εl,j,3)κ¯ε1,2 + κ¯ε4,3λi,jλk,l(κ¯ε1,i,k · κ¯εl,j,2)
+�
+.
+Using that κe2 = Φ(
+χ
+χ−3Θ) and carefully putting the graphs corresponding to the
+other terms into the normal form of Section 5.3 gives the identity in the statement
+of the theorem.
+□
+Our relation IHmod is graphically identical to the relation called IHbis
+0
+by
+Akazawa [Aka05, p. 100] and in the corrigendum [GN07] to the paper of Garo-
+ufalidis and Nakamura [GN98]. In those papers it is emphasised that IHbis
+0
+means
+this identity is imposed only when the 4 half-edges belong to distinct edges, but
+in fact this is redundant: if the 4 half-edges do not belong to distinct edges, then
+the identity already holds in Graphundec. So in fact imposing our relation IHmod
+is identical to imposing their relation IHbis
+0 .
+Theorem 5.7. Upon inverting χ − 2, χ − 3, and χ − 4, the maps
+Graphundec(S)
+(IHmod)
+⊗ det QS
+Φ
+−→ Graphtri(S)
+inc
+−→ Graph(S)
+are isomorphisms.
+Proof. Let R := Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1] and implicitly base change
+to this ring. We have already shown in Corollary 5.3 that the second map is an
+isomorphism, and Φ is certainly an epimorphism, so it remains to show that the
+composition is a monomorphism.
+
+28
+OSCAR RANDAL-WILLIAMS
+For an undecorated trivalent graph Γ, define a double edge to be an unordered
+pair of vertices which share precisely two edges, and a triple edge to be an unordered
+pair of vertices which share precisely three edges, i.e. form a theta-graph. Define
+µ(Γ) := 2 · #double edges of Γ + 3 · #triple edges of Γ,
+filter Graphundec by letting F kGraphundec be spanned by those Γ with µ(Γ) ≥ k,
+and give Graphundec/(IHmod) the induced filtration.
+If Γ = ΓH is a graph with µ(Γ) = k and a distinguished “H” subgraph, and ΓI
+is obtained by replacing this “H”-subgraph by “I”, then by applying the relation
+IHmod to this subgraph we find that
+(i) if the edge involved is not part of a double or triple edge then the relation
+gives ΓH − ΓI ∈ F k+1Graphundec/(IHmod),
+(ii) if the edge involved is part of a double or triple edge then the relation is trivial
+(i.e. already holds in Graphundec).
+Thus the associated graded of the induced filtration on Graphundec/(IHmod) can
+be described as Graphundec/(IH0), where as in [GN98] the relation IH0 means
+imposing the “I = H” relation when the four half-edges belong to different edges.
+Now IH0 is an equivalence relation on the set of isomorphism classes of trivalent
+graphs without loops, and similarly to [GN98, Proof of Proposition 2.3 (c)] it is
+easy to see that all connected trivalent graph without loops of the same rank and
+with the same legs are equivalent to each other: in other words the equivalences
+classes of such are given by partitions of S (the parts are the legs of each connected
+component) labelled by a power of e (recording the rank of the graph). It follows
+that the rank of Graphundec/(IHmod) in each degree, as an R-module, is at most that
+of Graph(∅) as determined in Lemma 5.1, and so the composition in the statement
+of the theorem, which is an epimorphism, must be an isomorphism.
+□
+5.5. On the work of Garoufalidis and Nakamura. The discussion of the last
+few sections can be used to complete the work of Garoufalidis and Nakamura [GN98,
+GN07], concerning the calculation of the invariants [Λ∗V13/(V22)]Sp in a stable
+range. Here we write Vλ for the irreducible Sp-representation corresponding to the
+partition λ, which was written as [λ]sp in those papers, and V22 denotes the unique
+copy of this irreducible in Λ2V13.
+Combining Theorem 1.1 and Proposition 2.3
+(c) of [GN98] was supposed to calculate [Λ∗V13/(V22)]Sp in a stable range, but for
+the corrected version of Theorem 1.1 in [GN07], which expresses these invariants as
+Graphundec(∅)g/(IHbis
+0 ), the authors say “it turns out that a simple stable structure
+of [these invariants] as in Proposition 2.3 (c) will not be easy to detect”. However
+Theorem 5.7 and equation (5.2) gives that
+[Λ∗V13/(V22)]Sp ∼= Graphundec(∅)g/(IHbis
+0 ) ∼= Graph(∅)g ∼= Q[Γ1, Γ2, . . .]
+in a stable range. Thus in fact Proposition 2.3 (c) of [GN98] is correct as stated.
+Remark 5.8. This can also be obtained from the work of Felder, Naef, and Willwacher
+[FNW21]. Specifically, the graded-commutative algebra A(g) defined just before
+Theorem 6 of that paper is Λ∗V13/(V22), and Theorem 6 together with Proposition
+36 (3) also gives the above.
+References
+[Aka05]
+H. Akazawa, Symplectic invariants arising from a Grassmann quotient and trivalent
+graphs, Math. J. Okayama Univ. 47 (2005), 99–117.
+[Bol12]
+S. K. Boldsen, Improved homological stability for the mapping class group with inte-
+gral or twisted coefficients, Math. Z. 270 (2012), no. 1-2, 297–329.
+[FNW21]
+M. Felder, F. Naef, and T. Willwacher, Stable cohomology of graph complexes,
+https://arxiv.org/abs/2106.12826, 2021.
+
+ON THE COHOMOLOGY OF TORELLI GROUPS. II
+29
+[GN98]
+S. Garoufalidis and H. Nakamura, Some IHX-type relations on trivalent graphs and
+symplectic representation theory, Math. Res. Lett. 5 (1998), no. 3, 391–402.
+[GN07]
+, Corrigendum: “Some IHX-type relations on trivalent graphs and symplectic
+representation theory” [Math. Res. Lett. 5 (1998), no. 3, 391–402], Math. Res. Lett.
+14 (2007), no. 4, 689–690.
+[GRW14]
+S. Galatius and O. Randal-Williams, Stable moduli spaces of high-dimensional man-
+ifolds, Acta Math. 212 (2014), no. 2, 257–377.
+[GRW17]
+, Homological stability for moduli spaces of high dimensional manifolds. II,
+Ann. of Math. (2) 186 (2017), no. 1, 127–204.
+[GRW18]
+, Homological stability for moduli spaces of high dimensional manifolds. I, J.
+Amer. Math. Soc. 31 (2018), no. 1, 215–264.
+[GRW19]
+, Moduli spaces of manifolds: a user’s guide, Handbook of homotopy theory,
+Chapman & Hall/CRC, CRC Press, Boca Raton, FL, 2019, pp. 445–487.
+[GTMW09] S. Galatius, U. Tillmann, I. Madsen, and M. Weiss, The homotopy type of the cobor-
+dism category, Acta Math. 202 (2009), no. 2, 195–239.
+[Hai20]
+R. Hain, Johnson homomorphisms, EMS Surv. Math. Sci. 7 (2020), no. 1, 33–116.
+[Joh85]
+D. Johnson, The structure of the Torelli group. III. The abelianization of T , Topology
+24 (1985), no. 2, 127–144.
+[KM96]
+N. Kawazumi and S. Morita, The primary approximation to the cohomology of the
+moduli space of curves and cocycles for the stable characteristic classes, Math. Res.
+Lett. 3 (1996), no. 5, 629–641.
+[KM01]
+,
+The
+primary
+approximation
+to
+the
+cohomology
+of
+the
+mod-
+uli
+space
+of
+curves
+and
+cocycles
+for
+the
+Mumford-Morita-Miller
+classes,
+www.ms.u-tokyo.ac.jp/preprint/pdf/2001-13.pdf, 2001.
+[Kre79]
+M.
+Kreck,
+Isotopy
+classes
+of
+diffeomorphisms
+of
+(k − 1)-connected
+almost-
+parallelizable 2k-manifolds, Algebraic topology, Aarhus 1978 (Proc. Sympos., Univ.
+Aarhus, Aarhus, 1978), Lecture Notes in Math., vol. 763, Springer, Berlin, 1979,
+pp. 643–663.
+[KRW20a]
+A. Kupers and O. Randal-Williams, The cohomology of Torelli groups is algebraic,
+Forum of Mathematics, Sigma 8 (2020), e64.
+[KRW20b]
+, On the cohomology of Torelli groups, Forum of Mathematics, Pi 8 (2020),
+e7.
+[KRW21]
+, On the Torelli Lie algebra, https://arxiv.org/abs/2106.16010, 2021.
+[Mor96]
+S. Morita, A linear representation of the mapping class group of orientable sur-
+faces and characteristic classes of surface bundles, Topology and Teichm¨uller spaces
+(Katinkulta, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 159–186.
+[Qui71]
+D. Quillen, The spectrum of an equivariant cohomology ring. I, Ann. of Math. (2) 94
+(1971), 549–572.
+[RW16]
+O. Randal-Williams, Resolutions of moduli spaces and homological stability, J. Eur.
+Math. Soc. (JEMS) 18 (2016), no. 1, 1–81.
+Email address: o.randal-williams@dpmms.cam.ac.uk
+Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
+
diff --git a/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..63e18c946ebe1bb327d4d2b9ec426408b629e4a3
--- /dev/null
+++ b/OtAzT4oBgHgl3EQfIftM/content/tmp_files/load_file.txt
@@ -0,0 +1,1166 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf,len=1165
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='01062v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='AT]  3 Jan 2023  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  OSCAR RANDAL-WILLIAMS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We describe the ring structure of the rational cohomology of the  Torelli groups of the manifolds #gSn × Sn in a stable range, for 2n ≥ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Some of our results are also valid for 2n = 2, where they are closely related to  unpublished results of Kawazumi and Morita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Introduction  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Characteristic classes  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Twisted cohomology of diffeomorphism groups  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Cohomology of Torelli groups  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The case 2n = 2  22  References  28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Introduction  This paper can be considered as a (somewhat extensive) addendum to our earlier  work with Kupers [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We shall be concerned with the manifold Wg :=  #gSn × Sn generalising to higher dimensions the orientable surface of genus g, its  topological group Diff+(Wg) of orientation-preserving diffeomorphisms, and various  subgroups of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The first kind of subgroups are Diff(Wg, D2n) ≤ Diff+(Wg, ∗) ≤  Diff+(Wg), the diffeomorphisms which fix a disc and a point respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The  second kind are their Torelli subgroups  Tor(Wg, D2n),  Tor+(Wg, ∗),  Tor+(Wg),  consisting of those diffeomorphisms which in addition act trivially on Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The intersection form on this middle cohomology group is nondegenerate and (−1)n-  symmetric, giving a homomorphism  αg : Diff+(Wg) −→ Gg :=  �  Sp2g(Z)  if n is odd,  Og,g(Z)  if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  This map is not always surjective, but its image is a certain finite-index subgroup  G′  g ≤ Gg, even when restricted to Diff(Wg, D2n), so there is an outer G′  g-action on  each of the Torelli subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This makes the rational cohomology of each of the  Torelli groups into G′  g-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In [KRW20b], for 2n ≥ 6 we determined H∗(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) as a ring and  as a G′  g-representation in a range of degrees tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Using  the Serre spectral sequence associated to various simple fibrations relating the dif-  ferent Torelli groups we were able to also determine H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) and  H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) as G′  g-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This kind of argument was not able to  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 55R40, 11F75, 57S05, 18D10, 20G05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Cohomology of diffeomorphism groups, Torelli groups, cohomology of  arithmetic groups, Miller-Morita-Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  1  2  OSCAR RANDAL-WILLIAMS  determine the ring structure, however, as multiplicative information gets lost when  passing to the associated graded of the Serre filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Here we shall determine  H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) and H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) as Q-algebras too: this is achieved in  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The statement given there is more powerful, but just as in [KRW20b,  Section 5] one can extract from it the following presentation for H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q),  which is easier to parse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (A presentation for H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) can be extracted  in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=')  Let us write H(g) := Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q), on which G′  g operates in the evident way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let  λ : H(g) ⊗ H(g) → Q denote the intersection form, and {ai}2g  i=1 be a basis of H(g)  with dual basis {a#  i }2g  i=1 in the sense that λ(a#  i , aj) = δij, so that the form dual  to the pairing λ is ω = �2g  i=1 ai ⊗ a#  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we will construct certain  “modified twisted Miller–Morita–Mumford classes”, which when restricted to the  Torelli group yield G′  g-equivariant maps  ¯κc : H(g)⊗r −→ Hn(r−2)+|c|(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  for each c ∈ Q[e, p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1] = H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) and each s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' When r = 0  we write ¯κc = ¯κc(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' these agree with the usual Miller–Morita–Mumford classes κc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If 2n ≥ 6 then, in a range of degrees tending to infinity with g,  H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is generated as a Q-algebra by the classes ¯κc(v1 ⊗ · · · ⊗ vr) for  c a monomial in e, p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1, and r ≥ 0, such that n(r −2)+|c| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' A complete  set of relations in this range is given by  (i) ¯κc(vσ(1) ⊗ · · · ⊗ vσ(r)) = sign(σ)n · ¯κc(v1 ⊗ · · · ⊗ vr),  (ii) ¯κe(v1) = 0,  (iii)  �  i  ¯κx(v ⊗ ai) · ¯κy(a#  i ⊗ w) = ¯κx·y(v ⊗ w) +  1  χ2 ¯κe2 · ¯κx(v) · ¯κy(w)  − 1  χ  �  ¯κe·x(v) · ¯κy(w) + ¯κx(v) · ¯κe·y(w)  �  ,  (iv)  �  i  ¯κx(v ⊗ ai ⊗ a#  i ) = χ−2  χ ¯κe·x(v) +  1  χ2 ¯κe2 · ¯κx(v),  (v) ¯κLi = 0,  for v ∈ H(g)⊗r and w ∈ H(g)⊗s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For 2n = 4 or 2n = 2 there is still a map from the Q-algebra given by this  presentation to H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If 2n = 2 then (in a stable range) this map  is an isomorphism onto the maximal algebraic subrepresentation in degrees ≤ N,  assuming that H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is finite-dimensional in degrees < N for all large  enough g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This is known to hold for N = 2 by work of Johnson [Joh85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The overall strategy is parallel to [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There we defined cer-  tain twisted Miller–Morita–Mumford classes and used them to describe the twisted  cohomology groups H∗(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s) in a stable range of degrees, where  H is the local coefficient system corresponding to Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) with the action by dif-  feomorphisms of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This calculation was valid for 2n = 2 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using that for  2n ≥ 6 the G′  g-representations H∗(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) extend to representations  of the ambient algebraic group (namely Sp2g or Og,g) by [KRW20a]1, the argument  was completed by establishing the degeneration of the Serre spectral sequence  Ep,q  2  = Hp(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Hq(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)⊗H⊗s) =⇒ Hp+q(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s)  using work of Borel, and then using a categorical form of Schur–Weyl duality to ex-  tract the structure of H∗(BTor+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) from the H∗(BDiff+(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s)  for all s’s and various structure maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  1In fact we did something more complicated in [KRW20b] because this algebraicity result was  not known at the time, but please allow some narrative leeway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  3  The twisted Miller–Morita–Mumford classes may be defined on BDiff+(Wg, ∗)  too, but not on BDiff+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Our first task will be to define so-called “modi-  fied twisted Miller–Morita–Mumford classes” in H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s) and analyse  their behaviour: it turns out that their behaviour is significantly more complicated  than the unmodified version, though still understandable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We will then use them  to describe the twisted cohomology groups H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s) in a stable range  of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This description will be in terms of a certain vector space of graphs  with vertices labelled by monomials in Euler and Pontrjagin classes, which play the  role here of the vector spaces of labelled partitions from [KRW20b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The passage  from this calculation to H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The case of dimension 2n = 2 is somewhat special, in precisely the same way as  it was in [KRW20b]: the calculation of H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗s) is valid in this case,  but as the cohomology of BTor+(Wg) is not even known to be finite-dimensional  in a stable range, we cannot make a conclusion about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (Instead one can make  a conclusion about the continuous cohomology of the Torelli group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' the Lie  algebra cohomology of its Mal’cev Lie algebra: see [KRW21], [FNW21], [Hai20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=')  In addition, in this case our modified twisted Miller–Morita–Mumford classes are  essentially the same as those that have been defined by Kawazumi and Morita  [Mor96, KM96, KM01], and the graphical calculus that we employ is similar to  theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In Section 5 we fully explain this connection, and also relate it to work of  Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  To avoid a great deal of repetition we have refrained from spelling out a lot of the  background that was given in [KRW20b], and from giving in detail arguments that  are very similar to those given there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As such this paper should not be considered  as attempting to be self-contained: given that its interest will be to readers of  [KRW20b] this should not present a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' I am grateful to Alexander Kupers for feedback on an  earlier draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' I was supported by the ERC under the European Union’s Horizon  2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 756444) and by a  Philip Leverhulme Prize from the Leverhulme Trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Characteristic classes  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Recollection on twisted Miller–Morita–Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If π′ : E′ →  X′ is an oriented smooth W 2n  g -bundle equipped with a section s : X′ → E′, and  H denotes the local coefficient system x �→ Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) on X′, then it is  explained in [KRW20b, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2] that there is a unique class ε = εs ∈ Hn(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H)  characterised by  (i) for each x ∈ X′ the element ε|(π′)−1(x) ∈ Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)⊗Hn((π′)−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  is coevaluation, and  (ii) s∗ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The Serre spectral sequence yields an exact sequence  0 → Hn(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H)  (π′)∗  → Hn(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) → H0(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H∨⊗H)  dn+1  → Hn+1(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H)  (π′)∗  → Hn+1(E′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H)  and the section s shows that the right-hand map (π′)∗ is injective, so that the map  dn+1 is zero, and splits the left-hand map (π′)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The class coev ∈ H0(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H∨ ⊗ H)  then gives rise to a unique ε satisfying the given properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  We then defined the twisted Miller–Morita–Mumford class  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1)  κεac = κεac(π′, s) :=  �  π′ εa · c(Tπ′E′) ∈ H(a−2)n+|c|(X′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  4  OSCAR RANDAL-WILLIAMS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Modified twisted Miller–Morita–Mumford classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If π : E → X is  an oriented smooth W 2n  g -bundle but is not equipped with a section then, as long  as χ := χ(Wg) = 2 + (−1)n2g ̸= 0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (n, g) ̸= (odd, 1), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1), the  cohomological role of the section can instead be played by the transfer map  1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · −) : H∗(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) −→ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H),  where e := e(TπE) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) denotes the Euler class of the vertical tangent  bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The projection formula  1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · π∗(x)) = 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e) · x = χ  χx = x  shows that this map splits π∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus in this situation there is a unique class ¯ε ∈  Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) characterised by  (i) for each x ∈ X the element ¯ε|π−1(x) ∈ Hn(π−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ Hn(π−1(x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is  coevaluation, and  (ii)  1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · ¯ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If (n, g) = (odd, 1) then there is no class ¯ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) satisfying (i)  and natural under pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' To see this it suffices to give one example of a smooth  oriented W1-bundle for which ¯ε does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Consider the Borel construction for  the evident action of S1 × S1 on W1 = Sn × Sn given by considering Sn as the unit  sphere in C(n+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This gives a smoth oriented W1-bundle over B(S1 × S1) with  total space E ≃ CP(n−1)/2 × CP(n−1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) = 0 as n is odd but E has  a cell structure with only even-dimensional cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  By analogy with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1) we may then define the modified twisted Miller–Morita–  Mumford class  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2)  κ¯εac = κ¯εac(π) :=  �  π  ¯εa · c(TπE) ∈ H(a−2)n+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If π : E → X does have a section s : X → E then the class ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) is also  defined, and we may compare it with ¯ε as follows:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If π : E → X has a section then ¯ε = ε − 1  χπ∗κεe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The classes ε, ¯ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) are both defined, and agree when restricted to  the fibres of the map π, so by considering the Serre spectral sequence for π we must  have ¯ε − ε = π∗(x) for some class x ∈ Hn(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Applying 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · −) we see that  x = 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · (¯ε − ε)) = 0 − 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · ε) = − 1  χκεe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (Here we have used, as we often will,  the fact that e has even degree to commute it past ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=')  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 (Splitting principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The pullback  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3)  E1 ×X E2  E2  E1  X,  pr1  pr2  π2  π1  where πi : Ei → X are copies of the map π, is equipped with a section given by the  diagonal map ∆ : E1 → E1 ×X E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As the maps π∗  1 and pr∗  2 are injective (they are  split by their corresponding transfer maps), for the purpose of establishing identities  between the characteristic classes we have discussed it suffices to do so for bundles  which do have a section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  There is another description of ¯ε which is sometimes useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let pr1 : E ×X E →  E be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3), which is an oriented Wg-bundle with section given by the diagonal  map ∆, and so has the class κεe(pr1, ∆) defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  5  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We have ¯ε = − 1  χκεe(pr1, ∆) ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 we may suppose without loss of generality that π : E → X  has a section s : X → E, defining a class ε = εs ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Consider the  pullback square (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' let ei = (pri)∗(e) ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) be the Euler class  of the vertical tangent bundle on the ith factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Considering pr1 as a Wg-bundle  with section given by the diagonal map ∆, there is a class ε∆ ∈ Hn(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H)  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As both ε∆ and pr∗  2(εs) restrict to coevaluation on the fibres of pr1, we  have ε∆ − pr∗  2(εs) = pr∗  1(x) for some class x ∈ Hn(E1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Pulling this equation  back along ∆ shows that x = −εs, so ε∆ = pr∗  2(εs) − pr∗  1(εs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then we have  κεe(pr1, ∆) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (ε∆ · e2)  = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' ((pr∗  2(εs) − pr∗  1(εs)) · e2)  = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (pr∗  2(εs · e)) − (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (pr∗  1(εs) · e2)  = π∗  1(π2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (εs · e) − χεs  = π∗  1κεe(π, s) − χεs  = −χ¯ε  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  The intersection form of the fibres of π : E → X provides a map of local coeffi-  cient systems λ : H⊗H → Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' as we will often be concerned with applying it to two  factors of a tensor power H⊗k and will have to specify which factors we apply it to,  we will denote λ by λ1,2 and more generally write λi,j : H⊗k → H⊗k−2 for the map  that applies λ to the ith and jth factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We call such operations contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If p : E1 ×X E2 → X denotes the fibre product of two copies of π : E → X, and  if this has a section s : X → E, then in [KRW20b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='9] we have established  the formula  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4)  λ1,2(ε × ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) − 1 × v − v × 1 + p∗s∗e ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q),  where v = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is the fibrewise Poincar´e dual to the section s, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KRW20b, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The analogue of this formula for ¯ε is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We have  λ1,2(¯ε × ¯ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) +  1  χ2 p∗κe2 − 1  χ(e × 1 + 1 × e) ∈ H2n(E1 ×X E2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 we may suppose without loss of generality that π : E → X  has a section s : X → E, so that ε ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we have ¯ε = ε − 1  χπ∗κεe ∈ Hn(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H), and so  λ1,2(¯ε × ¯ε) = λ1,2((ε − 1  χπ∗κe·ε) × (ε − 1  χπ∗κe·ε))  = λ1,2(ε × ε) − λ1,2( 1  χπ∗κεe × ε)  − λ1,2(ε × 1  χπ∗κεe) + λ1,2( 1  χπ∗κεe × 1  χπ∗κεe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The first term is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4), and using [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='10] the last  term is given by  λ1,2( 1  χπ∗κεe × 1  χπ∗κεe) =  1  χ2 p∗λ1,2(κεe · κεe) =  1  χ2 p∗(κe2 + (χ2 − 2χ)s∗e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For the middle two terms, note that  ε × 1  χπ∗κεe = 1  χ(ε × 1) · p∗(κe·ε) = 1  χ(ε · π∗κεe) × 1  6  OSCAR RANDAL-WILLIAMS  so we need to calculate λ1,2(ε · π∗κεe) ∈ H2n(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The class ε · κεe is the fibre  integral along pr1 : E1 ×X E2 → E1 of ε × (ε · e) = (ε × ε) · (1 × e), so  λ1,2(ε · κεe) = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (λ1,2(ε × ε) · (1 × e))  = (pr1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='((∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) − 1 × v − v × 1 + p∗s∗e) · (1 × e))  = e − π∗s∗e − χv + χπ∗s∗e  and hence  λ1,2(ε × 1  χκεe) = 1  χ(e − π∗s∗e − χv + χπ∗s∗e) × 1  = 1  χe × 1 + χ−1  χ p∗s∗e − v × 1  and similarly  λ1,2( 1  χκεe × ε) = 1  χ1 × e + χ−1  χ p∗s∗e − 1 × v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Combining these gives  λ1,2(¯ε × ¯ε) = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) − 1 × v − v × 1 + p∗s∗e  +  1  χ2 p∗κe2 + χ−2  χ p∗s∗e  − ( 1  χe × 1 + χ−1  χ p∗s∗e − v × 1)  − ( 1  χ1 × e + χ−1  χ p∗s∗e − 1 × v)  = ∆!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (1) +  1  χ2 p∗κe2 − 1  χ(e × 1 + 1 × e)  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  If in addition we have a lift ℓ : E → B of the fibrewise Gauss map along some  fibration θ : B → BSO(2n) then for any c ∈ H∗(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) we can define modified  twisted Miller–Morita–Mumford classes by the formula  κ¯εac := π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Under the action of a permutation σ ∈ Sa of the tensor factors these classes  transform as sign(σ)n, as ¯ε has degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Thus for any finite set S there is a  well-defined element  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5)  κ¯εSc := π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa · ℓ∗c) ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) ⊗ (det QS)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  To keep track of signs, for an ordered set S = {s1 < s2 < · · · < sa} we will often  write κ¯εs1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',sac ∈ Hn(a−2)+|c|(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) for the corresponding element, understand-  ing that if σ is a reordering of S then κ¯εσ(s1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',σ(sa)c = sign(σ)nκ¯εs1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',sa c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5 we immediately see that these characteristic classes satisfy the  following analogue of the contraction formula from [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6 (Modified contraction formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−) we have the  identities  λ1,2(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a · ℓ∗c)) = ( χ−2  χ )π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε3,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a · ℓ∗(e · c)) +  1  χ2 κe2 · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε3,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a · ℓ∗c)  and  λa,a+1(π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a+b · ℓ∗c′)) = π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a−1,a+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a+b · ℓ∗(c · c′))  +  1  χ2 κe2 · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a−1 · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a+b · ℓ∗c′)  − 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a−1 · ℓ∗(e · c)) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a+b · ℓ∗c′)  − 1  χπ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a−1 · ℓ∗c) · π!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (¯εa+2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',a+b · ℓ∗(e · c′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Similarly, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we immediately obtain the following:  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  7  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If the bundle π : E → X has a section, so that the class ε and  hence κεSc is defined, then  κ¯εSc =  �  I⊆S  κεIc(− 1  χκεe)S\\I ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) ⊗ (det QS)⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let us give an example of using the modified contraction formula to evaluate an  expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Consider the class λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then  λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = λ1,5λ2,6  �  κ¯ε1,2,5,6 +  1  χ2 κe2κ¯ε1,2κ¯ε5,6  − 1  χ(κ¯ε1,2eκ¯ε5,6 + κ¯ε1,2κ¯ε5,6e)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The first term is  λ1,5λ2,6(κ¯ε1,2,5,6) = (−1)nλ1,5λ2,6(κ¯ε1,5,2,6)  = (−1)nλ2,6( χ−2  χ κ¯ε2,6e +  1  χ2 κe2κ¯ε2,6)  = (−1)n χ−2  χ ( χ−2  χ κe2 +  1  χ2 κe2χ) + (−1)n 1  χ2 κe2( χ−2  χ χ)  = (−1)n( (χ−2)2  χ2  + 2 χ−2  χ2 )κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The second term is  1  χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε5,6) = (−1)n 1  χ2 κe2λ1,5λ2,6(κ¯ε1,2κ¯ε6,5)  = (−1)n 1  χ2 κe2λ1,5(κ¯ε1,5)  = (−1)n χ−2  χ2 κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The third term is  − 1  χλ1,5λ2,6(κ¯ε1,2eκ¯ε5,6) = (−1)n+1 1  χλ1,5λ2,6(κ¯ε1,2eκ¯ε6,5)  = (−1)n+1 1  χλ1,5(κ¯ε1,5e − 1  χκ¯ε1eκ¯ε5e)  = (−1)n+1 1  χ  �  ( χ−2  χ κe2 +  1  χ2 κe2χ)  − 1  χ(κe2 +  1  χ2 κe2χ2 − 1  χ(2χκe2))  �  = (−1)n+1( χ−2  χ2 +  1  χ2 −  1  χ2 −  1  χ2 +  2  χ2 )κe2  = (−1)n+1 χ−1  χ2 κe2  and the fourth term is the same as the third by the evident symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In total we have  λ1,5λ2,6λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6) = (−1)n( (χ−2)2  χ2  + 2 χ−2  χ2 + χ−2  χ2 − 2 χ−1  χ2 )κe2  = (−1)n χ−3  χ κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Graphical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In [KRW20b, Section 5] it was found to be very  convenient to adopt a graphical formalism where κεac corresponds to a vertex with  a half-edges incident to it and a formal label c, a product of κεac’s corresponds to  a disjoint union of such vertices, and applying the contraction λi,j corresponds to  pairing up the half-edges labelled i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  It will be convenient to adopt a similar formalism here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let S be a finite set, and  V be a graded Q-algebra with a distinguished element e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Slightly modifying2  2The difference is that we allow labelled vertices whose contribution to the degree is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  8  OSCAR RANDAL-WILLIAMS  the definition from [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1], a marked oriented graph with  legs S and labelled by V consists of the following data:  (i) a totally ordered finite set ⃗V (of vertices), a totally ordered finite set ⃗H (of  half-edges), and a monotone function a: ⃗H → ⃗V (encoding that a half-edge  h is incident to the vertex a(h)),  (ii) an ordered matching m = {(ai, bi)}i∈I of the set H ⊔ S (encoding the  oriented edges of the graph),  (iii) a function c: V → V with homogeneous values, such that |c(v)|+n(|a−1(v)|−  2) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Marked oriented graphs Γ = (⃗V , ⃗H, a, m, c) and Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) with the  same set of legs S are isomorphic if there are order-preserving bijections ⃗V  ∼  → ⃗V ′  and ⃗H  ∼  → ⃗H′ which intertwine a and a′, intertwine c and c′, and send m to m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' An  oriented graph is an isomorphism class [Γ] of marked oriented graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We assign to  a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) the degree  deg(Γ) :=  �  v∈V  �  |c(v)| + n(|a−1(v)| − 2)  �  = n(|H| − 2|V |) +  �  v∈V  |c(v)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let π : E → X be an oriented Wg-bundle with a lift ℓ : E → B of the map clas-  sifying the vertical tangent bundle along θ : B → BSO(2n), and let V := H∗(B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  and e := θ∗e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then given a marked oriented graph Γ = (⃗V , ⃗H, a, m, c) with  legs S we form a class  ¯κ(Γ) ∈ Hdeg(Γ)(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  by the following recipe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Firstly, we may form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6)  �  v∈V  κ¯εa−1(v)c(v) ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗H),  where we have used the ordering on ⃗V to order the product, and the ordering on  ⃗H to trivialise the factor of (det QH)⊗n = (�  v∈V det Qa−1(v))⊗n that arises from  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Secondly, taking two copies S1 and S2 of the set S and writing si ∈ Si for  the element corresponding to s ∈ S we can form  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7)  �  s∈S  κ¯εs1,s2 ∈ H∗(X, H⊗(S1⊔S2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  As each κ¯εs1,s2 has degree 0, the product does not depend on how the factors are  ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Taking the product of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7) and then applying λx,y for each  ordered pair (x, y) in the matching m on H ⊔ S = H ⊔ S1 gives the required class  ¯κ(Γ) ∈ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S2) = H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In this graphical interpretation we recognise the class evaluated  in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='8 as that associated to the theta-graph with a certain ordering and  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Clearly ¯κ(Γ) only depends on the underlying oriented graph [Γ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We now describe  how it transforms when the orderings on V , H, and the pairs m are changed, without  changing the underlying labelled graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If Γ′ = (⃗V ′, ⃗H′, a′, m′, c′) is another marked  oriented graph and there are bijections f : H → H′ and g : V → V ′ intertwining  a and a′ and c and c′ and such that under these bijections the matching m′ differs  from m by reversing the order of k pairs, then  ¯κ(Γ′) = (−1)nksign(f)sign(g) · ¯κ(Γ)  for certain signs described on [KRW20b, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 55-56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Graphs considered as representing ¯κ’s behave differently to those representing κ’s  described in [KRW20b, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' To distinguish them we will depict the graphs  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  9  representing κ’s in red, as we did in that paper, and the graphs representing ¯κ’s in  blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The contraction formula of [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='10] was interpreted in  [KRW20b, Section 5] as giving relations among red graphs which yield equivalent  κ-classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In the generality of a smooth oriented Wg-bundle π : E → X with section  s : X → E these may be depicted as follows:  c  =  ce  +s∗e  c  −2s∗c  c  c′  =  cc′  +s∗e  c  c′  −s∗c  c′  −s∗c′  c  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The contraction formula, displayed graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Here the negative terms only arise when they make sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' when the vertex  has valence 2 in the first case, when the vertex labelled c has valence 1 in the second  case, and when the vertex labelled c′ has valence 1 in the third case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In these and the following figures, to avoid clutter we have adopted the  following ordering conventions: vertices are numbered starting from 1 from left to  right, half-edges around each vertex are ordered clockwise starting from the marked  half-edge, and edges are oriented from the smaller half-edge to the larger one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Similarly, the modified contraction formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6 can be interpreted  as giving the following relations among blue graphs which yield equivalent ¯κ-classes:  c  =  χ−2  χ  ce  + 1  χ2  c  e2  c  c′  =  cc′  + 1  χ2  c  c′  e2  − 1  χ(  ce  c′  +  c c′e  )  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The modified contraction formula, displayed graphically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Twisted cohomology of diffeomorphism groups  The main goal of this section is to describe the twisted cohomology groups  H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) and H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  in a stable range of degrees, of the classifying space BDiff+(Wg) of the group of  orientation-preserving diffeomorphisms of Wg (which classifies oriented Wg-bundles),  and the classifying space BDiff+(Wg, ∗) of the group of orientation-preserving dif-  feomorphisms of Wg which fix a point ∗ ∈ Wg (which classifies oriented Wg-bundles  with section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15] the analogous calculation was given for  10  OSCAR RANDAL-WILLIAMS  the classifying space BDiff(Wg, D2n) of the group of diffeomorphisms of Wg which  fix a disc D2n ⊂ Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In order to do this we will also discuss the manifolds Wg equipped with θ-  structures for the tangential structure θ : BSO(2n)⟨n⟩ → BO(2n), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' the n-  connected cover of BO(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In this case we will consider the homotopy quotients  BDiffθ(Wg) := Bun(T Wg, θ∗γ2n)//Diff(Wg)  BDiffθ(Wg, ∗) := Bun(T Wg, θ∗γ2n)//Diff(Wg, ∗)  where Bun(T Wg, θ∗γ2n) denotes the space of vector bundle maps T Wg → θ∗γ2n  from the tangent bundle of Wg to the bundle classified by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The group Diff(Wg)  acts on the space of bundle maps by precomposing with the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  There is a factorisation θ : BSO(2n)⟨n⟩  θor  → BSO(2n)  σ→ BO(2n), and by ob-  struction theory one sees that the space Bun(T Wg, σ∗γ2n) has two contractible  path components corresponding to the two orientations of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In particular there  are equivalences  Bun(T Wg, σ∗γ2n)//Diff(Wg) ≃ BDiff+(Wg)  Bun(T Wg, σ∗γ2n)//Diff(Wg, ∗) ≃ BDiff+(Wg, ∗)  and so θor induces maps  BDiffθ(Wg) −→ BDiff+(Wg)  and  BDiffθ(Wg, ∗) −→ BDiff+(Wg, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  It is shown in [GRW19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2] that these are principal SO[0, n − 1]-fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In particular the spaces BDiffθ(Wg) and BDiffθ(Wg, ∗) are path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Spaces of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Our description of the twisted cohomology groups of  BDiff+(Wg), BDiff+(Wg, ∗), BDiff(Wg, D2n), BDiffθ(Wg) and BDiffθ(Wg, ∗) in a  stable range will be—via the graphical interpretation given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3—in terms  of graded vector spaces of labelled graphs, modulo certain relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (Readers of  [KRW20b] may have been expecting vector spaces of labelled partitions instead:  here we have found spaces of graphs more convenient for formulating results, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2, though spaces of labelled partitions will still play a role in the proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=')  To describe these spaces of graphs we will use the graded Q-algebras  V := H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = Q[p⌈ n+1  4  ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1, e]  W := H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = Q[p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1, e]  with distinguished elements e of degree 2n given by the Euler class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In order to work  in a way which is agnostic about the genus g of the manifold Wg under consideration,  we will work over the ring Q[χ±1] instead of Q, where χ is an invertible formal  parameter which will—later—be set to the Euler characteristic 2 + (−1)n2g of Wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (i) Let  Graph1(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼  where ∼  (a) imposes the sign rule for changing orderings of vertices and half-edges and  for reversing orientations of edges;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (b) imposes linearity in the labels, and sets a graph containing an a-valent  vertex labelled by c with |c| + n(a − 1) < 0 to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (c) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, and if 2n ≡ 0  mod 4 sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  11  (d) imposes the contraction relations3  c  =  ce  −  2c  c  c′  =  cc′  −  c  c′  −  c′  c  where the negative terms only arises when they makes sense, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' in the  first case when the vertex has valence 2 and its label c is a scalar multiple  of 1 ∈ V0, in the second case when c is a scalar multiple of 1 ∈ V0 and has  valence 1, and similarly in the third case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (ii) Let  Graphθ  ∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V} ⊗ V/ ∼  where ∼ imposes (a)–(c) as well as  (d′) imposes the contraction relations of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (iii) Let  Graph∗(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W} ⊗ W/ ∼  where ∼ imposes (a) and (b), as well as  (c′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent  vertex labelled by any degree 2n monomial in Pontrjagin classes equal  to 0, and for any 1 ≤ i ≤ ⌊n/4⌋ sets cpi  = 1  χ  c  ⊗pi  and (d′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (iv) Let  Graphθ(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in V}/ ∼  where ∼ imposes (a) and (b), as well as  (c′′′) sets the 0-valent vertex labelled by e ∈ V2n equal to χ, if 2n ≡ 0 mod 4  sets the 0-valent vertex labelled by pn/2 ∈ V2n equal to 0, and sets the  1-valent vertex labelled by e ∈ V2n equal to 0,  (d′′′) imposes the contraction relations of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (v) Let  Graph(S) := Q[χ±1]{Γ oriented graph with legs S, labelled in W}/ ∼  where ∼ imposes (a), (b), as well as  (c′′′′) sets the 0-valent vertex labelled by e ∈ W2n equal to χ, sets the 0-valent  vertex labelled by any degree 2n monomial in Pontrjagin classes equal  to 0, sets the 1-valent vertex labelled by e ∈ W2n equal to 0, and for  any 1 ≤ i ≤ ⌊n/4⌋ sets  3These are the relations from Figure 1 when s∗ kills all positive-degree classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  12  OSCAR RANDAL-WILLIAMS cpi  = 1  χ  c epi  and (d′′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 (Graphs and partitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In all cases one can apply the (modified)  contraction formula to pass from a graph to a sum of graphs with strictly fewer  edges, and so by iterating to a sum of graphs with no edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' These are disjoint  unions of labelled corollas, and so correspond to partitions of S with labels in V or  W, plus additional external labels in cases (ii) and (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There are two issues with  this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The first is that in cases (iv) and (v) it is not clear that the resulting sum of  disjoint unions of labelled corollas is unique, as one has to choose an order in which  to eliminate edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The second is that even if it is, then the functoriality on the  Brauer category which we describe below would involve gluing in edges and then  eliminating them, leading to a complicated formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This is why we have found it  convenient to work with spaces of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  We wish to consider each of the above as defining functors on the (signed) Brauer  category as in [KRW20b, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3], but to take into account the parameter χ we  must slightly generalise to a Q[χ]-linear version of the (signed) Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' For finite sets S and T let preBrχ(S, T ) be the free Q[χ]-module  on tuples (f, mS, mT ) of a bijection f from a subset S◦ ⊂ S to a subset T ◦ ⊂ T ,  an ordered matching mS of S \\ S◦, and an ordered matching mT of T \\ T ◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let Brχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −  (f, m′  S, m′  T ) whenever mS agrees with m′  S after reversing some pairs, and mT agrees  with m′  T after reversing some pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let sBrχ(S, T ) be the quotient of preBrχ(S, T ) by the span of (f, mS, mT ) −  (−1)kl(f, m′  S, m′  T ) whenever mS agrees with m′  S after reversing k pairs, and mT  agrees with m′  T after reversing l pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let (s)Brχ be the Q[χ]-linear category whose objects are finite sets, and whose  morphisms are the Q[χ]-modules (s)Brχ(S, T ) defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In the case of Brχ  we think of [f, mS, mT ] as representing 1-dimensional cobordisms with no closed  components: then the composition law is given by composing cobordisms and then  replacing each closed 1-manifold by a factor of χ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In the case of sBrχ we think  of (f, mS, mT ) as representing oriented 1-dimensional cobordisms with no closed  components: then the composition law is given by composing cobordisms and then  replacing each compatibly oriented closed 1-manifold by a factor of −(χ − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let d(s)Brχ denote the subcategories having all objects and morphisms spanned  by [f, mS, mT ] with T ◦ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For a central charge d ∈ Q let (d)(s)Brd denote the Q-linear category obtained  by specialising the Q[χ]-linear category (d)(s)Brχ to χ = 2+d for (d)Br or χ = 2−d  for (d)sBr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (This notation then agrees with [KRW20b, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='14, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=')  We consider the spaces of graphs above as defining Q[χ]-linear functors  Graph1(−), Graphθ  ∗(−), Graph∗(−), Graphθ(−), Graph(−) : (s)Brχ → Gr(Q[χ±1]-mod)  in the evident way, by gluing of oriented graphs (after orientations have been ar-  ranged to be compatible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We endow them with a lax symmetric monoidality by  disjoint union of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We write Graph1(−)g : (s)Br2g → Gr(Q-mod) and so on  for their specialisations at χ = 2 + (−1)n2g (defined for (n, g) ̸= (odd, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The isomorphism theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 below extends [KRW20b, Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15] to BDiffθ(Wg, ∗), BDiff+(Wg, ∗), BDiffθ(Wg), and BDiff+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  To formulate it we first observe that when π : E → X is a smooth oriented  Wg-bundle and H is the local coefficient system over X given by the fibrewise nth  homology of this bundle, the fibrewise intersection form λ : H⊗H → Q and its dual  ω : Q → H ⊗ H are (−1)n-symmetric and satisfy λ ◦ ω = (−1)n2g · Id, so provide a  Q-linear functor S �→ H⊗S from (s)Br2g to the category of local coefficient systems  of Q-modules over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (Strictly speaking our definitions require χ = 2 + (−1)n2g to  be invertible, so we omit the case (n, g) = (odd, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=') Composing this with taking  cohomology gives a functor  H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−) : (s)Br2g −→ Gr(Q-mod)  S �−→ H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The relations in the various spaces of graphs defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 were chosen  precisely to match the contraction formula of [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='10] (in the  case of Graph1) and the modified contraction formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6 (in the  other cases), so that assigning to a graph its associated κ- or ¯κ-class provides  natural transformations  (i) κ : Graph1(−)g → H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−),  (ii) κ : Graphθ  ∗(−)g → H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−),  (iii) κ : Graph∗(−)g → H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−),  (iv) ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−),  (v) ¯κ : Graph(−)g → H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−),  of functors (s)Br2g → Gr(Q-mod).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' For 2n = 2 or 2n ≥ 6 the maps (i)–(v) are isomorphisms in a  range of cohomological degrees tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  We will first give the proof in cases (i), (ii), (iii), and in case (v) assuming case  (iv);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' the much more involved case (iv) will be treated afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 (i), (ii), (iii), (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' For case (i) observe that Graph1(−)g is  naturally isomorphic to the functor G(−, V) from [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1],  which is shown there to be isomorphic to the functor P(−, V)≥0 ⊗ det⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This case  then follows from [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For case (ii) we first construct the homotopy fibre sequence  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1)  BDiff(Wg, D2n) −→ BDiffθ(Wg, ∗) −→ BSO(2n)⟨n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The left-hand term may be written as the homotopy quotient of Diff(Wg, ∗) acting  on the Stiefel manifold Fr(T∗Wg) given by the space of frames in the tangent space to  Wg at the point ∗ ∈ Wg, as this action is transitive and its stabiliser is the subgroup  which fixes a point and its tangent space, which is homotopy equivalent to fixing a  disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The middle term was defined as the homotopy quotient of Diff(Wg, ∗) acting  on Bun(T Wg, θ∗γ2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Evaluation at ∗ ∈ Wg defines a Diff(Wg, ∗)-invariant map  ev : Bun(T Wg, θ∗γ2n) −→ BSO(2n)⟨n⟩,  which is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If we choose a point x ∈ BSO(2n)⟨n⟩ and a framing ξ :  (θ∗γ2n)x  ∼  → R2n, then there is a map ξ∗ : ev−1(x) → Fr(T∗Wg) given by sending a  bundle map ˆℓ : T Wg → θ∗γ2n whose underlying map sends ∗ to x to the framing  ξ ◦ ˆℓx : T∗Wg → (θ∗γ2n)x → R2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' One verifies by obstruction theory that ξ∗ :  ev−1(x) → Fr(T∗Wg) is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Taking homotopy orbits for Diff(Wg, ∗)  then gives the required homotopy fibre sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  14  OSCAR RANDAL-WILLIAMS  As H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) is spanned by products of twisted Miller–Morita–  Mumford classes κεac with c ∈ V in a stable range by (i), and these classes may be  defined on BDiffθ(Wg, ∗), the Serre spectral sequence  H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) ⇒ H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  for the homotopy fibre sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1) collapses in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The result then  follows by observing that the analogue of the Serre filtration of Graphθ  ∗(−)g, induced  by the descending filtration by degrees of H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = V, has  gr(Graphθ  ∗(−)g) ∼= V ⊗ Graph1(−)g,  because modulo V>0 the formula of (d′) specialises to that of (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The induced map  gr(κ) : gr(Graphθ  ∗(−)g) −→ gr(H∗(BDiffθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−))  therefore has the form V ⊗ {the map κ in case (i)} so is an isomorphism in a stable  range by case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Case (ii) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Case (iii) is just like the above, using the homotopy fibre sequence  BDiff(Wg, D2n) −→ BDiff+(Wg, ∗) −→ BSO(2n)  instead, which is established in the analogous way, and W in place of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Case (v) can be deduced from case (iv) by applying the same method to the  homotopy fibre sequence  BDiffθ(Wg) −→ BDiff+(Wg)  ξ  −→ BSO[0, n]  established in [GRW19, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The filtration step is a little different, so we  give some details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It follows from (iv) that H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) is spanned by  products of twisted Miller–Morita–Mumford classes κ¯εac with c ∈ V in a stable  range, and these may be defined on BDiff+(Wg) (in fact they may be defined even  for c ∈ W) so the corresponding Serre spectral sequence  H∗(BSO[0, n];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) ⇒ H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  degenerates in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In this case the analogue of the Serre filtration on  Graph(−)g is induced by giving the graph Υi := ({0}, ∅, ∅ → {0}, ∅, c(0) = epi)  filtration 4i for 1 ≤ i ≤ ⌊n/4⌋, giving all other connected graphs filtration 0, and  extending multiplicatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The associated graded of this filtration has the form  gr(Graph(−)g) ∼= Q[Υ1, Υ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , Υ⌊n/4⌋] ⊗ Graphθ(−)g,  because the relation in (c′′′′) shows that any graph with a vertex labelled cpi for  1 ≤ i ≤ ⌊n/4⌋ is equivalent to a graph of strictly larger filtration, unless the vertex  is 0-valent and the label is epi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As ¯κ(Υi) = κepi = χ · ξ∗(pi) by [GRW19, Remark  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5] it follows that the induced map  gr(¯κ) : gr(Graph(−)g) −→ gr(H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S))  has the form {an isomorphism} ⊗ {the map ¯κ in case (iv)} so is an isomorphism in  a stable range by case (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 (iv) is of a less formal  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It will be parallel to that of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15], but algebraically  more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  An important tool will be the following lemma, inspired by  [Qui71, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 566].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let G be a topological group and p : P → X be a principal G-bundle  with action a : G×P → P, which satisfies the Leray–Hirsch property in cohomology  over a field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then  H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' F)  H∗(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' F)  H∗(G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' F) ⊗F H∗(P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' F)  p∗  a∗  1⊗Id  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  15  is an equaliser diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let us leave F implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  By the Leray–Hirsch property H∗(P) is a free  H∗(X)-module and hence is faithfully flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus it suffices to prove that the dia-  gram is an equaliser diagram after applying −⊗H∗(X) H∗(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By the Leray–Hirsch  property we also have H∗(P) ⊗H∗(X) H∗(P)  ∼  → H∗(P ×X P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus it suffices to  show that  H∗(P)  H∗(P ×X P)  H∗(G) ⊗ H∗(P ×X P)  pr∗  2  a∗  1⊗Id  is an equaliser diagram, which is the same question for the principal G-bundle  pr2 : P ×X P → P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' But this principal G-bundle has a section given by the diagonal  map, which trivialises it: this trivialisation identifies the diagram with  H∗(P)  H∗(G) ⊗ H∗(P)  H∗(G) ⊗ H∗(G) ⊗ H∗(P)  1⊗Id  µ∗⊗Id  1⊗Id  which is indeed an equaliser diagram as it has a contraction induced by a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  We adapt the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15], supposing for concreteness that  n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Consider the tangential structure θ × Y : BSO(2n)⟨n⟩ × Y → BSO(2n)  with Y = K(W ∨, n + 1) and W a generic rational vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then we have  H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ∼= Sym∗(W[n + 1]), the symmetric algebra on the vector space W places  in (even) degree n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If n is even then like at the end of the proof of [KRW20b,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15] we would take Y = K(W ∨, n + 2) instead, so H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) would still  be a symmetric algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Apart from this there is no essential difference, and we  will not comment further on the differences in the case n even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  There are associated universal Wg-bundles  π : Eθ −→ BDiffθ(Wg)  πY : Eθ×Y −→ BDiffθ×Y (Wg)  and an evaluation map ℓ : Eθ×Y → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Neglecting the “maps to Y ” part of the  tangential structure gives a homotopy fibre sequence  map(Wg, Y ) −→ BDiffθ×Y (Wg) −→ BDiffθ(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  We can take Y to be a topological abelian group, which then acts fibrewise on the  map θ × Y and hence acts on compatibly Eθ×Y and BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using this we  can form the homotopy fibre sequence  map(Wg, Y )//Y −→ BDiffθ×Y (Wg)//Y −→ BDiffθ(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The space map(Wg, Y )//Y is a K(Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)⊗W ∨, 1), so there is an identification  of graded local coefficient systems  H∗(map(Wg, Y )//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = Λ∗(H ⊗ W[1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  This is natural in the vector space W, and scaling by u ∈ Q× acts on Λk(H⊗ W[1])  by uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It follows that it acts this way on the kth row of the Serre spectral sequence  Ep,q  2  = Hp(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Λq(H ⊗ W[1])) ⇒ Hp+q(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  As the differentials in this spectral sequence must be equivariant for this Q×-action,  it follows that they must all be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Furthermore this action gives a weight  decomposition of both sides, which identifies  H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Λk(H ⊗ W)) ∼= H∗+k(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)(k),  the weight k-subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  To access the latter groups, we use that there is a map  α : BDiffθ×Y (Wg) −→ Ω∞  0 (MTθ ∧ Y+)  16  OSCAR RANDAL-WILLIAMS  which by the main theorems of [Bol12, RW16, GTMW09] (for 2n = 2) and [GRW18,  GRW14, GRW17] for (2n ≥ 6) is an isomorphism on cohomology in a stable  range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Here MTθ is the Thom spectrum of −θ∗γ2n, so writing u−2n ∈  H−2n(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) for its Thom class, by the Thom isomorphism we have  H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ∼= u−2n · H∗(BSO(2n)⟨n⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = u−2n · Q[p⌈n+1  4  ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1, e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The rational cohomology of Ω∞  0 (MTθ ∧ Y+) is then given by  Sym∗([H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ Sym∗(W[n + 1])]>0),  which can be considered as the free (graded-)commutative algebra on the even-  degree classes κc,w1···wr with c ∈ Q[p⌈ n+1  4  ⌉, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , pn−1, e] and wi ∈ W, modulo lin-  earity in c and in the wi, and modulo commutativity of the wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The pullbacks  of these classes along α we again denote κc,w1···wr, and they may be described  intrinsically as the fibre integrals πY  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (c(TπY Eθ×Y ) · ℓ∗(w1 · · · wr)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There are unique classes ¯κc,w1···wr ∈ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) which  pull back to  �  I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  κc,wI ·  �  j∈J  (− 1  χκe,wj) ∈ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q),  and in a stable range of degrees H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is the free graded-commutative  algebra on the classes ¯κc,w1···wr, modulo linearity in c and in the wi, commutativity  of the wi, and modulo ¯κe,w1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We wish to apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5 to the principal Y -bundle  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2)  BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg)//Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  First observe that the fibre inclusion j : Y → BDiffθ×Y (Wg) classifies the Wg-  bundle pr1 : Y × Wg → Y equipped with the product θ-structure and the map  ℓ = pr1 : Y × Wg → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus for any w ∈ W we have  j∗κe,w = χw ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q),  and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2) satisfies the Leray–Hirsch property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5 then describes H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) as the equaliser of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3)  H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  H∗(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  a∗  1⊗Id  In a stable range H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is described in terms of the classes κc,w1···wr,  so to make use of this equaliser description we must determine how these classes  pull back along the action map  a : Y × BDiffθ×Y (Wg) −→ BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  This map classifies the Wg-bundle Y × πY : Y × Eθ×Y → Y × BDiffθ×Y (Wg)  equipped with the structure map Y × Eθ×Y  Y ×ℓ  →  Y × Y  → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  As the wi ∈  W = Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) are primitive with respect to the coproduct induced by the  multiplication on Y , we have  a∗(κc,w1···wr) = (Y × πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' ((1 × c(TπY Eθ×Y )) ·  r  �  i=1  (wi × 1 + 1 × ℓ∗(wi)))  =  �  I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  wI × κc,wJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  17  Our goal now is to show that the classes defined by  ¯κc,w1···wr :=  �  I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  κc,wI ·  �  j∈J  (− 1  χκe,wj) ∈ H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  are equalised by the maps (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3), so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5 descend to unique  classes of the same name in H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' To see this, we calculate  using the formula above that  a∗(¯κc,w1···wr) =  �  I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  � �  S⊔T =I  wS × κc,wT  �  (−1)|J| �  j∈J  (wj × 1 + 1  χ1 × κe,wj)  =  �  S⊔T ⊔U⊔V ={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  (−1)|U|wS⊔U ×  �  κc,wT ·  �  v∈V  (− 1  χκe,wv)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For each A ⊆ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=', r} the coefficient of wA is  \uf8eb  \uf8ed �  U⊆A  (−1)|U|  \uf8f6  \uf8f8  \uf8eb  \uf8ed  �  T ⊔V ={1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}\\A  κc,wT ·  �  v∈V  (− 1  χκe,wv)  \uf8f6  \uf8f8  and �  U⊆A(−1)|U| vanishes if A ̸= ∅, and is 1 if A = ∅ (it is the binomial expansion  of (1 − 1)|A|), which shows that a∗(¯κc,w1···wr) = 1 × ¯κc,w1···wr as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Finally, that these classes (except ¯κe,w1 = 0) freely generate the Q-algebra  H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) in a stable range follows from the fact that the κc,w1···wr  freely generate H∗(BDiffθ×Y (Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) in a stable range, together with the observa-  tion that ¯κc,w1···wr ≡ κc,w1···wr modulo the ideal generated by classes κe,w and the  Leray–Hirsch property again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  Let us provide a “fibre-integral” interpretation of the classes we have just con-  structed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Consider the map of principal Y -bundles  Y  Eθ×Y  Eθ×Y //Y  Y  BDiffθ×Y (Wg)  BDiffθ×Y (Wg)//Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  i  πY  πY //Y  j  The composition ℓ ◦ i : Y → Y is the identity, so i∗ℓ∗(w) = w ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We  showed in the proof above that j∗κe,w = χw ∈ Hn+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q), so in particular both  these principal Y -bundles satisfy the Leray–Hirsch property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Together these give  that  i∗(ℓ∗(w) − 1  χ(πY )∗κe,w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  As Y is n-connected it follows from the Serre spectral sequence that there exists a  unique class ¯ℓ∗(w) ∈ Hn+1(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) which pulls back to ℓ∗(w) − 1  χ(πY )∗κe,w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We have  ¯κc,w1···wr = (πY //Y )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) ∈ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As the lower of the above principal Y -bundles satisfies the Leray–Hirsch  property, this identity may be verified after pulling back to BDiffθ×Y (Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In  H∗(Eθ×Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) we have ¯ℓ∗(w) = ℓ∗(w) − 1  χ(πY )∗κe,w, so expanding out gives  (πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (c · ¯ℓ∗(w1) · · · ¯ℓ∗(wr)) = (πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (c ·  r  �  i=1  (ℓ∗(wi) − 1  χ(πY )∗κe,wi))  =  �  I⊔J={1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',r}  κc,wI ·  �  j∈J  (− 1  χκe,wj)  18  OSCAR RANDAL-WILLIAMS  as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  The classes ¯κc,w1···wr provide an isomorphism  Sym∗  �[H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ Sym∗(W[n + 1])]>0  u−2n · e ⊗ W[n + 1]  �  −→ H∗(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  in a stable range, natural in W, which with the discussion above gives an identifi-  cation of graded vector spaces  H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Λ∗(H ⊗ W[1])) ∼= Sym∗  �[H∗(MTθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ Sym∗(W[n + 1])]>0  u−2n · e ⊗ W[n + 1]  �  natural in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Just as in the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15], and using its notation, this  implies that there is a natural transformation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4)  Pbis(−, V)≥0 ⊗ det⊗n −→ H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−)  of lax symmetric monoidal functors FB → Gr(Q-mod) which is an isomorphism in a  stable range, where P(−, V)≥0 → Pbis(−, V)≥0 is the quotient by those partitions  containing a part of size 1 labelled by e ∈ V2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Assigning to a labelled part the  corolla with that label gives a natural transformation  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5)  Pbis(−, V)≥0 ⊗ det⊗n −→ Graphθ(−)g,  of lax symmetric monoidal functors FB → Gr(Q-mod), and we claim that using this  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4) factors through the map ¯κ : Graphθ(−)g → H∗(BDiffθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Assuming  this claim for now, observe that using the contraction relations in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 (iv)  (d′′′) to contract all edges shows that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5) is surjective, which with the fact that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4) is an isomorphism in a stable range will show that the map ¯κ is an isomorphism  in a stable range too (as well as the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  It remains to show the factorisation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' that the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4) sends a part of size  a labelled by c ∈ V to the class κ¯εac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We again proceed as in the relevant step of  the proof of [KRW20b, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There is a fibration sequence  map(Wg, Y ) −→ Eθ×Y −→ Eθ  and so, taking homotopy orbits for the fibrewise Y -action, a fibration sequence  map(Wg, Y )//Y −→ Eθ×Y //Y −→ Eθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Again by functoriality in W the associated Serre spectral sequence collapses to  identify the weight decomposition as  H∗(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Λk(H ⊗ W)) ∼= H∗+k(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Given the description in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7 we must show that the map  ¯ℓ(−) : W −→ Hn+1(Eθ×Y //Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)(1) ∼= Hn(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H) ⊗ W  is given by w �→ ¯ε ⊗ w, which is the analogue of [KRW20b, Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As it is  natural in the vector space W it must certainly be given by ¯ℓ(w) = x ⊗ w for some  x ∈ Hn(Eθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H), and we must show that x = ¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' That the restriction of x to the  fibre Wg of π : Eθ → BDiffθ(Wg) is given by coevaluation may be done precisely  as in [KRW20b, Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By the characterisation of ¯ε it remains to check that  1  χ(πY )!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (e · ¯ℓ∗(w)) = 0 ∈ Hn+1(BDiffθ×Y (Wg)//Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  By the Leray–Hirsch property this may be checked after pulling back to BDiffθ×Y (Wg),  but as ¯ℓ∗(w) = ℓ∗(w) − 1  χ(πY )∗κe,w ∈ Hn+1(Eθ×Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) by definition, the vanishing  is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There are natural maps  BDiff(Wg, D2n)  BDiffθ(Wg, ∗)  BDiffθ(Wg)  BDiff(Wg, D2n)  BDiff+(Wg, ∗)  BDiff+(Wg)  a  b  c  d  e  f  which each induce maps on H∗(−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There are corresponding maps of spaces  of graphs  Graph1(−)g  Graphθ  ∗(−)g  Graphθ(−)g  Graph1(−)g  Graph∗(−)g  Graph(−)g  a∗  b∗  e∗  c∗  d∗  f ∗  given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The maps c∗ and d∗ are induced by the projections W → V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The  maps a∗ and e∗ are induced by applying the augmentations V → Q and W → Q  to the second tensor factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The maps b∗ and f ∗ are more subtle, as they involve  converting between blue graphs and red graphs, via the formula of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Graphically it is given by �→ − 1  χ( e e e  +  +  )  with certain orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The maps b and f are also oriented Wg-bundles, so they also induce fibre-  integration maps b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' and f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' on cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' These are b∗- and f ∗-linear respectively,  so are determined by the maps (of degree −2n)  b!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' : V −→ Graphθ(−)g  f!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' : W −→ Graph(−)g  which each send a monomial c in pi’s and e to the graph given by a single vertex  labelled by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Cohomology of Torelli groups  The isomorphisms provided by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 can be converted into information  about the spaces  BTor(Wg, D2n), BTorθ(Wg, ∗), BTor+(Wg, ∗), BTorθ(Wg), BTor+(Wg)  just as [KRW20a, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1] is deduced from [KRW20a, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let us  give the definition of these spaces and formulate the result: the following is largely  a reminder of some points from [KRW20a], and we do not spell out all details again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The group Diff+(Wg) acts on Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Z) preserving the nondegenerate (−1)n-  symmetric intersection form λ : Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Z) ⊗ Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Z) → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This provides a  homomorphism  αg : Diff+(Wg) −→ Gg :=  �  Sp2g(Z)  if n is odd,  Og,g(Z)  if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  This map is not always surjective, but its image is a certain finite-index subgroup  G′  g ≤ Gg, an arithmetic group associated to the algebraic group Sp2g or Og,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This  subgroup has been determined by Kreck [Kre79]: it is the whole of Gg if n is even  or n = 1, 3, 7, and otherwise is the subgroup Spq  2g(Z) ≤ Sp2g(Z) of those matrices  which preserve the standard quadratic refinement (of Arf invariant 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  20  OSCAR RANDAL-WILLIAMS  We define Tor+(Wg) to be the kernel of this homomorphism, and Tor+(Wg, ∗)  and Tor(Wg, D2n) to be the kernel of its restriction to the subgroups Diff+(Wg, ∗)  and Diff(Wg, D2n) respectively (these restrictions still have image G′  g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Further-  more, we define  BTorθ(Wg) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg)  BTorθ(Wg, ∗) := Bun+(T Wg, θ∗γ2n)//Tor+(Wg, ∗),  where Bun+(T Wg, θ∗γ2n) ⊂ Bun(T Wg, θ∗γ2n) consists of the orientation-preserving  bundle maps (for some choice of orientation of θ∗γ2n that we make once and for  all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By the discussion at the beginning of Section 3 the spaces Bun+(T Wg, θ∗γ2n)  are path-connected, so each of the BTor’s we have defined are principal G′  g-bundles  over the corresponding BDiff’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In particular, their rational cohomologies are both  Q-algebras and G′  g-representations, and we will describe them as such in a stable  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Before doing so, we recall that the work of Borel identifies  H∗(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) =  �  Q[σ2, σ6, σ10, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=']  if n is odd,  Q[σ4, σ8, σ12, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=']  if n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  in a stable range of degrees, where σ4i−2n may be chosen so that it pulls back to the  Miller–Morita–Mumford class κLi ∈ H4i−2n(BDiff+(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) associated to the ith  Hirzebruch L-class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In particular the κLi vanish in the cohomology of BTor+(Wg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let us write H(g) := Hn(Wg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q), which is the standard representation of G′  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Pulled back from BDiff+(Wg) to BTor+(Wg) the coefficient system H is canonically  trivialised, but has an action of G′  g: it can be identified with the dual H(g)∨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The  edge homomorphism of the Serre spectral sequence  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1)  H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) −→  �  H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ (H(g)∨)⊗S�G′  g  allows us to consider the modified twisted Miller–Morita–Mumford classes ¯κεSc as  providing G′  g-equivariant homomorphisms  ¯κc : H(g)⊗S −→ Hn(|S|−2)+|c|(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The identities from the modified contraction formula correspond to identities  among these maps: this will give relations analogous to [KRW20b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2],  which we will spell out after the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' First we explain how  these relations can be organised in a categorical way, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Considering (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1) as a natural transformation of functors on (s)Br2g, we may  precompose it with the map  ¯κ : Graphg(−) −→ H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−)  (which is an isomorphism in a stable range for n ̸= 2 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This gives  G′  g-equivariant maps H(g)⊗S ⊗ Graphg(S) → H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) which assemble  to a map  K∨ ⊗(s)Br2g Graphg(−) −→ H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  out of the coend, where K : (s)Br2g → Rep(G′  g) sends S to H(g)⊗S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The domain  obtains a graded-commutative Q-algebra structure coming from the lax symmetric  monoidality of Graphg(−) and strong symmetric monoidality of K(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 below will say that this is surjective in a stable range, with kernel the ideal  generated by the κLi, but before stating it we explain a simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let us write i : d(s)Br → (s)Br2g for the inclusion of the downward (signed)  Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus subcategory is independent if g, as no circles can be created  by composing morphisms in the downward Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Write Graph1(−)′ ⊂  i∗Graph1(−)g for the subfunctor where we forbid bivalent vertices labelled by 1 ∈ V  both of whose half-edges are legs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' similarly, this functor is independent of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Like  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  21  just after [KRW20b, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='11], Graph1(−)g is then the left Kan extension  i∗Graph1(−)′ of Graph1(−)′ along i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We similarly define Graphθ  ∗(−)′, Graph∗(−)′,  Graphθ(−)′, and Graph(−)′, whose left Kan extensions again recover the original  functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The following is the analogue of [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' There are G′  g-equivariant ring homomorphisms  i∗(K∨) ⊗d(s)Br Graph1(−)′  (κLi | 4i − 2n > 0)  −→ H∗(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  (i)  i∗(K∨) ⊗d(s)Br Graphθ  ∗(−)′  (κLi | 4i − 2n > 0)  −→ H∗(BTorθ(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  (ii)  i∗(K∨) ⊗d(s)Br Graph∗(−)′  (κLi | 4i − 2n > 0)  −→ H∗(BTor+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  (iii)  i∗(K∨) ⊗d(s)Br Graphθ(−)′  (κLi | 4i − 2n > 0)  −→ H∗(BTorθ(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  (iv)  i∗(K∨) ⊗d(s)Br Graph(−)′  (κLi | 4i − 2n > 0)  −→ H∗(BTor+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)  (v)  which for 2n ≥ 6 are isomorphisms in a stable range of degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If 2n = 2 then, in a stable range of degrees and assuming that the target is  finite-dimensional in degrees ∗ < N for all large enough g, these maps are iso-  morphisms onto the maximal algebraic subrepresentations in degrees ∗ ≤ N, and  monomorphisms in degrees ∗ ≤ N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By the main theorem of [KRW20a], as long as 2n ≥ 6 the G′  g-representations  Hi(BTor(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) are algebraic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using the inheritance properties for algebraic  representations from [KRW20a, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2], the Serre spectral sequences for the  homotopy fibre sequences  BTor(Wg, D2n) −→BTor+(Wg, ∗) −→ BSO(2n)  BTor(Wg, D2n) −→BTorθ(Wg, ∗) −→ BSO(2n)⟨n⟩  show that the cohomology groups of BTor+(Wg, ∗) and BTorθ(Wg, ∗) are also al-  gebraic G′  g-representations, and the same for the homotopy fibre sequences  Wg −→BTor+(Wg, ∗) −→ BTor+(Wg)  Wg −→BTorθ(Wg, ∗) −→ BTorθ(Wg)  show that the cohomology groups of BTor+(Wg) and BTorθ(Wg) are algebraic  G′  g-representations too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Using this algebraicity property, case (i) is precisely [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1],  using that by [KRW20b, Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1] Graph1(−)g is isomorphic to the  functor P(−, V)≥0⊗det⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The other cases follow in the same way, using [KRW20b,  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='16], from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4, with one elaboration which we describe below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The addendum in the case 2n = 2 is precisely as in [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The elaboration comes when verifying the first hypothesis of [KRW20b, Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3], which in case (v) for example requires us to know that H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  is a free H∗(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)-module in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' But by transfer H∗(BDiff+(Wg);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  is a summand of H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) (as H∗(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)-modules), and similarly  with θ-structures, so cases (ii) and (iii) imply cases (iv) and (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In the other  hand in case (iii) for example we have discussed in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 the  degeneration of the Serre spectral sequence in a stable range, giving  gr(H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)) ∼= H∗(BSO(2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) ⊗ H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  22  OSCAR RANDAL-WILLIAMS  The Serre filtration is one of H∗(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)-modules, so as the associated graded is a  free H∗(G′  g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q)-module in a stable range (because H∗(BDiff(Wg, D2n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S) is the  case treated in [KRW20b, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1]), it follows that H∗(BDiff+(Wg, ∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗S)  is too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The same argument applies in case (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  This quite categorical description can be used to get a more down-to-earth pre-  sentation for these cohomology rings: in case (v) this is the presentation we have  recorded in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This is deduced just as in [KRW20b, Section 5], though  most of the work has been done as we have already expressed things in terms of  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As in [KRW20b, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4] this is not the smallest possible presentation:  it can be simplified by manipulating graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' we leave the details to the interested  reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The case 2n = 2  Although Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 is only known to hold in a limited range of degrees in the  case 2n = 2 (N = 2 is currently the best known constant for g ≥ 3, using the work  of Johnson [Joh85]), Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 does hold in a range of cohomological degrees  tending to infinity with g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In this case our discussion is closely related to the work  of Kawazumi and Morita [Mor96, KM96, KM01], and in this section we we take  the opportunity to revisit that work from our perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Throughout this section  we assume that g ≥ 2, so that χ(Wg) = 2 − 2g ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In terms of Kawazumi and Morita’s notation we have  Mg := π0(Diff+(Wg))  Mg,∗ := π0(Diff+(Wg, ∗))  Mg,1 := π0(Diff+(Wg, D2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Under our assumption g ≥ 2 the groups Diff+(Wg), Diff+(Wg, ∗), and Diff+(Wg, D2)  all have contractible path-components, so the group cohomology of Mg is the co-  homology of BDiff+(Wg), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 gives a natural transformation  ¯κ : Graph(−)g −→ H∗(Mg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' H⊗−)  of functors sBr2g → Gr(Q-mod), which is an isomorphism in a stable range of  degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Note that in this case H∗(BSO(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) = Q[e] so V = W = Q[e] and there  is no difference between the tangential structure θ and an orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In particular  if we denote by Γi ∈ Graph(∅) the graph with a single vertex, no edges, and labelled  by ei+1, then ¯κ(Γi) = κi ∈ H2i(Mg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Q) is the usual Miller–Morita–Mumford class4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Our goal in Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 is to analyse Graph(−) in several ways, making  contact with the work of Kawazumi and Morita mentioned above as well as work  of Garoufalidis and Nakamura [GN98, GN07] and Akazawa [Aka05].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Reduction to corollas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The possible labels for the vertices of graphs in  Graph(S) are powers of the Euler class e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Given any graph we may iteratedly apply  the modified contraction formula to write it as a linear combination of graphs with  fewer edges, and hence any graph is equivalent to a linear combination of graphs  with no edges: these are disjoint unions of corollas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Of these, by definition of Graph:  the 0-valent corolla labelled by e is equal to the scalar χ, the 1-valent corolla labelled  by 1 ∈ V is trivial, and the 1-valent corolla labelled by e ∈ V is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Define a  labelled partition of a finite set S to be a partition {Sα}α∈I of S into (possibly  empty) subsets and a label enα for each part, such that  (i) If |Sα| = 0 then nα ≥ 2,  (ii) If |Sα| = 1 then nα ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  4Our κi is denoted ei in the work of Kawazumi and Morita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  23  We give a part (Sα, nα) degree 2nα + |Sα| − 2, and a labelled partition the degree  given by the sums of the degrees of its parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Similarly to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4  (iv) (particularly around equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5)), let Pbis(S, V)≥0 denote the free Q[χ±1]-  module with basis the set of labelled partitions of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Assigning to a labelled part  (Sα, enα) the corolla with legs Sα and label enα defines a map  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1)  Pbis(S, V)≥0 ⊗ det QS −→ Graph(S),  natural in S with respect to bijections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It is surjective, as explained above, by repeatedly applying the modified  contraction formula to express a graph in terms of graphs without edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If it were not injective then it would have some nontrivial Q[χ±1]-linear com-  bination of labelled partitions in its kernel, of a given degree d, and this would  remain a nontrivial Q-linear combination of labelled partitions when specialised to  χ = 2 − 2g for all g ≫ 0 (as a Laurent polynomial in χ has finitely-many roots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  But in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 (iv), in the discussion after equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5), it is  explained that when specialised to χ = 2 − 2g this map is an isomorphism in a  range of degrees tending to infinity with g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' for large enough g the degree d will be  in this stable range, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  In particular, for the graphs Γi described above there is an isomorphism  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2)  Q[χ±1][Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='] ∼= Graph(∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Reduction to trivalent graphs without labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In this section we will  prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using the modified contraction formula any marked oriented graph  is equivalent to a Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1]-linear combination of  trivalent graphs with all vertices labelled by 1 ∈ V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Let Graphtri(S) ≤ Graph(S) denote the sub-Q[χ±1]-module spanned by those  marked oriented graphs which are trivalent and all of whose labels are 1 ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The monomorphism i : Graphtri(−) → Graph(−) becomes an iso-  morphism upon inverting χ − 2, χ − 3, and χ − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In particular Graphtri(−)g =  Graph(−)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 (2-valent vertices labelled by 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using the relation  λ2,3(κ¯ε1,2κ¯ε3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',nc) = κ¯ε1,3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=',nc  we can always remove 2-valent vertices labelled by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It is sometimes convenient  when writing formulas for 3-valent graphs to also allow 2-valent vertices labelled  by 1: we allow ourselves to do so, noting that the above can always be used to  eliminate the 2-valent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As a matter of notation we will formally manipulate modi-  fied twisted Miller–Morita–Mumford classes, but this is equivalent to manipulating  marked oriented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Rearranging the first contraction formula gives  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3)  κ¯εaeb =  χ  χ−2  �  λ1,2κ¯ε2+aeb−1 −  1  χ2 κe2κ¯εaeb−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Rearranging the second contraction formula gives  κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −  1  χ2 (κe2 · κ¯εa · κ¯εb) + 1  χ(κ¯εae · κ¯εb + κ¯εa · κ¯εbe)  24  OSCAR RANDAL-WILLIAMS  and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3) to eliminate the Euler classes from the last two terms gives  κ¯εa+b = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) −  1  χ2 (κe2 · κ¯εa · κ¯εb)  +  1  χ−2((λ1,2(κ¯ε2+a) −  1  χ2 κe2κ¯εa) · κ¯εb + κ¯εa · (λ1,2(κ¯ε2+b) −  1  χ2 κe2κ¯εb))  = λa+1,a+2(κ¯εa+1 · κ¯ε1+b) +  1  χ−2 ((λ1,2(κ¯ε2+a) · κ¯εb + κ¯εa · λ1,2(κ¯ε2+b))  −  1  χ(χ−2)κe2 · κ¯εa · κ¯εb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  It suffices to show that each corolla κ¯εaeb may be represented by a linear combi-  nation of trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='8 the class κe2 may be represented by a  trivalent graph (after inverting χ − 3) so by iteratedly applying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3) it suffices to  show that each κ¯εn can too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' By Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4 we may as well show that classes can  be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' To get started we have κ¯ε = 0 as it has  negative degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Consider the class λ2,5λ3,4(κ¯ε1,2,3 ·κ¯ε4,5,6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using the form of the relations above,  which avoid creating Euler classes, this is  λ2,5(κ¯ε1,2,5,6 −  1  χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6 + κ¯ε1,2λu,v(κ¯εu,v,5,6)) +  1  χ(χ−2)(κe2κ¯ε1,2κ¯ε5,6))  = λ2,5(κ¯ε1,2,5,6) −  2  χ−2λu,v(κ¯εu,v,1,6) +  1  χ(χ−2)κe2κ¯ε1,6  = χ−4  χ−2λ2,5(κ¯ε1,2,5,6) +  1  χ(χ−2)κe2κ¯ε1,6  Renumbering legs and rearranging, this shows that λ1,2(κ¯ε4) may be represented  by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Applied with (a, b) = (2, 2) the second relation gives  κ¯ε4 = λ3,4(κ¯ε3 · κ¯ε3) +  1  χ−2 ((λ1,2(κ¯ε4) · κ¯ε2 + κ¯ε2 · λ1,2(κ¯ε4)) −  1  χ(χ−2)κe2 · κ¯ε2 · κ¯ε2,  which with the above shows that κ¯ε4 may be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Similarly to the above, consider λ2,5λ3,4(κ¯ε1,2,3 · κ¯ε4,5,6,7), which is  λ2,5(κ¯ε1,2,5,6,7 +  1  χ(χ−2)κe2κ¯ε1,2κ¯ε5,6,7 −  1  χ−2(λu,v(κ¯εu,v,1,2)κ¯ε5,6,7 + κ¯ε1,2λu,v(κ¯εu,v,5,6,7)))  = λ2,5(κ¯ε1,2,5,6,7) +  1  χ(χ−2)κe2κ¯ε1,6,7 −  1  χ−2(λ2,5λu,v(κ¯εu,v,1,2κ¯ε5,6,7) + λu,v(κ¯εu,v,1,6,7))  = χ−3  χ−2λ2,5(κ¯ε1,2,5,6,7) +  1  χ(χ−2)κe2κ¯ε1,6,7 −  1  χ−2λ2,5λu,v(κ¯εu,v,1,2)κ¯ε5,6,7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Renumbering legs and rearranging, this shows that λ1,2(κ¯ε5) may be represented by  2-, 3-, and 4-valent graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' with the above it follows that it can also be represented  by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Applied with (a, b) = (2, 3) the second relation gives  κ¯ε5 = λ3,4(κ¯ε3 · κ¯ε4) +  1  χ−2 ((λ1,2(κ¯ε4) · κ¯ε3 + κ¯ε2 · λ1,2(κ¯ε5)) −  1  χ(χ−2)κe2 · κ¯ε2 · κ¯ε3,  so it follows that κ¯ε5 may be represented by 2- and 3-valent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If n ≥ 6 then we can write n = a + b with a, b ≥ 3, so a + 2, b + 2 < n and so the  second relation expresses κ¯εn in terms of κ¯εm’s with m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus all κ¯εn’s may be  represented by 2- and 3-valent graphs as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  It is worth observing that we have the relation  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4)  λ1,2(κ¯ε3) = χ−2  χ κ¯εe +  1  χ2 κe2κ¯ε = 0,  using that κ¯εe = 0 (by definition) and that κ¯ε = 0 (as it has negative degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This  means that any graph having a trivalent vertex with a loop is trivial in Graph(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' A remark on orderings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' A curious normalisation is possible when consider-  ing trivalent graphs, allowing one to neglect the orderings of vertices, of half-edges,  and the orientations of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In [Mor96, KM96, KM01] this is implemented ab  initio and (marked) oriented graphs play no role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let us explain this normalisation,  extended to trivalent graphs with legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  A trivalent graph ˜Γ with legs S consists of a set V of vertices, a set H of half-  edges, a 3-to-1 map a : H → V recording to which vertex each half-edge is incident,  and an unordered matching µ on H ⊔ S recording which half-edges span an edge,  and which half-edges are connected to which legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Given a trivalent graph ˜Γ = (V, H, a : H → V, µ) with legs S, we may choose an  ordering of V and choose an ordering of H such that a : H → V is weakly monotone  (equivalently, choose an ordering of the half-edges incident at each vertex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We also  choose an ordering of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  There is an induced ordering of H ⊔ S by putting ⃗S  after ⃗H, and we form an ordered matching m of H ⊔ S by taking those pairs  (a, b) with a < b and {a, b} ∈ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using this we form an oriented trivalent graph  Γchoice = (⃗V , ⃗H, a : H → V, m), depending on these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The normalisation is  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let x1 < x2 < x3 < x4 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' < x2k ∈ H ⊔ S be the total order on  H ⊔ S, and let a1 < b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' , ak < bk be the ordered pairs which span an edge, with  a1 < a2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' < ak ∈ H ⊔ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Then there is a bijection given by  ρ :=  � a1 b1 a2 b2 a3 b3 ···  ak  bk  x1 x2 x3 x4 x5 x6 ··· x2k−1 x2k  �  and we define Γ := sign(ρ) · Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' As long as ˜Γ has no vertices with loops, the element Γ does not depend  on the choice of ordering of V or H, and depends on the ordering of S precisely as  the sign representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In particular if we set5  Graphundec(S) := Q[χ±1][˜Γ trivalent graph with legs S]/(graphs with loops)  then the Claim together with the relation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4) provides an epimorphism  Φ : Graphundec(S) ⊗ det QS −→ Graphtri(S)  of Q[χ±1]-modules, natural with respect to bijections in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This can be extended to  a natural transformation of functors on sBrχ by letting an ordered matching (a, b)  of elements of S act by adding an edge to the trivalent graph connecting a and b,  and contracting the determinant by a ∧ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Doing so might create a circle with no  vertices, which should be replaced by the scalar χ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof of Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' If (h1, h2, h3) are the half-edges incident at a vertex v and we  change their ordering to (hσ(1), hσ(2), hσ(3)) giving Γ′  choice, then (under the assump-  tion that Γ does not have loops) the relative ordering of half-edges forming an  edge has not changed, so m′ = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus Γ′  choice = sign(σ) · Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' On the other  hand ρ′ is obtained from ρ by postcomposing with σ, and precomposing with a  permutation which permutes some (ai < bi)’s, which is an even permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus  sign(ρ′) = sign(σ) · sign(ρ), so Γ′ = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Suppose a vertex v1 has half edges (h1  1, h1  2, h1  3) and v2 has half edges (h2  1, h2  2, h2  3),  and v1 < v2 ∈ ⃗V are adjacent in the ordering on V , and consider transposing the  ordering of these vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' For edges between a u < v1 and a vi or between a vi and  a u > v2 the relative ordering of their half-edges does not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Edges between  v1 and v2 have the relative ordering of their half-edges reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus if there are  N such edges we have Γ′  choice = (−1)1+N ·Γchoice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' But the permutation ρ is changed  5In [Mor96, KM96, KM01] they restrict to “trivalent graphs without loops”, however we find  it more natural to allow loops but set graphs with a loop to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  26  OSCAR RANDAL-WILLIAMS  by permuting (h2  1, h2  2, h2  3) past (h1  1, h1  2, h1  3), which has sign −1, and N transpositions  (aibi), which has sign (−1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus again Γ′ = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Finally, changing the order of S by a permutation τ changes ρ by postcomposition  with τ, so acts as sign(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' For the ordering of vertices and half-edges corresponding to the  theta-graph in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='8 the associated permutation is ρ = (1)(235)(46) which  is odd, so the undecorated theta-graph yields χ−3  χ κe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This is precisely minus the  evaluation of βΓ2 on [KM01, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 39] (unfortunately the theta-graph is denoted Γ2 in  that paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This minus comes from the use of a different sign convention, see the  discussion at [KRW20b, top of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Relations among trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The modified contraction formula de-  scribes relations among graphs involving contracting an edge, but this necessarily  involves graphs with vertices of different valencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we have ex-  plained that, in the case of surfaces, all graphs may be expressed purely in terms of  trivalent graphs: one may ask what relations among trivalent graphs Γ are imposed  by the contraction formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  For the unmodified contraction formula discussed in [KRW20b], the answer is  that it imposes the “I = H” relation among trivalent graphs: this is because  both the I- and H-graphs admit contractions to the X-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Furthermore, as all  connected trivalent graphs with the same number of legs and of the same genus are  equivalent under the “I = H” relation, and the contraction formula never changes  the genus or number of legs, there are no further relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  In the setting of the modified contraction formula discussed here it is more  complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It is best given in the setting of undecorated trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' After inverting χ−2, χ−3, and χ−4, undecorated trivalent graphs  which differ locally by  (IHmod)  =  +  1  (χ−4)(3−χ)(  −  )  +  1  χ−4(  +  −  −  )  give the same elements in Graphtri[(χ − 2)−1, (χ − 3)−1, (χ − 4)−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We establish this relation in Graphtri({a, b, c, d})⊗detQ{a,b,c,d}, and it then  follows in general using functoriality on the signed Brauer category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We order the  legs as a < b < c < d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  (i)  a  b  c  d  1  3  5  6 2  4  (ii)  a  b  c  d  1 2  6  3 4  5  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Some marked graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Consider first the H-shaped graph shown in Figure 3 (i), with the depicted names  of half edges, ordered as 3 < 1 < 5 < 6 < 2 < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Its corresponding permutation is  � 3 c 1 a 5 6 2 b 4 d  3 1 5 6 2 4 a b c d  �  which is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus this ordering data represents the underlying  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  27  undecorated H-shaped trivalent graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Ignoring for now the matchings to the legs  (which are given by matching 1 with a, 2 with b, and so on), it corresponds to  λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Using the form of the relations which avoid creating Euler  classes from the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we have  λ5,6(κ¯ε3,1,5 · κ¯ε6,2,4) = κ¯ε3,1,2,4 +  1  χ(χ−2)κe2κ¯ε3,1κ¯ε2,4  −  1  χ−2(λu,v(κ¯εu,v,3,1)κ¯ε2,4 + κ¯ε3,1λu,v(κ¯εu,v,2,4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Consider now the I-shaped graph shown in Figure 3 (ii), with the depicted names  of the half-edges, ordered as 4 < 3 < 5 < 6 < 1 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Its corresponding permutation  is  � 4 d 3 c 5 6 1 a 2 b  4 3 5 6 1 2 a b c d  �  which is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus this ordering data represents minus the  underlying undecorated I-shaped trivalent graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Ignoring again the matchings to  the legs, it corresponds to  λ5,6(κ¯ε4,3,5 · κ¯ε6,1,2) = κ¯ε4,3,1,2 +  1  χ(χ−2)κe2κ¯ε4,3κ¯ε1,2  −  1  χ−2(λu,v(κ¯εu,v,4,3)κ¯ε1,2 + κ¯ε4,3λu,v(κ¯εu,v,1,2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  The sum of these two expressions therefore represents the image under Φ of  the difference H − I of the underlying undecorated trivalent graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Furthermore,  κ¯ε4,3,1,2 = −κ¯ε3,1,2,4 so these terms cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  From the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2 we have the identity  λu,v(κ¯εu,v,s,t) = χ−2  χ−4λi,jλk,l(κ¯εs,i,k · κ¯εl,j,t) −  1  χ(χ−4)κe2κ¯εs,t,  expressing terms of the form λu,v(κ¯εu,v,s,t) in terms of (2- and) 3-valent vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Applying it to the sum of the two expressions above, and collecting terms, therefore  gives  Φ(H − I) =  1  χ(χ−4)κe2�  κ¯ε3,1κ¯ε2,4 + κ¯ε4,3κ¯ε2,1�  −  1  χ−4  �  λi,jλk,l(κ¯ε3,i,k · κ¯εl,j,1)κ¯ε2,4 + κ¯ε3,1λi,jλk,l(κ¯ε2,i,k · κ¯εl,j,4)  λi,jλk,l(κ¯ε4,i,k · κ¯εl,j,3)κ¯ε1,2 + κ¯ε4,3λi,jλk,l(κ¯ε1,i,k · κ¯εl,j,2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Using that κe2 = Φ(  χ  χ−3Θ) and carefully putting the graphs corresponding to the  other terms into the normal form of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 gives the identity in the statement  of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  Our relation IHmod is graphically identical to the relation called IHbis  0  by  Akazawa [Aka05, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 100] and in the corrigendum [GN07] to the paper of Garo-  ufalidis and Nakamura [GN98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' In those papers it is emphasised that IHbis  0  means  this identity is imposed only when the 4 half-edges belong to distinct edges, but  in fact this is redundant: if the 4 half-edges do not belong to distinct edges, then  the identity already holds in Graphundec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' So in fact imposing our relation IHmod  is identical to imposing their relation IHbis  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Upon inverting χ − 2, χ − 3, and χ − 4, the maps  Graphundec(S)  (IHmod)  ⊗ det QS  Φ  −→ Graphtri(S)  inc  −→ Graph(S)  are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Let R := Q[χ±1, (χ − 2)−1, (χ − 3)−1, (χ − 4)−1] and implicitly base change  to this ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' We have already shown in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 that the second map is an  isomorphism, and Φ is certainly an epimorphism, so it remains to show that the  composition is a monomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  28  OSCAR RANDAL-WILLIAMS  For an undecorated trivalent graph Γ, define a double edge to be an unordered  pair of vertices which share precisely two edges, and a triple edge to be an unordered  pair of vertices which share precisely three edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' form a theta-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Define  µ(Γ) := 2 · #double edges of Γ + 3 · #triple edges of Γ,  filter Graphundec by letting F kGraphundec be spanned by those Γ with µ(Γ) ≥ k,  and give Graphundec/(IHmod) the induced filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  If Γ = ΓH is a graph with µ(Γ) = k and a distinguished “H” subgraph, and ΓI  is obtained by replacing this “H”-subgraph by “I”, then by applying the relation  IHmod to this subgraph we find that  (i) if the edge involved is not part of a double or triple edge then the relation  gives ΓH − ΓI ∈ F k+1Graphundec/(IHmod),  (ii) if the edge involved is part of a double or triple edge then the relation is trivial  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' already holds in Graphundec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Thus the associated graded of the induced filtration on Graphundec/(IHmod) can  be described as Graphundec/(IH0), where as in [GN98] the relation IH0 means  imposing the “I = H” relation when the four half-edges belong to different edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Now IH0 is an equivalence relation on the set of isomorphism classes of trivalent  graphs without loops, and similarly to [GN98, Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 (c)] it is  easy to see that all connected trivalent graph without loops of the same rank and  with the same legs are equivalent to each other: in other words the equivalences  classes of such are given by partitions of S (the parts are the legs of each connected  component) labelled by a power of e (recording the rank of the graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' It follows  that the rank of Graphundec/(IHmod) in each degree, as an R-module, is at most that  of Graph(∅) as determined in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1, and so the composition in the statement  of the theorem, which is an epimorphism, must be an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' On the work of Garoufalidis and Nakamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The discussion of the last  few sections can be used to complete the work of Garoufalidis and Nakamura [GN98,  GN07], concerning the calculation of the invariants [Λ∗V13/(V22)]Sp in a stable  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Here we write Vλ for the irreducible Sp-representation corresponding to the  partition λ, which was written as [λ]sp in those papers, and V22 denotes the unique  copy of this irreducible in Λ2V13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Combining Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3  (c) of [GN98] was supposed to calculate [Λ∗V13/(V22)]Sp in a stable range, but for  the corrected version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='1 in [GN07], which expresses these invariants as  Graphundec(∅)g/(IHbis  0 ), the authors say “it turns out that a simple stable structure  of [these invariants] as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 (c) will not be easy to detect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' However  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='7 and equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='2) gives that  [Λ∗V13/(V22)]Sp ∼= Graphundec(∅)g/(IHbis  0 ) ∼= Graph(∅)g ∼= Q[Γ1, Γ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=']  in a stable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Thus in fact Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='3 (c) of [GN98] is correct as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' This can also be obtained from the work of Felder, Naef, and Willwacher  [FNW21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Specifically, the graded-commutative algebra A(g) defined just before  Theorem 6 of that paper is Λ∗V13/(V22), and Theorem 6 together with Proposition  36 (3) also gives the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  References  [Aka05]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Akazawa, Symplectic invariants arising from a Grassmann quotient and trivalent  graphs, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Okayama Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 47 (2005), 99–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Bol12]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Boldsen, Improved homological stability for the mapping class group with inte-  gral or twisted coefficients, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 270 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 1-2, 297–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [FNW21]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Felder, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Naef, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Willwacher, Stable cohomology of graph complexes,  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='12826, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  ON THE COHOMOLOGY OF TORELLI GROUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II  29  [GN98]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Garoufalidis and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Nakamura, Some IHX-type relations on trivalent graphs and  symplectic representation theory, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 5 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 3, 391–402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GN07]  , Corrigendum: “Some IHX-type relations on trivalent graphs and symplectic  representation theory” [Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 5 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 3, 391–402], Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  14 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 4, 689–690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GRW14]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Galatius and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Randal-Williams, Stable moduli spaces of high-dimensional man-  ifolds, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 212 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 2, 257–377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GRW17]  , Homological stability for moduli spaces of high dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' II,  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (2) 186 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 1, 127–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GRW18]  , Homological stability for moduli spaces of high dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' I, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 31 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 1, 215–264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GRW19]  , Moduli spaces of manifolds: a user’s guide, Handbook of homotopy theory,  Chapman & Hall/CRC, CRC Press, Boca Raton, FL, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 445–487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [GTMW09] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Galatius, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Tillmann, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Madsen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Weiss, The homotopy type of the cobor-  dism category, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 202 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 2, 195–239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Hai20]  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Hain, Johnson homomorphisms, EMS Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 7 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 1, 33–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Joh85]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Johnson, The structure of the Torelli group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' The abelianization of T , Topology  24 (1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 2, 127–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KM96]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Kawazumi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Morita, The primary approximation to the cohomology of the  moduli space of curves and cocycles for the stable characteristic classes, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 3 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 5, 629–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KM01]  ,  The  primary  approximation  to  the  cohomology  of  the  mod-  uli  space  of  curves  and  cocycles  for  the  Mumford-Morita-Miller  classes,  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='u-tokyo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='jp/preprint/pdf/2001-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='pdf, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Kre79]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Kreck,  Isotopy  classes  of  diffeomorphisms  of  (k − 1)-connected  almost-  parallelizable 2k-manifolds, Algebraic topology, Aarhus 1978 (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=', Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Aarhus, Aarhus, 1978), Lecture Notes in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 763, Springer, Berlin, 1979,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 643–663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KRW20a]  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Kupers and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Randal-Williams, The cohomology of Torelli groups is algebraic,  Forum of Mathematics, Sigma 8 (2020), e64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KRW20b]  , On the cohomology of Torelli groups, Forum of Mathematics, Pi 8 (2020),  e7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [KRW21]  , On the Torelli Lie algebra, https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='org/abs/2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='16010, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Mor96]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Morita, A linear representation of the mapping class group of orientable sur-  faces and characteristic classes of surface bundles, Topology and Teichm¨uller spaces  (Katinkulta, 1995), World Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=', River Edge, NJ, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 159–186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [Qui71]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Quillen, The spectrum of an equivariant cohomology ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' I, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (2) 94  (1971), 549–572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  [RW16]  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Randal-Williams, Resolutions of moduli spaces and homological stability, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' (JEMS) 18 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content=' 1, 1–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='  Email address: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='randal-williams@dpmms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
+page_content='uk  Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtAzT4oBgHgl3EQfIftM/content/2301.01062v1.pdf'}
diff --git a/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/2301.11456v1.pdf.txt b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/2301.11456v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..28c0af4e7b68211b56c092f0842e5e753bf1f4a9
--- /dev/null
+++ b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/2301.11456v1.pdf.txt
@@ -0,0 +1,3854 @@
+Graph Scattering beyond Wavelet Shackles
+Christian Koke
+Technical University of Munich &
+Ludwig Maximilian University Munich
+christian.koke@tum.de
+Gitta Kutyniok
+Ludwig Maximilian University Munich &
+University of Tromsø
+kutyniok@math.lmu.de
+Abstract
+This work develops a flexible and mathematically sound framework for the design
+and analysis of graph scattering networks with variable branching ratios and generic
+functional calculus filters. Spectrally-agnostic stability guarantees for node- and
+graph-level perturbations are derived; the vertex-set non-preserving case is treated
+by utilizing recently developed mathematical-physics based tools. Energy propaga-
+tion through the network layers is investigated and related to truncation stability.
+New methods of graph-level feature aggregation are introduced and stability of
+the resulting composite scattering architectures is established. Finally, scattering
+transforms are extended to edge- and higher order tensorial input. Theoretical
+results are complemented by numerical investigations: Suitably chosen scattering
+networks conforming to the developed theory perform better than traditional graph-
+wavelet based scattering approaches in social network graph classification tasks
+and significantly outperform other graph-based learning approaches to regression
+of quantum-chemical energies on QM7.
+1
+Introduction
+Euclidean wavelet scattering networks [22, 4] are deep convolutional architectures where output-
+features are generated in each layer. Employed filters are designed rather than learned and derive
+from a fixed (tight) wavelet frame, resulting in a tree structured network with constant branching ratio.
+Such networks provide state of the art methods in settings with limited data availability and serve
+as a mathematically tractable model of standard convolutional neural networks (CNNs). Rigorous
+investigations — establishing remarkable invariance- and stability properties of wavelet scattering
+networks — were initially carried out in [22]. The extensive mathematical analysis [38] generalized
+the term ’scattering network’ to include tree structured networks with varying branching rations and
+frames of convolutional filters, thus significantly narrowing the conceptual gap to general CNNs.
+With increasing interest in data on graph-structured domains, well performing networks generalizing
+Euclidean CNNs to this geometric setting emerged [18, 5, 9]. If efficiently implemented, such graph
+convolutional networks (GCNs) replace Euclidean convolutional filters by functional calculus filters;
+i.e. scalar functions applied to a suitably chosen graph-shift-oprator capturing the geometry of the
+underlying graph [18, 14, 9]. Almost immediately, proposals aimed at extending the success story of
+Euclidean scattering networks to the graph convolutional setting began appearing: In [48], the authors
+utilize dyadic graph wavelets (see e.g. [14]) based on the non-normalized graph Laplacian resulting
+in a norm preserving graph wavelet scattering transform. In [10], diffusion wavelets (see e.g. [8]) are
+used to construct a graph scattering transform enjoying spectrum-dependent stability guarantees to
+graph level perturbations. For scattering transforms with N layers and K distinct functional calculus
+filters, the work [11] derives node-level stability bounds of OpKN{2q and conducts corresponding
+numerical experiments choosing diffusion wavelets, monic cubic wavelets [14] and tight Hann
+wavelets [35] as filters. In [12] the authors, following [8], construct so called geometric wavelets and
+establish the expressivity of a scattering transform based on such a frame through extensive numerical
+Preprint. Under review.
+arXiv:2301.11456v1  [cs.LG]  26 Jan 2023
+
+experiments. A theoretical analysis of this and a closely related wavelet based scattering transform is
+the main focus of [28]. Additionally, graph-wavelet based scattering transforms have been extended
+to the spatio-temporal domain [27], utilized to overcome the problem of oversmoothing in GCNs
+[25] and pruned to deal with their exponential (in network depth) increase in needed resources [15].
+Common among all these contributions is the focus on graph wavelets, which are generically
+understood to derive in a scale-sampling procedure from a common wavelet generating kernel
+function g : R Ñ R satisfying various properties [14]. Established stability or expressivity properties
+— especially to structural perturbations — are then generally linked to the specific choice of the
+wavelet kernel g and utilized graph shift operator [10, 28]. This severely limits the diversity of
+available filter banks in the design of scattering networks and draws into question their validity as
+models for more general GCNs whose filters generically do not derive from a wavelet kernel.
+A primary focus of this work is to provide alleviation in this situation: After reviewing the graph signal
+processing setting in Section 2, we introduce a general framework for the construction of (generalized)
+graph scattering transforms beyond the wavelet setting in Section 3. Section 4 establishes spectrum-
+agnostic stability guarantees on the node signal level and for the first time also for graph-level
+perturbations. To handle the vertex-set non-preserving case, a new ’distance measure’ for operators
+capturing the geometry of varying graphs is utilized. After providing conditions for energy decay
+(with the layers) and relating it to truncation stability, we consider graph level feature aggregation
+and higher order inputs in Sections 5 and 6 respectively. In Section 7 we then provide numerical
+results indicating that general functional calculus filter based scattering is at least as expressive as
+standard wavelet based scattering in graph classification tasks and outperforms leading graph neural
+network approaches to regression of quantum chemical energies on QM7.
+2
+Graph Signal Processing
+Taking a signal processing approach, we consider signals on graphs as opposed to graph embeddings:
+Node-Signals:
+Given a graph pG, Eq, we are primarily interested in node-signals, which are
+functions from the node-set G to the complex numbers, modelled as elements of C|G|. We equip this
+space with an inner product according to xf, gy “ ř|G|
+i“1 figiµi (with all vertex weights µi ě 1) and
+denote the resulting inner product space by ℓ2pGq. We forego considering arbitrary inner products on
+C|G| solely in the interest of increased readability.
+Functional Calculus Filters:
+Our fundamental objects in investigating node-signals will be func-
+tional calculus filters based on a normal operator ∆ : ℓ2pGq Ñ ℓ2pGq. Prominent examples include
+the adjacency matrix W, the degree matrix D, normalized p1 ´ D´ 1
+2 WD´ 1
+2 q or un-normalized
+(L :“ D´W) graph Laplacians Writing normalized eigenvalue-eigenvector pairs of ∆ as pλi, φiq|G|
+i“1,
+the filter obtained from applying g : C Ñ C is given by gp∆qf “ ř|G|
+i“1 gpλiqxφi, fyℓ2pV qφi. The
+operator we utilize in our numerical investigations of Section 6, is given by L :“ L{λmaxpLq. We
+divide by the largest eigenvalue to ensure that the spectrum σpL q is contained in the interval r0, 1s,
+which aids in the choice of functions from which filters are derived.
+Generalized Frames:
+We are most interested in filters that arise from a collection of functions
+adequately covering the spectrum of the operator to which they are applied. To this end we call a
+collection tgip¨quiPI of functions a generalized frame if it satisfies the generalized frame condition
+A ď ř
+iPI |gipcq|2 ď B for any c in C for constants A; B ą 0. As proved in Appendix B, this
+condition is sufficient to guarantee that the associated operators form a frame:
+Theorem 2.1. Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal. If the family tgip¨quiPI of bounded functions
+satisfies A ď ř
+iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq)
+A}f}2
+ℓ2pGq ď
+ÿ
+iPI
+}gip∆qf}2
+ℓ2pGq ď B}f}2
+ℓ2pGq.
+Notably, the functions tgiuiPI need not be continuous: In fact, in our numerical implementations, we
+will – among other mappings – utilize the function δ0p¨q, defined by δ0p0q “ 1 and δ0pcq “ 0 for
+c ‰ 0 as well as a modified cosine, defined by cosp0q “ 0 and cospcq “ cospcq for c ‰ 0.
+2
+
+3
+The Generalized Graph Scattering Transform
+A generalized graph scattering transform is a non-linear map Φ based on a tree structured multilayer
+graph convolutional network with constant branching factor in each layer. For an input signal
+f P ℓ2pGq, outputs are generated in each layer of such a scattering network, and then concatenated to
+form a feature vector in a feature space F. The network is built up from three ingredients:
+Connecting Operators:
+To allow intermediate signal representations in the ’hidden’ network
+layers to be further processed with functional calculus filters based on varying operators, which might
+not all be normal for the same choice of node-weights, we allow these intermediate representations
+to live in varying graph signal spaces. In fact, we do not even assume that these signal spaces are
+based on a common vertex set. This is done to allow for modelling of recently proposed networks
+where input- and ’processing’ graphs are decoupled (see e.g. [1, 36]), as well as architectures
+incorporating graph pooling [20]. Instead, we associate one signal space ℓ2pGnq to each layer
+n. Connecting operators are then (not necessarily linear) operators Pn : ℓ2pGn´1q Ñ ℓ2pGnq
+connecting the signal spaces of subsequent layers. We assume them to be Lipschitz continuous
+(}Ppfq ´ Ppgq}ℓ2pGn´1q ď R`}f ´ g}ℓ2pGnqq and triviality preserving (Pp0q “ 0). For our original
+node-signal space we also write ℓ2pGq ” ℓ2pG0q .
+Non-Linearities:
+To each layer, we also associate a (possibly) non-linear function ρn : C Ñ C
+acting poinwise on signals in ℓ2pGnq. Similar to connecting operators, we assume ρn preserves zero
+and is Lipschitz-continuous with Lipschitz constant denoted by L`
+n . This definition allows for the
+absolute value non-linearity, but also ReLu or – trivially – the identity function.
+Operator Frames:
+Beyond these ingredients, the central building block of our scattering
+architecture is comprised of a family of functional calculus filters in each layer. That is, we
+assume that in each layer, the node signal space ℓ2pGnq carries a normal operator ∆n and an
+associated collection of functions comprised of an output generating function χnp¨q as well
+as a filter bank tgγnp¨quγnPΓn indexed by an index set Γn. As the network layer n varies (and
+in contrast to wavelet-scattering networks) we allow the index set Γn as well as the collection
+tχnp¨qu Ťtgγnp¨quγnPΓn of functions to vary. We only demand that in each layer the functions in the
+filter bank together with the output generating function constitute a generalized frame with frame
+constants An, Bn ě 0.
+We refer to the collection of functions ΩN
+:“ pρn, tχnp¨qu Ťtgγnp¨quγnPΓnqN
+n“1 as a mod-
+ule sequence and call DN :“ pPn, ∆nqN
+n“1 our operator collection. The generalized scattering
+transform is then constructed iteratively:
+Figure 1: Schematic Scattering Architecture
+To our initial signal f P ℓ2pGq we first apply
+the connecting operator P1, yielding a signal rep-
+resentation in ℓ2pG1q. Subsequently, we apply
+the pointwise non-linearity ρ1. Then we apply
+our graph filters tχ1p∆1qu Ťtgγ1p∆1quγ1PΓ1
+to ρ1pP1pfqq yielding the output V1pfq :“
+χ1p∆1qρ1pP1pfqq as well as the intermedi-
+ate hidden representations tU1rγ1spfq
+:“
+gγ1p∆1qρ1pP1pfqquγ1PΓ1 obtained in the first
+layer. Here we have introduced the one-step
+scattering propagator Unrγns : ℓ2pGn´1q Ñ
+ℓ2pGnq mapping f
+ÞÑ
+gγnp∆nqρnpPnpfqq
+as well as the output generating operator
+Vn
+: ℓ2pGn´1q Ñ ℓ2pGnq mapping f to
+χnp∆nqρnpPnpfqq.
+Upon defining the set
+ΓN´1 :“ ΓN´1 ˆ ... ˆ Γ1 of paths of length
+pN ´ 1q terminating in layer N ´ 1 (with Γ0
+taken to be the one-element set) and iterating the
+above procedure, we see that the outputs gener-
+ated in the N th-layer are indexed by paths ΓN´1
+terminating in the previous layer.
+3
+
+p3(P3()
+(△2)
+p2(P2())
+qa2
+9b2 (△2)
+P3(P3()
+X2(△2)
+gal
+P3(P3())
+(△2)
+IP1(Pi())/gbr (△1). [p2(P2())
+ga2
+9b2 (2)
+m
+P3(P3())
+l2(G)
+gc1
+(△1
+X2(△2)
+p3(P3())
+(△2)
+p2(P2())
+qa2
+9b2 (△2
+p3(P3())
+X1(△1)
+X2(△2)
+l2(G1)
+l2(G2)
+l2(G3)Outputs generated in the N th layer are thus given by tVN ˝UrγN´1s˝...˝Urγ1spfqupγN´1,...,γ1qPΓN´1.
+Concatenating the features obtained in the various layers of a network with depth N, our full feature
+vectors thus live in the feature space
+FN “ ‘N
+n“1
+`
+ℓ2pGnq
+˘|Γn´1| .
+(1)
+The associated canonical norm is denoted } ¨ }FN . For convenience, a brief review of direct sums of
+spaces, their associated norms and a discussion of corresponding direct sums of maps is provided in
+Appendix A. We denote the hence constructed generalized scattering transform of length N, based
+on a module sequence ΩN and operator collection DN by ΦN.
+In our numerical experiments in Section 7, we consider two particular
+instantiations of the above general architecture. In both cases the
+utilized shift-operator is L :“ L{λmaxpLq, node weights satisfy
+µi “ 1, the branching ratio in each layer is chosen as 4 and the
+depth is set to N “ 4 as well. The connecting operators are set to the
+identity and non-linearities are set to the modulus (|¨|). The two archi-
+tectures differ in the utilized filters, which are repeated in each layer
+and depicted in Fig. 2. Postponing a discussion of other parameter-
+choices, we note here that the filters tsinpπ{2¨q, cospπ{2¨qu provide
+a high and a low pass filter on the spectrum σpL q Ď r0, 1s, while
+tsinpπ¨q, cospπ¨qu provides a spectral refinement of the former two
+filters. The inner two elements of the filter bank in Architecture II
+thus separate an input signal into high- and low-lying spectral
+Figure 2: Filters of tested Ar-
+chitectures
+components. The outer two act similarly at a higher spectral scale. Additionally Architecture I –
+utilizing cos and δ0 as introduced Section 2 – prevents the lowest lying spectral information from
+propagating. Instead it is extracted via δ0p¨q in each layer. Note that Id arises from applying the
+constant-1 function to L . Normalizations are chosen to generate frames with upper bounds B ž 1.
+4
+Stability Guarantees
+In order to produce meaningful signal representations, a small change in input signal should produce
+a small change in the output of our generalized scattering transforms. This property is captured in the
+result below, which is proved in Appendix C.
+Theorem 4.1. With the notation of Section 3, we have for all f, h P ℓ2pGq:
+}ΦNpfq ´ ΦNphq}FN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBn ´ 1s, rBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+k“1
+Bk
+¸ 1
+2
+}f ´ h}ℓ2pGq
+In the case where upper frame bounds Bn and Lipschitz constants L`
+n and R`
+n are all smaller than or
+equal one, this statement reduces to the much nicer inequality:
+}ΦNpfq ´ ΦNphq}FN ď }f ´ h}ℓ2pGq.
+(2)
+Below, we always assume R`
+n , L`
+n ď 1 as this easily achievable through rescaling. We will keep Bn
+variable to demonstrate how filter size influences stability results. As for our experimentally tested
+architectures (cf. Fig. 2), we note for Architecture I that Bn “ 1{2 for all n, so that (2) applies. For
+Architecture II we have Bn “ 3, which yields a stability constant of
+?
+1 ` 2 ¨ 3 ` 2 ¨ 32 ` 2 ¨ 33 “ 9.
+Similar to other constants derived in this section, this bound is however not necessarily tight.
+Operators capturing graph geometries might only be known approximately in real world tasks; e.g. if
+edge weights are only known to a certain level of precision. Hence it is important that our scattering
+representation be insensitive to small perturbations in the underlying normal operators in each layer,
+which is captured by our next result, proved in Appendix D. Smallness here is measured in Frobenius
+norm } ¨ }F , which for convenience is briefly reviewed in Appendix A).
+Theorem 4.2. Let ΦN and rΦN be two scattering transforms based on the same module sequence
+ΩN and operator sequences DN, r
+DN with the same connecting operators (Pn “ rPn) in each
+layer. Assume R`
+n , L`
+n ď 1 and Bn ď B for some B and n ď N. Assume that the respective
+normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0. Further assume that the functions
+4
+
+()
+()SO0
+cOs()
+cos()
+ sin()
+sin()
+sin(π)
+()
+Id
+Architecture I
+Architecture IItgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants
+satisfying L2
+χn ` ř
+γnPΓn L2
+gγn ď D2 for all n ď N and some D ą 0. Then we have
+}rΦNpfq ´ ΦNpfq}FN ď
+b
+2p2N ´ 1q ¨
+b
+pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq
+for all f P ℓ2pGq. If B ď 1{2, the stability constant improves to
+a
+2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.
+The condition B ď 1
+2 is e.g. satisfied by our Architecture I, but –strictly speaking– we may not
+apply Theorem 4.2, since not all utilized filters are Lipschitz continuous. Remark D.3 in Appendix D
+however shows, that the above stability result remains applicable for this architecture as long as we
+demand that ∆ and r∆ are (potentially rescaled) graph Laplacians. For Architecture II we note that
+D “ π
+?
+10{2 and thus the stability constant is given by
+a
+2p24 ´ 1q ¨
+?
+33 ¨ π
+?
+10{2 “ 45π.
+We are also interested in perturbations that change the vertex set of the graphs in our architecture.
+This is important for example in the context of social networks, when passing from nodes representing
+individuals to nodes representing (close knit) groups of individuals. To investigate this setting, we
+utilize tools originally developed within the mathematical physics community [29]:
+Definition 4.3. Let H and r
+H be two finite dimensional Hilbert spaces. Let ∆ and r∆ be normal
+operators on these spaces. Let J : H Ñ r
+H and rJ : r
+H Ñ H be linear maps — called identification
+operators. We call the two spaces δ-quasi-unitarily-equivalent (with δ ě 0) if for any f P H and
+u P r
+H we have
+}Jf} r
+H ď 2}f}H,
+}pJ ´ rJ˚qf} r
+H ď δ}f}H,
+}f ´ rJJf}H ď δ
+b
+}f}2
+H ` xf, |∆| fyH,
+}u ´ J rJu} r
+H ď δ
+b
+}u}2
+r
+H ` xu, |r∆| uy r
+H.
+If, for some w P C the resolvent R :“ p∆ ´ ωq´1 satisfies }p rRJ ´ JRqf} r
+H ď δ}f}H for all f P H,
+we say that ∆ and r∆ are ω-δ-close with identification operator J.
+Absolute value |∆| and adjoint rJ˚ of operators are briefly reviewed in
+Appendix A. While the above definition might seem fairly abstract at
+first, it is in fact a natural setting to investigate structural perturbations
+as Figure 3 exemplifies. In our current setting, the Hilbert spaces
+in Definition 4.3 are node-signal spaces H “ ℓ2pGq, r
+H “ ℓ2p rGq
+of different graphs. The notion of ω-δ-closeness is then useful, as
+it allows to compare filters defined on different graphs but obtained
+from applying the same function to the respective graph-operators:
+Lemma 4.4. In the setting of Definition 4.3 let ∆ and r∆ be ω-δ-
+close and satisfy }∆}op, }r∆}op ď K for some K ą 0. If g : C Ñ C
+is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a
+constant Cg ě 0 so that
+}gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ
+with Cg depending on g, ω and K.
+An explicit characterization of Cg together with a proof of this result
+is presented in Appendix F. Lemma 4.4 is our main tool in establish-
+ing our next result, proved in Appendix G, which captures stability
+under vertex-set non-preserving perturbations:
+Figure 3: Prototypical Exam-
+ple of δ-unitary-equivalent
+Node
+Signal
+Spaces
+with
+p´1q-12δ-close
+Laplacians.
+Details in Appendix E.
+Theorem 4.5. Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and
+differing operator sequences DN, r
+DN. Assume R`
+n , L`
+n ď 1 and Bn ď B for some B and n ě 0.
+Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq
+(0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal
+operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting
+operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r
+Gnq “ 0. For the common module sequence ΩN assume
+that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r
+Gnq “ 0 and that the constants Cχn and
+5
+
+tCgγn uγnPΓN associated through Lemma 4.4 to the functions of the generalized frames in each layer
+satisfy C2
+χn ` ř
+γnPΓN C2
+gγn ď D2 for some D ą 0. Denote the operator that the family tJnun of
+identification operators induce on FN through concatenation by JN : FN Ñ Ă
+FN. Then, with
+KN “
+a
+p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “
+a
+2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2:
+}rΦNpJ0fq ´ JNΦNpfq} Ă
+FN ď KN ¨ δ ¨ }f}ℓ2pG,
+@f P ℓ2pGq.
+The stability result persists with slightly altered stability constants, if identification operators only
+almost commute with non-linearities and/or connecting operators, as Appendix G further elucidates.
+Theorem 4.5 is not applicable to Architecture I, where filters are not all holomorphic, but is directly
+applicable to Architecture II. Stability constants can be calculated in terms of D and B as before.
+Beyond these results, stability under truncation of the scattering transform is equally desirable: Given
+the energy WN :“ ř
+pγN,...,γ1qPΓN }UrγNs ˝ ... ˝ Urγ1spfq}2
+ℓ2pGNq stored in the network at layer N,
+it is not hard to see that after extending ΦNpfq by zero to match dimensions with ΦN`1pfq we have
+}ΦNpfq ´ ΦN`1pfq}2
+FN`1 ď
+`
+R`
+N`1L`
+N`1
+˘2 BN`1 ¨ WN (see Appendix H for more details). A
+bound for WN is then given as follows:
+Theorem 4.6. Let Φ8 be a generalized graph scattering transform based on a an operator sequence
+D8 “ pPn, ∆nq8
+n“1 and a module sequence Ω8 with each ρnp¨q ě 0. Assume in each layer n ě 1
+that there is an eigenvector ψn of ∆n with solely positive entries; denote the smallest entry by mn :“
+miniPGn ψnris and the eigenvalue corresponding to ψn by λn. Quantify the ’spectral-gap’ opened up
+at this eigenvalue through neglecting the output-generating function by ηn :“ ř
+γnPΓn |gγnpλnq|2
+and assume Bnmn ě ηn. We then have (with C`
+N :“ śN
+i“1 max
+␣
+1, BipL`
+i R`
+i q2(
+)
+WNpfq ď C`
+N ¨
+« N
+ź
+n“1
+ˆ
+1 ´
+ˆ
+mn ´ ηn
+Bn
+˙˙ff
+¨ }f}2
+ℓ2pGq.
+(3)
+The product in (3) decays if C`
+N Ñ C` converges and řN
+n“1pmn ´ ηn{Bnq Ñ 8 diverges as
+N Ñ 8. The positivity-assumptions on the eigenvectors ψn can e.g. always be ensured if they are
+chosen to lie in the lowest lying eigenspace of a graph Laplacian or normalized graph Laplacian
+(irrespective of the connectedness of the underlying graphs). As an example, we note that if we extend
+our Architecture I to infinite depth (recall from Section 3 that we are using the same filters, operators,
+etc. in each layer) we have upon choosing λn “ 0 and ψn to be the constant normalized vector that
+ηn “ 0, CN “ 1 and mn “ 1{
+a
+|G|, for a graph with |G| vertices. On a graph with 16 vertices, we
+then e.g. have WN ď p3{4qN}f}2
+ℓ2pGq and thus }ΦNpfq ´ ΦN`1pfq}FN`1 ď p3{4qN ¨ }f}2
+ℓ2pGq{2.
+As detailed in Appendix H, Theorem 4.6 also implies that under the given assumptions the scattering
+transform has trivial ’kernel’ for N Ñ 8, mapping only 0 to 0.
+5
+Graph-Level Feature Aggregation
+To solve tasks such as graph classification or regression over multiple graphs, we need to represent
+graphs of varying sizes in a common feature space. Given a scattering transform ΦN, we thus
+need to find a stability preserving map from the feature space FN to some Euclidean space that is
+independent of any vertex set cardinalities. Since FN is a large direct sum of smaller spaces (cf. (1)),
+we simply construct such maps on each summand independently and then concatenate them.
+General non-linear feature aggregation:
+Our main tool in passing to graph-level features is a
+non-linear map N G
+p : ℓ2pGq Ñ Rp given as
+N G
+p pfq “
+1
+?pp}f}ℓ1pGq{?µG, }f}ℓ2pGq, }f}ℓ3pGq, ..., }f}ℓppGqqJ,
+(4)
+with µG :“ ř
+iPG µi and }f}ℓqpGq :“ př
+iPG |fi|qµiq1{q. Our inspiration to use this map stems from
+the standard case where all µi “ 1: For p ě |G|, the vector |f| “ pp|f1|, ..., |fG|qJ can then be
+recovered from N G
+p pfq up to permutation of indices [23]. Hence, employing N G
+p (with p ě |G|) to
+aggregate node-information into graph-level information, we lose the minimal necessary information
+6
+
+about node permutation (clearly N G
+p pfq “ N G
+p pΠfq for any permutation matrix Π) and beyond that
+only information about the complex phase (respectively the sign in the real case) in each entry of f.
+Figure 4: Graph Level Scattering
+Given a scattering transform ΦN mapping from ℓ2pGq to
+the feature space FN “ ‘N
+n“1
+`
+ℓ2pGnq
+˘|Γn´1|, we ob-
+tain a corresponding map ΨN mapping from ℓ2pGq to
+RN “ ‘N
+n“1 pRpnq|Γn´1| by concatenating the feature
+map ΦN with the operator that the family of non-linear
+maps tN pn
+GnuN
+n“1 induces on FN by concatenation. Simi-
+larly we obtain the map rΨN : ℓ2p rGq Ñ RN by concatenat-
+ing the map rΦN : ℓ2p rGq Ñ ‘N
+n“1
+´
+ℓ2p rGnq
+¯|Γn´1|
+with
+the operator induced by the family tN pn
+r
+GnuN
+n“1. The feature
+space RN is completely determined by path-sets ΓN and
+used maximal p-norm indices pn. It no longer depends
+on cardinalities of vertex sets of any graphs, allowing to
+compare (signals on) varying graphs with each other. Most
+of the results of the previous sections then readily transfer
+to the graph-level-feature setting (c.f. Appendix I.1).
+Low-pass feature aggregation:
+The spectrum-free aggregation scheme of the previous paragraph
+is especially adapted to settings where there are no high-level spectral properties remaining constant
+under graph perturbations. However, many commonly utilized operators, such as normalized and
+un-normalized graph Laplacians, have a somewhat ’stable’ spectral theory: Eigenvalues are always
+real, non-negative, the lowest-lying eigenvalue equals zero and simple (if the graph is connected). In
+this section we shall thus assume that each mentioned normal operator ∆n (r∆n) has these spectral
+properties. We denote the lowest lying normalized eigenvector (which is generically determined up
+to a complex phase) by ψ∆n and denote by M |x¨,¨y|
+Gn
+: ℓ2pGnq Ñ C the map given by M |x¨,¨y|
+Gn
+pfq “
+|xψ∆n, fyℓ2pGnq|. The absolute value around the inner product is introduced to absorb the phase-
+ambiguity in the choice of ψ∆n. Given a scattering transform ΦN mapping from ℓ2pGq to the feature
+space FN, we obtain a corresponding map Ψ|x¨,¨y|
+N
+mapping from ℓ2pGq to CN “ ‘N
+n“1C|Γn´1| by
+concatenating the feature map ΦN with the operator that the family of maps tM |x¨,¨y|
+Gn
+uN
+n“1 induces on
+FN by concatenation. As detailed in Appendix I.2, this map inherits stability properties in complete
+analogy to the discussion of Section 4.
+6
+Higher Order Scattering
+Node signals capture information about nodes in isolation. However, one might be interested in
+binary, ternary or even higher order relations between nodes such as distances or angles in graphs
+representing molecules. In this section we focus on binary relations – i.e. edge level input – as this is
+the instantiation we also test in our regression experiment in Section 7. Appendix J.2 provides more
+details and extends these considerations beyond the binary setting. We equip the space of edge inputs
+with an inner product according to xf, gy “ ř|G|
+i,j“1 fijgijµij and denote the resulting inner-product
+space by ℓ2pEq with E “ GˆG the set of edges. Setting e.g. node-weights µi and edge weights µik to
+one, the adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self-
+adjoint operators on ℓ2pEq, where they act by matrix multiplication. Replacing the Gn of Section 3
+by En, we can then follow the recipe laid out there in constructing 2nd-order scattering transforms; all
+that we need are a module sequence ΩN and an operator sequence D2
+N :“ pP 2
+n, ∆2
+nqN
+n“1, where now
+P 2
+n : ℓ2pEn´1q Ñ ℓ2pEnq and ∆2
+n : ℓ2pEnq Ñ ℓ2pEnq. We denote the resulting feature map by Φ2
+N
+and write F 2
+N for the corresponding feature space. The map N G
+p introduced in (4) can also be adapted
+to aggregate higher-order features into graph level features: With }f}q :“ př
+ijPG |fij|qµijq1{q and
+µE :“ ř|G|
+ij“1 µij, we define N E
+p pfq “ p}f}ℓ1pEq{?µE, }f}ℓ2pEq, }f}ℓ3pEq, ..., }f}ℓppEqqJ{?p.
+Given a feature map Φ2
+N with feature space F 2
+N “ ‘N
+n“1
+`
+ℓ2pEnq
+˘|Γn´1|, we obtain a corresponding
+7
+
+f pi(Pi()
+qc1
+(△2)
+X1(△1)
+p2(P2())
+9b2 (△2
+X2(△2)
+12(G1)
+NG1
+NG2
+P1
+P2
+RP1map Ψ2
+N mapping from ℓ2pEq to RN “ ‘N
+n“1 pRpnq|Γn´1| by concatenating ΦE
+N with the map that
+the family of non-linear maps tN En
+pn uN
+n“1 induces on F N by concatenation. The stability results of
+the preceding sections then readily translate to Φ2
+N and Ψ2
+N (c.f. Appendix J).
+7
+Experimental Results
+We showcase that even upon selecting the fairly simple Architectures I and II introduced in Section
+3 (c.f. also Fig. 2), our generalized graph scattering networks are able to outperform both wavelet-
+based scattering transforms and leading graph-networks under different circumstances. To aid visual
+clarity when comparing results, we colour-code the best-performing method in green, the second-best
+performing in yellow and the third-best performing method in orange respectively.
+Social Network Graph Classification:
+To facilitate contact between our generalized graph scat-
+tering networks, and the wider literature, we combine a network conforming to our general theory
+namely Architecture I in Fig. 2 (as discussed in Section 3 with depth N “ 4, identity as connect-
+ing operators and | ¨ |-non-linearities) with the low pass aggregation scheme of Section 5 and a
+Euclidean support vector machine with RBF-kernel (GGSN+EK). The choice N “ 4 was made
+to keep computation-time palatable, while aggregation scheme and non-linearities were chosen
+to facilitate comparison with standard wavelet-scattering approaches. For this hybrid architecture
+(GGSN+EK), classification accuracies under the standard choice of 10-fold cross validation on five
+common social network graph datasets are compared with performances of popular graph kernel
+approaches, leading deep learning methods as well as geometric wavelet scattering (GS-SVM) [12].
+More details are provided in Appendix K. As evident from Table 1, our network consistently achieves
+higher accuracies than the geometric wavelet scattering transform of [12], with the performance gap
+becoming significant on the more complex REDDIT datasets, reaching a relative mean performance
+increase of more than 10% on REDDIT-12K. This indicates the liberating power of transcending
+the graph wavelet setting. While on comparatively smaller and somewhat simpler datasets there is
+a performance gap between our static architecture and fully trainable networks, this gap closes on
+more complex datasets: While P-Poinc e.g. outperforms our method on IMDB datasets, the roles
+are reversed on REDDIT datasets. On REDDIT-B our approach trails only GIN; with difference in
+accuracies insignificant. On REDDIT-5K our method comes in third, with the gap to the second best
+method (GIN) being statistically insignificant. On REDDIT-12K we generate state of the art results.
+Table 1: Classification Accuracies on Social Network Datasets
+Method
+Classification Accuracies r%s
+COLLAB
+IMDB- B
+IMDB-M
+REDDIT-B
+REDDIT-5K
+REDDIT-12K
+WL [33]
+77.82˘1.45
+71.60˘5.16
+N/A
+78.52˘2.01
+50.77 ˘ 2.02
+34.57 ˘ 1.32
+Graphlet [34]
+73.42˘2.43
+65.40˘5.95
+N/A
+77.26˘2.34
+39.75 ˘ 1.36
+25.98 ˘ 1.29
+DGK [42]
+73.00˘0.20
+66.90˘0.50
+44.50˘0.50
+78.00˘0.30
+41.20 ˘ 0.10
+32.20 ˘ 0.10
+DGCNN [46]
+73.76˘0.49
+70.03˘0.86
+47.83˘0.85
+N/A
+48.70 ˘ 4.54
+N/A
+PSCN [26]
+72.60˘2.15
+71.00˘2.29
+45.23˘2.84
+86.30˘1.58
+49.10 ˘ 0.70
+41.32 ˘ 0.42
+P-Poinc [19]
+N/A
+81.86˘4.26
+57.31˘4.27
+79.78˘3.21
+51.71 ˘ 3.01
+42.16 ˘ 3.41
+S2S-N2N-PP [16]
+81.75˘0.80
+73.80˘0.70
+51.19˘0.50
+86.50˘0.80
+52.28 ˘ 0.50
+42.47 ˘ 0.10
+GSN-e [3]
+85.5 ˘ 1.2
+77.8 ˘ 3.3
+54.3 ˘ 3.3
+N/A
+N/A
+N/A
+WKPI-kC[47]
+N/A
+75.1 ˘ 1.1
+49.5 ˘ 0.4
+N/A
+59.5 ˘ 0.6
+48.4 ˘ 0.5
+GIN [41]
+80.20˘1.90
+75.10˘5.10
+52.30˘2.80
+92.40˘2.50
+57.50 ˘ 1.50
+N/A
+GS-SVM [12]
+79.94˘1.61
+71.20˘3.25
+48.73˘2.32
+89.65˘1.94
+53.33 ˘ 1.37
+45.23 ˘ 1.25
+GGSN+EK [OURS]
+80.34˘1.68
+73.20˘3.76
+49.47˘2.27
+91.60˘1.97
+56.89 ˘ 2.24
+49.03 ˘ 1.58
+Regression of Quantum Chemical Energies:
+In order to showcase the prowess of both our higher
+order scattering scheme and our spectrum-agnostic aggregation method of Section 5, we combine
+these building blocks into a hybrid architecture which we then apply in combination with kernel
+methods (2GGST + EK) to the task of atomization energy regression on QM7. This is a comparatively
+small dataset of 7165 molecular graphs, taken from the 970 million strong molecular database GDB-
+13 [2]. Each graph in QM7 represents an organic molecule, with nodes corresponding to individual
+atoms. Beyond the node-level information of atomic charge, there is also edge level information
+characterising interaction strengths between individual nodes/atoms available. This is encoded into so
+called Coulomb matrices (see e.g. [31] or Appendix K) of molecular graphs, which for us serve a dual
+purpose: On the one hand we consider a Coulomb matrix as an edge-level input signal on a given graph.
+8
+
+On the other hand, we also treat it as an adjacency matrix from which we build up a graph Laplacian
+L. Our normal operator is then chosen as L “ L{λmaxpLq again. Connecting operators are set to
+the identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |. Filters are chosen as psinpπ{2 ¨ L q,
+cospπ{2 ¨ L q, sinpπ ¨ L q, cospπ ¨ L qq acting through matrix multiplication. Output generating
+functions are set to the identity and depth is N “ 4, so that we essentially recover Architecture II of
+Figure 5: Atomization Energy as a Function of pri-
+mary Principal Components of Scattering Features
+Fig.
+2; now applied to edge-level input.
+Graph level features are aggregated via the
+map N E
+5 of Section 6. We chose p “ 5
+(and not p " 5) for N E
+p to avoid overfitting.
+Generated feature vectors are combined with
+node level scattering features obtained from
+applying Architecture II of Fig. 2 to the in-
+put signal of atomic charge into composite
+feature vectors; plotted in Figure 5. As is
+visually evident, even when reduced to the
+low-dimensional subspace of their first three
+principal components, the generated scatter-
+ing features are able to aptly resolve the atom-
+ization energy of the molecules. This aptitude
+is also reflected in Table 2, comparing our
+approach with leading graph-based learning methods trained with ten-fold cross validation on node
+and (depending on the model) edge level information. Our
+method is the best performing. We significantly outperform
+the next best model (DTNN), producing less than half of its
+mean absolute error (MAE). Errors of other methods are at
+least one — sometimes two — orders of magnitude greater. In
+part, this performance discrepancy might be explained by the
+hightened suitability of our scattering transform for environ-
+ments with somewhat limited training-data availability. Here
+we speculate that the additional performance gap might be ex-
+plained by the fact that our graph shift operator ∆ carries the
+same information as the Coulomb matrix (a proven molecular
+graph descriptor in itself [31]). Additionally, our filters being
+infinite series’ in powers of the underlying normal operator
+allows for rapid dispersion of information across underlying
+molecular graphs, as opposed to e.g. the filters in GraphConv
+Table 2: Comparison of Methods
+Method
+MAE [kcal/mol]
+AttentiveFP [40]
+66.2 ˘ 2.8
+DMPNN [44]
+105.8 ˘ 13.2
+DTNN [39]
+8.2 ˘ 3.9
+GraphConv [18]
+118.9 ˘ 20.2
+GROVER (base)[30]
+72.5 ˘ 5.9
+MPNN [13]
+113.0 ˘ 17.2
+N-GRAM[21]
+125.6 ˘ 1.5
+PAGTN (global) [6]
+47.8 ˘ 3.0
+PhysChem [45]
+59.6 ˘ 2.3
+SchNet [32]
+74.2 ˘ 6.0
+Weave [17]
+59.6 ˘ 2.3
+GGST+EK [OURS]
+11.3 ˘ 0.6
+2GGST+EK [OURS]
+3.4 ˘ 0.3
+or SchNet, which do not incorporate such higher powers. To quantify the effect of including second
+order scattering coefficients, we also include the result of performing kernel-regression solely on
+first order features generated through Architecture II of Fig. 2 (GGST + EK). While results are still
+better than those of all but one leading approach, incorporating higher order scattering improves
+performance significantly.
+8
+Discussion
+Leaving behind the traditional reliance on graph wavelets, we developed a theoretically well founded
+framework for the design and analysis of (generalized) graph scattering networks; allowing for
+varying branching rations, non-linearities and filter banks. We provided spectrum independent
+stability guarantees, covering changes in input signals and for the first time also arbitrary normal
+perturbations in the underlying graph-shift-operators. After introducing a new framework to quantify
+vertex-set non-preserving changes in graph domains, we obtained spectrum-independent stability
+guarantees for this setting too. We provided conditions for energy decay and discussed implications
+for truncation stability. Then we introduced a new method of graph-level feature aggregation and
+extended scattering networks to higher order input data. Our numerical experiments showed that
+a simple scattering transform conforming to our framework is able to outperform the traditional
+graph-wavelet based approach to graph scattering in social network graph classification tasks. On
+complex datasets our method is also competitive with current fully trainable methods, ouperforming
+all competitors on REDDIT-12K. Additionally, higher order graph scattering transforms significantly
+outperform current leading graph-based learning methods in predicting atomization energies on QM7.
+A reasonable critique of scattering networks as tractable models for general graph convolutional
+9
+
+kcal
+mo]
+-100
+-600
+-800
+-50
+3rd eigenvector
+-1000
+0
+1200
+8
+50
+1400
+！
+100
+-1600
+-1800
+150
+2000
+-100
+-400
+0
+-200
+0
+100
+200
+200
+1st eigenvector
+400
+600
+300
+800
+2nd eigenvectornetworks is their inability to emulate non-tree-structured network topologies. While transcending
+the wavelet setting has arguably diminished the conceptual gap between the two architectures, this
+structural difference persists. Additionally we note that despite a provided promising example, it is
+not yet clear whether the newly introduced graph-perturbation framework can aptly provide stability
+guarantees to all reasonable coarse-graining procedures. Exploring this question is the subject of
+ongoing work.
+Broader Impact
+We caution against an over-interpretation of established mathematical guarantees: Such guarantees
+do not negate biases that may be inherent to utilized datasets.
+Disclosure of Funding
+Christian Koke acknowledges support from the German Research Foundation through the MIMO
+II-project (DFG SPP 1798, KU 1446/21-2). Gitta Kutyniok acknowledges support from the ONE
+Munich Strategy Forum (LMU Munich, TU Munich, and the Bavarian Ministery for Science and
+Art), the Konrad Zuse School of Excellence in Reliable AI (DAAD), the Munich Center for Machine
+Learning (BMBF) as well as the German Research Foundation under Grants DFG-SPP-2298, KU
+1446/31-1 and KU 1446/32-1 and under Grant DFG-SFB/TR 109, Project C09 and the Federal
+Ministry of Education and Research under Grant MaGriDo.
+References
+[1] Uri Alon and Eran Yahav.
+On the bottleneck of graph neural networks and its practical
+implications. In International Conference on Learning Representations, 2021.
+[2] L. C. Blum and J.-L. Reymond. 970 million druglike small molecules for virtual screening in
+the chemical universe database GDB-13. J. Am. Chem. Soc., 131:8732, 2009.
+[3] Giorgos Bouritsas, Fabrizio Frasca, Stefanos P Zafeiriou, and Michael Bronstein. Improving
+graph neural network expressivity via subgraph isomorphism counting. IEEE Transactions on
+Pattern Analysis and Machine Intelligence, 2022.
+[4] Joan Bruna and Stéphane Mallat. Invariant scattering convolution networks. IEEE Trans.
+Pattern Anal. Mach. Intell., 35(8):1872–1886, 2013.
+[5] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann Lecun. Spectral networks and locally
+connected networks on graphs. In International Conference on Learning Representations
+(ICLR2014), CBLS, April 2014, 2014.
+[6] Benson Chen, Regina Barzilay, and Tommi Jaakkola. Path-augmented graph transformer
+network, 2019.
+[7] Paul Civin, Nelson Dunford, and Jacob T. Schwartz. Linear operators. part i: General theory.
+American Mathematical Monthly, 67:199, 1960.
+[8] Ronald R. Coifman and Mauro Maggioni. Diffusion wavelets. Applied and Computational
+Harmonic Analysis, 21(1):53–94, 2006. Special Issue: Diffusion Maps and Wavelets.
+[9] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks
+on graphs with fast localized spectral filtering. Advances in neural information processing
+systems, 29, 2016.
+[10] Fernando Gama, Alejandro Ribeiro, and Joan Bruna. Diffusion scattering transforms on graphs.
+In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA,
+USA, May 6-9, 2019. OpenReview.net, 2019.
+10
+
+[11] Fernando Gama, Alejandro Ribeiro, and Joan Bruna. Stability of graph scattering transforms. In
+Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B. Fox,
+and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual
+Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14,
+2019, Vancouver, BC, Canada, pages 8036–8046, 2019.
+[12] Feng Gao, Guy Wolf, and Matthew J. Hirn. Geometric scattering for graph data analysis. In
+Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International
+Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA,
+volume 97 of Proceedings of Machine Learning Research, pages 2122–2131. PMLR, 2019.
+[13] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural
+message passing for quantum chemistry. In International conference on machine learning,
+pages 1263–1272. PMLR, 2017.
+[14] David K Hammond, Pierre Vandergheynst, and Rémi Gribonval. Wavelets on graphs via spectral
+graph theory. Applied and Computational Harmonic Analysis, 30(2):129–150, 2011.
+[15] Vassilis N. Ioannidis, Siheng Chen, and Georgios B. Giannakis. Pruned graph scattering
+transforms. In International Conference on Learning Representations, 2020.
+[16] Yu Jin and Joseph F. JáJá. Learning graph-level representations with gated recurrent neural
+networks. CoRR, abs/1805.07683, 2018.
+[17] Steven Kearnes, Kevin McCloskey, Marc Berndl, Vijay Pande, and Patrick Riley. Molecular
+graph convolutions: moving beyond fingerprints. Journal of computer-aided molecular design,
+30(8):595–608, 2016.
+[18] Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional
+networks. In 5th International Conference on Learning Representations, ICLR 2017, Toulon,
+France, April 24-26, 2017, Conference Track Proceedings. OpenReview.net, 2017.
+[19] Panagiotis Kyriakis, Iordanis Fostiropoulos, and Paul Bogdan. Learning hyperbolic repre-
+sentations of topological features. In International Conference on Learning Representations,
+2021.
+[20] Junhyun Lee, Inyeop Lee, and Jaewoo Kang. Self-attention graph pooling. In Kamalika
+Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference
+on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 3734–
+3743. PMLR, 09–15 Jun 2019.
+[21] Shengchao Liu, Mehmet F Demirel, and Yingyu Liang. N-gram graph: Simple unsupervised
+representation for graphs, with applications to molecules. Advances in neural information
+processing systems, 32, 2019.
+[22] Stéphane Mallat. Group invariant scattering. Communications on Pure and Applied Mathematics,
+65(10):1331–1398, 2012.
+[23] Hana Melánová, Bernd Sturmfels, and Rosa Winter. Recovery from power sums. Experimental
+Mathematics, 0(0):1–10, 2022.
+[24] Barry Simon Michael Reed. Methods of modern mathematical physics, Volume 1. Academic
+Press, 1981.
+[25] Yimeng Min, Frederik Wenkel, and Guy Wolf. Scattering gcn: Overcoming oversmoothness in
+graph convolutional networks. Advances in Neural Information Processing Systems, 33:14498–
+14508, 2020.
+[26] Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov. Learning convolutional neural
+networks for graphs. In Maria Florina Balcan and Kilian Q. Weinberger, editors, Proceedings of
+The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine
+Learning Research, pages 2014–2023, New York, New York, USA, 20–22 Jun 2016. PMLR.
+[27] Chao Pan, Siheng Chen, and Antonio Ortega. Spatio-temporal graph scattering transform, 2020.
+11
+
+[28] Michael Perlmutter, Feng Gao, Guy Wolf, and Matthew Hirn. Understanding graph neural
+networks with asymmetric geometric scattering transforms, 2019.
+[29] Olaf. Post. Spectral Analysis on Graph-like Spaces / by Olaf Post. Lecture Notes in Mathematics,
+2039. Springer Berlin Heidelberg, Berlin, Heidelberg, 1st ed. 2012. edition, 2012.
+[30] Yu Rong, Yatao Bian, Tingyang Xu, Weiyang Xie, Ying Wei, Wenbing Huang, and Junzhou
+Huang. Self-supervised graph transformer on large-scale molecular data. Advances in Neural
+Information Processing Systems, 33:12559–12571, 2020.
+[31] M. Rupp, A. Tkatchenko, K.-R. Müller, and O. A. von Lilienfeld. Fast and accurate modeling
+of molecular atomization energies with machine learning. Physical Review Letters, 108:058301,
+2012.
+[32] Kristof Schütt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre
+Tkatchenko, and Klaus-Robert Müller. Schnet: A continuous-filter convolutional neural network
+for modeling quantum interactions. Advances in neural information processing systems, 30,
+2017.
+[33] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M.
+Borgwardt.
+Weisfeiler-lehman graph kernels.
+Journal of Machine Learning Research,
+12(77):2539–2561, 2011.
+[34] Nino Shervashidze, SVN Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten Borgwardt.
+Efficient graphlet kernels for large graph comparison. In David van Dyk and Max Welling,
+editors, Proceedings of the Twelth International Conference on Artificial Intelligence and
+Statistics, volume 5 of Proceedings of Machine Learning Research, pages 488–495, Hilton
+Clearwater Beach Resort, Clearwater Beach, Florida USA, 16–18 Apr 2009. PMLR.
+[35] David I Shuman, Christoph Wiesmeyr, Nicki Holighaus, and Pierre Vandergheynst. Spectrum-
+adapted tight graph wavelet and vertex-frequency frames. IEEE Transactions on Signal Pro-
+cessing, 63(16):4223–4235, 2015.
+[36] Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and
+Michael M. Bronstein. Understanding over-squashing and bottlenecks on graphs via curvature,
+2021.
+[37] Wihler T.P. On the hölder continuity of matrix functions for normal matrices. Journal of
+inequalities in pure and applied mathematics, 10(4), Dec 2009.
+[38] Thomas Wiatowski and Helmut Bölcskei. A mathematical theory of deep convolutional neural
+networks for feature extraction. IEEE Transactions on Information Theory, 64:1845–1866,
+2018.
+[39] Zhenqin Wu, Bharath Ramsundar, Evan N. Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S.
+Pappu, Karl Leswing, and Vijay S. Pande. Moleculenet: a benchmark for molecular machine
+learning† †electronic supplementary information (esi) available. see doi: 10.1039/c7sc02664a.
+Chemical Science, 9:513 – 530, 2018.
+[40] Zhaoping Xiong, Dingyan Wang, Xiaohong Liu, Feisheng Zhong, Xiaozhe Wan, Xutong Li,
+Zhaojun Li, Xiaomin Luo, Kaixian Chen, Hualiang Jiang, and Mingyue Zheng. Pushing the
+boundaries of molecular representation for drug discovery with the graph attention mechanism.
+Journal of Medicinal Chemistry, 63(16):8749–8760, 2020. PMID: 31408336.
+[41] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural
+networks? In 7th International Conference on Learning Representations, ICLR 2019, New
+Orleans, LA, USA, May 6-9, 2019. OpenReview.net, 2019.
+[42] Pinar Yanardag and S.V.N. Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM
+SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’15, page
+1365–1374, New York, NY, USA, 2015. Association for Computing Machinery.
+12
+
+[43] Pinar Yanardag and S.V.N. Vishwanathan. Deep graph kernels. In Proceedings of the 21th ACM
+SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’15, page
+1365–1374, New York, NY, USA, 2015. Association for Computing Machinery.
+[44] Kevin Yang, Kyle Swanson, Wengong Jin, Connor Coley, Philipp Eiden, Hua Gao, Angel
+Guzman-Perez, Timothy Hopper, Brian Kelley, Miriam Mathea, Andrew Palmer, Volker Settels,
+Tommi Jaakkola, Klavs Jensen, and Regina Barzilay. Analyzing learned molecular representa-
+tions for property prediction. Journal of Chemical Information and Modeling, 59(8):3370–3388,
+2019. PMID: 31361484.
+[45] Shuwen Yang, Ziyao Li, Guojie Song, and Lingsheng Cai. Deep molecular representation
+learning via fusing physical and chemical information.
+Advances in Neural Information
+Processing Systems, 34, 2021.
+[46] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning
+architecture for graph classification. In Sheila A. McIlraith and Kilian Q. Weinberger, editors,
+Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the
+30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium
+on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA,
+February 2-7, 2018, pages 4438–4445. AAAI Press, 2018.
+[47] Qi Zhao and Yusu Wang. Learning metrics for persistence-based summaries and applications
+for graph classification. Advances in Neural Information Processing Systems, 32, 2019.
+[48] Dongmian Zou and Gilad Lerman. Graph convolutional neural networks via scattering. Applied
+and Computational Harmonic Analysis, 49(3):1046–1074, nov 2020.
+Checklist
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
+contributions and scope? [Yes] A discussion of how and in which order the main claims
+made in Abstract and Introduction were substantiated within the paper is a main focus
+of Section 8
+(b) Did you describe the limitations of your work? [Yes] This is a second focus of Section
+8.
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] This is
+part of Section 8.
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to
+them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] Every Theorem
+and Lemma is stated in a mathematically precise way, with all underlying assumptions
+included. Additionally, Appendix A briefly reviews some terminology that might not
+be immediately present in every readers mind, but is utilized in order to be able to state
+theoretical results in a precise and concise manner.
+(b) Did you include complete proofs of all theoretical results? [Yes] This is the Focus
+of Appendices B, C, D, F, G, H and I. Results in Appendix J are merely stated, as
+statements and corresponding proofs are in complete analogy (in fact almost verbatim
+the same) to previously discussed statements and proofs.
+3. If you ran experiments...
+(a) Did you include the code, data, and instructions needed to reproduce the main experi-
+mental results (either in the supplemental material or as a URL)? [Yes] Yes; please see
+the supplementary material.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
+were chosen)? [Yes] This is the main focus of Appendix K.
+13
+
+(c) Did you report error bars (e.g., with respect to the random seed after running experi-
+ments multiple times)? [Yes] Errors for results of conducted experiments are included
+in Table 1 and Table 2 respectively.
+(d) Did you include the total amount of compute and the type of resources used (e.g., type
+of GPUs, internal cluster, or cloud provider)? [Yes] This is described at the beginning
+of Appendix K
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes] All utilized datasets
+were matched to the papers that introduced them (cf. Section K). Additionally, we
+partially built on code corresponding to [12] which we mentioned in Appendix K.
+(b) Did you mention the license of the assets? [Yes] We mentioned in Appendix K that the
+code corresponding to [12] is freely available under an Apache License.
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]
+Please see supplementary material.
+(d) Did you discuss whether and how consent was obtained from people whose data you’re
+using/curating? [N/A]
+(e) Did you discuss whether the data you are using/curating contains personally identifiable
+information or offensive content? [Yes] For the utilized social network datasets, we
+mention in Appendix K that neither personally identifyable data nor content that might
+be considered offensive is utilised.
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if
+applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review
+Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount
+spent on participant compensation? [N/A]
+A
+Some Concepts in Linear Algebra
+In the interest of self-containedness, we provide a brief review of some concepts from linear algebra
+utilized in this work that might potentially be considered more advanced. Presented results are all
+standard; a very thorough reference is [24].
+Hilbert Spaces:
+To us, a Hilbert space — often denoted by H — is a vector space over the complex
+numbers which also has an inner product — often denoted by x¨, ¨yH. Prototypical examples are
+given by the Euclidean spaces Cd with inner product xx, yyCd :“ řd
+i“1 xiyi. Associated to an inner
+product is a norm, denoted by } ¨ }H and defined by }x}H :“
+a
+xx, xyH for x P H.
+Direct Sums of Spaces:
+Given two potentially different Hilbert spaces H and p
+H, one can form
+their direct sum H ‘ p
+H. Elements of H ‘ p
+H are vectors of the form pa, bq, with a P H and b P p
+H.
+Addition and scalar multiplication are defined in the obvious way by
+pa, bq ` λpc, dq :“ pa ` λc, b ` λdq
+for a, c P H, b, d P p
+H and λ P C. The inner product on the direct sum is defined by
+xpa, bq, pc, dqyH‘ p
+H :“ xa, cyH ` xb, dy p
+H.
+As is readily checked, this implies that the norm } ¨ }H‘ p
+H on the direct sum is given by
+}pa, bq}2
+H‘ p
+H :“ }a}2
+H ` }b}2
+p
+H.
+Standard examples of direct sums are again the Euclidean spaces, where one has Cd “ Cn ‘ Cm if
+m ` n “ d, as is easily checked. One might also consider direct sums with more than two summands,
+writing Cd “ ‘d
+i“1C for example. In fact, one might also consider infinite sums of Hilbert spaces:
+14
+
+The space ‘8
+i“1Hi is made up of those elements a “ pa1, a2, a3, ...q with ai P Hi for which the
+norm
+}a}2
+‘8
+i“1Hi :“
+8
+ÿ
+i“1
+}ai}2
+Hi
+is finite. This means for example that the vector p1, 0, 0, 0, ...q is in ‘8
+i“1C, while p1, 1, 1, 1, ...q is
+not.
+Direct Sums of Maps:
+Suppose we have two collections of Hilbert spaces tHiuΓ
+i“1, t r
+HiuΓ
+i“1 with
+Γ P N or Γ “ 8. Suppose further that for each i ď Γ (resp. i ă Γ) we have a (not necessarily linear)
+map Ji : Hi Ñ r
+Hi. Then the collection tJiuΓ
+i“1 of these ’component’ maps induce a ’composite’
+map
+J : ‘Γ
+i“1Hi ÝÑ ‘Γ
+i“1 r
+Hi
+between the direct sums. Its value on an element a “ pa1, a2, a3, ...q P ‘Γ
+i“1Hi is defined by
+J paq “ pJ1pa1q, J2pa2q, J3pa3q, ...q P ‘Γ
+i“1 r
+Hi.
+Strictly speaking, one has to be a bit more careful in the case where Γ “ 8 to ensure that
+}J paq}‘8
+i“1 r
+Hi ‰ 8. This can however be ensured if we have }Jipaiq} r
+Hi ď C}ai}Hi for all
+1 ď i and some C independent of all i, since then }J paq}‘8
+i“1 r
+Hi ď C}a}‘8
+i“1Hi ď 8. If each Ji is
+a linear operator, such a C exists precisely if the operator norms (defined below) of all Ji are smaller
+than some constant.
+Operator Norm:
+Let J : H Ñ r
+H be a linear operator between Hilbert spaces. We measure its
+’size’ by what is called the operator norm, denoted by } ¨ }op and defined by
+}J}op :“
+sup
+ψPH,}ψ}H“1
+}Aψ} r
+H
+}ψ}H
+.
+Adjoint Operators
+Let J : H Ñ r
+H be a linear operator from the Hilbert space H to the Hilbert
+space r
+H. Its adjoint J˚ : r
+H Ñ H is an operator mapping in the opposite direction. It is uniquely
+determined by demanding that
+xJf, uy r
+H “ xf, J˚uyH
+holds true for arbitrary f P H and u P r
+H.
+Normal Operators:
+If a linear operator ∆ : H Ñ H maps from and to the same Hilbert space,
+we can compare it directly with its adjoint. If ∆∆˚ “ ∆˚∆, we say that the operator ∆ is normal.
+Special instances of normal operators are self-adjoint operators, for which we have the stronger
+property ∆ “ ∆˚. If an operator is normal, there are unitary maps U : H Ñ H diagonalizing ∆ as
+U ˚∆U “ diagpλ1, ...λnq,
+with eigenvalues in C. We call the collection of eigenvalues the spectrum σp∆q of ∆. If dim H “ d,
+we may write σp∆q “ tλud
+i“1. It is a standard exercise to verify that each eigenvalue satisfies
+|λi| ď }∆}op. Associated to each eigenvalue is an eigenvector φi. The collection of all (normalized)
+eigenvectors forms an orthonormal basis of H. We may then write
+∆f “
+dÿ
+i“1
+λi xφi, fyHφi.
+Resolvent of a (normal) Operator:
+Given a normal operator ∆ on some Hilbert space H, we have
+that the operator p∆ ´ zq : H Ñ H is invertible precisely if z ‰ σp∆q. In this case we write
+Rpz, ∆q “ p∆ ´ zq´1
+and call this operator the resolvent of ∆ at z. It can be proved that the norm of the resolvent satisfies
+}Rpz, ∆q}op “
+1
+distpz, σp∆qq,
+where distpz, σp∆qq denotes the minimal distance between z and any eigenvalue of ∆.
+15
+
+Functional Calculus:
+Given a normal operator ∆ : H Ñ H on a Hilbert space of dimension d and
+a complex function g : C Ñ C, we can define another normal operator obtained from applying the
+function g to ∆ by
+gp∆qf “
+fÿ
+i“1
+gpλiqxφi, fyHφi.
+For example if gp¨q “ | ¨ |, we obtain the absolute value |∆| of ∆ by specifying for all f P H that
+|∆|f “
+dÿ
+i“1
+|λi|xφi, fyHφi.
+Similarly we find (if z R σp∆q and for f P H)
+1
+∆ ´ z “
+dÿ
+i“1
+1
+λi ´ z xφi, fyHφi “ p∆ ´ zq´1 “ Rpz, ∆q
+where we think of the left-hand-side as applying a function to ∆, while we think of the right-hand-side
+as inverting the operator p∆ ´ zq. This now allows us to apply tools from complex analysis also to
+operators: If a function g is analytic (i.e. can be expanded into a power series), we have
+gpλq “ ´ 1
+2πi
+¿
+S
+gpzq
+λ ´ z dz
+for any circle S Ď C encircling λ by Cauchy’s integral formula. Thus, if we chose S large enough to
+encircle the entire spectrum σp∆q, we have
+gp∆qf “ ´
+dÿ
+i“1
+1
+2πi
+¿
+S
+gpzq
+λi ´ z dzxφi, fyHφi “ ´ 1
+2πi
+¿
+S
+gpzqRpz, λqdz.
+Frobenius Norm:
+Given a finite dimensional Hilbert space H with inner product x¨, ¨yH, and an
+orthonormal basis tφiud
+i“1, we define the trace of an operator A : H Ñ H as
+TrpAq :“
+dÿ
+k“1
+xφk, AφkyH.
+It is a standard exercise to show that this is independent of the choice of orthonormal basis. The
+associated Frobenius inner product on the space of operators is then given as
+xB, AyF :“ TrpB˚Aq
+dÿ
+k“1
+xφk, B˚AφkyH.
+Hence the Frobenius norm of an operator is determined by
+}A}2
+F “ TrpA˚Aq “
+dÿ
+k“1
+xφk, A˚AφkyH.
+It is a standard exercise to verify that we have }A}op ď }A}F . Since the trace is independent of the
+choice of orthonormal basis, the Frobenius norm is invariant under unitary transformations. More
+precisely, if U, V : H Ñ H are unitary, we have
+}UAV }2
+F “ }A}2
+F .
+Frobenius norms can be used to transfer Lipschitz continuity properties of complex functions to the
+setting of functions applied to normal operators:
+Lemma A.1. Let g : C Ñ C be Lipschitz continuous with Lipschitz constant Dg. This implies
+}gpXq ´ gpY q}F ď Dg ¨ }X ´ Y }F .
+for normal operators X, Y on H.
+16
+
+Proof. This proof is taken (almost) verbatim from [37]. For an operator A : H Ñ H denote by Aij
+its matrix representation with respect to the orthonormal basis tφiud
+i“1:
+Aij :“ xφi, AφjyH.
+We then have
+}A}2
+F “
+dÿ
+i,j“1
+|Aij|2
+as a quick calculation shows. Let now U, W be unitary (with respect to the inner product x¨, ¨yH)
+operators diagonalizing the normal operators X and Y as
+V ˚XV “ diagpλ1, ...λnq “: DpXq
+W ˚Y W “ diagpµ1, ...µnq “: DpY q.
+Since the Frobenius norm is invariant under unitary transformations we find
+}gpXq ´ gpY q||2
+F “ ||gpV DpXqV ˚q ´ gpWDpY qW ˚q}2
+F
+“ }V gpDpXqqV ˚ ´ WgpDpY qqW ˚}2
+F
+“ }W ˚V gpDpXqq ´ gpDpY qqW ˚V }2
+F
+“
+dÿ
+i,j“1
+|pW ˚V gpDpXqq ´ gpDpY qqW ˚V qij|2
+“
+dÿ
+i,j“1
+ˇˇˇˇˇ
+n
+ÿ
+k“1
+rW ˚V sikrgpDpXqqskj ´ rgpDpY qqsikrW ˚V skj
+ˇˇˇˇˇ
+2
+“
+dÿ
+i,j“1
+|rW ˚V sij|2 |gpλjq ´ gpµiq|2
+ď
+dÿ
+i,j“1
+|rW ˚V sij|2 D2
+g|λj ´ µi|2
+“ D2
+g
+dÿ
+i,j“1
+ˇˇˇˇˇ
+n
+ÿ
+k“1
+rW ˚V sikrDpXqskj ´ rDpY qsikrW ˚V skj
+ˇˇˇˇˇ
+2
+“ D2
+g}X ´ Y }2
+F .
+B
+Proof of Theorem 2.1
+Theorem B.1. Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal. If the family tgip¨quiPI of bounded functions
+satisfies A ď ř
+iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq)
+A}f}2
+ℓ2pGq ď
+ÿ
+iPI
+}gip∆qf}2
+ℓ2pGq ď B}f}2
+ℓ2pGq.
+Proof. Writing the normalized eigenvalue-eigenvector sequence of ∆ as pλi, φiq|G|
+i“1, we simply note
+ÿ
+iPI
+|G|
+ÿ
+k“1
+|xgipλkqφk, fyℓ2pGq|2 “
+|G|
+ÿ
+k“1
+˜ÿ
+iPI
+|gipλkq|2
+¸
+|xφk, fyℓ2pGq|2.
+Now under the assumption, we can estimate the sum in brackets by A from below and by B from
+above. Then we need only use Bessel’s (in)equality to prove
+A||f||2 ď
+ÿ
+iPpI
+|G|
+ÿ
+k“1
+|xgipλkqφk, fyℓ2pGq|2 ď B||f||2.
+17
+
+C
+Proof of Theorem 4.1
+Theorem C.1. With the notation of Section 3 and setting B0 “ 1, we have:
+}ΦNpfq ´ ΦNphq}2
+FN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+k“0
+BkpR`
+k L`
+k q2
+¸
+}f ´ h}2
+ℓ2pGq
+To streamline the argumentation let us first introduce some notation:
+Notation C.2. Let us denote paths in ΓN as q :“ pγN, ..., γ1q. For f P ℓ2pGq let us write
+fq :“ UrγNs ˝ ... ˝ Urγ1spfq.
+Proof. By Definition, we have
+}ΦNpfq ´ ΦNpgq}2
+FN “
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}Vnpfqq ´ Vnphqq}2
+ℓ2pGnq
+˛
+‚
+“
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}χnp∆nqρnpPnpfqqq ´ χnp∆nqρnpPnphqqq}2
+ℓ2pGnq
+˛
+‚
+looooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooon
+“:an
+.
+We proceed in two steps:
+Our initial goal is to upper bound an as
+an ď BnpL`
+n R`
+n q2 ¨ bn´1 ´ bn ” pbn´1 ´ bnq `
+“
+BnpL`
+n R`
+n q2 ´ 1
+‰
+¨ bn´1
+(5)
+for bn :“ ř
+qPΓn }fq ´ hq}2
+ℓ2pGnq with b0 “ }f ´ h}2
+ℓ2pGq. To achieve this we note that (5) is
+equivalent to
+an ` bn ď BnpL`
+n R`
+n q2 ¨ bn´1
+which upon unraveling definitions may be written as
+ÿ
+qPΓn´1
+}χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2
+ℓ2pGnq `
+ÿ
+pqPΓn
+}fpq ´ hpq}2
+ℓ2pGnq
+ďBnpL`
+n R`
+n q2
+ÿ
+qPΓn´1
+}fq ´ hq}2
+ℓ2pGn´1q.
+(6)
+To establish (6), we note, that in the sum over paths of length n, any pq P Γn can uniquely be written
+as pq “ pγn, qq, with the path q P Γn´1 of length pn ´ 1q determined by
+pq “ pγn, γn´1, ..., γ1
+looooomooooon
+“:q
+q.
+With this we find
+ÿ
+pqPΓn
+}fpq ´ hpq}2
+ℓ2pGnq “
+ÿ
+γnPΓn
+ÿ
+qPΓn´1
+}gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2
+ℓ2pGnq.
+Thus we can rewrite the left hand side of (6) as
+ÿ
+qPΓn´1
+}χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2
+ℓ2pGnq `
+ÿ
+pqPΓn
+}fpq ´ hpq}2
+ℓ2pGnq
+“
+ÿ
+qPΓn´1
+ˆ
+}χnp∆nqρnpPnpfqq ´ χnp∆nqρnpPnphqq}2
+ℓ2pGnq
+`
+ÿ
+γnPΓn
+}gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2
+ℓ2pGnq
+¸
+“:‹
+18
+
+The fact that in each layer the function tχnp¨qu Ťtgγnp¨quγnPΓn form a generalized frame with upper
+frame constant Bn implies by Theorem 2.1, that we can further bound this as
+‹ ď Bn
+ÿ
+qPΓn´1
+}ρnpPnpfqq ´ ρnpPnphqq}2
+ℓ2pGnq.
+Using the Lipschitz continuity of ρn and Pn, we arrive at the desired expression (6).
+Having established that
+an ď pbn´1 ´ bnq `
+“
+BnpL`
+n R`
+n q2 ´ 1
+‰
+¨ bn´1
+holds true, we note that we can establish
+bn´1 ď
+n´1
+ź
+k“1
+BkpL`
+k R`
+k q2bn´2
+arguing similarly as in the case of (6) by using (for f P ℓ2pGn´1q)
+ÿ
+γn´1PΓn´1
+}gγn´1p∆n´1qf}2
+ℓ2pGn´1q ď }χn´1p∆n´1qf}2
+ℓ2pGn´1q `
+ÿ
+γPΓ
+}gγn´1p∆n´1qf}2
+ℓ2pGn´1q
+together with the frame property and Lipschitz continuities. We then iterate this inequality and recall
+that b0 “ }f ´ h}2
+ℓ2pGq. Using the fact that
+N
+ÿ
+n“1
+pbn´1 ´ bnq “ b0 ´ bN ď b0,
+we finally find
+}ΦNpfq ´ ΦNphq}2
+FN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+k“0
+BkpR`
+k L`
+k q2
+¸
+}f ´ h}2
+ℓ2pGq.
+D
+Proof or Theorem 4.2
+Theorem D.1. Let ΦN and rΦN be two scattering transforms based on the same module sequence
+ΩN and operator sequences DN, r
+DN with the same connecting operators (Pn “ rPn) in each
+layer. Assume R`
+n , L`
+n ď 1 and Bn ď B for some B and n ď N. Assume that the respective
+normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0. Further assume that the functions
+tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants
+satisfying L2
+χn ` ř
+γnPΓn L2
+gγn ď D2 for all n ď N and some D ą 0. Then we have
+}rΦNpfq ´ ΦNpfq}FN ď
+b
+2p2N ´ 1q ¨
+b
+pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq
+for all f P ℓ2pGq. If B ď 1{2, the stability constant improves to
+a
+2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.
+Notation D.2. Let us denote scattering propagators based on operators ∆n and connecting operators
+Pn by Un and scattering propagators based on operators r∆n by rUn. Similarly, to Notation C.2, let us
+then write (with q “ pγN, ..., γ1q)
+rfq :“ rUnrγns ˝ ... ˝ rU1rγ1spfq.
+Proof. By definition we have
+}ΦNpfq ´ rΦN}2
+FN “
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}χnp∆nqρnpPnppfqqqq ´ χnpr∆nqρnpPnp rfqqq}2
+ℓ2pGnq
+˛
+‚
+loooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooon
+“:an
+.
+19
+
+We define bn :“ ř
+qPΓn }fq ´ rfq}2
+ℓ2pGnq, with b0 “ }f ´ h}2
+ℓ2pGq “ 0 and note
+an ` bn “
+ÿ
+qPΓn´1
+ˆ
+}χnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2
+ℓ2pGnq
+`
+ÿ
+γnPΓn
+}gγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2
+ℓ2pGnq
+¸
+.
+Using (with |a ` b|2 ď 2p|a|2 ` |b|2q)
+1
+2}gγnp∆nqρnpPnpfqqq ´ gγnpr∆nqρnpPnp rfqqq}2
+ℓ2pGnq
+ď}rgγnp∆nq ´ gγnpr∆nqsρnpPnpfqqq}2
+ℓ2pGnq
+`}gγnpr∆nqrρnpPnppfqqqq ´ ρnpPnp rfqqqs}2
+ℓ2pGnq
+ď}rgγnp∆nq ´ gγnpr∆nqs}2
+8 ¨ }ρnpPnpfqqq}2
+ℓ2pGnq
+`}gγnpr∆nqrρnpPnpfqqq ´ ρnpPnp rfqqqs}2
+ℓ2pGnq,
+and
+}rgγnp∆nq ´ gγnpr∆nqs}2
+8 ď }rgγnp∆nq ´ gγnpr∆nqs}2
+F ď L2
+gγ ¨ δ2
+(c.f. Lemma A.1 ), we find
+an ` bn ď2
+ÿ
+qPΓn´1
+˜
+L2
+χn `
+ÿ
+γnPΓn
+Lg2γn
+¸
+pL`
+n R`
+n q2δ2||ρnpPnpfqqq||2
+ℓ2pGnq
+`2
+ÿ
+qPΓn´1
+Bn||ρnpPnpfqqq ´ ρnpPnp rfqqq||2
+ℓ2pGnq.
+Using L2
+χn ` ř
+γnPΓn L2
+γn ď D2, we then infer (using the assumption L`
+n , R`
+n ď 1)
+an ď pbn´1 ´ bnq ` r2B ´ 1sbn´1 ` Bn´12D2δ2||f||ℓ2pGq.
+Now if B ď 1
+2, we have
+an ď pbn´1 ´ bnq ` Bn´12D2δ2||f||ℓ2pGq
+and results of geometric sums leads to the desired bound after summing over n.
+Hence let us assume B ą 1
+2. Using similar arguments as before, we find
+bn´1 ďBn´22D2δ2||f||2
+ℓ2pGq ` 2Bbn´2 ď Bn´22D2δ2||f||2
+ℓ2pGq ` Bn´24D2δ2||f||2
+ℓ2pGq ` 4bn´3
+ďBn´2
+˜n´1
+ÿ
+k“1
+2k
+¸
+D2δ2||f||2
+ℓ2pGq “ Bn´2p2n ´ 2qD2δ2||f||2
+ℓ2pGq.
+Thus we now know
+an ď 2D2δ2Bn´1||f||2
+ℓ2pGq ` r2B ´ 1sp2n ´ 2qD2δ2Bn´2||f||2
+ℓ2pGq ` pbn´1 ´ bnq
+In total we find
+g
+f
+f
+e
+N
+ÿ
+n“1
+an ď
+b
+2p2N ´ 1q ¨
+?
+BN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq,
+where we have estimated the sum over pbn´1 ´ bnq by zero from above again. This establishes the
+claim.
+Remark D.3. To see that this also holds for our Architecture I of Fig. 2, we note that the critical step
+is establishing that Lemma A.1 also applies to δ0 and cos, as defined in Section 2. Here we establish
+that
+}δ0p∆q ´ δ0pr∆q}F “ 0
+20
+
+and
+}cosp∆q ´ cospr∆q}F ď Dcos}∆ ´ r∆}F .
+Indeed, since ∆ and r∆ are (possibly) rescaled graph Laplacians on the same graph, the spectral
+projections to their lowest lying eigen space, associated to the eigenvalue λmin “ 0 agree. Denoting
+this spectral projection by P, we have
+cosp∆q ´ cospr∆q “ rcosp∆q ´ Ps ´ rcospr∆q ´ Ps “ cosp∆q ´ cospr∆q
+and we can apply Lemma A.1. Similar considerations apply to δ0.
+E
+Prototypical Example illustrating ω-δ Closeness and δ-Unitary
+Equivalence
+To investigate the example of Figure 3, we label the vertices of
+the respective graphs as depicted in Figure 6. We denote the left
+graph by G and the right graph by rG. The node-weights on rG are
+given as rµi “ 1 for 1 ď i ď 7, while on G the weights are given
+as µi “ 1 for 1 ď i ď 5 while µ6 “ 2. We then consider the
+respective un-normalized graph Laplacians ∆ : ℓ2pGq Ñ ℓ2pGq and
+r∆ : ℓ2p rGq Ñ ℓ2p rGq, which for a given adjacency matrix W on a
+graph signal space ℓ2pGq with node weights tµiui is given as
+p∆fqi “ 1
+µi
+ÿ
+j
+Wijpfi ´ fjq.
+Such operators are positive and hence |∆| “ ∆ (similarly for r∆).
+We now need to find operators J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ
+ℓ2pGq satisfying the conditions of Definition 4.3. To construct J, we
+define a family tψiu6
+i“1 of vectors on ℓ2p rGq as
+ψ1 “ p1, 0, 0, 0, 0, 0, 0q, ψ2 “ p0, 1, 0, 0, 0, 0, 0q,
+ψ3 “ p0, 0, 1, 0, 0, 0, 0q, ψ4 “ p0, 0, 0, 1, 0, 0, 0q,
+ψ5 “ p0, 0, 0, 0, 1, 0, 0q, ψ6 “ p0, 0, 0, 0, 0, 1, 1q.
+Figure 6: Indexing on the re-
+spective graphs
+The map J : ℓ2pGq Ñ ℓ2p rGq is then defined as
+Jf :“
+6ÿ
+i“1
+fiψi,
+for any f P ℓ2pGq. We take rJ : ℓ2p rGq Ñ ℓ2pGq to be its adjoint ( rJ :“ J˚), which determined
+explicitly by
+p rJuqi “ 1
+µi
+xψi, uyℓ2p r
+Gq
+for any u P ℓ2p rGq We shall now first check the conditions for δ-quasi unitary equivalence, which we
+list again for convenience; now adapted to our current setting:
+}Jf}ℓ2p r
+Gq ď 2}f}ℓ2pGq,
+}pJ ´ rJ˚qf}ℓ2p r
+Gq ď δ}f}ℓ2pGq,
+}f ´ rJJf}2
+ℓ2pGq ď δ2 ´
+}f}2
+ℓ2pGq ` xf, ∆, fyℓ2pGq
+¯
+,
+}u ´ J rJu}2
+ℓ2p r
+Gq ď δ2 ´
+}u}2
+ℓ2p r
+Gq ` xu, r∆ uyℓ2p r
+Gq
+¯
+.
+We first note that since rJ “ J˚, we have }pJ ´ rJ˚qf}ℓ2p r
+Gq “ 0. Next we note
+}Jf}2
+ℓ2p r
+Gq “
+7ÿ
+i“1
+|pJfqi|2 “ |f6|2 `
+6ÿ
+i“1
+|fi|2 “
+6ÿ
+i“1
+µi “ }f}2
+ℓ2pGq.
+21
+
+Furthermore we note
+p rJJfqi “
+6ÿ
+k“1
+fk
+1
+µi
+xψi, ψkyℓ2p r
+Gq
+loooooooomoooooooon
+“δik
+“ fi
+and hence }f ´ rJJf}2
+ℓ2pGq “ 0. It remains to control }u ´ J rJu}2
+ℓ2p r
+Gq. We note
+rJu “ pu1, u2, u3, u4, u5, pu5 ` u6q{2qJ
+and thus
+J rJu “ pu1, u2, u3, u4, u5, pu6 ` u7q{2, pu6 ` u7q{2qJ,
+Which implies
+u ´ J rJu “ p0, 0, 0, 0, 0, pu7 ´ u6q{2, pu6 ´ u7q{2qJ,
+and thus
+}u ´ J rJu}2
+ℓ2p r
+Gq “ 2|u6 ´ u7|2
+4
+“ |u6 ´ u7|2
+2
+.
+We have
+xu, r∆ uyℓ2p r
+Gq “ 1
+2
+dÿ
+i,j“1
+Ă
+Wij|ui ´ uj|2.
+Since Ă
+W67 “ 1{δ2 by assumption, we have
+}u ´ J rJu}2
+ℓ2p r
+Gq “ 1
+2|u6 ´ u7|2 “ 1
+2
+δ2
+δ2 |u6 ´ u7|2 “ 1
+2δ2Ă
+W67|u6 ´ u7|2
+ď 1
+2δ2
+dÿ
+i,j“1
+Ă
+Wij|ui ´ uj|2 “ δ2xu, r∆ uyℓ2p r
+Gq
+ď δ2 ´
+}u}2
+ℓ2p r
+Gq ` xu, r∆ uyℓ2p r
+Gq
+¯
+.
+Thus we have proven δ-unitary-equivalence and it remains to establish p´1q-12δ closeness. Com-
+bining Proposition 4.4.12. and Theorem 4.4.15 of [29], instead of bounding }p rRJ ´ JRqf}ℓ2p r
+Gq ď
+12δ}f}ℓ2pGq directly, we may instead establish that there are operators J1 : ℓ2pGq Ñ ℓ2p rGq,
+Ă
+J1 : ℓ2p rGq Ñ ℓ2pGq satisfying
+}J1f ´ Jf}ℓ2p r
+Gq ď δ2 `
+}f}ℓ2pGq ` xf, ∆, fyℓ2pGq
+˘
+,
+(7)
+}Ă
+J1u ´ rJu}ℓ2pGq ď δ2 ´
+}u}ℓ2p r
+Gq ` xf, r∆, uyℓ2p r
+Gq
+¯
+,
+(8)
+and
+xJ1f, r∆ uyℓ2p r
+Gq “ xf, ∆ Ă
+J1uyℓ2pGq.
+(9)
+We chose J1 “ J and determine Ă
+J1 by setting (for (1 ď i ď 6))
+pĂ
+J1uqi “ ui.
+Thus (7) is clearly satisfied. For (8) we note that we have
+p rJu ´ Ă
+J1uq “ p0, 0, 0, 0, 0, pu7 ´ u6q{2q.
+Thus we have
+}Ă
+J1u ´ rJu}ℓ2pGq “ 1
+2|u6 ´ u7|2 ď δ2 ´
+}u}2
+ℓ2p r
+Gq ` xu, r∆ uyℓ2p r
+Gq
+¯
+as before. It remains to establish (9). We have
+xf, ∆ Ă
+J1uyℓ2pGq “
+6ÿ
+i,j“1
+fiWijpui ´ ujq,
+22
+
+while we have
+xJ1f, r∆ uyℓ2p r
+Gq “
+6ÿ
+i“1
+fi ¨ xψi, r∆ uyℓ2p r
+Gq
+“
+5ÿ
+i,j“1
+fiWijpUj ´ uiq ` f6 ¨ xψ6, r∆ uyℓ2p r
+Gq.
+We have (with all node-weights on ℓ2pGq equal to unity)
+xψ6, r∆ uyℓ2p r
+Gq “ p∆ uq6 ` p∆ uq7 “
+˜ÿ
+j
+W6jpf6 ´ fjq ` 1
+δ2 pf6 ´ f7q
+¸
+`
+ˆ 1
+δ2 pf7 ´ f6q
+˙
+“
+˜ÿ
+j
+W6jpf6 ´ fjq ` 1
+δ2 pf6 ´ f7q
+¸
+And thus
+xJ1f, r∆ uyℓ2p r
+Gq “
+6ÿ
+i,j“1
+fiWijpui ´ ujq “ xf, ∆ Ă
+J1uyℓ2pGq
+which proves the claim.
+F
+Proof of Lemma 4.4
+Lemma F.1. In the setting of Definition 4.3 let ∆ and r∆ be ω-δ-close and satisfy }∆}op, }r∆}op ď K
+for some K ą 0. If g : C Ñ C is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a
+constant Cg ě 0 so that
+}gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ
+with Cg depending on g , ω and K.
+Proof. Without loss of generality, let us assume that K ą |ω|. Let us denote the circle of radius r in
+C by Sr. For any holomorphic function g and (normal) operator ∆ whose spectrum is enclosed by
+the circle Sr, we can express the operator gp∆q as
+gp∆q “ ´ 1
+2πi
+¿
+Sr
+gpzq
+∆ ´ z dz
+as discussed in Appendix A (see also [7] for more details). Note that in our case the resolvent
+Rpz, ∆q “ p∆ ´ zq´1 is well defined for |z| ě K, since with our assumptions all eigenvalues are
+within the circle of radius K. Additionally note that we have
+distpz, σp∆qq ě distpz, SKq “ |z| ´ K
+if |z| ě K. The same holds true after replacing ∆ with r∆. Since for any normal operator ∆ we have
+}Rpz, ∆q}op “ 1{distpz, σp∆qq,
+we find
+|Rpz, r∆q}op, }Rpz, ∆q}op ď 1{p|z| ´ Kq.
+To quantify the difference }Rpz, r∆qJ ´ JRpz, ∆q}op in terms of the difference } rRpωqJ ´
+JRpωq}op ď δ, we define the function
+γ0pzq :“ 1 ` |z ´ ω|
+|z| ´ K ,
+for which
+}Rpz, r∆qJ ´ JRpz, ∆q}op ď γ0pzq2}Rpω, r∆qJ ´ JRpω, r∆q}op
+23
+
+holds, as proved (in more general form) in Lemma 4.5.9 in [29]. Since on SK`1 we have and
+|z ´ ω| ď 2K ` 1 hence γ0pzq ď 2pK ` 1q, we find
+}gpr∆qJ ´ Jgp∆q}op “
+›››››››
+1
+2πi
+¿
+SK`1
+gpzq
+´
+Rpz, r∆q ´ Rpz, ∆q
+¯
+dz
+›››››››
+op
+ď 1
+2π
+¿
+SK`1
+|gpzq|
+›››Rpz, r∆q ´ Rpz, ∆q
+›››
+op dz
+ď2pK ` 1q2
+π
+¨
+˚
+˝
+¿
+SK`1
+|gpzq|dz
+˛
+‹‚¨ }Rpω, r∆qJ ´ JRpω, r∆q}op.
+Thus we may set
+Cg :“ 2pK ` 1q2
+π
+¿
+SK`1
+|gpzq|dz.
+G
+Proof of Theorem 4.5
+We state and prove a somewhat more general theorem, incorporating also the case where the identifi-
+cation operators only almost commute with connecting operators or non-linearities. We also would
+like to point out that the constant 2 in Definition 4.3 is arbitrary and any constant larger than one
+would suffice. Much more details are provided in Chapter IV of [29].
+Theorem G.1. Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and
+differing operator sequences DN, r
+DN. Assume R`
+n , L`
+n ď 1 and Bn ď B for some B and n ě 0.
+Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq
+(0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal
+operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting
+operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r
+Gnq ď δ}f}ℓ2pGn´1q. For the common module sequence
+ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r
+Gnq ď δ}f}ℓ2pGnq and that the
+constants Cχn and tCgγn uγnPΓN associated through Lemma 4.4 to the functions of the generalized
+frames in each layer satisfy C2
+χn ` ř
+γnPΓN C2
+gγn ď D2 for some D ą 0. Denote the operator
+that the family tJnun of identification operators induce on FN through concatenation by JN :
+FN Ñ Ă
+FN. Then we have with KN “
+a
+p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and
+KN “
+a
+p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that
+}rΦNpJ0fq ´ JNΦNpfq} Ă
+FN ď KN ¨ δ ¨ }f}ℓ2pG,
+@f P ℓ2pGq.
+If additionally } rPnJn´1f ´ JnPnf}ℓ2p r
+Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r
+Gnq “ 0 holds in
+each layer, then we have KN “
+a
+p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “
+a
+p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.
+If both additional equations hold, we have
+KN “
+a
+p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “
+a
+2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.
+Notation G.2. Let us denote scattering propagators based on operators ∆n and Pn by Un and
+scattering propagators based on operators r∆n and rPn by rUn. Similarly, to Notation D.2 and , let us
+then write (with q “ pγN, ..., γ1q)
+rfq :“ rUnrγns ˝ ... ˝ rU1rγ1spJ0fq.
+24
+
+Proof. By definition we have
+}J ΦNpfq ´ rΦNpJ0fq}2
+Ă
+FN “
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}Jnχnp∆nqρnpPnpfqqq ´ χnpr∆nqρnpPnp rfqqq}2
+ℓ2p r
+Gnq
+˛
+‚
+looooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooon
+“:an
+.
+We define bn :“ ř
+qPΓn }Jnfq ´ rfq}2
+ℓ2p r
+Gnq, with b0 “ }J0f ´ J0f}2
+ℓ2p r
+Gq “ 0 and note
+an ` bn “
+ÿ
+qPΓn´1
+ˆ
+}Jnχnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2
+ℓ2p r
+Gnq
+`
+ÿ
+γnPΓn
+}Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2
+ℓ2p r
+Gnq
+¸
+.
+Using
+1
+2}Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2
+ℓ2p r
+Gnq
+ď}rJngγnp∆nq ´ gγnpr∆nqJnsρnpPnpfqqq}2
+ℓ2p r
+Gnq
+`}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2
+ℓ2p r
+Gnq
+ď}rJngγnp∆nq ´ gγnpr∆nqJns}op ¨ }ρnpPnpfqqq}2
+ℓ2p r
+Gnq
+`}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2
+ℓ2p r
+Gnq,
+and }rgγnp∆nq ´ gγnpr∆nqs}8 ď C2
+gγ ¨ δ2 (c.f. Lemma 4.4), we find
+an ` bn ď2
+ÿ
+qPΓn´1
+˜
+C2
+χn `
+ÿ
+γnPΓn
+Cg2γn
+¸
+pL`
+n R`
+n q2δ2||ρnpPnp rfqqq||2
+ℓ2p r
+Gnq
+`2
+ÿ
+qPΓn´1
+Bn||JnρnpPnpfqqq ´ ρnpPnp rfqqq||2
+ℓ2p r
+Gnq
+ď2
+ÿ
+qPΓn´1
+δ2
+˜
+C2
+χn `
+ÿ
+γnPΓn
+Cg2γn
+¸
+pL`
+n R`
+n q2||ρnpPnp rfqqq||2
+ℓ2p r
+Gnq
+`4B ¨ Bn´1||f||2
+ℓ2pGqδ2 ` 8B ¨ Bn´1||f||2
+ℓ2pGqδ2 ` 8Bbn´1,
+where the second inequality arises from permuting the identification operator Jn through non-linearity
+and connecting operator. Using C2
+χn ` ř
+γnPΓn C2
+γn ď D2, we then infer
+an ď pbn´1 ´ bnq ` r8B ´ 1sbn´1 ` p2D2 ` 12BqBn´1δ2||f||2
+ℓ2pGq.
+If B ď 1
+8, summing over n and using a geometric sum argument yields the desired stability constant.
+Hence let us assume B ą 1
+8. Using similar arguments as before, we find
+bn´1 ďp2D2 ` 12Bqδ2Bn´2||f||2
+ℓ2pGq ` 8Bbn´2
+ď
+˜n´1
+ÿ
+k“1
+8k´1
+¸
+Bn´2p2D2 ` 12Bqδ2||f||2
+ℓ2pGq “ 1
+56p8n ´ 8qp2D2 ` 12qδ2||f||2
+ℓ2pGq.
+In total we find
+N
+ÿ
+n“1
+an
+ď pb0 ´ bNq
+loooomoooon
+ď0
+`p2D2 ` 12BqBn´1δ2||f||2
+ℓ2pGq ` p8B ´ 1qp8n´1 ´ 1q{7Bn´2 ¨ p2D2 ` 12Bqδ2||f||2
+ℓ2pGq
+ďp8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1||f||2
+ℓ2pGq.
+25
+
+If one of the additional equations holds, we find
+an ` bn ď pbn´1 ´ bnq ` r4B ´ 1sbn´1 ` p2D2 ` 4Bqδ2||f||2
+ℓ2pGq.
+and
+bn´1 ďp2D2 ` 4Bqδ2Bn´2||f||2
+ℓ2pGq ` 4Bbn´2
+ď
+˜n´1
+ÿ
+k“1
+4k´1
+¸
+Bn´2p2D2 ` 4qδ2||f||2
+ℓ2pGq “ 1
+12p4n ´ 4qBn´2p2D2 ` 4qδ2||f||2
+ℓ2pGq.
+Arguing as previously yields the desired stability bounds.
+If both additional equations are satisfied the proof is virtually the same as the one for Theorem
+4.2.
+H
+Details on Energy Decay and Truncation Stability
+We first prove the statement made about the relation between truncation stability and energy:
+Lemma H.1. Given the energy WN :“ ř
+pγN,...,γ1qPΓN }UrγNs ˝ ... ˝ Urγ1spfq}2
+ℓ2pGNq stored in
+the network at layer N, we have after extending ΦNpfq by zero to match dimensions with ΦN`1pfq
+that
+}ΦNpfq ´ ΦN`1pfq}2
+FN`1 ď
+`
+R`
+N`1L`
+N`1
+˘2 BN`1 ¨ WN.
+Proof. We note
+}ΦNpfq ´ ΦN`1pfq}2
+FN`1 “
+ÿ
+pγN´1,...,γ1qPΓN
+}VN`1 ˝ UrγNs ˝ ... ˝ Urγ1spfq}2
+ℓ2pGN`1q
+ď
+`
+R`
+N`1L`
+N`1
+˘2 BN`1
+ÿ
+pγN´1,...,γ1qPΓN´1
+}UrγNs ˝ ... ˝ Urγ1spfq}2
+ℓ2pGN`1q.
+In fact one can prove even more:
+Lemma H.2. The energy WN stored in layer N satisfies
+C´
+N}f}2
+ℓ2pGq ď }ΦNpfq}FN ` WNpfq ď C`
+N}f}2
+ℓ2pGq,
+with constants C´
+N :“
+Nś
+i“1
+min
+␣
+1, AipL´
+i R´
+i q2(
+and C`
+N :“
+Nś
+i“1
+max
+␣
+1, BipL`
+i R`
+i q2(
+.
+Proof.
+min
+␣
+1, A1pL´
+1 R´
+1 q2(
+||f||2
+ℓ2pGq
+“A1pL´
+1 R´
+1 q2||f||2
+ℓ2pGq
+“A1||ρ1pP1pfqq||2
+ℓ2pG1q
+ď
+ÿ
+γ1PΓ1
+||gγ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q
+“
+ÿ
+qPΓ1
+||Urqspfq||2
+ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q
+“||χ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q ` W1pfq,
+and similarly
+||χ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q ` W1pfq
+“
+ÿ
+qPΓ1
+||Urqspfq||2
+ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2
+ℓ2pG1q
+ďB1pL`
+1 R`
+1 q2||f||2
+ℓ2pGq.
+26
+
+This yields the starting point for our induction. Now for the inductive step assume the claim holds up
+until layer N ´ 1. Then we have
+C´
+N´1||f||2
+ℓ2pGq ď
+N´1
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqfq||2
+ℓ2pGnq
+˛
+‚` WN´1pfq ď C`
+N´1||f||2
+ℓ2pGq.
+using Notation C.2. We note
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq
+˛
+‚` WN
+“
+N´1
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq
+˛
+‚`
+ÿ
+qPΓN´1
+||χNp∆NqρNpPNpfqqq||2
+ℓ2pGNq
+`
+ÿ
+qPΓN
+||fq||2
+ℓ2pGNq.
+Every path rq P ΓN may be written as q “ pγn, qq, for some γn P Γn and q P ΓN´1. Thus we have
+ÿ
+qPΓN
+||fq||2
+ℓ2pGNq “
+ÿ
+qPΓN´1
+ÿ
+γNPΓN
+||gγN p∆NqPNpρNpfqqq||2
+ℓ2pGNq
+Inserting this in the above equation yields
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq
+˛
+‚` WN
+“
+N´1
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq
+˛
+‚
+`
+ÿ
+qPΓN´1
+˜
+||χNp∆NqρNpPNpfqqq||2
+ℓ2pGn´1q `
+ÿ
+γnPΓN
+||gγN p∆NqPNpρNpfqq||2
+ℓ2pGNq
+¸
+looooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooon
+“:βpfqq
+.
+We have
+pL´
+NR´
+Nq2AN||fq||2
+ℓ2pGn´1q ď βpfqq ď pL`
+NR`
+Nq2BN||fq||2
+ℓ2pGn´1q,
+by the operator frame property. With this we find:
+mint1, pL´
+NR´
+Nq2ANu
+¨
+˝
+N´1
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq
+˛
+‚` WN´1
+˛
+‚
+ď
+N
+ÿ
+n“1
+ÿ
+qPΓn´1
+||χnp∆nqρnpPnpfqqq||2
+ℓ2pGnq ` WN
+ď maxt1, pL´
+NR´
+Nq2BNu
+˜N´1
+ÿ
+n“1
+˜ ÿ
+qPΓn
+||χnp∆nqUrqspfq||2
+ℓ2pGnq
+¸
+` WN´1
+¸
+,
+after unravelling the definition
+WN´1pfq ”
+ÿ
+qPΓN
+||fq||2
+ℓ2pGn´1q.
+The induction hypothesis together with the definition of C˘
+N now yields the claim.
+27
+
+With this we now prove our main theorem concerning energy decay.
+Theorem H.3. Let Φ8 be a generalized graph scattering transform based on a an operator sequence
+D8 “ pPn, ∆nq8
+n“1 and a module sequence Ω8 with each ρnp¨q ě 0. Assume in each layer
+n ě 1 that there is an eigenvector ψn of ∆n with solely positive entries; denote the smallest
+entry by mn :“ miniPGn ψnris. Denote the eigenvalue corresponding to ψn by λn. Quantify the
+’spectral-gap’ opened up at this eigenvalue through neglecting the output-generating function by
+ηn :“ ř
+γnPΓn |gγnpλnq|2 and assume Bnmn ě ηn. We then have
+WNpfq ď C`
+N ¨
+« N
+ź
+n“1
+ˆ
+1 ´
+ˆ
+m2
+n ´ ηn
+Bn
+˙˙ff
+¨ }f}2
+ℓ2pGq.
+Proof. Denote the spectral projection (i.e. the orthogonal projection projecting to the space of
+eigenvectors) onto the eigenspace corresponding to λn by P n
+c .
+Then we have
+WNpfq “
+ÿ
+qPΓN´1
+ÿ
+γNPΓN
+||gγN p∆NqρNpPNpfqqq||2
+ℓ2pGNq
+“
+ÿ
+qPΓN´1
+ÿ
+γNPΓN
+||gγN p∆Nqp1 ´ P N
+c qρNpPNpfqqq||2
+ℓ2pGNq
+`
+ÿ
+qPΓN´1
+ÿ
+γNPΓN
+||gγN p∆NqP N
+c ρNpPNpfqqq||2
+ℓ2pGNq
+ď
+ÿ
+qPΓN´1
+BN||p1 ´ P N
+c qρNpPNpfqqq||2
+ℓ2pGNq
+`
+ÿ
+qPΓN´1
+ηN||P N
+c ρNpPNpfqqq||2
+ℓ2pGNq
+ď
+ÿ
+qPΓN´1
+BN||p1 ´ P N
+c qρNpPNpfqqq||2
+ℓ2pGNq
+`
+ÿ
+qPΓN´1
+ηN||ρNpPNpfqqq||2
+ℓ2pGNq.
+By orthogonality of the spectral projection, we then have
+||p1 ´ P N
+c qρNpPnpfqqq||2
+ℓ2pGNq “ ||ρNpPNpfqqq||2
+ℓ2pGNq ´ ||P N
+c ρNpPnpfqqq||2
+ℓ2pGNq.
+Furthermore, we have
+|xψN, ρNpPnpfqqqyℓ2pGNq|2 ď ||P N
+c ρNpPnpfqqq||2
+ℓ2pGNq
+with equality if the multiplicity of λN is exactly one. With this we find
+||p1 ´ P N
+c qρNpPNpfqqq||2
+ℓ2pGNq “ ||ρNpPNpfqqq||2
+ℓ2pGNq ´ ||P N
+c ρNpPNpfqqq||2
+ℓ2pGNq
+ď ||ρNpPNpfqqq||2
+ℓ2pGNq ´ |xψN, ρNpPNpfqqqyℓ2pGNq|2.
+28
+
+Since the image of ρN is contained in R` by assumption, we have
+|xψN, ρNpPNpfqqqyℓ2pGNq|2 “
+ˇˇˇˇˇˇ
+|GN|
+ÿ
+i“1
+ρNpPNpfqqqipψNqiµi
+ˇˇˇˇˇˇ
+2
+ě
+ˇˇˇˇˇˇ
+|GN|
+ÿ
+i“1
+|ρNpPNpfqqqi|µi
+ˇˇˇˇˇˇ
+2
+¨ m2
+N
+ě
+ˇˇˇˇˇˇ
+|GN|
+ÿ
+i“1
+|ρNpPNpfqqqi|2µ2
+i
+ˇˇˇˇˇˇ
+¨ m2
+N
+ě
+ˇˇˇˇˇˇ
+|GN|
+ÿ
+i“1
+|ρNpPNpfqqqi|2µi
+ˇˇˇˇˇˇ
+¨ m2
+N
+ě ||ρNpPNpfqqq||2
+ℓ2pGNq ¨ m2
+N
+Here the second to last inequality follows since in any finite dimensional vector space, the 1-norm is
+larger than the 2-norm (||f||1 ě ||f||2) and all weights are assumed to satisfy µi ě 1. Thus we now
+know
+||p1 ´ P N
+c qρNpPNpfqqq||2
+ℓ2pGNq ď
+`
+1 ´ m2
+N
+˘
+||ρNpPNpfqqq||2
+ℓ2pGNq.
+Inserting this in our estimate for WNpfq we find
+WNpfq ď
+ˆ
+1 ´
+ˆ
+m2
+N ´ ηn
+Bn
+˙˙
+L`
+NR`
+NBN ¨ WN´1pfq
+ď C`
+N
+N
+ź
+n“1
+ˆ
+1 ´
+ˆ
+m2
+N ´ ηn
+Bn
+˙˙
+||f||2
+ℓ2pGq.
+Taking N to infinity, we know that C`
+N converges to something larger than zero by assumption.
+For products of the form
+Nś
+n“0
+p1 ´ qnq with 0 ď qn ă 1 it is a standard exercise to prove that the limit
+is non-zero precisely if the sum over the qn converges. Combining the above result with Lemma H.2,
+we obtain as an immediate Corollary:
+Corollary H.4. In the setting of Theorem 4.6, the generalized scattering transform satisfies Φ´1
+8 p0q “
+t0u if C˘
+N Ñ C˘ for some positive constants C˘ and řN
+n“1pmn ´ ηn{Bnq Ñ 8 as N Ñ 8.
+I
+Stability of Graph Level Feature Aggregation
+I.1
+General non-linear feature aggregation:
+Our main stability theorem for non-linear feature aggregation is as follows:
+Theorem I.1. We have
+}ΨNpfq´ΨNpgq}RN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBn ´ 1s, rBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+k“1
+Bk
+¸ 1
+2
+}f ´h}ℓ2pGq.
+With the conditions and notation of Theorem 4.2 we have
+}ΨNpfq ´ rΨNpfq}RN ď
+b
+2p2N ´ 1q ¨
+b
+pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq.
+29
+
+Additionally, in the setting of Theorem 4.5, assuming that for each n ď N the identification operator
+Jn satisfies
+ˇˇ}Jnf}ℓ1p r
+Gnq{?µ r
+Gn ´}f}ℓ1pGnq{?µGn
+ˇˇ,
+ˇˇ}Jnf}ℓkp r
+Gnq´}f}ℓkpGnq
+ˇˇ ď δ¨K ¨}f}ℓ2pGnq
+(2 ď k ď pn) implies (@f P ℓ2pGq)
+}rΨNpJ0fq ´ ΨNpfq}RN ď
+?
+2 ¨
+b
+K2
+N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.
+Furhermore, under the assumptions of Corollary H.4 Ψ8pfq “ 0 implies f “ 0.
+Proof. Let f, h P ℓ2pGq. To prove the first two claims, it suffices to prove
+}ΨNpfq ´ ΨNphq}RN ď }ΦNpfq ´ ΦNphq}FN ,
+and
+}ΨNpfq ´ rΨNpfq}RN ď }ΦNpfq ´ rΦNpfq}FN .
+Both statements follow immediately, as soon as we have proved
+}N G
+p pfq ´ N G
+p phq}Rp ď }f ´ h}ℓ2pGq
+for arbitrary choices of p and G. To this end we note that for p ě 2 we have }f}ℓppGq ď }f}ℓ2pGq
+by the monotonicity of p-norms, while we have }f}ℓ1pGq ď ?µG ¨ }f}ℓ2pGq by Hölder’s inequality.
+With this we find
+}N G
+p pfq ´ N G
+p phq}2
+Rp “ 1
+p
+˜
+1
+µG
+|}f}ℓ1pGq ´ }h}ℓ1pGq|2 `
+pÿ
+i“2
+|}f}ℓipGq ´ }h}ℓipGq|2
+¸
+ď 1
+p
+˜
+1
+µG
+|}f ´ h}ℓ1pGq|2 `
+pÿ
+i“2
+|}f ´ h}ℓipGq|2
+¸
+ď 1
+p ¨ p ¨ |}f ´ h}ℓ2pGq|2
+“ }f ´ h}ℓ2pGq.
+where we have employed the reverse triangle inequality in the first step.
+To prove the second claim, we note that we have
+}ΨNpfq ´ rΨNpJ0fq}2
+RN
+“
+N
+ÿ
+n“1
+¨
+˚
+˝
+ÿ
+qPΓn´1
+}N Gn
+pn pχnp∆nqρnpPnppfqqqq
+loooooooooooomoooooooooooon
+“:xq
+q ´ N
+r
+Gn
+pn pχnpr∆nqρnpPnp rfqqq
+loooooooooomoooooooooon
+“:rxq
+q}2
+Rpn
+˛
+‹‚
+ď2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}N
+r
+Gn
+pn pJnxqq ´ N
+r
+Gn
+pn prxqq}Rpn
+˛
+‚
+`2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}N
+r
+Gn
+pn pJnxqq ´ N Gn
+pn pxqq}Rpn
+˛
+‚
+“2}J ΦNpfq ´ rΦNpJ0fq}2
+FN
+`2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}N
+r
+Gn
+pn pJnxqq ´ N Gn
+pn pxqq}Rpn
+˛
+‚.
+Thus it remains to bound the last expression. We have
+}N
+r
+Gn
+p
+pJnxqq ´ N Gn
+pn pxqq}Rpn
+“ 1
+pn
+¨
+˝
+ˇˇˇˇˇ
+1
+?µGn
+}f}ℓ1pGq ´
+1
+?µ r
+Gn
+}Jnf}ℓ1p r
+Gq
+ˇˇˇˇˇ
+2
+`
+pn
+ÿ
+i“2
+|}f}ℓipGq ´ }Jnf}ℓip r
+Gq|2
+˛
+‚
+ďK2 ¨ δ2 ¨ }xq}2
+ℓ2pGnq.
+30
+
+By our results of Appendix C and since we assume admissibility, we have
+N
+ÿ
+n“1
+ÿ
+qPΓn´1
+}xq}2
+ℓ2pGnq ď }f}2
+ℓ2pGq.
+Thus in total
+}ΨNpfq ´ rΨNpJ0fq}2
+FN ď 2}J ΦNpfq ´ rΦNpJ0fq}2
+FN ` 2Kδ}f}ℓ2pGq,
+from which our stability claim follows.
+It remains to prove that the assumptions of Corollary H.4 Ψ8pfq “ 0 imply f “ 0. But since
+N G
+p pfq “ 0 implies f “ 0, this is clear.
+I.2
+Low-Pass feature Aggregation
+The main assumption we have in this section is that each operator ∆n (and r∆n) has a simple lowest
+lying eigenvalue equal to zero. We denote the associated eigenvector (determined up to a complex
+phase) by ψ∆n and the associated spectral projection to the lowest lying eigenvalue by P∆n. It acts
+as
+P∆nf ” ψ∆nxψ∆n, fyℓ2pGnq.
+Now we are ready to state our main stability result under these circumstances:
+Theorem I.2. We have
+}Ψ|x¨,¨y|
+N
+pfq´Ψ|x¨,¨y|
+N
+pgq}CN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBn ´ 1s, rBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+k“1
+Bk
+¸ 1
+2
+}f´h}ℓ2pGq.
+With the conditions and notation of Theorem 4.2 and under the additional assumption }pP∆n ´
+P r∆nq}op ď K ¨ δ for n ď N and some K ě 0, we have
+}Ψ|x¨,¨y|
+N
+pfq ´ rΨ|x¨,¨y|
+N
+pfq}CN ď
+?
+2 ¨
+b
+2p2N ´ 1qpmaxtB, 1{2uqN´1 ` K2 ¨ δ ¨ }f}ℓ2pGq.
+In the setting of Theorem 4.5 and under the additional assumption |}P∆nf}ℓ2pGnq ´
+}P r∆nJnf}ℓ2p r
+Gnq| ď Kδ||f||ℓ2pGnq for all f P ℓ2pGnq (n ď N), we have
+}rΨ|x¨,¨y|
+N
+pJ0fq ´ Ψ|x¨,¨y|
+N
+pfq}CN ď
+?
+2 ¨
+b
+K2
+N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.
+Proof. Let f, h P ℓ2pGq. To prove the first claim, it suffices to prove
+}Ψ|x¨,¨y|
+N
+pfq ´ Ψ|x¨,¨y|
+N
+phq}CN ď }ΦNpfq ´ ΦNphq}FN .
+This immediately follows from the fact that for all f P ℓ2pGnq
+|xψ∆n, fyℓ2pGnq|2 ď }ψ∆n}2
+ℓ2pGnq ¨ }f}2
+ℓ2pGnq
+by Hölder’s inequality.
+The next claim we want to prove is that we have for all f P ℓ2pGq
+}Ψ|x¨,¨y|
+N
+pfq ´ rΨ|x¨,¨y|
+N
+pfq}CN ď
+?
+2 ¨
+b
+2p2N ´ 1q ` K2 ¨ δ ¨ }f}ℓ2pGq.
+We note
+31
+
+}Ψ|x¨,¨y|
+N
+pfq ´ rΨ|x¨,¨y|
+N
+pfq}2
+CN
+“
+N
+ÿ
+n“1
+¨
+˚
+˝
+ÿ
+qPΓn´1
+ˇˇˇˇˇˇˇ
+|xψ∆n, χnp∆nqρnpPnpfqqq
+loooooooooomoooooooooon
+“:xq
+yℓ2pGnq| ´ |xψ r∆n, χnpr∆nqρnpPnp rfqqq
+loooooooooomoooooooooon
+rxq
+yℓ2pGnq|
+ˇˇˇˇˇˇˇ
+2˛
+‹‚
+“
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2pGnq
+ˇˇˇ
+2
+˛
+‚
+ď
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}P∆nxq ´ P r∆nrxq}2
+ℓ2pGnq
+˛
+‚
+ď2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}P r∆npxq ´ rxqq}2
+ℓ2pGnq
+˛
+‚` 2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}pP∆n ´ P r∆nqxq}2
+ℓ2pGnq
+˛
+‚
+ď2}ΦNpfq ´ ΦNphq}2
+FN ` 2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}pP∆n ´ P r∆nqxq}2
+ℓ2pGnq
+˛
+‚
+Hence we need to bound the expression "}pP∆n ´ P r∆nqxq}2
+ℓ2pGnq". We note
+}pP∆n ´ P r∆nqxq}2
+ℓ2pGnq ď }pP∆n ´ P r∆nq}op ¨ }xq}2
+ℓ2pGnq
+ď K2 ¨ δ2 ¨ }xq}2
+ℓ2pGnq
+and thus
+}Ψ|x¨,¨y|
+N
+pfq ´ rΨ|x¨,¨y|
+N
+pfq}2
+CN
+ď2}ΦNpfq ´ ΦNphq}2
+FN ` 2K2 ¨ δ2 ¨
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+}χnp∆nqρnpPnppfqqqq}2
+ℓ2pGnq
+˛
+‚
+ď2}ΦNpfq ´ ΦNphq}2
+FN ` 2K2 ¨ δ2 ¨ }f}2
+ℓ2pGq
+and the claim follows.
+Finally we want to prove
+}rΨ|x¨,¨y|
+N
+pJ0fq ´ Ψ|x¨,¨y|
+N
+pfq}CN ď
+?
+2 ¨
+b
+K2
+N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.
+We note
+32
+
+}Ψ|x¨,¨y|2
+N
+pfq ´ rΨ|x¨,¨y|
+N
+pfq}CN
+“
+N
+ÿ
+n“1
+¨
+˚
+˝
+ÿ
+qPΓn´1
+ˇˇˇˇˇˇˇ
+|xψ∆n, χnp∆nqρnpPnppfqqqq
+loooooooooooomoooooooooooon
+“:xq
+yℓ2pGnq| ´ |xψ r∆n, χnpr∆nqρnpPnp rfqqq
+loooooooooomoooooooooon
+rxq
+yℓ2p r
+Gnq|
+ˇˇˇˇˇˇˇ
+2˛
+‹‚
+“
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+ď
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r
+Gnq ` }P r∆nJnxq}ℓ2p r
+Gnq ´ }P r∆nrxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+ď2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P r∆nJnxq}ℓ2p r
+Gnq ´ }P r∆nrxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+`2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+ď2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P r∆nJnxq}ℓ2p r
+Gnq ´ }P r∆nrxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+`2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+ď2}J ΦNpfq ´ rΦNpJ0fq}2
+Ă
+FN ` 2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r
+Gnq
+ˇˇˇ
+2
+˛
+‚
+ď2}J ΦNpfq ´ rΦNpJ0fq}2
+Ă
+FN ` 2
+N
+ÿ
+n“1
+¨
+˝ ÿ
+qPΓn´1
+K2 ¨ δ2}xq}2
+ℓ2pGnq
+˛
+‚
+ď2}J ΦNpfq ´ rΦNpJ0fq}2
+Ă
+FN ` 2K2 ¨ δ2}f}2
+ℓ2pGq.
+which proves the claim.
+In establishing triviality of the ’kernel’, we have to be a tiny bit more careful:
+Theorem I.3. In the setting of of Corollary H.4, assume that in each layer n, the output generating
+function χn of the underlying scattering transform satisfies χnp0q ‰ 0 and χnpλiq “ 0 for ordered
+non-zero eigenvalues λ2 ď ... ď λ|Gn| of the operator ∆n. Then Ψ|x¨,¨y|
+8
+pfq “ 0 implies f “ 0.
+Proof. Under these assumptions, we do not lose any information by projecting to ψ∆n in each
+ℓ2pGnq, since the image of χnp∆nq is already contained in the one-dimensional space generated by
+the lowest lying eigenvector ψ∆n.
+J
+Details on Higher Order Scattering
+Node signals capture information about nodes in isolation. However, one might also want to analyse
+or incorporate information about binary, ternary or even higher order relations between nodes, such
+as distances or angles between nodes representing atoms in a molecule. This can be formalized by
+considering tensorial input signals:
+33
+
+Tensorial input:
+A 2-tensor on a graph G, as it was already utilized in Section 6, is simply an
+element of C|G|ˆ|G| or – equivalently – a map from GˆG to C, since it associates a complex number
+to each element pg1, g2q P G ˆ G. Since G ˆ G is precisely the set of (possible) edges E, we can
+equivalently think of 2-tensors edge-signals. A 3-tensor an element of C|G|ˆ|G|ˆ|G| or equivalently
+a map from G ˆ G ˆ G ” G3 to C. A 4-tensor then is a map from G4 ” G ˆ G ˆ G ˆ G to C
+or equivalenlty an element of C|G|ˆ|G|ˆ|G|ˆ|G| and so forth. Clearly the space of k-tensors forms a
+linear vector space. Addition and scalar multiplication by λ P C are given by
+pf ` λgqi1,...,ik :“ fi1,...,ik ` λgi1,...,ik
+with f and g being k-tensors. For fixed k, we equip the space of k-tensors with an inner product
+according to
+xf, gy “
+|G|
+ÿ
+i1,...,ik“1
+fi1,...,ikgi1,...,ikµi1,...,ik
+and denote the resulting inner-product space by ℓ2pGkq.
+Operators on Spaces of Tensors:
+Since for fixed k the space ℓ2pGkq is simply a |G|k-dimensional
+complex inner product space, there are exist normal operators ∆k : ℓ2pGkq Ñ ℓ2pGkq on this space.
+Note that the k in ∆k signifies on which space this operator acts. It does not signify that an operator
+is raised to the kth power. Setting for example node-weights µi and edge weights µik to one, the
+adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self-adjoint
+operators on ℓ2pG2q, where they act by matrix multiplication.
+Higher order Scattering Transforms:
+We can then follow the recipe laid out Section 3 in con-
+structing kth-order scattering transforms; all that we need are a module sequence ΩN and an operator
+sequence Dk
+N :“ pP k
+n, ∆k
+nqN
+n“1, where now P k
+n : ℓ2pGk
+n´1q Ñ ℓ2pGk
+nq and ∆k
+n : ℓ2pGk
+nq Ñ ℓ2pGk
+nq.
+Figure 7: Schematic Higher Order Scattering Ar-
+chitecture
+To our initial signal f P ℓ2pGkq we first apply
+the connecting operator P k
+1 , yielding a signal rep-
+resentation in ℓ2pGk
+1q. Subsequently, we apply
+the pointwise non-linearity ρ1. Then we apply
+our graph filters tχ1p∆k
+1qu Ťtgγ1p∆k
+1quγ1PΓ1
+to ρ1pP k
+1 pfqq yielding the output V1pfq :“
+χ1p∆k
+1qρ1pP k
+1 pfqq as well as the interme-
+diate hidden representations tU1rγ1spfq
+:“
+gγ1p∆k
+1qρ1pP k
+1 pfqquγ1PΓ1 obtained in the first
+layer. Here we have introduced the one-step
+scattering propagator Unrγns : ℓ2pGk
+n´1q Ñ
+ℓ2pGk
+nq mapping f
+ÞÑ
+gγnp∆nqρnpPnpfqq
+as well as the output generating operator
+Vn
+: ℓ2pGk
+n´1q Ñ ℓ2pGk
+nq mapping f to
+χnp∆k
+nqρnpP k
+npfqq.
+Upon defining the set
+ΓN´1 :“ ΓN´1 ˆ ... ˆ Γ1 of paths of length
+pN ´ 1q terminating in layer N ´ 1 (with Γ0
+taken to be the one-element set) and iterating the
+above procedure, we see that the outputs gener-
+ated in the N th-layer are indexed by paths ΓN´1
+terminating in the previous layer.
+We denote the resulting feature map by Φk
+N and write F k
+N for the corresponding feature space. The
+node-level stability results of the preceding sections then readily translate to higher order scattering
+transforms.
+As the respective proofs are identical to the corresponding results for the node setting, we do not
+repeat them here.
+Theorem J.1. With the notation of Section 4, we have for all f, h P ℓ2pGkq:
+}Φk
+Npfq ´ Φk
+Nphq}2
+F k
+N ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBn ´ 1s, rBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+ℓ“1
+Bℓ
+¸
+}f ´ h}2
+ℓ2pGkq
+34
+
+p3(Pk(.))
+p2(Pk(.))
+ga2
+9b2 (△)
+p3(Pk())
+X2(△)
+ga
+p3(Pk())
+P1(Pk())/gb1(△)、Ip2(Pk())
+qa?
+T
+p3(Pk())
+6
+JC1
+X2(△5)
+p3(Pk())
+(△)
+p2(Pk(.))
+qa2
+9b2 (△k
+p3(Pk())
+X1(△)
+X2(△))Theorem J.2. Let ΦN and rΦN be two scattering transforms based on the same module sequence
+ΩN and operator sequences Dk
+N, r
+Dk
+N with the same connecting operators (P k
+n “ rP k
+n) in each
+layer. Assume R`
+n , L`
+n ď 1 and Bn ď B for some B and n ď N. Assume that the respective
+normal operators satisfy }∆k
+n ´ r∆k
+n}F ď δ for some δ ą 0. Further assume that the functions
+tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants
+satisfying L2
+χn ` ř
+γnPΓn L2
+gγn ď D2 for all n ď N and some D ą 0. Then we have for all
+f P ℓ2pGkq
+}rΦk
+Npfq ´ Φk
+Npfq}FN ď
+b
+2p2N ´ 1q ¨
+b
+pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq
+Theorem J.3. Let Φk
+N, rΦk
+N be higher order scattering transforms based on a common module
+sequence ΩN and differing operator sequences Dk
+N, r
+Dk
+N. Assume R`
+n , L`
+n ď 1 and Bn ď B
+for some B and n ě 0. Assume that there are identification operators Jn : ℓ2pGk
+nq Ñ ℓ2p rGk
+nq,
+rJn : ℓ2p rGk
+nq Ñ ℓ2pGk
+nq (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent,
+the respective normal operators ∆k
+n, r∆k
+n are ω-δ-close as well as bounded (in norm) by K ą 0 and the
+connecting operators satisfy } rP k
+nJn´1f ´ JnP k
+nf}ℓ2p r
+Gknq ď δ}f}ℓ2pGk
+n´1q. For the common module
+sequence ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r
+Gknq ď δ}f}ℓ2pGknq
+and that the constants Cχn and tCgγn uγnPΓN associated through Lemma 4.4 to the functions of the
+generalized frames in each layer satisfy C2
+χn ` ř
+γnPΓN C2
+gγn ď D2 for some D ą 0. Denote the
+operator that the family tJnun of identification operators induce on F k
+N through concatenation by
+JN : F k
+N Ñ Ă
+F k
+N. Then we have with KN “
+a
+p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and
+KN “
+a
+p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that
+}rΦk
+NpJ0fq ´ JNΦk
+Npfq} Ă
+F k
+N ď KN ¨ δ ¨ }f}ℓ2pG,
+@f P ℓ2pGkq.
+If additionally } rP k
+nJn´1f ´ JnP k
+nf}ℓ2p r
+Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r
+Gknq “ 0 holds in
+each layer, then we have KN “
+a
+p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “
+a
+p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.
+If both additional equations hold, we have
+KN “
+a
+p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “
+a
+2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.
+The map N G
+p introduced in (4) can also be adapted to aggregate higher-order tensorial features into
+graph level features: With
+}f}q :“
+˜
+ÿ
+i1,...,ikPG
+|fi1,...,ik|qµi1,...,ik
+¸1{q
+and µGk :“ ř|G|
+i1...ik“1 µi1,...,ik, we define
+N Gk
+p
+pfq “ p}f}ℓ1pGkq{?µGk, }f}ℓ2pGkq, }f}ℓ3pGkq, ..., }f}ℓppGkqqJ{?p.
+Given a feature map Φk
+N with feature space
+FN “ ‘N
+n“1
+`
+ℓ2pGk
+nq
+˘|Γn´1| ,
+we obtain a corresponding map Ψk
+N mapping from ℓ2pGkq to
+RN “ ‘N
+n“1 pRpnq|Γn´1|
+by concatenating Φk
+N with the map that the family of non-linear maps tN pn
+GknuN
+n“1 induces on F N by
+concatenation. The resulting map Ψk
+N again has stability properties analogous to the node level case:
+Theorem J.4. Assuming admissibility, we have
+}Ψk
+Npfq ´ Ψk
+Nphq}RN ď
+˜
+1 `
+N
+ÿ
+n“1
+maxtrBn ´ 1s, rBnpL`
+n R`
+n q2 ´ 1s, 0u
+n´1
+ź
+ℓ“1
+Bℓ
+¸
+}f ´ h}2
+ℓ2pGkq
+35
+
+for all f, h P ℓ2pGq . With the conditions and notation of Theorem J.2 we have
+}Ψk
+Npfq ´ rΨk
+Npfq}RN ď
+b
+2p2N ´ 1q ¨
+b
+pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq.
+Additionally, in the setting of Theorem J.3, assuming that for each n ď N the identification operator
+Jn satisfies
+ˇˇ}Jnf}ℓ1p r
+Gknq{aµ r
+Gkn ´}f}ℓ1pGknq{?µGkn
+ˇˇ,
+ˇˇ}Jnf}ℓrp r
+Gknq´}f}ℓrpGknq
+ˇˇ ď δ¨K¨}f}ℓ2pGknq
+for 2 ď r ď pn implies (@f P ℓ2pGkq)
+}rΨNpJ0fq ´ ΨNpfq}RN ď
+?
+2 ¨
+b
+K2
+N ` K2 ¨ δ ¨ }f}ℓ2pGkq.
+As the proofs here are virtually the same as for the corresponding results in previous sections –
+essentially only replacing G by Gk, we omit a repetition of them here.
+K
+Additional Details on Experiments
+Here we provide additional details on utilized scattering architectures, training procedures, datasets
+and (performance of) other methods our approach is being compared to. Irrespective of task, our
+models are trained on an NVIDIA DGX A100 architecture utilizing between two and eight NVIDIA
+Tesla A100 GPUs with 80GB memory each. Running 10-fold cross validation for the respective
+experiments took at most 71 hours (which was needed for social network graph classification on
+REDDIT-12K).
+K.1
+Social Network Graph Classification
+Datasets:
+The data we are working with is taken from [43]. In particular this work introduced six
+social network datasets extracted from from scientific collaborations (COLLAB), movie collabora-
+tions (IMDB-B, IMDB-M) and Reddit discussion threads (REDDIT-B, REDDIT-5K, REDDIT-12K).
+Data is anonymised and contains no content that might be considered offensive. Each graph carries a
+class label, and the goal is to predict this label. Some basic properties of these datasets are listed in
+Table 3 below.
+Table 3: Social Network Dataset Characteristics
+Attributes:
+COLLAB
+IMDB- B
+IMDB-M
+REDDIT-B
+REDDIT-5K
+REDDIT-12K
+Graphs
+5K
+1K
+1.5K
+2K
+5K
+12K
+Nodes
+372.5K
+19.8K
+19.5K
+859.2K
+2.5M
+4.7M
+Edges
+49.1M
+386.1K
+395.6
+4M
+11.9M
+21.8M
+Maximum Degree
+2k
+540
+352
+12.2K
+8K
+12.2K
+Minimum Degree
+4
+4
+4
+4
+4
+4
+Average Degree
+263
+39
+40
+9
+9
+9
+Target Labels
+3
+2
+3
+2
+5
+11
+Disconnected Graphs
+No
+No
+No
+Yes
+Yes
+Yes
+These datasets contain graph structures, however they don’t contain associated weights or graph
+signals. Having unspecified weights simply means that the adjacency matrix W from which we
+construct the graph Laplacian
+L “ D ´ W
+on which our operator ∆ is based simply has each entry corresponding to an edge set to unity. If
+no edge is present between vertices i and j, the entry Wij is set to zero. It remains to solve the
+problem of the missing input signals. Our strategy is to generate signals reflecting the geometry
+of the underlying graph. We do this by utilizing features that associate to each node a number that
+characterizes its role or importance within its local environment or within the entire graph. We briefly
+describe them here:
+1. Degree: The degree of a node is the number of edges incident at this node.
+2. Eccentricity: For a connected graph, the eccentricity of a node is the maximum distance
+from this node to all other nodes. On a disconnected graph it is not defined.
+36
+
+3. Clustering: For unweighted graphs the clustering cpuq of a node u is the fraction of possible
+triangles through that node that actually exist. It is calculated as
+cpuq “
+2Tpuq
+degpuqpdegpuq ´ 1q.
+4. Number of triangles: The number of triangles containing the given node as a vertex.
+5. Core number: A k-core is a maximal subgraph that only contains nodes of degree k or
+more. The core number of a node is the largest value k of a k-core containing that node.
+6. Clique number: A clique is a subset of vertices of an undirected graph such that every two
+distinct vertices in the clique are adjacent. This input assigns the number of cliques the
+nodes participates in to each node.
+7. Pagerank: This returns the PageRank of the respective nodes in the graph. PageRank
+computes a ranking of the nodes in the graph based on the structure of the edges. Originally
+it was designed as an algorithm to rank web pages.
+For the first three datasets listed in Table 3 we utilize all listed input features. For the latter three
+datasets we have to refrain from using eccentricity as an input signal, as these datasets contain graphs
+that have multiple non-connected graph components.
+Scattering Architecture:
+We chose a generalized scattering architecture of depth N “ 4. As
+normal-operators, we utilize in each layer the un-normalized graph Laplacian L “ D ´ W scaled
+by its largest eigenvalue (∆ “ L{λmaxpLq). Filters are chosen as 1
+2psinpπ{2 ¨ ∆q, rcospπ{2 ¨ ∆q ´
+ψ∆ψJ
+∆s, sinpπ ¨ ∆q, rcospπ ¨ ∆q ´ ψ∆ψJ
+∆sq, which allows to specify the output generating function
+solely by demanding χp0q “ 1 and χpλq “ 0 on all other eigenvalues of ∆. Here ψ∆ is the
+normalized vector of all ones (satisfying ∆ψ∆ “ 0). Connecting operators are chosen as the identity,
+while we set ρně1p¨q “ | ¨ |. We note that for connected graphs, this recovers Architecture I of Fig. 2.
+On disconnected graphs (as they can appear in the REDDIT datasets), we however do not account
+for vectors other than ψ∆ in the lowest-lying eigenspace of the graph Laplacian. This scattering
+architecture is then applied to each of these input signal individually. For each input signal, this
+returns a feature vector with 1 ` 4 ` 16 ` 64 “ 85 entries. These individual feature vectors are
+then concatenated into one final composite feature vector for each graph. Concerning applicable
+theoretical results, we note the following:
+Training Procedure:
+We train RBF kernel support vector classifiers on our composite scattering
+features. We fix ϵ “ 0.1. The hyperparameter γ scaling the exponent is chosen from
+Gpool :“ t0.00001, 0.0001, 0.001, 0.01, 0.1, 1, 10, 100u,
+while we pick the C that controls the error our slack variables introduce among
+Cpool :“ t0.001, 0.01, 0.1, 1, 10, 25, 50, 100, 1000u.
+We chose these parameters in agreement with the choices of [12] to facilitate comparison between
+the two works.
+We could simply implement the training of the RBF-classifier on our composite scattering features
+by dividing each social-network dataset into 10 folds, then iteratively choosing one fold for testing
+and among the remaining 9 folds randomly choosing one for validation (i.e. for tuning the hyperpa-
+rameters). Instead, following [12] (whose code is released under an Apache license and on which
+we partially built), we take a slightly different approach: We still randomly split our dataset into 10
+folds. Among the 10 folds, we iteratively pick one for testing. Say we have picked the nth fold for
+testing. Then there are 9 remaining folds. We iteratively pick the mth
+n (with 1 ď mn ď 9) of the
+remaining 9 folds for choosing hyperparameters. This leaves 8 folds on which we train our model
+for each choice of hyper parameter in Cpool ˆ Gpool. The resulting classifiers are all evaluated on the
+mth
+n fold. The one that performs best is retained as classifier mn. As mn varies between 1 and 9
+(still for fixed n), this yields a set tfmn : 1 ď mn ď 9u of nine classifiers. From these we build the
+classifier fn, whose classification result is obtained from a majority vote among the nine classifiers
+in tfmn : 1 ď mn ď 9u. Then we evaluate the performance of fn on the nth fold to obtain the
+nth estimation of how well our model performs. As n varies from one to ten, we built the mean
+and variance of the performances of the classifiers fn on the nth fold expressed as the percentage of
+correct classifications.
+37
+
+Reference Methods:
+To allow for a comparison of our results to the literature, typical classification
+accuracies for graph algorithms on social network datasets are displayed in Table 1. Following the
+standard format of reporting classification accuracies, they are presented in the format (Accuracy ˘
+standard deviation). If results are not reported for a dataset, we denote this as not available (N/A).
+The first three rows of Table 1 display results for graph kernel methods; namely Weisfeiler-Lehman
+graph kernels (WL, [33]), Graphlet kernels (Graphlet, [34]) and deep graph kernels (DGK, [42]). The
+subsequent rows display results for geometric deep learning algorithms: Deep graph convolutional
+neural networks (DGCNN,[46]), Patchy-san (PSCN (with k=10), [26]), recurrent neural network
+autoencoders (S2S-N2N-PP, [16]) and graph isomorphism networks (GIN [41]). These results are
+taken from [12]. Additionally we compare with P-Poinc [19], which embeds nodes into a hyperbolic
+space (the Poincare ball, to be precise), GSN-e [3] which combines message passing with structural
+features extracted via subgraph isomorphism and WKPI-kC [47] which utilizes a weighted kernel
+within its metric learning framework. The second to last row (GS-SVM [12]) provides a result that is
+also based on a method that combines a static scattering architecture with a support vector machine.
+Its filters are based on graph wavelets built from differences between lazy random walks that have
+propagated at different time scales.
+K.2
+Regression of Quantum Chemical Energies
+Dataset:
+The dataset we consider is the QM7 dataset, introduced in [2, 31]. This dataset contains
+descriptions of 7165 organic molecules, each with up to seven heavy atoms, with all non-hydrogen
+atoms being considered heavy. A molecule is represented by its Coulomb matrix CClmb, whose
+off-diagonal elements
+CClmb
+ij
+“
+ZiZj
+|Ri ´ Rj|
+(10)
+correspond to the Coulomb-repulsion between atoms i and j, while diagonal elements encode a
+polynomial fit of atomic energies to nuclear charge [31]:
+CClmb
+ii
+“ 1
+2Z2.4
+i
+For each atom in any given molecular graph, the individual Cartesian coordinates Ri and the atomic
+charge Zi are also accessible individually. To each molecule an atomization energy - calculated via
+density functional theory - is associated. The objective is to predict this quantity, the performance
+metric is mean absolute error. Numerically, atomization energies are negative numbers in the range
+´600 to ´2200. The associated unit is rkcal/mols.
+Scattering Architecture:
+Off-diagonal entries in the Coulomb Matrix clearly represent an inverse
+distance. A weight of zero can then heuristically be thought of as the inverse distance between two
+infinitely separated atoms. After calculating the degree matrix D associated to C, we obtain the
+graph Laplacian once more as L “ D ´ C and set our normal operator to
+∆ “
+L
+λmaxpLq.
+If we continuously vary the distances in (10), staying clear of zero, then the adjacency matrix
+and hence the graph Laplacian L varies continuously. As long as we avoid complete degeneracy,
+the largest eigenvalue λmaxpLq will remain positive. This implies that our normal operator ∆
+varies continuously under changes of the inter-atomic distances, which implies that our feature
+vector also varies continuously, as distances are changed. Connecting operators are set to the
+identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |. Filters are chosen as psinpπ{2 ¨ ∆q,
+cospπ{2 ¨ ∆q, sinpπ ¨ ∆q, cospπ ¨ ∆qq acting through matrix multiplication. The output generating
+functions are set to the identity as well. Graph level features are aggregated via the map N E
+5 p¨q
+of Section 6; slightly modified to neglect the normalizing factor in the first entry for improved
+convenience in numerical implementability. As weights µij for our second-order feature space are set
+to unity and molecular graphs in QM7 contain at most 23-molecules, we note that ?µG2 ď
+?
+232 “
+23. Going through the proofs of our graph-level stability results, we see that they remain valid after
+multiplying each stability constant by 23. The Coulomb matrix (divided by a factor of 10 as this
+empirically improved performance) is then also utilized as an edge level input signal. Node level
+38
+
+features are obtained by applying the above architecture to the node level information provided by
+the respective atomic charges tZiu on each graph. We aggregate to graph level features using N G
+5
+(cf. Section 5), again neglecting the normalizing factor in the first entry for improved convenience in
+implementing. The network depth is set to N “ 4 in both cases. We then concatenate graph level
+features obtained from node- and edge level input into a composite scattering feature vector.
+Training Procedure:
+The QM7 dataset comes with a precomputed partition into five subsets; each
+containing a representative amount of heavy and light molecules covering the entire complexity range
+of QM7. To allow for 10-fold cross validation, we further dissect each of these subsets into two
+smaller datasets, one containing graphs indexed by an even number, one containing graphs indexed
+by an odd number. On these 10-subsets, we then perform 10-fold cross validation. Among the 10
+folds, we iteratively pick one for testing. Say we have picked the nth fold for testing. Then there
+are 9 remaining folds. We iteratively pick the mth
+n (with 1 ď mn ď 9) of the remaining 9 folds for
+choosing hyperparameters. This leaves 8 folds on which we train our model for each choice of hyper
+parameter in Cpool ˆ Gpool. This yields 8 regression models, which we average to built our final
+predictor for the nth run. This mean absolute error of this predictor is then evaluated on the nth fold
+which was retained for testing. As n varies from one to ten, we built the mean and variance of the
+performances of the generated regression models. We chose log-linear equidistant hyperparameters
+from
+Gpool :“ t0.00003, 0.0003, 0.003, 0.03, 0.3, 3, 30u,
+and
+Cpool :“ t400000, 40000, 4000, 400, 40, 4, 0.4u.
+Reference Methods:
+We comprehensively evaluate our method against 11 popular baselines and
+state of the art approaches. Among these methods are graph convolutional methods such as GraphConv
+[18], Weave [17] or SchNet [32]. MPNN [13] and its variant DMPNN [44] are models considering
+edge features during message passing. AttentiveFP [40] is an extension of the graph attention
+framework, while N-Gram [21] is a pretrained method. Results for these methods as well as for
+GROVER are taken from [30]. PhysChem [45] learns molecular representations via fusing physical
+and chemical information. Deep Tensor Neural Networks (DTNN [39]) are adaptable extensions of the
+Coulomb Matrix featurizer mapping atom numbers to trainable embeddings which are then updated
+based on distance information and other (node-level) atomic features. Finally Path-Augmented Graph
+Transformer Networks (PATGN, [6]) exploit the connectivity structure of the data in a global attention
+mechanism.
+39
+
diff --git a/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bf1e24ba606cc708a209aec0dfef22e3bf79ab7d
--- /dev/null
+++ b/PNFJT4oBgHgl3EQfIiwq/content/tmp_files/load_file.txt
@@ -0,0 +1,1629 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf,len=1628
+page_content='Graph Scattering beyond Wavelet Shackles  Christian Koke  Technical University of Munich &  Ludwig Maximilian University Munich  christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='koke@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='de  Gitta Kutyniok  Ludwig Maximilian University Munich &  University of Tromsø  kutyniok@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='de  Abstract  This work develops a flexible and mathematically sound framework for the design  and analysis of graph scattering networks with variable branching ratios and generic  functional calculus filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Spectrally-agnostic stability guarantees for node- and  graph-level perturbations are derived;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' the vertex-set non-preserving case is treated  by utilizing recently developed mathematical-physics based tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Energy propaga-  tion through the network layers is investigated and related to truncation stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  New methods of graph-level feature aggregation are introduced and stability of  the resulting composite scattering architectures is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Finally, scattering  transforms are extended to edge- and higher order tensorial input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Theoretical  results are complemented by numerical investigations: Suitably chosen scattering  networks conforming to the developed theory perform better than traditional graph-  wavelet based scattering approaches in social network graph classification tasks  and significantly outperform other graph-based learning approaches to regression  of quantum-chemical energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  1  Introduction  Euclidean wavelet scattering networks [22, 4] are deep convolutional architectures where output-  features are generated in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Employed filters are designed rather than learned and derive  from a fixed (tight) wavelet frame, resulting in a tree structured network with constant branching ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Such networks provide state of the art methods in settings with limited data availability and serve  as a mathematically tractable model of standard convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Rigorous  investigations — establishing remarkable invariance- and stability properties of wavelet scattering  networks — were initially carried out in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The extensive mathematical analysis [38] generalized  the term ’scattering network’ to include tree structured networks with varying branching rations and  frames of convolutional filters, thus significantly narrowing the conceptual gap to general CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  With increasing interest in data on graph-structured domains, well performing networks generalizing  Euclidean CNNs to this geometric setting emerged [18, 5, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If efficiently implemented, such graph  convolutional networks (GCNs) replace Euclidean convolutional filters by functional calculus filters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' scalar functions applied to a suitably chosen graph-shift-oprator capturing the geometry of the  underlying graph [18, 14, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Almost immediately, proposals aimed at extending the success story of  Euclidean scattering networks to the graph convolutional setting began appearing: In [48], the authors  utilize dyadic graph wavelets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [14]) based on the non-normalized graph Laplacian resulting  in a norm preserving graph wavelet scattering transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In [10], diffusion wavelets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [8]) are  used to construct a graph scattering transform enjoying spectrum-dependent stability guarantees to  graph level perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For scattering transforms with N layers and K distinct functional calculus  filters, the work [11] derives node-level stability bounds of OpKN{2q and conducts corresponding  numerical experiments choosing diffusion wavelets, monic cubic wavelets [14] and tight Hann  wavelets [35] as filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In [12] the authors, following [8], construct so called geometric wavelets and  establish the expressivity of a scattering transform based on such a frame through extensive numerical  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='11456v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='LG]  26 Jan 2023  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A theoretical analysis of this and a closely related wavelet based scattering transform is  the main focus of [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally, graph-wavelet based scattering transforms have been extended  to the spatio-temporal domain [27], utilized to overcome the problem of oversmoothing in GCNs  [25] and pruned to deal with their exponential (in network depth) increase in needed resources [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Common among all these contributions is the focus on graph wavelets, which are generically  understood to derive in a scale-sampling procedure from a common wavelet generating kernel  function g : R Ñ R satisfying various properties [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Established stability or expressivity properties  — especially to structural perturbations — are then generally linked to the specific choice of the  wavelet kernel g and utilized graph shift operator [10, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This severely limits the diversity of  available filter banks in the design of scattering networks and draws into question their validity as  models for more general GCNs whose filters generically do not derive from a wavelet kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  A primary focus of this work is to provide alleviation in this situation: After reviewing the graph signal  processing setting in Section 2, we introduce a general framework for the construction of (generalized)  graph scattering transforms beyond the wavelet setting in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Section 4 establishes spectrum-  agnostic stability guarantees on the node signal level and for the first time also for graph-level  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To handle the vertex-set non-preserving case, a new ’distance measure’ for operators  capturing the geometry of varying graphs is utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' After providing conditions for energy decay  (with the layers) and relating it to truncation stability, we consider graph level feature aggregation  and higher order inputs in Sections 5 and 6 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Section 7 we then provide numerical  results indicating that general functional calculus filter based scattering is at least as expressive as  standard wavelet based scattering in graph classification tasks and outperforms leading graph neural  network approaches to regression of quantum chemical energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2  Graph Signal Processing  Taking a signal processing approach, we consider signals on graphs as opposed to graph embeddings:  Node-Signals:  Given a graph pG, Eq, we are primarily interested in node-signals, which are  functions from the node-set G to the complex numbers, modelled as elements of C|G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We equip this  space with an inner product according to xf, gy “ ř|G|  i“1 figiµi (with all vertex weights µi ě 1) and  denote the resulting inner product space by ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We forego considering arbitrary inner products on  C|G| solely in the interest of increased readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Functional Calculus Filters:  Our fundamental objects in investigating node-signals will be func-  tional calculus filters based on a normal operator ∆ : ℓ2pGq Ñ ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Prominent examples include  the adjacency matrix W, the degree matrix D, normalized p1 ´ D´ 1  2 WD´ 1  2 q or un-normalized  (L :“ D´W) graph Laplacians Writing normalized eigenvalue-eigenvector pairs of ∆ as pλi, φiq|G|  i“1,  the filter obtained from applying g : C Ñ C is given by gp∆qf “ ř|G|  i“1 gpλiqxφi, fyℓ2pV qφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The  operator we utilize in our numerical investigations of Section 6, is given by L :“ L{λmaxpLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We  divide by the largest eigenvalue to ensure that the spectrum σpL q is contained in the interval r0, 1s,  which aids in the choice of functions from which filters are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Generalized Frames:  We are most interested in filters that arise from a collection of functions  adequately covering the spectrum of the operator to which they are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To this end we call a  collection tgip¨quiPI of functions a generalized frame if it satisfies the generalized frame condition  A ď ř  iPI |gipcq|2 ď B for any c in C for constants A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' B ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As proved in Appendix B, this  condition is sufficient to guarantee that the associated operators form a frame:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If the family tgip¨quiPI of bounded functions  satisfies A ď ř  iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq)  A}f}2  ℓ2pGq ď  ÿ  iPI  }gip∆qf}2  ℓ2pGq ď B}f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Notably, the functions tgiuiPI need not be continuous: In fact, in our numerical implementations, we  will – among other mappings – utilize the function δ0p¨q, defined by δ0p0q “ 1 and δ0pcq “ 0 for  c ‰ 0 as well as a modified cosine, defined by cosp0q “ 0 and cospcq “ cospcq for c ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2  3  The Generalized Graph Scattering Transform  A generalized graph scattering transform is a non-linear map Φ based on a tree structured multilayer  graph convolutional network with constant branching factor in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For an input signal  f P ℓ2pGq, outputs are generated in each layer of such a scattering network, and then concatenated to  form a feature vector in a feature space F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The network is built up from three ingredients:  Connecting Operators:  To allow intermediate signal representations in the ’hidden’ network  layers to be further processed with functional calculus filters based on varying operators, which might  not all be normal for the same choice of node-weights, we allow these intermediate representations  to live in varying graph signal spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In fact, we do not even assume that these signal spaces are  based on a common vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This is done to allow for modelling of recently proposed networks  where input- and ’processing’ graphs are decoupled (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [1, 36]), as well as architectures  incorporating graph pooling [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Instead, we associate one signal space ℓ2pGnq to each layer  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Connecting operators are then (not necessarily linear) operators Pn : ℓ2pGn´1q Ñ ℓ2pGnq  connecting the signal spaces of subsequent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We assume them to be Lipschitz continuous  (}Ppfq ´ Ppgq}ℓ2pGn´1q ď R`}f ´ g}ℓ2pGnqq and triviality preserving (Pp0q “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For our original  node-signal space we also write ℓ2pGq ” ℓ2pG0q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Non-Linearities:  To each layer, we also associate a (possibly) non-linear function ρn : C Ñ C  acting poinwise on signals in ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Similar to connecting operators, we assume ρn preserves zero  and is Lipschitz-continuous with Lipschitz constant denoted by L`  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This definition allows for the  absolute value non-linearity, but also ReLu or – trivially – the identity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Operator Frames:  Beyond these ingredients, the central building block of our scattering  architecture is comprised of a family of functional calculus filters in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' That is, we  assume that in each layer, the node signal space ℓ2pGnq carries a normal operator ∆n and an  associated collection of functions comprised of an output generating function χnp¨q as well  as a filter bank tgγnp¨quγnPΓn indexed by an index set Γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As the network layer n varies (and  in contrast to wavelet-scattering networks) we allow the index set Γn as well as the collection  tχnp¨qu Ťtgγnp¨quγnPΓn of functions to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We only demand that in each layer the functions in the  filter bank together with the output generating function constitute a generalized frame with frame  constants An, Bn ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We refer to the collection of functions ΩN  :“ pρn, tχnp¨qu Ťtgγnp¨quγnPΓnqN  n“1 as a mod-  ule sequence and call DN :“ pPn, ∆nqN  n“1 our operator collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The generalized scattering  transform is then constructed iteratively:  Figure 1: Schematic Scattering Architecture  To our initial signal f P ℓ2pGq we first apply  the connecting operator P1, yielding a signal rep-  resentation in ℓ2pG1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Subsequently, we apply  the pointwise non-linearity ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we apply  our graph filters tχ1p∆1qu Ťtgγ1p∆1quγ1PΓ1  to ρ1pP1pfqq yielding the output V1pfq :“  χ1p∆1qρ1pP1pfqq as well as the intermedi-  ate hidden representations tU1rγ1spfq  :“  gγ1p∆1qρ1pP1pfqquγ1PΓ1 obtained in the first  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Here we have introduced the one-step  scattering propagator Unrγns : ℓ2pGn´1q Ñ  ℓ2pGnq mapping f  ÞÑ  gγnp∆nqρnpPnpfqq  as well as the output generating operator  Vn  : ℓ2pGn´1q Ñ ℓ2pGnq mapping f to  χnp∆nqρnpPnpfqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Upon defining the set  ΓN´1 :“ ΓN´1 ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˆ Γ1 of paths of length  pN ´ 1q terminating in layer N ´ 1 (with Γ0  taken to be the one-element set) and iterating the  above procedure, we see that the outputs gener-  ated in the N th-layer are indexed by paths ΓN´1  terminating in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  3  p3(P3()  (△2)  p2(P2())  qa2  9b2 (△2)  P3(P3()  X2(△2)  gal  P3(P3())  (△2)  IP1(Pi())/gbr (△1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [p2(P2())  ga2  9b2 (2)  m  P3(P3())  l2(G)  gc1  (△1  X2(△2)  p3(P3())  (△2)  p2(P2())  qa2  9b2 (△2  p3(P3())  X1(△1)  X2(△2)  l2(G1)  l2(G2)  l2(G3)Outputs generated in the N th layer are thus given by tVN ˝UrγN´1s˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝Urγ1spfqupγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',γ1qPΓN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Concatenating the features obtained in the various layers of a network with depth N, our full feature  vectors thus live in the feature space  FN “ ‘N  n“1  `  ℓ2pGnq  ˘|Γn´1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (1)  The associated canonical norm is denoted } ¨ }FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For convenience, a brief review of direct sums of  spaces, their associated norms and a discussion of corresponding direct sums of maps is provided in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We denote the hence constructed generalized scattering transform of length N, based  on a module sequence ΩN and operator collection DN by ΦN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In our numerical experiments in Section 7, we consider two particular  instantiations of the above general architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In both cases the  utilized shift-operator is L :“ L{λmaxpLq, node weights satisfy  µi “ 1, the branching ratio in each layer is chosen as 4 and the  depth is set to N “ 4 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The connecting operators are set to the  identity and non-linearities are set to the modulus (|¨|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The two archi-  tectures differ in the utilized filters, which are repeated in each layer  and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Postponing a discussion of other parameter-  choices, we note here that the filters tsinpπ{2¨q, cospπ{2¨qu provide  a high and a low pass filter on the spectrum σpL q Ď r0, 1s, while  tsinpπ¨q, cospπ¨qu provides a spectral refinement of the former two  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The inner two elements of the filter bank in Architecture II  thus separate an input signal into high- and low-lying spectral  Figure 2: Filters of tested Ar-  chitectures  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The outer two act similarly at a higher spectral scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally Architecture I –  utilizing cos and δ0 as introduced Section 2 – prevents the lowest lying spectral information from  propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Instead it is extracted via δ0p¨q in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Note that Id arises from applying the  constant-1 function to L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Normalizations are chosen to generate frames with upper bounds B ž 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  4  Stability Guarantees  In order to produce meaningful signal representations, a small change in input signal should produce  a small change in the output of our generalized scattering transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This property is captured in the  result below, which is proved in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With the notation of Section 3, we have for all f, h P ℓ2pGq:  }ΦNpfq ´ ΦNphq}FN ď  ˜  1 `  N  ÿ  n“1  maxtrBn ´ 1s, rBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  k“1  Bk  ¸ 1  2  }f ´ h}ℓ2pGq  In the case where upper frame bounds Bn and Lipschitz constants L`  n and R`  n are all smaller than or  equal one, this statement reduces to the much nicer inequality:  }ΦNpfq ´ ΦNphq}FN ď }f ´ h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (2)  Below, we always assume R`  n , L`  n ď 1 as this easily achievable through rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We will keep Bn  variable to demonstrate how filter size influences stability results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As for our experimentally tested  architectures (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2), we note for Architecture I that Bn “ 1{2 for all n, so that (2) applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For  Architecture II we have Bn “ 3, which yields a stability constant of  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  1 ` 2 ¨ 3 ` 2 ¨ 32 ` 2 ¨ 33 “ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Similar to other constants derived in this section, this bound is however not necessarily tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Operators capturing graph geometries might only be known approximately in real world tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' if  edge weights are only known to a certain level of precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Hence it is important that our scattering  representation be insensitive to small perturbations in the underlying normal operators in each layer,  which is captured by our next result, proved in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Smallness here is measured in Frobenius  norm } ¨ }F , which for convenience is briefly reviewed in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence  ΩN and operator sequences DN, r  DN with the same connecting operators (Pn “ rPn) in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume that the respective  normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Further assume that the functions  4  ()  ()SO0  cOs()  cos()  sin()  sin()  sin(π)  ()  Id  Architecture I  Architecture IItgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants  satisfying L2  χn ` ř  γnPΓn L2  gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have  }rΦNpfq ´ ΦNpfq}FN ď  b  2p2N ´ 1q ¨  b  pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq  for all f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If B ď 1{2, the stability constant improves to  a  2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The condition B ď 1  2 is e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' satisfied by our Architecture I, but –strictly speaking– we may not  apply Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2, since not all utilized filters are Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Remark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 in Appendix D  however shows, that the above stability result remains applicable for this architecture as long as we  demand that ∆ and r∆ are (potentially rescaled) graph Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For Architecture II we note that  D “ π  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  10{2 and thus the stability constant is given by  a  2p24 ´ 1q ¨  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  33 ¨ π  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  10{2 “ 45π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We are also interested in perturbations that change the vertex set of the graphs in our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  This is important for example in the context of social networks, when passing from nodes representing  individuals to nodes representing (close knit) groups of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To investigate this setting, we  utilize tools originally developed within the mathematical physics community [29]:  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let H and r  H be two finite dimensional Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ∆ and r∆ be normal  operators on these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let J : H Ñ r  H and rJ : r  H Ñ H be linear maps — called identification  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We call the two spaces δ-quasi-unitarily-equivalent (with δ ě 0) if for any f P H and  u P r  H we have  }Jf} r  H ď 2}f}H,  }pJ ´ rJ˚qf} r  H ď δ}f}H,  }f ´ rJJf}H ď δ  b  }f}2  H ` xf, |∆| fyH,  }u ´ J rJu} r  H ď δ  b  }u}2  r  H ` xu, |r∆| uy r  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If, for some w P C the resolvent R :“ p∆ ´ ωq´1 satisfies }p rRJ ´ JRqf} r  H ď δ}f}H for all f P H,  we say that ∆ and r∆ are ω-δ-close with identification operator J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Absolute value |∆| and adjoint rJ˚ of operators are briefly reviewed in  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' While the above definition might seem fairly abstract at  first, it is in fact a natural setting to investigate structural perturbations  as Figure 3 exemplifies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In our current setting, the Hilbert spaces  in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 are node-signal spaces H “ ℓ2pGq, r  H “ ℓ2p rGq  of different graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The notion of ω-δ-closeness is then useful, as  it allows to compare filters defined on different graphs but obtained  from applying the same function to the respective graph-operators:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In the setting of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 let ∆ and r∆ be ω-δ-  close and satisfy }∆}op, }r∆}op ď K for some K ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If g : C Ñ C  is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a  constant Cg ě 0 so that  }gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ  with Cg depending on g, ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  An explicit characterization of Cg together with a proof of this result  is presented in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 is our main tool in establish-  ing our next result, proved in Appendix G, which captures stability  under vertex-set non-preserving perturbations:  Figure 3: Prototypical Exam-  ple of δ-unitary-equivalent  Node  Signal  Spaces  with  p´1q-12δ-close  Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Details in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and  differing operator sequences DN, r  DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq  (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal  operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting  operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r  Gnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For the common module sequence ΩN assume  that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r  Gnq “ 0 and that the constants Cχn and  5  tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 to the functions of the generalized frames in each layer  satisfy C2  χn ` ř  γnPΓN C2  gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denote the operator that the family tJnun of  identification operators induce on FN through concatenation by JN : FN Ñ Ă  FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then, with  KN “  a  p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “  a  2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2:  }rΦNpJ0fq ´ JNΦNpfq} Ă  FN ď KN ¨ δ ¨ }f}ℓ2pG,  @f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The stability result persists with slightly altered stability constants, if identification operators only  almost commute with non-linearities and/or connecting operators, as Appendix G further elucidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 is not applicable to Architecture I, where filters are not all holomorphic, but is directly  applicable to Architecture II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Stability constants can be calculated in terms of D and B as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Beyond these results, stability under truncation of the scattering transform is equally desirable: Given  the energy WN :“ ř  pγN,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',γ1qPΓN }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ Urγ1spfq}2  ℓ2pGNq stored in the network at layer N,  it is not hard to see that after extending ΦNpfq by zero to match dimensions with ΦN`1pfq we have  }ΦNpfq ´ ΦN`1pfq}2  FN`1 ď  `  R`  N`1L`  N`1  ˘2 BN`1 ¨ WN (see Appendix H for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A  bound for WN is then given as follows:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let Φ8 be a generalized graph scattering transform based on a an operator sequence  D8 “ pPn, ∆nq8  n“1 and a module sequence Ω8 with each ρnp¨q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume in each layer n ě 1  that there is an eigenvector ψn of ∆n with solely positive entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' denote the smallest entry by mn :“  miniPGn ψnris and the eigenvalue corresponding to ψn by λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Quantify the ’spectral-gap’ opened up  at this eigenvalue through neglecting the output-generating function by ηn :“ ř  γnPΓn |gγnpλnq|2  and assume Bnmn ě ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We then have (with C`  N :“ śN  i“1 max  ␣  1, BipL`  i R`  i q2(  )  WNpfq ď C`  N ¨  « N  ź  n“1  ˆ  1 ´  ˆ  mn ´ ηn  Bn  ˙˙ff  ¨ }f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (3)  The product in (3) decays if C`  N Ñ C` converges and řN  n“1pmn ´ ηn{Bnq Ñ 8 diverges as  N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The positivity-assumptions on the eigenvectors ψn can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' always be ensured if they are  chosen to lie in the lowest lying eigenspace of a graph Laplacian or normalized graph Laplacian  (irrespective of the connectedness of the underlying graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As an example, we note that if we extend  our Architecture I to infinite depth (recall from Section 3 that we are using the same filters, operators,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' in each layer) we have upon choosing λn “ 0 and ψn to be the constant normalized vector that  ηn “ 0, CN “ 1 and mn “ 1{  a  |G|, for a graph with |G| vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On a graph with 16 vertices, we  then e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' have WN ď p3{4qN}f}2  ℓ2pGq and thus }ΦNpfq ´ ΦN`1pfq}FN`1 ď p3{4qN ¨ }f}2  ℓ2pGq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  As detailed in Appendix H, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6 also implies that under the given assumptions the scattering  transform has trivial ’kernel’ for N Ñ 8, mapping only 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  5  Graph-Level Feature Aggregation  To solve tasks such as graph classification or regression over multiple graphs, we need to represent  graphs of varying sizes in a common feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Given a scattering transform ΦN, we thus  need to find a stability preserving map from the feature space FN to some Euclidean space that is  independent of any vertex set cardinalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Since FN is a large direct sum of smaller spaces (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' (1)),  we simply construct such maps on each summand independently and then concatenate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  General non-linear feature aggregation:  Our main tool in passing to graph-level features is a  non-linear map N G  p : ℓ2pGq Ñ Rp given as  N G  p pfq “  1  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pp}f}ℓ1pGq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µG, }f}ℓ2pGq, }f}ℓ3pGq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', }f}ℓppGqqJ,  (4)  with µG :“ ř  iPG µi and }f}ℓqpGq :“ př  iPG |fi|qµiq1{q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Our inspiration to use this map stems from  the standard case where all µi “ 1: For p ě |G|, the vector |f| “ pp|f1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', |fG|qJ can then be  recovered from N G  p pfq up to permutation of indices [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Hence, employing N G  p (with p ě |G|) to  aggregate node-information into graph-level information, we lose the minimal necessary information  6  about node permutation (clearly N G  p pfq “ N G  p pΠfq for any permutation matrix Π) and beyond that  only information about the complex phase (respectively the sign in the real case) in each entry of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Figure 4: Graph Level Scattering  Given a scattering transform ΦN mapping from ℓ2pGq to  the feature space FN “ ‘N  n“1  `  ℓ2pGnq  ˘|Γn´1|, we ob-  tain a corresponding map ΨN mapping from ℓ2pGq to  RN “ ‘N  n“1 pRpnq|Γn´1| by concatenating the feature  map ΦN with the operator that the family of non-linear  maps tN pn  GnuN  n“1 induces on FN by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Simi-  larly we obtain the map rΨN : ℓ2p rGq Ñ RN by concatenat-  ing the map rΦN : ℓ2p rGq Ñ ‘N  n“1  ´  ℓ2p rGnq  ¯|Γn´1|  with  the operator induced by the family tN pn  r  GnuN  n“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The feature  space RN is completely determined by path-sets ΓN and  used maximal p-norm indices pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It no longer depends  on cardinalities of vertex sets of any graphs, allowing to  compare (signals on) varying graphs with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Most  of the results of the previous sections then readily transfer  to the graph-level-feature setting (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Low-pass feature aggregation:  The spectrum-free aggregation scheme of the previous paragraph  is especially adapted to settings where there are no high-level spectral properties remaining constant  under graph perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' However, many commonly utilized operators, such as normalized and  un-normalized graph Laplacians, have a somewhat ’stable’ spectral theory: Eigenvalues are always  real, non-negative, the lowest-lying eigenvalue equals zero and simple (if the graph is connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In  this section we shall thus assume that each mentioned normal operator ∆n (r∆n) has these spectral  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We denote the lowest lying normalized eigenvector (which is generically determined up  to a complex phase) by ψ∆n and denote by M |x¨,¨y|  Gn  : ℓ2pGnq Ñ C the map given by M |x¨,¨y|  Gn  pfq “  |xψ∆n, fyℓ2pGnq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The absolute value around the inner product is introduced to absorb the phase-  ambiguity in the choice of ψ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Given a scattering transform ΦN mapping from ℓ2pGq to the feature  space FN, we obtain a corresponding map Ψ|x¨,¨y|  N  mapping from ℓ2pGq to CN “ ‘N  n“1C|Γn´1| by  concatenating the feature map ΦN with the operator that the family of maps tM |x¨,¨y|  Gn  uN  n“1 induces on  FN by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As detailed in Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2, this map inherits stability properties in complete  analogy to the discussion of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  6  Higher Order Scattering  Node signals capture information about nodes in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' However, one might be interested in  binary, ternary or even higher order relations between nodes such as distances or angles in graphs  representing molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In this section we focus on binary relations – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' edge level input – as this is  the instantiation we also test in our regression experiment in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 provides more  details and extends these considerations beyond the binary setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We equip the space of edge inputs  with an inner product according to xf, gy “ ř|G|  i,j“1 fijgijµij and denote the resulting inner-product  space by ℓ2pEq with E “ GˆG the set of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Setting e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' node-weights µi and edge weights µik to  one, the adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self-  adjoint operators on ℓ2pEq, where they act by matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Replacing the Gn of Section 3  by En, we can then follow the recipe laid out there in constructing 2nd-order scattering transforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' all  that we need are a module sequence ΩN and an operator sequence D2  N :“ pP 2  n, ∆2  nqN  n“1, where now  P 2  n : ℓ2pEn´1q Ñ ℓ2pEnq and ∆2  n : ℓ2pEnq Ñ ℓ2pEnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We denote the resulting feature map by Φ2  N  and write F 2  N for the corresponding feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The map N G  p introduced in (4) can also be adapted  to aggregate higher-order features into graph level features: With }f}q :“ př  ijPG |fij|qµijq1{q and  µE :“ ř|G|  ij“1 µij, we define N E  p pfq “ p}f}ℓ1pEq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µE, }f}ℓ2pEq, }f}ℓ3pEq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', }f}ℓppEqqJ{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Given a feature map Φ2  N with feature space F 2  N “ ‘N  n“1  `  ℓ2pEnq  ˘|Γn´1|, we obtain a corresponding  7  f pi(Pi()  qc1  (△2)  X1(△1)  p2(P2())  9b2 (△2  X2(△2)  12(G1)  NG1  NG2  P1  P2  RP1map Ψ2  N mapping from ℓ2pEq to RN “ ‘N  n“1 pRpnq|Γn´1| by concatenating ΦE  N with the map that  the family of non-linear maps tN En  pn uN  n“1 induces on F N by concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The stability results of  the preceding sections then readily translate to Φ2  N and Ψ2  N (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Appendix J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  7  Experimental Results  We showcase that even upon selecting the fairly simple Architectures I and II introduced in Section  3 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2), our generalized graph scattering networks are able to outperform both wavelet-  based scattering transforms and leading graph-networks under different circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To aid visual  clarity when comparing results, we colour-code the best-performing method in green, the second-best  performing in yellow and the third-best performing method in orange respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Social Network Graph Classification:  To facilitate contact between our generalized graph scat-  tering networks, and the wider literature, we combine a network conforming to our general theory  namely Architecture I in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2 (as discussed in Section 3 with depth N “ 4, identity as connect-  ing operators and | ¨ |-non-linearities) with the low pass aggregation scheme of Section 5 and a  Euclidean support vector machine with RBF-kernel (GGSN+EK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The choice N “ 4 was made  to keep computation-time palatable, while aggregation scheme and non-linearities were chosen  to facilitate comparison with standard wavelet-scattering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For this hybrid architecture  (GGSN+EK), classification accuracies under the standard choice of 10-fold cross validation on five  common social network graph datasets are compared with performances of popular graph kernel  approaches, leading deep learning methods as well as geometric wavelet scattering (GS-SVM) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  More details are provided in Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As evident from Table 1, our network consistently achieves  higher accuracies than the geometric wavelet scattering transform of [12], with the performance gap  becoming significant on the more complex REDDIT datasets, reaching a relative mean performance  increase of more than 10% on REDDIT-12K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This indicates the liberating power of transcending  the graph wavelet setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' While on comparatively smaller and somewhat simpler datasets there is  a performance gap between our static architecture and fully trainable networks, this gap closes on  more complex datasets: While P-Poinc e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' outperforms our method on IMDB datasets, the roles  are reversed on REDDIT datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On REDDIT-B our approach trails only GIN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' with difference in  accuracies insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On REDDIT-5K our method comes in third, with the gap to the second best  method (GIN) being statistically insignificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On REDDIT-12K we generate state of the art results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Table 1: Classification Accuracies on Social Network Datasets  Method  Classification Accuracies r%s  COLLAB  IMDB- B  IMDB-M  REDDIT-B  REDDIT-5K  REDDIT-12K  WL [33]  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='82˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='45  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='60˘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='16  N/A  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='52˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='01  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='77 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='02  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='57 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='32  Graphlet [34]  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='42˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='43  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='40˘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='95  N/A  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='26˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='34  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='75 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='36  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='98 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='29  DGK [42]  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='00˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='90˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='00˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='30  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10  DGCNN [46]  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='76˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='49  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='03˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='86  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='83˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='85  N/A  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='70 ˘ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='54  N/A  PSCN [26]  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='60˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='15  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='00˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='29  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='23˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='84  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='30˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='58  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='70  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='32 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='42  P-Poinc [19]  N/A  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='86˘4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='26  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='31˘4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='27  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='78˘3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='21  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='71 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='01  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='16 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='41  S2S-N2N-PP [16]  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='75˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='80  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='80˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='70  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='19˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50˘0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='80  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='28 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='47 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10  GSN-e [3]  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3  N/A  N/A  N/A  WKPI-kC[47]  N/A  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4  N/A  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5  GIN [41]  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='90  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10˘5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='10  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='30˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='80  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='40˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='50  N/A  GS-SVM [12]  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='94˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='61  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20˘3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='25  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='73˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='32  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='65˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='94  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='33 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='37  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='23 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='25  GGSN+EK [OURS]  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='34˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='68  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='20˘3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='76  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='47˘2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='27  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='60˘1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='97  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='89 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='24  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='03 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='58  Regression of Quantum Chemical Energies:  In order to showcase the prowess of both our higher  order scattering scheme and our spectrum-agnostic aggregation method of Section 5, we combine  these building blocks into a hybrid architecture which we then apply in combination with kernel  methods (2GGST + EK) to the task of atomization energy regression on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This is a comparatively  small dataset of 7165 molecular graphs, taken from the 970 million strong molecular database GDB-  13 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Each graph in QM7 represents an organic molecule, with nodes corresponding to individual  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Beyond the node-level information of atomic charge, there is also edge level information  characterising interaction strengths between individual nodes/atoms available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This is encoded into so  called Coulomb matrices (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [31] or Appendix K) of molecular graphs, which for us serve a dual  purpose: On the one hand we consider a Coulomb matrix as an edge-level input signal on a given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  8  On the other hand, we also treat it as an adjacency matrix from which we build up a graph Laplacian  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Our normal operator is then chosen as L “ L{λmaxpLq again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Connecting operators are set to  the identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Filters are chosen as psinpπ{2 ¨ L q,  cospπ{2 ¨ L q, sinpπ ¨ L q, cospπ ¨ L qq acting through matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Output generating  functions are set to the identity and depth is N “ 4, so that we essentially recover Architecture II of  Figure 5: Atomization Energy as a Function of pri-  mary Principal Components of Scattering Features  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' now applied to edge-level input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Graph level features are aggregated via the  map N E  5 of Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We chose p “ 5  (and not p " 5) for N E  p to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Generated feature vectors are combined with  node level scattering features obtained from  applying Architecture II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2 to the in-  put signal of atomic charge into composite  feature vectors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' plotted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As is  visually evident, even when reduced to the  low-dimensional subspace of their first three  principal components, the generated scatter-  ing features are able to aptly resolve the atom-  ization energy of the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This aptitude  is also reflected in Table 2, comparing our  approach with leading graph-based learning methods trained with ten-fold cross validation on node  and (depending on the model) edge level information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Our  method is the best performing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We significantly outperform  the next best model (DTNN), producing less than half of its  mean absolute error (MAE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Errors of other methods are at  least one — sometimes two — orders of magnitude greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In  part, this performance discrepancy might be explained by the  hightened suitability of our scattering transform for environ-  ments with somewhat limited training-data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Here  we speculate that the additional performance gap might be ex-  plained by the fact that our graph shift operator ∆ carries the  same information as the Coulomb matrix (a proven molecular  graph descriptor in itself [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally, our filters being  infinite series’ in powers of the underlying normal operator  allows for rapid dispersion of information across underlying  molecular graphs, as opposed to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' the filters in GraphConv  Table 2: Comparison of Methods  Method  MAE [kcal/mol]  AttentiveFP [40]  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8  DMPNN [44]  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8 ˘ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  DTNN [39]  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='9  GraphConv [18]  118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='9 ˘ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  GROVER (base)[30]  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 ˘ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='9  MPNN [13]  113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='0 ˘ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  N-GRAM[21]  125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6 ˘ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5  PAGTN (global) [6]  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8 ˘ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='0  PhysChem [45]  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3  SchNet [32]  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 ˘ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='0  Weave [17]  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6 ˘ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3  GGST+EK [OURS]  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6  2GGST+EK [OURS]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 ˘ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3  or SchNet, which do not incorporate such higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To quantify the effect of including second  order scattering coefficients, we also include the result of performing kernel-regression solely on  first order features generated through Architecture II of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2 (GGST + EK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' While results are still  better than those of all but one leading approach, incorporating higher order scattering improves  performance significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  8  Discussion  Leaving behind the traditional reliance on graph wavelets, we developed a theoretically well founded  framework for the design and analysis of (generalized) graph scattering networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' allowing for  varying branching rations, non-linearities and filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We provided spectrum independent  stability guarantees, covering changes in input signals and for the first time also arbitrary normal  perturbations in the underlying graph-shift-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' After introducing a new framework to quantify  vertex-set non-preserving changes in graph domains, we obtained spectrum-independent stability  guarantees for this setting too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We provided conditions for energy decay and discussed implications  for truncation stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we introduced a new method of graph-level feature aggregation and  extended scattering networks to higher order input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Our numerical experiments showed that  a simple scattering transform conforming to our framework is able to outperform the traditional  graph-wavelet based approach to graph scattering in social network graph classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On  complex datasets our method is also competitive with current fully trainable methods, ouperforming  all competitors on REDDIT-12K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally, higher order graph scattering transforms significantly  outperform current leading graph-based learning methods in predicting atomization energies on QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  A reasonable critique of scattering networks as tractable models for general graph convolutional  9  kcal  mo]  100  600  800  50  3rd eigenvector  1000  0  1200  8  50  1400  ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  100  1600  1800  150  2000  100  400  0  200  0  100  200  200  1st eigenvector  400  600  300  800  2nd eigenvectornetworks is their inability to emulate non-tree-structured network topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' While transcending  the wavelet setting has arguably diminished the conceptual gap between the two architectures, this  structural difference persists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally we note that despite a provided promising example, it is  not yet clear whether the newly introduced graph-perturbation framework can aptly provide stability  guarantees to all reasonable coarse-graining procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Exploring this question is the subject of  ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Broader Impact  We caution against an over-interpretation of established mathematical guarantees: Such guarantees  do not negate biases that may be inherent to utilized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Disclosure of Funding  Christian Koke acknowledges support from the German Research Foundation through the MIMO  II-project (DFG SPP 1798, KU 1446/21-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Gitta Kutyniok acknowledges support from the ONE  Munich Strategy Forum (LMU Munich, TU Munich, and the Bavarian Ministery for Science and  Art), the Konrad Zuse School of Excellence in Reliable AI (DAAD), the Munich Center for Machine  Learning (BMBF) as well as the German Research Foundation under Grants DFG-SPP-2298, KU  1446/31-1 and KU 1446/32-1 and under Grant DFG-SFB/TR 109, Project C09 and the Federal  Ministry of Education and Research under Grant MaGriDo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  References  [1] Uri Alon and Eran Yahav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  On the bottleneck of graph neural networks and its practical  implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Blum and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Reymond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 970 million druglike small molecules for virtual screening in  the chemical universe database GDB-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', 131:8732, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [3] Giorgos Bouritsas, Fabrizio Frasca, Stefanos P Zafeiriou, and Michael Bronstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Improving  graph neural network expressivity via subgraph isomorphism counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' IEEE Transactions on  Pattern Analysis and Machine Intelligence, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [4] Joan Bruna and Stéphane Mallat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Invariant scattering convolution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Pattern Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', 35(8):1872–1886, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [5] Joan Bruna, Wojciech Zaremba, Arthur Szlam, and Yann Lecun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Spectral networks and locally  connected networks on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In International Conference on Learning Representations  (ICLR2014), CBLS, April 2014, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [6] Benson Chen, Regina Barzilay, and Tommi Jaakkola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Path-augmented graph transformer  network, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [7] Paul Civin, Nelson Dunford, and Jacob T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Linear operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' part i: General theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  American Mathematical Monthly, 67:199, 1960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [8] Ronald R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Coifman and Mauro Maggioni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Diffusion wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Applied and Computational  Harmonic Analysis, 21(1):53–94, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Special Issue: Diffusion Maps and Wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [9] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Convolutional neural networks  on graphs with fast localized spectral filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in neural information processing  systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [10] Fernando Gama, Alejandro Ribeiro, and Joan Bruna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Diffusion scattering transforms on graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA,  USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='net, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  10  [11] Fernando Gama, Alejandro Ribeiro, and Joan Bruna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Stability of graph scattering transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In  Hanna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Wallach, Hugo Larochelle, Alina Beygelzimer, Florence d’Alché-Buc, Emily B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Fox,  and Roman Garnett, editors, Advances in Neural Information Processing Systems 32: Annual  Conference on Neural Information Processing Systems 2019, NeurIPS 2019, December 8-14,  2019, Vancouver, BC, Canada, pages 8036–8046, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [12] Feng Gao, Guy Wolf, and Matthew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Hirn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Geometric scattering for graph data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In  Kamalika Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International  Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA,  volume 97 of Proceedings of Machine Learning Research, pages 2122–2131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMLR, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [13] Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Neural  message passing for quantum chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In International conference on machine learning,  pages 1263–1272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [14] David K Hammond, Pierre Vandergheynst, and Rémi Gribonval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Wavelets on graphs via spectral  graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Applied and Computational Harmonic Analysis, 30(2):129–150, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [15] Vassilis N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Ioannidis, Siheng Chen, and Georgios B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Giannakis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Pruned graph scattering  transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In International Conference on Learning Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [16] Yu Jin and Joseph F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' JáJá.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Learning graph-level representations with gated recurrent neural  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' CoRR, abs/1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='07683, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [17] Steven Kearnes, Kevin McCloskey, Marc Berndl, Vijay Pande, and Patrick Riley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Molecular  graph convolutions: moving beyond fingerprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Journal of computer-aided molecular design,  30(8):595–608, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [18] Thomas N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Kipf and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Semi-supervised classification with graph convolutional  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In 5th International Conference on Learning Representations, ICLR 2017, Toulon,  France, April 24-26, 2017, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='net, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [19] Panagiotis Kyriakis, Iordanis Fostiropoulos, and Paul Bogdan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Learning hyperbolic repre-  sentations of topological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In International Conference on Learning Representations,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [20] Junhyun Lee, Inyeop Lee, and Jaewoo Kang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Self-attention graph pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Kamalika  Chaudhuri and Ruslan Salakhutdinov, editors, Proceedings of the 36th International Conference  on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages 3734–  3743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMLR, 09–15 Jun 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [21] Shengchao Liu, Mehmet F Demirel, and Yingyu Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' N-gram graph: Simple unsupervised  representation for graphs, with applications to molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in neural information  processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [22] Stéphane Mallat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Group invariant scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Communications on Pure and Applied Mathematics,  65(10):1331–1398, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [23] Hana Melánová, Bernd Sturmfels, and Rosa Winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Recovery from power sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Experimental  Mathematics, 0(0):1–10, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [24] Barry Simon Michael Reed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Methods of modern mathematical physics, Volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Academic  Press, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [25] Yimeng Min, Frederik Wenkel, and Guy Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Scattering gcn: Overcoming oversmoothness in  graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 33:14498–  14508, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [26] Mathias Niepert, Mohamed Ahmed, and Konstantin Kutzkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Learning convolutional neural  networks for graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Maria Florina Balcan and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Weinberger, editors, Proceedings of  The 33rd International Conference on Machine Learning, volume 48 of Proceedings of Machine  Learning Research, pages 2014–2023, New York, New York, USA, 20–22 Jun 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [27] Chao Pan, Siheng Chen, and Antonio Ortega.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Spatio-temporal graph scattering transform, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  11  [28] Michael Perlmutter, Feng Gao, Guy Wolf, and Matthew Hirn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Understanding graph neural  networks with asymmetric geometric scattering transforms, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [29] Olaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Spectral Analysis on Graph-like Spaces / by Olaf Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Lecture Notes in Mathematics,  2039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Springer Berlin Heidelberg, Berlin, Heidelberg, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' edition, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [30] Yu Rong, Yatao Bian, Tingyang Xu, Weiyang Xie, Ying Wei, Wenbing Huang, and Junzhou  Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Self-supervised graph transformer on large-scale molecular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in Neural  Information Processing Systems, 33:12559–12571, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Rupp, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Tkatchenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Müller, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' von Lilienfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Fast and accurate modeling  of molecular atomization energies with machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Physical Review Letters, 108:058301,  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [32] Kristof Schütt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre  Tkatchenko, and Klaus-Robert Müller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Schnet: A continuous-filter convolutional neural network  for modeling quantum interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in neural information processing systems, 30,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [33] Nino Shervashidze, Pascal Schweitzer, Erik Jan van Leeuwen, Kurt Mehlhorn, and Karsten M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Weisfeiler-lehman graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Journal of Machine Learning Research,  12(77):2539–2561, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [34] Nino Shervashidze, SVN Vishwanathan, Tobias Petri, Kurt Mehlhorn, and Karsten Borgwardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Efficient graphlet kernels for large graph comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In David van Dyk and Max Welling,  editors, Proceedings of the Twelth International Conference on Artificial Intelligence and  Statistics, volume 5 of Proceedings of Machine Learning Research, pages 488–495, Hilton  Clearwater Beach Resort, Clearwater Beach, Florida USA, 16–18 Apr 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [35] David I Shuman, Christoph Wiesmeyr, Nicki Holighaus, and Pierre Vandergheynst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Spectrum-  adapted tight graph wavelet and vertex-frequency frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' IEEE Transactions on Signal Pro-  cessing, 63(16):4223–4235, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [36] Jake Topping, Francesco Di Giovanni, Benjamin Paul Chamberlain, Xiaowen Dong, and  Michael M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Bronstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Understanding over-squashing and bottlenecks on graphs via curvature,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [37] Wihler T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On the hölder continuity of matrix functions for normal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Journal of  inequalities in pure and applied mathematics, 10(4), Dec 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [38] Thomas Wiatowski and Helmut Bölcskei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A mathematical theory of deep convolutional neural  networks for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' IEEE Transactions on Information Theory, 64:1845–1866,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [39] Zhenqin Wu, Bharath Ramsundar, Evan N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Feinberg, Joseph Gomes, Caleb Geniesse, Aneesh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Pappu, Karl Leswing, and Vijay S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Pande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Moleculenet: a benchmark for molecular machine  learning† †electronic supplementary information (esi) available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' see doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1039/c7sc02664a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Chemical Science, 9:513 – 530, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [40] Zhaoping Xiong, Dingyan Wang, Xiaohong Liu, Feisheng Zhong, Xiaozhe Wan, Xutong Li,  Zhaojun Li, Xiaomin Luo, Kaixian Chen, Hualiang Jiang, and Mingyue Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Pushing the  boundaries of molecular representation for drug discovery with the graph attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Journal of Medicinal Chemistry, 63(16):8749–8760, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMID: 31408336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [41] Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' How powerful are graph neural  networks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In 7th International Conference on Learning Representations, ICLR 2019, New  Orleans, LA, USA, May 6-9, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='net, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [42] Pinar Yanardag and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Vishwanathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Deep graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Proceedings of the 21th ACM  SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’15, page  1365–1374, New York, NY, USA, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  12  [43] Pinar Yanardag and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Vishwanathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Deep graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Proceedings of the 21th ACM  SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’15, page  1365–1374, New York, NY, USA, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [44] Kevin Yang, Kyle Swanson, Wengong Jin, Connor Coley, Philipp Eiden, Hua Gao, Angel  Guzman-Perez, Timothy Hopper, Brian Kelley, Miriam Mathea, Andrew Palmer, Volker Settels,  Tommi Jaakkola, Klavs Jensen, and Regina Barzilay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Analyzing learned molecular representa-  tions for property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Journal of Chemical Information and Modeling, 59(8):3370–3388,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PMID: 31361484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [45] Shuwen Yang, Ziyao Li, Guojie Song, and Lingsheng Cai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Deep molecular representation  learning via fusing physical and chemical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Advances in Neural Information  Processing Systems, 34, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [46] Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' An end-to-end deep learning  architecture for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In Sheila A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' McIlraith and Kilian Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Weinberger, editors,  Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence, (AAAI-18), the  30th innovative Applications of Artificial Intelligence (IAAI-18), and the 8th AAAI Symposium  on Educational Advances in Artificial Intelligence (EAAI-18), New Orleans, Louisiana, USA,  February 2-7, 2018, pages 4438–4445.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' AAAI Press, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [47] Qi Zhao and Yusu Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Learning metrics for persistence-based summaries and applications  for graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Advances in Neural Information Processing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  [48] Dongmian Zou and Gilad Lerman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Graph convolutional neural networks via scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Applied  and Computational Harmonic Analysis, 49(3):1046–1074, nov 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Checklist  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s  contributions and scope?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] A discussion of how and in which order the main claims  made in Abstract and Introduction were substantiated within the paper is a main focus  of Section 8  (b) Did you describe the limitations of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] This is a second focus of Section  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (c) Did you discuss any potential negative societal impacts of your work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] This is  part of Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (d) Have you read the ethics review guidelines and ensured that your paper conforms to  them?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If you are including theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (a) Did you state the full set of assumptions of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] Every Theorem  and Lemma is stated in a mathematically precise way, with all underlying assumptions  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally, Appendix A briefly reviews some terminology that might not  be immediately present in every readers mind, but is utilized in order to be able to state  theoretical results in a precise and concise manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (b) Did you include complete proofs of all theoretical results?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] This is the Focus  of Appendices B, C, D, F, G, H and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Results in Appendix J are merely stated, as  statements and corresponding proofs are in complete analogy (in fact almost verbatim  the same) to previously discussed statements and proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If you ran experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (a) Did you include the code, data, and instructions needed to reproduce the main experi-  mental results (either in the supplemental material or as a URL)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] Yes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' please see  the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (b) Did you specify all the training details (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', data splits, hyperparameters, how they  were chosen)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] This is the main focus of Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  13  (c) Did you report error bars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', with respect to the random seed after running experi-  ments multiple times)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] Errors for results of conducted experiments are included  in Table 1 and Table 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (d) Did you include the total amount of compute and the type of resources used (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', type  of GPUs, internal cluster, or cloud provider)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] This is described at the beginning  of Appendix K  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If you are using existing assets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', code, data, models) or curating/releasing new assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (a) If your work uses existing assets, did you cite the creators?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] All utilized datasets  were matched to the papers that introduced them (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Section K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally, we  partially built on code corresponding to [12] which we mentioned in Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (b) Did you mention the license of the assets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] We mentioned in Appendix K that the  code corresponding to [12] is freely available under an Apache License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (c) Did you include any new assets either in the supplemental material or as a URL?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes]  Please see supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (d) Did you discuss whether and how consent was obtained from people whose data you’re  using/curating?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [N/A]  (e) Did you discuss whether the data you are using/curating contains personally identifiable  information or offensive content?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [Yes] For the utilized social network datasets, we  mention in Appendix K that neither personally identifyable data nor content that might  be considered offensive is utilised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If you used crowdsourcing or conducted research with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (a) Did you include the full text of instructions given to participants and screenshots, if  applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [N/A]  (b) Did you describe any potential participant risks, with links to Institutional Review  Board (IRB) approvals, if applicable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [N/A]  (c) Did you include the estimated hourly wage paid to participants and the total amount  spent on participant compensation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' [N/A]  A  Some Concepts in Linear Algebra  In the interest of self-containedness, we provide a brief review of some concepts from linear algebra  utilized in this work that might potentially be considered more advanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Presented results are all  standard;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' a very thorough reference is [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Hilbert Spaces:  To us, a Hilbert space — often denoted by H — is a vector space over the complex  numbers which also has an inner product — often denoted by x¨, ¨yH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Prototypical examples are  given by the Euclidean spaces Cd with inner product xx, yyCd :“ řd  i“1 xiyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Associated to an inner  product is a norm, denoted by } ¨ }H and defined by }x}H :“  a  xx, xyH for x P H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Direct Sums of Spaces:  Given two potentially different Hilbert spaces H and p  H, one can form  their direct sum H ‘ p  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Elements of H ‘ p  H are vectors of the form pa, bq, with a P H and b P p  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Addition and scalar multiplication are defined in the obvious way by  pa, bq ` λpc, dq :“ pa ` λc, b ` λdq  for a, c P H, b, d P p  H and λ P C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The inner product on the direct sum is defined by  xpa, bq, pc, dqyH‘ p  H :“ xa, cyH ` xb, dy p  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  As is readily checked, this implies that the norm } ¨ }H‘ p  H on the direct sum is given by  }pa, bq}2  H‘ p  H :“ }a}2  H ` }b}2  p  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Standard examples of direct sums are again the Euclidean spaces, where one has Cd “ Cn ‘ Cm if  m ` n “ d, as is easily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' One might also consider direct sums with more than two summands,  writing Cd “ ‘d  i“1C for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In fact, one might also consider infinite sums of Hilbert spaces:  14  The space ‘8  i“1Hi is made up of those elements a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q with ai P Hi for which the  norm  }a}2  ‘8  i“1Hi :“  8  ÿ  i“1  }ai}2  Hi  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This means for example that the vector p1, 0, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q is in ‘8  i“1C, while p1, 1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q is  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Direct Sums of Maps:  Suppose we have two collections of Hilbert spaces tHiuΓ  i“1, t r  HiuΓ  i“1 with  Γ P N or Γ “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Suppose further that for each i ď Γ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' i ă Γ) we have a (not necessarily linear)  map Ji : Hi Ñ r  Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then the collection tJiuΓ  i“1 of these ’component’ maps induce a ’composite’  map  J : ‘Γ  i“1Hi ÝÑ ‘Γ  i“1 r  Hi  between the direct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Its value on an element a “ pa1, a2, a3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q P ‘Γ  i“1Hi is defined by  J paq “ pJ1pa1q, J2pa2q, J3pa3q, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q P ‘Γ  i“1 r  Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Strictly speaking, one has to be a bit more careful in the case where Γ “ 8 to ensure that  }J paq}‘8  i“1 r  Hi ‰ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This can however be ensured if we have }Jipaiq} r  Hi ď C}ai}Hi for all  1 ď i and some C independent of all i, since then }J paq}‘8  i“1 r  Hi ď C}a}‘8  i“1Hi ď 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If each Ji is  a linear operator, such a C exists precisely if the operator norms (defined below) of all Ji are smaller  than some constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Operator Norm:  Let J : H Ñ r  H be a linear operator between Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We measure its  ’size’ by what is called the operator norm, denoted by } ¨ }op and defined by  }J}op :“  sup  ψPH,}ψ}H“1  }Aψ} r  H  }ψ}H  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Adjoint Operators  Let J : H Ñ r  H be a linear operator from the Hilbert space H to the Hilbert  space r  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Its adjoint J˚ : r  H Ñ H is an operator mapping in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It is uniquely  determined by demanding that  xJf, uy r  H “ xf, J˚uyH  holds true for arbitrary f P H and u P r  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Normal Operators:  If a linear operator ∆ : H Ñ H maps from and to the same Hilbert space,  we can compare it directly with its adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If ∆∆˚ “ ∆˚∆, we say that the operator ∆ is normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Special instances of normal operators are self-adjoint operators, for which we have the stronger  property ∆ “ ∆˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If an operator is normal, there are unitary maps U : H Ñ H diagonalizing ∆ as  U ˚∆U “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='λnq,  with eigenvalues in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We call the collection of eigenvalues the spectrum σp∆q of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If dim H “ d,  we may write σp∆q “ tλud  i“1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It is a standard exercise to verify that each eigenvalue satisfies  |λi| ď }∆}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Associated to each eigenvalue is an eigenvector φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The collection of all (normalized)  eigenvectors forms an orthonormal basis of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We may then write  ∆f “  dÿ  i“1  λi xφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Resolvent of a (normal) Operator:  Given a normal operator ∆ on some Hilbert space H, we have  that the operator p∆ ´ zq : H Ñ H is invertible precisely if z ‰ σp∆q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In this case we write  Rpz, ∆q “ p∆ ´ zq´1  and call this operator the resolvent of ∆ at z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It can be proved that the norm of the resolvent satisfies  }Rpz, ∆q}op “  1  distpz, σp∆qq,  where distpz, σp∆qq denotes the minimal distance between z and any eigenvalue of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  15  Functional Calculus:  Given a normal operator ∆ : H Ñ H on a Hilbert space of dimension d and  a complex function g : C Ñ C, we can define another normal operator obtained from applying the  function g to ∆ by  gp∆qf “  fÿ  i“1  gpλiqxφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  For example if gp¨q “ | ¨ |, we obtain the absolute value |∆| of ∆ by specifying for all f P H that  |∆|f “  dÿ  i“1  |λi|xφi, fyHφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Similarly we find (if z R σp∆q and for f P H)  1  ∆ ´ z “  dÿ  i“1  1  λi ´ z xφi, fyHφi “ p∆ ´ zq´1 “ Rpz, ∆q  where we think of the left-hand-side as applying a function to ∆, while we think of the right-hand-side  as inverting the operator p∆ ´ zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This now allows us to apply tools from complex analysis also to  operators: If a function g is analytic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' can be expanded into a power series), we have  gpλq “ ´ 1  2πi  ¿  S  gpzq  λ ´ z dz  for any circle S Ď C encircling λ by Cauchy’s integral formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Thus, if we chose S large enough to  encircle the entire spectrum σp∆q, we have  gp∆qf “ ´  dÿ  i“1  1  2πi  ¿  S  gpzq  λi ´ z dzxφi, fyHφi “ ´ 1  2πi  ¿  S  gpzqRpz, λqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Frobenius Norm:  Given a finite dimensional Hilbert space H with inner product x¨, ¨yH, and an  orthonormal basis tφiud  i“1, we define the trace of an operator A : H Ñ H as  TrpAq :“  dÿ  k“1  xφk, AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  It is a standard exercise to show that this is independent of the choice of orthonormal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The  associated Frobenius inner product on the space of operators is then given as  xB, AyF :“ TrpB˚Aq  dÿ  k“1  xφk, B˚AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Hence the Frobenius norm of an operator is determined by  }A}2  F “ TrpA˚Aq “  dÿ  k“1  xφk, A˚AφkyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  It is a standard exercise to verify that we have }A}op ď }A}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Since the trace is independent of the  choice of orthonormal basis, the Frobenius norm is invariant under unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' More  precisely, if U, V : H Ñ H are unitary, we have  }UAV }2  F “ }A}2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Frobenius norms can be used to transfer Lipschitz continuity properties of complex functions to the  setting of functions applied to normal operators:  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let g : C Ñ C be Lipschitz continuous with Lipschitz constant Dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This implies  }gpXq ´ gpY q}F ď Dg ¨ }X ´ Y }F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  for normal operators X, Y on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This proof is taken (almost) verbatim from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For an operator A : H Ñ H denote by Aij  its matrix representation with respect to the orthonormal basis tφiud  i“1:  Aij :“ xφi, AφjyH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We then have  }A}2  F “  dÿ  i,j“1  |Aij|2  as a quick calculation shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let now U, W be unitary (with respect to the inner product x¨, ¨yH)  operators diagonalizing the normal operators X and Y as  V ˚XV “ diagpλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='λnq “: DpXq  W ˚Y W “ diagpµ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µnq “: DpY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Since the Frobenius norm is invariant under unitary transformations we find  }gpXq ´ gpY q||2  F “ ||gpV DpXqV ˚q ´ gpWDpY qW ˚q}2  F  “ }V gpDpXqqV ˚ ´ WgpDpY qqW ˚}2  F  “ }W ˚V gpDpXqq ´ gpDpY qqW ˚V }2  F  “  dÿ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='j“1  |pW ˚V gpDpXqq ´ gpDpY qqW ˚V qij|2  “  dÿ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='j“1  ˇˇˇˇˇ  n  ÿ  k“1  rW ˚V sikrgpDpXqqskj ´ rgpDpY qqsikrW ˚V skj  ˇˇˇˇˇ  2  “  dÿ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='j“1  |rW ˚V sij|2 |gpλjq ´ gpµiq|2  ď  dÿ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='j“1  |rW ˚V sij|2 D2  g|λj ´ µi|2  “ D2  g  dÿ  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='j“1  ˇˇˇˇˇ  n  ÿ  k“1  rW ˚V sikrDpXqskj ´ rDpY qsikrW ˚V skj  ˇˇˇˇˇ  2  “ D2  g}X ´ Y }2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  B  Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ∆ : ℓ2pGq Ñ ℓ2pGq be normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If the family tgip¨quiPI of bounded functions  satisfies A ď ř  iPI |gipcq|2 ď B for all c in the spectrum σp∆q, we have (@f P ℓ2pGq)  A}f}2  ℓ2pGq ď  ÿ  iPI  }gip∆qf}2  ℓ2pGq ď B}f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Writing the normalized eigenvalue-eigenvector sequence of ∆ as pλi, φiq|G|  i“1, we simply note  ÿ  iPI  |G|  ÿ  k“1  |xgipλkqφk, fyℓ2pGq|2 “  |G|  ÿ  k“1  ˜ÿ  iPI  |gipλkq|2  ¸  |xφk, fyℓ2pGq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Now under the assumption, we can estimate the sum in brackets by A from below and by B from  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we need only use Bessel’s (in)equality to prove  A||f||2 ď  ÿ  iPpI  |G|  ÿ  k“1  |xgipλkqφk, fyℓ2pGq|2 ď B||f||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  17  C  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With the notation of Section 3 and setting B0 “ 1, we have:  }ΦNpfq ´ ΦNphq}2  FN ď  ˜  1 `  N  ÿ  n“1  maxtrBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  k“0  BkpR`  k L`  k q2  ¸  }f ´ h}2  ℓ2pGq  To streamline the argumentation let us first introduce some notation:  Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let us denote paths in ΓN as q :“ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', γ1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For f P ℓ2pGq let us write  fq :“ UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ Urγ1spfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' By Definition, we have  }ΦNpfq ´ ΦNpgq}2  FN “  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  }Vnpfqq ´ Vnphqq}2  ℓ2pGnq  ˛  ‚  “  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  }χnp∆nqρnpPnpfqqq ´ χnp∆nqρnpPnphqqq}2  ℓ2pGnq  ˛  ‚  looooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooon  “:an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We proceed in two steps:  Our initial goal is to upper bound an as  an ď BnpL`  n R`  n q2 ¨ bn´1 ´ bn ” pbn´1 ´ bnq `  “  BnpL`  n R`  n q2 ´ 1  ‰  ¨ bn´1  (5)  for bn :“ ř  qPΓn }fq ´ hq}2  ℓ2pGnq with b0 “ }f ´ h}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To achieve this we note that (5) is  equivalent to  an ` bn ď BnpL`  n R`  n q2 ¨ bn´1  which upon unraveling definitions may be written as  ÿ  qPΓn´1  }χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2  ℓ2pGnq `  ÿ  pqPΓn  }fpq ´ hpq}2  ℓ2pGnq  ďBnpL`  n R`  n q2  ÿ  qPΓn´1  }fq ´ hq}2  ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (6)  To establish (6), we note, that in the sum over paths of length n, any pq P Γn can uniquely be written  as pq “ pγn, qq, with the path q P Γn´1 of length pn ´ 1q determined by  pq “ pγn, γn´1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', γ1  looooomooooon  “:q  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  With this we find  ÿ  pqPΓn  }fpq ´ hpq}2  ℓ2pGnq “  ÿ  γnPΓn  ÿ  qPΓn´1  }gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2  ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus we can rewrite the left hand side of (6) as  ÿ  qPΓn´1  }χnp∆nqρnpPnppfqqqq ´ χnp∆nqρnpPnphqq}2  ℓ2pGnq `  ÿ  pqPΓn  }fpq ´ hpq}2  ℓ2pGnq  “  ÿ  qPΓn´1  ˆ  }χnp∆nqρnpPnpfqq ´ χnp∆nqρnpPnphqq}2  ℓ2pGnq  `  ÿ  γnPΓn  }gγnp∆nqρnpPnppfqqqq ´ gγnp∆nqρnpPnphqqq}2  ℓ2pGnq  ¸  “:‹  18  The fact that in each layer the function tχnp¨qu Ťtgγnp¨quγnPΓn form a generalized frame with upper  frame constant Bn implies by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1, that we can further bound this as  ‹ ď Bn  ÿ  qPΓn´1  }ρnpPnpfqq ´ ρnpPnphqq}2  ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Using the Lipschitz continuity of ρn and Pn, we arrive at the desired expression (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Having established that  an ď pbn´1 ´ bnq `  “  BnpL`  n R`  n q2 ´ 1  ‰  ¨ bn´1  holds true, we note that we can establish  bn´1 ď  n´1  ź  k“1  BkpL`  k R`  k q2bn´2  arguing similarly as in the case of (6) by using (for f P ℓ2pGn´1q)  ÿ  γn´1PΓn´1  }gγn´1p∆n´1qf}2  ℓ2pGn´1q ď }χn´1p∆n´1qf}2  ℓ2pGn´1q `  ÿ  γPΓ  }gγn´1p∆n´1qf}2  ℓ2pGn´1q  together with the frame property and Lipschitz continuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We then iterate this inequality and recall  that b0 “ }f ´ h}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Using the fact that  N  ÿ  n“1  pbn´1 ´ bnq “ b0 ´ bN ď b0,  we finally find  }ΦNpfq ´ ΦNphq}2  FN ď  ˜  1 `  N  ÿ  n“1  maxtrBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  k“0  BkpR`  k L`  k q2  ¸  }f ´ h}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  D  Proof or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence  ΩN and operator sequences DN, r  DN with the same connecting operators (Pn “ rPn) in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume that the respective  normal operators satisfy }∆n ´ r∆n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Further assume that the functions  tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants  satisfying L2  χn ` ř  γnPΓn L2  gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have  }rΦNpfq ´ ΦNpfq}FN ď  b  2p2N ´ 1q ¨  b  pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq  for all f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If B ď 1{2, the stability constant improves to  a  2p1 ´ BNq{p1 ´ Bq ¨ D ď 2 ¨ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Notation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let us denote scattering propagators based on operators ∆n and connecting operators  Pn by Un and scattering propagators based on operators r∆n by rUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Similarly, to Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2, let us  then write (with q “ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', γ1q)  rfq :“ rUnrγns ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ rU1rγ1spfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' By definition we have  }ΦNpfq ´ rΦN}2  FN “  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  }χnp∆nqρnpPnppfqqqq ´ χnpr∆nqρnpPnp rfqqq}2  ℓ2pGnq  ˛  ‚  loooooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooooon  “:an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  19  We define bn :“ ř  qPΓn }fq ´ rfq}2  ℓ2pGnq, with b0 “ }f ´ h}2  ℓ2pGq “ 0 and note  an ` bn “  ÿ  qPΓn´1  ˆ  }χnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2  ℓ2pGnq  `  ÿ  γnPΓn  }gγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2  ℓ2pGnq  ¸  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Using (with |a ` b|2 ď 2p|a|2 ` |b|2q)  1  2}gγnp∆nqρnpPnpfqqq ´ gγnpr∆nqρnpPnp rfqqq}2  ℓ2pGnq  ď}rgγnp∆nq ´ gγnpr∆nqsρnpPnpfqqq}2  ℓ2pGnq  `}gγnpr∆nqrρnpPnppfqqqq ´ ρnpPnp rfqqqs}2  ℓ2pGnq  ď}rgγnp∆nq ´ gγnpr∆nqs}2  8 ¨ }ρnpPnpfqqq}2  ℓ2pGnq  `}gγnpr∆nqrρnpPnpfqqq ´ ρnpPnp rfqqqs}2  ℓ2pGnq,  and  }rgγnp∆nq ´ gγnpr∆nqs}2  8 ď }rgγnp∆nq ´ gγnpr∆nqs}2  F ď L2  gγ ¨ δ2  (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1 ), we find  an ` bn ď2  ÿ  qPΓn´1  ˜  L2  χn `  ÿ  γnPΓn  Lg2γn  ¸  pL`  n R`  n q2δ2||ρnpPnpfqqq||2  ℓ2pGnq  `2  ÿ  qPΓn´1  Bn||ρnpPnpfqqq ´ ρnpPnp rfqqq||2  ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Using L2  χn ` ř  γnPΓn L2  γn ď D2, we then infer (using the assumption L`  n , R`  n ď 1)  an ď pbn´1 ´ bnq ` r2B ´ 1sbn´1 ` Bn´12D2δ2||f||ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Now if B ď 1  2, we have  an ď pbn´1 ´ bnq ` Bn´12D2δ2||f||ℓ2pGq  and results of geometric sums leads to the desired bound after summing over n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Hence let us assume B ą 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Using similar arguments as before, we find  bn´1 ďBn´22D2δ2||f||2  ℓ2pGq ` 2Bbn´2 ď Bn´22D2δ2||f||2  ℓ2pGq ` Bn´24D2δ2||f||2  ℓ2pGq ` 4bn´3  ďBn´2  ˜n´1  ÿ  k“1  2k  ¸  D2δ2||f||2  ℓ2pGq “ Bn´2p2n ´ 2qD2δ2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus we now know  an ď 2D2δ2Bn´1||f||2  ℓ2pGq ` r2B ´ 1sp2n ´ 2qD2δ2Bn´2||f||2  ℓ2pGq ` pbn´1 ´ bnq  In total we find  g  f  f  e  N  ÿ  n“1  an ď  b  2p2N ´ 1q ¨  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  BN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq,  where we have estimated the sum over pbn´1 ´ bnq by zero from above again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This establishes the  claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Remark D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To see that this also holds for our Architecture I of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2, we note that the critical step  is establishing that Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1 also applies to δ0 and cos, as defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Here we establish  that  }δ0p∆q ´ δ0pr∆q}F “ 0  20  and  }cosp∆q ´ cospr∆q}F ď Dcos}∆ ´ r∆}F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Indeed, since ∆ and r∆ are (possibly) rescaled graph Laplacians on the same graph, the spectral  projections to their lowest lying eigen space, associated to the eigenvalue λmin “ 0 agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denoting  this spectral projection by P, we have  cosp∆q ´ cospr∆q “ rcosp∆q ´ Ps ´ rcospr∆q ´ Ps “ cosp∆q ´ cospr∆q  and we can apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Similar considerations apply to δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  E  Prototypical Example illustrating ω-δ Closeness and δ-Unitary  Equivalence  To investigate the example of Figure 3, we label the vertices of  the respective graphs as depicted in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We denote the left  graph by G and the right graph by rG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The node-weights on rG are  given as rµi “ 1 for 1 ď i ď 7, while on G the weights are given  as µi “ 1 for 1 ď i ď 5 while µ6 “ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We then consider the  respective un-normalized graph Laplacians ∆ : ℓ2pGq Ñ ℓ2pGq and  r∆ : ℓ2p rGq Ñ ℓ2p rGq, which for a given adjacency matrix W on a  graph signal space ℓ2pGq with node weights tµiui is given as  p∆fqi “ 1  µi  ÿ  j  Wijpfi ´ fjq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Such operators are positive and hence |∆| “ ∆ (similarly for r∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We now need to find operators J : ℓ2pGq Ñ ℓ2p rGq and rJ : ℓ2p rGq Ñ  ℓ2pGq satisfying the conditions of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To construct J, we  define a family tψiu6  i“1 of vectors on ℓ2p rGq as  ψ1 “ p1, 0, 0, 0, 0, 0, 0q, ψ2 “ p0, 1, 0, 0, 0, 0, 0q,  ψ3 “ p0, 0, 1, 0, 0, 0, 0q, ψ4 “ p0, 0, 0, 1, 0, 0, 0q,  ψ5 “ p0, 0, 0, 0, 1, 0, 0q, ψ6 “ p0, 0, 0, 0, 0, 1, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Figure 6: Indexing on the re-  spective graphs  The map J : ℓ2pGq Ñ ℓ2p rGq is then defined as  Jf :“  6ÿ  i“1  fiψi,  for any f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We take rJ : ℓ2p rGq Ñ ℓ2pGq to be its adjoint ( rJ :“ J˚), which determined  explicitly by  p rJuqi “ 1  µi  xψi, uyℓ2p r  Gq  for any u P ℓ2p rGq We shall now first check the conditions for δ-quasi unitary equivalence, which we  list again for convenience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' now adapted to our current setting:  }Jf}ℓ2p r  Gq ď 2}f}ℓ2pGq,  }pJ ´ rJ˚qf}ℓ2p r  Gq ď δ}f}ℓ2pGq,  }f ´ rJJf}2  ℓ2pGq ď δ2 ´  }f}2  ℓ2pGq ` xf, ∆, fyℓ2pGq  ¯  ,  }u ´ J rJu}2  ℓ2p r  Gq ď δ2 ´  }u}2  ℓ2p r  Gq ` xu, r∆ uyℓ2p r  Gq  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We first note that since rJ “ J˚, we have }pJ ´ rJ˚qf}ℓ2p r  Gq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Next we note  }Jf}2  ℓ2p r  Gq “  7ÿ  i“1  |pJfqi|2 “ |f6|2 `  6ÿ  i“1  |fi|2 “  6ÿ  i“1  µi “ }f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  21  Furthermore we note  p rJJfqi “  6ÿ  k“1  fk  1  µi  xψi, ψkyℓ2p r  Gq  loooooooomoooooooon  “δik  “ fi  and hence }f ´ rJJf}2  ℓ2pGq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It remains to control }u ´ J rJu}2  ℓ2p r  Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We note  rJu “ pu1, u2, u3, u4, u5, pu5 ` u6q{2qJ  and thus  J rJu “ pu1, u2, u3, u4, u5, pu6 ` u7q{2, pu6 ` u7q{2qJ,  Which implies  u ´ J rJu “ p0, 0, 0, 0, 0, pu7 ´ u6q{2, pu6 ´ u7q{2qJ,  and thus  }u ´ J rJu}2  ℓ2p r  Gq “ 2|u6 ´ u7|2  4  “ |u6 ´ u7|2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We have  xu, r∆ uyℓ2p r  Gq “ 1  2  dÿ  i,j“1  Ă  Wij|ui ´ uj|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Since Ă  W67 “ 1{δ2 by assumption, we have  }u ´ J rJu}2  ℓ2p r  Gq “ 1  2|u6 ´ u7|2 “ 1  2  δ2  δ2 |u6 ´ u7|2 “ 1  2δ2Ă  W67|u6 ´ u7|2  ď 1  2δ2  dÿ  i,j“1  Ă  Wij|ui ´ uj|2 “ δ2xu, r∆ uyℓ2p r  Gq  ď δ2 ´  }u}2  ℓ2p r  Gq ` xu, r∆ uyℓ2p r  Gq  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus we have proven δ-unitary-equivalence and it remains to establish p´1q-12δ closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Com-  bining Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='15 of [29], instead of bounding }p rRJ ´ JRqf}ℓ2p r  Gq ď  12δ}f}ℓ2pGq directly, we may instead establish that there are operators J1 : ℓ2pGq Ñ ℓ2p rGq,  Ă  J1 : ℓ2p rGq Ñ ℓ2pGq satisfying  }J1f ´ Jf}ℓ2p r  Gq ď δ2 `  }f}ℓ2pGq ` xf, ∆, fyℓ2pGq  ˘  ,  (7)  }Ă  J1u ´ rJu}ℓ2pGq ď δ2 ´  }u}ℓ2p r  Gq ` xf, r∆, uyℓ2p r  Gq  ¯  ,  (8)  and  xJ1f, r∆ uyℓ2p r  Gq “ xf, ∆ Ă  J1uyℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  (9)  We chose J1 “ J and determine Ă  J1 by setting (for (1 ď i ď 6))  pĂ  J1uqi “ ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus (7) is clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For (8) we note that we have  p rJu ´ Ă  J1uq “ p0, 0, 0, 0, 0, pu7 ´ u6q{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus we have  }Ă  J1u ´ rJu}ℓ2pGq “ 1  2|u6 ´ u7|2 ď δ2 ´  }u}2  ℓ2p r  Gq ` xu, r∆ uyℓ2p r  Gq  ¯  as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It remains to establish (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We have  xf, ∆ Ă  J1uyℓ2pGq “  6ÿ  i,j“1  fiWijpui ´ ujq,  22  while we have  xJ1f, r∆ uyℓ2p r  Gq “  6ÿ  i“1  fi ¨ xψi, r∆ uyℓ2p r  Gq  “  5ÿ  i,j“1  fiWijpUj ´ uiq ` f6 ¨ xψ6, r∆ uyℓ2p r  Gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We have (with all node-weights on ℓ2pGq equal to unity)  xψ6, r∆ uyℓ2p r  Gq “ p∆ uq6 ` p∆ uq7 “  ˜ÿ  j  W6jpf6 ´ fjq ` 1  δ2 pf6 ´ f7q  ¸  `  ˆ 1  δ2 pf7 ´ f6q  ˙  “  ˜ÿ  j  W6jpf6 ´ fjq ` 1  δ2 pf6 ´ f7q  ¸  And thus  xJ1f, r∆ uyℓ2p r  Gq “  6ÿ  i,j“1  fiWijpui ´ ujq “ xf, ∆ Ă  J1uyℓ2pGq  which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  F  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4  Lemma F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In the setting of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 let ∆ and r∆ be ω-δ-close and satisfy }∆}op, }r∆}op ď K  for some K ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If g : C Ñ C is holomorphic on the disk BK`1p0q of radius pK ` 1q, there is a  constant Cg ě 0 so that  }gpr∆qJ ´ Jgp∆q}op ď Cg ¨ δ  with Cg depending on g , ω and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Without loss of generality, let us assume that K ą |ω|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let us denote the circle of radius r in  C by Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For any holomorphic function g and (normal) operator ∆ whose spectrum is enclosed by  the circle Sr, we can express the operator gp∆q as  gp∆q “ ´ 1  2πi  ¿  Sr  gpzq  ∆ ´ z dz  as discussed in Appendix A (see also [7] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Note that in our case the resolvent  Rpz, ∆q “ p∆ ´ zq´1 is well defined for |z| ě K, since with our assumptions all eigenvalues are  within the circle of radius K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally note that we have  distpz, σp∆qq ě distpz, SKq “ |z| ´ K  if |z| ě K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The same holds true after replacing ∆ with r∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Since for any normal operator ∆ we have  }Rpz, ∆q}op “ 1{distpz, σp∆qq,  we find  |Rpz, r∆q}op, }Rpz, ∆q}op ď 1{p|z| ´ Kq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  To quantify the difference }Rpz, r∆qJ ´ JRpz, ∆q}op in terms of the difference } rRpωqJ ´  JRpωq}op ď δ, we define the function  γ0pzq :“ 1 ` |z ´ ω|  |z| ´ K ,  for which  }Rpz, r∆qJ ´ JRpz, ∆q}op ď γ0pzq2}Rpω, r∆qJ ´ JRpω, r∆q}op  23  holds, as proved (in more general form) in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='9 in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Since on SK`1 we have and  |z ´ ω| ď 2K ` 1 hence γ0pzq ď 2pK ` 1q, we find  }gpr∆qJ ´ Jgp∆q}op “  ›››››››  1  2πi  ¿  SK`1  gpzq  ´  Rpz, r∆q ´ Rpz, ∆q  ¯  dz  ›››››››  op  ď 1  2π  ¿  SK`1  |gpzq|  ›››Rpz, r∆q ´ Rpz, ∆q  ›››  op dz  ď2pK ` 1q2  π  ¨  ˚  ˝  ¿  SK`1  |gpzq|dz  ˛  ‹‚¨ }Rpω, r∆qJ ´ JRpω, r∆q}op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus we may set  Cg :“ 2pK ` 1q2  π  ¿  SK`1  |gpzq|dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  G  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5  We state and prove a somewhat more general theorem, incorporating also the case where the identifi-  cation operators only almost commute with connecting operators or non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We also would  like to point out that the constant 2 in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3 is arbitrary and any constant larger than one  would suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Much more details are provided in Chapter IV of [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ΦN, rΦN be scattering transforms based on a common module sequence ΩN and  differing operator sequences DN, r  DN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Assume that there are identification operators Jn : ℓ2pGnq Ñ ℓ2p rGnq, rJn : ℓ2p rGnq Ñ ℓ2pGnq  (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent, the respective normal  operators ∆n, r∆n are ω-δ-close as well as bounded (in norm) by K ą 0 and the connecting  operators satisfy } rPnJn´1f ´ JnPnf}ℓ2p r  Gnq ď δ}f}ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For the common module sequence  ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r  Gnq ď δ}f}ℓ2pGnq and that the  constants Cχn and tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 to the functions of the generalized  frames in each layer satisfy C2  χn ` ř  γnPΓN C2  gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denote the operator  that the family tJnun of identification operators induce on FN through concatenation by JN :  FN Ñ Ă  FN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have with KN “  a  p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and  KN “  a  p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that  }rΦNpJ0fq ´ JNΦNpfq} Ă  FN ď KN ¨ δ ¨ }f}ℓ2pG,  @f P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If additionally } rPnJn´1f ´ JnPnf}ℓ2p r  Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r  Gnq “ 0 holds in  each layer, then we have KN “  a  p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “  a  p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If both additional equations hold, we have  KN “  a  p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “  a  2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Notation G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let us denote scattering propagators based on operators ∆n and Pn by Un and  scattering propagators based on operators r∆n and rPn by rUn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Similarly, to Notation D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 and , let us  then write (with q “ pγN, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', γ1q)  rfq :“ rUnrγns ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ rU1rγ1spJ0fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  24  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' By definition we have  }J ΦNpfq ´ rΦNpJ0fq}2  Ă  FN “  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  }Jnχnp∆nqρnpPnpfqqq ´ χnpr∆nqρnpPnp rfqqq}2  ℓ2p r  Gnq  ˛  ‚  looooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooon  “:an  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We define bn :“ ř  qPΓn }Jnfq ´ rfq}2  ℓ2p r  Gnq, with b0 “ }J0f ´ J0f}2  ℓ2p r  Gq “ 0 and note  an ` bn “  ÿ  qPΓn´1  ˆ  }Jnχnp∆nqρnpPnpfqq ´ χnpr∆nqρnpPnp rfqq}2  ℓ2p r  Gnq  `  ÿ  γnPΓn  }Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2  ℓ2p r  Gnq  ¸  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Using  1  2}Jngγnp∆nqρnpPnppfqqqq ´ gγnpr∆nqρnpPnp rfqqq}2  ℓ2p r  Gnq  ď}rJngγnp∆nq ´ gγnpr∆nqJnsρnpPnpfqqq}2  ℓ2p r  Gnq  `}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2  ℓ2p r  Gnq  ď}rJngγnp∆nq ´ gγnpr∆nqJns}op ¨ }ρnpPnpfqqq}2  ℓ2p r  Gnq  `}gγnpr∆nqrJnρnpPnppfqqqq ´ ρnpPnp rfqqqs}2  ℓ2p r  Gnq,  and }rgγnp∆nq ´ gγnpr∆nqs}8 ď C2  gγ ¨ δ2 (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4), we find  an ` bn ď2  ÿ  qPΓn´1  ˜  C2  χn `  ÿ  γnPΓn  Cg2γn  ¸  pL`  n R`  n q2δ2||ρnpPnp rfqqq||2  ℓ2p r  Gnq  `2  ÿ  qPΓn´1  Bn||JnρnpPnpfqqq ´ ρnpPnp rfqqq||2  ℓ2p r  Gnq  ď2  ÿ  qPΓn´1  δ2  ˜  C2  χn `  ÿ  γnPΓn  Cg2γn  ¸  pL`  n R`  n q2||ρnpPnp rfqqq||2  ℓ2p r  Gnq  `4B ¨ Bn´1||f||2  ℓ2pGqδ2 ` 8B ¨ Bn´1||f||2  ℓ2pGqδ2 ` 8Bbn´1,  where the second inequality arises from permuting the identification operator Jn through non-linearity  and connecting operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Using C2  χn ` ř  γnPΓn C2  γn ď D2, we then infer  an ď pbn´1 ´ bnq ` r8B ´ 1sbn´1 ` p2D2 ` 12BqBn´1δ2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If B ď 1  8, summing over n and using a geometric sum argument yields the desired stability constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Hence let us assume B ą 1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Using similar arguments as before, we find  bn´1 ďp2D2 ` 12Bqδ2Bn´2||f||2  ℓ2pGq ` 8Bbn´2  ď  ˜n´1  ÿ  k“1  8k´1  ¸  Bn´2p2D2 ` 12Bqδ2||f||2  ℓ2pGq “ 1  56p8n ´ 8qp2D2 ` 12qδ2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In total we find  N  ÿ  n“1  an  ď pb0 ´ bNq  loooomoooon  ď0  `p2D2 ` 12BqBn´1δ2||f||2  ℓ2pGq ` p8B ´ 1qp8n´1 ´ 1q{7Bn´2 ¨ p2D2 ` 12Bqδ2||f||2  ℓ2pGq  ďp8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  25  If one of the additional equations holds, we find  an ` bn ď pbn´1 ´ bnq ` r4B ´ 1sbn´1 ` p2D2 ` 4Bqδ2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  and  bn´1 ďp2D2 ` 4Bqδ2Bn´2||f||2  ℓ2pGq ` 4Bbn´2  ď  ˜n´1  ÿ  k“1  4k´1  ¸  Bn´2p2D2 ` 4qδ2||f||2  ℓ2pGq “ 1  12p4n ´ 4qBn´2p2D2 ` 4qδ2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Arguing as previously yields the desired stability bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If both additional equations are satisfied the proof is virtually the same as the one for Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  H  Details on Energy Decay and Truncation Stability  We first prove the statement made about the relation between truncation stability and energy:  Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Given the energy WN :“ ř  pγN,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',γ1qPΓN }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ Urγ1spfq}2  ℓ2pGNq stored in  the network at layer N, we have after extending ΦNpfq by zero to match dimensions with ΦN`1pfq  that  }ΦNpfq ´ ΦN`1pfq}2  FN`1 ď  `  R`  N`1L`  N`1  ˘2 BN`1 ¨ WN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We note  }ΦNpfq ´ ΦN`1pfq}2  FN`1 “  ÿ  pγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',γ1qPΓN  }VN`1 ˝ UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ Urγ1spfq}2  ℓ2pGN`1q  ď  `  R`  N`1L`  N`1  ˘2 BN`1  ÿ  pγN´1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',γ1qPΓN´1  }UrγNs ˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˝ Urγ1spfq}2  ℓ2pGN`1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In fact one can prove even more:  Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The energy WN stored in layer N satisfies  C´  N}f}2  ℓ2pGq ď }ΦNpfq}FN ` WNpfq ď C`  N}f}2  ℓ2pGq,  with constants C´  N :“  Nś  i“1  min  ␣  1, AipL´  i R´  i q2(  and C`  N :“  Nś  i“1  max  ␣  1, BipL`  i R`  i q2(  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  min  ␣  1, A1pL´  1 R´  1 q2(  ||f||2  ℓ2pGq  “A1pL´  1 R´  1 q2||f||2  ℓ2pGq  “A1||ρ1pP1pfqq||2  ℓ2pG1q  ď  ÿ  γ1PΓ1  ||gγ1p∆1qρ1pP1pfqq||2  ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2  ℓ2pG1q  “  ÿ  qPΓ1  ||Urqspfq||2  ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2  ℓ2pG1q  “||χ1p∆1qρ1pP1pfqq||2  ℓ2pG1q ` W1pfq,  and similarly  ||χ1p∆1qρ1pP1pfqq||2  ℓ2pG1q ` W1pfq  “  ÿ  qPΓ1  ||Urqspfq||2  ℓ2pG1q ` ||χ1p∆1qρ1pP1pfqq||2  ℓ2pG1q  ďB1pL`  1 R`  1 q2||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  26  This yields the starting point for our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Now for the inductive step assume the claim holds up  until layer N ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have  C´  N´1||f||2  ℓ2pGq ď  N´1  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqfq||2  ℓ2pGnq  ˛  ‚` WN´1pfq ď C`  N´1||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  using Notation C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We note  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq  ˛  ‚` WN  “  N´1  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq  ˛  ‚`  ÿ  qPΓN´1  ||χNp∆NqρNpPNpfqqq||2  ℓ2pGNq  `  ÿ  qPΓN  ||fq||2  ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Every path rq P ΓN may be written as q “ pγn, qq, for some γn P Γn and q P ΓN´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Thus we have  ÿ  qPΓN  ||fq||2  ℓ2pGNq “  ÿ  qPΓN´1  ÿ  γNPΓN  ||gγN p∆NqPNpρNpfqqq||2  ℓ2pGNq  Inserting this in the above equation yields  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq  ˛  ‚` WN  “  N´1  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq  ˛  ‚  `  ÿ  qPΓN´1  ˜  ||χNp∆NqρNpPNpfqqq||2  ℓ2pGn´1q `  ÿ  γnPΓN  ||gγN p∆NqPNpρNpfqq||2  ℓ2pGNq  ¸  looooooooooooooooooooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooooooooooooooooooon  “:βpfqq  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We have  pL´  NR´  Nq2AN||fq||2  ℓ2pGn´1q ď βpfqq ď pL`  NR`  Nq2BN||fq||2  ℓ2pGn´1q,  by the operator frame property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With this we find:  mint1, pL´  NR´  Nq2ANu  ¨  ˝  N´1  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq  ˛  ‚` WN´1  ˛  ‚  ď  N  ÿ  n“1  ÿ  qPΓn´1  ||χnp∆nqρnpPnpfqqq||2  ℓ2pGnq ` WN  ď maxt1, pL´  NR´  Nq2BNu  ˜N´1  ÿ  n“1  ˜ ÿ  qPΓn  ||χnp∆nqUrqspfq||2  ℓ2pGnq  ¸  ` WN´1  ¸  ,  after unravelling the definition  WN´1pfq ”  ÿ  qPΓN  ||fq||2  ℓ2pGn´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The induction hypothesis together with the definition of C˘  N now yields the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  27  With this we now prove our main theorem concerning energy decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let Φ8 be a generalized graph scattering transform based on a an operator sequence  D8 “ pPn, ∆nq8  n“1 and a module sequence Ω8 with each ρnp¨q ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume in each layer  n ě 1 that there is an eigenvector ψn of ∆n with solely positive entries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' denote the smallest  entry by mn :“ miniPGn ψnris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denote the eigenvalue corresponding to ψn by λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Quantify the  ’spectral-gap’ opened up at this eigenvalue through neglecting the output-generating function by  ηn :“ ř  γnPΓn |gγnpλnq|2 and assume Bnmn ě ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We then have  WNpfq ď C`  N ¨  « N  ź  n“1  ˆ  1 ´  ˆ  m2  n ´ ηn  Bn  ˙˙ff  ¨ }f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denote the spectral projection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' the orthogonal projection projecting to the space of  eigenvectors) onto the eigenspace corresponding to λn by P n  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Then we have  WNpfq “  ÿ  qPΓN´1  ÿ  γNPΓN  ||gγN p∆NqρNpPNpfqqq||2  ℓ2pGNq  “  ÿ  qPΓN´1  ÿ  γNPΓN  ||gγN p∆Nqp1 ´ P N  c qρNpPNpfqqq||2  ℓ2pGNq  `  ÿ  qPΓN´1  ÿ  γNPΓN  ||gγN p∆NqP N  c ρNpPNpfqqq||2  ℓ2pGNq  ď  ÿ  qPΓN´1  BN||p1 ´ P N  c qρNpPNpfqqq||2  ℓ2pGNq  `  ÿ  qPΓN´1  ηN||P N  c ρNpPNpfqqq||2  ℓ2pGNq  ď  ÿ  qPΓN´1  BN||p1 ´ P N  c qρNpPNpfqqq||2  ℓ2pGNq  `  ÿ  qPΓN´1  ηN||ρNpPNpfqqq||2  ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  By orthogonality of the spectral projection, we then have  ||p1 ´ P N  c qρNpPnpfqqq||2  ℓ2pGNq “ ||ρNpPNpfqqq||2  ℓ2pGNq ´ ||P N  c ρNpPnpfqqq||2  ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Furthermore, we have  |xψN, ρNpPnpfqqqyℓ2pGNq|2 ď ||P N  c ρNpPnpfqqq||2  ℓ2pGNq  with equality if the multiplicity of λN is exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With this we find  ||p1 ´ P N  c qρNpPNpfqqq||2  ℓ2pGNq “ ||ρNpPNpfqqq||2  ℓ2pGNq ´ ||P N  c ρNpPNpfqqq||2  ℓ2pGNq  ď ||ρNpPNpfqqq||2  ℓ2pGNq ´ |xψN, ρNpPNpfqqqyℓ2pGNq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  28  Since the image of ρN is contained in R` by assumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' we have  |xψN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ρNpPNpfqqqyℓ2pGNq|2 “  ˇˇˇˇˇˇ  |GN|  ÿ  i“1  ρNpPNpfqqqipψNqiµi  ˇˇˇˇˇˇ  2  ě  ˇˇˇˇˇˇ  |GN|  ÿ  i“1  |ρNpPNpfqqqi|µi  ˇˇˇˇˇˇ  2  ¨ m2  N  ě  ˇˇˇˇˇˇ  |GN|  ÿ  i“1  |ρNpPNpfqqqi|2µ2  i  ˇˇˇˇˇˇ  ¨ m2  N  ě  ˇˇˇˇˇˇ  |GN|  ÿ  i“1  |ρNpPNpfqqqi|2µi  ˇˇˇˇˇˇ  ¨ m2  N  ě ||ρNpPNpfqqq||2  ℓ2pGNq ¨ m2  N  Here the second to last inequality follows since in any finite dimensional vector space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' the 1-norm is  larger than the 2-norm (||f||1 ě ||f||2) and all weights are assumed to satisfy µi ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Thus we now  know  ||p1 ´ P N  c qρNpPNpfqqq||2  ℓ2pGNq ď  `  1 ´ m2  N  ˘  ||ρNpPNpfqqq||2  ℓ2pGNq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Inserting this in our estimate for WNpfq we find  WNpfq ď  ˆ  1 ´  ˆ  m2  N ´ ηn  Bn  ˙˙  L`  NR`  NBN ¨ WN´1pfq  ď C`  N  N  ź  n“1  ˆ  1 ´  ˆ  m2  N ´ ηn  Bn  ˙˙  ||f||2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Taking N to infinity, we know that C`  N converges to something larger than zero by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  For products of the form  Nś  n“0  p1 ´ qnq with 0 ď qn ă 1 it is a standard exercise to prove that the limit  is non-zero precisely if the sum over the qn converges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Combining the above result with Lemma H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2,  we obtain as an immediate Corollary:  Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6, the generalized scattering transform satisfies Φ´1  8 p0q “  t0u if C˘  N Ñ C˘ for some positive constants C˘ and řN  n“1pmn ´ ηn{Bnq Ñ 8 as N Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  I  Stability of Graph Level Feature Aggregation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1  General non-linear feature aggregation:  Our main stability theorem for non-linear feature aggregation is as follows:  Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We have  }ΨNpfq´ΨNpgq}RN ď  ˜  1 `  N  ÿ  n“1  maxtrBn ´ 1s, rBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  k“1  Bk  ¸ 1  2  }f ´h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  With the conditions and notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 we have  }ΨNpfq ´ rΨNpfq}RN ď  b  2p2N ´ 1q ¨  b  pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  29  Additionally, in the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5, assuming that for each n ď N the identification operator  Jn satisfies  ˇˇ}Jnf}ℓ1p r  Gnq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µ r  Gn ´}f}ℓ1pGnq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µGn  ˇˇ,  ˇˇ}Jnf}ℓkp r  Gnq´}f}ℓkpGnq  ˇˇ ď δ¨K ¨}f}ℓ2pGnq  (2 ď k ď pn) implies (@f P ℓ2pGq)  }rΨNpJ0fq ´ ΨNpfq}RN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  K2  N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Furhermore, under the assumptions of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 Ψ8pfq “ 0 implies f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let f, h P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To prove the first two claims, it suffices to prove  }ΨNpfq ´ ΨNphq}RN ď }ΦNpfq ´ ΦNphq}FN ,  and  }ΨNpfq ´ rΨNpfq}RN ď }ΦNpfq ´ rΦNpfq}FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Both statements follow immediately, as soon as we have proved  }N G  p pfq ´ N G  p phq}Rp ď }f ´ h}ℓ2pGq  for arbitrary choices of p and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To this end we note that for p ě 2 we have }f}ℓppGq ď }f}ℓ2pGq  by the monotonicity of p-norms, while we have }f}ℓ1pGq ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µG ¨ }f}ℓ2pGq by Hölder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  With this we find  }N G  p pfq ´ N G  p phq}2  Rp “ 1  p  ˜  1  µG  |}f}ℓ1pGq ´ }h}ℓ1pGq|2 `  pÿ  i“2  |}f}ℓipGq ´ }h}ℓipGq|2  ¸  ď 1  p  ˜  1  µG  |}f ´ h}ℓ1pGq|2 `  pÿ  i“2  |}f ´ h}ℓipGq|2  ¸  ď 1  p ¨ p ¨ |}f ´ h}ℓ2pGq|2  “ }f ´ h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  where we have employed the reverse triangle inequality in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  To prove the second claim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' we note that we have  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}ΨNpfq ´ rΨNpJ0fq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='RN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}N Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pχnp∆nqρnpPnppfqqqq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='loooooooooooomoooooooooooon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“:xq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q ´ N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pχnpr∆nqρnpPnp rfqqq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='loooooooooomoooooooooon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“:rxq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='q}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Rpn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‹‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pJnxqq ´ N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn prxqq}Rpn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='`2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pJnxqq ´ N Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pxqq}Rpn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“2}J ΦNpfq ´ rΦNpJ0fq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='FN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='`2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pJnxqq ´ N Gn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='pn pxqq}Rpn  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus it remains to bound the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We have  }N  r  Gn  p  pJnxqq ´ N Gn  pn pxqq}Rpn  “ 1  pn  ¨  ˝  ˇˇˇˇˇ  1  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µGn  }f}ℓ1pGq ´  1  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µ r  Gn  }Jnf}ℓ1p r  Gq  ˇˇˇˇˇ  2  `  pn  ÿ  i“2  |}f}ℓipGq ´ }Jnf}ℓip r  Gq|2  ˛  ‚  ďK2 ¨ δ2 ¨ }xq}2  ℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  30  By our results of Appendix C and since we assume admissibility, we have  N  ÿ  n“1  ÿ  qPΓn´1  }xq}2  ℓ2pGnq ď }f}2  ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Thus in total  }ΨNpfq ´ rΨNpJ0fq}2  FN ď 2}J ΦNpfq ´ rΦNpJ0fq}2  FN ` 2Kδ}f}ℓ2pGq,  from which our stability claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  It remains to prove that the assumptions of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 Ψ8pfq “ 0 imply f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' But since  N G  p pfq “ 0 implies f “ 0, this is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  Low-Pass feature Aggregation  The main assumption we have in this section is that each operator ∆n (and r∆n) has a simple lowest  lying eigenvalue equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We denote the associated eigenvector (determined up to a complex  phase) by ψ∆n and the associated spectral projection to the lowest lying eigenvalue by P∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It acts  as  P∆nf ” ψ∆nxψ∆n, fyℓ2pGnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Now we are ready to state our main stability result under these circumstances:  Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We have  }Ψ|x¨,¨y|  N  pfq´Ψ|x¨,¨y|  N  pgq}CN ď  ˜  1 `  N  ÿ  n“1  maxtrBn ´ 1s, rBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  k“1  Bk  ¸ 1  2  }f´h}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  With the conditions and notation of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 and under the additional assumption }pP∆n ´  P r∆nq}op ď K ¨ δ for n ď N and some K ě 0, we have  }Ψ|x¨,¨y|  N  pfq ´ rΨ|x¨,¨y|  N  pfq}CN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  2p2N ´ 1qpmaxtB, 1{2uqN´1 ` K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In the setting of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5 and under the additional assumption |}P∆nf}ℓ2pGnq ´  }P r∆nJnf}ℓ2p r  Gnq| ď Kδ||f||ℓ2pGnq for all f P ℓ2pGnq (n ď N), we have  }rΨ|x¨,¨y|  N  pJ0fq ´ Ψ|x¨,¨y|  N  pfq}CN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  K2  N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let f, h P ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To prove the first claim, it suffices to prove  }Ψ|x¨,¨y|  N  pfq ´ Ψ|x¨,¨y|  N  phq}CN ď }ΦNpfq ´ ΦNphq}FN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  This immediately follows from the fact that for all f P ℓ2pGnq  |xψ∆n, fyℓ2pGnq|2 ď }ψ∆n}2  ℓ2pGnq ¨ }f}2  ℓ2pGnq  by Hölder’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The next claim we want to prove is that we have for all f P ℓ2pGq  }Ψ|x¨,¨y|  N  pfq ´ rΨ|x¨,¨y|  N  pfq}CN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  2p2N ´ 1q ` K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We note  31  }Ψ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨y|  N  pfq ´ rΨ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨y|  N  pfq}2  CN  “  N  ÿ  n“1  ¨  ˚  ˝  ÿ  qPΓn´1  ˇˇˇˇˇˇˇ  |xψ∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' χnp∆nqρnpPnpfqqq  loooooooooomoooooooooon  “:xq  yℓ2pGnq| ´ |xψ r∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' χnpr∆nqρnpPnp rfqqq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='loooooooooomoooooooooon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='rxq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='yℓ2pGnq|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇˇˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‹‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}P∆nxq ´ P r∆nrxq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}P r∆npxq ´ rxqq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚` 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}pP∆n ´ P r∆nqxq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2}ΦNpfq ´ ΦNphq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='FN ` 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='}pP∆n ´ P r∆nqxq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Hence we need to bound the expression "}pP∆n ´ P r∆nqxq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We note  }pP∆n ´ P r∆nqxq}2  ℓ2pGnq ď }pP∆n ´ P r∆nq}op ¨ }xq}2  ℓ2pGnq  ď K2 ¨ δ2 ¨ }xq}2  ℓ2pGnq  and thus  }Ψ|x¨,¨y|  N  pfq ´ rΨ|x¨,¨y|  N  pfq}2  CN  ď2}ΦNpfq ´ ΦNphq}2  FN ` 2K2 ¨ δ2 ¨  N  ÿ  n“1  ¨  ˝ ÿ  qPΓn´1  }χnp∆nqρnpPnppfqqqq}2  ℓ2pGnq  ˛  ‚  ď2}ΦNpfq ´ ΦNphq}2  FN ` 2K2 ¨ δ2 ¨ }f}2  ℓ2pGq  and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Finally we want to prove  }rΨ|x¨,¨y|  N  pJ0fq ´ Ψ|x¨,¨y|  N  pfq}CN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  K2  N ¨ `K2 ¨ δ ¨ }f}ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We note  32  }Ψ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨y|2  N  pfq ´ rΨ|x¨,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨y|  N  pfq}CN  “  N  ÿ  n“1  ¨  ˚  ˝  ÿ  qPΓn´1  ˇˇˇˇˇˇˇ  |xψ∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' χnp∆nqρnpPnppfqqqq  loooooooooooomoooooooooooon  “:xq  yℓ2pGnq| ´ |xψ r∆n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' χnpr∆nqρnpPnp rfqqq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='loooooooooomoooooooooon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='rxq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='yℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇˇˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‹‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='“  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nrxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq ` }P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq ´ }P r∆nrxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq ´ }P r∆nrxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='`2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq ´ }P r∆nrxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='`2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Ă  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='FN ` 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ}P∆nxq}ℓ2pGnq ´ }P r∆nJnxq}ℓ2p r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Gnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ˇˇˇ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Ă  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='FN ` 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='n“1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='¨  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˝ ÿ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='qPΓn´1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='K2 ¨ δ2}xq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGnq  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='˛  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='‚  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ď2}J ΦNpfq ´ rΦNpJ0fq}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='Ă  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='FN ` 2K2 ¨ δ2}f}2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ℓ2pGq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  which proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  In establishing triviality of the ’kernel’, we have to be a tiny bit more careful:  Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In the setting of of Corollary H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4, assume that in each layer n, the output generating  function χn of the underlying scattering transform satisfies χnp0q ‰ 0 and χnpλiq “ 0 for ordered  non-zero eigenvalues λ2 ď .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ď λ|Gn| of the operator ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then Ψ|x¨,¨y|  8  pfq “ 0 implies f “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Under these assumptions, we do not lose any information by projecting to ψ∆n in each  ℓ2pGnq, since the image of χnp∆nq is already contained in the one-dimensional space generated by  the lowest lying eigenvector ψ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  J  Details on Higher Order Scattering  Node signals capture information about nodes in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' However, one might also want to analyse  or incorporate information about binary, ternary or even higher order relations between nodes, such  as distances or angles between nodes representing atoms in a molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This can be formalized by  considering tensorial input signals:  33  Tensorial input:  A 2-tensor on a graph G, as it was already utilized in Section 6, is simply an  element of C|G|ˆ|G| or – equivalently – a map from GˆG to C, since it associates a complex number  to each element pg1, g2q P G ˆ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Since G ˆ G is precisely the set of (possible) edges E, we can  equivalently think of 2-tensors edge-signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A 3-tensor an element of C|G|ˆ|G|ˆ|G| or equivalently  a map from G ˆ G ˆ G ” G3 to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A 4-tensor then is a map from G4 ” G ˆ G ˆ G ˆ G to C  or equivalenlty an element of C|G|ˆ|G|ˆ|G|ˆ|G| and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Clearly the space of k-tensors forms a  linear vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Addition and scalar multiplication by λ P C are given by  pf ` λgqi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik :“ fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik ` λgi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik  with f and g being k-tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For fixed k, we equip the space of k-tensors with an inner product  according to  xf, gy “  |G|  ÿ  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik“1  fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ikgi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ikµi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik  and denote the resulting inner-product space by ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Operators on Spaces of Tensors:  Since for fixed k the space ℓ2pGkq is simply a |G|k-dimensional  complex inner product space, there are exist normal operators ∆k : ℓ2pGkq Ñ ℓ2pGkq on this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Note that the k in ∆k signifies on which space this operator acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It does not signify that an operator  is raised to the kth power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Setting for example node-weights µi and edge weights µik to one, the  adjacency matrix W as well as normalized or un-normalized graph Laplacians constitute self-adjoint  operators on ℓ2pG2q, where they act by matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Higher order Scattering Transforms:  We can then follow the recipe laid out Section 3 in con-  structing kth-order scattering transforms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' all that we need are a module sequence ΩN and an operator  sequence Dk  N :“ pP k  n, ∆k  nqN  n“1, where now P k  n : ℓ2pGk  n´1q Ñ ℓ2pGk  nq and ∆k  n : ℓ2pGk  nq Ñ ℓ2pGk  nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Figure 7: Schematic Higher Order Scattering Ar-  chitecture  To our initial signal f P ℓ2pGkq we first apply  the connecting operator P k  1 , yielding a signal rep-  resentation in ℓ2pGk  1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Subsequently, we apply  the pointwise non-linearity ρ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we apply  our graph filters tχ1p∆k  1qu Ťtgγ1p∆k  1quγ1PΓ1  to ρ1pP k  1 pfqq yielding the output V1pfq :“  χ1p∆k  1qρ1pP k  1 pfqq as well as the interme-  diate hidden representations tU1rγ1spfq  :“  gγ1p∆k  1qρ1pP k  1 pfqquγ1PΓ1 obtained in the first  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Here we have introduced the one-step  scattering propagator Unrγns : ℓ2pGk  n´1q Ñ  ℓ2pGk  nq mapping f  ÞÑ  gγnp∆nqρnpPnpfqq  as well as the output generating operator  Vn  : ℓ2pGk  n´1q Ñ ℓ2pGk  nq mapping f to  χnp∆k  nqρnpP k  npfqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Upon defining the set  ΓN´1 :“ ΓN´1 ˆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' ˆ Γ1 of paths of length  pN ´ 1q terminating in layer N ´ 1 (with Γ0  taken to be the one-element set) and iterating the  above procedure, we see that the outputs gener-  ated in the N th-layer are indexed by paths ΓN´1  terminating in the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We denote the resulting feature map by Φk  N and write F k  N for the corresponding feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The  node-level stability results of the preceding sections then readily translate to higher order scattering  transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  As the respective proofs are identical to the corresponding results for the node setting, we do not  repeat them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With the notation of Section 4, we have for all f, h P ℓ2pGkq:  }Φk  Npfq ´ Φk  Nphq}2  F k  N ď  ˜  1 `  N  ÿ  n“1  maxtrBn ´ 1s, rBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  ℓ“1  Bℓ  ¸  }f ´ h}2  ℓ2pGkq  34  p3(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='))  p2(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='))  ga2  9b2 (△)  p3(Pk())  X2(△)  ga  p3(Pk())  P1(Pk())/gb1(△)、Ip2(Pk())  qa?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  T  p3(Pk())  6  JC1  X2(△5)  p3(Pk())  (△)  p2(Pk(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='))  qa2  9b2 (△k  p3(Pk())  X1(△)  X2(△))Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let ΦN and rΦN be two scattering transforms based on the same module sequence  ΩN and operator sequences Dk  N, r  Dk  N with the same connecting operators (P k  n “ rP k  n) in each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B for some B and n ď N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume that the respective  normal operators satisfy }∆k  n ´ r∆k  n}F ď δ for some δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Further assume that the functions  tgγnuγnPΓn and χn in each layer are Lipschitz continuous with associated Lipschitz constants  satisfying L2  χn ` ř  γnPΓn L2  gγn ď D2 for all n ď N and some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have for all  f P ℓ2pGkq  }rΦk  Npfq ´ Φk  Npfq}FN ď  b  2p2N ´ 1q ¨  b  pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq  Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Let Φk  N, rΦk  N be higher order scattering transforms based on a common module  sequence ΩN and differing operator sequences Dk  N, r  Dk  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume R`  n , L`  n ď 1 and Bn ď B  for some B and n ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assume that there are identification operators Jn : ℓ2pGk  nq Ñ ℓ2p rGk  nq,  rJn : ℓ2p rGk  nq Ñ ℓ2pGk  nq (0 ď n ď N) so that the respective signal spaces are δ-unitarily equivalent,  the respective normal operators ∆k  n, r∆k  n are ω-δ-close as well as bounded (in norm) by K ą 0 and the  connecting operators satisfy } rP k  nJn´1f ´ JnP k  nf}ℓ2p r  Gknq ď δ}f}ℓ2pGk  n´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For the common module  sequence ΩN assume that the non-linearities satisfy }ρnpJnfq ´ Jnρnpfq}ℓ2p r  Gknq ď δ}f}ℓ2pGknq  and that the constants Cχn and tCgγn uγnPΓN associated through Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4 to the functions of the  generalized frames in each layer satisfy C2  χn ` ř  γnPΓN C2  gγn ď D2 for some D ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Denote the  operator that the family tJnun of identification operators induce on F k  N through concatenation by  JN : F k  N Ñ Ă  F k  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we have with KN “  a  p8N ´ 1qp2D2 ` 12Bq{7 ¨ BN´1 if B ą 1{8 and  KN “  a  p2D2 ` 12Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{8 that  }rΦk  NpJ0fq ´ JNΦk  Npfq} Ă  F k  N ď KN ¨ δ ¨ }f}ℓ2pG,  @f P ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If additionally } rP k  nJn´1f ´ JnP k  nf}ℓ2p r  Gnq “ 0 or }ρnpJnfq ´ Jnρnpfq}ℓ2p r  Gknq “ 0 holds in  each layer, then we have KN “  a  p4N ´ 1qp2D2 ` 4Bq{3 ¨ BN´1 if B ą 1{4 and KN “  a  p2D2 ` 4Bq ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If both additional equations hold, we have  KN “  a  p2N ´ 1q2D2 ¨ BN´1 if B ą 1{2 and KN “  a  2D2 ¨ p1 ´ BNq{p1 ´ Bq if B ď 1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The map N G  p introduced in (4) can also be adapted to aggregate higher-order tensorial features into  graph level features: With  }f}q :“  ˜  ÿ  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ikPG  |fi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik|qµi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik  ¸1{q  and µGk :“ ř|G|  i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='ik“1 µi1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=',ik, we define  N Gk  p  pfq “ p}f}ℓ1pGkq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µGk, }f}ℓ2pGkq, }f}ℓ3pGkq, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=', }f}ℓppGkqqJ{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Given a feature map Φk  N with feature space  FN “ ‘N  n“1  `  ℓ2pGk  nq  ˘|Γn´1| ,  we obtain a corresponding map Ψk  N mapping from ℓ2pGkq to  RN “ ‘N  n“1 pRpnq|Γn´1|  by concatenating Φk  N with the map that the family of non-linear maps tN pn  GknuN  n“1 induces on F N by  concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The resulting map Ψk  N again has stability properties analogous to the node level case:  Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Assuming admissibility, we have  }Ψk  Npfq ´ Ψk  Nphq}RN ď  ˜  1 `  N  ÿ  n“1  maxtrBn ´ 1s, rBnpL`  n R`  n q2 ´ 1s, 0u  n´1  ź  ℓ“1  Bℓ  ¸  }f ´ h}2  ℓ2pGkq  35  for all f, h P ℓ2pGq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' With the conditions and notation of Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2 we have  }Ψk  Npfq ´ rΨk  Npfq}RN ď  b  2p2N ´ 1q ¨  b  pmaxtB, 1{2uqN´1 ¨ D ¨ δ ¨ }f}ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Additionally, in the setting of Theorem J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3, assuming that for each n ď N the identification operator  Jn satisfies  ˇˇ}Jnf}ℓ1p r  Gknq{aµ r  Gkn ´}f}ℓ1pGknq{?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µGkn  ˇˇ,  ˇˇ}Jnf}ℓrp r  Gknq´}f}ℓrpGknq  ˇˇ ď δ¨K¨}f}ℓ2pGknq  for 2 ď r ď pn implies (@f P ℓ2pGkq)  }rΨNpJ0fq ´ ΨNpfq}RN ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2 ¨  b  K2  N ` K2 ¨ δ ¨ }f}ℓ2pGkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  As the proofs here are virtually the same as for the corresponding results in previous sections –  essentially only replacing G by Gk, we omit a repetition of them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  K  Additional Details on Experiments  Here we provide additional details on utilized scattering architectures, training procedures, datasets  and (performance of) other methods our approach is being compared to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Irrespective of task, our  models are trained on an NVIDIA DGX A100 architecture utilizing between two and eight NVIDIA  Tesla A100 GPUs with 80GB memory each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Running 10-fold cross validation for the respective  experiments took at most 71 hours (which was needed for social network graph classification on  REDDIT-12K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1  Social Network Graph Classification  Datasets:  The data we are working with is taken from [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' In particular this work introduced six  social network datasets extracted from from scientific collaborations (COLLAB), movie collabora-  tions (IMDB-B, IMDB-M) and Reddit discussion threads (REDDIT-B, REDDIT-5K, REDDIT-12K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Data is anonymised and contains no content that might be considered offensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Each graph carries a  class label, and the goal is to predict this label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Some basic properties of these datasets are listed in  Table 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Table 3: Social Network Dataset Characteristics  Attributes:  COLLAB  IMDB- B  IMDB-M  REDDIT-B  REDDIT-5K  REDDIT-12K  Graphs  5K  1K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5K  2K  5K  12K  Nodes  372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5K  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8K  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5K  859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2K  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='5M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='7M  Edges  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1M  386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1K  395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='6  4M  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='9M  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='8M  Maximum Degree  2k  540  352  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2K  8K  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2K  Minimum Degree  4  4  4  4  4  4  Average Degree  263  39  40  9  9  9  Target Labels  3  2  3  2  5  11  Disconnected Graphs  No  No  No  Yes  Yes  Yes  These datasets contain graph structures, however they don’t contain associated weights or graph  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Having unspecified weights simply means that the adjacency matrix W from which we  construct the graph Laplacian  L “ D ´ W  on which our operator ∆ is based simply has each entry corresponding to an edge set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If  no edge is present between vertices i and j, the entry Wij is set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It remains to solve the  problem of the missing input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Our strategy is to generate signals reflecting the geometry  of the underlying graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We do this by utilizing features that associate to each node a number that  characterizes its role or importance within its local environment or within the entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We briefly  describe them here:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Degree: The degree of a node is the number of edges incident at this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Eccentricity: For a connected graph, the eccentricity of a node is the maximum distance  from this node to all other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On a disconnected graph it is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  36  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Clustering: For unweighted graphs the clustering cpuq of a node u is the fraction of possible  triangles through that node that actually exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' It is calculated as  cpuq “  2Tpuq  degpuqpdegpuq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Number of triangles: The number of triangles containing the given node as a vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Core number: A k-core is a maximal subgraph that only contains nodes of degree k or  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The core number of a node is the largest value k of a k-core containing that node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Clique number: A clique is a subset of vertices of an undirected graph such that every two  distinct vertices in the clique are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This input assigns the number of cliques the  nodes participates in to each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Pagerank: This returns the PageRank of the respective nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PageRank  computes a ranking of the nodes in the graph based on the structure of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Originally  it was designed as an algorithm to rank web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  For the first three datasets listed in Table 3 we utilize all listed input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For the latter three  datasets we have to refrain from using eccentricity as an input signal, as these datasets contain graphs  that have multiple non-connected graph components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Scattering Architecture:  We chose a generalized scattering architecture of depth N “ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As  normal-operators, we utilize in each layer the un-normalized graph Laplacian L “ D ´ W scaled  by its largest eigenvalue (∆ “ L{λmaxpLq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Filters are chosen as 1  2psinpπ{2 ¨ ∆q, rcospπ{2 ¨ ∆q ´  ψ∆ψJ  ∆s, sinpπ ¨ ∆q, rcospπ ¨ ∆q ´ ψ∆ψJ  ∆sq, which allows to specify the output generating function  solely by demanding χp0q “ 1 and χpλq “ 0 on all other eigenvalues of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Here ψ∆ is the  normalized vector of all ones (satisfying ∆ψ∆ “ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Connecting operators are chosen as the identity,  while we set ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We note that for connected graphs, this recovers Architecture I of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  On disconnected graphs (as they can appear in the REDDIT datasets), we however do not account  for vectors other than ψ∆ in the lowest-lying eigenspace of the graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This scattering  architecture is then applied to each of these input signal individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' For each input signal, this  returns a feature vector with 1 ` 4 ` 16 ` 64 “ 85 entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' These individual feature vectors are  then concatenated into one final composite feature vector for each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Concerning applicable  theoretical results, we note the following:  Training Procedure:  We train RBF kernel support vector classifiers on our composite scattering  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We fix ϵ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The hyperparameter γ scaling the exponent is chosen from  Gpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='00001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='0001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1, 1, 10, 100u,  while we pick the C that controls the error our slack variables introduce among  Cpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='1, 1, 10, 25, 50, 100, 1000u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We chose these parameters in agreement with the choices of [12] to facilitate comparison between  the two works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  We could simply implement the training of the RBF-classifier on our composite scattering features  by dividing each social-network dataset into 10 folds, then iteratively choosing one fold for testing  and among the remaining 9 folds randomly choosing one for validation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' for tuning the hyperpa-  rameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Instead, following [12] (whose code is released under an Apache license and on which  we partially built), we take a slightly different approach: We still randomly split our dataset into 10  folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Among the 10 folds, we iteratively pick one for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Say we have picked the nth fold for  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then there are 9 remaining folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We iteratively pick the mth  n (with 1 ď mn ď 9) of the  remaining 9 folds for choosing hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This leaves 8 folds on which we train our model  for each choice of hyper parameter in Cpool ˆ Gpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The resulting classifiers are all evaluated on the  mth  n fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The one that performs best is retained as classifier mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As mn varies between 1 and 9  (still for fixed n), this yields a set tfmn : 1 ď mn ď 9u of nine classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' From these we build the  classifier fn, whose classification result is obtained from a majority vote among the nine classifiers  in tfmn : 1 ď mn ď 9u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then we evaluate the performance of fn on the nth fold to obtain the  nth estimation of how well our model performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As n varies from one to ten, we built the mean  and variance of the performances of the classifiers fn on the nth fold expressed as the percentage of  correct classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  37  Reference Methods:  To allow for a comparison of our results to the literature, typical classification  accuracies for graph algorithms on social network datasets are displayed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Following the  standard format of reporting classification accuracies, they are presented in the format (Accuracy ˘  standard deviation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' If results are not reported for a dataset, we denote this as not available (N/A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  The first three rows of Table 1 display results for graph kernel methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' namely Weisfeiler-Lehman  graph kernels (WL, [33]), Graphlet kernels (Graphlet, [34]) and deep graph kernels (DGK, [42]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The  subsequent rows display results for geometric deep learning algorithms: Deep graph convolutional  neural networks (DGCNN,[46]), Patchy-san (PSCN (with k=10), [26]), recurrent neural network  autoencoders (S2S-N2N-PP, [16]) and graph isomorphism networks (GIN [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' These results are  taken from [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Additionally we compare with P-Poinc [19], which embeds nodes into a hyperbolic  space (the Poincare ball, to be precise), GSN-e [3] which combines message passing with structural  features extracted via subgraph isomorphism and WKPI-kC [47] which utilizes a weighted kernel  within its metric learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The second to last row (GS-SVM [12]) provides a result that is  also based on a method that combines a static scattering architecture with a support vector machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Its filters are based on graph wavelets built from differences between lazy random walks that have  propagated at different time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='2  Regression of Quantum Chemical Energies  Dataset:  The dataset we consider is the QM7 dataset, introduced in [2, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This dataset contains  descriptions of 7165 organic molecules, each with up to seven heavy atoms, with all non-hydrogen  atoms being considered heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A molecule is represented by its Coulomb matrix CClmb, whose  off-diagonal elements  CClmb  ij  “  ZiZj  |Ri ´ Rj|  (10)  correspond to the Coulomb-repulsion between atoms i and j, while diagonal elements encode a  polynomial fit of atomic energies to nuclear charge [31]:  CClmb  ii  “ 1  2Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4  i  For each atom in any given molecular graph, the individual Cartesian coordinates Ri and the atomic  charge Zi are also accessible individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To each molecule an atomization energy - calculated via  density functional theory - is associated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The objective is to predict this quantity, the performance  metric is mean absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Numerically, atomization energies are negative numbers in the range  ´600 to ´2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The associated unit is rkcal/mols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Scattering Architecture:  Off-diagonal entries in the Coulomb Matrix clearly represent an inverse  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' A weight of zero can then heuristically be thought of as the inverse distance between two  infinitely separated atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' After calculating the degree matrix D associated to C, we obtain the  graph Laplacian once more as L “ D ´ C and set our normal operator to  ∆ “  L  λmaxpLq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  If we continuously vary the distances in (10), staying clear of zero, then the adjacency matrix  and hence the graph Laplacian L varies continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As long as we avoid complete degeneracy,  the largest eigenvalue λmaxpLq will remain positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This implies that our normal operator ∆  varies continuously under changes of the inter-atomic distances, which implies that our feature  vector also varies continuously, as distances are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Connecting operators are set to the  identity, while non-linearities are fixed to ρně1p¨q “ | ¨ |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Filters are chosen as psinpπ{2 ¨ ∆q,  cospπ{2 ¨ ∆q, sinpπ ¨ ∆q, cospπ ¨ ∆qq acting through matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The output generating  functions are set to the identity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Graph level features are aggregated via the map N E  5 p¨q  of Section 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' slightly modified to neglect the normalizing factor in the first entry for improved  convenience in numerical implementability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As weights µij for our second-order feature space are set  to unity and molecular graphs in QM7 contain at most 23-molecules, we note that ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='µG2 ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  232 “  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Going through the proofs of our graph-level stability results, we see that they remain valid after  multiplying each stability constant by 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The Coulomb matrix (divided by a factor of 10 as this  empirically improved performance) is then also utilized as an edge level input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Node level  38  features are obtained by applying the above architecture to the node level information provided by  the respective atomic charges tZiu on each graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We aggregate to graph level features using N G  5  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Section 5), again neglecting the normalizing factor in the first entry for improved convenience in  implementing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' The network depth is set to N “ 4 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We then concatenate graph level  features obtained from node- and edge level input into a composite scattering feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Training Procedure:  The QM7 dataset comes with a precomputed partition into five subsets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' each  containing a representative amount of heavy and light molecules covering the entire complexity range  of QM7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' To allow for 10-fold cross validation, we further dissect each of these subsets into two  smaller datasets, one containing graphs indexed by an even number, one containing graphs indexed  by an odd number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' On these 10-subsets, we then perform 10-fold cross validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Among the 10  folds, we iteratively pick one for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Say we have picked the nth fold for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Then there  are 9 remaining folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We iteratively pick the mth  n (with 1 ď mn ď 9) of the remaining 9 folds for  choosing hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This leaves 8 folds on which we train our model for each choice of hyper  parameter in Cpool ˆ Gpool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This yields 8 regression models, which we average to built our final  predictor for the nth run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' This mean absolute error of this predictor is then evaluated on the nth fold  which was retained for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' As n varies from one to ten, we built the mean and variance of the  performances of the generated regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' We chose log-linear equidistant hyperparameters  from  Gpool :“ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='00003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='0003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='003, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='3, 3, 30u,  and  Cpool :“ t400000, 40000, 4000, 400, 40, 4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='4u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  Reference Methods:  We comprehensively evaluate our method against 11 popular baselines and  state of the art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Among these methods are graph convolutional methods such as GraphConv  [18], Weave [17] or SchNet [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' MPNN [13] and its variant DMPNN [44] are models considering  edge features during message passing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' AttentiveFP [40] is an extension of the graph attention  framework, while N-Gram [21] is a pretrained method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Results for these methods as well as for  GROVER are taken from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' PhysChem [45] learns molecular representations via fusing physical  and chemical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Deep Tensor Neural Networks (DTNN [39]) are adaptable extensions of the  Coulomb Matrix featurizer mapping atom numbers to trainable embeddings which are then updated  based on distance information and other (node-level) atomic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content=' Finally Path-Augmented Graph  Transformer Networks (PATGN, [6]) exploit the connectivity structure of the data in a global attention  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
+page_content='  39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFJT4oBgHgl3EQfIiwq/content/2301.11456v1.pdf'}
diff --git a/QNFPT4oBgHgl3EQfojUh/content/tmp_files/2301.13134v1.pdf.txt b/QNFPT4oBgHgl3EQfojUh/content/tmp_files/2301.13134v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f0eb51a236f48a7928090cadf24c42abd06c6d11
--- /dev/null
+++ b/QNFPT4oBgHgl3EQfojUh/content/tmp_files/2301.13134v1.pdf.txt
@@ -0,0 +1,4138 @@
+arXiv:2301.13134v1  [math.RA]  30 Jan 2023
+The fundamental theorem of calculus in
+differential rings
+Clemens G. Raaba,∗ and Georg Regensburgera,b
+aInstitute for Algebra, Johannes Kepler University Linz, Austria
+bInstitute of Mathematics, University of Kassel, Germany
+clemensr@algebra.uni-linz.ac.at
+regensburger@mathematik.uni-kassel.de
+Abstract
+In this paper, we study the fundamental theorem of calculus and its
+consequences from an algebraic point of view. For functions with singu-
+larities, this leads to a generalized notion of evaluation. We investigate
+properties of such integro-differential rings and discuss many examples.
+We also construct corresponding integro-differential operators and pro-
+vide normal forms via rewrite rules. They are then used to derive several
+identities and properties in a purely algebraic way, generalizing well-known
+results from analysis. In identities like shuffle relations for nested integrals
+and the Taylor formula, additional terms are obtained that take singular-
+ities into account. Another focus lies on treating basics of linear ODEs
+in this framework of integro-differential operators. These operators can
+have matrix coefficients, which allow to treat systems of arbitrary size in a
+unified way. In the appendix, using tensor reduction systems, we give the
+technical details of normal forms and prove them for operators including
+other functionals besides evaluation.
+Keywords
+Integro-differential rings, integro-differential operators, normal
+forms, generalized shuffle relations, generalized Taylor formula
+1
+Introduction
+Differential rings are a well-established algebraic structure for modelling dif-
+ferentiation by derivations, i.e. linear operations satisfying the Leibniz rule.
+More recently, integro-differential algebras have been introduced to additionally
+model integration and point evaluation of continuous univariate functions by
+∗Corresponding author
+1
+
+linear operations satisfying corresponding algebraic identities. In contrast, the
+integro-differential rings introduced in this paper only use the two identities
+d
+dx
+� x
+a
+f(t) dt = f(x)
+and
+� x
+a
+f ′(t) dt = f(x) − f(a)
+of the fundamental theorem of calculus and the Leibniz rule as axioms.
+In
+particular, this results in a generalized notion of evaluation that is only required
+to map to constants. This allows to deal with evaluations of functions even if
+singularities or discontinuities are present. For example, it is natural to consider
+integro-differential rings containing the rational functions leading to so-called
+hyperlogarithms.
+It turns out that many analytic identities and general statements about
+ODEs, like variation of constants, have a purely algebraic proof in integro-
+differential rings that is independent of the analytic properties of concrete
+functions. Likewise, computations with and transformations of linear integro-
+differential equations and initial conditions can be done in this algebraic setting
+as well.
+Moreover, exploring algebraic consequences of the Leibniz rule and
+the fundamental theorem of calculus, we also find new results that introduce
+additional evaluation terms into identities like shuffle relations for nested inte-
+grals and the Taylor formula. In short, we investigate the analytic operations
+of differentiation, integration, and evaluation from a purely algebraic point of
+view. In this context, we implement in some sense the somewhat provocative
+statement of Rota [33, p. 57] that “the algebraic structure sooner or later comes
+to dominate [. . . ]. Algebra dictates the analysis.”
+In order to study linear integro-differential equations and identities, we use
+the operator point of view, which treats identities of functions as identities of
+corresponding linear operators acting on them. To this end, we algebraically
+construct the ring of integro-differential operators. For simplifying expressions
+for operators, we use identities as rewrite rules. As a key result, we work out a
+particular rewrite system that can be applied straightforwardly to obtain nor-
+mal forms and to prove any algebraic identity of integro-differential operators.
+Our construction of the ring of operators can be used for operators with scalar
+coefficients and for operators with matrix coefficients. In particular, it allows to
+uniformly deal with scalar equations as well as with systems, even of undeter-
+mined size. Preliminary versions of some results presented in this paper have
+already been presented by the authors at the conference “Differential Algebra
+and Related Topics” (DART VII) in 2016.
+A related approach was taken in the work of Danuta Przeworska-Rolewicz,
+see for example [23]. In short, she considers a right invertible linear operator on
+a vector subspace as a generalization of derivation. Right inverses and projec-
+tions onto the kernel take the role of integration and evaluation, respectively. In
+this linear setting, she develops algebraic generalizations of results for calculus
+and linear differential equations. She refers to this as algebraic analysis, see
+also [24] for historic context and references. Several results, e.g. a Taylor for-
+mula, can be formulated already in this purely linear setting. For other results,
+multiplication in a commutative algebra is considered. In some statements, the
+2
+
+Leibniz rule or weakened versions of it are needed. However, she does not con-
+sider shuffle relations for nested integrals or additional evaluation terms in the
+Taylor formula. Her treatment of operators is limited to properties and iden-
+tities of given operators and she does not consider rings generated by them or
+normal forms.
+Integro-differential algebras and operators over a field of constants were al-
+ready introduced in [29, 30], see [31] for a detailed overview and further refer-
+ences. More general differential algebras with integration over rings were intro-
+duced in [11], see [12] for a unified presentation and comparison. In contrast, the
+integro-differential rings defined in [14, 13] require the integration to be linear
+over all constants, but the construction of corresponding operators introduced
+there allows noncommutative coefficients and constants. Further references to
+the literature can be found in the respective sections of the present paper.
+So far, all algebraic treatments of integro-differential operators in the litera-
+ture restrict to multiplicative evaluations, i.e. evaluation of a product is the prod-
+uct of the individual evaluations. Assuming only the Leibniz rule and the iden-
+tities of the fundamental theorem of calculus, we deal with non-multiplicative
+evaluations quite naturally in this paper. In Section 2, we introduce (general-
+ized) integro-differential rings following this principle. Analyzing the relations
+imposed, we show for example that any linear projection onto constants may be
+used as the evaluation of such an integro-differential ring. We also present many
+examples, both with multiplicative and with non-multiplicative evaluation. Re-
+mark 2.12 discusses the differences among our definition and the definitions in
+the literature.
+In Section 3, we investigate identities satisfied by integrals and their prod-
+ucts. This reveals a generalization of the Rota-Baxter identity for integration
+that contains an additional evaluation term. Focusing on nested integrals, we
+discover generalized shuffle relations with elaborate additional terms involving
+nested integrals of lower depth. We also characterize properties of repeated in-
+tegrals of 1, which form the smallest integro-differential ring with given ring of
+constants.
+Linear integro-differential operators with coefficients in arbitrary (gener-
+alized) integro-differential rings are constructed algebraically in Section 4 by
+generators and relations. Working at the operator level enables statements of
+broader applicability, since operators not only act on integro-differential rings
+but also on more general modules. We compute a complete set of rewrite rules
+to simplify such operators to normal form. A precise analysis of the uniqueness
+of these normal forms is presented in the appendix only, since it requires a re-
+fined construction relying on tensor rings (like in [14, 13] for the multiplicative
+case). After a largely self-contained introduction to tensor reduction systems in
+the appendix, this is carried out in Section A.4 allowing also other functionals
+besides evaluation. In Section 4.1, we collect properties of integro-differential
+operators with coefficients from an integral domain (e.g. analytic functions). In
+particular, we also characterize the action of such operators when evaluation is
+multiplicative.
+In the remaining sections, we illustrate how computations in the ring of
+3
+
+integro-differential operators can be used to prove and generalize well-known
+results from analysis. For example, variation of constants remains valid for ar-
+bitrary integro-differential rings, which is the focus of Section 5. In particular,
+we also detail how integro-differential operators with noncommutative coeffi-
+cients can be used for proving statements about systems of arbitrary or even
+undetermined size. In Section 6, we discuss how results from analysis need to
+be modified for allowing the induced evaluation to be non-multiplicative. First,
+we look at the formula for variation of constants, which for multiplicative eval-
+uations automatically satisfies homogeneous initial conditions, and include an
+extra term to retain this property in general. Finally, we present a version of the
+Taylor formula with integral remainder term that is valid also for generalized
+evaluations.
+Conventions
+Throughout the paper, rings are implicitly assumed to have
+a unit element and to be different from the zero ring.
+Unless stated other-
+wise, rings are not assumed to be commutative and can be of arbitrary char-
+acteristic. Nevertheless, for easier reading, we use the notions of modules and
+linear maps from the commutative setting to refer to bimodules and bimodule-
+homomorphisms over noncommutative rings. In addition, we use operator no-
+tation for linear maps, e.g. the Leibniz rule for the derivative of products then
+reads ∂fg = (∂f)g + f∂g.
+2
+Integro-differential rings
+To uniformly deal with differentiation of various kinds of functions, we use a
+few basic abstract notions. Recall from differential algebra that a derivation on
+a ring R is an additive map ∂ : R → R that satisfies the Leibniz rule
+∂fg = (∂f)g + f∂g
+(1)
+for all f, g ∈ R. Then, (R, ∂) is called a differential ring and f ∈ R is called
+a constant in this differential ring if and only if ∂f = 0. It is easy to see that
+the set of constants forms a subring of R and ∂ is linear w.r.t. the ring of its
+constants.
+For further theory of differential rings see e.g. [15].
+In addition,
+we introduce the following notions of integration and evaluation in differential
+rings.
+Definition 2.1. Let (R, ∂) be a differential ring and let C be its ring of con-
+stants. We call a C-linear map
+�
+: R → R an integration on R, if
+∂
+�
+f = f
+(2)
+holds for all f ∈ R. A C-linear functional e: R → C which acts on C as the
+identity is called an evaluation on R.
+In other words, integrations on differential rings are right inverses of the
+derivation that are linear over the constants and evaluations on differential rings
+are C-linear projectors onto the ring of constants C.
+4
+
+Definition 2.2. Let (R, ∂) be a differential ring and let
+�
+: R → R be an
+integration on R. We call (R, ∂,
+�
+) a (generalized) integro-differential ring and
+we define the (induced) evaluation E on R by
+Ef := f −
+�
+∂f.
+(3)
+If in addition R is a field or skew field, then we also call (R, ∂,
+�
+) a (generalized)
+integro-differential (skew) field, respectively.
+This extends the definition of integro-differential rings in [14] by dropping
+the additional requirement that the induced evaluation should be multiplicative.
+In the present paper, the notion of integro-differential rings always refers to
+Definition 2.2. The following lemma shows that in any integro-differential ring,
+the (induced) evaluation E is indeed an evaluation as defined in Definition 2.1.
+Moreover, the ring R can be decomposed as direct sum of constant and non-
+constant “functions”.
+Lemma 2.3. Let (R, ∂,
+�
+) be an integro-differential ring with constants C.
+Then, for all f ∈ R and c ∈ C, we have Ef ∈ C, E
+�
+f = 0, and Ec = c.
+Moreover,
+R = C ⊕
+�
+R
+as direct sum of C-modules.
+Proof. First, we compute ∂Ef = ∂(f − � ∂f) = ∂f − ∂f = 0 and E� f =
+�
+f −
+�
+∂
+�
+f = 0 for f ∈ R as well as Ec = c−
+�
+∂c = c for c ∈ C. For any f ∈ R,
+we have f = Ef + f − Ef = Ef +
+�
+∂f and hence R = C +
+�
+R. Let f ∈ C ∩
+�
+R
+and g ∈ R such that f =
+�
+g. Then, 0 = ∂f = ∂
+�
+g = g, which implies f = 0.
+Hence, the sum R = C +
+�
+R is direct.
+By the previous lemma, any integration induces an evaluation by id −
+�
+∂.
+Conversely, any evaluation e can be used to define an integration that has e as
+its induced evaluation, as the following theorem shows. It is easy to see that
+two different integrations cannot have the same induced evaluation. Altogether,
+on differential rings with ∂R = R, there is a one-to-one correspondence of
+integrations and evaluations.
+Theorem 2.4. Let (R, ∂) be a differential ring such that ∂R = R and let e be
+an evaluation on R. Define
+�
+e : R → R by
+�
+ef := g − eg
+for all f ∈ R, where g ∈ R is such that ∂g = f. Then (R, ∂,
+�
+e) is an integro-
+differential ring and the induced evaluation is E = e.
+Moreover, any integration
+�
+on R can be obtained from its induced evaluation
+E via this construction:
+�
+=
+�
+E.
+Proof. Let C be the ring of constants of (R, ∂). First, we show that
+�
+e is well-
+defined.
+If g, ˜g ∈ R are such that ∂g = ∂˜g, then with c := ˜g − g ∈ C we
+5
+
+have ˜g − e˜g = g + c − eg − ec = g − eg, since ec = c by definition of e. For
+showing C-linearity of
+�
+e, we let c ∈ C and f1, f2, g1, g2 ∈ R with ∂gi = fi.
+Then, ∂(cg1 + g2) = cf1 + f2 together with C-linearity of id − e implies C-
+linearity of
+�
+e.
+Consequently, (R, ∂,
+�
+e) is an integro-differential ring, since
+we also have ∂
+�
+ef = f by construction. The induced evaluation is given by
+Ef = f −
+�
+e∂f = f − (f − ef) = ef for f ∈ R.
+Let
+�
+: R → R be any C-linear right inverse of ∂, E its induced evaluation,
+and f ∈ R. Then, we have �
+Ef = � f − E� f = � f by definition of �
+E and by
+Lemma 2.3.
+In particular, if (R, ∂,
+�
+) is an integro-differential ring and e is any evaluation
+on R, then the integration that induces e can be given in terms of
+�
+by
+�
+e :=
+�
+− e
+�
+.
+(4)
+This implies that the difference of two integrations
+�
+1,
+�
+2 on the same differential
+ring can be given as
+�
+1−
+�
+2 = E2
+�
+1 = −E1
+�
+2 in terms of the induced evaluations
+E1, E2. More generally, if (R, ∂,
+�
+) is an integro-differential ring with constants
+C and e : R → C is only C-linear, then
+�
+e defined by (4) can be easily seen to be
+an integration on R and its induced evaluation is given by e + (id − e)E, which
+agrees with e if and only if the latter is an evaluation (i.e. e1 = 1).
+By Lemma 2.3,
+�
+R is a direct complement of C in an integro-differential ring
+R. Conversely, any direct complement of C gives rise to an evaluation on R,
+which in turn induces an integration by Theorem 2.4. More precisely, we have
+the following characterization of integro-differential rings.
+Corollary 2.5. Let (R, ∂) be a differential ring with ring of constants C. Then,
+(R, ∂) can be enriched into an integro-differential ring if and only if ∂R = R and
+C is a complemented C-module in R. Moreover, if ∂R = R, there exists a one-
+to-one correspondence between direct complements of C in R and integrations
+on (R, ∂).
+This characterization shows that on an integro-differential ring (R, ∂,
+�
+),
+in general, there are many other integrations that make R into an integro-
+differential ring with the same derivation. In contrast, the following character-
+ization shows that, in general, ∂ is the only derivation that turns R into an
+integro-differential ring with the same integration.
+Lemma 2.6. Let R be a ring, let C be a subring of R, and let
+�
+: R → R be a
+C-linear map. Then, there exists a derivation ∂ on R such that (R, ∂, � ) is an
+integro-differential ring with constants C if and only if the following conditions
+hold.
+1.
+�
+is injective.
+2. R = C ⊕
+�
+R
+3. (
+�
+f)
+�
+g −
+�
+(
+�
+f)g −
+�
+f
+�
+g ∈ C for all f, g ∈ R.
+6
+
+Moreover, this derivation is unique if it exists.
+Proof. First, it is easy to see, from injectivity of
+�
+and by R = C ⊕
+�
+R, that
+there exists a C-linear map ∂ : R → R such that ker ∂ = C and ∂
+�
+= id and
+that this map is unique. To show that ∂ is indeed a derivation, we verify the
+Leibniz rule on two arbitrary elements of R. By R = C ⊕
+�
+R, we write these
+two elements as c+
+�
+f and d+
+�
+g with c, d ∈ C and f, g ∈ R. Now, we compute
+∂(c + � f)(d + � g) − (∂(c + � f))(d + � g) − (c + � f)∂(d + � g)
+= ∂(
+�
+f)
+�
+g − f
+�
+g − (
+�
+f)g
+= ∂
+�
+(
+�
+f)
+�
+g −
+�
+(
+�
+f)g −
+�
+f
+�
+g
+�
+,
+which is zero by ker ∂ = C and the last assumption on
+�
+.
+Conversely, if (R, ∂,
+�
+) is an integro-differential ring with constants C, then
+injectivity of
+�
+follows from the definition (2) and the other two conditions on
+�
+follow from Lemma 2.3 and Theorem 3.1.
+It is straightforward to equip the matrix ring over an integro-differential
+ring with an integro-differential ring structure. Such noncommutative integro-
+differential rings are relevant when working with linear systems, see Section 5.2.
+Lemma 2.7. Let (R, ∂,
+�
+) be an integro-differential ring with constants C and
+let n ≥ 1.
+Then, (Rn×n, ∂, � ) is an integro-differential ring with constants
+Cn×n, where operations ∂,
+�
+, E act on matrices by applying the corresponding
+operation entrywise in R.
+Proof. Clearly, ∂,
+�
+, E are Cn×n-linear on Rn×n satisfying ∂
+�
+A = A and
+�
+∂A =
+A − EA for all A ∈ Rn×n. Moreover, it is straightforward to verify that ∂ is a
+derivation on R with constants Cn×n.
+Example 2.8. Basic examples for commutative integro-differential rings are
+univariate polynomials C[x] and formal power series C[[x]] over a commutative
+ring C with Q ⊆ C. The derivation is given by ∂ =
+d
+dx with ring of constants C
+and integration is defined C-linearly by
+�
+xn = xn+1
+n + 1
+for all n ∈ N.
+The induced evaluation extracts the constant coefficient and
+corresponds to evaluation at 0.
+If C ⊆ C, the integration
+�
+corresponds to
+integration
+� x
+0 from 0. Also the rings of complex-valued smooth or analytic
+functions on a (possibly unbounded) interval I ⊆ R together with derivation
+∂ =
+d
+dx and integration
+�
+=
+� x
+a , for fixed a ∈ I, are integro-differential rings.
+Then, the induced evaluation is the evaluation of functions at the point a.
+In particular, the ring of exponential polynomials on the real line generated
+by polynomials and exponential functions is closed under differentiation and
+integration and hence is an integro-differential ring as well. Algebraically, for any
+field C of characteristic zero, we can consider the ring of exponential polynomials
+7
+
+C[x, eCx], where ecxedx = e(c+d)x for all c, d ∈ C and e0x = 1, together with
+derivation ∂ =
+d
+dx and with the integration that is induced by evaluation at 0
+based on Theorem 2.4.
+Example 2.9. A basic example of integro-differential rings of arbitrary charac-
+teristic are Hurwitz series, which are closely related to formal power series and
+have been defined in [16, 17], with derivation ∂(a0, a1, . . . ) = (a1, a2, . . . ) and
+integration given by
+�
+(a0, a1, . . . ) = (0, a0, a1, . . . ), see also [18].
+The examples mentioned so far have the special property that the evalua-
+tion of the integro-differential ring is multiplicative, as in the usual definition of
+integro-differential algebras. However, for certain differential rings (in particu-
+lar for differential fields, cf. Corollary 5 in [31]), it is not possible to define a
+multiplicative evaluation for the following reason.
+Remark 2.10. If the induced evaluation is multiplicative, one can see easily
+that no element of
+�
+R can have a multiplicative inverse. Since otherwise we
+would have Ef 1
+f = E1 = 1 and (Ef)E 1
+f = 0E 1
+f = 0 for such f ∈ � R.
+Analytically, if evaluation should correspond to evaluation of functions at a
+fixed point a for functions that are continuous at a, then requiring multiplica-
+tivity of evaluation means that functions with poles at a cannot be considered.
+In particular, if a function f has a pole of order m at a, then evaluation of the
+product (x − a)mf gives a nonzero value, but the factor (x − a)m evaluates to
+zero at a.
+In the following theorem, based on results from the literature, we briefly
+characterize when the induced evaluation is multiplicative. From (3) it imme-
+diately follows that the identity (7) is equivalent to multiplicativity of E. Since
+in integro-differential rings
+�
+is C-linear by definition, we have the following
+characterization of integro-differential rings with multiplicative evaluation.
+Theorem 2.11. Let (R, ∂,
+�
+) be an integro-differential ring. Then the following
+properties are equivalent.
+1. E is multiplicative, i.e. for all f, g ∈ R we have
+Efg = (Ef)Eg.
+(5)
+2.
+�
+satisfies the Rota-Baxter identity, i.e. for all f, g ∈ R we have
+(� f)� g = � (� f)g + � f� g.
+(6)
+3. The hybrid Rota-Baxter identity holds, i.e. for all f, g ∈ R we have
+(
+�
+∂f)
+�
+∂g = (
+�
+∂f)g + f
+�
+∂g −
+�
+∂fg.
+(7)
+Proof. Since (R, ∂) is a differential Z-algebra of weight 0 and
+�
+is Z-linear, this
+immediately follows from items (b), (g), and (a) of Theorem 2.5 in [12].
+8
+
+Remark 2.12. In the literature, integro-differential K-algebras (of weight 0)
+over a commutative ring K with unit element are defined as differential K-
+algebras where the additional map
+�
+is only required to be K-linear, but has
+to satisfy the hybrid Rota-Baxter axiom (7) in addition to (2), see [12]. Analo-
+gously, the more general notion of differential Rota-Baxter K-algebras (of weight
+0) imposes the Rota-Baxter identity (6) instead of (7) in addition to (2). On a
+differential Rota-Baxter K-algebra with constants C, a K-linear map E is defined
+by (3) as well and, by Theorem 2.5 in [12], properties (5), (7), and C-linearity
+of
+�
+are equivalent, see also Proposition 10 in [31].
+By the previous theorem, any (generalized) integro-differential ring with
+constants C is an integro-differential K-algebra for K = Z and for K = C ∩Z(R)
+(i.e. K = C, if R is commutative) if any of the equivalent conditions holds.
+Conversely, any differential Rota-Baxter K-algebra (of weight 0) is an integro-
+differential ring if and only if
+�
+is linear over the constants C. In particular,
+this is automatically the case for integro-differential K-algebras (of weight 0) by
+Proposition 10 in [31]. For concrete differential Rota-Baxter algebras (of weight
+0) where
+�
+is not C-linear, see Example 3 in [30] and the algebraic analog of
+piecewise functions constructed in [32].
+As a basic example for integro-differential rings with non-multiplicative eval-
+uation, we extend the polynomial ring C[x] over a commutative ring C with
+Q ⊆ C by adjoining the multiplicative inverse x−1. In order to have a surjective
+derivation ∂ =
+d
+dx, we also need to adjoin the logarithm ln(x) as in the following
+example.
+Example 2.13. On C[x, x−1, ln(x)], with Q ⊆ C and ∂ =
+d
+dx, we can define the
+C-linear integration recursively as follows.
+� xk ln(x)n :=
+
+
+
+
+
+
+
+xk+1
+k+1
+k ̸= −1 ∧ n = 0
+xk+1
+k+1 ln(x)n −
+n
+k+1
+�
+xk ln(x)n−1
+k ̸= −1 ∧ n > 0
+ln(x)n+1
+n+1
+k = −1
+The same recursive definition also works on the larger ring C((x))[ln(x)] of formal
+Laurent series with logarithms, where every element can be written in the form
+�∞
+k=−m
+�m
+n=0 ck,nxk ln(x)n for some m ∈ N and ck,n ∈ C. In both cases, the
+induced evaluation acts by
+E
+∞
+�
+k=−m
+m
+�
+n=0
+ck,nxk ln(x)n = c0,0
+and is not multiplicative as expected by Remark 2.10. Moreover, for the integro-
+differential subrings of polynomials or formal power series, this evaluation cor-
+responds to the usual multiplicative evaluation at 0.
+Example 2.14. Rational functions together with nested integrals of rational
+functions also form an integro-differential ring with non-multiplicative evalua-
+tion. Algebraically, if C is a field of characteristic zero, this can be understood
+9
+
+as an integro-differential subring of C((x))[ln(x)] with integration
+�
+as in the
+previous example. In fact, this ring is the smallest integro-differential ring con-
+taining C(x) and is generated as a C(x)-vector space by 1 and all nested inte-
+grals
+�
+f1
+�
+f2
+�
+. . .
+�
+fn of arbitrary depth n ≥ 1, where fi ∈ C(x) are proper
+and have irreducible denominators. In particular, if C = C, the integrands can
+be chosen as fi =
+1
+x−ai with ai ∈ C. These kind of nested integrals are called
+hyperlogarithms [21] and have been investigated already in [20]. In [12], the
+free integro-differential algebra (having multiplicative evaluation) generated by
+C(x) has been constructed at the somewhat unnatural expense that
+�
+1 ̸∈ C(x)
+and the constants of the resulting differential ring contain much more than just
+C.
+Example 2.15. Another example of an integro-differential ring that contains
+the rational functions are the D-finite functions [35]. They are characterized
+as solutions of linear differential equations with rational function coefficients
+and indeed form a differential ring with surjective derivation
+d
+dx.
+Since the
+constants of this differential ring are given by C, there exists an integration by
+Corollary 2.5.
+Example 2.16. All the examples considered above are just rings, not fields.
+In contrast, transseries R[[[x]]] are an explicit construction of a differential
+field (with field of constants R) that is closed under taking antiderivatives,
+see [36, 8, 1] and references therein.
+Thus, R[[[x]]] can be turned into an
+integro-differential field by Corollary 2.5 whose evaluation necessarily is non-
+multiplicative by Remark 2.10.
+As shown by Corollary 2.5, in general, there are many different choices for
+an integration
+�
+in order to turn a differential ring with ∂R = R into an
+integro-differential ring. On the same differential ring, for some integrations
+the induced evaluation is multiplicative and for others it is not. It may even
+be the case that a canonical choice of
+�
+yields a non-multiplicative evaluation
+while there are other choices that would give a multiplicative evaluation. In
+particular, this is the case for exponential polynomials, for example. They were
+mentioned above with a multiplicative evaluation, while the following canonical
+definition of
+�
+gives rise to a non-multiplicative one.
+Example 2.17. The ring of exponential polynomials C[x, eCx] over a field C of
+characteristic zero is C-linearly generated by terms of the form xkecx with k ∈ N
+and c ∈ C. Apart from the evaluation-based integration on C[x, eCx] mentioned
+above, it is quite natural to define a C-linear integration
+�
+recursively as follows.
+�
+xkecx :=
+
+
+
+
+
+
+
+xk+1
+k+1
+c = 0
+1
+cecx
+k = 0 ∧ c ̸= 0
+1
+cxkecx − k
+c
+�
+xk−1ecx
+k > 0 ∧ c ̸= 0
+In terms of the Pochhammer symbol (a)k := a·(a + 1)· . . . ·(a + k − 1),
+�
+can be
+10
+
+given explicitly as
+�
+xkecx =
+k
+�
+i=0
+(−k)k−ici−k−1xiecx.
+Then, the induced evaluation Ef := f −
+�
+∂f acts by
+Exkecx =
+�
+1
+k = c = 0
+0
+otherwise
+and is not multiplicative since we have Eecxe−cx = E1 = 1 but Ee±cx = 0 for any
+c ̸= 0, for example. On the other hand, C[x, ex] is an integro-differential subring
+with multiplicative evaluation, for example. With this integration, for instance,
+the subset C[x, ex]ex is closed under addition, multiplication, derivation, and
+integration and, hence, could be viewed as an integro-differential subring with-
+out unit element and having multiplicative evaluation and the zero ring as its
+constants.
+All the examples with explicit integration discussed so far contain an integro-
+differential subring on which the induced evaluation is multiplicative. In general,
+however, this need not be the case as the following example shows.
+Example 2.18. On C[x] with the usual derivation and Q ⊆ C, for example,
+we can define a C-linear integration by
+�
+xn = xn+1
+n+1 + c for all n ∈ N for any
+fixed c ∈ Z(C). Such an integration induces the evaluation Ef = f(0) − cf ′(1)
+on C[x], which is not multiplicative if c ̸= 0 (e.g. f = x and g = x2 − 2x yield
+Eg = 0 and Efg = c).
+3
+Products of nested integrals
+In integro-differential rings with multiplicative evaluation the standard Rota-
+Baxter identity (6) allows to write the product of integrals as a sum of two
+nested integrals. For nested integrals, this leads to shuffle identities [27] where
+a product of two nested integrals is expressed as a sum of nested integrals.
+More generally, for Rota-Baxter operators with weight and corresponding shuffle
+products involving additional terms, see [10] and references therein. In general
+integro-differential rings, i.e. if E is not multiplicative, additional terms involving
+the evaluation arise in the identities (6) and (7) and also in the shuffle identi-
+ties. Note that all evaluation terms are evaluations of products of integrals.
+Therefore, they vanish if E is multiplicative, since E� f = 0 for all f ∈ R.
+Theorem 3.1. Let (R, ∂,
+�
+) be an integro-differential ring. Then the Rota-
+Baxter identity with evaluation
+(
+�
+f)
+�
+g =
+�
+f
+�
+g +
+�
+(
+�
+f)g + E(
+�
+f)
+�
+g
+(8)
+holds for all f, g ∈ R as well as
+(
+�
+∂f)
+�
+∂g = (
+�
+∂f)g + f
+�
+∂g −
+�
+∂fg − E(
+�
+∂f)
+�
+∂g.
+(9)
+11
+
+Proof. Using (3), we can effect the decomposition of R shown in Lemma 2.3.
+For (
+�
+f)
+�
+g, we thereby obtain the decomposition
+�
+∂(
+�
+f)
+�
+g + E(
+�
+f)
+�
+g, which
+implies (8) by the Leibniz rule for the derivation ∂. By
+�
+∂f = f − Ef,
+�
+∂g =
+g − Eg, and
+�
+∂fg = fg − Efg, one can write (9) in a form that can be easily
+verified using the fact that E is an evaluation.
+As a first application, we show that in every integro-differential ring the re-
+peated integrals of 1 give rise to an integro-differential subring. It is the smallest
+integro-differential ring with the same constants. In characteristic zero, this ring
+consists of the univariate polynomials with coefficients in the constants and, for
+nonzero characteristic, it consists of a finite version of Hurwitz series [17].
+Theorem 3.2. Let (R, ∂,
+�
+) be an integro-differential ring with constants C.
+For all n ≥ 1, let xn :=
+� n1 ∈ R and let x0 := 1. Then, 1, x1, x2, . . . commute
+with all elements of C and are C-linearly independent. The C-module
+P := spanC{1, x1, x2, . . .}
+is an integro-differential subring of R. If Q ⊆ R, then P = C[x1].
+Moreover, E is multiplicative on P if and only if Exmxn = 0 for m, n ≥ 1.
+Assuming E is multiplicative on P, then we have xmxn =
+�m+n
+m
+�
+xm+n and, if
+in addition Q ⊆ R, xn = 1
+n!xn
+1 for m, n ∈ N.
+Proof. Since
+�
+is C-linear, every element of C commutes with xi for every i ∈ N,
+even if R or C is noncommutative.
+So, every element of P is of the form
+�n
+i=0 cixi, for some ci ∈ C. To show C-linear independence of x0, x1, . . . , let n ∈
+N be minimal such that there are c0, . . . , cn ∈ C with cn ̸= 0 and �n
+i=0 cixi = 0.
+Then, �n−1
+i=0 ci+1xi = ∂ �n
+i=0 cixi = 0 would imply n = 0 by minimality of
+n. Because this would yield c0 = 0, we conclude that x0, x1, . . . are C-linearly
+independent.
+Obviously, P is closed under ∂ and
+�
+since both operations are C-linear with
+∂xn ∈ P and
+�
+xn = xn+1 for all n ∈ N. For showing that P is closed under
+multiplication, it suffices to show that xmxn ∈ P for all m, n ≥ 1. We proceed
+by induction on the sum n + m. For m = n = 1 we have x2
+1 = 2x2 + Ex2
+1 by (8).
+For m + n > 2 we have
+xmxn =
+�
+xm−1xn +
+�
+xmxn−1 + Exmxn
+by (8). By the induction hypothesis, xm−1xn and xmxn−1 are in P. Hence, the
+same is also true after applying
+�
+, which completes the induction. Altogether,
+P is an integro-differential subring of R.
+For showing P = C[x1], it is sufficient to prove that every xn is contained
+in C[x1]. By (8), we obtain xn+1 = x1xn −
+�
+xn−1x1 − Ex1xn for all n ≥ 1.
+Therefore, xn ∈ C[x1] follows by induction, if C[x1] is closed under
+�
+. Assuming
+Q ⊆ R, we verify that
+�
+xn
+1 −
+1
+n+1xn+1
+1
+∈ C for all n ≥ 1 by applying ∂ to it,
+which shows
+�
+xn
+1 ∈ C[x1] for all n ≥ 1.
+Moreover, any xn with n ≥ 1 satisfies Exn = 0. So, if E is multiplicative on
+P, then trivially Exmxn = (Exm)Exn = 0 for m, n ≥ 1. Conversely, since P
+12
+
+is generated by 1, x1, x2, . . . as a C-module, we know that
+�
+P is generated by
+x1, x2, . . . as a C-module. Hence, applying E to the product of two elements of
+�
+P gives 0, if Exmxn = 0 for all m, n ≥ 1. Altogether, using the decomposition
+P = C ⊕
+�
+P given by Lemma 2.3, we conclude that E is multiplicative on P, if
+Exmxn = 0 for all m, n ≥ 1.
+Now, we assume E is multiplicative on P and let m, n ∈ N. If m + n ≤ 1,
+then xmxn =
+�m+n
+m
+�
+xm+n and xn =
+1
+n!xn
+1 hold trivially. For m + n ≥ 2, it fol-
+lows inductively by (8) that xmxn = � �m+n−1
+m−1
+�
+xm+n−1 + � �m+n−1
+m
+�
+xm+n−1 =
+�m+n
+m
+�
+xm+n. In particular, for n ≥ 2, x1xn−1 = nxn implies xn =
+1
+n!xn
+1 induc-
+tively, if Q ⊆ R.
+Even if E is not multiplicative on P, we can analyze some properties of the
+sequence of constants cm,n := Exmxn with n, m ≥ 1. By C-linearity of � and
+E, it trivially follows that all cm,n are in Z(C). It can be shown that all xn
+commute with each other w.r.t. multiplication if and only if the sequence cm,n
+is symmetric.
+If Q ⊆ R, it can be shown by lengthy computation that the
+constants cm,n are determined by all c1,n via the recursion
+cm,n = 1
+m
+��m+n−1
+m−1
+�
+c1,m+n−1 +
+m−2
+�
+j=0
+n−1
+�
+k=1
+�j+k
+j
+�
+c1,j+kcm−j−1,n−k
+�
+.
+To express xn in terms of powers of x1 in general, for Q ⊆ R, we obtain the
+recursion xn = 1
+n!xn
+1 − �n
+i=2
+1
+i!xn−iExi
+1 from Theorem 6.2.2 in [23].
+3.1
+Generalized shuffle relations
+In this section, we let (R, ∂,
+�
+) be a commutative integro-differential ring. Iter-
+ating the standard Rota-Baxter identity (6) leads to shuffle relations for nested
+integrals expressing a product of two nested integrals of depth m and n as a
+sum of nested integrals of depth exactly m + n. Also by recursively applying
+the Rota-Baxter identity with evaluation (8), products of nested integrals can
+be rewritten in R as sums of nested integrals where also terms of lower depth
+may occur. For convenient notation of the formulae involved, it is standard to
+work in tensor products of R and to use the shuffle product, which we recall in
+the following (see [10] for example).
+We consider the C-module C⟨R⟩ := �∞
+n=0 R⊗n, where the tensor prod-
+uct is taken over C and the empty tensor is denoted by ε. Let pure tensors
+a1⊗ . . . ⊗an ∈ C⟨R⟩ represent nested integrals
+�
+a1
+�
+a2 . . .
+�
+an ∈ R. More for-
+mally, by C-linearity of
+�
+, we consider the unique C-module homomorphism
+ϕ : C⟨R⟩ → R such that
+ϕ(a1⊗ . . . ⊗an) =
+�
+a1
+�
+a2 . . .
+�
+an ∈ R
+and ϕ(ε) = 1 ∈ R. For pure tensors a ∈ C⟨R⟩, we denote shortened versions
+of them by aj
+i := ai⊗ai+1⊗ . . . ⊗aj, where aj
+i := ε ∈ R⊗0 if i = j + 1. The
+13
+
+shuffle product on C⟨R⟩ can be recursively defined as follows. For pure tensors
+a, b ∈ C⟨R⟩ of length m and n, respectively, we set
+a
+� b :=
+�
+a ⊗ b
+if m = 0 ∨ n = 0
+a1 ⊗ (am
+2
+� b) + b1 ⊗ (a
+� bn
+2)
+otherwise
+in R⊗(m+n). Extending this definition to C⟨R⟩ by C-linearity, the shuffle product
+turns C⟨R⟩ into a commutative C-algebra.
+Using the shuffle product for pure tensors, the product of nested integrals in
+R can now be represented as sum of nested integrals as follows. The constant
+coefficients of nested integrals of lower depth are evaluations of products of
+integrals. Consequently, if E is multiplicative, then we recover the standard
+shuffle relations [27] with all these constant coefficients equal zero.
+Theorem 3.3. Let (R, ∂,
+�
+) be a commutative integro-differential ring with
+constants C. Let f, g ∈ C⟨R⟩ be pure tensors of length m and n, respectively.
+Then, the product of the nested integrals ϕ(f) =
+�
+f1
+�
+f2 . . .
+�
+fm and ϕ(g) =
+�
+g1
+�
+g2 . . .
+�
+gn is given by
+ϕ(f)ϕ(g) = ϕ(f
+� g) +
+m−1
+�
+i=0
+n−1
+�
+j=0
+e(f m
+i+1, gn
+j+1)ϕ(f i
+1
+� gj
+1) ∈ R
+(10)
+with constants e(f m
+i+1, gn
+j+1) := Eϕ(f m
+i+1)ϕ(gn
+j+1) ∈ C.
+Proof. Without loss of generality, assume m ≤ n. We proceed by induction
+on m. If m = 0, then f = cε for some c ∈ C and the equation (10) reads
+cϕ(g) = ϕ(cε ⊗ g), which is trivially true since cε ⊗ g = cg and ϕ is C-linear.
+For m ≥ 1, we proceed by induction on n. By virtue of (8), for n ≥ m, we have
+ϕ(f)ϕ(g) = � f1ϕ(f m
+2 )ϕ(g) + � ϕ(f)g1ϕ(gn
+2 ) + e(f, g). The product ϕ(f m
+2 )ϕ(g)
+is covered by the induction hypothesis on m so that we obtain
+�
+f1ϕ(f m
+2 )ϕ(g) =
+�
+f1ϕ(f m
+2
+� g) +
+m−1
+�
+i=1
+n−1
+�
+j=0
+e(f m
+i+1, gn
+j+1)
+�
+f1ϕ(f i
+2
+� gj
+1)
+by (10). The product ϕ(f)ϕ(gn
+2 ) is covered by the induction hypothesis on n
+(or on m, if n = m) so that (10) yields
+� ϕ(f)g1ϕ(gn
+2 ) = � g1ϕ(f
+� gn
+2 ) +
+m−1
+�
+i=0
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)� g1ϕ(f i
+1
+� gj
+2).
+By definition of ϕ and
+�, we have
+�
+f1ϕ(f m
+2
+� g) +
+�
+g1ϕ(f
+� gn
+2 ) = ϕ(f
+� g)
+and similarly
+�
+f1ϕ(f i
+2
+� gj
+1) +
+�
+g1ϕ(f i
+1
+� gj
+2) = ϕ(f i
+1
+� gj
+1) for i, j ≥ 1 as well
+as
+�
+f1ϕ(f i
+2
+� g0
+1) = ϕ(f m
+1
+� g0
+1) and
+�
+g1ϕ(f 0
+1
+� gj
+2) = ϕ(f 0
+1
+� gj
+1). Altogether,
+14
+
+this yields
+ϕ(f)ϕ(g) = ϕ(f
+� g) +
+m−1
+�
+i=1
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)ϕ(f i
+1
+� gj
+1)
++
+m−1
+�
+i=1
+e(f m
+i+1, gn
+1 )ϕ(f m
+1
+� g0
+1) +
+n−1
+�
+j=1
+e(f m
+i+1, gn
+j+1)ϕ(f 0
+1
+� gj
+1) + e(f, g),
+which proves (10).
+4
+Integro-differential operators
+In the following, starting from a given integro-differential ring (R, ∂,
+�
+), we
+define the corresponding ring of operators by generators ∂,
+�
+, E and relations.
+As additive maps on R, any f ∈ R acts as multiplication operator g �→ fg and
+satisfies certain identities together with the maps ∂,
+�
+, E. Those identities of
+additive maps that correspond to the defining properties of the operations on
+R will be used as defining relations for the abstract ring of operators below.
+In particular, the Leibniz rule ∂fg = f∂g + (∂f)g of the derivation ∂ on R
+implies the identity ∂ ◦ f = f ◦ ∂ + ∂f of additive maps for every multiplication
+operator f ∈ R. This motivates the identity (11) in the definition below. Sim-
+ilarly, the identities (2) and (3) in R give rise to the identities (12) and (13).
+Moreover, from C-linearity of the operations ∂,
+�
+, E we obtain
+�
+cg = c
+�
+g for all
+c ∈ C and g ∈ R, for example. In addition, we also obtain
+�
+fEg = (
+�
+f)Eg for
+all f, g ∈ R, since Eg ∈ C. Hence, we also impose commutativity of ∂,
+�
+, E with
+elements of C and the identities (14)–(16) in the following definition.
+Definition 4.1. Let (R, ∂,
+�
+) be an integro-differential ring and let C be its ring
+of constants. We let
+R⟨∂,
+�
+, E⟩
+be the (noncommutative unital) ring extension of R generated by indeterminates
+∂,
+�
+, E, where ∂,
+�
+, E commute with all elements of C and the following identities
+hold for all f ∈ R.
+∂ · f = f · ∂ + ∂f
+(11)
+∂ ·
+�
+= 1
+(12)
+�
+· ∂ = 1 − E
+(13)
+∂ · f · E = ∂f · E
+(14)
+�
+· f · E =
+�
+f · E
+(15)
+E · f · E = Ef · E
+(16)
+We call R⟨∂,
+�
+, E⟩ the ring of (generalized) integro-differential operators (IDO).
+Note that, for multiplication in R⟨∂,
+�
+, E⟩, we always explicitly write · when
+one of ∂,
+�
+, E is involved. This is necessary in order to distinguish the product
+15
+
+∂ · f of operators from the multiplication operator ∂f, for example. By con-
+struction, the ring R⟨∂,
+�
+, E⟩ has a natural action on R, where the elements of
+R act as multiplication operators and ∂,
+�
+, E act as the corresponding opera-
+tions. With this action, R becomes a left R⟨∂,
+�
+, E⟩-module and multiplication
+in R⟨∂,
+�
+, E⟩ corresponds to composition of additive maps on R.
+Since elements of C always commute with ∂,
+�
+, E in R⟨∂,
+�
+, E⟩, it follows
+that R⟨∂,
+�
+, E⟩ is a C-algebra whenever R is commutative. For computing in
+the ring R⟨∂, � , E⟩ of IDO, we use the identities (11)–(16) as rewrite rules in
+the following way. If the left hand side of one of these identities appears in an
+expression of an operator, we replace it by the right hand side to obtain a new
+expression for the same operator.
+If rewrite rules can be applied to a given expression in different ways, then
+it may happen that useful consequences of the defining relations (11)–(16) are
+discovered. A simple instance starts with the expression
+�
+· ∂ ·
+�
+, to which we
+can apply either (12) or (13) to obtain the expressions
+�
+and
+�
+− E ·
+�
+for the
+same operator. Hence, by taking their difference, we see that the identity
+E ·
+�
+= 0
+(17)
+holds in R⟨∂,
+�
+, E⟩. Moreover, for every f ∈ R, the expression
+�
+· ∂ · f can be
+rewritten by (11) and by (13). Thereby we obtain the expressions
+�
+·f ·∂+
+�
+·∂f
+and f − E · f for the same operator, which implies the identity
+�
+· f · ∂ = f − E · f −
+�
+· ∂f.
+(18)
+By letting both sides of this identity in R⟨∂,
+�
+, E⟩ act on any g ∈ R, we show
+that integration by parts
+�
+f∂g = fg − Efg −
+�
+(∂f)g
+holds in R. Furthermore, by considering also the newly obtained identity (18)
+as rewrite rule, for every f ∈ R, we can rewrite the expression
+�
+·f ·∂ ·
+�
+by (12)
+and by (18). Substituting f by � f in the difference f ·� −E·f ·� −� ·∂f ·� −� ·f
+of the results, we obtain the identity
+�
+· f ·
+�
+=
+�
+f ·
+�
+−
+�
+·
+�
+f − E ·
+�
+f ·
+�
+.
+(19)
+By acting with both sides of this identity on any g ∈ R, we obtain an alternative
+proof for the Rota-Baxter identity with evaluation (8). Note that, for f = 1, we
+also obtain the following identities from (14)–(16) and (19).
+∂ · E = 0,
+�
+· E =
+�
+1 · E,
+E · E = E,
+(20)
+�
+·
+�
+=
+�
+1 ·
+�
+−
+�
+·
+�
+1 − E ·
+�
+1 ·
+�
+(21)
+In Table 1, we collect the identities (11)–(21) as a rewrite system for expres-
+sions of operators in the ring of IDO. In fact, we drop (14) since it is redundant
+in the presence of (11) and (20).
+16
+
+Theorem 4.2. Let (R, ∂,
+�
+) be an integro-differential ring. Then, by repeatedly
+applying the rewrite rules of Table 1 in any order, every element of the ring
+R⟨∂,
+�
+, E⟩ can be written as a sum of expressions of the form
+f · ∂j,
+f ·
+�
+· g,
+f · E · g · ∂j,
+or
+f · E · h ·
+�
+· g
+where j ∈ N0, f, g ∈ R, and h ∈
+�
+R.
+Note that the expressions specified in the above theorem are irreducible in
+the sense that they cannot be rewritten any further by any rewrite rules from
+Table 1. The above derivation of identities (17)–(21) is similar to Knuth-Bendix
+completion [19] and Buchberger’s algorithm for computing Gröbner bases [4, 22].
+So, one can show that Table 1 represents all consequences of Definition 4.1
+in the sense that every identity in R⟨∂,
+�
+, E⟩ can be proven by applying the
+rewrite rules in the table and by exploiting identities in R.
+Moreover, the
+irreducible forms of operators specified in the above theorem are unique up to
+multiadditivity and commutativity.
+In the appendix, we will give a precise statement (Theorem A.3) of this
+by giving an explicit construction of the ring R⟨∂,
+�
+, E⟩ as a quotient of an
+appropriate tensor ring. Then, the translation of Table 1 into a tensor reduction
+system facilitates the proof. In fact, this proof is carried out in Theorem A.5 for
+a more general class of operators including linear functionals, which are useful
+for dealing with boundary problems, for instance.
+Remark 4.3. In the literature, integro-differential operators were considered
+only with multiplicative evaluation so far. Integro-differential operators were
+first introduced in [29, 30] over a field of constants using a parametrized Gröb-
+ner basis in infinitely many variables and a basis of the commutative coefficient
+algebra. Integro-differential operators with polynomial coefficients over a field
+of characteristic zero were also studied using generalized Weyl algebras [2], skew
+polynomials [28], and noncommutative Gröbner bases [26]. A general construc-
+tion of rings of linear operators over commutative operated algebras is presented
+in [9]. In particular, integro-differential operators are discussed in that setting
+and also differential Rota-Baxter operators are investigated. Tensor reduction
+systems have already been used in [14, 13] for the construction of IDO includ-
+ing functionals in the special case that the evaluation and all functionals are
+multiplicative.
+There, also additional operators arising from linear substitu-
+tions were included.
+In particular, these cover integro-differential-time-delay
+operators, which were already constructed algebraically in [25], see also [5].
+∂ · f = f · ∂ + ∂f
+�
+· f · ∂ = f − E · f −
+�
+· ∂f
+∂ · E = 0
+� · f · E = � f · E
+∂ ·
+�
+= 1
+�
+· f ·
+�
+=
+�
+f ·
+�
+−
+�
+·
+�
+f − E ·
+�
+f ·
+�
+E · f · E = Ef · E
+�
+· ∂ = 1 − E
+E · E = E
+�
+· E =
+�
+1 · E
+E ·
+�
+= 0
+�
+·
+�
+=
+�
+1 ·
+�
+−
+�
+·
+�
+1 − E ·
+�
+1 ·
+�
+Table 1: Rewrite rules for operator expressions
+17
+
+To refer to integro-differential operators of special form, we use the following
+notions.
+Definition 4.4. Let L ∈ R⟨∂,
+�
+, E⟩, then we call L
+1. a differential operator, if there are f0, . . . , fn ∈ R such that
+L =
+n
+�
+i=0
+fi · ∂i,
+where we call L monic, if fn = 1,
+2. an integral operator, if there are f1, . . . , fn, g1, . . . , gn ∈ R such that
+L =
+n
+�
+i=1
+fi ·
+�
+· gi,
+3. an initial operator, if there are f1, . . . , fn ∈ R and differential and integral
+operators L1, . . . , Ln such that
+L =
+n
+�
+i=1
+fi · E · Li.
+In particular, we call an initial operator L monic, if there is a differential
+operator L1 and an integral operator L2 such that
+L = E · (L1 + L2).
+In R⟨∂, � , E⟩, using Table 1, one can check that the differential operators
+are the elements of the subring generated by R and ∂, the integral operators
+are the elements of the R-bimodule generated by
+�
+, the initial operators are the
+elements of the two-sided ideal generated by E, and the monic initial operators
+are the elements of the right ideal generated by E. Theorem 4.2 says that every
+integro-differential operator can be written as the sum of a differential operator,
+an integral operator, and an initial operator. In fact, by the stronger results in
+the appendix, this decomposition of integro-differential operators even is unique.
+Remark 4.5. We outline how the construction of integro-differential opera-
+tors changes when the evaluation is multiplicative, see also [31, 14, 13]. For
+multiplicative evaluation, we have Efg = (Ef)Eg for all f, g ∈ R. So, for the
+algebraic construction of corresponding operators as in Definition 4.1, we need
+to impose in addition that
+E · f = Ef · E
+(22)
+for all f ∈ R. This does not give rise to new consequences other than (17)–(21),
+but together with (20) it makes (16) redundant. Hence, in Table 1, we can
+replace (16) by (22). Moreover, (22) allows to reduce the evaluation term in
+(18) and, since E
+�
+f = 0 for all f ∈ R, to omit the evaluation terms in (19) and
+18
+
+(21). Consequently, the irreducible forms of operators given in Theorem 4.2 can
+be simplified to
+f · ∂j,
+f ·
+�
+· g,
+f · E · ∂j,
+where j ∈ N0 and f, g ∈ R. Evidently, this ring of operators is isomorphic to
+R⟨∂,
+�
+, E⟩ factored by the two-sided ideal (E · f − Ef · E | f ∈ R).
+4.1
+IDO over integral domains
+In many concrete situations, when computing with differential operators or dif-
+ferential equations with scalar coefficients, the order of a product of differential
+operators is the sum of the orders of the factors.
+This is equivalent to the
+ring R of coefficients being an integral domain, i.e. a commutative ring without
+nontrivial zero divisors. For the rest of this section, we only consider integro-
+differential rings that are integral domains. Under this assumption, we can in-
+vestigate further properties of computations in the ring of IDO. For instance, an
+integro-differential equation can be reduced to a differential equation by differ-
+entiation, see Lemma 4.8 below. Additionally, investigating the action of IDOs
+on integro-differential rings algebraically leads to analyzing two-sided ideals.
+As explained earlier, elements of R⟨∂,
+�
+, E⟩ naturally act as additive maps
+on R. Likewise, they act naturally on any integro-differential ring extension of
+R. In Theorem 4.11, we also provide conditions when this action is faithful, i.e.
+0 ∈ R⟨∂,
+�
+, E⟩ is the only element that induces the zero map.
+Let (S,
+�
+, ∂) be the integro-differential ring defined in Example 2.13 and as-
+sume C is an integral domain. Then, S⟨∂, � , E⟩ does not act faithfully on S, since
+E · ln(x) acts like zero. However, with the integro-differential subring R := C[x]
+of S, R⟨∂,
+�
+, E⟩ acts faithfully on S by Theorem 4.11 below. To see this, let
+L := �n
+i=0 fi ·∂i ∈ R⟨∂,
+�
+, E⟩ and let k ∈ N such that coeff(fn, xk) ̸= 0. Apply-
+ing E·L to xn−k ln(x)n ∈ S, we obtain (E·L)xn−k ln(x)n = n! coeff(fn, xk) ̸= 0.
+As a preparatory step, we need some basic statements involving multiplica-
+tion with constants that are valid for integro-differential rings that are integral
+domains. Linear independence over constants is tied to the Wronskian. Follow-
+ing the analytic definition, the Wronskian of elements f1, . . . , fn of a commuta-
+tive differential ring is defined by
+W(f1, . . . , fn) := det
+��
+∂i−1fj
+�
+i,j=1,...,n
+�
+.
+Lemma 4.6. Let (R, ∂) be a differential ring that is an integral domain with
+ring of constants C and let f1, . . . , fn ∈ R, n ≥ 1.
+1. f1, . . . , fn are linearly independent over C if and only if their Wronskian
+W(f1, . . . , fn) is zero.
+2. If f1, . . . , fn are linearly independent over C, then g1, . . . , gn−1 are linearly
+independent over C, where gi := W(fi, fn).
+Proof. For showing the first statement, we note that neither linear independence
+nor zeroness of the Wronskian changes, if we replace R and hence C by their
+19
+
+quotient fields. For differential fields, a proof can be found in [15], which implies
+the statement given here.
+For showing the second statement, we assume that f1, . . . , fn are linearly
+independent over C and we let c1, . . . , cn−1 ∈ C such that �n−1
+i=1 cigi = 0.
+By definition of gi and multilinearity of the Wronskian over C, we conclude
+W(�n−1
+i=1 cifi, fn) = 0. By assumption on f1, . . . , fn, this implies �n−1
+i=1 cifi = 0
+and hence c1 = . . . = cn−1 = 0.
+Lemma 4.7. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain and let C be its ring of constants. Then, elements of C commute with
+all elements of R⟨∂,
+�
+, E⟩. Moreover, for any L ∈ R⟨∂,
+�
+, E⟩ and any nonzero
+c ∈ C, we have that L = 0 if and only if c · L = 0.
+Proof. By construction, constants commute with ∂,
+�
+, E ∈ R⟨∂,
+�
+, E⟩. Since
+R is commutative, it follows that elements of C commute with all elements
+of R⟨∂,
+�
+, E⟩. Therefore, R⟨∂,
+�
+, E⟩ is a unital C-algebra and hence Q(C) ⊗C
+R⟨∂,
+�
+, E⟩ is a unital C-algebra as well, with multiplication (c1⊗L1)·(c2⊗L2) =
+(c1c2) ⊗ (L1 · L2). Now, 1 ⊗ L = c−1 ⊗ (c · L) implies L = 0 if c · L = 0.
+Now, we are ready to have a closer look at certain computations with IDO.
+The following two lemmas construct left or right multiples of IDO by differential
+operators such that the product does not involve the integration operator any-
+more. The tricky part will be to ensure that the product is a nonzero operator
+again.
+Lemma 4.8. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain.
+Let C be the ring of constants of R and let L ∈ R⟨∂, � , E⟩ be not
+an initial operator.
+Then, there exist nonzero h1, . . . , hn ∈ R such that the
+following product is a nonzero differential operator.
+(h1 · ∂ − ∂h1) · . . . · (hn · ∂ − ∂hn) · L
+Proof. There are n ∈ N, a nonzero c ∈ C, differential operators L0, . . . , Ln ∈
+R⟨∂,
+�
+, E⟩, and f1, . . . , fn, g1, . . . , gn ∈ R such that f1, . . . , fn are linearly inde-
+pendent over C and c · L = L0 + �n
+i=1 fi ·
+��
+· gi + E · Li
+�
+. Since L is not an
+initial operator, L0 and g1, . . . , gn are not all zero by Lemma 4.7. If L0 = 0,
+then we assume without loss of generality that g1 ̸= 0.
+To remove the sum �n
+i=1 fi·
+��
+· gi + E · Li
+�
+, we will multiply c·L iteratively
+by n first-order differential operators from the left as follows. If n > 0, then by
+(11) and (12) we have (fn ·∂ − ∂fn)·fi ·
+�
+·gi = fnfigi + (fn∂fi − (∂fn)fi)·
+�
+·gi
+for all i. Therefore, using (14), we obtain
+(fn · ∂ − ∂fn) · c · L = (fn · ∂ − ∂fn) · L0 +
+n
+�
+i=1
+fnfigi
++
+n−1
+�
+i=1
+(fn∂fi − (∂fn)fi) ·
+��
+· gi + E · Li
+�
+,
+20
+
+which has the form ˜L0+�n−1
+i=1 ˜fi ·
+��
+· gi + E · Li
+�
+similar to c·L. Note that also
+the following two properties of the above representation of c · L are preserved.
+First, if L0 is nonzero, the differential operator ˜L0 := (fn · ∂ − ∂fn) · L0 +
+�n
+i=1 fnfigi is nonzero too, since fn ̸= 0. Second, ˜fi := fn∂fi − (∂fn)fi, for
+i ∈ {1, . . . , n − 1}, are again linearly independent over C by Lemma 4.6. Now,
+we let hn := fnc such that (hn · ∂ − ∂hn) · L = (fn · ∂ − ∂fn) · c · L.
+If n = 1, we only need to show that (hn · ∂ − ∂hn) · L = ˜L0 is nonzero. As
+remarked above, if L0 ̸= 0 then ˜L0 ̸= 0. On the other hand, if L0 = 0, then
+˜L0 = f 2
+1 g1, which is nonzero as well due to the assumptions.
+If n > 1, we continue by repeating what we did with c · L above. That is,
+noting ˜f1, . . . , ˜fn−1 are linearly independent over C, we multiply (hn · ∂ − ∂hn) ·
+L = ˜L0 + �n−1
+i=1 ˜fi ·
+��
+· gi + E · Li
+�
+from the left by hn−1 · ∂ − ∂hn−1, where
+hn−1 := ˜fn−1. We iterate this a total of n − 1 times, the result is a differential
+operator. To see that it is also nonzero, we focus on the last step, i.e. we refer
+to the case n = 1 above.
+An immediate consequence of the previous lemma is that a left ideal in
+R⟨∂,
+�
+, E⟩ is nontrivial if and only if it contains a nonzero differential operator
+or a nonzero initial operator. Left ideals in R⟨∂,
+�
+, E⟩ arise, for instance, as sets
+of operators annihilating a given subset of a left R⟨∂, � , E⟩-module. As another
+consequence, for two-sided ideals, we obtain Lemma 4.10 below.
+Lemma 4.9. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain. Let C be the ring of constants of R and let L ∈ R⟨∂,
+�
+, E⟩ be a nonzero
+monic initial operator. Then, there exist nonzero h1, . . . , hn ∈ R and a nonzero
+differential operator ˜L ∈ R⟨∂,
+�
+, E⟩ such that
+L · (hn · ∂ + 2∂hn) · . . . · (h1 · ∂ + 2∂h1) = E · ˜L.
+Proof. There are n ∈ N, a nonzero c ∈ C, a differential operator L0 ∈ R⟨∂,
+�
+, E⟩,
+nonzero f1, . . . , fn ∈ � R, and nonzero g1, . . . , gn ∈ R such that g1, . . . , gn are
+linearly independent over C and L·c = E·L0+�n
+i=1 E·fi·
+�
+·gi. By Lemma 4.7,
+L · c = c · L is nonzero.
+To remove the sum �n
+i=1 E · fi ·
+�
+· gi, we will multiply L · c iteratively by
+n first-order differential operators from the right as follows. If n > 0, then we
+have
+E · fi ·
+�
+· gign · ∂ = E · figign − Efi · E · gign − E · fi ·
+�
+· ((∂gi)gn + gi∂gn)
+for all i by (18) and by (16). Therefore, by Efi = 0, we obtain
+L · c · (gn · ∂ + 2∂gn) = E · L0 · (gn · ∂ + 2∂gn) +
+n
+�
+i=1
+E · fi ·
+�
+· (gign · ∂ + 2gi∂gn)
+= E · L1 +
+n−1
+�
+i=1
+E · fi ·
+�
+· (gi∂gn − (∂gi)gn),
+21
+
+with L1 := L0 · (gn · ∂ + 2∂gn) + �n
+i=1 figign.
+Note that gi∂gn − (∂gi)gn,
+for i ∈ {1, . . . , n − 1}, are again linearly independent over C by Lemma 4.6.
+Consequently, with hn := cgn being nonzero, the operator L · (hn · ∂ + 2∂hn) is
+expressed like L · c above, just with L0 replaced by L1, with different gi, and
+without the last summand.
+By iterating this process of multiplying by a differential operator that is
+chosen as above, we obtain a right multiple of L that is of the form E · Ln with
+a differential operator Ln. It remains to show that Ln is nonzero. After the
+last step, i.e. passing from E · Ln−1 + E · f1 ·
+�
+· ˜g1, with differential operator
+Ln−1 and nonzero ˜g1 ∈ R, to E · Ln, we have Ln = Ln−1 · (˜g1 · ∂ + 2∂˜g1) +
+f1˜g2
+1. If the differential operator Ln−1 is nonzero, then also Ln is nonzero, since
+˜g1 ̸= 0. Otherwise, we have Ln = f1˜g2
+1, which is nonzero as well due to the
+assumptions.
+Lemma 4.10. Let (R, ∂,
+�
+) be an integro-differential ring that is an integral
+domain and let I ⊆ R⟨∂,
+�
+, E⟩ be a two-sided ideal. If I contains an operator
+that is not an initial operator, then I contains an operator of the form f ·E with
+nonzero f ∈ R.
+Proof. By Lemma 4.8, I contains a nonzero differential operator L = �n
+i=0 fi·∂i.
+Let k ∈ {0, . . ., n} be minimal such that fk ̸= 0. Then, fk · E ∈ I since
+L ·
+� k · E =
+n
+�
+i=k
+fi · ∂i ·
+� k · E =
+n
+�
+i=k
+fi · ∂i−k · E = fk · E.
+Based on the above lemmas, we can find conditions when the action of
+R⟨∂,
+�
+, E⟩ on an integro-differential ring is faithful. In this case, the annihilator
+AnnR⟨∂,�
+,E⟩(S) = {L ∈ R⟨∂,
+�
+, E⟩ | ∀f ∈ S : Lf = 0}
+is a two-sided ideal in R⟨∂,
+�
+, E⟩. In particular, we show that the annihilator
+only contains initial operators and, if C is a field, can be generated by monic
+initial operators.
+Theorem 4.11. Let (S, ∂,
+�
+) be an integro-differential ring with its ring of
+constants C being an integral domain. Let R be an integro-differential subring of
+S that is an integral domain and contains C. Then, R⟨∂,
+�
+, E⟩ acts faithfully on
+S if and only if there is no nonzero differential operator L ∈ R⟨∂,
+�
+, E⟩ such that
+E · L vanishes on all of S. In any case, the two-sided ideal I = AnnR⟨∂,�
+,E⟩(S)
+only contains initial operators. Moreover, if C is a field, I has a generating set
+consisting only of monic initial operators.
+Proof. Since operators of the form f · E, with nonzero f ∈ R, do not vanish on
+1 ∈ S, the ideal I only contains initial operators by Lemma 4.10. If there is a
+nonzero differential operator L ∈ R⟨∂,
+�
+, E⟩ such that E · L vanishes on all of
+S, then R⟨∂,
+�
+, E⟩ obviously does not act faithfully on S, since E · L is nonzero
+as well.
+22
+
+Now, we assume that there is no nonzero differential operator L such that
+E · L ∈ I.
+Let L ∈ I, then there are nonzero c ∈ C, f1, . . . , fn ∈ R, and
+nonzero monic initial operators L1, . . . , Ln ∈ R⟨∂,
+�
+, E⟩ such that f1, . . . , fn are
+C-linearly independent and c · L = �n
+i=1 fi · Li. If we would have n ̸= 0, then
+we would conclude L1, . . . , Ln ∈ I, since �n
+i=1 fiLig = cLg = 0 and Lig ∈ C
+for all g ∈ S. Therefore, by Lemma 4.9, there would be a nonzero differential
+operator ˜L ∈ R⟨∂,
+�
+, E⟩ such that E · ˜L ∈ I in contradiction to the assumption.
+Hence, n = 0, which implies c · L = 0. By Lemma 4.7, it follows that L = 0,
+which shows that R⟨∂,
+�
+, E⟩ acts faithfully on S.
+Finally, if C is a field, we show that I has a generating set consisting only
+of monic initial operators. Let L ∈ I, then L is an initial operator. Now, there
+are nonzero f1, . . . , fn ∈ R, and nonzero monic initial operators L1, . . . , Ln ∈
+R⟨∂,
+�
+, E⟩ such that f1, . . . , fn are C-linearly independent and L = �n
+i=1 fi · Li.
+For all g ∈ S, we have Lig ∈ C and �n
+i=1 fiLig = Lg = 0, which implies Lig = 0
+for all i. Therefore, L1, . . . , Ln ∈ I, which shows that I is generated by nonzero
+monic initial operators.
+Assuming additional properties of R resp. of the action of R⟨∂, � , E⟩, gen-
+erating sets of annihilators can be narrowed down even more. In particular,
+the following two corollaries consider the situation when R is a field or E is
+multiplicative.
+Corollary 4.12. Let (S, ∂, � ) be an integro-differential ring with its ring of
+constants C being a field. Let R be an integro-differential subring of S that is a
+field and contains C. Then, the two-sided ideal AnnR⟨∂,
+�
+,E⟩(S) has a generating
+set consisting only of operators of the form E · �n
+i=0 fi · ∂i, with f0, . . . , fn ∈ R
+where f0 ∈
+�
+R is nonzero and fn = 1.
+Proof. By Theorem 4.11, I := AnnR⟨∂,
+�
+,E⟩(S) has a generating set consisting
+only of monic initial operators. Now, let L ∈ I be a monic initial operator and
+let A be the set of all elements in I that have the form specified in the statement
+of the corollary. By Lemma 4.9, we obtain nonzero h1, . . . , hm ∈ R such that
+L · M = E · ˜L with M := (hm · ∂ + 2∂hm) · . . . · (h1 · ∂ + 2∂h1) ∈ R⟨∂,
+�
+, E⟩
+and some nonzero differential operator ˜L = �n
+i=0 ˜fi · ∂i ∈ R⟨∂,
+�
+, E⟩ with
+˜fn ̸= 0. Then, by repeated use of (11), there are f0, . . . , fn ∈ R with fn = 1
+such that ˜L · ˜f −1
+n
+= �n
+i=0 fi · ∂i. Let k be minimal such that fk ̸= 0, then
+˜L · ˜f −1
+n
+· � k = �n−k
+i=0 fi+k · ∂i. Since E · ˜L · ˜f −1
+n
+· � k ∈ I vanishes on 1 ∈ S,
+we conclude fk ∈
+�
+R and hence E · �n−k
+i=0 fi+k · ∂i ∈ A. Since R is a field, we
+can verify by (11) and (12) that h−1
+i
+·
+�
+· hi ∈ R⟨∂,
+�
+, E⟩ is a right inverse of
+hi·∂+2∂hi for every i. Hence, there exists ˜
+M ∈ R⟨∂,
+�
+, E⟩ such that M · ˜
+M = 1.
+Then, E ·
+��n−k
+i=0 fi+k · ∂i�
+· ∂k · ˜fn · ˜
+M = E · ˜L · ˜
+M = L · M · ˜
+M = L, i.e. L is in
+the ideal generated by A. Altogether, this shows that I is generated by A.
+Whenever E is multiplicative on R, the action of R⟨∂,
+�
+, E⟩ cannot be faithful
+on R since the operator E · f acts as zero map for any f ∈ R with Ef = 0.
+23
+
+By Lemma 2.3, Ef = 0 is equivalent to f ∈
+�
+R and one can always choose
+f =
+�
+1, for example. In Remark 4.5, we already discussed factoring the ring of
+operators by the relations (22) immediately arising from multiplicativity of E.
+The following corollary shows that the resulting quotient always acts faithfully.
+Corollary 4.13. Let (S, ∂,
+�
+) be an integro-differential ring with its ring of
+constants C being an integral domain. Let R be an integro-differential subring
+of S that is an integral domain and contains C. If Efg = (Ef)Eg for all f ∈ R
+and g ∈ S, then the two-sided ideal AnnR⟨∂,
+�
+,E⟩(S) is generated by the set
+{E · f − Ef · E | f ∈ R}.
+Proof. Let I := AnnR⟨∂,�
+,E⟩(S) and let J ⊆ R⟨∂,
+�
+, E⟩ be the two-sided ideal
+generated by the set {E · f − Ef · E | f ∈ R}. By assumption on E, we have
+that J ⊆ I, so it only remains to show I ⊆ J.
+By Theorem 4.11, every L ∈ I is an initial operator. Following the form
+of initial operators in Remark 4.5, there are f0, . . . , fn ∈ R such that L =
+�n
+i=0 fi ·E·∂i modulo J. By J ⊆ I, we have that �n
+i=0 fi ·E·∂i ∈ I. Therefore,
+�n
+i=0 fi·E·∂i vanishes on
+� k1 ∈ S for all k ∈ {0, . . . , n}. This implies inductively
+that f0, . . . , fn are all zero. Consequently, we have L ∈ J.
+5
+Equational prover in calculus
+In this section, we illustrate how results from analysis can be proven via compu-
+tations in the ring of integro-differential operators. The general approach con-
+sists in formulating an analytic statement as an identity of integro-differential
+operators and then prove this identity algebraically in R⟨∂,
+�
+, E⟩, with minimal
+assumptions on the integro-differential ring (R, ∂,
+�
+) used in the coefficients.
+Note that we prove results directly by a computation with integro-differential
+operators, instead of doing the whole computation with elements of R or any
+other left R⟨∂,
+�
+, E⟩-module.
+An identity in R⟨∂,
+�
+, E⟩ can be proven by comparing irreducible forms
+of the left hand side and right hand side. Recall that such irreducible forms
+are given by Theorem 4.2 and can be computed systematically by the rewrite
+rules in Table 1. In this sense, the rewrite rules provide an equational prover
+for integro-differential operators provided one can decide equality of irreducible
+forms in R⟨∂,
+�
+, E⟩, which includes deciding identities in R. In practice, this is
+often possible for concrete irreducible forms.
+Once an identity L1 = L2 is proven in R⟨∂,
+�
+, E⟩, we immediately infer the
+corresponding identity in R, i.e. L1f = L2f for all f ∈ R, by the canonical
+action of R⟨∂,
+�
+, E⟩ on R. Moreover, by acting on any other left R⟨∂,
+�
+, E⟩-
+module, we obtain the analogous identity also in those modules. Furthermore,
+any concrete computation with operators acting on functions uses only finitely
+many derivatives. Consequently, by inspecting every step of the computation,
+an identity proven for infinitely differentiable functions can also be proven for
+functions that are only sufficiently often differentiable.
+24
+
+5.1
+Variation of constants for scalar equations
+In this section, we deal with the method of variation of constants for comput-
+ing solutions of inhomogeneous ODEs in terms of integro-differential operators.
+First, we recall the analytic statement, see e.g. Theorem 6.4 in Chapter 3 of [6].
+Consider the inhomogeneous linear ODE
+y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).
+Assume that z1(x), . . . , zn(x) is a fundamental system of the homogeneous equa-
+tion, i.e. z(n)
+i
+(x) + an−1(x)z(n−1)
+i
+(x) + · · · + a0(x)zi(x) = 0 and the Wronskian
+w(x) := W(z1(x), . . . , zn(x)) is nonzero. Then,
+z∗(x) :=
+n
+�
+i=1
+(−1)n−izi(x)
+� x
+x0
+W(z1(t), . . . , zi−1(t), zi+1(t), . . . , zn(t))
+w(t)
+f(t) dt
+(23)
+is a particular solution of the inhomogeneous equation above.
+Algebraically, we model scalar functions by fixing a commutative integro-
+differential ring (R, ∂,
+�
+) and, as indicated above, we model computations with
+scalar equations by computations in the corresponding ring of integro-differential
+operators R⟨∂,
+�
+, E⟩. From the operator viewpoint, mapping the inhomogeneous
+part f to a solution of the equation Ly = f amounts to constructing a right
+inverse of the differential operator L.
+In general, for an operator L and a
+right inverse H, a particular solution of the inhomogeneous equation is given by
+z∗ = Hf, since
+Lz∗ = L(Hf) = (L · H)f = 1f = f.
+Recall that
+�
+is a right inverse of ∂ by definition.
+Now, we consider an
+arbitrary monic first-order differential operator
+L = ∂ + a,
+with a ∈ R, and assume that z ∈ R is a solution of Ly = 0, i.e. ∂z + az = 0.
+Then, using (11), we compute
+(∂ + a) · z = ∂ · z + a · z = z · ∂ + ∂z + az = z · ∂.
+If, moreover, we assume that z has a multiplicative inverse z−1 ∈ R, we obtain
+(∂ + a) · (z ·
+�
+· z−1) = z · ∂ ·
+�
+· z−1 = z · z−1 = 1.
+Hence
+H = z ·
+�
+· z−1
+is a right inverse of L in R⟨∂,
+�
+, E⟩.
+We also outline the computation for a second order differential operator
+L = ∂2 + a1 · ∂ + a0,
+25
+
+with a1, a0 ∈ R. If z ∈ R is a solution of Ly = 0, then
+L · z = z · ∂2 + (2∂z + a1z) · ∂.
+Assume that there exist two solutions z1, z2 ∈ R of Ly = 0 such that their
+Wronskian
+w = z1∂z2 − z2∂z1
+has a multiplicative inverse 1
+w ∈ R. Let
+H = −z1 ·
+�
+· z2
+w + z2 ·
+�
+· z1
+w .
+In the ring of operators, we can compute
+∂ · zi
+w = zi
+w · ∂ + ∂zi
+w − zi∂w
+w2 .
+Then, using the normal forms of L · zi and ∂ · zi
+w , we obtain
+L · H = −z1 · ∂ · z2
+w − 2(∂z1) z2
+w − a1z1 z2
+w + z2 · ∂ · z1
+w + 2(∂z2) z1
+w + a1z2 z1
+w
+= −z1 · ∂ · z2
+w + z2 · ∂ · z1
+w + 2 w
+w = −z1∂ z2
+w + z2∂ z1
+w + 2 = 1.
+Hence H is a right inverse of L in R⟨∂,
+�
+, E⟩.
+More generally, we have the following formulation of the method of variation
+of constants for integro-differential operators.
+Recall from Section 4.1, that
+the Wronskian W(f1, . . . , fn) of elements f1, . . . , fn ∈ R is defined completely
+analogous to the analytic situation.
+Theorem 5.1. Let (R, ∂,
+�
+) be a commutative integro-differential ring and let
+L = ∂n + �n−1
+i=0 ai · ∂i ∈ R⟨∂,
+�
+, E⟩, n ≥ 1, with a0, . . . , an−1 ∈ R. Assume that
+z1, . . . , zn ∈ R are such that Lzi = 0 and w := W(z1, . . . , zn) has a multiplicative
+inverse
+1
+w ∈ R. Then, with
+H :=
+n
+�
+i=1
+(−1)n−izi ·
+�
+· W(z1, . . . , zi−1, zi+1, . . . , zn)
+w
+we have that L · H = 1 in R⟨∂,
+�
+, E⟩.
+Proof. The cases n = 1 and n = 2 have been shown above. In principle, an
+analogous computation could be done for any concrete n ≥ 3 as well. To obtain
+a finite proof for all n ≥ 3 at once, we will utilize a more general framework in
+Section 5.3.
+5.2
+Linear systems and operators with matrix coefficients
+Algebraically, computing with linear systems of differential equations can be
+modelled by integro-differential operators over some noncommutative integro-
+differential ring, whose elements correspond to matrices. Such noncommutative
+integro-differential rings can be obtained from any scalar integro-differential
+26
+
+ring, since one can equip the ring of n × n matrices with a derivation and an
+integration defined entrywise, see Lemma 2.7.
+Recall that Definition 4.1 and Theorem 4.2 hold also for operators with
+coefficients in noncommutative integro-differential rings, in particular for oper-
+ators with matrix coefficients. Consequently, any computation using a general
+noncommutative integro-differential ring is automatically valid for concrete ma-
+trices of any size, without the need for entrywise computations. This allows for
+compact proofs for matrices of general size. For switching to entrywise com-
+putations, one can view operators with matrix coefficients equivalently also as
+matrices of operators with scalar coefficients by identifying operators ∂,
+�
+, E
+with corresponding diagonal matrices of operators. The following lemma shows
+that also computations are equivalent in both viewpoints. Here, we use the
+notation Ei,j(L) := (δi,kδj,lL)k,l=1,...,n for matrices with only one nonzero entry
+L ∈ R⟨∂,
+�
+, E⟩.
+Lemma 5.2. Let (R, ∂,
+�
+) an integro-differential ring and let n ≥ 1. Then,
+there is exactly one ring homomorphism
+ϕ : Rn×n⟨∂,
+�
+, E⟩ → R⟨∂,
+�
+, E⟩n×n
+with ϕ(A) = A for A ∈ Rn×n and ϕ(L) = diag(L, . . . , L) for L ∈ {∂,
+�
+, E}.
+This ϕ is an isomorphism and its inverse homomorphism
+ψ : R⟨∂, � , E⟩n×n → Rn×n⟨∂, � , E⟩
+can be given by ψ(Ei,j(f)) = Ei,j(f) and ψ(Ei,j(L)) = Ei,j(1) · L for all f ∈ R
+resp. L ∈ {∂,
+�
+, E}.
+Proof. First, we check that the definition of ϕ indeed provides a unique and
+well-defined homomorphism of rings. Uniqueness follows from the fact that ϕ is
+defined on a generating set of Rn×n⟨∂,
+�
+, E⟩. For proving well-definedness, we
+need to verify that the definition of ϕ respects all identities of generators given
+in Definition 4.1 as follows. Evidently, we have ϕ(1) = In. It is immediate to see
+that ϕ(∂) · ϕ(
+�
+) = ϕ(1) − ϕ(E) and ϕ(
+�
+) · ϕ(∂) = ϕ(1) hold. For L ∈ {∂,
+�
+, E},
+we verify in R⟨∂,
+�
+, E⟩n×n that ϕ(L) commutes with all elements of Cn×n and
+that, for all A ∈ Rn×n, we have ϕ(L) · ϕ(A) · ϕ(E) = ϕ(LA) · ϕ(E).
+More
+explicitly, using (14)–(16) in R⟨∂,
+�
+, E⟩, we compute
+ϕ(L) · ϕ(A) · ϕ(E) = diag(L, . . . , L) · A · diag(E, . . . , E)
+= (L · ai,j · E)i,j=1,...,n = (Lai,j · E)i,j=1,...,n = ϕ(LA) · ϕ(E).
+Analogously, using (11) in R⟨∂,
+�
+, E⟩, we can also verify that ϕ(∂) · ϕ(A) =
+ϕ(A) · ϕ(∂) + ϕ(∂A) for all A ∈ Rn×n.
+Similarly, we need to verify that the definition of ψ on a generating set of
+R⟨∂,
+�
+, E⟩n×n gives rise to a well-defined homomorphism. For example, using
+27
+
+(14)–(16) in Rn×n⟨∂,
+�
+, E⟩, we compute
+ψ(Ei,j(L)) · ψ(Ep,q(f)) · ψ(Ek,l(E)) = Ei,j(1) · L · Ep,q(f) · Ek,l(1) · E
+= Ei,j(1) · L · δq,kEp,l(f) · E = Ei,j(1) · δq,kLEp,l(f) · E = δj,pδq,kEi,l(Lf) · E
+= Ei,q(δj,pLf) · Ek,l(1) · E = ψ(Ei,q(δj,pLf)) · ψ(Ek,l(E))
+for all f ∈ R, L ∈ {∂,
+�
+, E}, and all i, j, k, l, p, q ∈ {1, . . ., n}.
+Finally, ψ is the inverse of ϕ, since we have ψ(ϕ(A)) = A, ϕ(ψ(Ei,j(f))) =
+Ei,j(f), ψ(ϕ(L)) = L, and ϕ(ψ(Ei,j(L))) = Ei,j(L) for the generators.
+Integro-differential operators with coefficients from Rn×n constructed this
+way have natural actions on Rn×n and on Rn.
+By the above lemma, for
+any left R⟨∂,
+�
+, E⟩-module M, there is a natural way of viewing M n as a
+left Rn×n⟨∂, � , E⟩-module.
+It is straightforward to show that the action of
+Rn×n⟨∂,
+�
+, E⟩ on M n is faithful if and only if R⟨∂,
+�
+, E⟩ acts faithfully on M.
+5.3
+Variation of constants for first-order systems
+Instead of higher order scalar ODEs, we now consider variation of constants for
+first-order systems, see Theorem 3.1 in Chapter 3 of [6] for example. Analyti-
+cally, it can be stated as follows. For an n × n matrix A(x) and a vector f(x)
+of size n, we consider the first-order system given by
+y′(x) + A(x)y(x) = f(x).
+If Φ(x) is a fundamental matrix of the homogeneous system, i.e. it satisfies
+Φ′(x) + A(x)Φ(x) = 0 and det(Φ(x)) ̸= 0, then a particular solution of the
+inhomogeneous system is given by
+z∗(x) = Φ(x)
+� x
+x0
+Φ(t)−1f(t) dt.
+Theorem 5.3. Let (R, ∂,
+�
+) be an integro-differential ring. Assume that a ∈ R
+is such that there exists z ∈ R that satisfies ∂z +az = 0 and has a multiplicative
+right inverse z−1 ∈ R.
+Then, in R⟨∂,
+�
+, E⟩, the operators L := ∂ + a and
+H := z ·
+�
+· z−1 satisfy
+L · H = 1.
+Proof. The same computation as in the commutative case above can be done
+also for noncommutative R without any changes. Note that zz−1 = 1 was the
+only property of z−1 used there, so it suffices that z−1 is a right inverse of z.
+Observe that from this statement over an abstract integro-differential ring
+(R, ∂,
+�
+), which was proven without referring to matrices at all, the analytic
+statement follows for arbitrary size n of the matrix A(x), provided we assume
+sufficient regularity of the functions involved.
+Based on Theorem 5.3, we now can complete the proof of Theorem 5.1 for
+arbitrary n ≥ 3. Before doing so, we first detail the required translation from
+28
+
+a first-order system to the scalar equation entirely at the operator level. For
+shorter notation, in the following, we again use the symbol ∂ also for the matrix
+diag(∂, . . . , ∂) ∈ R⟨∂,
+�
+, E⟩n×n.
+Lemma 5.4. Let (R, ∂,
+�
+) be an integro-differential ring and let
+L = ∂n +
+n−1
+�
+i=0
+ai · ∂i ∈ R⟨∂,
+�
+, E⟩
+with a0, . . . , an−1 ∈ R. Moreover, let H ∈ R⟨∂,
+�
+, E⟩n×n satisfy (∂+A)·H = In
+in R⟨∂,
+�
+, E⟩n×n, where
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+.
+Then, the upper right entry H1,n of H satisfies L · H1,n = 1 in R⟨∂,
+�
+, E⟩ and
+Hi,n = ∂i−1 · H1,n for i ∈ {1, . . . , n}.
+Proof. For n = 1, the statement is trivial. So, we let n ≥ 2 in the following. In
+short, starting from the identity (∂ + A)·H = In of n× n matrices of operators,
+we multiply both sides from the left with a suitable S ∈ R⟨∂,
+�
+, E⟩n×n and
+inspect the last column. In particular, for elimination in the last n − 1 columns
+of ∂ + A, we use
+S :=
+
+
+
+
+
+
+
+
+
+
+
+1
+0
+· · ·
+· · ·
+· · ·
+0
+∂
+...
+...
+...
+∂2
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+∂n−2
+· · ·
+∂2
+∂
+1
+0
+s1
+· · ·
+· · ·
+· · ·
+sn−1
+1
+
+
+
+
+
+
+
+
+
+
+
+with sj := ∂n−j+�n−j−1
+i=0
+ai+j ·∂i ∈ R⟨∂,
+�
+, E⟩ for j ∈ {1, . . ., n−1}. Observing
+that, in R⟨∂, � , E⟩, we have −sj + sj+1 · ∂ = −aj for j ∈ {1, . . ., n − 2}, we
+29
+
+compute S · (∂ + A) explicitly:
+S ·
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+∂
+−1
+a0
+a1
+· · ·
+an−2
+∂ + an−1
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+0
+· · ·
+0
+−1
+s1 · ∂ + a0
+0
+· · ·
+0
+−sn−1 + ∂ + an−1
+
+
+
+
+
+
+
+
+=
+
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+...
+...
+−1
+L
+0
+· · ·
+· · ·
+0
+
+
+
+
+
+
+
+
+
+.
+Altogether, we obtain that
+
+
+
+
+
+
+
+
+
+∂
+−1
+0
+· · ·
+0
+∂2
+0
+−1
+...
+...
+...
+...
+...
+...
+0
+∂n−1
+...
+...
+−1
+L
+0
+· · ·
+· · ·
+0
+
+
+
+
+
+
+
+
+
+· H = S · (∂ + A) · H = S,
+where comparison of the entries in the last column of the left hand side and
+right hand side yields ∂ · H1,n − H2,n = 0, . . . , ∂n−1 · H1,n − Hn,n = 0 and
+L · H1,n = 0.
+Finally, we proceed with the proof of Theorem 5.1. Recalling the assump-
+tions, we fix a commutative integro-differential ring (R, ∂,
+�
+) and we let L =
+∂n+�n−1
+i=0 ai·∂i ∈ R⟨∂,
+�
+, E⟩ with a0, . . . , an−1 ∈ R. We also let z1, . . . , zn ∈ R
+such that Lzi = 0 and such that w := W(z1, . . . , zn) has a multiplicative inverse
+1
+w ∈ R.
+Proof of Theorem 5.1. Let n ≥ 3. In the (noncommutative) integro-differential
+30
+
+ring (Rn×n, ∂,
+�
+), we consider the matrices
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+and
+Z :=
+
+
+
+
+
+z1
+· · ·
+zn
+∂z1
+· · ·
+∂zn
+...
+...
+∂n−1z1
+· · ·
+∂n−1zn
+
+
+
+
+
+and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w
+was assumed to be invertible in R.
+Then, we apply Theorem 5.3 to obtain
+(∂ + A) · (Z ·
+�
+· Z−1) = 1 in Rn×n⟨∂,
+�
+, E⟩.
+Instead of integro-differential
+operators with matrix coefficients in Rn×n, we now consider the objects as
+n × n matrices whose entries are integro-differential operators with coefficients
+in R, cf. Lemma 5.2. Then, by Lemma 5.4, we obtain that L·(Z ·� ·Z−1)1,n = 1
+holds in R⟨∂,
+�
+, E⟩. In order to compute the entry (Z ·
+�
+·Z−1)1,n, we can easily
+determine a general form of the entries of the matrix product Z ·
+�
+· Z−1 ∈
+R⟨∂,
+�
+, E⟩n×n:
+Z ·
+
+
+
+
+
+
+�
+0
+· · ·
+0
+0
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+0
+�
+
+
+
+
+
+
+· Z−1 = Z ·
+
+
+
+�
+· (Z−1)1,1
+· · ·
+�
+· (Z−1)1,n
+...
+...
+�
+· (Z−1)n,1
+· · ·
+�
+· (Z−1)n,n
+
+
+
+=
+� n
+�
+k=1
+(∂i−1zk) · � · (Z−1)k,j
+�
+i,j=1,...,n
+.
+Then, we compute all entries
+(Z−1)i,n = (−1)n+i W(z1, . . . , zi−1, zi+1, . . . , zn)
+w
+in the last column of Z−1 ∈ Rn×n via Cramer’s rule (or via the cofactor matrix)
+so that we recognize H ∈ R⟨∂,
+�
+, E⟩ defined in the statement of Theorem 5.1
+as the top right entry (Z ·
+�
+· Z−1)1,n of the above matrix. This concludes the
+proof that L · H = 1.
+6
+Generalizing identities from calculus
+The generalizations of well-known identities presented in this section introduce
+additional terms involving the induced evaluation, which vanish if the evaluation
+is multiplicative. As explained in the previous section, the proofs mostly rely
+on computing irreducible forms for IDO.
+31
+
+6.1
+Initial value problems
+Recall that the formula given in (23) provides a particular solution of the inho-
+mogeneous linear ODE
+y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).
+Moreover, this particular solution also satisfies the homogeneous initial condi-
+tions y(x0) = 0, y′(x0) = 0, . . . , y(n−1)(x0) = 0, see e.g. Theorem 6.4 in Chap-
+ter 3 of [6].
+The induced evaluation of an integro-differential ring allows us
+to model such properties algebraically. Similarly to the general proof of Theo-
+rem 5.1, we first investigate the general solution formula for homogeneous initial
+value problems of first-order systems. To this end, we fix a (not necessarily com-
+mutative) integro-differential ring R and we work in the corresponding ring of
+IDO R⟨∂,
+�
+, E⟩.
+As a general principle, not only in the ring R⟨∂,
+�
+, E⟩ but in any ring with
+unit element, if some H is a right inverse of some L and B, P are such that
+L · P = 0 and B · P = B, then G := (1 − P) · H satisfies
+L · G = 1
+and
+B · G = 0.
+So, by choosing an appropriate operator P ∈ R⟨∂,
+�
+, E⟩ that satisfies (∂+a)·P =
+0 and E · P = E, we can obtain the following version of Theorem 5.3 that also
+solves the homogeneous initial condition.
+Theorem 6.1. Let (R, ∂,
+�
+) be an integro-differential ring and let L = ∂ + a ∈
+R⟨∂,
+�
+, E⟩ with a ∈ R. Assume that z ∈ R is such that ∂z + az = 0 and in
+addition to a multiplicative right inverse z−1 ∈ R also a right inverse (Ez)−1 ∈ C
+exists. Then, in R⟨∂, � , E⟩, the operator
+G := (1 − z(Ez)−1 · E) · z ·
+�
+· z−1
+satisfies L · G = 1 and E · G = 0.
+Proof. By Theorem 5.3, we have that H := z ·
+�
+· z−1 satisfies L · H = 1. Now,
+letting P := z(Ez)−1 · E, we compute the normal forms of L · P and E · P:
+(∂ + a) · z(Ez)−1 · E =
+�
+z(Ez)−1 · ∂ + ∂z(Ez)−1�
+· E + az(Ez)−1 · E
+= z(Ez)−1 · ∂ · E = 0,
+E · z(Ez)−1 · E = Ez(Ez)−1 · E = E.
+Then, from L · P = 0 and E · P = E, it follows straightforwardly that G =
+(1 − P) · H has the properties claimed.
+Remark 6.2. If the evaluation E is multiplicative, then the existence of (Ez)−1 ∈
+C follows form the existence of z−1 ∈ R and, with (22), we also obtain E·z ·
+�
+=
+(Ez)E ·
+�
+= 0, which implies P · H = 0 and hence G = H. Therefore, the right
+inverse H obtained in Theorem 5.3 satisfies the initial value condition E ·H = 0
+already.
+32
+
+For conversion to the scalar case, we need to compute the element in the top
+right corner of G = (1 − Z(EZ)−1 · E) · Z ·
+�
+· Z−1 and show that it has the
+desired properties.
+Theorem 6.3. Let (R, ∂,
+�
+) be a commutative integro-differential ring and let
+L = ∂n + �n−1
+i=0 ai · ∂i ∈ R⟨∂,
+�
+, E⟩, n ≥ 1, with a0, . . . , an−1 ∈ R. Assume
+that z1, . . . , zn ∈ R are such that Lzi = 0 and w := W(z1, . . . , zn) has a mul-
+tiplicative inverse
+1
+w ∈ R. Moreover, assume that there are ci,j ∈ C such that
+E∂k �n
+i=1 ci,jzi = δj,k for all j, k ∈ {0, . . . , n − 1}. Then, with these ci,j and
+G :=
+n
+�
+k=1
+(−1)n−k
+�
+zk −
+n
+�
+i,j=1
+zici,j−1 · E · (∂j−1zk)
+�
+·
+�
+· W(z1,...,zk−1,zk+1,...,zn)
+w
+we have that L · G = 1 and E · G = 0 in R⟨∂,
+�
+, E⟩.
+Proof. As in the proof of Theorem 5.1, we consider the matrices
+A :=
+
+
+
+
+
+
+
+
+0
+−1
+0
+· · ·
+0
+...
+...
+...
+...
+...
+...
+...
+...
+0
+0
+· · ·
+· · ·
+0
+−1
+a0
+a1
+· · ·
+· · ·
+an−1
+
+
+
+
+
+
+
+
+and
+Z :=
+
+
+
+
+
+z1
+· · ·
+zn
+∂z1
+· · ·
+∂zn
+...
+...
+∂n−1z1
+· · ·
+∂n−1zn
+
+
+
+
+
+and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w was
+assumed to be invertible in R. Furthermore, we note that
+
+
+
+c1,0
+· · ·
+c1,n−1
+...
+...
+cn,0
+· · ·
+cn,n−1
+
+
+
+is the multiplicative (right) inverse of EZ in Cn×n. By Theorem 6.1, we conclude
+that ˜G := (1−Z(EZ)−1 ·E)·Z ·
+�
+·Z−1 ∈ Rn×n⟨∂,
+�
+, E⟩ satisfies (∂ +A)· ˜G = 1
+and E · ˜G = 0. Passing to R⟨∂,
+�
+, E⟩n×n via Lemma 5.2, the latter identity
+implies E · ˜Gi,j = 0 and the former implies L · ˜G1,n = 1 and ˜Gi,n = ∂i−1 · ˜G1,n
+by Lemma 5.4. Hence, we also have E · ∂i−1 · ˜G1,n = 0 for i ∈ {1, . . . , n}.
+Finally, we verify that ˜G1,n = G. With H := Z ·
+�
+· Z−1 ∈ R⟨∂,
+�
+, E⟩n×n,
+we have ˜G1,n = H1,n − �n
+i,j=1 Z1,i((EZ)−1)i,j · E · Hj,n. From the proof of
+Theorem 5.1, we obtain that Hj,n = (−1)n+k(∂j−1zk)·
+�
+· W(z1,...,zi−1,zi+1,...,zn)
+w
+.
+Altogether, this yields ˜G1,n = G.
+6.2
+Taylor formula
+Usually, Taylor’s theorem is only considered for sufficiently smooth functions.
+In integro-differential rings, an analog of the Taylor formula with integral re-
+mainder term
+f(x) =
+n
+�
+k=0
+f (k)(x0)
+k!
+xk +
+� x
+x0
+(x − t)n
+n!
+f (n+1)(t) dt
+33
+
+can be formulated. While the formula arising from the identity of operators in
+Theorem 6.6 is more complicated, it is also valid if singularities are present.
+We start by giving a first version of the Taylor formula where the remainder
+term is given as repeated integral. It simply follows by iterating (3) resp. (13).
+See also Corollary 2.2.1 in [23].
+Lemma 6.4. Let (R, ∂,
+�
+) be an integro-differential ring. In R⟨∂,
+�
+, E⟩ we have
+for any n ∈ N that
+1 =
+n
+�
+i=0
+� i · E · ∂i +
+� n+1 · ∂n+1.
+Proof. For any n ∈ N, we can use (13) to rewrite the right hand side:
+n
+�
+i=0
+� i · E · ∂i +
+� n+1 · ∂n+1 =
+n
+�
+i=0
+� i · E · ∂i +
+� n · (1 − E) · ∂n
+=
+n−1
+�
+i=0
+� i · E · ∂i +
+� n · ∂n.
+Setting n = 0 here gives E+
+�
+·∂ = 1. Hence, the claim follows by induction.
+Using (15) and the related identity in (20), we can always write operators
+� i · E as xi · E, where xi := � i1 as in Theorem 3.2. By (19) and (21), we can
+always write
+� n+1 without higher powers of
+�
+. To make the resulting expressions
+simpler, we restrict to the case that E is multiplicative on polynomials, i.e.
+Exmxn = 0 for all m, n ≥ 1.
+Lemma 6.5. Let (R, ∂,
+�
+) be an integro-differential ring such that the induced
+evaluation E satisfies Exmxn = 0 for all m, n ≥ 1. Then, in R⟨∂,
+�
+, E⟩, we
+have for any n ∈ N that
+� n+1 =
+n
+�
+k=0
+(−1)n−kxk ·
+�
+· xn−k −
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk · E · xj ·
+�
+· xn−k−j.
+Proof. We prove this identity by induction on n ∈ N. For n = 0, the right
+hand side directly yields
+�
+in agreement with the left hand side. Assuming the
+identity holds for some n ∈ N, we multiply both sides by
+�
+from the left and we
+rewrite the right hand side using (19) and (15) to obtain
+� n+2 =
+n
+�
+k=0
+(−1)n−k�� xk · � − � · � xk − E · � xk · � �
+· xn−k
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−j�
+xk · E · xj ·
+�
+· xn−k−j.
+34
+
+We have xk+1xn−k =
+�n+1
+k+1
+�
+xn+1 by Theorem 3.2, so we can expand the right
+hand side into
+n
+�
+k=0
+(−1)n−kxk+1 ·
+�
+· xn−k +
+n
+�
+k=0
+(−1)n−k+1
+�n + 1
+k + 1
+��
+· xn+1
+−
+n
+�
+k=0
+(−1)n−kE · xk+1 · � · xn−k −
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk+1 · E · xj · � · xn−k−j.
+Exploiting �n
+k=0(−1)n−k+1�n+1
+k+1
+�
+= (−1)n+1 in the second sum, we can regroup
+terms to obtain the right hand side of the claimed identity for n + 1.
+This
+completes the induction.
+Altogether, we obtain the following identity in R⟨∂,
+�
+, E⟩ generalizing the
+usual Taylor formula with integral remainder term. This identity holds over any
+integro-differential ring in which the induced evaluation is multiplicative on the
+integro-differential subring generated by 1.
+Theorem 6.6 (Taylor formula). Let (R, ∂,
+�
+) be an integro-differential ring
+such that E is multiplicative on the integro-differential subring generated by 1,
+i.e. Exmxn = 0 for all m, n ≥ 1. Then, for all f ∈ R and all n ∈ N we have
+1 =
+n
+�
+k=0
+xk · E · ∂k +
+n
+�
+k=0
+(−1)n−kxk ·
+�
+· xn−k · ∂n+1
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxk · E · xj · � · xn−k−j · ∂n+1
+Proof. Follows from Lemma 6.4 using
+� i·E = xi·E and Lemma 6.5, as explained
+above.
+In particular, if in addition to the assumptions of the theorem we have
+Q ⊆ R, then, with operators acting on some f, we have that
+f =
+n
+�
+k=0
+xk
+1
+k! E∂kf +
+n
+�
+k=0
+(−1)n−k
+k!(n − k)!xk
+1
+�
+xn−k
+1
+∂n+1f
+−
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−j
+k!j!(n − k − j)!xk
+1Exj
+1
+�
+xn−k−j
+1
+∂n+1f
+While the first and the second sum correspond to the Taylor polynomial and the
+integral remainder term, the third sum corresponds to an additional polynomial
+that arises from our general setting allowing non-multiplicative evaluations. It
+can be viewed as an integro-differential algebraic version of the analytic formula
+−
+n−1
+�
+k=0
+(x − x0)k
+k!
+�� x
+x0
+(x − t)n−k − (x0 − t)n−k
+(n − k)!
+f (n+1)(t)dt
+�
+x=x0
+,
+35
+
+which vanishes for smooth functions f(x). Although in practice these additional
+terms yield zero even for many elements f that model singular functions in con-
+crete integro-differential rings, they cannot be dropped in general, as illustrated
+in C[x, x−1, ln(x)] by considering n = 1 and f = ln(x), for example. With in-
+tegration defined as in Example 2.13, we have x1 = x and with f = ln(x) the
+Taylor polynomial Ef + xE∂f vanishes. By ∂2f = − 1
+x2 , the second sum yields
+−
+�
+x∂2f + x
+�
+∂2f = ln(x) + 1 and the third sum −Ex
+�
+∂2f = −1 compensates
+the constant term. More generally, we can characterize the integro-differential
+rings where the additional polynomial does not play a role in the Taylor formula.
+Corollary 6.7. Let (R, ∂,
+�
+) be an integro-differential ring such that E is mul-
+tiplicative on the integro-differential subring generated by 1, i.e. Exmxn = 0 for
+all m, n ≥ 1. Then, we have Exng = 0 for all n ≥ 1 and g ∈ R if and only if
+we have
+f =
+n
+�
+k=0
+xkE∂kf +
+n
+�
+k=0
+(−1)n−kxk
+�
+xn−k∂n+1f
+for all n ∈ N and f ∈ R.
+Proof. If Exng = 0 for all n ≥ 1 and g ∈ R, then we have in particular
+Exj
+�
+xn−k−j∂n+1f = 0 for all j, k, n ∈ N and f ∈ R with 1 ≤ j ≤ n − k.
+Hence, Theorem 6.6 implies the claimed identity for all n ∈ N and f ∈ R.
+For the converse, let n ≥ 1 be minimal such that Exng ̸= 0 for some g ∈ R.
+With such g, we let f :=
+� ng and, by minimality of n, we obtain that
+n−1
+�
+k=0
+n−k
+�
+j=1
+(−1)n−k−jxkExj
+�
+xn−k−j∂n+1f = Exn
+�
+∂g = Exng − ExnEg = Exng
+is nonzero. Consequently, Theorem 6.6 implies that the claimed identity does
+not hold for this n and f.
+Acknowledgements
+This work was supported by the Austrian Science Fund (FWF): P 27229,
+P 31952, and P 32301. Part of this work was done while both authors were
+at the Radon Institute of Computational and Applied Mathematics (RICAM)
+of the Austrian Academy of Sciences. The authors would like to thank Alban
+Quadrat for bringing the book [23] to the attention of the second author during
+his stay at INRIA Saclay.
+A
+Normal forms for IDO in tensor rings
+The goal of this appendix is to state and prove a refinement of Theorem 4.2
+providing uniqueness of normal forms. Uniqueness is achieved by representing
+operators by elements of a tensor ring, which is formed on a module of basic
+operators generated by R, ∂,
+�
+, E. The ring of operators can be constructed
+36
+
+as quotient of the tensor ring, where relations of basic operators are encoded by
+tensor reduction rules. The main technical tool for proving uniqueness of nor-
+mal forms is a generalization of Bergman’s Diamond Lemma in tensor rings [3].
+For the convenience of the reader, we give a formal and largely self-contained
+summary of tensor reduction systems and we explain the translation of identi-
+ties of operators into this framework. In this appendix, K denotes a ring (not
+necessarily commutative) with unit element.
+In Section A.1, we start by recalling basic properties of bimodules and tensor
+rings on them. For further details on tensor rings and proofs see, for example,
+[34, 7]. Then, we recall decompositions with specialization, which are used for
+defining reduction rules. In Section A.2, we state the Diamond Lemma for tensor
+reduction systems with specialization from [14] and provide a summary of the
+relevant notions. In Section A.3, we construct an appropriate tensor ring along
+with a tensor reduction system for dealing with IDO. We use these to state The-
+orem A.3, which provides a precise formulation of uniqueness of normal forms
+of IDO in terms of tensors. In Section A.4, we give a complete proof of the
+theorem, where the necessary computations for verifying uniqueness of normal
+forms in the tensor ring are done in an automated way by our package TenReS in
+the computer algebra system Mathematica. These computations are contained
+in the Mathematica file accompanying this paper, which includes a log of the re-
+duction steps and is available at http://gregensburger.com/softw/tenres/.
+The theorem proved in that section even covers more general rings of operators,
+which allow to deal with additional functionals besides the induced evaluation.
+A.1
+Tensor rings on bimodules and decompositions
+A K-bimodule is a left K-module M which is also a right K-module satisfying
+the associativity condition (km)l = k(ml) for all m ∈ M and k, l ∈ K. A ring R
+that is a K-bimodule such that (xy)z = x(yz) for any x, y, z in R or K is called
+a K-ring. In particular, if K is a subring of some ring R, then R is a K-ring.
+We first recall basic properties of the tensor product on K-bimodules. For
+K-bimodules M, N, their K-tensor product M ⊗ N is a K-bimodule generated
+by the pure tensors {m ⊗ n | m ∈ M, n ∈ N} with relations
+(m + ˜m) ⊗ n = m ⊗ n + ˜m ⊗ n,
+m ⊗ (n + ˜n) = m ⊗ n + m ⊗ ˜n,
+and
+mk ⊗ n = m ⊗ kn
+having scalar multiplications
+k(m ⊗ n) = (km) ⊗ n
+and
+(m ⊗ n)k = m ⊗ (nk)
+for all m, ˜m ∈ M, n, ˜n ∈ N, and k ∈ K.
+We denote the tensor product of M with itself over K by M ⊗n = M ⊗· · ·⊗M
+(n factors). In particular, M ⊗1 = M and we interpret M ⊗0 as the K-module
+Kε, where ε denotes the empty tensor and right scalar multiplication satisfies
+(k1ε)k2 = (k1k2)ε for k1, k2 ∈ K. As a K-bimodule, the tensor ring K⟨M⟩ is
+37
+
+defined as the direct sum
+K⟨M⟩ =
+∞
+�
+n=0
+M ⊗n.
+It can be turned into a K-ring with unit element ε where multiplication M ⊗r ×
+M ⊗s → M ⊗(r+s) is defined via
+(m1 ⊗ · · · ⊗ mr, ˜m1 ⊗ · · · ⊗ ˜ms) �→ m1 ⊗ · · · ⊗ mr ⊗ ˜m1 ⊗ · · · ⊗ ˜ms.
+Via the tensor product, any decomposition of the module M carries over to
+a decomposition of the tensor ring K⟨M⟩. In particular, we use decompositions
+with specialization, which were introduced in [14]. These are given by a family
+(Mz)z∈Z of K-subbimodules of M and a subset X ⊆ Z with M = �
+z∈Z Mz =
+�
+x∈X Mx such that every module Mz, z ∈ Z, satisfies
+Mz =
+�
+x∈S(z)
+Mx
+where S(z) := {x ∈ X | Mx ⊆ Mz} is the set of specializations of z. Note that
+this definition implies S(x) = {x} for x ∈ X. For words W = w1 . . . wn in the
+word monoid ⟨Z⟩, we define the corresponding K-subbimodule of K⟨M⟩ by
+MW := Mw1 ⊗ · · · ⊗ Mwn.
+The notion of specialization extends from the alphabet Z to the whole word
+monoid ⟨Z⟩ by
+S(W) := {v1 . . . vn ∈ ⟨X⟩ | ∀i : vi ∈ S(wi)}
+such that S(W) = {V ∈ ⟨X⟩ | MV ⊆ MW }. We have the following generaliza-
+tion
+MW =
+�
+V ∈S(W)
+MV
+of the direct sum above and the decomposition
+K⟨M⟩ =
+�
+W∈⟨Z⟩
+MW =
+�
+W∈⟨X⟩
+MW
+of the tensor ring.
+A.2
+Tensor reduction systems with specialization
+Fixing a decomposition with specialization of M, a reduction rule for K⟨M⟩ is
+given by a pair r = (W, h) of a word W ∈ ⟨Z⟩ and a K-bimodule homomorphism
+h: MW → K⟨M⟩. It acts on tensors of the form a⊗w⊗b with a ∈ MA, w ∈ MW ,
+and b ∈ MB for some A, B ∈ ⟨Z⟩ by a ⊗ w ⊗ b →r a ⊗ h(w) ⊗ b. Later, we will
+specify homomorphisms h in concrete reduction rules (W, h) via their values on
+a generating set of MW . Formally, well-definedness of such homomorphisms can
+38
+
+be ensured by the universal property of the tensor product. A set Σ of such
+reduction rules is called a reduction system over Z on K⟨M⟩ and induces the
+two-sided reduction ideal
+IΣ := (t − h(t) | (W, h) ∈ Σ and t ∈ MW ) ⊆ K⟨M⟩.
+For computing in the factor ring K⟨M⟩/IΣ, we apply the reduction relation →Σ
+induced by Σ on K⟨M⟩. It reduces a tensor t ∈ K⟨M⟩ to a tensor s ∈ K⟨M⟩
+if there is an r ∈ Σ such that t →r s. We say that t can be reduced to s by Σ
+if t = s or there exists a finite sequence of reduction rules r1, . . . , rn in Σ such
+that
+t →r1 t1 →r2 · · · →rn−1 tn−1 →rn s.
+If one tensor can be reduced to another, then their difference is contained in
+IΣ and they represent the same element of K⟨M⟩/IΣ. The K-subbimodule of
+irreducible tensors
+K⟨M⟩irr =
+�
+W∈⟨X⟩irr
+MW
+can be characterized by the set of irreducible words ⟨X⟩irr ⊆ ⟨X⟩, which consists
+of those words that avoid subwords arising as specializations S(W) of words
+occurring in reduction rules (W, h) ∈ Σ. The irreducible tensors to which a
+given tensor t can be reduced, are called its normal forms. If t has a unique
+normal form, it is denoted by t↓Σ.
+An ambiguity is a minimal situation where two (not necessarily distinct) re-
+duction rules can be applied to tensors in different ways. For each pure tensor
+of this kind, the corresponding S-polynomial is the difference of the results of
+the two reduction steps. For example, an overlap ambiguity arises from two
+reduction rules (AB1, h), (B2C, ˜h) ∈ Σ, where A, B1, B2, C ∈ ⟨Z⟩ are nonempty
+such that B1, B2 are equal or have a common specialization, and corresponding
+S-polynomials are referred to by SP(AB1, B2C). An ambiguity is called resolv-
+able, if all its S-polynomials can be reduced to zero by Σ. If all ambiguities of Σ
+are resolvable, then the reduction relation induced by Σ on K⟨M⟩ is confluent
+and, by abuse of language, we also call Σ confluent. This means that there are
+no hidden consequences implied in K⟨M⟩/IΣ by the identities explicitly specified
+by Σ.
+The following theorem relies on the existence of a partial order of words in
+⟨Z⟩ that has certain properties, which are briefly explained now. A partial order
+≤ on ⟨Z⟩ is called a semigroup partial order if it is compatible with concatenation
+of words. If in addition the empty word ǫ is the least element of ⟨Z⟩, then ≤
+is called a monoid partial order. It is called Noetherian if there are no infinite
+descending chains. We call a partial order ≤ on ⟨Z⟩ consistent with specialization
+if every strict inequality V < W implies ˜V < ˜W for all specializations ˜V ∈ S(V )
+and ˜W ∈ S(W). A partial order ≤ on ⟨Z⟩ is compatible with a reduction system
+Σ over Z on K⟨M⟩ if for all (W, h) ∈ Σ the image of h is contained in the sum
+of modules MV where V ∈ ⟨Z⟩ satisfies V < W.
+39
+
+Theorem A.1. [14, Thm. 20] Let M be a K-bimodule and let (Mz)z∈Z be
+a decomposition with specialization.
+Let Σ be a reduction system over Z on
+K⟨M⟩ and let ≤ be a Noetherian semigroup partial order on ⟨Z⟩ consistent with
+specialization and compatible with Σ. Then, the following are equivalent:
+1. All ambiguities of Σ are resolvable.
+2. Every t ∈ K⟨M⟩ has a unique normal form t↓Σ.
+3. K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic as K-bimodules.
+Moreover, if these conditions are satisfied, then we can define a multiplication
+on K⟨M⟩irr by s · t := (s ⊗ t)↓Σ so that K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic
+as K-rings.
+A.3
+Tensor reduction systems for IDO
+Before using the tensor setting to construct the ring of integro-differential op-
+erators R⟨∂,
+�
+, E⟩, we illustrate this construction on the well-known ring of
+differential operators R⟨∂⟩ to highlight some of the special properties of the
+construction. Usually, the ring of differential operators with coefficients from R
+is constructed via skew polynomials �
+i fi∂i over R in one indeterminate ∂, with
+commutation rule ∂ · f = f · ∂ + ∂f. Let (R, ∂,
+�
+) be an integro-differential ring
+and let C denote its ring of constants. In the following, we consider K-tensor
+rings with K := C.
+Example A.2. The module of basic operators that generates all differential
+operators is given by
+M := MR ⊕ MD,
+with K-bimodules
+MR := R
+(24)
+and MD defined as the free left K-module
+MD := K∂
+(25)
+generated by the symbol ∂, which we view as a K-bimodule with the right
+multiplication c∂ · d = cd∂ for all c, d ∈ K. This definition is based on left
+K-linearity of the derivation ∂ on R. The commutation rule ∂ · f = f · ∂ + ∂f
+coming from the Leibniz rule in R translates to the tensor reduction rule
+(DR, ∂⊗f �→ f⊗∂ + ∂f).
+This rule is formalized by the K-bimodule homomorphism MDR = MD ⊗ MR →
+K⟨M⟩ defined by ∂⊗f �→ f⊗∂ + ∂f on tensors ∂⊗f generating MDR. Note that
+this homomorphism represents a parameterized family of identities. Reduction
+of tensors by the rule above allows to syntactically move all multiplication op-
+erators to the left of any differentiation operator.
+40
+
+Note that, in order to correctly model differential operators as equivalence
+classes of tensors in K⟨M⟩, other relations among operators need to be phrased
+as tensor reduction rules as well. This is because the tensor ring K⟨M⟩ itself
+is constructed without respecting relations coming from multiplication in R.
+For instance, the composition of two multiplication operators f and g is a mul-
+tiplication operator again, which leads to the reduction rule (RR, f⊗g �→ fg)
+defined on the module MRR = MR ⊗ MR. Moreover, the multiplication operator
+that multiplies by 1 acts like the identity operator, which is represented by the
+empty tensor ε. To define reduction rules that act only on C instead of all of R,
+we need a direct decomposition of the K-bimodule
+MR = MK ⊕ M˜R,
+(26)
+which in our case can be given by
+MK := K
+and
+M˜R :=
+�
+R
+(27)
+based on Lemma 2.3. Then, we can define a reduction rule on MK = C by 1 �→ ε.
+Altogether, the tensor reduction system ΣDiff = {rK, rRR, rDR} for differential
+operators is given by the three reduction rules
+{(K, 1 �→ ε),
+(RR, f⊗g �→ fg),
+(DR, ∂⊗f �→ f⊗∂ + ∂f)}
+defined above. It induces the two-sided ideal
+IDiff := (t − h(t) | (W, h) ∈ ΣDiff and t ∈ MW )
+= (1 − ε, f⊗g − fg, ∂⊗f − f⊗∂ − ∂f | f, g ∈ R)
+in the ring K⟨M⟩ and computations with differential operators are modelled in
+the quotient ring
+R⟨∂⟩ := K⟨M⟩/IDiff.
+Tensors that are not reducible w.r.t. ΣDiff are precisely K-linear combinations
+of pure tensors of the form ∂⊗i and f ⊗ ∂⊗i, where i ∈ N0 and f ∈
+�
+R. With
+alphabets X = {K, ˜R, D} and Z = X ∪{R}, one can check that all ambiguities of
+ΣDiff are resolvable. Using an appropriate ordering of words, one can show that
+the conditions of Theorem A.1 are satisfied by this construction of R⟨∂⟩.
+Proceeding to an analogous construction of the ring R⟨∂,
+�
+, E⟩ from Defini-
+tion 4.1, we also require tensor reduction rules corresponding to the identities
+(12)–(16), in addition to the three rules from the example above. To this end,
+analogous to MD above, we introduce the K-bimodules
+MI := K
+�
+and
+ME := KE,
+(28)
+which are freely generated as left K-modules by the symbols
+�
+and E, respec-
+tively. Then, we consider the K-tensor ring on the K-bimodule
+M := MR ⊕ MD ⊕ MI ⊕ ME.
+41
+
+K
+1 �→ ε
+ID
+�
+⊗∂ �→ ε − E
+RR
+f⊗g �→ fg
+DRE
+∂⊗f⊗E �→ ∂f⊗E
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRE
+�
+⊗f⊗E �→
+�
+f⊗E
+DI
+∂⊗
+�
+�→ ε
+ERE
+E⊗f⊗E �→ (Ef)E
+Table 2: Defining reduction system for integro-differential operators
+Altogether, we obtain the tensor reduction system given in Table 2. The two-
+sided ideal IIDO induced by it allows to construct the ring R⟨∂,
+�
+, E⟩ as the
+quotient K⟨M⟩/IIDO. However, this reduction system is not confluent. In order
+to obtain normal forms that are unique as tensors in K⟨M⟩, we need a confluent
+tensor reduction system on K⟨M⟩ that induces the same ideal IIDO. The con-
+fluent tensor reduction system given in Table 3 can be obtained by turning the
+identities of Table 1 into tensor reduction rules and including the rules rK and
+rRR from above. This allows us to state the following more precise version of
+K
+1 �→ ε
+EI
+E⊗
+�
+�→ 0
+RR
+f⊗g �→ fg
+IRD
+� ⊗f⊗∂ �→ f − E⊗f − � ⊗∂f
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRE
+�
+⊗f⊗E �→
+�
+f⊗E
+DE
+∂⊗E �→ 0
+IRI
+�
+⊗f⊗
+�
+�→
+�
+f⊗
+�
+− E⊗
+�
+f⊗
+�
+−
+�
+⊗
+�
+f
+DI
+∂⊗
+�
+�→ ε
+ID
+�
+⊗∂ �→ ε − E
+ERE
+E⊗f⊗E �→ (Ef)E
+IE
+�
+⊗E �→
+�
+1⊗E
+EE
+E⊗E �→ E
+II
+�
+⊗
+�
+�→
+�
+1⊗
+�
+− E⊗
+�
+1⊗
+�
+−
+�
+⊗
+�
+1
+Table 3: Confluent reduction system ΣIDO for integro-differential operators
+Theorem 4.2. Note that multiples like f · ∂ are treated differently now, due to
+the fact that we are working in the tensor ring K⟨M⟩ and we have the reduction
+rule (K, 1 �→ ε), which splits f ∈ R according to Lemma 2.3.
+Theorem A.3. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C. Let M be given as above in terms of the modules defined in Eqs. (26), (27),
+(25), and (28) and let the tensor reduction system ΣIDO be defined by Table 3.
+Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDO∈ K⟨M⟩, which can
+be written as a K-linear combination of pure tensors of the form
+f ⊗ ∂⊗j,
+f ⊗
+�
+⊗ g,
+f ⊗ E ⊗ g ⊗ ∂⊗j,
+or
+f ⊗ E ⊗ h ⊗
+�
+⊗ g
+where j ∈ N0, f, g, h ∈
+�
+R, and each f and g may be absent. Moreover,
+R⟨∂,
+�
+, E⟩ ∼= K⟨M⟩irr
+as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDO.
+Instead of proving this theorem, we will prove a more general one below,
+which allows to include additional functionals into the construction of the ring
+of operators. Theorem A.3 follows from Theorem A.5 by specializing Φ = {E}.
+42
+
+A.4
+Proof of normal forms for IDO with functionals
+To treat more general problems than the initial value problems in Section 6.1, it
+is useful to include additional functionals into the ring of operators. For instance,
+dealing with boundary problems requires evaluations at more than one point.
+In general, we consider a set Φ of K-linear functionals R → K including E. We
+consider the K-bimodule MΦ defined as free left K-module
+MΦ := KΦ
+(29)
+generated by the elements of Φ, where we define right multiplication in terms
+of Φ by
+��
+ϕ∈Φ cϕϕ
+�
+· d = �
+ϕ∈Φ cϕdϕ for cϕ, d ∈ K.
+Note that K-linear
+combinations of K-linear maps are not necessarily K-linear again, if K is not
+commutative.
+Since ∂,
+�
+, and all elements of Φ are K-linear, we have, for
+instance, that ϕfψg = (ϕf)ψg for all f, g ∈ R and ϕ, ψ ∈ Φ.
+This allows
+to extend the last three reduction rules of Table 2 to cover all elements of Φ
+instead of the evaluation E only. Altogether, we consider the K-tensor ring on
+the K-bimodule
+M := MR ⊕ MD ⊕ MI ⊕ MΦ
+(30)
+and we have the defining reduction system given in Table 4.
+K
+1 �→ ε
+ID
+� ⊗∂ �→ ε − E
+RR
+f⊗g �→ fg
+DRΦ
+∂⊗f⊗ϕ �→ ∂f⊗ϕ
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRΦ
+�
+⊗f⊗ϕ �→
+�
+f⊗ϕ
+DI
+∂⊗
+�
+�→ ε
+ΦRΦ
+ϕ⊗f⊗ψ �→ (ϕf)ψ
+Table 4: Defining reduction system for integro-differential operators with func-
+tionals
+Definition A.4. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C and let Φ be a set of K-linear functionals R → K including E. Let the K-
+bimodule M be defined as above in Eqs. (24), (25), (28), (29), and (30). We
+call
+R⟨∂,
+�
+, Φ⟩ := K⟨M⟩/IIDOΦ
+the ring of integro-differential operators with functionals Φ, where IIDOΦ is the
+two-sided ideal induced by the reduction system obtained from Table 4 using also
+submodules of M defined in Eq. (27).
+We use Theorem A.1 above to determine unique normal forms of tensors.
+Again, the reduction system given by Table 4 is not confluent and we need
+to construct a confluent reduction system, like ΣIDOΦ given in Table 5, by a
+completion process similar to how Table 3 was obtained. Observe that, whenever
+Φ = {E}, the ring R⟨∂, � , Φ⟩ and the relation →ΣIDOΦ specialize to R⟨∂, � , E⟩
+and →ΣIDO, respectively.
+43
+
+K
+1 �→ ε
+EI
+E⊗
+�
+�→ 0
+RR
+f⊗g �→ fg
+IRD
+�
+⊗f⊗∂ �→ f − E⊗f −
+�
+⊗∂f
+DR
+∂⊗f �→ f⊗∂ + ∂f
+IRΦ
+�
+⊗f⊗ϕ �→
+�
+f⊗ϕ
+DΦ
+∂⊗ϕ �→ 0
+IRI
+�
+⊗f⊗
+�
+�→
+�
+f⊗
+�
+− E⊗
+�
+f⊗
+�
+−
+�
+⊗
+�
+f
+DI
+∂⊗
+�
+�→ ε
+ID
+�
+⊗∂ �→ ε − E
+ΦRΦ
+ϕ⊗f⊗ψ �→ (ϕf)ψ
+IΦ
+�
+⊗ϕ �→
+�
+1⊗ϕ
+ΦΦ
+ϕ⊗ψ �→ (ϕ1)ψ
+II
+�
+⊗
+�
+�→
+�
+1⊗
+�
+− E⊗
+�
+1⊗
+�
+−
+�
+⊗
+�
+1
+Table 5: Confluent reduction system ΣIDOΦ for integro-differential operators
+with functionals
+Theorem A.5. Let (R, ∂,
+�
+) be an integro-differential ring with constants K =
+C and let Φ be a set of K-linear functionals R → K including E. Let M and
+R⟨∂,
+�
+, Φ⟩ be defined as in Definition A.4 above and let the tensor reduction
+system ΣIDOΦ be defined by Table 5.
+Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDOΦ∈ K⟨M⟩, which can
+be written as a K-linear combination of pure tensors of the form
+f ⊗ ∂⊗j,
+f ⊗
+�
+⊗ g,
+f ⊗ ϕ ⊗ h ⊗ ∂⊗j,
+or
+f ⊗ ϕ ⊗ h ⊗
+�
+⊗ g
+where j ∈ N0, f, g, h ∈
+�
+R, ϕ ∈ Φ, and each f, g, h may be absent such that
+ϕ ⊗ h ⊗
+�
+does not specialize to E ⊗
+�
+. Moreover,
+R⟨∂,
+�
+, Φ⟩ ∼= K⟨M⟩irr
+as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDOΦ.
+Proof. We use the alphabets X := {K, ˜R, D, I, E, ˜Φ} and Z := X ∪ {R, Φ}, which
+turns (Mz)z∈Z into a decomposition with specialization for the module M,
+where S(R) = {K, ˜R} and S(Φ) = {E, ˜Φ}. For defining a Noetherian monoid
+partial order ≤ on ⟨Z⟩ consistent with specialization that is compatible with
+ΣIDOΦ, it is sufficient to require the order to satisfy
+DR > RD
+and
+I > E˜R.
+For instance, we could first define a monoid total order on ⟨{R, D, I, Φ}⟩ ⊆
+⟨Z⟩ by counting occurrences of the letter I and breaking ties with any degree-
+lexicographic order satisfying D > R and then generate from it a partial order
+on ⟨Z⟩ that is consistent with specialization.
+By our Mathematica package TenReS, we generate all ambiguities of ΣIDOΦ
+and verify that they are resolvable.
+There are 54 ambiguities and indeed
+all S-polynomials reduce to zero, see the accompanying Mathematica file at
+http://gregensburger.com/softw/tenres/. Here, we just give two short ex-
+amples. The first one illustrates in its last step of computation that also iden-
+44
+
+tities in MR, like the Leibniz rule, need to be used.
+SP(IRD, DR) = (f − E ⊗ f −
+�
+⊗ ∂f) ⊗ g −
+�
+⊗ f ⊗ (g ⊗ ∂ + ∂g)
+→rRR fg − E ⊗ fg −
+�
+⊗ (∂f)g −
+�
+⊗ fg ⊗ ∂ −
+�
+⊗ f∂g
+→rIRD −
+�
+⊗ (∂f)g +
+�
+⊗ ∂fg −
+�
+⊗ f∂g
+=
+�
+⊗ (−(∂f)g + ∂fg − f∂g) = 0
+The second one illustrates that ambiguities involving specialization also need to
+be considered.
+SP(ΦΦ, EI) = (ϕ1)E ⊗
+�
+− ϕ ⊗ 0 →rEI 0
+Since all ambiguities are resolvable, by Theorem A.1 every element in K⟨M⟩ has
+a unique normal form and R⟨∂,
+�
+, Φ⟩ ∼= K⟨M⟩irr as K-rings.
+It remains to determine the explicit form of elements in K⟨M⟩irr. In order to
+do so, we determine the set of irreducible words ⟨X⟩irr in ⟨X⟩. Irreducible words
+containing only the letters K, ˜R, E, ˜Φ have to avoid the subwords arising from
+the reduction rules K, S(RR) = {KK, K˜R, ˜RK, ˜R˜R}, S(ΦΦ) = {EE, E˜Φ, ˜ΦE, ˜Φ˜Φ},
+and S(ΦRΦ). Hence they are given by
+ǫ, ˜R, E, ˜Φ, ˜RE, ˜R˜Φ, E˜R, ˜Φ˜R, ˜RE˜R, ˜R˜Φ˜R
+Allowing also the letter D, we have to avoid the subwords coming from S(DR) =
+{DK, D˜R} and S(DΦ) = {DE, D˜Φ}. Therefore, we can only append words Dj
+with j ∈ N0 to the irreducible words determined so far, in order to obtain all
+elements of ⟨X⟩irr not containing the letter I. Finally, we also consider the letter
+I. Since we have to avoid the subwords S(IΦ) = {IE, I˜Φ}, ID, and II, any letter
+immediately following I has to be ˜R. In addition, we have to avoid the subwords
+S(IRΦ) = {IKE, IK˜Φ, I˜RE, I˜R˜Φ}, S(IRD) = {IKD, I˜RD}, and S(IRI) = {IKI, I˜RI},
+so the letter I cannot be followed by a subword of length greater than one.
+Therefore, the letter I can appear at most once in an element of ⟨X⟩irr and, since
+subwords EI and DI have to be avoided, it can only be immediately preceded by
+the letters ˜R or ˜Φ. Altogether, the elements of ⟨X⟩irr are precisely of the form
+˜RUDj
+or
+˜RV I˜R,
+where j ∈ N0 and each of ˜R and U ∈ {E, ˜Φ, E˜R, ˜Φ˜R} and V ∈ {˜Φ, E˜R, ˜Φ˜R} may
+be absent.
+Remark A.6. As discussed in Remark 2.12, the induced evaluation of an
+integro-differential ring often is multiplicative in concrete examples. Also when
+considering several point evaluations of regular functions, the resulting func-
+tionals are multiplicative, i.e. ϕfg = (ϕf)ϕg for all f, g ∈ R. We discuss how
+reflecting this additional property of some functionals in the ring of operators
+influences the normal forms of operators. To this end, we consider a subset
+Φm ⊆ {ϕ ∈ Φ | ϕ is multiplicative and ϕ1 = 1}
+45
+
+of Φ, which may or may not include the evaluation E. For the corresponding
+elements ϕ ∈ Φm in R⟨∂,
+�
+, Φ⟩, we impose
+ϕ · f = (ϕf)ϕ
+for all f ∈ R.
+To model this identity by reduction rules, we consider the
+submodule
+MΦm := KΦm
+of MΦ. Evidently, we have the decomposition
+MΦ = ME ⊕ M˜Φm ⊕ M˜Φ
+where the submodules M˜Φm and M˜Φ are generated by Φm \ {E} and Φ \ ({E} ∪
+Φm), respectively. We include the reduction rule
+(ΦmR, ϕ⊗f �→ (ϕf)ϕ)
+into Tables 4 and 5, where ϕ in the formula defining this K-bimodule homomor-
+phism on MΦmR is not a general element of MΦm but of Φm and the definition
+needs to be extended by left K-linearity to all of MΦmR.
+Theorem A.5 generalizes to this situation with the additional restriction on
+normal forms in K⟨M⟩ that h needs to be absent whenever ϕ ∈ Φm.
+The
+proof is analogous by adapting the alphabets and the order accordingly. The
+additional reduction rule gives rise to additional ambiguities, whose resolvability
+is also checked in the Mathematica file. Determination of irreducible words also
+needs to be adapted accordingly. The resulting theorem includes Theorem A.5
+for Φm = ∅ and it includes Theorem 27 from [14] for Φm = Φ, i.e. when all
+functionals are multiplicative.
+References
+[1] Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven,
+Asymptotic differential algebra and model theory of transseries, Annals of
+Mathematics Studies, vol. 195, Princeton University Press, Princeton, NJ,
+2017.
+[2] Vladimir V. Bavula, The algebra of integro-differential operators on a poly-
+nomial algebra, J. Lond. Math. Soc. (2) 83 (2011), 517–543.
+[3] George M. Bergman, The diamond lemma for ring theory, Adv. in Math.
+29 (1978), 178–218.
+[4] Bruno Buchberger, An algorithm for finding the bases elements of the
+residue class ring modulo a zero dimensional polynomial ideal (German),
+Ph.D. thesis, University of Innsbruck, 1965.
+46
+
+[5] Thomas Cluzeau, Jamal Hossein Poor, Alban Quadrat, Clemens G. Raab,
+and Georg Regensburger, Symbolic computation for integro-differential-
+time-delay operators with matrix coefficients, IFAC-PapersOnLine 51
+(2018), no. 14, 153–158.
+[6] Earl A. Coddington and Norman Levinson, Theory of ordinary differential
+equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955.
+[7] Paul M. Cohn, Further algebra and applications, Springer-Verlag London,
+Ltd., London, 2003.
+[8] Gerald A. Edgar, Transseries for beginners, Real Anal. Exchange 35 (2010),
+no. 2, 253–309.
+[9] Xing Gao, Li Guo, and Markus Rosenkranz, On rings of differential Rota-
+Baxter operators, Internat. J. Algebra Comput. 28 (2018), no. 1, 1–36.
+[10] Li Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Math-
+ematics, vol. 4, International Press, Somerville, MA; Higher Education
+Press, Beijing, 2012.
+[11] Li Guo and William Keigher, On differential Rota-Baxter algebras, J. Pure
+Appl. Algebra 212 (2008), 522–540.
+[12] Li Guo, Georg Regensburger, and Markus Rosenkranz, On integro-
+differential algebras, J. Pure Appl. Algebra 218 (2014), 456–473.
+[13] Jamal Hossein Poor, Tensor reduction systems for rings of linear operators,
+Ph.D. thesis, Johannes Kepler University Linz, Austria, 2018.
+[14] Jamal Hossein Poor, Clemens G. Raab, and Georg Regensburger, Algo-
+rithmic operator algebras via normal forms in tensor rings, J. Symbolic
+Comput. 85 (2018), 247–274.
+[15] Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math.
+Univ. Nancago, No. 5, Hermann, Paris, 1957.
+[16] William F. Keigher, Adjunctions and comonads in differential algebra, Pa-
+cific J. Math. 59 (1975), no. 1, 99–112.
+[17]
+, On the ring of Hurwitz series, Comm. Algebra 25 (1997), no. 6,
+1845–1859.
+[18] William F. Keigher and F. Leon Pritchard, Hurwitz series as formal func-
+tions, J. Pure Appl. Algebra 146 (2000), no. 3, 291–304.
+[19] Donald E. Knuth and Peter B. Bendix, Simple word problems in universal
+algebras, Computational Problems in Abstract Algebra, Pergamon, Oxford,
+1970, pp. 263–297.
+47
+
+[20] Ernst E. Kummer, Ueber die Transcendenten, welche aus wiederholten Inte-
+grationen rationaler Formeln entstehen, J. Reine Angew. Math. 21 (1840),
+74–90, 193–225, and 328–371.
+[21] Ivan A. Lappo-Danilevski, Résolution algorithmique des problèmes réguliers
+de Poincaré et de riemann (Mémoire premier: Le problème de Poincaré,
+concernant de la construction d’un groupe de monodromie d’un système
+donné d’équations différentielles linéaires aux intégrales régulières), J. Soc.
+Phys.-Math. Léningrade 2 (1928), no. 1, 94–120.
+[22] Teo Mora, An introduction to commutative and noncommutative Gröbner
+bases, Theoret. Comput. Sci. 134 (1994), no. 1, 131–173.
+[23] Danuta Przeworska-Rolewicz, Algebraic analysis, PWN—Polish Scientific
+Publishers, Warsaw; D. Reidel Publishing Co., Dordrecht, 1988.
+[24]
+, Two centuries of the term “algebraic analysis”, Algebraic analysis
+and related topics (Warsaw, 1999), Banach Center Publ., vol. 53, Polish
+Acad. Sci. Inst. Math., Warsaw, 2000, pp. 47–70.
+[25] Alban Quadrat, A constructive algebraic analysis approach to Artstein’s
+reduction of linear time-delay systems, IFAC-PapersOnLine 48 (2015),
+no. 12, 209–214.
+[26] Alban Quadrat and Georg Regensburger, Computing polynomial solutions
+and annihilators of integro-differential operators with polynomial coeffi-
+cients, Algebraic and symbolic computation methods in dynamical systems,
+Adv. Delays Dyn., vol. 9, Springer, Cham, 2020, pp. 87–114.
+[27] Rimhak Ree, Lie elements and an algebra associated with shuffles, Ann. of
+Math. (2) 68 (1958), 210–220.
+[28] Georg Regensburger, Markus Rosenkranz, and Johannes Middeke, A skew
+polynomial approach to integro-differential operators, Proceedings of ISSAC
+’09 (New York, NY, USA), ACM, 2009, pp. 287–294.
+[29] Markus Rosenkranz, A new symbolic method for solving linear two-point
+boundary value problems on the level of operators, J. Symbolic Comput. 39
+(2005), 171–199.
+[30] Markus Rosenkranz and Georg Regensburger, Solving and factoring bound-
+ary problems for linear ordinary differential equations in differential alge-
+bras, J. Symbolic Comput. 43 (2008), 515–544.
+[31] Markus Rosenkranz, Georg Regensburger, Loredana Tec, and Bruno
+Buchberger, Symbolic analysis for boundary problems: From rewriting to
+parametrized Gröbner bases, Numerical and Symbolic Scientific Computing:
+Progress and Prospects (Ulrich Langer and Peter Paule, eds.), Springer Vi-
+enna, Vienna, 2012, pp. 273–331.
+48
+
+[32] Markus Rosenkranz and Nitin Serwa, An integro-differential structure for
+Dirac distributions, J. Symbolic Comput. 92 (2019), 156–189.
+[33] Gian-Carlo Rota, Twelve problems in probability no one likes to bring up,
+Algebraic combinatorics and computer science, Springer Italia, Milan, 2001,
+pp. 57–93.
+[34] Louis H. Rowen, Ring theory, student ed., Academic Press, Inc., Boston,
+MA, 1991.
+[35] Richard P. Stanley, Differentiably finite power series, European J. Combin.
+1 (1980), no. 2, 175–188.
+[36] Joris van der Hoeven, Transseries and real differential algebra, Lecture
+Notes in Mathematics, vol. 1888, Springer-Verlag, Berlin, 2006.
+49
+
diff --git a/QNFPT4oBgHgl3EQfojUh/content/tmp_files/load_file.txt b/QNFPT4oBgHgl3EQfojUh/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cadc9eb49de3f6c022a991693b3bb5dc366c164a
--- /dev/null
+++ b/QNFPT4oBgHgl3EQfojUh/content/tmp_files/load_file.txt
@@ -0,0 +1,1782 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf,len=1781
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='13134v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='RA]  30 Jan 2023  The fundamental theorem of calculus in  differential rings  Clemens G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Raaba,∗ and Georg Regensburgera,b  aInstitute for Algebra, Johannes Kepler University Linz, Austria  bInstitute of Mathematics, University of Kassel, Germany  clemensr@algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='uni-linz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='at  regensburger@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='uni-kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='de  Abstract  In this paper, we study the fundamental theorem of calculus and its  consequences from an algebraic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For functions with singu-  larities, this leads to a generalized notion of evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We investigate  properties of such integro-differential rings and discuss many examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We also construct corresponding integro-differential operators and pro-  vide normal forms via rewrite rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' They are then used to derive several  identities and properties in a purely algebraic way, generalizing well-known  results from analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In identities like shuffle relations for nested integrals  and the Taylor formula, additional terms are obtained that take singular-  ities into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Another focus lies on treating basics of linear ODEs  in this framework of integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' These operators can  have matrix coefficients, which allow to treat systems of arbitrary size in a  unified way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In the appendix, using tensor reduction systems, we give the  technical details of normal forms and prove them for operators including  other functionals besides evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Keywords  Integro-differential rings, integro-differential operators, normal  forms, generalized shuffle relations, generalized Taylor formula  1  Introduction  Differential rings are a well-established algebraic structure for modelling dif-  ferentiation by derivations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' linear operations satisfying the Leibniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  More recently, integro-differential algebras have been introduced to additionally  model integration and point evaluation of continuous univariate functions by  ∗Corresponding author  1  linear operations satisfying corresponding algebraic identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In contrast, the  integro-differential rings introduced in this paper only use the two identities  d  dx  � x  a  f(t) dt = f(x)  and  � x  a  f ′(t) dt = f(x) − f(a)  of the fundamental theorem of calculus and the Leibniz rule as axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In  particular, this results in a generalized notion of evaluation that is only required  to map to constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This allows to deal with evaluations of functions even if  singularities or discontinuities are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For example, it is natural to consider  integro-differential rings containing the rational functions leading to so-called  hyperlogarithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  It turns out that many analytic identities and general statements about  ODEs, like variation of constants, have a purely algebraic proof in integro-  differential rings that is independent of the analytic properties of concrete  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Likewise, computations with and transformations of linear integro-  differential equations and initial conditions can be done in this algebraic setting  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, exploring algebraic consequences of the Leibniz rule and  the fundamental theorem of calculus, we also find new results that introduce  additional evaluation terms into identities like shuffle relations for nested inte-  grals and the Taylor formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In short, we investigate the analytic operations  of differentiation, integration, and evaluation from a purely algebraic point of  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In this context, we implement in some sense the somewhat provocative  statement of Rota [33, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 57] that “the algebraic structure sooner or later comes  to dominate [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra dictates the analysis.”  In order to study linear integro-differential equations and identities, we use  the operator point of view, which treats identities of functions as identities of  corresponding linear operators acting on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To this end, we algebraically  construct the ring of integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For simplifying expressions  for operators, we use identities as rewrite rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As a key result, we work out a  particular rewrite system that can be applied straightforwardly to obtain nor-  mal forms and to prove any algebraic identity of integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Our construction of the ring of operators can be used for operators with scalar  coefficients and for operators with matrix coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, it allows to  uniformly deal with scalar equations as well as with systems, even of undeter-  mined size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Preliminary versions of some results presented in this paper have  already been presented by the authors at the conference “Differential Algebra  and Related Topics” (DART VII) in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  A related approach was taken in the work of Danuta Przeworska-Rolewicz,  see for example [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In short, she considers a right invertible linear operator on  a vector subspace as a generalization of derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Right inverses and projec-  tions onto the kernel take the role of integration and evaluation, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In  this linear setting, she develops algebraic generalizations of results for calculus  and linear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' She refers to this as algebraic analysis, see  also [24] for historic context and references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Several results, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' a Taylor for-  mula, can be formulated already in this purely linear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For other results,  multiplication in a commutative algebra is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In some statements, the  2  Leibniz rule or weakened versions of it are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' However, she does not con-  sider shuffle relations for nested integrals or additional evaluation terms in the  Taylor formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Her treatment of operators is limited to properties and iden-  tities of given operators and she does not consider rings generated by them or  normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Integro-differential algebras and operators over a field of constants were al-  ready introduced in [29, 30], see [31] for a detailed overview and further refer-  ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' More general differential algebras with integration over rings were intro-  duced in [11], see [12] for a unified presentation and comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In contrast, the  integro-differential rings defined in [14, 13] require the integration to be linear  over all constants, but the construction of corresponding operators introduced  there allows noncommutative coefficients and constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Further references to  the literature can be found in the respective sections of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  So far, all algebraic treatments of integro-differential operators in the litera-  ture restrict to multiplicative evaluations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' evaluation of a product is the prod-  uct of the individual evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assuming only the Leibniz rule and the iden-  tities of the fundamental theorem of calculus, we deal with non-multiplicative  evaluations quite naturally in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section 2, we introduce (general-  ized) integro-differential rings following this principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Analyzing the relations  imposed, we show for example that any linear projection onto constants may be  used as the evaluation of such an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We also present many  examples, both with multiplicative and with non-multiplicative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Re-  mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='12 discusses the differences among our definition and the definitions in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In Section 3, we investigate identities satisfied by integrals and their prod-  ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This reveals a generalization of the Rota-Baxter identity for integration  that contains an additional evaluation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Focusing on nested integrals, we  discover generalized shuffle relations with elaborate additional terms involving  nested integrals of lower depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We also characterize properties of repeated in-  tegrals of 1, which form the smallest integro-differential ring with given ring of  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Linear integro-differential operators with coefficients in arbitrary (gener-  alized) integro-differential rings are constructed algebraically in Section 4 by  generators and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Working at the operator level enables statements of  broader applicability, since operators not only act on integro-differential rings  but also on more general modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We compute a complete set of rewrite rules  to simplify such operators to normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A precise analysis of the uniqueness  of these normal forms is presented in the appendix only, since it requires a re-  fined construction relying on tensor rings (like in [14, 13] for the multiplicative  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' After a largely self-contained introduction to tensor reduction systems in  the appendix, this is carried out in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4 allowing also other functionals  besides evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we collect properties of integro-differential  operators with coefficients from an integral domain (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' analytic functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In  particular, we also characterize the action of such operators when evaluation is  multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In the remaining sections, we illustrate how computations in the ring of  3  integro-differential operators can be used to prove and generalize well-known  results from analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For example, variation of constants remains valid for ar-  bitrary integro-differential rings, which is the focus of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular,  we also detail how integro-differential operators with noncommutative coeffi-  cients can be used for proving statements about systems of arbitrary or even  undetermined size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section 6, we discuss how results from analysis need to  be modified for allowing the induced evaluation to be non-multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' First,  we look at the formula for variation of constants, which for multiplicative eval-  uations automatically satisfies homogeneous initial conditions, and include an  extra term to retain this property in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Finally, we present a version of the  Taylor formula with integral remainder term that is valid also for generalized  evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Conventions  Throughout the paper, rings are implicitly assumed to have  a unit element and to be different from the zero ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Unless stated other-  wise, rings are not assumed to be commutative and can be of arbitrary char-  acteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Nevertheless, for easier reading, we use the notions of modules and  linear maps from the commutative setting to refer to bimodules and bimodule-  homomorphisms over noncommutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In addition, we use operator no-  tation for linear maps, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' the Leibniz rule for the derivative of products then  reads ∂fg = (∂f)g + f∂g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  2  Integro-differential rings  To uniformly deal with differentiation of various kinds of functions, we use a  few basic abstract notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Recall from differential algebra that a derivation on  a ring R is an additive map ∂ : R → R that satisfies the Leibniz rule  ∂fg = (∂f)g + f∂g  (1)  for all f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, (R, ∂) is called a differential ring and f ∈ R is called  a constant in this differential ring if and only if ∂f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It is easy to see that  the set of constants forms a subring of R and ∂ is linear w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' the ring of its  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For further theory of differential rings see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In addition,  we introduce the following notions of integration and evaluation in differential  rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂) be a differential ring and let C be its ring of con-  stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We call a C-linear map  �  : R → R an integration on R, if  ∂  �  f = f  (2)  holds for all f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A C-linear functional e: R → C which acts on C as the  identity is called an evaluation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In other words, integrations on differential rings are right inverses of the  derivation that are linear over the constants and evaluations on differential rings  are C-linear projectors onto the ring of constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  4  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂) be a differential ring and let  �  : R → R be an  integration on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We call (R, ∂,  �  ) a (generalized) integro-differential ring and  we define the (induced) evaluation E on R by  Ef := f −  �  ∂f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (3)  If in addition R is a field or skew field, then we also call (R, ∂,  �  ) a (generalized)  integro-differential (skew) field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This extends the definition of integro-differential rings in [14] by dropping  the additional requirement that the induced evaluation should be multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In the present paper, the notion of integro-differential rings always refers to  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The following lemma shows that in any integro-differential ring,  the (induced) evaluation E is indeed an evaluation as defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, the ring R can be decomposed as direct sum of constant and non-  constant “functions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, for all f ∈ R and c ∈ C, we have Ef ∈ C, E  �  f = 0, and Ec = c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover,  R = C ⊕  �  R  as direct sum of C-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' First, we compute ∂Ef = ∂(f − � ∂f) = ∂f − ∂f = 0 and E� f =  �  f −  �  ∂  �  f = 0 for f ∈ R as well as Ec = c−  �  ∂c = c for c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For any f ∈ R,  we have f = Ef + f − Ef = Ef +  �  ∂f and hence R = C +  �  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let f ∈ C ∩  �  R  and g ∈ R such that f =  �  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, 0 = ∂f = ∂  �  g = g, which implies f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Hence, the sum R = C +  �  R is direct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By the previous lemma, any integration induces an evaluation by id −  �  ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Conversely, any evaluation e can be used to define an integration that has e as  its induced evaluation, as the following theorem shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It is easy to see that  two different integrations cannot have the same induced evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether,  on differential rings with ∂R = R, there is a one-to-one correspondence of  integrations and evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂) be a differential ring such that ∂R = R and let e be  an evaluation on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Define  �  e : R → R by  �  ef := g − eg  for all f ∈ R, where g ∈ R is such that ∂g = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then (R, ∂,  �  e) is an integro-  differential ring and the induced evaluation is E = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, any integration  �  on R can be obtained from its induced evaluation  E via this construction:  �  =  �  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let C be the ring of constants of (R, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' First, we show that  �  e is well-  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If g, ˜g ∈ R are such that ∂g = ∂˜g, then with c := ˜g − g ∈ C we  5  have ˜g − e˜g = g + c − eg − ec = g − eg, since ec = c by definition of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For  showing C-linearity of  �  e, we let c ∈ C and f1, f2, g1, g2 ∈ R with ∂gi = fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, ∂(cg1 + g2) = cf1 + f2 together with C-linearity of id − e implies C-  linearity of  �  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Consequently, (R, ∂,  �  e) is an integro-differential ring, since  we also have ∂  �  ef = f by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The induced evaluation is given by  Ef = f −  �  e∂f = f − (f − ef) = ef for f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let  �  : R → R be any C-linear right inverse of ∂, E its induced evaluation,  and f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, we have �  Ef = � f − E� f = � f by definition of �  E and by  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, if (R, ∂,  �  ) is an integro-differential ring and e is any evaluation  on R, then the integration that induces e can be given in terms of  �  by  �  e :=  �  − e  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (4)  This implies that the difference of two integrations  �  1,  �  2 on the same differential  ring can be given as  �  1−  �  2 = E2  �  1 = −E1  �  2 in terms of the induced evaluations  E1, E2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' More generally, if (R, ∂,  �  ) is an integro-differential ring with constants  C and e : R → C is only C-linear, then  �  e defined by (4) can be easily seen to be  an integration on R and its induced evaluation is given by e + (id − e)E, which  agrees with e if and only if the latter is an evaluation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' e1 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3,  �  R is a direct complement of C in an integro-differential ring  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Conversely, any direct complement of C gives rise to an evaluation on R,  which in turn induces an integration by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' More precisely, we have  the following characterization of integro-differential rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂) be a differential ring with ring of constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then,  (R, ∂) can be enriched into an integro-differential ring if and only if ∂R = R and  C is a complemented C-module in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, if ∂R = R, there exists a one-  to-one correspondence between direct complements of C in R and integrations  on (R, ∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This characterization shows that on an integro-differential ring (R, ∂,  �  ),  in general, there are many other integrations that make R into an integro-  differential ring with the same derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In contrast, the following character-  ization shows that, in general, ∂ is the only derivation that turns R into an  integro-differential ring with the same integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let R be a ring, let C be a subring of R, and let  �  : R → R be a  C-linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, there exists a derivation ∂ on R such that (R, ∂, � ) is an  integro-differential ring with constants C if and only if the following conditions  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' R = C ⊕  �  R  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (  �  f)  �  g −  �  (  �  f)g −  �  f  �  g ∈ C for all f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  6  Moreover, this derivation is unique if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' First, it is easy to see, from injectivity of  �  and by R = C ⊕  �  R, that  there exists a C-linear map ∂ : R → R such that ker ∂ = C and ∂  �  = id and  that this map is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To show that ∂ is indeed a derivation, we verify the  Leibniz rule on two arbitrary elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By R = C ⊕  �  R, we write these  two elements as c+  �  f and d+  �  g with c, d ∈ C and f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now, we compute  ∂(c + � f)(d + � g) − (∂(c + � f))(d + � g) − (c + � f)∂(d + � g)  = ∂(  �  f)  �  g − f  �  g − (  �  f)g  = ∂  �  (  �  f)  �  g −  �  (  �  f)g −  �  f  �  g  �  ,  which is zero by ker ∂ = C and the last assumption on  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Conversely, if (R, ∂,  �  ) is an integro-differential ring with constants C, then  injectivity of  �  follows from the definition (2) and the other two conditions on  �  follow from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  It is straightforward to equip the matrix ring over an integro-differential  ring with an integro-differential ring structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Such noncommutative integro-  differential rings are relevant when working with linear systems, see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants C and  let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, (Rn×n, ∂, � ) is an integro-differential ring with constants  Cn×n, where operations ∂,  �  , E act on matrices by applying the corresponding  operation entrywise in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Clearly, ∂,  �  , E are Cn×n-linear on Rn×n satisfying ∂  �  A = A and  �  ∂A =  A − EA for all A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, it is straightforward to verify that ∂ is a  derivation on R with constants Cn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Basic examples for commutative integro-differential rings are  univariate polynomials C[x] and formal power series C[[x]] over a commutative  ring C with Q ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The derivation is given by ∂ =  d  dx with ring of constants C  and integration is defined C-linearly by  �  xn = xn+1  n + 1  for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The induced evaluation extracts the constant coefficient and  corresponds to evaluation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If C ⊆ C, the integration  �  corresponds to  integration  � x  0 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Also the rings of complex-valued smooth or analytic  functions on a (possibly unbounded) interval I ⊆ R together with derivation  ∂ =  d  dx and integration  �  =  � x  a , for fixed a ∈ I, are integro-differential rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, the induced evaluation is the evaluation of functions at the point a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, the ring of exponential polynomials on the real line generated  by polynomials and exponential functions is closed under differentiation and  integration and hence is an integro-differential ring as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebraically, for any  field C of characteristic zero, we can consider the ring of exponential polynomials  7  C[x, eCx], where ecxedx = e(c+d)x for all c, d ∈ C and e0x = 1, together with  derivation ∂ =  d  dx and with the integration that is induced by evaluation at 0  based on Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A basic example of integro-differential rings of arbitrary charac-  teristic are Hurwitz series, which are closely related to formal power series and  have been defined in [16, 17], with derivation ∂(a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ) = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ) and  integration given by  �  (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ) = (0, a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ), see also [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The examples mentioned so far have the special property that the evalua-  tion of the integro-differential ring is multiplicative, as in the usual definition of  integro-differential algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' However, for certain differential rings (in particu-  lar for differential fields, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Corollary 5 in [31]), it is not possible to define a  multiplicative evaluation for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If the induced evaluation is multiplicative, one can see easily  that no element of  �  R can have a multiplicative inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since otherwise we  would have Ef 1  f = E1 = 1 and (Ef)E 1  f = 0E 1  f = 0 for such f ∈ � R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Analytically, if evaluation should correspond to evaluation of functions at a  fixed point a for functions that are continuous at a, then requiring multiplica-  tivity of evaluation means that functions with poles at a cannot be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, if a function f has a pole of order m at a, then evaluation of the  product (x − a)mf gives a nonzero value, but the factor (x − a)m evaluates to  zero at a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In the following theorem, based on results from the literature, we briefly  characterize when the induced evaluation is multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' From (3) it imme-  diately follows that the identity (7) is equivalent to multiplicativity of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since  in integro-differential rings  �  is C-linear by definition, we have the following  characterization of integro-differential rings with multiplicative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then the following  properties are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' E is multiplicative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' for all f, g ∈ R we have  Efg = (Ef)Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (5)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  satisfies the Rota-Baxter identity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' for all f, g ∈ R we have  (� f)� g = � (� f)g + � f� g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (6)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The hybrid Rota-Baxter identity holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' for all f, g ∈ R we have  (  �  ∂f)  �  ∂g = (  �  ∂f)g + f  �  ∂g −  �  ∂fg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since (R, ∂) is a differential Z-algebra of weight 0 and  �  is Z-linear, this  immediately follows from items (b), (g), and (a) of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  8  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In the literature, integro-differential K-algebras (of weight 0)  over a commutative ring K with unit element are defined as differential K-  algebras where the additional map  �  is only required to be K-linear, but has  to satisfy the hybrid Rota-Baxter axiom (7) in addition to (2), see [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Analo-  gously, the more general notion of differential Rota-Baxter K-algebras (of weight  0) imposes the Rota-Baxter identity (6) instead of (7) in addition to (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On a  differential Rota-Baxter K-algebra with constants C, a K-linear map E is defined  by (3) as well and, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 in [12], properties (5), (7), and C-linearity  of  �  are equivalent, see also Proposition 10 in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By the previous theorem, any (generalized) integro-differential ring with  constants C is an integro-differential K-algebra for K = Z and for K = C ∩Z(R)  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' K = C, if R is commutative) if any of the equivalent conditions holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Conversely, any differential Rota-Baxter K-algebra (of weight 0) is an integro-  differential ring if and only if  �  is linear over the constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular,  this is automatically the case for integro-differential K-algebras (of weight 0) by  Proposition 10 in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For concrete differential Rota-Baxter algebras (of weight  0) where  �  is not C-linear, see Example 3 in [30] and the algebraic analog of  piecewise functions constructed in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As a basic example for integro-differential rings with non-multiplicative eval-  uation, we extend the polynomial ring C[x] over a commutative ring C with  Q ⊆ C by adjoining the multiplicative inverse x−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In order to have a surjective  derivation ∂ =  d  dx, we also need to adjoin the logarithm ln(x) as in the following  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On C[x, x−1, ln(x)], with Q ⊆ C and ∂ =  d  dx, we can define the  C-linear integration recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  � xk ln(x)n :=  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  xk+1  k+1  k ̸= −1 ∧ n = 0  xk+1  k+1 ln(x)n −  n  k+1  �  xk ln(x)n−1  k ̸= −1 ∧ n > 0  ln(x)n+1  n+1  k = −1  The same recursive definition also works on the larger ring C((x))[ln(x)] of formal  Laurent series with logarithms, where every element can be written in the form  �∞  k=−m  �m  n=0 ck,nxk ln(x)n for some m ∈ N and ck,n ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In both cases, the  induced evaluation acts by  E  ∞  �  k=−m  m  �  n=0  ck,nxk ln(x)n = c0,0  and is not multiplicative as expected by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, for the integro-  differential subrings of polynomials or formal power series, this evaluation cor-  responds to the usual multiplicative evaluation at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Rational functions together with nested integrals of rational  functions also form an integro-differential ring with non-multiplicative evalua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebraically, if C is a field of characteristic zero, this can be understood  9  as an integro-differential subring of C((x))[ln(x)] with integration  �  as in the  previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In fact, this ring is the smallest integro-differential ring con-  taining C(x) and is generated as a C(x)-vector space by 1 and all nested inte-  grals  �  f1  �  f2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  fn of arbitrary depth n ≥ 1, where fi ∈ C(x) are proper  and have irreducible denominators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, if C = C, the integrands can  be chosen as fi =  1  x−ai with ai ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' These kind of nested integrals are called  hyperlogarithms [21] and have been investigated already in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In [12], the  free integro-differential algebra (having multiplicative evaluation) generated by  C(x) has been constructed at the somewhat unnatural expense that  �  1 ̸∈ C(x)  and the constants of the resulting differential ring contain much more than just  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Another example of an integro-differential ring that contains  the rational functions are the D-finite functions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' They are characterized  as solutions of linear differential equations with rational function coefficients  and indeed form a differential ring with surjective derivation  d  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Since the  constants of this differential ring are given by C, there exists an integration by  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' All the examples considered above are just rings, not fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In contrast, transseries R[[[x]]] are an explicit construction of a differential  field (with field of constants R) that is closed under taking antiderivatives,  see [36, 8, 1] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Thus, R[[[x]]] can be turned into an  integro-differential field by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 whose evaluation necessarily is non-  multiplicative by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As shown by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5, in general, there are many different choices for  an integration  �  in order to turn a differential ring with ∂R = R into an  integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On the same differential ring, for some integrations  the induced evaluation is multiplicative and for others it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It may even  be the case that a canonical choice of  �  yields a non-multiplicative evaluation  while there are other choices that would give a multiplicative evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In  particular, this is the case for exponential polynomials, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' They were  mentioned above with a multiplicative evaluation, while the following canonical  definition of  �  gives rise to a non-multiplicative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The ring of exponential polynomials C[x, eCx] over a field C of  characteristic zero is C-linearly generated by terms of the form xkecx with k ∈ N  and c ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Apart from the evaluation-based integration on C[x, eCx] mentioned  above, it is quite natural to define a C-linear integration  �  recursively as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  xkecx :=  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  xk+1  k+1  c = 0  1  cecx  k = 0 ∧ c ̸= 0  1  cxkecx − k  c  �  xk−1ecx  k > 0 ∧ c ̸= 0  In terms of the Pochhammer symbol (a)k := a·(a + 1)· .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ·(a + k − 1),  �  can be  10  given explicitly as  �  xkecx =  k  �  i=0  (−k)k−ici−k−1xiecx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, the induced evaluation Ef := f −  �  ∂f acts by  Exkecx =  �  1  k = c = 0  0  otherwise  and is not multiplicative since we have Eecxe−cx = E1 = 1 but Ee±cx = 0 for any  c ̸= 0, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On the other hand, C[x, ex] is an integro-differential subring  with multiplicative evaluation, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' With this integration, for instance,  the subset C[x, ex]ex is closed under addition, multiplication, derivation, and  integration and, hence, could be viewed as an integro-differential subring with-  out unit element and having multiplicative evaluation and the zero ring as its  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  All the examples with explicit integration discussed so far contain an integro-  differential subring on which the induced evaluation is multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In general,  however, this need not be the case as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On C[x] with the usual derivation and Q ⊆ C, for example,  we can define a C-linear integration by  �  xn = xn+1  n+1 + c for all n ∈ N for any  fixed c ∈ Z(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Such an integration induces the evaluation Ef = f(0) − cf ′(1)  on C[x], which is not multiplicative if c ̸= 0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' f = x and g = x2 − 2x yield  Eg = 0 and Efg = c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  3  Products of nested integrals  In integro-differential rings with multiplicative evaluation the standard Rota-  Baxter identity (6) allows to write the product of integrals as a sum of two  nested integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For nested integrals, this leads to shuffle identities [27] where  a product of two nested integrals is expressed as a sum of nested integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  More generally, for Rota-Baxter operators with weight and corresponding shuffle  products involving additional terms, see [10] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In general  integro-differential rings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' if E is not multiplicative, additional terms involving  the evaluation arise in the identities (6) and (7) and also in the shuffle identi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that all evaluation terms are evaluations of products of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Therefore, they vanish if E is multiplicative, since E� f = 0 for all f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then the Rota-  Baxter identity with evaluation  (  �  f)  �  g =  �  f  �  g +  �  (  �  f)g + E(  �  f)  �  g  (8)  holds for all f, g ∈ R as well as  (  �  ∂f)  �  ∂g = (  �  ∂f)g + f  �  ∂g −  �  ∂fg − E(  �  ∂f)  �  ∂g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (9)  11  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Using (3), we can effect the decomposition of R shown in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For (  �  f)  �  g, we thereby obtain the decomposition  �  ∂(  �  f)  �  g + E(  �  f)  �  g, which  implies (8) by the Leibniz rule for the derivation ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By  �  ∂f = f − Ef,  �  ∂g =  g − Eg, and  �  ∂fg = fg − Efg, one can write (9) in a form that can be easily  verified using the fact that E is an evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As a first application, we show that in every integro-differential ring the re-  peated integrals of 1 give rise to an integro-differential subring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It is the smallest  integro-differential ring with the same constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In characteristic zero, this ring  consists of the univariate polynomials with coefficients in the constants and, for  nonzero characteristic, it consists of a finite version of Hurwitz series [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For all n ≥ 1, let xn :=  � n1 ∈ R and let x0 := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, 1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' commute  with all elements of C and are C-linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The C-module  P := spanC{1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='}  is an integro-differential subring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If Q ⊆ R, then P = C[x1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, E is multiplicative on P if and only if Exmxn = 0 for m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Assuming E is multiplicative on P, then we have xmxn =  �m+n  m  �  xm+n and, if  in addition Q ⊆ R, xn = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xn  1 for m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since  �  is C-linear, every element of C commutes with xi for every i ∈ N,  even if R or C is noncommutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  So, every element of P is of the form  �n  i=0 cixi, for some ci ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To show C-linear independence of x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , let n ∈  N be minimal such that there are c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , cn ∈ C with cn ̸= 0 and �n  i=0 cixi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, �n−1  i=0 ci+1xi = ∂ �n  i=0 cixi = 0 would imply n = 0 by minimality of  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Because this would yield c0 = 0, we conclude that x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' are C-linearly  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Obviously, P is closed under ∂ and  �  since both operations are C-linear with  ∂xn ∈ P and  �  xn = xn+1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For showing that P is closed under  multiplication, it suffices to show that xmxn ∈ P for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We proceed  by induction on the sum n + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For m = n = 1 we have x2  1 = 2x2 + Ex2  1 by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For m + n > 2 we have  xmxn =  �  xm−1xn +  �  xmxn−1 + Exmxn  by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By the induction hypothesis, xm−1xn and xmxn−1 are in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, the  same is also true after applying  �  , which completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether,  P is an integro-differential subring of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For showing P = C[x1], it is sufficient to prove that every xn is contained  in C[x1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By (8), we obtain xn+1 = x1xn −  �  xn−1x1 − Ex1xn for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Therefore, xn ∈ C[x1] follows by induction, if C[x1] is closed under  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assuming  Q ⊆ R, we verify that  �  xn  1 −  1  n+1xn+1  1  ∈ C for all n ≥ 1 by applying ∂ to it,  which shows  �  xn  1 ∈ C[x1] for all n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, any xn with n ≥ 1 satisfies Exn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' So, if E is multiplicative on  P, then trivially Exmxn = (Exm)Exn = 0 for m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Conversely, since P  12  is generated by 1, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' as a C-module, we know that  �  P is generated by  x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' as a C-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, applying E to the product of two elements of  �  P gives 0, if Exmxn = 0 for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether, using the decomposition  P = C ⊕  �  P given by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, we conclude that E is multiplicative on P, if  Exmxn = 0 for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Now, we assume E is multiplicative on P and let m, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If m + n ≤ 1,  then xmxn =  �m+n  m  �  xm+n and xn =  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xn  1 hold trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For m + n ≥ 2, it fol-  lows inductively by (8) that xmxn = � �m+n−1  m−1  �  xm+n−1 + � �m+n−1  m  �  xm+n−1 =  �m+n  m  �  xm+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, for n ≥ 2, x1xn−1 = nxn implies xn =  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xn  1 induc-  tively, if Q ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Even if E is not multiplicative on P, we can analyze some properties of the  sequence of constants cm,n := Exmxn with n, m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By C-linearity of � and  E, it trivially follows that all cm,n are in Z(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It can be shown that all xn  commute with each other w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' multiplication if and only if the sequence cm,n  is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If Q ⊆ R, it can be shown by lengthy computation that the  constants cm,n are determined by all c1,n via the recursion  cm,n = 1  m  ��m+n−1  m−1  �  c1,m+n−1 +  m−2  �  j=0  n−1  �  k=1  �j+k  j  �  c1,j+kcm−j−1,n−k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  To express xn in terms of powers of x1 in general, for Q ⊆ R, we obtain the  recursion xn = 1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xn  1 − �n  i=2  1  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xn−iExi  1 from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  Generalized shuffle relations  In this section, we let (R, ∂,  �  ) be a commutative integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Iter-  ating the standard Rota-Baxter identity (6) leads to shuffle relations for nested  integrals expressing a product of two nested integrals of depth m and n as a  sum of nested integrals of depth exactly m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Also by recursively applying  the Rota-Baxter identity with evaluation (8), products of nested integrals can  be rewritten in R as sums of nested integrals where also terms of lower depth  may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For convenient notation of the formulae involved, it is standard to  work in tensor products of R and to use the shuffle product, which we recall in  the following (see [10] for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We consider the C-module C⟨R⟩ := �∞  n=0 R⊗n, where the tensor prod-  uct is taken over C and the empty tensor is denoted by ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let pure tensors  a1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ⊗an ∈ C⟨R⟩ represent nested integrals  �  a1  �  a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  an ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' More for-  mally, by C-linearity of  �  , we consider the unique C-module homomorphism  ϕ : C⟨R⟩ → R such that  ϕ(a1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ⊗an) =  �  a1  �  a2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  an ∈ R  and ϕ(ε) = 1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For pure tensors a ∈ C⟨R⟩, we denote shortened versions  of them by aj  i := ai⊗ai+1⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ⊗aj, where aj  i := ε ∈ R⊗0 if i = j + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The  13  shuffle product on C⟨R⟩ can be recursively defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For pure tensors  a, b ∈ C⟨R⟩ of length m and n, respectively, we set  a  � b :=  �  a ⊗ b  if m = 0 ∨ n = 0  a1 ⊗ (am  2  � b) + b1 ⊗ (a  � bn  2)  otherwise  in R⊗(m+n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Extending this definition to C⟨R⟩ by C-linearity, the shuffle product  turns C⟨R⟩ into a commutative C-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Using the shuffle product for pure tensors, the product of nested integrals in  R can now be represented as sum of nested integrals as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The constant  coefficients of nested integrals of lower depth are evaluations of products of  integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, if E is multiplicative, then we recover the standard  shuffle relations [27] with all these constant coefficients equal zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be a commutative integro-differential ring with  constants C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let f, g ∈ C⟨R⟩ be pure tensors of length m and n, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, the product of the nested integrals ϕ(f) =  �  f1  �  f2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  fm and ϕ(g) =  �  g1  �  g2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  gn is given by  ϕ(f)ϕ(g) = ϕ(f  � g) +  m−1  �  i=0  n−1  �  j=0  e(f m  i+1, gn  j+1)ϕ(f i  1  � gj  1) ∈ R  (10)  with constants e(f m  i+1, gn  j+1) := Eϕ(f m  i+1)ϕ(gn  j+1) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Without loss of generality, assume m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We proceed by induction  on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If m = 0, then f = cε for some c ∈ C and the equation (10) reads  cϕ(g) = ϕ(cε ⊗ g), which is trivially true since cε ⊗ g = cg and ϕ is C-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For m ≥ 1, we proceed by induction on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By virtue of (8), for n ≥ m, we have  ϕ(f)ϕ(g) = � f1ϕ(f m  2 )ϕ(g) + � ϕ(f)g1ϕ(gn  2 ) + e(f, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The product ϕ(f m  2 )ϕ(g)  is covered by the induction hypothesis on m so that we obtain  �  f1ϕ(f m  2 )ϕ(g) =  �  f1ϕ(f m  2  � g) +  m−1  �  i=1  n−1  �  j=0  e(f m  i+1, gn  j+1)  �  f1ϕ(f i  2  � gj  1)  by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The product ϕ(f)ϕ(gn  2 ) is covered by the induction hypothesis on n  (or on m, if n = m) so that (10) yields  � ϕ(f)g1ϕ(gn  2 ) = � g1ϕ(f  � gn  2 ) +  m−1  �  i=0  n−1  �  j=1  e(f m  i+1, gn  j+1)� g1ϕ(f i  1  � gj  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By definition of ϕ and  �, we have  �  f1ϕ(f m  2  � g) +  �  g1ϕ(f  � gn  2 ) = ϕ(f  � g)  and similarly  �  f1ϕ(f i  2  � gj  1) +  �  g1ϕ(f i  1  � gj  2) = ϕ(f i  1  � gj  1) for i, j ≥ 1 as well  as  �  f1ϕ(f i  2  � g0  1) = ϕ(f m  1  � g0  1) and  �  g1ϕ(f 0  1  � gj  2) = ϕ(f 0  1  � gj  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether,  14  this yields  ϕ(f)ϕ(g) = ϕ(f  � g) +  m−1  �  i=1  n−1  �  j=1  e(f m  i+1, gn  j+1)ϕ(f i  1  � gj  1)  +  m−1  �  i=1  e(f m  i+1, gn  1 )ϕ(f m  1  � g0  1) +  n−1  �  j=1  e(f m  i+1, gn  j+1)ϕ(f 0  1  � gj  1) + e(f, g),  which proves (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  4  Integro-differential operators  In the following, starting from a given integro-differential ring (R, ∂,  �  ), we  define the corresponding ring of operators by generators ∂,  �  , E and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As additive maps on R, any f ∈ R acts as multiplication operator g �→ fg and  satisfies certain identities together with the maps ∂,  �  , E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Those identities of  additive maps that correspond to the defining properties of the operations on  R will be used as defining relations for the abstract ring of operators below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, the Leibniz rule ∂fg = f∂g + (∂f)g of the derivation ∂ on R  implies the identity ∂ ◦ f = f ◦ ∂ + ∂f of additive maps for every multiplication  operator f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This motivates the identity (11) in the definition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Sim-  ilarly, the identities (2) and (3) in R give rise to the identities (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, from C-linearity of the operations ∂,  �  , E we obtain  �  cg = c  �  g for all  c ∈ C and g ∈ R, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In addition, we also obtain  �  fEg = (  �  f)Eg for  all f, g ∈ R, since Eg ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, we also impose commutativity of ∂,  �  , E with  elements of C and the identities (14)–(16) in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring and let C be its ring  of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We let  R⟨∂,  �  , E⟩  be the (noncommutative unital) ring extension of R generated by indeterminates  ∂,  �  , E, where ∂,  �  , E commute with all elements of C and the following identities  hold for all f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂ · f = f · ∂ + ∂f  (11)  ∂ ·  �  = 1  (12)  �  ∂ = 1 − E  (13)  ∂ · f · E = ∂f · E  (14)  �  f · E =  �  f · E  (15)  E · f · E = Ef · E  (16)  We call R⟨∂,  �  , E⟩ the ring of (generalized) integro-differential operators (IDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Note that, for multiplication in R⟨∂,  �  , E⟩, we always explicitly write · when  one of ∂,  �  , E is involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This is necessary in order to distinguish the product  15  ∂ · f of operators from the multiplication operator ∂f, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By con-  struction, the ring R⟨∂,  �  , E⟩ has a natural action on R, where the elements of  R act as multiplication operators and ∂,  �  , E act as the corresponding opera-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' With this action, R becomes a left R⟨∂,  �  , E⟩-module and multiplication  in R⟨∂,  �  , E⟩ corresponds to composition of additive maps on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Since elements of C always commute with ∂,  �  , E in R⟨∂,  �  , E⟩, it follows  that R⟨∂,  �  , E⟩ is a C-algebra whenever R is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For computing in  the ring R⟨∂, � , E⟩ of IDO, we use the identities (11)–(16) as rewrite rules in  the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If the left hand side of one of these identities appears in an  expression of an operator, we replace it by the right hand side to obtain a new  expression for the same operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If rewrite rules can be applied to a given expression in different ways, then  it may happen that useful consequences of the defining relations (11)–(16) are  discovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A simple instance starts with the expression  �  ∂ ·  �  , to which we  can apply either (12) or (13) to obtain the expressions  �  and  �  − E ·  �  for the  same operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, by taking their difference, we see that the identity  E ·  �  = 0  (17)  holds in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, for every f ∈ R, the expression  �  ∂ · f can be  rewritten by (11) and by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Thereby we obtain the expressions  �  f ·∂+  �  ∂f  and f − E · f for the same operator, which implies the identity  �  f · ∂ = f − E · f −  �  ∂f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (18)  By letting both sides of this identity in R⟨∂,  �  , E⟩ act on any g ∈ R, we show  that integration by parts  �  f∂g = fg − Efg −  �  (∂f)g  holds in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Furthermore, by considering also the newly obtained identity (18)  as rewrite rule, for every f ∈ R, we can rewrite the expression  �  f ·∂ ·  �  by (12)  and by (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Substituting f by � f in the difference f ·� −E·f ·� −� ·∂f ·� −� ·f  of the results, we obtain the identity  �  f ·  �  =  �  f ·  �  −  � �  f − E ·  �  f ·  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (19)  By acting with both sides of this identity on any g ∈ R, we obtain an alternative  proof for the Rota-Baxter identity with evaluation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that, for f = 1, we  also obtain the following identities from (14)–(16) and (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂ · E = 0,  �  E =  �  1 · E,  E · E = E,  (20)  � �  =  �  1 ·  �  −  � �  1 − E ·  �  1 ·  �  (21)  In Table 1, we collect the identities (11)–(21) as a rewrite system for expres-  sions of operators in the ring of IDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In fact, we drop (14) since it is redundant  in the presence of (11) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  16  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, by repeatedly  applying the rewrite rules of Table 1 in any order, every element of the ring  R⟨∂,  �  , E⟩ can be written as a sum of expressions of the form  f · ∂j,  f ·  �  g,  f · E · g · ∂j,  or  f · E · h ·  �  g  where j ∈ N0, f, g ∈ R, and h ∈  �  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Note that the expressions specified in the above theorem are irreducible in  the sense that they cannot be rewritten any further by any rewrite rules from  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The above derivation of identities (17)–(21) is similar to Knuth-Bendix  completion [19] and Buchberger’s algorithm for computing Gröbner bases [4, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  So, one can show that Table 1 represents all consequences of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  in the sense that every identity in R⟨∂,  �  , E⟩ can be proven by applying the  rewrite rules in the table and by exploiting identities in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, the  irreducible forms of operators specified in the above theorem are unique up to  multiadditivity and commutativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In the appendix, we will give a precise statement (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3) of this  by giving an explicit construction of the ring R⟨∂,  �  , E⟩ as a quotient of an  appropriate tensor ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, the translation of Table 1 into a tensor reduction  system facilitates the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In fact, this proof is carried out in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 for  a more general class of operators including linear functionals, which are useful  for dealing with boundary problems, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In the literature, integro-differential operators were considered  only with multiplicative evaluation so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Integro-differential operators were  first introduced in [29, 30] over a field of constants using a parametrized Gröb-  ner basis in infinitely many variables and a basis of the commutative coefficient  algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Integro-differential operators with polynomial coefficients over a field  of characteristic zero were also studied using generalized Weyl algebras [2], skew  polynomials [28], and noncommutative Gröbner bases [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A general construc-  tion of rings of linear operators over commutative operated algebras is presented  in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, integro-differential operators are discussed in that setting  and also differential Rota-Baxter operators are investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Tensor reduction  systems have already been used in [14, 13] for the construction of IDO includ-  ing functionals in the special case that the evaluation and all functionals are  multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  There, also additional operators arising from linear substitu-  tions were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, these cover integro-differential-time-delay  operators, which were already constructed algebraically in [25], see also [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂ · f = f · ∂ + ∂f  �  f · ∂ = f − E · f −  �  ∂f  ∂ · E = 0  � · f · E = � f · E  ∂ ·  �  = 1  �  f ·  �  =  �  f ·  �  −  � �  f − E ·  �  f ·  �  E · f · E = Ef · E  �  ∂ = 1 − E  E · E = E  �  E =  �  1 · E  E ·  �  = 0  � �  =  �  1 ·  �  −  � �  1 − E ·  �  1 ·  �  Table 1: Rewrite rules for operator expressions  17  To refer to integro-differential operators of special form, we use the following  notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let L ∈ R⟨∂,  �  , E⟩, then we call L  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' a differential operator, if there are f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R such that  L =  n  �  i=0  fi · ∂i,  where we call L monic, if fn = 1,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' an integral operator, if there are f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn ∈ R such that  L =  n  �  i=1  fi ·  �  gi,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' an initial operator, if there are f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R and differential and integral  operators L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln such that  L =  n  �  i=1  fi · E · Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, we call an initial operator L monic, if there is a differential  operator L1 and an integral operator L2 such that  L = E · (L1 + L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In R⟨∂, � , E⟩, using Table 1, one can check that the differential operators  are the elements of the subring generated by R and ∂, the integral operators  are the elements of the R-bimodule generated by  �  , the initial operators are the  elements of the two-sided ideal generated by E, and the monic initial operators  are the elements of the right ideal generated by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2 says that every  integro-differential operator can be written as the sum of a differential operator,  an integral operator, and an initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In fact, by the stronger results in  the appendix, this decomposition of integro-differential operators even is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We outline how the construction of integro-differential opera-  tors changes when the evaluation is multiplicative, see also [31, 14, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For  multiplicative evaluation, we have Efg = (Ef)Eg for all f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' So, for the  algebraic construction of corresponding operators as in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we need  to impose in addition that  E · f = Ef · E  (22)  for all f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This does not give rise to new consequences other than (17)–(21),  but together with (20) it makes (16) redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, in Table 1, we can  replace (16) by (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, (22) allows to reduce the evaluation term in  (18) and, since E  �  f = 0 for all f ∈ R, to omit the evaluation terms in (19) and  18  (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, the irreducible forms of operators given in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2 can  be simplified to  f · ∂j,  f ·  �  g,  f · E · ∂j,  where j ∈ N0 and f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Evidently, this ring of operators is isomorphic to  R⟨∂,  �  , E⟩ factored by the two-sided ideal (E · f − Ef · E | f ∈ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  IDO over integral domains  In many concrete situations, when computing with differential operators or dif-  ferential equations with scalar coefficients, the order of a product of differential  operators is the sum of the orders of the factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This is equivalent to the  ring R of coefficients being an integral domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' a commutative ring without  nontrivial zero divisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For the rest of this section, we only consider integro-  differential rings that are integral domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Under this assumption, we can in-  vestigate further properties of computations in the ring of IDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For instance, an  integro-differential equation can be reduced to a differential equation by differ-  entiation, see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Additionally, investigating the action of IDOs  on integro-differential rings algebraically leads to analyzing two-sided ideals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As explained earlier, elements of R⟨∂,  �  , E⟩ naturally act as additive maps  on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Likewise, they act naturally on any integro-differential ring extension of  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11, we also provide conditions when this action is faithful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0 ∈ R⟨∂,  �  , E⟩ is the only element that induces the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let (S,  �  , ∂) be the integro-differential ring defined in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='13 and as-  sume C is an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, S⟨∂, � , E⟩ does not act faithfully on S, since  E · ln(x) acts like zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' However, with the integro-differential subring R := C[x]  of S, R⟨∂,  �  , E⟩ acts faithfully on S by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To see this, let  L := �n  i=0 fi ·∂i ∈ R⟨∂,  �  , E⟩ and let k ∈ N such that coeff(fn, xk) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Apply-  ing E·L to xn−k ln(x)n ∈ S, we obtain (E·L)xn−k ln(x)n = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' coeff(fn, xk) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As a preparatory step, we need some basic statements involving multiplica-  tion with constants that are valid for integro-differential rings that are integral  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Linear independence over constants is tied to the Wronskian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Follow-  ing the analytic definition, the Wronskian of elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn of a commuta-  tive differential ring is defined by  W(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn) := det  ��  ∂i−1fj  �  i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂) be a differential ring that is an integral domain with  ring of constants C and let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are linearly independent over C if and only if their Wronskian  W(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn) is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are linearly independent over C, then g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn−1 are linearly  independent over C, where gi := W(fi, fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For showing the first statement, we note that neither linear independence  nor zeroness of the Wronskian changes, if we replace R and hence C by their  19  quotient fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For differential fields, a proof can be found in [15], which implies  the statement given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For showing the second statement, we assume that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are linearly  independent over C and we let c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , cn−1 ∈ C such that �n−1  i=1 cigi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By definition of gi and multilinearity of the Wronskian over C, we conclude  W(�n−1  i=1 cifi, fn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By assumption on f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn, this implies �n−1  i=1 cifi = 0  and hence c1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' = cn−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring that is an integral  domain and let C be its ring of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, elements of C commute with  all elements of R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, for any L ∈ R⟨∂,  �  , E⟩ and any nonzero  c ∈ C, we have that L = 0 if and only if c · L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By construction, constants commute with ∂,  �  , E ∈ R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since  R is commutative, it follows that elements of C commute with all elements  of R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, R⟨∂,  �  , E⟩ is a unital C-algebra and hence Q(C) ⊗C  R⟨∂,  �  , E⟩ is a unital C-algebra as well, with multiplication (c1⊗L1)·(c2⊗L2) =  (c1c2) ⊗ (L1 · L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now, 1 ⊗ L = c−1 ⊗ (c · L) implies L = 0 if c · L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Now, we are ready to have a closer look at certain computations with IDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The following two lemmas construct left or right multiples of IDO by differential  operators such that the product does not involve the integration operator any-  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The tricky part will be to ensure that the product is a nonzero operator  again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring that is an integral  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let C be the ring of constants of R and let L ∈ R⟨∂, � , E⟩ be not  an initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, there exist nonzero h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , hn ∈ R such that the  following product is a nonzero differential operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  (h1 · ∂ − ∂h1) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' · (hn · ∂ − ∂hn) · L  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' There are n ∈ N, a nonzero c ∈ C, differential operators L0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln ∈  R⟨∂,  �  , E⟩, and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn ∈ R such that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are linearly inde-  pendent over C and c · L = L0 + �n  i=1 fi ·  ��  gi + E · Li  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since L is not an  initial operator, L0 and g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn are not all zero by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If L0 = 0,  then we assume without loss of generality that g1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  To remove the sum �n  i=1 fi·  ��  gi + E · Li  �  , we will multiply c·L iteratively  by n first-order differential operators from the left as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If n > 0, then by  (11) and (12) we have (fn ·∂ − ∂fn)·fi ·  �  gi = fnfigi + (fn∂fi − (∂fn)fi)·  �  gi  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, using (14), we obtain  (fn · ∂ − ∂fn) · c · L = (fn · ∂ − ∂fn) · L0 +  n  �  i=1  fnfigi  +  n−1  �  i=1  (fn∂fi − (∂fn)fi) ·  ��  gi + E · Li  �  ,  20  which has the form ˜L0+�n−1  i=1 ˜fi ·  ��  gi + E · Li  �  similar to c·L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that also  the following two properties of the above representation of c · L are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  First, if L0 is nonzero, the differential operator ˜L0 := (fn · ∂ − ∂fn) · L0 +  �n  i=1 fnfigi is nonzero too, since fn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Second, ˜fi := fn∂fi − (∂fn)fi, for  i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n − 1}, are again linearly independent over C by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now,  we let hn := fnc such that (hn · ∂ − ∂hn) · L = (fn · ∂ − ∂fn) · c · L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If n = 1, we only need to show that (hn · ∂ − ∂hn) · L = ˜L0 is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As  remarked above, if L0 ̸= 0 then ˜L0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' On the other hand, if L0 = 0, then  ˜L0 = f 2  1 g1, which is nonzero as well due to the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If n > 1, we continue by repeating what we did with c · L above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' That is,  noting ˜f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , ˜fn−1 are linearly independent over C, we multiply (hn · ∂ − ∂hn) ·  L = ˜L0 + �n−1  i=1 ˜fi ·  ��  gi + E · Li  �  from the left by hn−1 · ∂ − ∂hn−1, where  hn−1 := ˜fn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We iterate this a total of n − 1 times, the result is a differential  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To see that it is also nonzero, we focus on the last step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' we refer  to the case n = 1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  An immediate consequence of the previous lemma is that a left ideal in  R⟨∂,  �  , E⟩ is nontrivial if and only if it contains a nonzero differential operator  or a nonzero initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Left ideals in R⟨∂,  �  , E⟩ arise, for instance, as sets  of operators annihilating a given subset of a left R⟨∂, � , E⟩-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As another  consequence, for two-sided ideals, we obtain Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring that is an integral  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let C be the ring of constants of R and let L ∈ R⟨∂,  �  , E⟩ be a nonzero  monic initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, there exist nonzero h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , hn ∈ R and a nonzero  differential operator ˜L ∈ R⟨∂,  �  , E⟩ such that  L · (hn · ∂ + 2∂hn) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' · (h1 · ∂ + 2∂h1) = E · ˜L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' There are n ∈ N, a nonzero c ∈ C, a differential operator L0 ∈ R⟨∂,  �  , E⟩,  nonzero f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ � R, and nonzero g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn ∈ R such that g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , gn are  linearly independent over C and L·c = E·L0+�n  i=1 E·fi·  �  gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7,  L · c = c · L is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  To remove the sum �n  i=1 E · fi ·  �  gi, we will multiply L · c iteratively by  n first-order differential operators from the right as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If n > 0, then we  have  E · fi ·  �  gign · ∂ = E · figign − Efi · E · gign − E · fi ·  �  ((∂gi)gn + gi∂gn)  for all i by (18) and by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, by Efi = 0, we obtain  L · c · (gn · ∂ + 2∂gn) = E · L0 · (gn · ∂ + 2∂gn) +  n  �  i=1  E · fi ·  �  (gign · ∂ + 2gi∂gn)  = E · L1 +  n−1  �  i=1  E · fi ·  �  (gi∂gn − (∂gi)gn),  21  with L1 := L0 · (gn · ∂ + 2∂gn) + �n  i=1 figign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Note that gi∂gn − (∂gi)gn,  for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n − 1}, are again linearly independent over C by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Consequently, with hn := cgn being nonzero, the operator L · (hn · ∂ + 2∂hn) is  expressed like L · c above, just with L0 replaced by L1, with different gi, and  without the last summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By iterating this process of multiplying by a differential operator that is  chosen as above, we obtain a right multiple of L that is of the form E · Ln with  a differential operator Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It remains to show that Ln is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' After the  last step, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' passing from E · Ln−1 + E · f1 ·  �  ˜g1, with differential operator  Ln−1 and nonzero ˜g1 ∈ R, to E · Ln, we have Ln = Ln−1 · (˜g1 · ∂ + 2∂˜g1) +  f1˜g2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If the differential operator Ln−1 is nonzero, then also Ln is nonzero, since  ˜g1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Otherwise, we have Ln = f1˜g2  1, which is nonzero as well due to the  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring that is an integral  domain and let I ⊆ R⟨∂,  �  , E⟩ be a two-sided ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If I contains an operator  that is not an initial operator, then I contains an operator of the form f ·E with  nonzero f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='8, I contains a nonzero differential operator L = �n  i=0 fi·∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', n} be minimal such that fk ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, fk · E ∈ I since  L ·  � k · E =  n  �  i=k  fi · ∂i ·  � k · E =  n  �  i=k  fi · ∂i−k · E = fk · E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Based on the above lemmas, we can find conditions when the action of  R⟨∂,  �  , E⟩ on an integro-differential ring is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In this case, the annihilator  AnnR⟨∂,�  ,E⟩(S) = {L ∈ R⟨∂,  �  , E⟩ | ∀f ∈ S : Lf = 0}  is a two-sided ideal in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, we show that the annihilator  only contains initial operators and, if C is a field, can be generated by monic  initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (S, ∂,  �  ) be an integro-differential ring with its ring of  constants C being an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let R be an integro-differential subring of  S that is an integral domain and contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, R⟨∂,  �  , E⟩ acts faithfully on  S if and only if there is no nonzero differential operator L ∈ R⟨∂,  �  , E⟩ such that  E · L vanishes on all of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In any case, the two-sided ideal I = AnnR⟨∂,�  ,E⟩(S)  only contains initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, if C is a field, I has a generating set  consisting only of monic initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since operators of the form f · E, with nonzero f ∈ R, do not vanish on  1 ∈ S, the ideal I only contains initial operators by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If there is a  nonzero differential operator L ∈ R⟨∂,  �  , E⟩ such that E · L vanishes on all of  S, then R⟨∂,  �  , E⟩ obviously does not act faithfully on S, since E · L is nonzero  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  22  Now, we assume that there is no nonzero differential operator L such that  E · L ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let L ∈ I, then there are nonzero c ∈ C, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R, and  nonzero monic initial operators L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln ∈ R⟨∂,  �  , E⟩ such that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are  C-linearly independent and c · L = �n  i=1 fi · Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If we would have n ̸= 0, then  we would conclude L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln ∈ I, since �n  i=1 fiLig = cLg = 0 and Lig ∈ C  for all g ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='9, there would be a nonzero differential  operator ˜L ∈ R⟨∂,  �  , E⟩ such that E · ˜L ∈ I in contradiction to the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Hence, n = 0, which implies c · L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7, it follows that L = 0,  which shows that R⟨∂,  �  , E⟩ acts faithfully on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Finally, if C is a field, we show that I has a generating set consisting only  of monic initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let L ∈ I, then L is an initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now, there  are nonzero f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R, and nonzero monic initial operators L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln ∈  R⟨∂,  �  , E⟩ such that f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are C-linearly independent and L = �n  i=1 fi · Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For all g ∈ S, we have Lig ∈ C and �n  i=1 fiLig = Lg = 0, which implies Lig = 0  for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , Ln ∈ I, which shows that I is generated by nonzero  monic initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Assuming additional properties of R resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' of the action of R⟨∂, � , E⟩, gen-  erating sets of annihilators can be narrowed down even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular,  the following two corollaries consider the situation when R is a field or E is  multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (S, ∂, � ) be an integro-differential ring with its ring of  constants C being a field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let R be an integro-differential subring of S that is a  field and contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, the two-sided ideal AnnR⟨∂,  �  ,E⟩(S) has a generating  set consisting only of operators of the form E · �n  i=0 fi · ∂i, with f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R  where f0 ∈  �  R is nonzero and fn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11, I := AnnR⟨∂,  �  ,E⟩(S) has a generating set consisting  only of monic initial operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now, let L ∈ I be a monic initial operator and  let A be the set of all elements in I that have the form specified in the statement  of the corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='9, we obtain nonzero h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , hm ∈ R such that  L · M = E · ˜L with M := (hm · ∂ + 2∂hm) · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' · (h1 · ∂ + 2∂h1) ∈ R⟨∂,  �  , E⟩  and some nonzero differential operator ˜L = �n  i=0 ˜fi · ∂i ∈ R⟨∂,  �  , E⟩ with  ˜fn ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, by repeated use of (11), there are f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R with fn = 1  such that ˜L · ˜f −1  n  = �n  i=0 fi · ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let k be minimal such that fk ̸= 0, then  ˜L · ˜f −1  n  � k = �n−k  i=0 fi+k · ∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since E · ˜L · ˜f −1  n  � k ∈ I vanishes on 1 ∈ S,  we conclude fk ∈  �  R and hence E · �n−k  i=0 fi+k · ∂i ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since R is a field, we  can verify by (11) and (12) that h−1  i �  hi ∈ R⟨∂,  �  , E⟩ is a right inverse of  hi·∂+2∂hi for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, there exists ˜  M ∈ R⟨∂,  �  , E⟩ such that M · ˜  M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, E ·  ��n−k  i=0 fi+k · ∂i�  ∂k · ˜fn · ˜  M = E · ˜L · ˜  M = L · M · ˜  M = L, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' L is in  the ideal generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether, this shows that I is generated by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Whenever E is multiplicative on R, the action of R⟨∂,  �  , E⟩ cannot be faithful  on R since the operator E · f acts as zero map for any f ∈ R with Ef = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  23  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, Ef = 0 is equivalent to f ∈  �  R and one can always choose  f =  �  1, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5, we already discussed factoring the ring of  operators by the relations (22) immediately arising from multiplicativity of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The following corollary shows that the resulting quotient always acts faithfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (S, ∂,  �  ) be an integro-differential ring with its ring of  constants C being an integral domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let R be an integro-differential subring  of S that is an integral domain and contains C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If Efg = (Ef)Eg for all f ∈ R  and g ∈ S, then the two-sided ideal AnnR⟨∂,  �  ,E⟩(S) is generated by the set  {E · f − Ef · E | f ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let I := AnnR⟨∂,�  ,E⟩(S) and let J ⊆ R⟨∂,  �  , E⟩ be the two-sided ideal  generated by the set {E · f − Ef · E | f ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By assumption on E, we have  that J ⊆ I, so it only remains to show I ⊆ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='11, every L ∈ I is an initial operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Following the form  of initial operators in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5, there are f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R such that L =  �n  i=0 fi ·E·∂i modulo J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By J ⊆ I, we have that �n  i=0 fi ·E·∂i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore,  �n  i=0 fi·E·∂i vanishes on  � k1 ∈ S for all k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This implies inductively  that f0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn are all zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, we have L ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  5  Equational prover in calculus  In this section, we illustrate how results from analysis can be proven via compu-  tations in the ring of integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The general approach con-  sists in formulating an analytic statement as an identity of integro-differential  operators and then prove this identity algebraically in R⟨∂,  �  , E⟩, with minimal  assumptions on the integro-differential ring (R, ∂,  �  ) used in the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Note that we prove results directly by a computation with integro-differential  operators, instead of doing the whole computation with elements of R or any  other left R⟨∂,  �  , E⟩-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  An identity in R⟨∂,  �  , E⟩ can be proven by comparing irreducible forms  of the left hand side and right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Recall that such irreducible forms  are given by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2 and can be computed systematically by the rewrite  rules in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In this sense, the rewrite rules provide an equational prover  for integro-differential operators provided one can decide equality of irreducible  forms in R⟨∂,  �  , E⟩, which includes deciding identities in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In practice, this is  often possible for concrete irreducible forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Once an identity L1 = L2 is proven in R⟨∂,  �  , E⟩, we immediately infer the  corresponding identity in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' L1f = L2f for all f ∈ R, by the canonical  action of R⟨∂,  �  , E⟩ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, by acting on any other left R⟨∂,  �  , E⟩-  module, we obtain the analogous identity also in those modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Furthermore,  any concrete computation with operators acting on functions uses only finitely  many derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, by inspecting every step of the computation,  an identity proven for infinitely differentiable functions can also be proven for  functions that are only sufficiently often differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  Variation of constants for scalar equations  In this section, we deal with the method of variation of constants for comput-  ing solutions of inhomogeneous ODEs in terms of integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  First, we recall the analytic statement, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4 in Chapter 3 of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Consider the inhomogeneous linear ODE  y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Assume that z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn(x) is a fundamental system of the homogeneous equa-  tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' z(n)  i  (x) + an−1(x)z(n−1)  i  (x) + · · · + a0(x)zi(x) = 0 and the Wronskian  w(x) := W(z1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn(x)) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then,  z∗(x) :=  n  �  i=1  (−1)n−izi(x)  � x  x0  W(z1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zi−1(t), zi+1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn(t))  w(t)  f(t) dt  (23)  is a particular solution of the inhomogeneous equation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Algebraically, we model scalar functions by fixing a commutative integro-  differential ring (R, ∂,  �  ) and, as indicated above, we model computations with  scalar equations by computations in the corresponding ring of integro-differential  operators R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' From the operator viewpoint, mapping the inhomogeneous  part f to a solution of the equation Ly = f amounts to constructing a right  inverse of the differential operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In general, for an operator L and a  right inverse H, a particular solution of the inhomogeneous equation is given by  z∗ = Hf, since  Lz∗ = L(Hf) = (L · H)f = 1f = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Recall that  �  is a right inverse of ∂ by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Now, we consider an  arbitrary monic first-order differential operator  L = ∂ + a,  with a ∈ R, and assume that z ∈ R is a solution of Ly = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ∂z + az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, using (11), we compute  (∂ + a) · z = ∂ · z + a · z = z · ∂ + ∂z + az = z · ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If, moreover, we assume that z has a multiplicative inverse z−1 ∈ R, we obtain  (∂ + a) · (z ·  �  z−1) = z · ∂ ·  �  z−1 = z · z−1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Hence  H = z ·  �  z−1  is a right inverse of L in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We also outline the computation for a second order differential operator  L = ∂2 + a1 · ∂ + a0,  25  with a1, a0 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If z ∈ R is a solution of Ly = 0, then  L · z = z · ∂2 + (2∂z + a1z) · ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Assume that there exist two solutions z1, z2 ∈ R of Ly = 0 such that their  Wronskian  w = z1∂z2 − z2∂z1  has a multiplicative inverse 1  w ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let  H = −z1 ·  �  z2  w + z2 ·  �  z1  w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In the ring of operators, we can compute  ∂ · zi  w = zi  w · ∂ + ∂zi  w − zi∂w  w2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, using the normal forms of L · zi and ∂ · zi  w , we obtain  L · H = −z1 · ∂ · z2  w − 2(∂z1) z2  w − a1z1 z2  w + z2 · ∂ · z1  w + 2(∂z2) z1  w + a1z2 z1  w  = −z1 · ∂ · z2  w + z2 · ∂ · z1  w + 2 w  w = −z1∂ z2  w + z2∂ z1  w + 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Hence H is a right inverse of L in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  More generally, we have the following formulation of the method of variation  of constants for integro-differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Recall from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, that  the Wronskian W(f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn) of elements f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , fn ∈ R is defined completely  analogous to the analytic situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be a commutative integro-differential ring and let  L = ∂n + �n−1  i=0 ai · ∂i ∈ R⟨∂,  �  , E⟩, n ≥ 1, with a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , an−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assume that  z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn ∈ R are such that Lzi = 0 and w := W(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn) has a multiplicative  inverse  1  w ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, with  H :=  n  �  i=1  (−1)n−izi ·  �  W(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zi−1, zi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn)  w  we have that L · H = 1 in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The cases n = 1 and n = 2 have been shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In principle, an  analogous computation could be done for any concrete n ≥ 3 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To obtain  a finite proof for all n ≥ 3 at once, we will utilize a more general framework in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2  Linear systems and operators with matrix coefficients  Algebraically, computing with linear systems of differential equations can be  modelled by integro-differential operators over some noncommutative integro-  differential ring, whose elements correspond to matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Such noncommutative  integro-differential rings can be obtained from any scalar integro-differential  26  ring, since one can equip the ring of n × n matrices with a derivation and an  integration defined entrywise, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Recall that Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2 hold also for operators with  coefficients in noncommutative integro-differential rings, in particular for oper-  ators with matrix coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, any computation using a general  noncommutative integro-differential ring is automatically valid for concrete ma-  trices of any size, without the need for entrywise computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This allows for  compact proofs for matrices of general size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For switching to entrywise com-  putations, one can view operators with matrix coefficients equivalently also as  matrices of operators with scalar coefficients by identifying operators ∂,  �  , E  with corresponding diagonal matrices of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The following lemma shows  that also computations are equivalent in both viewpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Here, we use the  notation Ei,j(L) := (δi,kδj,lL)k,l=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',n for matrices with only one nonzero entry  L ∈ R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) an integro-differential ring and let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then,  there is exactly one ring homomorphism  ϕ : Rn×n⟨∂,  �  , E⟩ → R⟨∂,  �  , E⟩n×n  with ϕ(A) = A for A ∈ Rn×n and ϕ(L) = diag(L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , L) for L ∈ {∂,  �  , E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This ϕ is an isomorphism and its inverse homomorphism  ψ : R⟨∂, � , E⟩n×n → Rn×n⟨∂, � , E⟩  can be given by ψ(Ei,j(f)) = Ei,j(f) and ψ(Ei,j(L)) = Ei,j(1) · L for all f ∈ R  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' L ∈ {∂,  �  , E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' First, we check that the definition of ϕ indeed provides a unique and  well-defined homomorphism of rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Uniqueness follows from the fact that ϕ is  defined on a generating set of Rn×n⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For proving well-definedness, we  need to verify that the definition of ϕ respects all identities of generators given  in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Evidently, we have ϕ(1) = In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It is immediate to see  that ϕ(∂) · ϕ(  �  ) = ϕ(1) − ϕ(E) and ϕ(  �  ) · ϕ(∂) = ϕ(1) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For L ∈ {∂,  �  , E},  we verify in R⟨∂,  �  , E⟩n×n that ϕ(L) commutes with all elements of Cn×n and  that, for all A ∈ Rn×n, we have ϕ(L) · ϕ(A) · ϕ(E) = ϕ(LA) · ϕ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  More  explicitly, using (14)–(16) in R⟨∂,  �  , E⟩, we compute  ϕ(L) · ϕ(A) · ϕ(E) = diag(L, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , L) · A · diag(E, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , E)  = (L · ai,j · E)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',n = (Lai,j · E)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',n = ϕ(LA) · ϕ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Analogously, using (11) in R⟨∂,  �  , E⟩, we can also verify that ϕ(∂) · ϕ(A) =  ϕ(A) · ϕ(∂) + ϕ(∂A) for all A ∈ Rn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Similarly, we need to verify that the definition of ψ on a generating set of  R⟨∂,  �  , E⟩n×n gives rise to a well-defined homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For example, using  27  (14)–(16) in Rn×n⟨∂,  �  , E⟩, we compute  ψ(Ei,j(L)) · ψ(Ep,q(f)) · ψ(Ek,l(E)) = Ei,j(1) · L · Ep,q(f) · Ek,l(1) · E  = Ei,j(1) · L · δq,kEp,l(f) · E = Ei,j(1) · δq,kLEp,l(f) · E = δj,pδq,kEi,l(Lf) · E  = Ei,q(δj,pLf) · Ek,l(1) · E = ψ(Ei,q(δj,pLf)) · ψ(Ek,l(E))  for all f ∈ R, L ∈ {∂,  �  , E}, and all i, j, k, l, p, q ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Finally, ψ is the inverse of ϕ, since we have ψ(ϕ(A)) = A, ϕ(ψ(Ei,j(f))) =  Ei,j(f), ψ(ϕ(L)) = L, and ϕ(ψ(Ei,j(L))) = Ei,j(L) for the generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Integro-differential operators with coefficients from Rn×n constructed this  way have natural actions on Rn×n and on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By the above lemma, for  any left R⟨∂,  �  , E⟩-module M, there is a natural way of viewing M n as a  left Rn×n⟨∂, � , E⟩-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  It is straightforward to show that the action of  Rn×n⟨∂,  �  , E⟩ on M n is faithful if and only if R⟨∂,  �  , E⟩ acts faithfully on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3  Variation of constants for first-order systems  Instead of higher order scalar ODEs, we now consider variation of constants for  first-order systems, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 in Chapter 3 of [6] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Analyti-  cally, it can be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For an n × n matrix A(x) and a vector f(x)  of size n, we consider the first-order system given by  y′(x) + A(x)y(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If Φ(x) is a fundamental matrix of the homogeneous system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' it satisfies  Φ′(x) + A(x)Φ(x) = 0 and det(Φ(x)) ̸= 0, then a particular solution of the  inhomogeneous system is given by  z∗(x) = Φ(x)  � x  x0  Φ(t)−1f(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assume that a ∈ R  is such that there exists z ∈ R that satisfies ∂z +az = 0 and has a multiplicative  right inverse z−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, in R⟨∂,  �  , E⟩, the operators L := ∂ + a and  H := z ·  �  z−1 satisfy  L · H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The same computation as in the commutative case above can be done  also for noncommutative R without any changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that zz−1 = 1 was the  only property of z−1 used there, so it suffices that z−1 is a right inverse of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Observe that from this statement over an abstract integro-differential ring  (R, ∂,  �  ), which was proven without referring to matrices at all, the analytic  statement follows for arbitrary size n of the matrix A(x), provided we assume  sufficient regularity of the functions involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Based on Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, we now can complete the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 for  arbitrary n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Before doing so, we first detail the required translation from  28  a first-order system to the scalar equation entirely at the operator level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For  shorter notation, in the following, we again use the symbol ∂ also for the matrix  diag(∂, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , ∂) ∈ R⟨∂,  �  , E⟩n×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring and let  L = ∂n +  n−1  �  i=0  ai · ∂i ∈ R⟨∂,  �  , E⟩  with a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , an−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, let H ∈ R⟨∂,  �  , E⟩n×n satisfy (∂+A)·H = In  in R⟨∂,  �  , E⟩n×n, where  A :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −1  0  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  0  · ·  · ·  0  −1  a0  a1  · ·  · ·  an−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, the upper right entry H1,n of H satisfies L · H1,n = 1 in R⟨∂,  �  , E⟩ and  Hi,n = ∂i−1 · H1,n for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For n = 1, the statement is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' So, we let n ≥ 2 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In  short, starting from the identity (∂ + A)·H = In of n× n matrices of operators,  we multiply both sides from the left with a suitable S ∈ R⟨∂,  �  , E⟩n×n and  inspect the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, for elimination in the last n − 1 columns  of ∂ + A, we use  S :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  0  · ·  · ·  · ·  0  ∂  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂n−2  · ·  ∂2  ∂  1  0  s1  · ·  · ·  · ·  sn−1  1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  with sj := ∂n−j+�n−j−1  i=0  ai+j ·∂i ∈ R⟨∂,  �  , E⟩ for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Observing  that, in R⟨∂, � , E⟩, we have −sj + sj+1 · ∂ = −aj for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', n − 2}, we  29  compute S · (∂ + A) explicitly:  S ·  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ∂  −1  0  · ·  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  0  · ·  0  ∂  −1  a0  a1  · ·  an−2  ∂ + an−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ∂  −1  0  · ·  0  ∂2  0  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  ∂n−1  0  · ·  0  −1  s1 · ∂ + a0  0  · ·  0  −sn−1 + ∂ + an−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ∂  −1  0  · ·  0  ∂2  0  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  ∂n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  −1  L  0  · ·  · ·  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Altogether, we obtain that  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  ∂  −1  0  · ·  0  ∂2  0  −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  ∂n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  −1  L  0  · ·  · ·  0  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  H = S · (∂ + A) · H = S,  where comparison of the entries in the last column of the left hand side and  right hand side yields ∂ · H1,n − H2,n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , ∂n−1 · H1,n − Hn,n = 0 and  L · H1,n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Finally, we proceed with the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Recalling the assump-  tions, we fix a commutative integro-differential ring (R, ∂,  �  ) and we let L =  ∂n+�n−1  i=0 ai·∂i ∈ R⟨∂,  �  , E⟩ with a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , an−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We also let z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn ∈ R  such that Lzi = 0 and such that w := W(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn) has a multiplicative inverse  1  w ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In the (noncommutative) integro-differential  30  ring (Rn×n, ∂,  �  ), we consider the matrices  A :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −1  0  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  0  · ·  · ·  0  −1  a0  a1  · ·  · ·  an−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  and  Z :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  z1  · ·  zn  ∂z1  · ·  ∂zn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂n−1z1  · ·  ∂n−1zn  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8  and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w  was assumed to be invertible in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, we apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3 to obtain  (∂ + A) · (Z ·  �  Z−1) = 1 in Rn×n⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Instead of integro-differential  operators with matrix coefficients in Rn×n, we now consider the objects as  n × n matrices whose entries are integro-differential operators with coefficients  in R, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4, we obtain that L·(Z ·� ·Z−1)1,n = 1  holds in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In order to compute the entry (Z ·  �  Z−1)1,n, we can easily  determine a general form of the entries of the matrix product Z ·  �  Z−1 ∈  R⟨∂,  �  , E⟩n×n:  Z ·  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  �  0  · ·  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  0  · ·  0  �  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  Z−1 = Z ·  \uf8eb  \uf8ec  \uf8ed  �  (Z−1)1,1  · ·  �  (Z−1)1,n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �  (Z−1)n,1  · ·  �  (Z−1)n,n  \uf8f6  \uf8f7  \uf8f8  =  � n  �  k=1  (∂i−1zk) · � · (Z−1)k,j  �  i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, we compute all entries  (Z−1)i,n = (−1)n+i W(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zi−1, zi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn)  w  in the last column of Z−1 ∈ Rn×n via Cramer’s rule (or via the cofactor matrix)  so that we recognize H ∈ R⟨∂,  �  , E⟩ defined in the statement of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  as the top right entry (Z ·  �  Z−1)1,n of the above matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This concludes the  proof that L · H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  6  Generalizing identities from calculus  The generalizations of well-known identities presented in this section introduce  additional terms involving the induced evaluation, which vanish if the evaluation  is multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As explained in the previous section, the proofs mostly rely  on computing irreducible forms for IDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  31  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  Initial value problems  Recall that the formula given in (23) provides a particular solution of the inho-  mogeneous linear ODE  y(n)(x) + an−1(x)y(n−1)(x) + · · · + a0(x)y(x) = f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, this particular solution also satisfies the homogeneous initial condi-  tions y(x0) = 0, y′(x0) = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , y(n−1)(x0) = 0, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4 in Chap-  ter 3 of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The induced evaluation of an integro-differential ring allows us  to model such properties algebraically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Similarly to the general proof of Theo-  rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we first investigate the general solution formula for homogeneous initial  value problems of first-order systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To this end, we fix a (not necessarily com-  mutative) integro-differential ring R and we work in the corresponding ring of  IDO R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  As a general principle, not only in the ring R⟨∂,  �  , E⟩ but in any ring with  unit element, if some H is a right inverse of some L and B, P are such that  L · P = 0 and B · P = B, then G := (1 − P) · H satisfies  L · G = 1  and  B · G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  So, by choosing an appropriate operator P ∈ R⟨∂,  �  , E⟩ that satisfies (∂+a)·P =  0 and E · P = E, we can obtain the following version of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3 that also  solves the homogeneous initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring and let L = ∂ + a ∈  R⟨∂,  �  , E⟩ with a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assume that z ∈ R is such that ∂z + az = 0 and in  addition to a multiplicative right inverse z−1 ∈ R also a right inverse (Ez)−1 ∈ C  exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, in R⟨∂, � , E⟩, the operator  G := (1 − z(Ez)−1 · E) · z ·  �  z−1  satisfies L · G = 1 and E · G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, we have that H := z ·  �  z−1 satisfies L · H = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Now,  letting P := z(Ez)−1 · E, we compute the normal forms of L · P and E · P:  (∂ + a) · z(Ez)−1 · E =  �  z(Ez)−1 · ∂ + ∂z(Ez)−1�  E + az(Ez)−1 · E  = z(Ez)−1 · ∂ · E = 0,  E · z(Ez)−1 · E = Ez(Ez)−1 · E = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then, from L · P = 0 and E · P = E, it follows straightforwardly that G =  (1 − P) · H has the properties claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If the evaluation E is multiplicative, then the existence of (Ez)−1 ∈  C follows form the existence of z−1 ∈ R and, with (22), we also obtain E·z ·  �  =  (Ez)E ·  �  = 0, which implies P · H = 0 and hence G = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, the right  inverse H obtained in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3 satisfies the initial value condition E ·H = 0  already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  32  For conversion to the scalar case, we need to compute the element in the top  right corner of G = (1 − Z(EZ)−1 · E) · Z ·  �  Z−1 and show that it has the  desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be a commutative integro-differential ring and let  L = ∂n + �n−1  i=0 ai · ∂i ∈ R⟨∂,  �  , E⟩, n ≥ 1, with a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , an−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assume  that z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn ∈ R are such that Lzi = 0 and w := W(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , zn) has a mul-  tiplicative inverse  1  w ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, assume that there are ci,j ∈ C such that  E∂k �n  i=1 ci,jzi = δj,k for all j, k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, with these ci,j and  G :=  n  �  k=1  (−1)n−k  �  zk −  n  �  i,j=1  zici,j−1 · E · (∂j−1zk)  � �  W(z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',zk−1,zk+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',zn)  w  we have that L · G = 1 and E · G = 0 in R⟨∂,  �  , E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As in the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we consider the matrices  A :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  0  −1  0  · ·  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  0  0  · ·  · ·  0  −1  a0  a1  · ·  · ·  an−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  and  Z :=  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ed  z1  · ·  zn  ∂z1  · ·  ∂zn  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  ∂n−1z1  · ·  ∂n−1zn  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f8  and note that (∂ + A)Z = 0 and that Z−1 ∈ Rn×n exists, since det(Z) = w was  assumed to be invertible in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Furthermore, we note that  \uf8eb  \uf8ec  \uf8ed  c1,0  · ·  c1,n−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  cn,0  · ·  cn,n−1  \uf8f6  \uf8f7  \uf8f8  is the multiplicative (right) inverse of EZ in Cn×n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we conclude  that ˜G := (1−Z(EZ)−1 ·E)·Z ·  �  Z−1 ∈ Rn×n⟨∂,  �  , E⟩ satisfies (∂ +A)· ˜G = 1  and E · ˜G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Passing to R⟨∂,  �  , E⟩n×n via Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2, the latter identity  implies E · ˜Gi,j = 0 and the former implies L · ˜G1,n = 1 and ˜Gi,n = ∂i−1 · ˜G1,n  by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, we also have E · ∂i−1 · ˜G1,n = 0 for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Finally, we verify that ˜G1,n = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' With H := Z ·  �  Z−1 ∈ R⟨∂,  �  , E⟩n×n,  we have ˜G1,n = H1,n − �n  i,j=1 Z1,i((EZ)−1)i,j · E · Hj,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' From the proof of  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we obtain that Hj,n = (−1)n+k(∂j−1zk)·  �  W(z1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',zi−1,zi+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=',zn)  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Altogether, this yields ˜G1,n = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2  Taylor formula  Usually, Taylor’s theorem is only considered for sufficiently smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In integro-differential rings, an analog of the Taylor formula with integral re-  mainder term  f(x) =  n  �  k=0  f (k)(x0)  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  xk +  � x  x0  (x − t)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  f (n+1)(t) dt  33  can be formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' While the formula arising from the identity of operators in  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6 is more complicated, it is also valid if singularities are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We start by giving a first version of the Taylor formula where the remainder  term is given as repeated integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It simply follows by iterating (3) resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  See also Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In R⟨∂,  �  , E⟩ we have  for any n ∈ N that  1 =  n  �  i=0  � i · E · ∂i +  � n+1 · ∂n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For any n ∈ N, we can use (13) to rewrite the right hand side:  n  �  i=0  � i · E · ∂i +  � n+1 · ∂n+1 =  n  �  i=0  � i · E · ∂i +  � n · (1 − E) · ∂n  =  n−1  �  i=0  � i · E · ∂i +  � n · ∂n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Setting n = 0 here gives E+  �  ∂ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence, the claim follows by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Using (15) and the related identity in (20), we can always write operators  � i · E as xi · E, where xi := � i1 as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By (19) and (21), we can  always write  � n+1 without higher powers of  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To make the resulting expressions  simpler, we restrict to the case that E is multiplicative on polynomials, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Exmxn = 0 for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring such that the induced  evaluation E satisfies Exmxn = 0 for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, in R⟨∂,  �  , E⟩, we  have for any n ∈ N that  � n+1 =  n  �  k=0  (−1)n−kxk ·  �  xn−k −  n−1  �  k=0  n−k  �  j=1  (−1)n−k−jxk · E · xj ·  �  xn−k−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We prove this identity by induction on n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For n = 0, the right  hand side directly yields  �  in agreement with the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Assuming the  identity holds for some n ∈ N, we multiply both sides by  �  from the left and we  rewrite the right hand side using (19) and (15) to obtain  � n+2 =  n  �  k=0  (−1)n−k�� xk · � − � · � xk − E · � xk · � �  xn−k  −  n−1  �  k=0  n−k  �  j=1  (−1)n−k−j�  xk · E · xj ·  �  xn−k−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  34  We have xk+1xn−k =  �n+1  k+1  �  xn+1 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2, so we can expand the right  hand side into  n  �  k=0  (−1)n−kxk+1 ·  �  xn−k +  n  �  k=0  (−1)n−k+1  �n + 1  k + 1  ��  xn+1  −  n  �  k=0  (−1)n−kE · xk+1 · � · xn−k −  n−1  �  k=0  n−k  �  j=1  (−1)n−k−jxk+1 · E · xj · � · xn−k−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Exploiting �n  k=0(−1)n−k+1�n+1  k+1  �  = (−1)n+1 in the second sum, we can regroup  terms to obtain the right hand side of the claimed identity for n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This  completes the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Altogether, we obtain the following identity in R⟨∂,  �  , E⟩ generalizing the  usual Taylor formula with integral remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This identity holds over any  integro-differential ring in which the induced evaluation is multiplicative on the  integro-differential subring generated by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6 (Taylor formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring  such that E is multiplicative on the integro-differential subring generated by 1,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Exmxn = 0 for all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, for all f ∈ R and all n ∈ N we have  1 =  n  �  k=0  xk · E · ∂k +  n  �  k=0  (−1)n−kxk ·  �  xn−k · ∂n+1  −  n−1  �  k=0  n−k  �  j=1  (−1)n−k−jxk · E · xj · � · xn−k−j · ∂n+1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4 using  � i·E = xi·E and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5, as explained  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In particular, if in addition to the assumptions of the theorem we have  Q ⊆ R, then, with operators acting on some f, we have that  f =  n  �  k=0  xk  1  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' E∂kf +  n  �  k=0  (−1)n−k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xk  1  �  xn−k  1  ∂n+1f  −  n−1  �  k=0  n−k  �  j=1  (−1)n−k−j  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (n − k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='xk  1Exj  1  �  xn−k−j  1  ∂n+1f  While the first and the second sum correspond to the Taylor polynomial and the  integral remainder term, the third sum corresponds to an additional polynomial  that arises from our general setting allowing non-multiplicative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It  can be viewed as an integro-differential algebraic version of the analytic formula  −  n−1  �  k=0  (x − x0)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  �� x  x0  (x − t)n−k − (x0 − t)n−k  (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  f (n+1)(t)dt  �  x=x0  ,  35  which vanishes for smooth functions f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Although in practice these additional  terms yield zero even for many elements f that model singular functions in con-  crete integro-differential rings, they cannot be dropped in general, as illustrated  in C[x, x−1, ln(x)] by considering n = 1 and f = ln(x), for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' With in-  tegration defined as in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='13, we have x1 = x and with f = ln(x) the  Taylor polynomial Ef + xE∂f vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' By ∂2f = − 1  x2 , the second sum yields  −  �  x∂2f + x  �  ∂2f = ln(x) + 1 and the third sum −Ex  �  ∂2f = −1 compensates  the constant term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' More generally, we can characterize the integro-differential  rings where the additional polynomial does not play a role in the Taylor formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring such that E is mul-  tiplicative on the integro-differential subring generated by 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Exmxn = 0 for  all m, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, we have Exng = 0 for all n ≥ 1 and g ∈ R if and only if  we have  f =  n  �  k=0  xkE∂kf +  n  �  k=0  (−1)n−kxk  �  xn−k∂n+1f  for all n ∈ N and f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If Exng = 0 for all n ≥ 1 and g ∈ R, then we have in particular  Exj  �  xn−k−j∂n+1f = 0 for all j, k, n ∈ N and f ∈ R with 1 ≤ j ≤ n − k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Hence, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6 implies the claimed identity for all n ∈ N and f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For the converse, let n ≥ 1 be minimal such that Exng ̸= 0 for some g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  With such g, we let f :=  � ng and, by minimality of n, we obtain that  n−1  �  k=0  n−k  �  j=1  (−1)n−k−jxkExj  �  xn−k−j∂n+1f = Exn  �  ∂g = Exng − ExnEg = Exng  is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Consequently, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6 implies that the claimed identity does  not hold for this n and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Acknowledgements  This work was supported by the Austrian Science Fund (FWF): P 27229,  P 31952, and P 32301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Part of this work was done while both authors were  at the Radon Institute of Computational and Applied Mathematics (RICAM)  of the Austrian Academy of Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The authors would like to thank Alban  Quadrat for bringing the book [23] to the attention of the second author during  his stay at INRIA Saclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  A  Normal forms for IDO in tensor rings  The goal of this appendix is to state and prove a refinement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2  providing uniqueness of normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Uniqueness is achieved by representing  operators by elements of a tensor ring, which is formed on a module of basic  operators generated by R, ∂,  �  , E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The ring of operators can be constructed  36  as quotient of the tensor ring, where relations of basic operators are encoded by  tensor reduction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The main technical tool for proving uniqueness of nor-  mal forms is a generalization of Bergman’s Diamond Lemma in tensor rings [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For the convenience of the reader, we give a formal and largely self-contained  summary of tensor reduction systems and we explain the translation of identi-  ties of operators into this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In this appendix, K denotes a ring (not  necessarily commutative) with unit element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we start by recalling basic properties of bimodules and tensor  rings on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For further details on tensor rings and proofs see, for example,  [34, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, we recall decompositions with specialization, which are used for  defining reduction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2, we state the Diamond Lemma for tensor  reduction systems with specialization from [14] and provide a summary of the  relevant notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, we construct an appropriate tensor ring along  with a tensor reduction system for dealing with IDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We use these to state The-  orem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3, which provides a precise formulation of uniqueness of normal forms  of IDO in terms of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4, we give a complete proof of the  theorem, where the necessary computations for verifying uniqueness of normal  forms in the tensor ring are done in an automated way by our package TenReS in  the computer algebra system Mathematica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' These computations are contained  in the Mathematica file accompanying this paper, which includes a log of the re-  duction steps and is available at http://gregensburger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='com/softw/tenres/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The theorem proved in that section even covers more general rings of operators,  which allow to deal with additional functionals besides the induced evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1  Tensor rings on bimodules and decompositions  A K-bimodule is a left K-module M which is also a right K-module satisfying  the associativity condition (km)l = k(ml) for all m ∈ M and k, l ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A ring R  that is a K-bimodule such that (xy)z = x(yz) for any x, y, z in R or K is called  a K-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, if K is a subring of some ring R, then R is a K-ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We first recall basic properties of the tensor product on K-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For  K-bimodules M, N, their K-tensor product M ⊗ N is a K-bimodule generated  by the pure tensors {m ⊗ n | m ∈ M, n ∈ N} with relations  (m + ˜m) ⊗ n = m ⊗ n + ˜m ⊗ n,  m ⊗ (n + ˜n) = m ⊗ n + m ⊗ ˜n,  and  mk ⊗ n = m ⊗ kn  having scalar multiplications  k(m ⊗ n) = (km) ⊗ n  and  (m ⊗ n)k = m ⊗ (nk)  for all m, ˜m ∈ M, n, ˜n ∈ N, and k ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We denote the tensor product of M with itself over K by M ⊗n = M ⊗· · ·⊗M  (n factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, M ⊗1 = M and we interpret M ⊗0 as the K-module  Kε, where ε denotes the empty tensor and right scalar multiplication satisfies  (k1ε)k2 = (k1k2)ε for k1, k2 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As a K-bimodule, the tensor ring K⟨M⟩ is  37  defined as the direct sum  K⟨M⟩ =  ∞  �  n=0  M ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  It can be turned into a K-ring with unit element ε where multiplication M ⊗r ×  M ⊗s → M ⊗(r+s) is defined via  (m1 ⊗ · · · ⊗ mr, ˜m1 ⊗ · · · ⊗ ˜ms) �→ m1 ⊗ · · · ⊗ mr ⊗ ˜m1 ⊗ · · · ⊗ ˜ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Via the tensor product, any decomposition of the module M carries over to  a decomposition of the tensor ring K⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In particular, we use decompositions  with specialization, which were introduced in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' These are given by a family  (Mz)z∈Z of K-subbimodules of M and a subset X ⊆ Z with M = �  z∈Z Mz =  �  x∈X Mx such that every module Mz, z ∈ Z, satisfies  Mz =  �  x∈S(z)  Mx  where S(z) := {x ∈ X | Mx ⊆ Mz} is the set of specializations of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that  this definition implies S(x) = {x} for x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For words W = w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' wn in the  word monoid ⟨Z⟩, we define the corresponding K-subbimodule of K⟨M⟩ by  MW := Mw1 ⊗ · · · ⊗ Mwn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The notion of specialization extends from the alphabet Z to the whole word  monoid ⟨Z⟩ by  S(W) := {v1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' vn ∈ ⟨X⟩ | ∀i : vi ∈ S(wi)}  such that S(W) = {V ∈ ⟨X⟩ | MV ⊆ MW }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We have the following generaliza-  tion  MW =  �  V ∈S(W)  MV  of the direct sum above and the decomposition  K⟨M⟩ =  �  W∈⟨Z⟩  MW =  �  W∈⟨X⟩  MW  of the tensor ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2  Tensor reduction systems with specialization  Fixing a decomposition with specialization of M, a reduction rule for K⟨M⟩ is  given by a pair r = (W, h) of a word W ∈ ⟨Z⟩ and a K-bimodule homomorphism  h: MW → K⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It acts on tensors of the form a⊗w⊗b with a ∈ MA, w ∈ MW ,  and b ∈ MB for some A, B ∈ ⟨Z⟩ by a ⊗ w ⊗ b →r a ⊗ h(w) ⊗ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Later, we will  specify homomorphisms h in concrete reduction rules (W, h) via their values on  a generating set of MW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Formally, well-definedness of such homomorphisms can  38  be ensured by the universal property of the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A set Σ of such  reduction rules is called a reduction system over Z on K⟨M⟩ and induces the  two-sided reduction ideal  IΣ := (t − h(t) | (W, h) ∈ Σ and t ∈ MW ) ⊆ K⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For computing in the factor ring K⟨M⟩/IΣ, we apply the reduction relation →Σ  induced by Σ on K⟨M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It reduces a tensor t ∈ K⟨M⟩ to a tensor s ∈ K⟨M⟩  if there is an r ∈ Σ such that t →r s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We say that t can be reduced to s by Σ  if t = s or there exists a finite sequence of reduction rules r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' , rn in Σ such  that  t →r1 t1 →r2 · · · →rn−1 tn−1 →rn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  If one tensor can be reduced to another, then their difference is contained in  IΣ and they represent the same element of K⟨M⟩/IΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The K-subbimodule of  irreducible tensors  K⟨M⟩irr =  �  W∈⟨X⟩irr  MW  can be characterized by the set of irreducible words ⟨X⟩irr ⊆ ⟨X⟩, which consists  of those words that avoid subwords arising as specializations S(W) of words  occurring in reduction rules (W, h) ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The irreducible tensors to which a  given tensor t can be reduced, are called its normal forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If t has a unique  normal form, it is denoted by t↓Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  An ambiguity is a minimal situation where two (not necessarily distinct) re-  duction rules can be applied to tensors in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For each pure tensor  of this kind, the corresponding S-polynomial is the difference of the results of  the two reduction steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For example, an overlap ambiguity arises from two  reduction rules (AB1, h), (B2C, ˜h) ∈ Σ, where A, B1, B2, C ∈ ⟨Z⟩ are nonempty  such that B1, B2 are equal or have a common specialization, and corresponding  S-polynomials are referred to by SP(AB1, B2C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' An ambiguity is called resolv-  able, if all its S-polynomials can be reduced to zero by Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If all ambiguities of Σ  are resolvable, then the reduction relation induced by Σ on K⟨M⟩ is confluent  and, by abuse of language, we also call Σ confluent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This means that there are  no hidden consequences implied in K⟨M⟩/IΣ by the identities explicitly specified  by Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The following theorem relies on the existence of a partial order of words in  ⟨Z⟩ that has certain properties, which are briefly explained now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A partial order  ≤ on ⟨Z⟩ is called a semigroup partial order if it is compatible with concatenation  of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' If in addition the empty word ǫ is the least element of ⟨Z⟩, then ≤  is called a monoid partial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It is called Noetherian if there are no infinite  descending chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We call a partial order ≤ on ⟨Z⟩ consistent with specialization  if every strict inequality V < W implies ˜V < ˜W for all specializations ˜V ∈ S(V )  and ˜W ∈ S(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' A partial order ≤ on ⟨Z⟩ is compatible with a reduction system  Σ over Z on K⟨M⟩ if for all (W, h) ∈ Σ the image of h is contained in the sum  of modules MV where V ∈ ⟨Z⟩ satisfies V < W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  39  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' [14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 20] Let M be a K-bimodule and let (Mz)z∈Z be  a decomposition with specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Let Σ be a reduction system over Z on  K⟨M⟩ and let ≤ be a Noetherian semigroup partial order on ⟨Z⟩ consistent with  specialization and compatible with Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, the following are equivalent:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' All ambiguities of Σ are resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Every t ∈ K⟨M⟩ has a unique normal form t↓Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic as K-bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Moreover, if these conditions are satisfied, then we can define a multiplication  on K⟨M⟩irr by s · t := (s ⊗ t)↓Σ so that K⟨M⟩/IΣ and K⟨M⟩irr are isomorphic  as K-rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3  Tensor reduction systems for IDO  Before using the tensor setting to construct the ring of integro-differential op-  erators R⟨∂,  �  , E⟩, we illustrate this construction on the well-known ring of  differential operators R⟨∂⟩ to highlight some of the special properties of the  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Usually, the ring of differential operators with coefficients from R  is constructed via skew polynomials �  i fi∂i over R in one indeterminate ∂, with  commutation rule ∂ · f = f · ∂ + ∂f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring  and let C denote its ring of constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In the following, we consider K-tensor  rings with K := C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The module of basic operators that generates all differential  operators is given by  M := MR ⊕ MD,  with K-bimodules  MR := R  (24)  and MD defined as the free left K-module  MD := K∂  (25)  generated by the symbol ∂, which we view as a K-bimodule with the right  multiplication c∂ · d = cd∂ for all c, d ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This definition is based on left  K-linearity of the derivation ∂ on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The commutation rule ∂ · f = f · ∂ + ∂f  coming from the Leibniz rule in R translates to the tensor reduction rule  (DR, ∂⊗f �→ f⊗∂ + ∂f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This rule is formalized by the K-bimodule homomorphism MDR = MD ⊗ MR →  K⟨M⟩ defined by ∂⊗f �→ f⊗∂ + ∂f on tensors ∂⊗f generating MDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that  this homomorphism represents a parameterized family of identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Reduction  of tensors by the rule above allows to syntactically move all multiplication op-  erators to the left of any differentiation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  40  Note that, in order to correctly model differential operators as equivalence  classes of tensors in K⟨M⟩, other relations among operators need to be phrased  as tensor reduction rules as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This is because the tensor ring K⟨M⟩ itself  is constructed without respecting relations coming from multiplication in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For instance, the composition of two multiplication operators f and g is a mul-  tiplication operator again, which leads to the reduction rule (RR, f⊗g �→ fg)  defined on the module MRR = MR ⊗ MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover, the multiplication operator  that multiplies by 1 acts like the identity operator, which is represented by the  empty tensor ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To define reduction rules that act only on C instead of all of R,  we need a direct decomposition of the K-bimodule  MR = MK ⊕ M˜R,  (26)  which in our case can be given by  MK := K  and  M˜R :=  �  R  (27)  based on Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, we can define a reduction rule on MK = C by 1 �→ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Altogether, the tensor reduction system ΣDiff = {rK, rRR, rDR} for differential  operators is given by the three reduction rules  {(K, 1 �→ ε),  (RR, f⊗g �→ fg),  (DR, ∂⊗f �→ f⊗∂ + ∂f)}  defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' It induces the two-sided ideal  IDiff := (t − h(t) | (W, h) ∈ ΣDiff and t ∈ MW )  = (1 − ε, f⊗g − fg, ∂⊗f − f⊗∂ − ∂f | f, g ∈ R)  in the ring K⟨M⟩ and computations with differential operators are modelled in  the quotient ring  R⟨∂⟩ := K⟨M⟩/IDiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Tensors that are not reducible w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ΣDiff are precisely K-linear combinations  of pure tensors of the form ∂⊗i and f ⊗ ∂⊗i, where i ∈ N0 and f ∈  �  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' With  alphabets X = {K, ˜R, D} and Z = X ∪{R}, one can check that all ambiguities of  ΣDiff are resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Using an appropriate ordering of words, one can show that  the conditions of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 are satisfied by this construction of R⟨∂⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proceeding to an analogous construction of the ring R⟨∂,  �  , E⟩ from Defini-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, we also require tensor reduction rules corresponding to the identities  (12)–(16), in addition to the three rules from the example above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To this end,  analogous to MD above, we introduce the K-bimodules  MI := K  �  and  ME := KE,  (28)  which are freely generated as left K-modules by the symbols  �  and E, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Then, we consider the K-tensor ring on the K-bimodule  M := MR ⊕ MD ⊕ MI ⊕ ME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  41  K  1 �→ ε  ID  �  ⊗∂ �→ ε − E  RR  f⊗g �→ fg  DRE  ∂⊗f⊗E �→ ∂f⊗E  DR  ∂⊗f �→ f⊗∂ + ∂f  IRE  �  ⊗f⊗E �→  �  f⊗E  DI  ∂⊗  �  �→ ε  ERE  E⊗f⊗E �→ (Ef)E  Table 2: Defining reduction system for integro-differential operators  Altogether, we obtain the tensor reduction system given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The two-  sided ideal IIDO induced by it allows to construct the ring R⟨∂,  �  , E⟩ as the  quotient K⟨M⟩/IIDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' However, this reduction system is not confluent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In order  to obtain normal forms that are unique as tensors in K⟨M⟩, we need a confluent  tensor reduction system on K⟨M⟩ that induces the same ideal IIDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The con-  fluent tensor reduction system given in Table 3 can be obtained by turning the  identities of Table 1 into tensor reduction rules and including the rules rK and  rRR from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' This allows us to state the following more precise version of  K  1 �→ ε  EI  E⊗  �  �→ 0  RR  f⊗g �→ fg  IRD  � ⊗f⊗∂ �→ f − E⊗f − � ⊗∂f  DR  ∂⊗f �→ f⊗∂ + ∂f  IRE  �  ⊗f⊗E �→  �  f⊗E  DE  ∂⊗E �→ 0  IRI  �  ⊗f⊗  �  �→  �  f⊗  �  − E⊗  �  f⊗  �  −  �  ⊗  �  f  DI  ∂⊗  �  �→ ε  ID  �  ⊗∂ �→ ε − E  ERE  E⊗f⊗E �→ (Ef)E  IE  �  ⊗E �→  �  1⊗E  EE  E⊗E �→ E  II  �  ⊗  �  �→  �  1⊗  �  − E⊗  �  1⊗  �  −  �  ⊗  �  1  Table 3: Confluent reduction system ΣIDO for integro-differential operators  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Note that multiples like f · ∂ are treated differently now, due to  the fact that we are working in the tensor ring K⟨M⟩ and we have the reduction  rule (K, 1 �→ ε), which splits f ∈ R according to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants K =  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let M be given as above in terms of the modules defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (26), (27),  (25), and (28) and let the tensor reduction system ΣIDO be defined by Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDO∈ K⟨M⟩, which can  be written as a K-linear combination of pure tensors of the form  f ⊗ ∂⊗j,  f ⊗  �  ⊗ g,  f ⊗ E ⊗ g ⊗ ∂⊗j,  or  f ⊗ E ⊗ h ⊗  �  ⊗ g  where j ∈ N0, f, g, h ∈  �  R, and each f and g may be absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover,  R⟨∂,  �  , E⟩ ∼= K⟨M⟩irr  as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Instead of proving this theorem, we will prove a more general one below,  which allows to include additional functionals into the construction of the ring  of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='3 follows from Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 by specializing Φ = {E}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  42  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4  Proof of normal forms for IDO with functionals  To treat more general problems than the initial value problems in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1, it  is useful to include additional functionals into the ring of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For instance,  dealing with boundary problems requires evaluations at more than one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  In general, we consider a set Φ of K-linear functionals R → K including E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We  consider the K-bimodule MΦ defined as free left K-module  MΦ := KΦ  (29)  generated by the elements of Φ, where we define right multiplication in terms  of Φ by  ��  ϕ∈Φ cϕϕ  �  d = �  ϕ∈Φ cϕdϕ for cϕ, d ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Note that K-linear  combinations of K-linear maps are not necessarily K-linear again, if K is not  commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Since ∂,  �  , and all elements of Φ are K-linear, we have, for  instance, that ϕfψg = (ϕf)ψg for all f, g ∈ R and ϕ, ψ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  This allows  to extend the last three reduction rules of Table 2 to cover all elements of Φ  instead of the evaluation E only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether, we consider the K-tensor ring on  the K-bimodule  M := MR ⊕ MD ⊕ MI ⊕ MΦ  (30)  and we have the defining reduction system given in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  K  1 �→ ε  ID  � ⊗∂ �→ ε − E  RR  f⊗g �→ fg  DRΦ  ∂⊗f⊗ϕ �→ ∂f⊗ϕ  DR  ∂⊗f �→ f⊗∂ + ∂f  IRΦ  �  ⊗f⊗ϕ �→  �  f⊗ϕ  DI  ∂⊗  �  �→ ε  ΦRΦ  ϕ⊗f⊗ψ �→ (ϕf)ψ  Table 4: Defining reduction system for integro-differential operators with func-  tionals  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants K =  C and let Φ be a set of K-linear functionals R → K including E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let the K-  bimodule M be defined as above in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (24), (25), (28), (29), and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We  call  R⟨∂,  �  , Φ⟩ := K⟨M⟩/IIDOΦ  the ring of integro-differential operators with functionals Φ, where IIDOΦ is the  two-sided ideal induced by the reduction system obtained from Table 4 using also  submodules of M defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  We use Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 above to determine unique normal forms of tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Again, the reduction system given by Table 4 is not confluent and we need  to construct a confluent reduction system, like ΣIDOΦ given in Table 5, by a  completion process similar to how Table 3 was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Observe that, whenever  Φ = {E}, the ring R⟨∂, � , Φ⟩ and the relation →ΣIDOΦ specialize to R⟨∂, � , E⟩  and →ΣIDO, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  43  K  1 �→ ε  EI  E⊗  �  �→ 0  RR  f⊗g �→ fg  IRD  �  ⊗f⊗∂ �→ f − E⊗f −  �  ⊗∂f  DR  ∂⊗f �→ f⊗∂ + ∂f  IRΦ  �  ⊗f⊗ϕ �→  �  f⊗ϕ  DΦ  ∂⊗ϕ �→ 0  IRI  �  ⊗f⊗  �  �→  �  f⊗  �  − E⊗  �  f⊗  �  −  �  ⊗  �  f  DI  ∂⊗  �  �→ ε  ID  �  ⊗∂ �→ ε − E  ΦRΦ  ϕ⊗f⊗ψ �→ (ϕf)ψ  IΦ  �  ⊗ϕ �→  �  1⊗ϕ  ΦΦ  ϕ⊗ψ �→ (ϕ1)ψ  II  �  ⊗  �  �→  �  1⊗  �  − E⊗  �  1⊗  �  −  �  ⊗  �  1  Table 5: Confluent reduction system ΣIDOΦ for integro-differential operators  with functionals  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let (R, ∂,  �  ) be an integro-differential ring with constants K =  C and let Φ be a set of K-linear functionals R → K including E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Let M and  R⟨∂,  �  , Φ⟩ be defined as in Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='4 above and let the tensor reduction  system ΣIDOΦ be defined by Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Then every t ∈ K⟨M⟩ has a unique normal form t↓ΣIDOΦ∈ K⟨M⟩, which can  be written as a K-linear combination of pure tensors of the form  f ⊗ ∂⊗j,  f ⊗  �  ⊗ g,  f ⊗ ϕ ⊗ h ⊗ ∂⊗j,  or  f ⊗ ϕ ⊗ h ⊗  �  ⊗ g  where j ∈ N0, f, g, h ∈  �  R, ϕ ∈ Φ, and each f, g, h may be absent such that  ϕ ⊗ h ⊗  �  does not specialize to E ⊗  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Moreover,  R⟨∂,  �  , Φ⟩ ∼= K⟨M⟩irr  as K-rings, where multiplication on K⟨M⟩irr is defined by s · t := (s ⊗ t)↓ΣIDOΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We use the alphabets X := {K, ˜R, D, I, E, ˜Φ} and Z := X ∪ {R, Φ}, which  turns (Mz)z∈Z into a decomposition with specialization for the module M,  where S(R) = {K, ˜R} and S(Φ) = {E, ˜Φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For defining a Noetherian monoid  partial order ≤ on ⟨Z⟩ consistent with specialization that is compatible with  ΣIDOΦ, it is sufficient to require the order to satisfy  DR > RD  and  I > E˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  For instance, we could first define a monoid total order on ⟨{R, D, I, Φ}⟩ ⊆  ⟨Z⟩ by counting occurrences of the letter I and breaking ties with any degree-  lexicographic order satisfying D > R and then generate from it a partial order  on ⟨Z⟩ that is consistent with specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  By our Mathematica package TenReS, we generate all ambiguities of ΣIDOΦ  and verify that they are resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  There are 54 ambiguities and indeed  all S-polynomials reduce to zero, see the accompanying Mathematica file at  http://gregensburger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='com/softw/tenres/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Here, we just give two short ex-  amples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The first one illustrates in its last step of computation that also iden-  44  tities in MR, like the Leibniz rule, need to be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  SP(IRD, DR) = (f − E ⊗ f −  �  ⊗ ∂f) ⊗ g −  �  ⊗ f ⊗ (g ⊗ ∂ + ∂g)  →rRR fg − E ⊗ fg −  �  ⊗ (∂f)g −  �  ⊗ fg ⊗ ∂ −  �  ⊗ f∂g  →rIRD −  �  ⊗ (∂f)g +  �  ⊗ ∂fg −  �  ⊗ f∂g  =  �  ⊗ (−(∂f)g + ∂fg − f∂g) = 0  The second one illustrates that ambiguities involving specialization also need to  be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  SP(ΦΦ, EI) = (ϕ1)E ⊗  �  − ϕ ⊗ 0 →rEI 0  Since all ambiguities are resolvable, by Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='1 every element in K⟨M⟩ has  a unique normal form and R⟨∂,  �  , Φ⟩ ∼= K⟨M⟩irr as K-rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  It remains to determine the explicit form of elements in K⟨M⟩irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In order to  do so, we determine the set of irreducible words ⟨X⟩irr in ⟨X⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Irreducible words  containing only the letters K, ˜R, E, ˜Φ have to avoid the subwords arising from  the reduction rules K, S(RR) = {KK, K˜R, ˜RK, ˜R˜R}, S(ΦΦ) = {EE, E˜Φ, ˜ΦE, ˜Φ˜Φ},  and S(ΦRΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Hence they are given by  ǫ, ˜R, E, ˜Φ, ˜RE, ˜R˜Φ, E˜R, ˜Φ˜R, ˜RE˜R, ˜R˜Φ˜R  Allowing also the letter D, we have to avoid the subwords coming from S(DR) =  {DK, D˜R} and S(DΦ) = {DE, D˜Φ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Therefore, we can only append words Dj  with j ∈ N0 to the irreducible words determined so far, in order to obtain all  elements of ⟨X⟩irr not containing the letter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Finally, we also consider the letter  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Since we have to avoid the subwords S(IΦ) = {IE, I˜Φ}, ID, and II, any letter  immediately following I has to be ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' In addition, we have to avoid the subwords  S(IRΦ) = {IKE, IK˜Φ, I˜RE, I˜R˜Φ}, S(IRD) = {IKD, I˜RD}, and S(IRI) = {IKI, I˜RI},  so the letter I cannot be followed by a subword of length greater than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Therefore, the letter I can appear at most once in an element of ⟨X⟩irr and, since  subwords EI and DI have to be avoided, it can only be immediately preceded by  the letters ˜R or ˜Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Altogether, the elements of ⟨X⟩irr are precisely of the form  ˜RUDj  or  ˜RV I˜R,  where j ∈ N0 and each of ˜R and U ∈ {E, ˜Φ, E˜R, ˜Φ˜R} and V ∈ {˜Φ, E˜R, ˜Φ˜R} may  be absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' As discussed in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='12, the induced evaluation of an  integro-differential ring often is multiplicative in concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Also when  considering several point evaluations of regular functions, the resulting func-  tionals are multiplicative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ϕfg = (ϕf)ϕg for all f, g ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We discuss how  reflecting this additional property of some functionals in the ring of operators  influences the normal forms of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' To this end, we consider a subset  Φm ⊆ {ϕ ∈ Φ | ϕ is multiplicative and ϕ1 = 1}  45  of Φ, which may or may not include the evaluation E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' For the corresponding  elements ϕ ∈ Φm in R⟨∂,  �  , Φ⟩, we impose  ϕ · f = (ϕf)ϕ  for all f ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  To model this identity by reduction rules, we consider the  submodule  MΦm := KΦm  of MΦ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Evidently, we have the decomposition  MΦ = ME ⊕ M˜Φm ⊕ M˜Φ  where the submodules M˜Φm and M˜Φ are generated by Φm \\ {E} and Φ \\ ({E} ∪  Φm), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' We include the reduction rule  (ΦmR, ϕ⊗f �→ (ϕf)ϕ)  into Tables 4 and 5, where ϕ in the formula defining this K-bimodule homomor-  phism on MΦmR is not a general element of MΦm but of Φm and the definition  needs to be extended by left K-linearity to all of MΦmR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5 generalizes to this situation with the additional restriction on  normal forms in K⟨M⟩ that h needs to be absent whenever ϕ ∈ Φm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  The  proof is analogous by adapting the alphabets and the order accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The  additional reduction rule gives rise to additional ambiguities, whose resolvability  is also checked in the Mathematica file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Determination of irreducible words also  needs to be adapted accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' The resulting theorem includes Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='5  for Φm = ∅ and it includes Theorem 27 from [14] for Φm = Φ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' when all  functionals are multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  References  [1] Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven,  Asymptotic differential algebra and model theory of transseries, Annals of  Mathematics Studies, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 195, Princeton University Press, Princeton, NJ,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [2] Vladimir V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Bavula, The algebra of integro-differential operators on a poly-  nomial algebra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (2) 83 (2011), 517–543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [3] George M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Bergman, The diamond lemma for ring theory, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  29 (1978), 178–218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [4] Bruno Buchberger, An algorithm for finding the bases elements of the  residue class ring modulo a zero dimensional polynomial ideal (German),  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' thesis, University of Innsbruck, 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  46  [5] Thomas Cluzeau, Jamal Hossein Poor, Alban Quadrat, Clemens G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Raab,  and Georg Regensburger, Symbolic computation for integro-differential-  time-delay operators with matrix coefficients, IFAC-PapersOnLine 51  (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 14, 153–158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [6] Earl A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Coddington and Norman Levinson, Theory of ordinary differential  equations, McGraw-Hill Book Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', New York-Toronto-London, 1955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [7] Paul M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Cohn, Further algebra and applications, Springer-Verlag London,  Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', London, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [8] Gerald A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Edgar, Transseries for beginners, Real Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Exchange 35 (2010),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 2, 253–309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [9] Xing Gao, Li Guo, and Markus Rosenkranz, On rings of differential Rota-  Baxter operators, Internat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 28 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 1, 1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [10] Li Guo, An introduction to Rota-Baxter algebra, Surveys of Modern Math-  ematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 4, International Press, Somerville, MA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Higher Education  Press, Beijing, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [11] Li Guo and William Keigher, On differential Rota-Baxter algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Pure  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra 212 (2008), 522–540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [12] Li Guo, Georg Regensburger, and Markus Rosenkranz, On integro-  differential algebras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra 218 (2014), 456–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [13] Jamal Hossein Poor, Tensor reduction systems for rings of linear operators,  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' thesis, Johannes Kepler University Linz, Austria, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [14] Jamal Hossein Poor, Clemens G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Raab, and Georg Regensburger, Algo-  rithmic operator algebras via normal forms in tensor rings, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Symbolic  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 85 (2018), 247–274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [15] Irving Kaplansky, An introduction to differential algebra, Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Nancago, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 5, Hermann, Paris, 1957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [16] William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Keigher, Adjunctions and comonads in differential algebra, Pa-  cific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 59 (1975), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 1, 99–112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [17]  , On the ring of Hurwitz series, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra 25 (1997), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 6,  1845–1859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [18] William F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Keigher and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Leon Pritchard, Hurwitz series as formal func-  tions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Algebra 146 (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 3, 291–304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [19] Donald E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Knuth and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Bendix, Simple word problems in universal  algebras, Computational Problems in Abstract Algebra, Pergamon, Oxford,  1970, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 263–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  47  [20] Ernst E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Kummer, Ueber die Transcendenten, welche aus wiederholten Inte-  grationen rationaler Formeln entstehen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 21 (1840),  74–90, 193–225, and 328–371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [21] Ivan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Lappo-Danilevski, Résolution algorithmique des problèmes réguliers  de Poincaré et de riemann (Mémoire premier: Le problème de Poincaré,  concernant de la construction d’un groupe de monodromie d’un système  donné d’équations différentielles linéaires aux intégrales régulières), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='-Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Léningrade 2 (1928), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 1, 94–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [22] Teo Mora, An introduction to commutative and noncommutative Gröbner  bases, Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 134 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 1, 131–173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [23] Danuta Przeworska-Rolewicz, Algebraic analysis, PWN—Polish Scientific  Publishers, Warsaw;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Reidel Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', Dordrecht, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [24]  , Two centuries of the term “algebraic analysis”, Algebraic analysis  and related topics (Warsaw, 1999), Banach Center Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 53, Polish  Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', Warsaw, 2000, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 47–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [25] Alban Quadrat, A constructive algebraic analysis approach to Artstein’s  reduction of linear time-delay systems, IFAC-PapersOnLine 48 (2015),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 12, 209–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [26] Alban Quadrat and Georg Regensburger, Computing polynomial solutions  and annihilators of integro-differential operators with polynomial coeffi-  cients, Algebraic and symbolic computation methods in dynamical systems,  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Delays Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 9, Springer, Cham, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 87–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [27] Rimhak Ree, Lie elements and an algebra associated with shuffles, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' of  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' (2) 68 (1958), 210–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [28] Georg Regensburger, Markus Rosenkranz, and Johannes Middeke, A skew  polynomial approach to integro-differential operators, Proceedings of ISSAC  ’09 (New York, NY, USA), ACM, 2009, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 287–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [29] Markus Rosenkranz, A new symbolic method for solving linear two-point  boundary value problems on the level of operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Symbolic Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 39  (2005), 171–199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [30] Markus Rosenkranz and Georg Regensburger, Solving and factoring bound-  ary problems for linear ordinary differential equations in differential alge-  bras, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Symbolic Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 43 (2008), 515–544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [31] Markus Rosenkranz, Georg Regensburger, Loredana Tec, and Bruno  Buchberger, Symbolic analysis for boundary problems: From rewriting to  parametrized Gröbner bases, Numerical and Symbolic Scientific Computing:  Progress and Prospects (Ulrich Langer and Peter Paule, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' ), Springer Vi-  enna, Vienna, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 273–331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  48  [32] Markus Rosenkranz and Nitin Serwa, An integro-differential structure for  Dirac distributions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Symbolic Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 92 (2019), 156–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [33] Gian-Carlo Rota, Twelve problems in probability no one likes to bring up,  Algebraic combinatorics and computer science, Springer Italia, Milan, 2001,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 57–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [34] Louis H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Rowen, Ring theory, student ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', Academic Press, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=', Boston,  MA, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [35] Richard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Stanley, Differentiably finite power series, European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  1 (1980), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 2, 175–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  [36] Joris van der Hoeven, Transseries and real differential algebra, Lecture  Notes in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content=' 1888, Springer-Verlag, Berlin, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
+page_content='  49' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QNFPT4oBgHgl3EQfojUh/content/2301.13134v1.pdf'}
diff --git a/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/2301.04940v1.pdf.txt b/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/2301.04940v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9def23414d6aad3e6fcf1fca3a3e8b49738764dd
--- /dev/null
+++ b/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/2301.04940v1.pdf.txt
@@ -0,0 +1,262 @@
+arXiv:2301.04940v1  [math.AG]  12 Jan 2023
+FAILURE OF LEFSCHETZ HYPERPLANE THEOREM
+ANANYO DAN
+Abstract. In this article, we give a counterexample to the Lefschetz hyperplane theorem
+for non-singular quasi-projective varieties. A classical result of Hamm-Lˆe shows that Lefschetz
+hyperplane theorem can hold for hyperplanes in general position. We observe that the condition
+of “hyperplane” is strict in the sense that it is not possible to replace it by higher degree
+hypersurfaces. The counterexample is very simple: projective space minus finitely many points.
+Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails
+in the quasi-projective case.
+1. Introduction
+The underlying field will always be C.
+Consider a non-singular, projective variety Y of
+dimension n. The Lefschetz hyperplane theorem (LHT) states that for any hypersurface X ⊂ Y
+with OX(Y ) very ample, the restriction morphism
+Hk(Y, Z) → Hk(X, Z) is an isomorphism for all k < n − 1 and injective for k = n − 1.
+(1.1)
+If Y is the projective space, then the theorem extends further. In particular, the restriction from
+Hn−1(Pn) to Hn−1(X) is an isomorphism for a very general hypersurface X. The geometry of
+the locus of hypersurfaces where this isomorphism fails (also known as the Noether-Lefschetz
+locus), has been extensively studied [1–4, 11, 12].
+It is therefore evident that the failure of
+the Lefschetz hyperplane theorem can give rise to important questions in Hodge theory and
+deformation theory. The goal of this article is to investigate the failure of this theorem in the
+quasi-projective case.
+It was observed by Hamm and Lˆe [6, 7] that if a hyperplane section X in a quasi-projective
+variety Y is in “general” position, then (1.1) holds true. The criterion for general position, is
+given explicitly in terms of a Whitney stratification of Y (see §2.2). This leads to the natural
+question:
+Question: Is the Hamm-Lˆe theorem (Theorem 2.1) true if we replace “hyperplane” by higher
+degree hypersurface?
+This is true in the case when Y is a projective, non-singular variety. Surprisingly, this can
+fail even if Y is the complement of a single point in a projective space. In particular, we give
+an example of a higher degree hypersurface which satisfies all the conditions in the Hamm-Lˆe
+theorem except for being a hyperplane. Yet, in this case LHT fails. We now discuss this in
+details. Recall, a projective variety X is called non-factorial if the rank of the divisor class
+group Div(X) (i.e., the free abelian group of divisors on X modulo linear equivalence) is not
+the same as the rank of the Picard group Pic(X). We prove:
+Date: January 13, 2023.
+2020 Mathematics Subject Classification. 14C30, 32S35, 32S50.
+Key words and phrases. Hodge theory, Lefschetz hyperplane theorem, quasi-projective varieties, factoriality,
+Grothendieck-Lefschetz theorem, Picard group.
+1
+
+2
+ANANYO DAN
+Theorem 1.1. Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with
+n ≥ 4. Denote by Xsing the singular locus of X. Then, the natural restriction morphism
+H2(Pn\Xsing, Z) → H2(X\Xsing, Z)
+is not surjective.
+Using this theorem we now give an explicit example.
+Example 1.2. Let X ⊂ P4 be a hypersurface defined by the equation X2
+0 + X2
+1 + X2
+2 + X2
+3,
+where X0, ..., X4 are the coordinates on P4. Clearly, X has exactly one singular point x = [0 :
+0 : 0 : 0 : 1]. The divisor class group Div(X) is isomorphic to Z ⊕ Z (see [8, Ex. II.6.5]). By
+Lefschetz hyperplane theorem, we have H2(X, Z) ∼= Z. Using the exponential exact sequence,
+one can check that Pic(X) ∼= Z. Hence, X is non-factorial. Theorem 1.1 then implies that the
+restriction morphism from H2(P4\{x}, Z) to H2(X\{x}, Z) is not surjective.
+As an intermediate step we show that the Grothendieck-Lefschetz theorem [5] fails in the
+quasi-projective case (see Remark 3.2).
+Acknowledgement: I thank Dr. I. Kaur for discussions. The author was funded by EPSRC
+grant number EP/T019379/1.
+2. On the Hamm-Lˆe result
+In [7], Hamm and Lˆe proved a version of the Lefschetz hyperplane theorem for quasi-projective
+varieties (see Theorem 2.1 below). The proof follows in two stages. We use notations as in §2.1
+below. The first step is to check that for all i ≤ dim(Y ) − 2, Hi(Y \Z) (resp. Hm−1(Y \Z)) is
+isomorphic to (resp. contained in) the i-th (resp. (m − 1)-th) cohomology of Vr(L) ∩ (Y \Z),
+for some neighbourhood Vr(L) of L of “radius” r, for almost all r > 0 (see [7, Theorem 1.1.1]).
+The second step is to check whether L ∩ (Y \Z) is a deformation retract of Vr(L) ∩ (Y \Z). One
+observes that this holds true if L is in a “general” position. An explicit description of the general
+position will be mentioned in Theorem 2.1 below.
+2.1. Setup. Let Y be a projective subvariety of dimension m in Pn, Z ⊂ Y be an algebraic
+subspace and L ⊂ Pn a hyperplane in Pn such that Y \(Z ∪ L) is non-singular. Consider a
+stratification {Yi}i∈I of Y satisfying the following conditions:
+(1) each Yi is a real semi-algebraic subset of Y ,
+(2) {Yi} is a Whitney stratification,
+(3) Z is a union of some of the strata,
+(4) the stratification satisfies the Thom condition for the following function:
+τ : Y → R, sending y ∈ Y to
+k�
+i=1
+|fi(y)|2d/di
+n�
+i=0
+|yi|2d
+, where y = (y1, ..., yn),
+Z is defined by the homogeneous polynomials f1, ..., fk of degrees di, respectively and d
+is the l.c.m. of the di’s. See [10, §1.4.4] for the precise definition.
+2.2. On the Hamm-Lˆe result. Let Ω be the set of complex projective hyperplanes of Pn
+transverse to all the strata Yi.
+
+LEFSCHETZ THEOREM
+3
+Theorem 2.1. (Hamm-Lˆe [7, Theorem 1.1.3]) Assume that Y \Z is non-singular. Then, for any
+L ∈ Ω we have
+Hk(Y \Z, L ∩ (Y \Z)) = 0 for all k ≤ m − 1.
+In other words, the natural morphism from Hk(Y \Z, Z) to Hk(L∩(Y \Z), Z) is an isomorphism
+for all k ≤ m − 2 and injective for k = m − 1.
+We now write the stratification relevant to Example 1.2.
+Remark 2.2. Take Y = P4 ⊂ P5 defined by z5 = 0, where zi are the coordinates on P5. Take
+Z := [0, 0, 0, 0, 1, 0] the closed point in Y . Take the stratification of Y consisting of
+(Y \Z)
+�
+Z.
+Then, the equations defining Z in P5 are given by fi := zi for 0 ≤ i ≤ 3 and f5 := z5. The
+function τ is simply
+τ :=
+|z5|2 +
+3�
+i=0
+|zi|2
+5�
+i=0
+|zi|2
+.
+Note that this stratification satisfies conditions (1)-(4) in §2.1 above, with the stratification on R
+given by R\{0} �{0}. Finally, note that the hypersurface X in P5 defined by z2
+0 +z2
+1 +z2
+2 +z2
+3+z2
+5
+is singular at the point Z. As a result X is transverse to all the strata of Y . We will observe in
+Theorem 1.1 that if we replace L in Theorem 2.1 above by X, then the conclusion fails.
+3. Proof of Main theorem
+We will assume that the reader has basic familiarity with local cohomology. See [9] for basic
+definitions and results in this topic.
+Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with n ≥ 4. Denote by
+Xsing the singular locus of X, Y := Pn\Xsing and Xsm := X\Xsing. We first show:
+Proposition 3.1. The cohomology groups H1(OY ), H2(OY ) and H1(OXsm) all vanish, in both
+analytic as well as Zariski topology.
+Proof. Recall, the long exact sequence for local cohomology groups, which exists in both topolo-
+gies (see [9, Corollary 1.9]):
+... → H1(OPn) → H1(OY ) → H2
+Xsing(OPn) → H2(OPn) → H2(OY ) → H3
+Xsing(OPn) → ...
+Recall, H1(OPn) = 0 = H2(OPn). By Serre’s GAGA, H1(O
+an
+Pn) = 0 = H2(O
+an
+Pn). To prove the
+vanishing of H1(OY ) and H2(OY ), we simply need to prove the vanishing of Hi
+Xsing(OPn) for
+i = 2, 3 in both topologies.
+Consider the spectral sequence (see [9, Proposition 1.4]):
+Ep,q
+2
+= Hp(Pn, Hq
+Xsing(OPn)) ⇒ Hp+q
+Xsing(OPn).
+(3.1)
+We are interested in the cases when p + q equals 2 or 3. Since n ≥ 4 and Xsing are closed points,
+we have (see [13, Proposition 1.2])
+Hq
+Xsing(OPn) = 0 for q ≤ 3.
+This implies that Ep,q
+2
+= 0 for p + q equals 2 or 3. Hence the spectral sequence degenerates at
+E2 in this case and Hi
+Xsing(OPn) = 0 in both topologies. This proves the vanishing of H1(OY )
+and H2(OY ).
+
+4
+ANANYO DAN
+The proof for the vanishing of H1(OXsm) follows similarly. In particular, using [9, Corollary
+1.9], it suffices to check the vanishing of H1(OX) and H2
+Xsing(OX). Since X is a hypersurface
+in Pn and n ≥ 4, H1(OX) = 0.
+By Serre’s GAGA, H1(O
+an
+X ) = 0. To prove the vanishing
+of H2
+sing(OX) use the spectral sequence (3.1) above after replacing Pn by X and p + q = 2.
+Since dim X ≥ 3, [13, Proposition 1.2] implies that Hq
+Xsing(OX) = 0 for q ≤ 2. This implies
+that the spectral sequence degenerates at E2 and H2
+Xsing(OX) = 0 in both topologies. Hence,
+H1(OXsm) = 0 in both topologies. This proves the proposition.
+□
+Proof of the main theorem. We prove the theorem by contradiction. Suppose that the restric-
+tion morphism from H2(Y, Z) to H2(X, Z) is surjective. Comparing the long exact sequences
+associated to the exponential exact sequence for Y and Xsm we get the following diagram where
+the horizontal rows are exact:
+H1(OY )
+✲ H1(O∗
+Y )
+∂1✲ H2(Y, Z)
+✲ H2(OY )
+⟲
+⟲
+⟲
+H1(OXsm)
+❄
+✲ H1(O∗
+Xsm)
+ρ′
+❄
+∂2✲ H2(Xsm, Z)
+ρ
+❄
+✲ H2(OXsm)
+❄
+(3.2)
+Using the vanishing results from Proposition 3.1, we conclude that ∂1 is an isomorphism and ∂2
+is injective. By assumption, ρ is surjective. We claim that ρ′ is surjective. Indeed, given α ∈
+H1(O∗
+Xsm), the surjectivity of ρ implies that there exists β ∈ H2(Y, Z) such that ρ(β) = ∂2(α).
+Since ∂1 is an isomorphism, there exist α′ ∈ H1(O∗
+Y ) mapping to β via ∂1. Using the injectivity
+of ∂2 and the commutativity of the middle square, we have ρ′(α′) = α. This proves the claim.
+Since ρ′ is surjective, we have the following surjective morphism:
+Z = Pic(Pn) ∼= Pic(Y )
+ρ′
+։ Pic(Xsm) ∼= Div(X)
+(3.3)
+where the second and the last isomorphisms follow from the fact that Xsing is of codimensional
+at least 2 in X and Pn. By Lefschetz hyperplane theorem, we have H2(X, Z) ∼= H2(Pn, Z) = Z,
+generated by the class of the hyperplane section. Note that, H1(OX) and H2(OX) vanish (use [8,
+Ex. III.5.5] and n ≥ 4). Using the exponential short exact sequence for X, we conclude that
+Pic(X) ∼= Z. Combining with (3.3), this implies rk Div(X) = rk Pic(X). But this contradicts
+the fact that X is non-factorial. Hence, the restriction morphism from H2(Y, Z) to H2(X, Z)
+cannot be surjective. This proves the theorem.
+□
+Remark 3.2. Let X be as in Theorem 1.1. Then, the restriction morphism
+Pic(Pn\Xsing) → Pic(X\Xsing)
+is not surjective. Indeed,
+Pic(Pn\Xsing) ∼= Pic(Pn) ∼= Z and Pic(X\Xsing) ∼= Div(X).
+By Lefschetz hyperplane theorem for projective hypersurfaces, we have Pic(X) ∼= Z. Since X is
+non-factorial, the rank of Div(X) is not the same as that of Pic(X). Therefore, Pic(Pn\Xsing)
+cannot be isomorphic to Pic(X\Xsing).
+References
+[1] C. Ciliberto, J. Harris, and R. Miranda. General components of the Noether-Lefschetz locus and their density
+in the space of all surfaces. Mathematische Annalen, 282(4):667–680, 1988.
+[2] A. Dan. On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci. Communications
+in Contemporary Mathematics, 17(5):1550002, 2015.
+[3] A. Dan. On a conjecture of Harris. Communications in Contemporary Mathematics, 23(07):2050028, 2021.
+
+LEFSCHETZ THEOREM
+5
+[4] M. Green. A new proof of the explicit Noether-Lefschetz theorem. J. Differential Geometry, 27:155–159, 1988.
+[5] A. Grothendieck. SGA 2. S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie-1962-Cohomologie locale des
+faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (North-Holland, Amsterdam), 1968.
+[6] H. Hamm. Lefschetz theorems for singular varieties. In Proceedings of symposia in pure mathematics, vol-
+ume 40, pages 547–557. AMS, 1983.
+[7] H. Hamm and D. T. Lˆe. Lefschetz theorems on quasi-projective varieties. Bulletin de la Soci´et´e math´ematique
+de France, 113:123–142, 1985.
+[8] R. Hartshorne. Algebraic Geometry. Graduate text in Mathematics-52. Springer-Verlag, 1977.
+[9] R. Hartshorne. Local Cohomology: A Seminar Given by A. Groethendieck, Harvard University. Fall, 1961,
+volume 41. Springer, 2006.
+[10] D. T. Lˆe and B. Teissier. Cycles ´evanescents, sections planes et conditions de whitney ii, singularities, part
+2 (arcata, calif., 1981), 65-103. In Proc. Sympos. Pure Math, volume 40.
+[11] C. Voisin. Une pr´ecision concernant le th´eor`eme de Noether. Math. Ann., 280(4):605–611, 1988.
+[12] C. Voisin. Sur le lieu de Noether-Lefschetz en degr´es 6 et 7. Compositio Mathematica, 75(1):47–68, 1990.
+[13] Y. Yoshino. Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings, volume 146. Cambridge Uni-
+versity Press, 1990.
+School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road,
+S3 7RH, UK
+Email address: a.dan@sheffield.ac.uk
+
diff --git a/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/load_file.txt b/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1d1f35d9769cbdc6d2c7a670c5103d3a527ca2b1
--- /dev/null
+++ b/TdE4T4oBgHgl3EQfLwwK/content/tmp_files/load_file.txt
@@ -0,0 +1,267 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf,len=266
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='04940v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='AG]  12 Jan 2023  FAILURE OF LEFSCHETZ HYPERPLANE THEOREM  ANANYO DAN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In this article, we give a counterexample to the Lefschetz hyperplane theorem  for non-singular quasi-projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' A classical result of Hamm-Lˆe shows that Lefschetz  hyperplane theorem can hold for hyperplanes in general position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We observe that the condition  of “hyperplane” is strict in the sense that it is not possible to replace it by higher degree  hypersurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The counterexample is very simple: projective space minus finitely many points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Moreover, as an intermediate step we prove that the Grothendieck-Lefschetz theorem also fails  in the quasi-projective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Introduction  The underlying field will always be C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Consider a non-singular, projective variety Y of  dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The Lefschetz hyperplane theorem (LHT) states that for any hypersurface X ⊂ Y  with OX(Y ) very ample, the restriction morphism  Hk(Y, Z) → Hk(X, Z) is an isomorphism for all k < n − 1 and injective for k = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1)  If Y is the projective space, then the theorem extends further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, the restriction from  Hn−1(Pn) to Hn−1(X) is an isomorphism for a very general hypersurface X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The geometry of  the locus of hypersurfaces where this isomorphism fails (also known as the Noether-Lefschetz  locus), has been extensively studied [1–4, 11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  It is therefore evident that the failure of  the Lefschetz hyperplane theorem can give rise to important questions in Hodge theory and  deformation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The goal of this article is to investigate the failure of this theorem in the  quasi-projective case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  It was observed by Hamm and Lˆe [6, 7] that if a hyperplane section X in a quasi-projective  variety Y is in “general” position, then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The criterion for general position, is  given explicitly in terms of a Whitney stratification of Y (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This leads to the natural  question:  Question: Is the Hamm-Lˆe theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) true if we replace “hyperplane” by higher  degree hypersurface?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  This is true in the case when Y is a projective, non-singular variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Surprisingly, this can  fail even if Y is the complement of a single point in a projective space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, we give  an example of a higher degree hypersurface which satisfies all the conditions in the Hamm-Lˆe  theorem except for being a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Yet, in this case LHT fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We now discuss this in  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Recall, a projective variety X is called non-factorial if the rank of the divisor class  group Div(X) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', the free abelian group of divisors on X modulo linear equivalence) is not  the same as the rank of the Picard group Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We prove:  Date: January 13, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' 14C30, 32S35, 32S50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hodge theory, Lefschetz hyperplane theorem, quasi-projective varieties, factoriality,  Grothendieck-Lefschetz theorem, Picard group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  1  2  ANANYO DAN  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with  n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Denote by Xsing the singular locus of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, the natural restriction morphism  H2(Pn\\Xsing, Z) → H2(X\\Xsing, Z)  is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Using this theorem we now give an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X ⊂ P4 be a hypersurface defined by the equation X2  0 + X2  1 + X2  2 + X2  3,  where X0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', X4 are the coordinates on P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Clearly, X has exactly one singular point x = [0 :  0 : 0 : 0 : 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The divisor class group Div(X) is isomorphic to Z ⊕ Z (see [8, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By  Lefschetz hyperplane theorem, we have H2(X, Z) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the exponential exact sequence,  one can check that Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence, X is non-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 then implies that the  restriction morphism from H2(P4\\{x}, Z) to H2(X\\{x}, Z) is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  As an intermediate step we show that the Grothendieck-Lefschetz theorem [5] fails in the  quasi-projective case (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Acknowledgement: I thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Kaur for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The author was funded by EPSRC  grant number EP/T019379/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On the Hamm-Lˆe result  In [7], Hamm and Lˆe proved a version of the Lefschetz hyperplane theorem for quasi-projective  varieties (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The proof follows in two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We use notations as in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The first step is to check that for all i ≤ dim(Y ) − 2, Hi(Y \\Z) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hm−1(Y \\Z)) is  isomorphic to (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' contained in) the i-th (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' (m − 1)-th) cohomology of Vr(L) ∩ (Y \\Z),  for some neighbourhood Vr(L) of L of “radius” r, for almost all r > 0 (see [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  The second step is to check whether L ∩ (Y \\Z) is a deformation retract of Vr(L) ∩ (Y \\Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' One  observes that this holds true if L is in a “general” position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' An explicit description of the general  position will be mentioned in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let Y be a projective subvariety of dimension m in Pn, Z ⊂ Y be an algebraic  subspace and L ⊂ Pn a hyperplane in Pn such that Y \\(Z ∪ L) is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Consider a  stratification {Yi}i∈I of Y satisfying the following conditions:  (1) each Yi is a real semi-algebraic subset of Y ,  (2) {Yi} is a Whitney stratification,  (3) Z is a union of some of the strata,  (4) the stratification satisfies the Thom condition for the following function:  τ : Y → R, sending y ∈ Y to  k�  i=1  |fi(y)|2d/di  n�  i=0  |yi|2d  , where y = (y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', yn),  Z is defined by the homogeneous polynomials f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', fk of degrees di, respectively and d  is the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' of the di’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' See [10, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4] for the precise definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On the Hamm-Lˆe result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let Ω be the set of complex projective hyperplanes of Pn  transverse to all the strata Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  LEFSCHETZ THEOREM  3  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' (Hamm-Lˆe [7, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3]) Assume that Y \\Z is non-singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, for any  L ∈ Ω we have  Hk(Y \\Z, L ∩ (Y \\Z)) = 0 for all k ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  In other words, the natural morphism from Hk(Y \\Z, Z) to Hk(L∩(Y \\Z), Z) is an isomorphism  for all k ≤ m − 2 and injective for k = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  We now write the stratification relevant to Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take Y = P4 ⊂ P5 defined by z5 = 0, where zi are the coordinates on P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take  Z := [0, 0, 0, 0, 1, 0] the closed point in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Take the stratification of Y consisting of  (Y \\Z)  �  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Then, the equations defining Z in P5 are given by fi := zi for 0 ≤ i ≤ 3 and f5 := z5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The  function τ is simply  τ :=  |z5|2 +  3�  i=0  |zi|2  5�  i=0  |zi|2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Note that this stratification satisfies conditions (1)-(4) in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 above, with the stratification on R  given by R\\{0} �{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Finally, note that the hypersurface X in P5 defined by z2  0 +z2  1 +z2  2 +z2  3+z2  5  is singular at the point Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' As a result X is transverse to all the strata of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We will observe in  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 that if we replace L in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1 above by X, then the conclusion fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Proof of Main theorem  We will assume that the reader has basic familiarity with local cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' See [9] for basic  definitions and results in this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Let X ⊂ Pn be a non-factorial hypersurface with isolated singularities with n ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Denote by  Xsing the singular locus of X, Y := Pn\\Xsing and Xsm := X\\Xsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We first show:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' The cohomology groups H1(OY ), H2(OY ) and H1(OXsm) all vanish, in both  analytic as well as Zariski topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Recall, the long exact sequence for local cohomology groups, which exists in both topolo-  gies (see [9, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='9]):  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' → H1(OPn) → H1(OY ) → H2  Xsing(OPn) → H2(OPn) → H2(OY ) → H3  Xsing(OPn) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Recall, H1(OPn) = 0 = H2(OPn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By Serre’s GAGA, H1(O  an  Pn) = 0 = H2(O  an  Pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' To prove the  vanishing of H1(OY ) and H2(OY ), we simply need to prove the vanishing of Hi  Xsing(OPn) for  i = 2, 3 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Consider the spectral sequence (see [9, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='4]):  Ep,q  2  = Hp(Pn, Hq  Xsing(OPn)) ⇒ Hp+q  Xsing(OPn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1)  We are interested in the cases when p + q equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since n ≥ 4 and Xsing are closed points,  we have (see [13, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2])  Hq  Xsing(OPn) = 0 for q ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  This implies that Ep,q  2  = 0 for p + q equals 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence the spectral sequence degenerates at  E2 in this case and Hi  Xsing(OPn) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the vanishing of H1(OY )  and H2(OY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  4  ANANYO DAN  The proof for the vanishing of H1(OXsm) follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In particular, using [9, Corollary  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='9], it suffices to check the vanishing of H1(OX) and H2  Xsing(OX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since X is a hypersurface  in Pn and n ≥ 4, H1(OX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  By Serre’s GAGA, H1(O  an  X ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' To prove the vanishing  of H2  sing(OX) use the spectral sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1) above after replacing Pn by X and p + q = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since dim X ≥ 3, [13, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2] implies that Hq  Xsing(OX) = 0 for q ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This implies  that the spectral sequence degenerates at E2 and H2  Xsing(OX) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence,  H1(OXsm) = 0 in both topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  □  Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We prove the theorem by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Suppose that the restric-  tion morphism from H2(Y, Z) to H2(X, Z) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Comparing the long exact sequences  associated to the exponential exact sequence for Y and Xsm we get the following diagram where  the horizontal rows are exact:  H1(OY )  ✲ H1(O∗  Y )  ∂1✲ H2(Y, Z)  ✲ H2(OY )  ⟲  ⟲  ⟲  H1(OXsm)  ❄  ✲ H1(O∗  Xsm)  ρ′  ❄  ∂2✲ H2(Xsm, Z)  ρ  ❄  ✲ H2(OXsm)  ❄  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2)  Using the vanishing results from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1, we conclude that ∂1 is an isomorphism and ∂2  is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By assumption, ρ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' We claim that ρ′ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Indeed, given α ∈  H1(O∗  Xsm), the surjectivity of ρ implies that there exists β ∈ H2(Y, Z) such that ρ(β) = ∂2(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since ∂1 is an isomorphism, there exist α′ ∈ H1(O∗  Y ) mapping to β via ∂1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the injectivity  of ∂2 and the commutativity of the middle square, we have ρ′(α′) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  Since ρ′ is surjective, we have the following surjective morphism:  Z = Pic(Pn) ∼= Pic(Y )  ρ′  ։ Pic(Xsm) ∼= Div(X)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3)  where the second and the last isomorphisms follow from the fact that Xsing is of codimensional  at least 2 in X and Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' By Lefschetz hyperplane theorem, we have H2(X, Z) ∼= H2(Pn, Z) = Z,  generated by the class of the hyperplane section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Note that, H1(OX) and H2(OX) vanish (use [8,  Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='5] and n ≥ 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Using the exponential short exact sequence for X, we conclude that  Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Combining with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='3), this implies rk Div(X) = rk Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' But this contradicts  the fact that X is non-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hence, the restriction morphism from H2(Y, Z) to H2(X, Z)  cannot be surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' This proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Let X be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Then, the restriction morphism  Pic(Pn\\Xsing) → Pic(X\\Xsing)  is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Indeed,  Pic(Pn\\Xsing) ∼= Pic(Pn) ∼= Z and Pic(X\\Xsing) ∼= Div(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  By Lefschetz hyperplane theorem for projective hypersurfaces, we have Pic(X) ∼= Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Since X is  non-factorial, the rank of Div(X) is not the same as that of Pic(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Therefore, Pic(Pn\\Xsing)  cannot be isomorphic to Pic(X\\Xsing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Ciliberto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Harris, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Miranda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' General components of the Noether-Lefschetz locus and their density  in the space of all surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Mathematische Annalen, 282(4):667–680, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On a conjecture by Griffiths and Harris concerning certain Noether–Lefschetz loci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Communications  in Contemporary Mathematics, 17(5):1550002, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' On a conjecture of Harris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Communications in Contemporary Mathematics, 23(07):2050028, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  LEFSCHETZ THEOREM  5  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' A new proof of the explicit Noether-Lefschetz theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Differential Geometry, 27:155–159, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Grothendieck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' SGA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' S´eminaire de G´eom´etrie Alg´ebrique du Bois Marie-1962-Cohomologie locale des  faisceaux coh´erents et th´eoremes de Lefschetz locaux et globaux (North-Holland, Amsterdam), 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [6] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hamm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lefschetz theorems for singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In Proceedings of symposia in pure mathematics, vol-  ume 40, pages 547–557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' AMS, 1983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hamm and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lˆe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lefschetz theorems on quasi-projective varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Bulletin de la Soci´et´e math´ematique  de France, 113:123–142, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Algebraic Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Graduate text in Mathematics-52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Springer-Verlag, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [9] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Hartshorne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Local Cohomology: A Seminar Given by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Groethendieck, Harvard University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Fall, 1961,  volume 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Springer, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [10] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Lˆe and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Teissier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Cycles ´evanescents, sections planes et conditions de whitney ii, singularities, part  2 (arcata, calif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', 1981), 65-103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Pure Math, volume 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Une pr´ecision concernant le th´eor`eme de Noether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=', 280(4):605–611, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Voisin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Sur le lieu de Noether-Lefschetz en degr´es 6 et 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Compositio Mathematica, 75(1):47–68, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Yoshino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Maximal Cohen-Macaulay Modules Over Cohen-Macaulay Rings, volume 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content=' Cambridge Uni-  versity Press, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='  School of Mathematics and Statistics, University of Sheffield, Hicks building, Hounsfield Road,  S3 7RH, UK  Email address: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='dan@sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
+page_content='uk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE4T4oBgHgl3EQfLwwK/content/2301.04940v1.pdf'}
diff --git a/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/2301.02498v1.pdf.txt b/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/2301.02498v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..025530b669aea57fbd8ac7ac86949d32b967bca9
--- /dev/null
+++ b/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/2301.02498v1.pdf.txt
@@ -0,0 +1,3030 @@
+Dynamics and stability of the two-body problem with Yukawa
+correction to Newton’s gravity, revisited and applied
+numerically to the solar system
+Nawras Abo Hasan1∗, Nabil Joudieh1†and Nidal Chamoun2‡
+1 Physics Department, Damascus University, Damascus, Syria
+2 Physics Department, HIAST, P.O. Box 31983, Damascus, Syria
+Abstract
+In this manuscript, we review the motion of two-body celestial system (planet-sun) for
+a Yukawa-type correction on Newton’s gravitational potential using Hamilton’s formulation.
+We reexamine the stability using the corresponding linearization Jacobian matrix, and verify
+that the Bertrand’s theorem conditions are met for radii ≪ 1015m, and so bound closed orbits
+are expected. Applied to the solar system, we present the equation of motion of the planet,
+then solve it both analytically and numerically. Making use of the analytical expression of
+the orbit, we estimate the Yukawa strength α, and find it larger than the nominal value
+(10−8) adopted in previous studies, in that it is of order (α = 10−4 − 10−5) for terrestrial
+planets (Mercury, Venus, earth, Mars and Pluto) whereas it is even larger (α = 10−3) for
+the Giant planets (Jupiter, Saturn, Uranus and Neptune). Taking as inputs (rmin, vmas, e)
+observed by NASA, we analyse the orbits analytically and numerically for both the estimated
+and nominal values of α, and determine the corresponding trajectories. For each obtained
+orbit we recalculate the characterizing parameters (rmin, rmax, a, b, e) and compare their
+values according to the used potential (Newton with/without Yukawa correction) and to the
+method used (analytical and/or numerical). When compared to the observational data, we
+conclude that the correction on the path due to Yukawa correction is of order of and up to
+80 million km (20 million km) as a maximum deviation occurring for Neptune (Pluto) for
+nominal (estimated) value of α.
+Keywords: gravitational two-body problem, Yukawa potential, closed orbit
+’
+0
+Introduction
+The past several years have witnessed a resurgence of interest in experimental testing of gravity,
+particularly in the possibility of deviations from the predictions of Newtonian gravity, which is
+considered as an excellent approximation of General Relativity (GR) on large distance scale [1].
+Many theoretical models suggest the existence of new, relatively weak, intermediate-range force
+coexisting with gravity such that the net resulting interaction would behave like a new correction
+∗seagull18990@gmail.com
+†njoudieh@yahoo.fr
+‡nidal.chamoun@hiast.edu.sy
+1
+arXiv:2301.02498v1  [astro-ph.EP]  6 Jan 2023
+
+to the potentials defining the gravitational field. It is known [2] that there are only two types
+of central potentials, namely the Newton 1r and the Harmonic r2 potentials, where ANY finite
+motion of an object, subject to this central potential, leads to a closed path (Bertrands theorem).
+There are some ‘exceptions’ to this statement, in the sense that there might be closed bound
+trajectories for a central potential different from the Newton and Harmonic ones, which have
+been studied in [3, 4]. In this contribution, we revisit the effect of a Yukawa correction to the
+gravitational force over large distances.
+Theories of massive gravity [5,6], adding a mass term to the graviton (the carrier of gravity),
+have raised a wide interest and the Yukawa potential is the popular parametrization of such
+theories. Actually, many works describing deviations from Newtons inverse square law have
+addressed the Yukawa-type correction. Assuming gravity is exerted by exchanges of gravitons,
+it is clear that a test for the graviton mass (µg) is to ask whether the Newton (1/r) potential
+shows any evidence of dying at large distances because of Yukawa exponential cutoff (e−µgr).
+Since the seventies [7], bounds on the gravitons mass (µg ≤ 1.1 × 10−29 eV) were used to
+put a bound on its compton wavelength considered as a distance scale for Yukawa correction
+(λ ∼ 2π
+µg ≥ 3.7 MPc). The authors of [8] gave a bound on the Yukawa range (λ) in the order of
+(101 − 104 AU) corresponding to (µg ≤ 10−24) eV.
+Theories like Scalar-Tensor-Vector Gravity Theory [9] predict a Yukawa-like fifth force. The
+authors of [10], showed that screened modified gravity can suppress the fifth force in dense regions
+and allow theories to evade the solar system and laboratory tests of the weak equivalence princi-
+ple. In [11], an extended theory of gravity, with a modified potential including post-Newtonian
+terms, whose expansion is different from that of Yukawa correction, called ‘vacuum bootstrapped
+Newtonian gravity’, was subjected to solar system tests, through a procedure which was applied
+to Yukawa corrections at the Galactic center [12], with no significant deviations from GR found.
+In [13], a Keplerian-type parametrization was shown as a solution of the equations of motion
+for a Yukawa-type potential between two bodies. In fact, the two-body solution for alternative
+theories yield a strong constraint for solar system [14, 15], whereas several analyses of Yukawa
+potential for a 2-body system in different contexts were carried out [16, 17]. The orbit of a
+single particle moving under Yukawa potential was studied in [18], and the precessing ellipse
+type orbits were observed. In [19], it was noted that the modified gravity with Yukawa-like
+long-range potential was (un)successful on astrophysical scales (in solar system), whereas an
+analysis of Yukawa potential in f(R) gravity was given in [20]. The work of [21] showed that
+a Yukawa fifth force is expected to be sub-dominant in satellite dynamics and space geodesy
+experiments, as long as they are performed at altitudes greater than a few hundred kilometres.
+The Yukawa strength was estimated in [22] to be (α < 10−5-10−8) for distances of order 109
+cm, whereas the use of laser data from LAGEOS satellites yield a constraint on α of the order
+of 10−12.
+In this letter, we build on work from [23], in which the dynamics and stability of the two body
+problem with a Newtonian potential corrected by a Yukawa term were explored. In particular,
+we reproduced their analytical results and applied them to the study of all the planets of our solar
+system. Solving analytically the planet equation of motion, one finds an elliptical trajectory,
+which one can also obtain numerically using Runge-Kutta method. Starting from the observed
+values of the perihelion distance and velocity (rmin, vmax) and of the tranjectory eccentricity e,
+stated in NASA public results [24]∗, one could determine the ellipsis equation and estimate, for
+∗Although the standard deviations of the planetary trajectories are not quoted in the NASA public website,
+however one can consider that the corresponding error equals to the last digit of the quoted significant numbers.
+2
+
+Yukawa corrected potential, the Yukawa strength α. One can use this estimated value, or another
+nominal value taken from other studies, to either draw the analytical trajectory and recalculate
+the characterizing parameters: the shortest (longest) distance to the Sun rmin(rmax), the semi
+major (minor) axis a(b) and the eccentricity e, or to solve numerically the equations of motion
+with the Yukawa-corrected potential in order to check the closedness of the resulting trajectory,
+whose characteristics are to be reevaluated again. Later, we compared these results with those
+calculated for the Keplerian motion of planets subject to the pure Newtonian potential, and, in
+addition, showed the compatibility of the results with the observational NASA data.
+More specifically, for the two-body system (planet-sun), the Newtonian potential is given by:
+VN(r) = −GmpM⊙
+r
+(1)
+where G = 6.674×10−11 Nm2
+Kg2 is the gravitational Newton constant, mp (M⊙) is the planet (sun)
+mass. With a Yukawa correction, the gravitational potential becomes
+V (r) = −GmpM⊙
+r
+�
+1 + αe− r
+λ
+�
+= VN(r) + VY k(r)
+(2)
+where VY k is the Yukawa correction to the Newtonian potential and α (λ) represents the
+strength (range) of the Yukawa correction. Previous studies [23, 25] gave the nominal values
+(α = 10−8(λ = 103AU = 1015m). However, our estimations gave a larger order of magnitude for
+the Yukawa strength: α ∼ 10−4 − 10−5 for terrestrial planets (Mercury, Venus, Earth, Mars and
+Pluto) and α ∼ 10−3 for the remaining Giant planets (Jupiter, Saturn, Uranus and Neptune),
+which are in line with [13].
+We saw that for estimated α, the maximum deviation from observed data, which increases
+the further the planet is (20 million km in Pluto), is less than that of the α nominal value
+(80 million km in Neptune), which is plausible considering that the estimation of α is done by
+identifying the factor containing it to observational data.
+For each of the nominal and estimated values of α, we analysed the planet’s trajectory
+both analytically and numerically. Analytics wise, we started from the observational data of
+NASA (rmin, vmax, e) and reconstructed the closed ellipse trajectory of which we re-evaluated
+the characteristics (rmin, rmax, a, b, e) and compared with the pure Newton case and with the
+observational data. Numerics wise, the α determines the potential under which the planet moves,
+and so one can solve the equations of motion numerically using Runge-Kutta method taking as
+initial conditions the observed data of (rmin, vmax), to check that one gets closed trajectories in
+excellent agreement with the elliptical shapes, of which we can evaluate the characteristics that
+one compares to the pure Newtonian case, to the analytical method results and to the observed
+data.
+The manuscript is organized as follows.
+In section (1), we revise the system dynamics
+using Hamilton’s method. In section(2), we state the types of stability and determine the one
+corresponding to the system under study. We discuss, in section(3) and following [23], Bertrand’s
+theorem and get the analytical solution to the equation of motion. Finally, we apply in section
+(4) the obtained approximative analytical results to the study of the solar system planets in
+order to estimate the Yukawa strength and re-determine the trajectory characteristics for both
+estimated and nominal values of α, as well as solve numerically the equations. The results,
+of comparing the analytical/numerical outputs with the observed data according to the used
+In our computations, we used the whole digits allowed by machine precision, however the results in the appendices
+tables showed only significant digits equal to those of the observed data.
+3
+
+potential, are presented in form of plots for all the planets, whereas the corresponding tables
+are given in an appendix. We end up with conclusions in section (5).
+1
+Hamiltonian formulation
+We start with the Hamiltonian H = T + V where T is the kinetic energy of both masses and V
+is the Gravitational potential energy.
+H =
+⃗p2
+1
+2mp
++
+⃗p2
+2
+2M⊙
+−
+K
+|⃗r2 − ⃗r1|
+�
+1 + αe− |⃗r2−⃗r1|
+λ
+�
+(3)
+where ⃗ri, (⃗vi), i = 1, 2 are the positions (velocities) of the two masses with corresponding mo-
+menta p1 = M⊙v1, p2 = mpv2, K = GmpM⊙. Changing to the center of mass frame (c.o.m),
+with
+⃗r1 = +
+mp
+mp + M⊙
+⃗r = + µ
+M⊙
+⃗r + ⃗R
+,
+⃗r2 = −
+M⊙
+mp + M⊙
+⃗r = − µ
+mp
+⃗r + ⃗R
+(4)
+⃗r = ⃗r1 − ⃗r2
+,
+⃗R = M⊙⃗r1 + mp⃗r2
+mp + M⊙
+(5)
+⃗v1 = ˙⃗R +
+µ
+M⊙
+⃗v
+,
+⃗v2 = ˙⃗R + −µ
+mp
+⃗v
+(6)
+⃗v = ˙⃗r
+,
+⃗p = µ⃗v,
+(7)
+¨⃗R = ⃗0
+,
+µ¨⃗r = M⊙¨⃗r1 = −mp¨⃗r2,
+(8)
+we get
+H = 1
+2(M⊙ + mp) ˙⃗R2 + H
+:
+H = p2
+2µ − K
+r
+�
+1 + αe− r
+λ
+�
+(9)
+Here we have defined µ =
+mpM⊙
+mp+M⊙ as the reduced mass of the system and r = |⃗r|. We switch to
+polar coordinates in the c.o.m to get
+H = 1
+2µ
+�
+p2
+r + p2
+ϕ
+r2
+�
+− K
+r
+�
+1 + αe− r
+λ
+�
+(10)
+From the canonical equations ( [26]): ˙qi =
+�
+∂H
+∂pi
+�
+, ˙pi = −
+�
+∂H
+∂qi
+�
+, and since the Hamiltonian is
+cyclic in ϕ (i.e. it does not depend explicitly on ϕ), we have:
+˙ϕ = ∂H
+∂pϕ
+= pϕ
+µr2
+(11)
+˙pϕ = −∂H
+∂ϕ = 0 ⇒ pϕ = µr2 ˙ϕ = ℓ = constant
+(12)
+where ℓ is the angular momentum of the two-body system, and therefore Hamiltons equations
+for r become:
+˙r = ∂H
+∂pr
+= pr
+µ
+(13)
+˙pr = −∂H
+∂r = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+�
+e− r
+λ
+�
+(14)
+4
+
+Figure 1:
+The reduced potential (red line) given for fixed angular momentum (Eq. 16). The
+pink line denotes the magnitude of the purely Yukawa term (
+���− αK
+r e− r
+λ
+���), whereas the blue line
+represents the Keplerian reduced potential, i.e. Eq. 16 without the Yukawa term.
+Again, and since H(t) = H(t0) = h is constant during the motion of the masses [26], and since
+p2
+r = µ2 ˙r2 ≥ 0 we get a lower bound for the total energy of the system:
+h ≥
+ℓ2
+2µr2 − K
+r
+�
+1 + αe− r
+λ
+�
+(15)
+The right hand side of eq. (15) is defined to be the “reduced potential”, which is common in the
+Kepler problem moving from two degrees of freedom to only one (with the Yukawa correction)
+Vred(r) =
+ℓ2
+2µr2 − K
+r
+�
+1 + αe− r
+λ
+�
+(16)
+One can draw the function for fixed ℓ giving the allowed regions of motion (look at figure 1).
+Note that µ > 0, λ > 0 and α > 0.
+2
+The linearization matrix
+Following [27], in order to determine the stability of the equilibrium points of the system, we
+must form a matrix differential equation using the system equations of motion (Hamiltons Eqs.
+13 and 14 for r, p). The linear system has the form:
+d
+dt
+�
+r
+pr
+�
+=
+�
+f(r, p)
+g(r, p)
+�
+=
+�
+f0
+g0
+�
+eq
++
+� ∂f
+∂r
+∂f
+∂pr
+∂g
+∂r
+∂g
+∂pr
+� �
+r
+pr
+�
+where f (r, pr) = pr
+µ ,
+g (r, pr) = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+�
+e− r
+λ
+�
+(17)
+Given that λ = 1015m for orbits of size comparable to the solar system dimensions [28], one can
+assume that r
+λ is small enough that one can Taylor expand the exponential and ignore terms of
+5
+
+70
+Vred
+09
+Vkep
+-Vyuk
+50
+40
+>
+30
+20
+10
+0
+-10�
+r2
+λ2
+�
+, leading to:
+e− r
+λ ≈ 1 − r
+λ + O
+� r2
+λ2
+�
+≈ 1 − r
+λ
+(18)
+Thus
+g (r, pr) = ℓ2
+µr3 − K
+r2
+�
+1 + α
+�
+1 + r
+λ
+� �
+1 − r
+λ
+��
+≈ ℓ2
+µr3 − K
+r2 (1 + α),
+with the Yukawa effect within this approximation being limited to replacing K by K(1+α), which
+tells that the potential shape is still Newtonian (1/r), and according to Bertrand’s theorem every
+bound trajectory is thus closed for small r/λ. One can see this fact directly from Eq. (16) as
+it gives, compared to the Keplerian potential, within the approximation just a shift, in addition
+to the replacement (K → K(1 + α)), which does not interfere in the equations of motion:
+Vred(r)
+≈
+ℓ2
+2µr2 − K
+r (1 + α) + Kα
+λ .
+(19)
+Consequently, the Jacobian matrix takes the form:
+�
+˙r
+˙pr
+�
+=
+�
+0
+1
+µ
+−3ℓ2
+µr4 + 2K
+r3 (1 + α)
+0
+� �
+r
+pr
+�
+(20)
+where terms of order O
+�
+r2
+λ2
+�
+were ignored, and where the equilibrium point (r, pr)eq satisfies
+feq(r, pr) = geq(r, pr) = 0. We can determine the r at equilibrium using (eq. 14) to get upto
+leading order:
+req =
+ℓ2
+µK(1 + α)
+(21)
+We can now test for stability by choosing values of (α, µ, K, ℓ, λ) and finding the eigenvalues
+of the Jacobian matrix (20) after substituting the equilibrium solution found above (eq. 21).
+Recall that the eigenvalues β1, β2 are found by solving the following equation:
+det |J − βI2×2| = 0
+(22)
+with I2×2 referring to the 2 × 2 identity matrix. Thus we have
+�����
+−β
+1
+µ
+−3ℓ2
+µr4 + 2K
+r3 (1 + α)
+−β
+����� = 0
+(23)
+The characteristic equation (the eigenvalue equation) becomes:
+β1,2 = 1
+2
+�
+τ ±
+�
+τ 2 − 4∆
+�
+(24)
+τ = trace(J) = 0
+(25)
+∆ = det(J) = µ2K4(1 + α)4
+ℓ6
+(26)
+Following [29], the stability is determined by the sign of the eigenvalues. Since ∆ > 0, we have
+the following cases:
+• τ < 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) a stable node.
+• τ < 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a stable spiral.
+6
+
+• τ > 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) an unstable node.
+• τ > 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) an unstable spiral.
+• τ = 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a neutrally stable center (which is our case).
+Actually, the stability refers to how the solution behaves near the equilibrium point; in that
+unstable solutions grow to infinity, whereas stable solutions tend to zero. Also, it is the imaginary
+cases which are the ones giving bound orbital solutions (specifically the center case, whereas the
+stable and unstable imaginary cases are bound solutions tending towards or away from zero).
+3
+Stability & Bertrands theorem
+First, we rewrite the eigenvalue equation in the form
+β2 + µ2K4(1 + α)4
+ℓ6
+= 0
+(27)
+leading to:
+β = ±iµK2(1 + α)2
+ℓ3
+(28)
+We note that one can study the case for a purely Newtonian Potential by letting α → 0.
+Similarly, by ignoring the terms derived from the Newtonian potential, one can single out the
+pure Yukawa contribution. In these two extreme cases, the characteristic equations becomes
+Pure Newtonian: β2 + µ2K4
+ℓ6
+= 0
+(29)
+Pure Yukawa: β2 + µ2K4α4
+ℓ6
+= 0
+(30)
+giving
+Pure Newtonian: β = ±iµK2
+ℓ3
+(31)
+Pure Yukawa: β = ±iµK2α2
+ℓ3
+(32)
+Thus, the equilibrium points for the purely Newtonian, the purely Yukawa, and the Newton
+plus Yukawa Potentials remain center solutions. This implies that the motion would remain
+restricted to ellipses about the equilibrium point; and so, orbits near the equilibrium point are
+possible (further away from the equilibrium point one would have unbounded solutions, as Fig.
+1 shows). This proves that for small r/λ we have stable, closed orbits.
+For the Keplerian orbit equation, it can be written as:
+d2u
+dϕ2 + u = − µ
+ℓ2
+d
+duV
+�1
+u
+�
+(33)
+where u = 1r denotes the Binet transformation, giving, for small r/λ, the following differential
+equation:
+d2u
+dϕ2 + u = +µK
+ℓ2 (1 + α)
+(34)
+whose solution is given by
+u(ϕ) = 1
+r = A [1 + e cos (ϕ − ϕ0)] : A = µK
+ℓ2 (1 + α)
+(35)
+7
+
+with e is the eccentricity of the orbit. The purely Newtonian and purely Yukawa cases follow
+respectively from (34)
+Newtonoian: u(ϕ) = 1
+r = µK
+ℓ2 [1 + e cos (ϕ − ϕ0)]
+(36)
+Purely Yukawa: u(ϕ) = 1
+r = µKα
+ℓ2
+[1 + e cos (ϕ − ϕ0)]
+(37)
+Finally, in order to satisfy Bertrands theorem, the following condition should be satisfied
+d2Vred(r)
+dr2
+����
+r=r0
+> 0
+(38)
+where the reduced potential is given by (16). With the approximations of (eq. 18)) and ignoring
+terms of order O
+�
+r2
+λ2
+�
+this condition becomes
+d2Vred(r)
+dr2
+����
+r=r0
+= µ2K4(1 + α)4
+ℓ6
+> 0
+(39)
+which is true, since α, µ, K, ℓ > 0, in general and in the special cases of Newtonian (α = 0)
+and purely Yukawa potentials. This shows that the Yukawa plus Newtonian potential satisfies
+Bertrands theorem for small rλ.
+4
+Application to the solar system
+We present here our results consisting of determining first the parameters of the models (rmin, rmax, a, b, e)
+by comparing the previous approximative analytical solutions with the NASA data. Then, we
+solved the equations of motion numerically using Matlab and the fourth-order Runge-Kutta
+method with no approximation so that to be compared with the analytical solutions and with
+the observed NASA data. We applied this for all the planets of the solar system. For each pair
+(sun-planet) we used the following values M⊙ = 1.9885 × 1030kg, αnominal = 10−8, λ = 1015m.
+We list in Table (1) the initial conditions used in the analytical and numerical calculations (the
+period τ is used only in the numerical solution to determine the corresponding ‘step’):
+MERCURY
+VENUS
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+NEPTUNE
+PLUTO
+mp(×1024kg)0.3302
+4.8673
+5.9722
+0.64169
+1898.13
+568.32
+86.811
+102.409
+0.01303
+τ (days)
+87.969
+224.701
+365.256
+686.98
+4332.589
+10832.33
+30685.4
+60189
+90560
+rmin
+(×106km)
+0.046
+0.10748
+0.147095
+0.20665
+0.740595
+1.357554
+2.732696
+4.47105
+4.434987
+vmax
+(×103m/s)
+58.98
+35.26
+30.29
+26.5
+13.72
+10.18
+7.11
+5.5
+6.1
+eccentricity
+0.20563
+0.00677
+0.01671
+0.09341
+0.04839
+0.05415
+0.04717
+0.00859
+0.24881
+Table 1: Initial conditions used in the calculations where mp denotes the planet mass, τ is the
+orbit period, rmin is the perihelion and vmax denotes the perihelion velocity
+4.1
+Analytical Method
+The analytical ellipsis equation is of the form
+1
+r ≡ u
+=
+a
+b2 (1 + e cos ϕ) ,
+(40)
+8
+
+where (for a y-axis perpendicular to the polar axis in the orbit plane)
+rmin = a(1 − e)
+,
+rmax = a(1 + e),
+(41)
+e = c
+a =
+�
+1 − b2
+a2
+:
+c2 = a2 − b2,
+(42)
+a = rmin + rmax
+2
+,
+b = ymax − ymin
+2
+(43)
+Thus, analytically one can start with (rmin, vmax, e) observed by NASA in [24] to compute†:
+a = rmin
+1 − e
+,
+b = a
+�
+1 − e2,
+(44)
+and estimate the strength α from
+µK
+ℓ2 (1 + α) = a
+b2
+using
+ℓ = rminvmax.
+(45)
+Once the analytical equation is determined, then one can plot the trajectory and recompute the
+characteristics (rmin, rmax, a, b, e) using Eqs (41,42). We call this procedure the “analytical-α-
+estimated” approach.
+One can also use the nominal value of α = 10−8, and plug it in Eq. (40), where ℓ, e are taken
+from the observed data, to re-evaluate (rmin, rmax, a, b, e) from
+a =
+1
+A(1 − e2),
+A = µK(1+α)
+ℓ2
+,
+b =
+1
+A
+√
+1 − e2 .
+(46)
+We call this procedure the “analytical-α-nominal” approach, which can be looked at as a method
+with three inputs (α, ℓ, e) instead of the three inputs (rmin, vmax, e) used in the other approach.
+4.2
+Numerical Method
+Here, we just solve numerically, using the fourth-order Runge-Kutta method, the Newton’s law
+equation of motion in the c.o.m frame with initial conditions taken from NASA. Thus we solve
+the equations:
+¨⃗r1 = Gmp
+⃗r2 − ⃗r1
+r3
+, Newton,
+¨⃗r2 = GM⊙
+⃗r1 − ⃗r2
+r3
+= −M⊙
+mp
+¨⃗r1,
+(47)
+¨⃗r1 = Gmp
+�
+(1 + αe− r
+λ )1
+r + α
+λe− r
+λ
+� ⃗r2 − ⃗r1
+r2
+, Newton+Yukawa,
+¨⃗r2 = −M⊙
+mp
+¨⃗r1,
+(48)
+under the initial conditions given by NASA data of (rmin, vmax):
+⃗r1(t = tmin) =
+mp
+mp + M⊙
+⃗rmin
+,
+⃗v1(t = tmin) =
+mp
+mp + M⊙
+⃗vmax,
+(49)
+⃗r2(t = tmin) = −
+M⊙
+mp + M⊙
+⃗rmin
+,
+⃗v2(t = tmin) = −
+M⊙
+mp + M⊙
+⃗vmax.
+(50)
+Once the trajectory is solved numerically, we check that it is closed, as the Fig.
+(2) shows
+for both the pure Newton and that with the Yukawa corrections (since the differences are not
+visible on the figure scale containing all the planets). For each obtained orbit, we recalculate
+the corresponding characteristics (rmin, rmax, a, b, e).
+†Due to measurement errors and orbits not being perfectly elliptical, the NASA data may give slightly different
+values of a using Eq. 43 or Eq. 44.
+9
+
+Figure 2: Closed bound planets’ trajectories with and without Yukawa corrections with strength
+α nominal.
+4.3
+Results
+We report in the Tables of Appendix A (from A1 to A18), the calculated characteristics of the
+resulting trajectories for all the planets in the solar system, corresponding to the pure Newton
+and the Newton corrected with Yukawa potentials, both in the analytical and the numerical
+approaches. The odd (even) numbered tables correspond to the nominal (estimated) Yukawa
+strength α. The number of moons of each planet is determined according to [30]. Below we
+explain the meanings of the symbols used in the tables.
+• Nnum: Numerical calculations using the Newtonian potential.
+• Nanal: Analytical calculations using the Newtonian potential.
+• RN = Nnum
+Nanal %: The percentage ratio of the numerical to the analytical results for Newton
+potential.
+• (N + Y K)num : Numerical calculations using the modified potential.
+• (N + Y K)anal: Analytical calculations using the modified potential.
+• RN+Y K = (N+Y K)num
+(N+Y K)anal %: The percentage ratio of the numerical to the analytical results
+for modified potential.
+• RN−Obs
+num
+= Nnum/Obs%: Percentage ratio of the numerical results, using the Newtonian
+potential, to the observed results.
+• RN−Obs
+anal
+= Nanal/Obs %: Percentage ratio the analytical results, using the Newtonian
+potential, to the observed results.
+• RY K−Obs
+num
+= (N + Y K)num/Obs %: Percentage ratio of the numerical results, using the
+modified potential, to the observed results.
+10
+
+(km)• RY K−Obs
+anal
+= (N + Y K)anal/Obs %: Percentage ratio of the analytical results, using the
+modified potential, to the observed results.
+In order to summarize the findings of the Tables, we present in Fig. (3) plots showing, for each
+planet and at every polar angle, the deviation from unity of the ratio between two quantities of
+the following, allowing thus to compare the effects of the considered potential (Newton vs New-
+ton+Yuakawa) and/or the used method (numerical vs analytical) and/or the Yukawa strength
+determination (nominal vs estimated):
+• rn(num) representing the trajectory equation of the numerical approach with Newton
+potential,
+• rn(anl) representing the trajectory equation of the analytical approach with Newton po-
+tential,
+• ryk(num) representing the trajectory equation of the numerical approach with New-
+ton+Yukawa potential and nominal α,
+• ryk(anl) representing the trajectory equation of the analytical approach with Newton+Yukawa
+potential and nominal α,
+• ryka(num) representing the trajectory equation of the numerical approach with New-
+ton+Yukawa potential and estimated α,
+• ryka(anl) representing the trajectory equation of the analytical approach with New-
+ton+Yukawa potential and estimated α.
+We see that some ratios (e.g. the dashed red and sky blue) do coincide near zero deviation
+from one, meaning no tangible effect of adding the Yukawa correction, be it in the analytic or
+the numeric method, as long as one takes the nominal value of α. Also,we note local extremums
+for the deviations from unity at polar angles multiples of π/2 as a generic feature in many plots.
+One can interpret the large values of the deviations for the nearest (Mercury) and the farthest
+(Pluto) planet, in that for the former; the perturbative effect of solar winds, important as we
+approach the sun, was not taken into consideration, whereas for the furthest; accumulating
+gravitational screening effects of the other planets and their moons, which were not considered
+in the study, are becoming important especially for a small sized- planetoid like Pluto.
+In order to show the effects of the separating distance effect, one should compute the absolute
+deviations from observed data for each planet. In appendix B, the Tables B1, B2 (B3, B4), report
+the deviation from observation for each planet of rmax, rmin respectively, in the case of nominal
+(estimated) α. We summarize these findings in Fig. (4). We see that the agreement between
+the numerical and analytical solutions is excellent in both estimated and nominal α cases. We
+see that the deviations due to Yukawa correction are not large, but note the following:
+1. For estimated α:
+• rmax-deviation: The numerical deviation is larger by about 103 times the analytical
+deviation. In general, it increases the further the planet is, and reaches a maximum
+of order (−25 million km) (less than the observed value) in Pluto.
+• rmin-deviation: Again, the numerical deviation is larger by about (101-102)-order of
+magnitude than analytical deviation, where it is largest in Neptune (−8 million km),
+however it reverses sign and becomes (+5 million km) more than the observation in
+Pluto.
+11
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  1-{ rn(num)/ryka(anl) }        ____ 
+1-{ryka(anl)/ryk(num)}      ____ 
+1-{ryka(anl)/ryka(num)}    ____ 
+ 1-{ rn(anl)/ryka(anl)}           ____ 
+1-{ryka(anl)/ryk(anl)}         ____ 
+1-{ rn(num)/ryk(num)}    ----- 
+1-{ryka(num)/rn(num)}   ----- 
+1-{ryk(num)/ryka(num)} ----- 
+1-{ryk(anl)/rn(anl)}          ----- 
+ 
+ 
+Figure 3:
+Deviations from Unity for Ratios of Computed trajectories at each polar angle,
+according to the considered potential (Newton vs Newton+Yuakawa) and/or to the used method
+(numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).
+We show in a zoomed region, for one planet (Earth) generic case, that the dashed red and sky
+blue curves are very near each other (the same applies to the green and blue curves in Mercury
+case).
+12
+
+VENUS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.03
+0.02
+1-Percentage Ratio(%)
+0.01
+0.01
+-0.02
+-0.03
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)EARTH
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.08
+0.06
+1-Percentage Ratio(%)
+0.04
+0.02
+0.02
+-0.04
+-0.06
+-0.08
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)MARS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+2
+1.5
+0.5
+0.5
+1
+-1.5
+-2
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)JUPITER
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+0.6
+0.4
+1-Percentage Ratio(%)
+0.2
+-0.2
+-0.4
+-0.6
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)SATURN
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+3
+2
+-1
+-2
+-3
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)URANUS
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+1.5
+1-Percentage Ratio(%)
+0.5
+0.5
+1
+-1.5
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)Neptune
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+2 
+1.5
+0.5
+0.5
+-1.5
+-2
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)Pluto
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+15
+10
+1-Percentage Ratio(%)
+-5
+-10
+-15
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg)MERCURY
+Deviationsfrom unityforRatiosof ComputedTrajectories(%)
+10
+8
+6
+1-PercentageRatio(%)
+4
+N
+A
+-6
+-8
+-10
+0
+50
+100
+150
+200
+250
+300
+350
+400
+angle (deg) 
+ 
+ 
+ 
+ 
+-50
+0
+50
+100
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmax; alpha nominal
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-20
+0
+20
+40
+60
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmin; alpha nominal
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-30
+-20
+-10
+0
+10
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmax; alpha estimated
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+-10
+-5
+0
+5
+10
+MERCURY
+Venus
+EARTH
+MARS
+JUPITER
+SATURN
+URANUS
+Neptune
+Pluto
+Deviations from the observed values for rmin; alpha estimated
+Absolute deviation from observation numerically
+×10^6(km)
+Absolute deviation from  observation analytically
+×10^6(km)
+Figure 4: Absolute deviations from observed data for each planet, according to the used method
+(numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).
+13
+
+2. For nominal α:
+• rmax-deviation: The numerical deviation is larger than the analytical one, but are of
+the same order reaching a maximum of +80 (+40) million km using the numerical
+(analytical) method in Neptune. For Pluto and Uranus, we get (−40) million km in
+the numerical method (less than observed).
+• rmin-deviation: The analytical deviation is larger, and sometimes reverses sign com-
+pared to the numerical. For example, in Neptune the analytical approach gives a
+deviation of (+40 million km) from observation, whereas the numerical one gives a
+deviation of −5 million km (less than the observed value).
+Actually, the disagreements with observations are due to several reasons: the first one is physical
+in nature, in that it results from neglecting the perturbation due to third bodies, or, more
+generally, the effect of the natural satellites, such as moons or asteroids.
+Also, we did not
+either take into account the radiation and the solar wind physical effects. Moreover, the results
+were obtained as a 2-body problem, and hence the movement of more distant planets might be
+affected by planets closer to the sun, which can be present not in the dominant term, but in
+higher orders of expansion. The second factor lies in the computational side, and concerns the
+numerical method used, the value of the step size, and the high sensitivity of the problem to
+the initial conditions. One should also mention that for the analytical solution we restricted the
+study to leading order neglecting higher orders in the expansion of exponentials, whereas for the
+numerical solution the entire exponential is considered.
+5
+Summary and Conclusion
+In this work, we followed [23] and used the Hamilton’s formulation in order to obtain the
+differential equation of motion and the path equation for the gravitational two-body system.
+The developments are carried out in the case of the pure Newtonian potential, the Newtonian
+corrected with Yukawa type potential and the pure Yukawa potential.
+As in [23], we have
+reviewed the stability problem, constructed the linearization matrix and tested the stability
+of the system for a Yukawa correction, and found that it is of a central solution type, which
+implies stable solutions near the fixed point. We repeated the analysis for a purely Yukawa force
+and found similar results. We also confirmed that the modified potential obeys the Bertrands
+theorem.
+Then, we determined the parameters’ set corresponding to the planets of the solar system
+starting from the observed (rmin, vmax, e) estimating α. For both the estimated and nominal
+values of α, we determined the characteristics of the trajectories numerically and analytically,
+and compared between the methods and with the observed data. We explained the extent to
+which these results are consistent with the observational data, presenting in form of histograms
+the absolute deviations from observations, which were found to give an upper deviation of order
+80 million km in Neptune using nominal α, and 20 million km in Pluto using estimated α.
+Acknowledgments:
+N. Chamoun acknowledges support from the ICTP-Associate pro-
+gram, from the Humboldt Foundation and from the CAS-PIFI scholarship.
+14
+
+Appendices
+A. Tables of Calculated/Observed Parameters of the Planets
+15
+
+Mercury
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+46
+69.832
+57.916
+56.67679158
+0.205744228
+Nanal
+47
+72.043
+59.756
+58.47840586
+0.205646344
+RN = Nnum
+Nanal %
+97
+96.930
+96.921
+96.91918025
+99.95242442
+(N + Y K)num
+46
+69.831
+57.916
+56.67678243
+0.205738221
+(N + Y K)anal
+47
+72.043
+59.756
+58.47840528
+0.205646344
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+97
+96.930
+96.921
+96.91916556
+99.95534277
+Observation
+46
+69.818
+57.909
+0.20563069
+RN−Obs
+num
+= Nnum/Obs %
+100
+99.980
+99.988
+99.94481595
+RN−Obs
+anal
+= Nanal/Obs %
+97
+96.911
+96.909
+99.9923879
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100
+99.981
+99.988
+99.94773407
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+97
+96.911
+96.909
+99.9923879
+nominal α = 10−8
+Table A1: The values of the calculated and observational astronomical parameters of the planet
+Mercury whose number of moons is 0
+Mercury
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+46
+69.623
+57.826
+56.65144795
+0.2022
+Nanal
+46
+69.819
+57.912
+56.67470066
+0.2056
+RN = Nnum
+Nanal %
+100
+99.719
+99.851
+99.95897162
+98.3743
+(N + Y K)num
+46
+69.613
+57.820
+56.64729064
+0.2022
+(N + Y K)anal
+46
+69.815
+57.908
+56.6714469
+0.2056
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100
+99.711
+99.847
+99.9573749
+98.3408
+Observation
+46
+69.818
+57.909
+0.2056
+RN−Obs
+num
+= Nnum/Obs %
+100
+99.721
+99.856
+98.3817
+RN−Obs
+anal
+= Nanal/Obs %
+100
+100.001
+100.005
+100.0075
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100
+99.706
+99.847
+98.3482
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100
+99.995
+99.999
+100.0075
+estimated α = 5.741444131301954 × 10−5
+Table A2: The values of the calculated and observational astronomical parameters of the planet
+Mercury whose number of moons is 0
+16
+
+Venus
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+107.30
+108.689
+107.99
+107.9982364
+0.0044
+Nanal
+107.48
+108.961
+108.22
+108.2222348
+0.0072
+RN = Nnum
+Nanal %
+99.83
+99.750
+99.79
+99.79301998
+60.8187
+(N + Y K)num
+107.30
+108.689
+107.99
+107.9982353
+0.0044
+(N + Y K)anal
+107.48
+108.961
+108.22
+108.2222337
+0.0072
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.83
+99.750
+99.79
+99.79301998
+60.8189
+Observation
+107.48
+108.941
+108.21
+0.0068
+RN−Obs
+num
+= Nnum/Obs %
+99.83
+99.769
+99.80
+64.7150
+RN−Obs
+anal
+= Nanal/Obs %
+100.00
+100.018
+100.01
+106.4063
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.83
+99.769
+99.80
+64.7152
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.00
+100.018
+100.01
+106.4063
+nominal α = 10−8
+Table A3: The values of the calculated and observational astronomical parameters of the planet
+Venus whose number of moons is 0
+Venus
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+107.30
+108.689
+107.99
+107.9982364
+0.0044
+Nanal
+107.48
+108.961
+108.22
+108.2222348
+0.0072
+RN = Nnum
+Nanal %
+99.83
+99.750
+99.79
+99.79301998
+60.8187
+(N + Y K)num
+107.30
+108.658
+107.98
+107.9821956
+0.0045
+(N + Y K)anal
+107.47
+108.945
+108.20
+108.2068155
+0.0072
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.84
+99.736
+99.78
+99.79241613
+63.0300
+Observation
+107.48
+108.941
+108.21
+0.0068
+RN−Obs
+num
+= Nnum/Obs %
+99.83
+99.769
+99.80
+64.7150
+RN−Obs
+anal
+= Nanal/Obs %
+100.00
+100.018
+100.01
+106.4063
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.83
+99.740
+99.78
+67.0680
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.99
+100.004
+99.99
+106.4063
+estimated α = 1.424988220126711 × 10−4
+Table A4: The values of the calculated and observational astronomical parameters of the planet
+Venus whose number of moons is 0
+17
+
+EARTH
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+146.884
+151.7
+149.336
+149.319847
+0.0156
+Nanal
+147.126
+152.1
+149.625
+149.6034965
+0.0168
+RN = Nnum
+Nanal %
+99.835
+99.7
+99.806
+99.81039915
+92.5721
+(N + Y K)num
+146.884
+151.7
+149.336
+149.3198455
+0.0156
+(N + Y K)anal
+147.126
+152.1
+149.625
+149.603495
+0.0168
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.835
+99.7
+99.806
+99.81039915
+92.5720
+Observation
+147.095
+152.1
+149.598
+0.0167
+RN−Obs
+num
+= Nnum/Obs %
+99.8572
+99.7
+99.825
+93.5903
+RN−Obs
+anal
+= Nanal/Obs %
+99.978
+99.9
+99.981
+98.9120
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.857
+99.7
+99.825
+93.5902
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.978
+99.9
+99.981
+98.9120
+nominal α = 10−8
+Table A5: The values of the calculated and observational astronomical parameters of the planet
+Earth whose number of moons is 0
+EARTH
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+146.884
+151.7
+149.336
+149.319847
+0.0156
+Nanal
+147.126
+152.1
+149.625
+149.6034965
+0.0168
+RN = Nnum
+Nanal %
+99.835
+99.7
+99.806
+99.81039915
+92.5721
+(N + Y K)num
+146.883
+151.7
+149.307
+149.2910008
+0.0154
+(N + Y K)anal
+147.099
+152.0
+149.597
+149.5762082
+0.01688
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.853
+99.7
+99.805
+99.80932302
+91.4700
+Observation
+147.095
+152.1
+149.598
+0.0167
+RN−Obs
+num
+= Nnum/Obs %
+99.8572
+99.7
+99.825
+93.5903
+RN−Obs
+anal
+= Nanal/Obs %
+99.978
+99.9
+99.981
+98.9120
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.856
+99.7
+99.805
+92.4761
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.003
+99.9
+99.999
+101.0999
+estimated α = 1.824376359731428 × 10−4
+Table A6: The values of the calculated and observational astronomical parameters of the planet
+Earth whose number of moons is 0
+18
+
+MARS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+206.57
+248.480
+227.52
+226.6509159
+0.0898
+Nanal
+206.64
+249.277
+227.96
+226.9631182
+0.0935
+RN = Nnum
+Nanal %
+99.96
+99.680
+99.80
+99.8624436
+96.0965
+(N + Y K)num
+206.57
+248.480
+227.52
+226.6509134
+0.0898
+(N + Y K)anal
+206.64
+249.277
+227.96
+226.9631159
+0.0935
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.96
+99.680
+99.80
+99.86244351
+96.0965
+Observation
+206.65
+249.261
+227.94
+0.0935
+RN−Obs
+num
+= Nnum/Obs %
+99.96
+99.687
+99.81
+96.1252
+RN−Obs
+anal
+= Nanal/Obs %
+99.99
+100.006
+100.01
+100.0298
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.96
+99.687
+99.81
+96.1252
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.99
+100.006
+100.01
+100.0298
+nominal α = 10−8
+Table A7: The values of the calculated and observational astronomical parameters of the planet
+Mars whose number of moons is 0
+MARS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+206.57
+248.480
+227.52
+226.6509159
+0.0898
+Nanal
+206.64
+249.277
+227.96
+226.9631182
+0.0935
+RN = Nnum
+Nanal %
+99.96
+99.680
+99.80
+99.8624436
+96.0965
+(N + Y K)num
+206.57
+248.425
+227.49
+226.6249874
+0.0897
+(N + Y K)anal
+206.62
+249.252
+227.93
+226.9402451
+0.0935
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.97
+99.668
+99.80
+99.86108339
+95.9861
+Observation
+206.65
+249.261
+227.94
+0.0935
+RN−Obs
+num
+= Nnum/Obs %
+99.96
+99.687
+99.81
+96.1252
+RN−Obs
+anal
+= Nanal/Obs %
+99.99
+100.006
+100.01
+100.0298
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.96
+99.664
+99.80
+96.0147
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.98
+99.996
+99.99
+100.0298
+estimated α = 1.007889331583467 × 10−4
+Table A8: The values of the calculated and observational astronomical parameters of the planet
+Mars whose number of moons is 0
+19
+
+JUPITER
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+739.902
+815.533
+777.717
+776.9190412
+0.0469
+Nanal
+742.542
+818.568
+780.555
+779.626266
+0.04873
+RN = Nnum
+Nanal %
+99.644
+99.629
+99.636
+99.65275352
+96.3711
+(N + Y K)num
+739.902
+815.533
+777.717
+776.9190329
+0.0469
+(N + Y K)anal
+742.542
+818.568
+780.555
+779.6262582
+0.0487
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.644
+99.629
+99.636
+99.65275345
+96.3711
+Observation
+740.595
+816.363
+778.479
+0.0487
+RN−Obs
+num
+= Nnum/Obs %
+99.906
+99.898
+99.902
+96.4399
+RN−Obs
+anal
+= Nanal/Obs %
+100.262
+100.270
+100.266
+100.0714
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.906
+99.898
+99.902
+96.4399
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.262
+100.270
+100.266
+100.0714
+nominal α = 10−8
+Table A9: The values of the calculated and observational astronomical parameters of the planet
+Jupiter whose number of moons is 0
+JUPITER
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+739.902
+815.533
+777.717
+776.9190412
+0.0469
+Nanal
+742.542
+818.568
+780.555
+779.626266
+0.04873
+RN = Nnum
+Nanal %
+99.644
+99.629
+99.636
+99.65275352
+96.3711
+(N + Y K)num
+739.837
+810.932
+775.385
+774.6852056
+0.0441
+(N + Y K)anal
+740.567
+816.390
+778.478
+777.5526264
+0.0487
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.901
+99.331
+99.602
+99.63122486
+90.6263
+Observation
+740.595
+816.363
+778.479
+0.0487
+RN−Obs
+num
+= Nnum/Obs %
+99.906
+99.898
+99.902
+96.4399
+RN−Obs
+anal
+= Nanal/Obs %
+100.262
+100.270
+100.266
+100.0714
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.897
+99.334
+99.602
+90.6911
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.996
+100.003
+99.999
+100.0714
+estimated α = 2.666880127522 × 10−3
+Table A10: The values of the calculated and observational astronomical parameters of the planet
+Jupiter whose number of moons is 0
+20
+
+SATURN
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+1355.461
+1523.344
+1439.403
+1437.455093
+0.055
+Nanal
+1368.378
+1518.496
+1443.437
+1441.481829
+0.052
+RN = Nnum
+Nanal %
+99.056
+100.319
+99.720
+99.72065302
+106.042
+(N + Y K)num
+1355.461
+1523.344
+1439.403
+1437.455078
+0.055
+(N + Y K)anal
+1368.378
+1518.496
+1443.437
+1441.481815
+0.052
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.056
+100.319
+99.720
+99.72065294
+106.042
+Observation
+1357.554
+1506.527
+1432.041
+0.052
+RN−Obs
+num
+= Nnum/Obs %
+99.845
+101.116
+100.514
+106.081
+RN−Obs
+anal
+= Nanal/Obs %
+100.797
+100.794
+100.795
+100.036
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.845
+101.116
+100.514
+106.081
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.797
+100.794
+100.795
+100.036
+nominal α = 10−8
+Table A11: The values of the calculated and observational astronomical parameters of the planet
+Saturn whose number of moons is 0
+SATURN
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+1355.461
+1523.344
+1439.403
+1437.455093
+0.055
+Nanal
+1368.378
+1518.496
+1443.437
+1441.481829
+0.052
+RN = Nnum
+Nanal %
+99.056
+100.319
+99.720
+99.72065302
+106.042
+(N + Y K)num
+1354.869
+1497.652
+1426.261
+1424.954776
+0.046
+(N + Y K)anal
+1357.574
+1506.507
+1432.040
+1430.100672
+0.052
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.800
+99.412
+99.596
+99.64017246
+89.244
+Observation
+1357.554
+1506.527
+1432.041
+0.052
+RN−Obs
+num
+= Nnum/Obs %
+99.845
+101.116
+100.514
+106.081
+RN−Obs
+anal
+= Nanal/Obs %
+100.797
+100.794
+100.795
+100.036
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.802
+99.410
+99.596
+89.277
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.001
+99.998
+99.999
+100.036
+estimated α = 7.958291053541 × 10−3
+Table A12: The values of the calculated and observational astronomical parameters of the planet
+Saturn whose number of moons is 0
+21
+
+URANUS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+2729.595
+2957.44
+2843.519
+2841.649275
+0.0381
+Nanal
+2717.213
+2984.63
+2850.921
+2847.766462
+0.0469
+RN = Nnum
+Nanal %
+100.455
+99.08
+99.740
+99.78519352
+81.2504
+(N + Y K)num
+2729.595
+2957.44
+2843.519
+2841.649245
+0.0381
+(N + Y K)anal
+2717.213
+2984.63
+2850.921
+2847.766434
+0.0469
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.455
+99.08
+99.740
+99.78519344
+81.2504
+Observation
+2732.696
+3001.39
+2867.043
+0.0469
+RN−Obs
+num
+= Nnum/Obs %
+99.886
+98.53
+99.179
+81.3684
+RN−Obs
+anal
+= Nanal/Obs %
+99.433
+99.44
+99.437
+100.1452
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.886
+98.53
+99.179
+81.3684
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.433
+99.44
+99.437
+100.1452
+nominal α = 10−8
+Table A13: The values of the calculated and observational astronomical parameters of the planet
+Uranus whose number of moons is 0
+URANUS
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+2729.595
+2957.44
+2843.519
+2841.649275
+0.0381
+Nanal
+2717.213
+2984.63
+2850.921
+2847.766462
+0.0469
+RN = Nnum
+Nanal %
+100.455
+99.08
+99.740
+99.78519352
+81.2504
+(N + Y K)num
+2730.116
+2992.91
+2861.516
+2858.935401
+0.0441
+(N + Y K)anal
+2732.578
+3001.50
+2867.042
+2863.869614
+0.0469
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.909
+99.71
+99.807
+99.82770818
+94.0932
+Observation
+2732.696
+3001.39
+2867.043
+0.0469
+RN−Obs
+num
+= Nnum/Obs %
+99.886
+98.53
+99.179
+81.3684
+RN−Obs
+anal
+= Nanal/Obs %
+99.433
+99.44
+99.437
+100.1452
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.905
+99.71
+99.807
+94.2299
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.995
+100.00
+99.999
+100.1452
+estimated α = −5.622864957252 × 10−3
+Table A14: The values of the calculated and observational astronomical parameters of the planet
+Uranus whose number of moons is 0
+22
+
+Neptune
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4464.81
+4634.099
+4549.454
+4548.810665
+0.0177
+Nanal
+4512.97
+4601.381
+4557.176
+4556.953752
+0.0097
+RN = Nnum
+Nanal %
+98.93
+100.711
+99.830
+99.82130416
+180.8744
+(N + Y K)num
+4464.81
+4634.098
+4549.454
+4548.810617
+0.0177
+(N + Y K)anal
+4512.97
+4601.381
+4557.176
+4556.953706
+0.0097
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+98.93
+100.711
+99.830
+99.82130411
+180.8743
+Observation
+4471.05
+4558.857
+4514.953
+0.0097
+RN−Obs
+num
+= Nnum/Obs %
+99.86
+101.650
+100.764
+182.6474
+RN−Obs
+anal
+= Nanal/Obs %
+100.93
+100.932
+100.935
+100.9802
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.86
+101.650
+100.764
+182.6473
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.93
+100.932
+100.935
+100.9802
+nominal α = 10−8
+Table A15: The values of the calculated and observational astronomical parameters of the planet
+Neptune whose number of moons is 0
+Neptune
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4464.81
+4634.099
+4549.454
+4548.810665
+0.0177
+Nanal
+4512.97
+4601.381
+4557.176
+4556.953752
+0.0097
+RN = Nnum
+Nanal %
+98.93
+100.711
+99.830
+99.82130416
+180.8744
+(N + Y K)num
+4463.01
+4546.479
+4504.745
+4504.517794
+0.0096
+(N + Y K)anal
+4471.15
+4558.747
+4514.952
+4514.73215
+0.0097
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+99.81
+99.730
+99.773
+99.77375499
+98.6587
+Observation
+4471.05
+4558.857
+4514.953
+0.0097
+RN−Obs
+num
+= Nnum/Obs %
+99.86
+101.650
+100.764
+182.6474
+RN−Obs
+anal
+= Nanal/Obs %
+100.93
+100.932
+100.935
+100.9802
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+99.82
+99.728
+99.773
+99.6259
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.00
+99.997
+99.999
+100.9802
+estimated α = 9.351961741362 × 10−3
+Table A16: The values of the calculated and observational astronomical parameters of the planet
+Neptune whose number of moons is 0
+23
+
+Pluto
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4439.709
+7265.423
+5852.566
+5684.326067
+0.2397
+Nanal
+4431.722
+7298.614
+5865.168
+5687.267307
+0.2444
+RN = Nnum
+Nanal %
+100.180
+99.545
+99.785
+99.94828377
+98.0832
+(N + Y K)num
+4439.709
+7265.423
+5852.566
+5684.325992
+0.2397
+(N + Y K)anal
+4431.722
+7298.614
+5865.168
+5687.26725
+0.2444
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.180
+99.545
+99.785
+99.94828346
+98.0832
+Observation
+4434.987
+7304.326
+5869.656
+0.2444
+RN−Obs
+num
+= Nnum/Obs %
+100.106
+99.467
+99.708
+98.0882
+RN−Obs
+anal
+= Nanal/Obs %
+99.926
+99.921
+99.923
+100.0051
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100.106
+99.467
+99.708
+98.0882
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+99.926
+99.921
+99.923
+100.0051
+nominal α = 10−8
+Table A17: The values of the calculated and observational astronomical parameters of the planet
+Pluto whose number of moons is 0
+Pluto
+rmin(×106km)
+rmax(×106km)
+a(×106km)
+b(×106km)
+eccentricity
+Nnum
+4439.709
+7265.423
+5852.566
+5684.326067
+0.2397
+Nanal
+4431.722
+7298.614
+5865.168
+5687.267307
+0.2444
+RN = Nnum
+Nanal %
+100.180
+99.545
+99.785
+99.94828377
+98.0832
+(N + Y K)num
+4439.740
+7280.242
+5859.991
+5690.112819
+0.2407
+(N + Y K)anal
+4435.112
+7304.196
+5869.654
+5691.616958
+0.2444
+RN+Y K
+= (N+Y K)num
+(N+Y K)anal %
+100.104
+99.672
+99.835
+99.97357273
+98.4812
+Observation
+4434.987
+7304.326
+5869.656
+0.2444
+RN−Obs
+num
+= Nnum/Obs %
+100.106
+99.467
+99.708
+98.0882
+RN−Obs
+anal
+= Nanal/Obs %
+99.926
+99.921
+99.923
+100.0051
+RY K−Obs
+num
+=
+(N + Y K)num/Obs %
+100.107
+99.670
+99.835
+98.4862
+RY K−Obs
+anal
+=
+(N + Y K)anal/Obs %
+100.002
+99.998
+99.999
+100.0051
+estimated α = −7.642205983339201 × 10−4
+Table A18: The values of the calculated and observational astronomical parameters of the planet
+Pluto whose number of moons is 0
+24
+
+B. Tables of Absolute Deviations from Observation of the Planets
+25
+
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmax
+rnum
+max − Obs
+ranal
+max − Obs
+MERCURY
+99.721
+100.001
+69.818
+-0.194
+0.001
+Venus
+99.769
+100.018
+108.941
+-0.251
+0.020
+EARTH
+99.794
+99.984
+152.100
+-0.312
+-0.024
+MARS
+99.687
+100.006
+249.261
+-0.780
+0.016
+JUPITER
+99.898
+100.270
+816.363
+-0.829
+2.205
+SATURN
+101.116
+100.794
+1506.527
+16.817
+11.969
+URANUS
+98.535
+99.441
+3001.390
+-43.947
+-16.759
+Neptune
+101.650
+100.932
+4558.857
+75.241
+42.524
+Pluto
+99.467
+99.921
+7304.326
+-38.902
+-5.711
+Table B1: Absolute deviations, with nominal α, of rmax from observation, evaluated in (106
+km).
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmin
+rnum
+min − Obs
+ranal
+min − Obs
+MERCURY
+100.062
+100.012
+46.000
+0.028
+0.005
+Venus
+99.839
+100.008
+107.480
+-0.172
+0.009
+EARTH
+99.857
+99.978
+147.095
+-0.210
+-0.031
+MARS
+99.962
+99.999
+206.650
+-0.077
+-0.001
+JUPITER
+99.906
+100.262
+740.595
+-0.692
+1.947
+SATURN
+99.845
+100.797
+1357.554
+-2.092
+10.824
+URANUS
+99.886
+99.433
+2732.696
+-3.100
+-15.482
+Neptune
+99.860
+100.937
+4471.050
+-6.239
+41.922
+Pluto
+100.106
+99.926
+4434.987
+4.722
+-3.264
+Table B2: Absolute deviations, with nominal α, of rmin from observation, evaluated in (106 km).
+26
+
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmax
+rnum
+max − Obs
+ranal
+max − Obs
+MERCURY
+99.706
+99.995
+69.818
+-0.204
+-0.002
+Venus
+99.740
+100.004
+108.941
+-0.282
+0.004
+EARTH
+99.757
+99.997
+152.100
+-0.369
+-0.003
+MARS
+99.664
+99.996
+249.261
+-0.835
+-0.008
+JUPITER
+99.334
+100.003
+816.363
+-5.430
+0.027
+SATURN
+99.410
+99.998
+1506.527
+-8.874
+-0.019
+URANUS
+99.717
+100.003
+3001.390
+-8.472
+0.117
+Neptune
+99.728
+99.997
+4558.857
+-12.377
+-0.109
+Pluto
+99.670
+99.998
+7304.326
+-24.083
+-0.12907
+Table B3: Absolute deviations, with estimated α, of rmax from observation, evaluated in (106
+km).
+RY K−Obs
+num
+RY K−Obs
+anal
+Observed rmin
+rnum
+min − Obs
+ranal
+min − Obs
+MERCURY
+100.062
+100.006
+46.000
+0.028
+0.002
+Venus
+99.838
+99.994
+107.480
+-0.173
+-0.005
+EARTH
+99.856
+100.003
+147.095
+-0.211
+0.004
+MARS
+99.962
+99.989
+206.650
+-0.077
+-0.022
+JUPITER
+99.897
+99.996
+740.595
+-0.757
+-0.027
+SATURN
+99.802
+100.001
+1357.554
+-2.684
+0.020
+URANUS
+99.905
+99.995
+2732.696
+-2.579
+-0.117
+Neptune
+99.820
+100.002
+4471.050
+-8.037
+0.107
+Pluto
+100.107
+100.002
+4434.987
+4.753
+0.125
+Table B4: Absolute deviations, with estimated α, of rmin from observation, evaluated in (106
+km).
+27
+
+References
+[1] Ephraim Fischbach and Carrick L. Talmadge, The Search for Non-Newtonian Gravity,
+AIP Press, Springer (1999),
+[2] L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics (Mechanics), Vols. 1 Ch
+3, Sec 14, p 32, (Pergamon Press : Oxford), (1969).
+[3] I. Rodriguez and J. L. Brun, ”Closed orbits in central forces distinct from Coulomb or
+harmonic oscillator type,” European Journal of Physics, vol. 19, pp. 41-49, 1998.
+[4] . J. L. Brun and A. F. Pacheco, ”On closed but non-geometrically similar orbits,” Celestial
+Mech Dyn Astr, pp. 311-316, 2006.
+[5] K. Hinterbichler, “Theoretical Aspects of Massive Gravity”, Rev. Mod. Phys. 84, 671
+(2012), arXiv:1105.3735 [hep-th].
+[6] Claudia de Rham, “Massive Gravity”, Living Reviews in Relativity, 17 (2014) 7,
+arXiv:1401.4173 [hep-th].
+[7] A. S. Goldhabert and M. M. Nieto, “Mass of the graviton”, PRD9, 1119 (1974)
+[8] Yiming Dong, Lijing Shao, Zexin Hu, Xueli Miaoc and Ziming Wang, “Prospects for
+Constraining the Yukawa Gravity with Pulsars around Sagittarius A*”, JCAP-11-(2022)-
+051, arXiv:2210.16130 [astro-ph.HE].
+[9] J. W. Moffat, ”Scalar-tensor-vector gravity theory,” Journal of Cosmology and Astropar-
+ticle Physics, 2006.
+[10] Xing Zhang, Tan Liu and Wen Zhao, “Gravitational radiation from compact binary sys-
+tems in screened modified gravity”, Phys. Rev.D95, 104027 (2017), arXiv: 1702.08752
+[gr-qc].
+[11] A. DAddio, R. Casadio, A. Giusti, and M. De Laurentis, “Orbits in bootstrapped Newto-
+nian gravity”, Phys. Rev. D 105, 104010 (2021), arXiv:2110.08379 [gr-qc]
+[12] R. Della Monica, I. de Martino, M. De Laurentis, “Orbital precession of the S2 star in
+ScalarTensorVector Gravity”, Monthly Notices of the Royal Astronomical Society, Volume
+510, Issue 4, March 2022, Pages 47574766, arXiv:2105.12687 [gr-qc]
+[13] David Benisty, “Testing modified gravity via Yukawa potential in two body problem:
+Analytical solution and observational constraints”, Phys. Rev. D106, 043001 (2022).
+[14] I. Banik and H. Zhao, Mon. Not. Roy. Astron. Soc. 480, 2660 (2018), [Erratum:
+Mon.Not.Roy.Astron.Soc. 482, 3453 (2019), Erratum: Mon.Not.Roy.Astron.Soc. 484, 1589
+(2019)], arXiv:1805.12273 [astro-ph.GA].
+[15] Q. Yu, F. Zhang, and Y. Lu, Astrophys. J. 827, 114 (2016), arXiv:1606.07725 [astro-
+ph.HE].
+[16] D. Pricopi, Astrophys. Space Sci. 361, 277 (2016).
+[17] J. P. Edwards, U. Gerber, C. Schubert, M. A. Trejo, and A.Weber, PTEP 2017, 083A01
+(2017), arXiv:1706.09979 [physics.atom-ph].
+28
+
+[18] R. Mukherjee and S. Sounda, Indian Journal of Physics 92, 197 (2018), arXiv:1705.02444
+[physics.plasm-ph].
+[19] Lorenzo Iorio, “Putting Yukawa-LikeModified Gravity (MOG) on the Test in the Solar
+System”, Scholarly Research Exchange, Volume 2008, Article ID 238385
+[20] M. De Laurentis, I. De Martino, and R. Lazkoz, “Analysis of the Yukawa gravitational
+potential in f(R) gravity II: relativistic periastron advance”, Phys. Rev. D 97, 104068
+(2018), arXiv:1801.08136 [gr-qc].
+[21] J. Berg, P. Brax, M. Pernot-Borrs, and J.-P. Uzan, “Interpretation of geodesy exper-
+iments in non-Newtonian theories of gravity”, Class. Quant. Grav. 35, 234001 (2018),
+arXiv:1808.00340 [gr-qc].
+[22] Lorenzo Iorio, “Constraints on a Yukawa gravitational potential from laser data of LA-
+GEOS satellites”, Physics Letters A 298 (2002) 315318
+[23] E. Cavan, I. Haranas, I. Gkigkitzis and K. Cobbett, ”Dynamics and stability of the two
+body problem with Yukawa correction,” Astrophysics and Space Science, 2020.
+[24] ”https://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html,” [Online].
+[25] A. Rujula, dedicated to viktor weisskopf on the occasion of the viki-fest, erice, 1986.
+[26] H. Goldstein, Classical Mechanics, SECOND EDITION ed., ADDISON-WESLEY PUB-
+LISHING COMPANY , 1980.
+[27] J. D. Meiss, Differential Dynamical Systems, 2007.
+[28] O. FACKLER and J. T. T. VAN, 5th FORCE NEUTRINO PHYSICS, 1988.
+[29] S. L. Ross, Differential Equation (John Willey & Sons), 1984, p. 661.
+[30] K. Wakker, Fundamentals of Astrodynamics, 2015.
+29
+
diff --git a/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/load_file.txt b/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fc7cdffad5e41cc8f84cdd473746375deabf7298
--- /dev/null
+++ b/XdE0T4oBgHgl3EQfmgEE/content/tmp_files/load_file.txt
@@ -0,0 +1,1586 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf,len=1585
+page_content='Dynamics and stability of the two-body problem with Yukawa  correction to Newton’s gravity, revisited and applied  numerically to the solar system  Nawras Abo Hasan1∗, Nabil Joudieh1†and Nidal Chamoun2‡  1 Physics Department, Damascus University, Damascus, Syria  2 Physics Department, HIAST, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Box 31983, Damascus, Syria  Abstract  In this manuscript, we review the motion of two-body celestial system (planet-sun) for  a Yukawa-type correction on Newton’s gravitational potential using Hamilton’s formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  We reexamine the stability using the corresponding linearization Jacobian matrix, and verify  that the Bertrand’s theorem conditions are met for radii ≪ 1015m, and so bound closed orbits  are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Applied to the solar system, we present the equation of motion of the planet,  then solve it both analytically and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Making use of the analytical expression of  the orbit, we estimate the Yukawa strength α, and find it larger than the nominal value  (10−8) adopted in previous studies, in that it is of order (α = 10−4 − 10−5) for terrestrial  planets (Mercury, Venus, earth, Mars and Pluto) whereas it is even larger (α = 10−3) for  the Giant planets (Jupiter, Saturn, Uranus and Neptune).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Taking as inputs (rmin, vmas, e)  observed by NASA, we analyse the orbits analytically and numerically for both the estimated  and nominal values of α, and determine the corresponding trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For each obtained  orbit we recalculate the characterizing parameters (rmin, rmax, a, b, e) and compare their  values according to the used potential (Newton with/without Yukawa correction) and to the  method used (analytical and/or numerical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' When compared to the observational data, we  conclude that the correction on the path due to Yukawa correction is of order of and up to  80 million km (20 million km) as a maximum deviation occurring for Neptune (Pluto) for  nominal (estimated) value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Keywords: gravitational two-body problem, Yukawa potential, closed orbit  ’  0  Introduction  The past several years have witnessed a resurgence of interest in experimental testing of gravity,  particularly in the possibility of deviations from the predictions of Newtonian gravity, which is  considered as an excellent approximation of General Relativity (GR) on large distance scale [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Many theoretical models suggest the existence of new, relatively weak, intermediate-range force  coexisting with gravity such that the net resulting interaction would behave like a new correction  ∗seagull18990@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='com  †njoudieh@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='fr  ‡nidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='chamoun@hiast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='sy  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02498v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='EP]  6 Jan 2023  to the potentials defining the gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' It is known [2] that there are only two types  of central potentials, namely the Newton 1r and the Harmonic r2 potentials, where ANY finite  motion of an object, subject to this central potential, leads to a closed path (Bertrands theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  There are some ‘exceptions’ to this statement, in the sense that there might be closed bound  trajectories for a central potential different from the Newton and Harmonic ones, which have  been studied in [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In this contribution, we revisit the effect of a Yukawa correction to the  gravitational force over large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Theories of massive gravity [5,6], adding a mass term to the graviton (the carrier of gravity),  have raised a wide interest and the Yukawa potential is the popular parametrization of such  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Actually, many works describing deviations from Newtons inverse square law have  addressed the Yukawa-type correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Assuming gravity is exerted by exchanges of gravitons,  it is clear that a test for the graviton mass (µg) is to ask whether the Newton (1/r) potential  shows any evidence of dying at large distances because of Yukawa exponential cutoff (e−µgr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Since the seventies [7], bounds on the gravitons mass (µg ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1 × 10−29 eV) were used to  put a bound on its compton wavelength considered as a distance scale for Yukawa correction  (λ ∼ 2π  µg ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7 MPc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The authors of [8] gave a bound on the Yukawa range (λ) in the order of  (101 − 104 AU) corresponding to (µg ≤ 10−24) eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Theories like Scalar-Tensor-Vector Gravity Theory [9] predict a Yukawa-like fifth force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The  authors of [10], showed that screened modified gravity can suppress the fifth force in dense regions  and allow theories to evade the solar system and laboratory tests of the weak equivalence princi-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In [11], an extended theory of gravity, with a modified potential including post-Newtonian  terms, whose expansion is different from that of Yukawa correction, called ‘vacuum bootstrapped  Newtonian gravity’, was subjected to solar system tests, through a procedure which was applied  to Yukawa corrections at the Galactic center [12], with no significant deviations from GR found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  In [13], a Keplerian-type parametrization was shown as a solution of the equations of motion  for a Yukawa-type potential between two bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In fact, the two-body solution for alternative  theories yield a strong constraint for solar system [14, 15], whereas several analyses of Yukawa  potential for a 2-body system in different contexts were carried out [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The orbit of a  single particle moving under Yukawa potential was studied in [18], and the precessing ellipse  type orbits were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In [19], it was noted that the modified gravity with Yukawa-like  long-range potential was (un)successful on astrophysical scales (in solar system), whereas an  analysis of Yukawa potential in f(R) gravity was given in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The work of [21] showed that  a Yukawa fifth force is expected to be sub-dominant in satellite dynamics and space geodesy  experiments, as long as they are performed at altitudes greater than a few hundred kilometres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  The Yukawa strength was estimated in [22] to be (α < 10−5-10−8) for distances of order 109  cm, whereas the use of laser data from LAGEOS satellites yield a constraint on α of the order  of 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  In this letter, we build on work from [23], in which the dynamics and stability of the two body  problem with a Newtonian potential corrected by a Yukawa term were explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In particular,  we reproduced their analytical results and applied them to the study of all the planets of our solar  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Solving analytically the planet equation of motion, one finds an elliptical trajectory,  which one can also obtain numerically using Runge-Kutta method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Starting from the observed  values of the perihelion distance and velocity (rmin, vmax) and of the tranjectory eccentricity e,  stated in NASA public results [24]∗, one could determine the ellipsis equation and estimate, for  ∗Although the standard deviations of the planetary trajectories are not quoted in the NASA public website,  however one can consider that the corresponding error equals to the last digit of the quoted significant numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  2  Yukawa corrected potential, the Yukawa strength α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' One can use this estimated value, or another  nominal value taken from other studies, to either draw the analytical trajectory and recalculate  the characterizing parameters: the shortest (longest) distance to the Sun rmin(rmax), the semi  major (minor) axis a(b) and the eccentricity e, or to solve numerically the equations of motion  with the Yukawa-corrected potential in order to check the closedness of the resulting trajectory,  whose characteristics are to be reevaluated again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Later, we compared these results with those  calculated for the Keplerian motion of planets subject to the pure Newtonian potential, and, in  addition, showed the compatibility of the results with the observational NASA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  More specifically, for the two-body system (planet-sun), the Newtonian potential is given by:  VN(r) = −GmpM⊙  r  (1)  where G = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='674×10−11 Nm2  Kg2 is the gravitational Newton constant, mp (M⊙) is the planet (sun)  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' With a Yukawa correction, the gravitational potential becomes  V (r) = −GmpM⊙  r  �  1 + αe− r  λ  �  = VN(r) + VY k(r)  (2)  where VY k is the Yukawa correction to the Newtonian potential and α (λ) represents the  strength (range) of the Yukawa correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Previous studies [23, 25] gave the nominal values  (α = 10−8(λ = 103AU = 1015m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' However, our estimations gave a larger order of magnitude for  the Yukawa strength: α ∼ 10−4 − 10−5 for terrestrial planets (Mercury, Venus, Earth, Mars and  Pluto) and α ∼ 10−3 for the remaining Giant planets (Jupiter, Saturn, Uranus and Neptune),  which are in line with [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  We saw that for estimated α, the maximum deviation from observed data, which increases  the further the planet is (20 million km in Pluto), is less than that of the α nominal value  (80 million km in Neptune), which is plausible considering that the estimation of α is done by  identifying the factor containing it to observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  For each of the nominal and estimated values of α, we analysed the planet’s trajectory  both analytically and numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Analytics wise, we started from the observational data of  NASA (rmin, vmax, e) and reconstructed the closed ellipse trajectory of which we re-evaluated  the characteristics (rmin, rmax, a, b, e) and compared with the pure Newton case and with the  observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Numerics wise, the α determines the potential under which the planet moves,  and so one can solve the equations of motion numerically using Runge-Kutta method taking as  initial conditions the observed data of (rmin, vmax), to check that one gets closed trajectories in  excellent agreement with the elliptical shapes, of which we can evaluate the characteristics that  one compares to the pure Newtonian case, to the analytical method results and to the observed  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  The manuscript is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  In section (1), we revise the system dynamics  using Hamilton’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In section(2), we state the types of stability and determine the one  corresponding to the system under study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We discuss, in section(3) and following [23], Bertrand’s  theorem and get the analytical solution to the equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Finally, we apply in section  (4) the obtained approximative analytical results to the study of the solar system planets in  order to estimate the Yukawa strength and re-determine the trajectory characteristics for both  estimated and nominal values of α, as well as solve numerically the equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The results,  of comparing the analytical/numerical outputs with the observed data according to the used  In our computations, we used the whole digits allowed by machine precision, however the results in the appendices  tables showed only significant digits equal to those of the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  3  potential, are presented in form of plots for all the planets, whereas the corresponding tables  are given in an appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We end up with conclusions in section (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  1  Hamiltonian formulation  We start with the Hamiltonian H = T + V where T is the kinetic energy of both masses and V  is the Gravitational potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  H =  ⃗p2  1  2mp  +  ⃗p2  2  2M⊙  −  K  |⃗r2 − ⃗r1|  �  1 + αe− |⃗r2−⃗r1|  λ  �  (3)  where ⃗ri, (⃗vi), i = 1, 2 are the positions (velocities) of the two masses with corresponding mo-  menta p1 = M⊙v1, p2 = mpv2, K = GmpM⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Changing to the center of mass frame (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='m),  with  ⃗r1 = +  mp  mp + M⊙  ⃗r = + µ  M⊙  ⃗r + ⃗R  ,  ⃗r2 = −  M⊙  mp + M⊙  ⃗r = − µ  mp  ⃗r + ⃗R  (4)  ⃗r = ⃗r1 − ⃗r2  ,  ⃗R = M⊙⃗r1 + mp⃗r2  mp + M⊙  (5)  ⃗v1 = ˙⃗R +  µ  M⊙  ⃗v  ,  ⃗v2 = ˙⃗R + −µ  mp  ⃗v  (6)  ⃗v = ˙⃗r  ,  ⃗p = µ⃗v,  (7)  ¨⃗R = ⃗0  ,  µ¨⃗r = M⊙¨⃗r1 = −mp¨⃗r2,  (8)  we get  H = 1  2(M⊙ + mp) ˙⃗R2 + H  :  H = p2  2µ − K  r  �  1 + αe− r  λ  �  (9)  Here we have defined µ =  mpM⊙  mp+M⊙ as the reduced mass of the system and r = |⃗r|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We switch to  polar coordinates in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='m to get  H = 1  2µ  �  p2  r + p2  ϕ  r2  �  − K  r  �  1 + αe− r  λ  �  (10)  From the canonical equations ( [26]): ˙qi =  �  ∂H  ∂pi  �  , ˙pi = −  �  ∂H  ∂qi  �  , and since the Hamiltonian is  cyclic in ϕ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' it does not depend explicitly on ϕ), we have:  ˙ϕ = ∂H  ∂pϕ  = pϕ  µr2  (11)  ˙pϕ = −∂H  ∂ϕ = 0 ⇒ pϕ = µr2 ˙ϕ = ℓ = constant  (12)  where ℓ is the angular momentum of the two-body system, and therefore Hamiltons equations  for r become:  ˙r = ∂H  ∂pr  = pr  µ  (13)  ˙pr = −∂H  ∂r = ℓ2  µr3 − K  r2  �  1 + α  �  1 + r  λ  �  e− r  λ  �  (14)  4  Figure 1:  The reduced potential (red line) given for fixed angular momentum (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The  pink line denotes the magnitude of the purely Yukawa term (  ���− αK  r e− r  λ  ���), whereas the blue line  represents the Keplerian reduced potential, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 16 without the Yukawa term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Again, and since H(t) = H(t0) = h is constant during the motion of the masses [26], and since  p2  r = µ2 ˙r2 ≥ 0 we get a lower bound for the total energy of the system:  h ≥  ℓ2  2µr2 − K  r  �  1 + αe− r  λ  �  (15)  The right hand side of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' (15) is defined to be the “reduced potential”, which is common in the  Kepler problem moving from two degrees of freedom to only one (with the Yukawa correction)  Vred(r) =  ℓ2  2µr2 − K  r  �  1 + αe− r  λ  �  (16)  One can draw the function for fixed ℓ giving the allowed regions of motion (look at figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Note that µ > 0, λ > 0 and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  2  The linearization matrix  Following [27], in order to determine the stability of the equilibrium points of the system, we  must form a matrix differential equation using the system equations of motion (Hamiltons Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  13 and 14 for r, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The linear system has the form:  d  dt  �  r  pr  �  =  �  f(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' p)  g(r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' p)  �  =  �  f0  g0  �  eq  +  � ∂f  ∂r  ∂f  ∂pr  ∂g  ∂r  ∂g  ∂pr  � �  r  pr  �  where f (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' pr) = pr  µ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  g (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' pr) = ℓ2  µr3 − K  r2  �  1 + α  �  1 + r  λ  �  e− r  λ  �  (17)  Given that λ = 1015m for orbits of size comparable to the solar system dimensions [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' one can  assume that r  λ is small enough that one can Taylor expand the exponential and ignore terms of  5  70  Vred  09  Vkep  Vyuk  50  40  >  30  20  10  0  10�  r2  λ2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' leading to:  e− r  λ ≈ 1 − r  λ + O  � r2  λ2  �  ≈ 1 − r  λ  (18)  Thus  g (r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' pr) = ℓ2  µr3 − K  r2  �  1 + α  �  1 + r  λ  � �  1 − r  λ  ��  ≈ ℓ2  µr3 − K  r2 (1 + α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  with the Yukawa effect within this approximation being limited to replacing K by K(1+α),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' which  tells that the potential shape is still Newtonian (1/r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' and according to Bertrand’s theorem every  bound trajectory is thus closed for small r/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' One can see this fact directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' (16) as  it gives, compared to the Keplerian potential, within the approximation just a shift, in addition  to the replacement (K → K(1 + α)), which does not interfere in the equations of motion:  Vred(r)  ≈  ℓ2  2µr2 − K  r (1 + α) + Kα  λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (19)  Consequently, the Jacobian matrix takes the form:  �  ˙r  ˙pr  �  =  �  0  1  µ  −3ℓ2  µr4 + 2K  r3 (1 + α)  0  � �  r  pr  �  (20)  where terms of order O  �  r2  λ2  �  were ignored, and where the equilibrium point (r, pr)eq satisfies  feq(r, pr) = geq(r, pr) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We can determine the r at equilibrium using (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 14) to get upto  leading order:  req =  ℓ2  µK(1 + α)  (21)  We can now test for stability by choosing values of (α, µ, K, ℓ, λ) and finding the eigenvalues  of the Jacobian matrix (20) after substituting the equilibrium solution found above (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Recall that the eigenvalues β1, β2 are found by solving the following equation:  det |J − βI2×2| = 0  (22)  with I2×2 referring to the 2 × 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Thus we have  �����  −β  1  µ  −3ℓ2  µr4 + 2K  r3 (1 + α)  −β  ����� = 0  (23)  The characteristic equation (the eigenvalue equation) becomes:  β1,2 = 1  2  �  τ ±  �  τ 2 − 4∆  �  (24)  τ = trace(J) = 0  (25)  ∆ = det(J) = µ2K4(1 + α)4  ℓ6  (26)  Following [29], the stability is determined by the sign of the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Since ∆ > 0, we have  the following cases:  τ < 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) a stable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  τ < 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a stable spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  6  τ > 0, τ 2 − 4∆ > 0 ⇒ (r0, pr0) an unstable node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  τ > 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) an unstable spiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  τ = 0, τ 2 − 4∆ < 0 ⇒ (r0, pr0) a neutrally stable center (which is our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Actually, the stability refers to how the solution behaves near the equilibrium point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' in that  unstable solutions grow to infinity, whereas stable solutions tend to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Also, it is the imaginary  cases which are the ones giving bound orbital solutions (specifically the center case, whereas the  stable and unstable imaginary cases are bound solutions tending towards or away from zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  3  Stability & Bertrands theorem  First, we rewrite the eigenvalue equation in the form  β2 + µ2K4(1 + α)4  ℓ6  = 0  (27)  leading to:  β = ±iµK2(1 + α)2  ℓ3  (28)  We note that one can study the case for a purely Newtonian Potential by letting α → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Similarly, by ignoring the terms derived from the Newtonian potential, one can single out the  pure Yukawa contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In these two extreme cases, the characteristic equations becomes  Pure Newtonian: β2 + µ2K4  ℓ6  = 0  (29)  Pure Yukawa: β2 + µ2K4α4  ℓ6  = 0  (30)  giving  Pure Newtonian: β = ±iµK2  ℓ3  (31)  Pure Yukawa: β = ±iµK2α2  ℓ3  (32)  Thus, the equilibrium points for the purely Newtonian, the purely Yukawa, and the Newton  plus Yukawa Potentials remain center solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' This implies that the motion would remain  restricted to ellipses about the equilibrium point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' and so, orbits near the equilibrium point are  possible (further away from the equilibrium point one would have unbounded solutions, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  1 shows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' This proves that for small r/λ we have stable, closed orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  For the Keplerian orbit equation, it can be written as:  d2u  dϕ2 + u = − µ  ℓ2  d  duV  �1  u  �  (33)  where u = 1r denotes the Binet transformation, giving, for small r/λ, the following differential  equation:  d2u  dϕ2 + u = +µK  ℓ2 (1 + α)  (34)  whose solution is given by  u(ϕ) = 1  r = A [1 + e cos (ϕ − ϕ0)] : A = µK  ℓ2 (1 + α)  (35)  7  with e is the eccentricity of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The purely Newtonian and purely Yukawa cases follow  respectively from (34)  Newtonoian: u(ϕ) = 1  r = µK  ℓ2 [1 + e cos (ϕ − ϕ0)]  (36)  Purely Yukawa: u(ϕ) = 1  r = µKα  ℓ2  [1 + e cos (ϕ − ϕ0)]  (37)  Finally, in order to satisfy Bertrands theorem, the following condition should be satisfied  d2Vred(r)  dr2  ����  r=r0  > 0  (38)  where the reduced potential is given by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' With the approximations of (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 18)) and ignoring  terms of order O  �  r2  λ2  �  this condition becomes  d2Vred(r)  dr2  ����  r=r0  = µ2K4(1 + α)4  ℓ6  > 0  (39)  which is true, since α, µ, K, ℓ > 0, in general and in the special cases of Newtonian (α = 0)  and purely Yukawa potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' This shows that the Yukawa plus Newtonian potential satisfies  Bertrands theorem for small rλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  4  Application to the solar system  We present here our results consisting of determining first the parameters of the models (rmin, rmax, a, b, e)  by comparing the previous approximative analytical solutions with the NASA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Then, we  solved the equations of motion numerically using Matlab and the fourth-order Runge-Kutta  method with no approximation so that to be compared with the analytical solutions and with  the observed NASA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We applied this for all the planets of the solar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For each pair  (sun-planet) we used the following values M⊙ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9885 × 1030kg, αnominal = 10−8, λ = 1015m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  We list in Table (1) the initial conditions used in the analytical and numerical calculations (the  period τ is used only in the numerical solution to determine the corresponding ‘step’):  MERCURY  VENUS  EARTH  MARS  JUPITER  SATURN  URANUS  NEPTUNE  PLUTO  mp(×1024kg)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3302  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8673  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9722  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64169  1898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='13  568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='32  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='811  102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='409  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01303  τ (days)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='969  224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='701  365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='256  686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='98  4332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='589  10832.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='33  30685.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4  60189  90560  rmin  (×106km)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='10748  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='147095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='20665  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740595  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='357554  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='732696  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='47105  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='434987  vmax  (×103m/s)  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='98  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='26  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='29  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='72  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='18  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='11  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  eccentricity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='20563  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00677  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01671  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='09341  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04839  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='05415  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04717  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00859  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='24881  Table 1: Initial conditions used in the calculations where mp denotes the planet mass, τ is the  orbit period, rmin is the perihelion and vmax denotes the perihelion velocity  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  Analytical Method  The analytical ellipsis equation is of the form  1  r ≡ u  =  a  b2 (1 + e cos ϕ) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (40)  8  where (for a y-axis perpendicular to the polar axis in the orbit plane)  rmin = a(1 − e)  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  rmax = a(1 + e),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (41)  e = c  a =  �  1 − b2  a2  :  c2 = a2 − b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (42)  a = rmin + rmax  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  b = ymax − ymin  2  (43)  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' analytically one can start with (rmin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' vmax,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' e) observed by NASA in [24] to compute†:  a = rmin  1 − e  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  b = a  �  1 − e2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (44)  and estimate the strength α from  µK  ℓ2 (1 + α) = a  b2  using  ℓ = rminvmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (45)  Once the analytical equation is determined, then one can plot the trajectory and recompute the  characteristics (rmin, rmax, a, b, e) using Eqs (41,42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We call this procedure the “analytical-α-  estimated” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  One can also use the nominal value of α = 10−8, and plug it in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' (40), where ℓ, e are taken  from the observed data, to re-evaluate (rmin, rmax, a, b, e) from  a =  1  A(1 − e2),  A = µK(1+α)  ℓ2  ,  b =  1  A  √  1 − e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (46)  We call this procedure the “analytical-α-nominal” approach, which can be looked at as a method  with three inputs (α, ℓ, e) instead of the three inputs (rmin, vmax, e) used in the other approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2  Numerical Method  Here, we just solve numerically, using the fourth-order Runge-Kutta method, the Newton’s law  equation of motion in the c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='m frame with initial conditions taken from NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Thus we solve  the equations:  ¨⃗r1 = Gmp  ⃗r2 − ⃗r1  r3  , Newton,  ¨⃗r2 = GM⊙  ⃗r1 − ⃗r2  r3  = −M⊙  mp  ¨⃗r1,  (47)  ¨⃗r1 = Gmp  �  (1 + αe− r  λ )1  r + α  λe− r  λ  � ⃗r2 − ⃗r1  r2  , Newton+Yukawa,  ¨⃗r2 = −M⊙  mp  ¨⃗r1,  (48)  under the initial conditions given by NASA data of (rmin, vmax):  ⃗r1(t = tmin) =  mp  mp + M⊙  ⃗rmin  ,  ⃗v1(t = tmin) =  mp  mp + M⊙  ⃗vmax,  (49)  ⃗r2(t = tmin) = −  M⊙  mp + M⊙  ⃗rmin  ,  ⃗v2(t = tmin) = −  M⊙  mp + M⊙  ⃗vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (50)  Once the trajectory is solved numerically, we check that it is closed, as the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (2) shows  for both the pure Newton and that with the Yukawa corrections (since the differences are not  visible on the figure scale containing all the planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For each obtained orbit, we recalculate  the corresponding characteristics (rmin, rmax, a, b, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  †Due to measurement errors and orbits not being perfectly elliptical, the NASA data may give slightly different  values of a using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 43 or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  9  Figure 2: Closed bound planets’ trajectories with and without Yukawa corrections with strength  α nominal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3  Results  We report in the Tables of Appendix A (from A1 to A18), the calculated characteristics of the  resulting trajectories for all the planets in the solar system, corresponding to the pure Newton  and the Newton corrected with Yukawa potentials, both in the analytical and the numerical  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The odd (even) numbered tables correspond to the nominal (estimated) Yukawa  strength α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The number of moons of each planet is determined according to [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Below we  explain the meanings of the symbols used in the tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Nnum: Numerical calculations using the Newtonian potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Nanal: Analytical calculations using the Newtonian potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RN = Nnum  Nanal %: The percentage ratio of the numerical to the analytical results for Newton  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (N + Y K)num : Numerical calculations using the modified potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  (N + Y K)anal: Analytical calculations using the modified potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RN+Y K = (N+Y K)num  (N+Y K)anal %: The percentage ratio of the numerical to the analytical results  for modified potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RN−Obs  num  = Nnum/Obs%: Percentage ratio of the numerical results, using the Newtonian  potential, to the observed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RN−Obs  anal  = Nanal/Obs %: Percentage ratio the analytical results, using the Newtonian  potential, to the observed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RY K−Obs  num  = (N + Y K)num/Obs %: Percentage ratio of the numerical results, using the  modified potential, to the observed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  10  (km)• RY K−Obs  anal  = (N + Y K)anal/Obs %: Percentage ratio of the analytical results, using the  modified potential, to the observed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  In order to summarize the findings of the Tables, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' (3) plots showing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' for each  planet and at every polar angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' the deviation from unity of the ratio between two quantities of  the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' allowing thus to compare the effects of the considered potential (Newton vs New-  ton+Yuakawa) and/or the used method (numerical vs analytical) and/or the Yukawa strength  determination (nominal vs estimated):  rn(num) representing the trajectory equation of the numerical approach with Newton  potential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  rn(anl) representing the trajectory equation of the analytical approach with Newton po-  tential,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  ryk(num) representing the trajectory equation of the numerical approach with New-  ton+Yukawa potential and nominal α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  ryk(anl) representing the trajectory equation of the analytical approach with Newton+Yukawa  potential and nominal α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  ryka(num) representing the trajectory equation of the numerical approach with New-  ton+Yukawa potential and estimated α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  ryka(anl) representing the trajectory equation of the analytical approach with New-  ton+Yukawa potential and estimated α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  We see that some ratios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' the dashed red and sky blue) do coincide near zero deviation  from one, meaning no tangible effect of adding the Yukawa correction, be it in the analytic or  the numeric method, as long as one takes the nominal value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Also,we note local extremums  for the deviations from unity at polar angles multiples of π/2 as a generic feature in many plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  One can interpret the large values of the deviations for the nearest (Mercury) and the farthest  (Pluto) planet, in that for the former;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' the perturbative effect of solar winds, important as we  approach the sun, was not taken into consideration, whereas for the furthest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' accumulating  gravitational screening effects of the other planets and their moons, which were not considered  in the study, are becoming important especially for a small sized- planetoid like Pluto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  In order to show the effects of the separating distance effect, one should compute the absolute  deviations from observed data for each planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In appendix B, the Tables B1, B2 (B3, B4), report  the deviation from observation for each planet of rmax, rmin respectively, in the case of nominal  (estimated) α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We summarize these findings in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We see that the agreement between  the numerical and analytical solutions is excellent in both estimated and nominal α cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We  see that the deviations due to Yukawa correction are not large, but note the following:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For estimated α:  rmax-deviation: The numerical deviation is larger by about 103 times the analytical  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' In general, it increases the further the planet is, and reaches a maximum  of order (−25 million km) (less than the observed value) in Pluto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  rmin-deviation: Again, the numerical deviation is larger by about (101-102)-order of  magnitude than analytical deviation, where it is largest in Neptune (−8 million km),  however it reverses sign and becomes (+5 million km) more than the observation in  Pluto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  11  1-{ rn(num)/ryka(anl) }        ____  1-{ryka(anl)/ryk(num)}      ____  1-{ryka(anl)/ryka(num)}    ____  1-{ rn(anl)/ryka(anl)}           ____  1-{ryka(anl)/ryk(anl)}         ____  1-{ rn(num)/ryk(num)}    -----  1-{ryka(num)/rn(num)}   -----  1-{ryk(num)/ryka(num)} -----  1-{ryk(anl)/rn(anl)}          -----  Figure 3:  Deviations from Unity for Ratios of Computed trajectories at each polar angle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  according to the considered potential (Newton vs Newton+Yuakawa) and/or to the used method  (numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  We show in a zoomed region, for one planet (Earth) generic case, that the dashed red and sky  blue curves are very near each other (the same applies to the green and blue curves in Mercury  case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  12  VENUS  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02  1-Percentage Ratio(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='03  0  50  100  150  200  250  300  350  400  angle (deg)EARTH  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='06  1-Percentage Ratio(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08  0  50  100  150  200  250  300  350  400  angle (deg)MARS  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  2  0  50  100  150  200  250  300  350  400  angle (deg)JUPITER  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4  1-Percentage Ratio(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6  0  50  100  150  200  250  300  350  400  angle (deg)SATURN  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  3  2  1  2  3  0  50  100  150  200  250  300  350  400  angle (deg)URANUS  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  1-Percentage Ratio(%)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0  50  100  150  200  250  300  350  400  angle (deg)Neptune  Deviationsfrom unityforRatiosof ComputedTrajectories(%)  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='angle (deg)Pluto  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Deviationsfrom unityforRatiosof ComputedTrajectories(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1-Percentage Ratio(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='angle (deg)MERCURY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Deviationsfrom unityforRatiosof ComputedTrajectories(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1-PercentageRatio(%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='350  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='angle (deg)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='MERCURY  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Venus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='EARTH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='MARS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='JUPITER  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='SATURN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='URANUS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Neptune  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Pluto  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Deviations from the observed values for rmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' alpha nominal  Absolute deviation from observation numerically  ×10^6(km)  Absolute deviation from  observation analytically  ×10^6(km)  20  0  20  40  60  MERCURY  Venus  EARTH  MARS  JUPITER  SATURN  URANUS  Neptune  Pluto  Deviations from the observed values for rmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' alpha nominal  Absolute deviation from observation numerically  ×10^6(km)  Absolute deviation from  observation analytically  ×10^6(km)  30  20  10  0  10  MERCURY  Venus  EARTH  MARS  JUPITER  SATURN  URANUS  Neptune  Pluto  Deviations from the observed values for rmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' alpha estimated  Absolute deviation from observation numerically  ×10^6(km)  Absolute deviation from  observation analytically  ×10^6(km)  10  5  0  5  10  MERCURY  Venus  EARTH  MARS  JUPITER  SATURN  URANUS  Neptune  Pluto  Deviations from the observed values for rmin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' alpha estimated  Absolute deviation from observation numerically  ×10^6(km)  Absolute deviation from  observation analytically  ×10^6(km)  Figure 4: Absolute deviations from observed data for each planet, according to the used method  (numerical vs analytical) and/or to the Yukawa strength determination (nominal vs estimated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For nominal α:  rmax-deviation: The numerical deviation is larger than the analytical one, but are of  the same order reaching a maximum of +80 (+40) million km using the numerical  (analytical) method in Neptune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For Pluto and Uranus, we get (−40) million km in  the numerical method (less than observed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  rmin-deviation: The analytical deviation is larger, and sometimes reverses sign com-  pared to the numerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For example, in Neptune the analytical approach gives a  deviation of (+40 million km) from observation, whereas the numerical one gives a  deviation of −5 million km (less than the observed value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Actually, the disagreements with observations are due to several reasons: the first one is physical  in nature, in that it results from neglecting the perturbation due to third bodies, or, more  generally, the effect of the natural satellites, such as moons or asteroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Also, we did not  either take into account the radiation and the solar wind physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Moreover, the results  were obtained as a 2-body problem, and hence the movement of more distant planets might be  affected by planets closer to the sun, which can be present not in the dominant term, but in  higher orders of expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' The second factor lies in the computational side, and concerns the  numerical method used, the value of the step size, and the high sensitivity of the problem to  the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' One should also mention that for the analytical solution we restricted the  study to leading order neglecting higher orders in the expansion of exponentials, whereas for the  numerical solution the entire exponential is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  5  Summary and Conclusion  In this work, we followed [23] and used the Hamilton’s formulation in order to obtain the  differential equation of motion and the path equation for the gravitational two-body system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  The developments are carried out in the case of the pure Newtonian potential, the Newtonian  corrected with Yukawa type potential and the pure Yukawa potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  As in [23], we have  reviewed the stability problem, constructed the linearization matrix and tested the stability  of the system for a Yukawa correction, and found that it is of a central solution type, which  implies stable solutions near the fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We repeated the analysis for a purely Yukawa force  and found similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We also confirmed that the modified potential obeys the Bertrands  theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Then, we determined the parameters’ set corresponding to the planets of the solar system  starting from the observed (rmin, vmax, e) estimating α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' For both the estimated and nominal  values of α, we determined the characteristics of the trajectories numerically and analytically,  and compared between the methods and with the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' We explained the extent to  which these results are consistent with the observational data, presenting in form of histograms  the absolute deviations from observations, which were found to give an upper deviation of order  80 million km in Neptune using nominal α, and 20 million km in Pluto using estimated α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  Acknowledgments:  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Chamoun acknowledges support from the ICTP-Associate pro-  gram, from the Humboldt Foundation and from the CAS-PIFI scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  14  Appendices  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Tables of Calculated/Observed Parameters of the Planets  15  Mercury  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='832  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='916  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='67679158  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='205744228  Nanal  47  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='043  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='756  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='47840586  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='205646344  RN = Nnum  Nanal %  97  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='930  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='91918025  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='95242442  (N + Y K)num  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='831  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='916  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='67678243  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='205738221  (N + Y K)anal  47  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='043  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='756  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='47840528  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='205646344  RN+Y K  = (N+Y K)num  (N+Y K)anal %  97  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='930  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='91916556  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='95534277  Observation  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='818  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='909  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='20563069  RN−Obs  num  = Nnum/Obs %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='980  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='988  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94481595  RN−Obs  anal  = Nanal/Obs %  97  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='911  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='909  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9923879  RY K−Obs  num  =  (N + Y K)num/Obs %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='981  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='988  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94773407  RY K−Obs  anal  =  (N + Y K)anal/Obs %  97  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='911  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='909  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9923879  nominal α = 10−8  Table A1: The values of the calculated and observational astronomical parameters of the planet  Mercury whose number of moons is 0  Mercury  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='623  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='826  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65144795  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2022  Nanal  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='819  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='912  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='67470066  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2056  RN = Nnum  Nanal %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='719  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='851  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='95897162  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3743  (N + Y K)num  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='613  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='820  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64729064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2022  (N + Y K)anal  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='815  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='908  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6714469  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2056  RN+Y K  = (N+Y K)num  (N+Y K)anal %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='711  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='847  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9573749  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3408  Observation  46  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='818  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='909  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2056  RN−Obs  num  = Nnum/Obs %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='721  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='856  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3817  RN−Obs  anal  = Nanal/Obs %  100  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='005  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0075  RY K−Obs  num  =  (N + Y K)num/Obs %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='706  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='847  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3482  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='995  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0075  estimated α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='741444131301954 × 10−5  Table A2: The values of the calculated and observational astronomical parameters of the planet  Mercury whose number of moons is 0  16  Venus  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='30  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='689  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9982364  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0044  Nanal  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='48  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='961  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='22  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2222348  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0072  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='750  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79301998  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8187  (N + Y K)num  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='30  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='689  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9982353  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0044  (N + Y K)anal  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='48  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='961  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='22  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2222337  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0072  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='750  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79301998  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8189  Observation  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='48  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='941  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0068  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='769  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7150  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='018  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4063  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='769  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7152  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='018  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4063  nominal α = 10−8  Table A3: The values of the calculated and observational astronomical parameters of the planet  Venus whose number of moons is 0  Venus  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='30  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='689  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9982364  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0044  Nanal  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='48  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='961  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='22  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2222348  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0072  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='750  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79301998  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8187  (N + Y K)num  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='30  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='658  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='98  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9821956  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0045  (N + Y K)anal  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='47  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='945  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='20  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2068155  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0072  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='84  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='736  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='78  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='79241613  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0300  Observation  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='48  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='941  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0068  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='769  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7150  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='018  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4063  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='83  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='78  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0680  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='004  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4063  estimated α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='424988220126711 × 10−4  Table A4: The values of the calculated and observational astronomical parameters of the planet  Venus whose number of moons is 0  17  EARTH  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='884  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='336  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='319847  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0156  Nanal  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='126  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='625  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6034965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0168  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='806  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81039915  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5721  (N + Y K)num  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='884  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='336  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3198455  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0156  (N + Y K)anal  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='126  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='625  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='603495  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0168  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='806  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81039915  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5720  Observation  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='095  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='598  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0167  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8572  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='825  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5903  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='978  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='981  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9120  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='825  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5902  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='978  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='981  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9120  nominal α = 10−8  Table A5: The values of the calculated and observational astronomical parameters of the planet  Earth whose number of moons is 0  EARTH  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='884  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='336  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='319847  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0156  Nanal  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='126  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='625  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6034965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0168  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='806  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81039915  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5721  (N + Y K)num  146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='883  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='307  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2910008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0154  (N + Y K)anal  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='099  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='597  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5762082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01688  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='853  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='805  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80932302  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4700  Observation  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='095  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1  149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='598  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0167  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8572  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='825  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5903  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='978  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='981  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9120  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='856  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='7  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='805  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4761  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0999  estimated α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='824376359731428 × 10−4  Table A6: The values of the calculated and observational astronomical parameters of the planet  Earth whose number of moons is 0  18  MARS  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='57  248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='480  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='52  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6509159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0898  Nanal  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='277  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9631182  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='680  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8624436  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0965  (N + Y K)num  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='57  248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='480  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='52  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6509134  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0898  (N + Y K)anal  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='277  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9631159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='680  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='86244351  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0965  Observation  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='261  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='687  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1252  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='006  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0298  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='687  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1252  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='006  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0298  nominal α = 10−8  Table A7: The values of the calculated and observational astronomical parameters of the planet  Mars whose number of moons is 0  MARS  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='57  248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='480  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='52  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6509159  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0898  Nanal  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='277  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9631182  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='680  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8624436  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0965  (N + Y K)num  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='57  248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='425  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='49  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6249874  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0897  (N + Y K)anal  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='62  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='252  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9402451  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='97  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='668  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='86108339  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9861  Observation  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='261  227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0935  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='687  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1252  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='006  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0298  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='96  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='664  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='80  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0147  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='98  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='996  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='99  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0298  estimated α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='007889331583467 × 10−4  Table A8: The values of the calculated and observational astronomical parameters of the planet  Mars whose number of moons is 0  19  JUPITER  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='533  777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='717  776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9190412  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  Nanal  742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='542  818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='568  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='555  779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='626266  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04873  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='644  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='629  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='636  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65275352  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3711  (N + Y K)num  739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='533  777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='717  776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9190329  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  (N + Y K)anal  742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='542  818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='568  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='555  779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6262582  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0487  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='644  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='629  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='636  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65275345  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3711  Observation  740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='363  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='479  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0487  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='906  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='898  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4399  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='262  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='270  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='266  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0714  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='906  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='898  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4399  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='262  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='270  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='266  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0714  nominal α = 10−8  Table A9: The values of the calculated and observational astronomical parameters of the planet  Jupiter whose number of moons is 0  JUPITER  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='533  777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='717  776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9190412  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  Nanal  742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='542  818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='568  780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='555  779.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='626266  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='04873  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='644  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='629  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='636  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='65275352  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3711  (N + Y K)num  739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='837  810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='932  775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='385  774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6852056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0441  (N + Y K)anal  740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='567  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='390  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='478  777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='5526264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0487  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='901  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='331  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='602  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='63122486  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6263  Observation  740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='363  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='479  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0487  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='906  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='898  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4399  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='262  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='270  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='266  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0714  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='897  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='334  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='602  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6911  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='996  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0714  estimated α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='666880127522 × 10−3  Table A10: The values of the calculated and observational astronomical parameters of the planet  Jupiter whose number of moons is 0  20  SATURN  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  1355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='461  1523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='344  1439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='403  1437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455093  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='055  Nanal  1368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='378  1518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='496  1443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='481829  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='056  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='319  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='720  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='72065302  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='042  (N + Y K)num  1355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='461  1523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='344  1439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='403  1437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455078  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='055  (N + Y K)anal  1368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='378  1518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='496  1443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='481815  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='056  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='319  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='720  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='72065294  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='042  Observation  1357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='554  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='527  1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='041  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='845  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='116  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='514  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='081  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='797  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='794  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='795  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='036  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='845  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='116  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='514  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='081  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='797  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='794  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='795  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='036  nominal α = 10−8  Table A11: The values of the calculated and observational astronomical parameters of the planet  Saturn whose number of moons is 0  SATURN  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  1355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='461  1523.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='344  1439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='403  1437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455093  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='055  Nanal  1368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='378  1518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='496  1443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  1441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='481829  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN = Nnum  Nanal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='056  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='319  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='720  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='72065302  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='042  (N + Y K)num  1354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='869  1497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='652  1426.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='261  1424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='954776  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='046  (N + Y K)anal  1357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='574  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='507  1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='040  1430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100672  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='800  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='412  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='596  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='64017246  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='244  Observation  1357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='554  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='527  1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='041  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='052  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='845  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='116  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='514  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='081  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='797  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='794  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='795  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='036  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='802  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='410  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='596  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='277  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='998  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='036  estimated α = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='958291053541 × 10−3  Table A12: The values of the calculated and observational astronomical parameters of the planet  Saturn whose number of moons is 0  21  URANUS  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  2957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  2843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='519  2841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='649275  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0381  Nanal  2717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='213  2984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='63  2850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  2847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='766462  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN = Nnum  Nanal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='78519352  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2504  (N + Y K)num  2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  2957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  2843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='519  2841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='649245  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0381  (N + Y K)anal  2717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='213  2984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='63  2850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  2847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='766434  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN+Y K  = (N+Y K)num  (N+Y K)anal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='78519344  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2504  Observation  2732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='696  3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='39  2867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='043  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='886  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='53  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='179  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3684  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='433  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1452  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='886  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='53  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='179  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3684  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='433  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1452  nominal α = 10−8  Table A13: The values of the calculated and observational astronomical parameters of the planet  Uranus whose number of moons is 0  URANUS  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  2729.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  2957.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  2843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='519  2841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='649275  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0381  Nanal  2717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='213  2984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='63  2850.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  2847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='766462  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN = Nnum  Nanal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='455  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='78519352  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2504  (N + Y K)num  2730.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='116  2992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='91  2861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='516  2858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='935401  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0441  (N + Y K)anal  2732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='578  3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='50  2867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='042  2863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='869614  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='909  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='71  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='807  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='82770818  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0932  Observation  2732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='696  3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='39  2867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='043  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0469  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='886  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='53  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='179  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3684  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='433  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='44  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='437  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1452  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='905  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='71  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='807  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2299  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='995  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='1452  estimated α = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='622864957252 × 10−3  Table A14: The values of the calculated and observational astronomical parameters of the planet  Uranus whose number of moons is 0  22  Neptune  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  4464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  4634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='099  4549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='454  4548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='810665  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0177  Nanal  4512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='97  4601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='381  4557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='176  4556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='953752  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN = Nnum  Nanal %  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='711  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='830  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='82130416  180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8744  (N + Y K)num  4464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  4634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='098  4549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='454  4548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='810617  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0177  (N + Y K)anal  4512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='97  4601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='381  4557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='176  4556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='953706  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN+Y K  = (N+Y K)num  (N+Y K)anal %  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='711  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='830  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='82130411  180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8743  Observation  4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='05  4558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  4514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='953  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='86  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='764  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6474  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='932  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='935  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9802  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='86  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='764  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6473  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='932  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='935  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9802  nominal α = 10−8  Table A15: The values of the calculated and observational astronomical parameters of the planet  Neptune whose number of moons is 0  Neptune  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  4464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  4634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='099  4549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='454  4548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='810665  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0177  Nanal  4512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='97  4601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='381  4557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='176  4556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='953752  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN = Nnum  Nanal %  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='711  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='830  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='82130416  180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='8744  (N + Y K)num  4463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='01  4546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='479  4504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='745  4504.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='517794  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0096  (N + Y K)anal  4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='15  4558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='747  4514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='952  4514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='73215  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN+Y K  = (N+Y K)num  (N+Y K)anal %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='81  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='730  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='773  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='77375499  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6587  Observation  4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='05  4558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  4514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='953  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0097  RN−Obs  num  = Nnum/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='86  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='764  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6474  RN−Obs  anal  = Nanal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='93  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='932  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='935  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9802  RY K−Obs  num  =  (N + Y K)num/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='82  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='728  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='773  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='6259  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='997  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='9802  estimated α = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='351961741362 × 10−3  Table A16: The values of the calculated and observational astronomical parameters of the planet  Neptune whose number of moons is 0  23  Pluto  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  4439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='709  7265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='423  5852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='566  5684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326067  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2397  Nanal  4431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='722  7298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='614  5865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='168  5687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='267307  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN = Nnum  Nanal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='180  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='545  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='785  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94828377  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0832  (N + Y K)num  4439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='709  7265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='423  5852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='566  5684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='325992  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2397  (N + Y K)anal  4431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='722  7298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='614  5865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='168  5687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='26725  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN+Y K  = (N+Y K)num  (N+Y K)anal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='180  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='545  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='785  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94828346  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0832  Observation  4434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='987  7304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326  5869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='656  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN−Obs  num  = Nnum/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='106  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='467  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='708  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0882  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='926  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='923  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0051  RY K−Obs  num  =  (N + Y K)num/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='106  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='467  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='708  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0882  RY K−Obs  anal  =  (N + Y K)anal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='926  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='923  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0051  nominal α = 10−8  Table A17: The values of the calculated and observational astronomical parameters of the planet  Pluto whose number of moons is 0  Pluto  rmin(×106km)  rmax(×106km)  a(×106km)  b(×106km)  eccentricity  Nnum  4439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='709  7265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='423  5852.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='566  5684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326067  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2397  Nanal  4431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='722  7298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='614  5865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='168  5687.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='267307  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN = Nnum  Nanal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='180  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='545  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='785  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='94828377  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0832  (N + Y K)num  4439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  7280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='242  5859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='991  5690.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='112819  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2407  (N + Y K)anal  4435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='112  7304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='196  5869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='654  5691.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='616958  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN+Y K  = (N+Y K)num  (N+Y K)anal %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='104  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='672  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='97357273  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4812  Observation  4434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='987  7304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326  5869.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='656  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='2444  RN−Obs  num  = Nnum/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='106  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='467  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='708  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0882  RN−Obs  anal  = Nanal/Obs %  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='926  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='923  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0051  RY K−Obs  num  =  (N + Y K)num/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='107  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='670  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4862  RY K−Obs  anal  =  (N + Y K)anal/Obs %  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='002  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='998  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='0051  estimated α = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='642205983339201 × 10−4  Table A18: The values of the calculated and observational astronomical parameters of the planet  Pluto whose number of moons is 0  24  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Tables of Absolute Deviations from Observation of the Planets  25  RY K−Obs  num  RY K−Obs  anal  Observed rmax  rnum  max − Obs  ranal  max − Obs  MERCURY  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='721  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='818  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='194  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  Venus  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='769  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='018  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='941  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='251  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='020  EARTH  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='794  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='984  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='312  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='024  MARS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='687  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='006  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='261  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='780  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='016  JUPITER  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='898  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='270  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='363  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='829  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='205  SATURN  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='116  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='794  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='527  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='817  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='969  URANUS  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='535  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='441  3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='390  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='947  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='759  Neptune  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='932  4558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='241  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='524  Pluto  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='467  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='921  7304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='902  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='711  Table B1: Absolute deviations, with nominal α, of rmax from observation, evaluated in (106  km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RY K−Obs  num  RY K−Obs  anal  Observed rmin  rnum  min − Obs  ranal  min − Obs  MERCURY  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='062  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='012  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='005  Venus  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='839  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='008  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='480  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='172  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='009  EARTH  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='978  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='210  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='031  MARS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='962  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='999  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  JUPITER  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='906  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='262  740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='692  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='947  SATURN  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='845  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='797  1357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='554  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='092  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='824  URANUS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='886  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='433  2732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='696  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='482  Neptune  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='860  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='937  4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='050  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='239  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='922  Pluto  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='106  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='926  4434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='987  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='722  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='264  Table B2: Absolute deviations, with nominal α, of rmin from observation, evaluated in (106 km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  26  RY K−Obs  num  RY K−Obs  anal  Observed rmax  rnum  max − Obs  ranal  max − Obs  MERCURY  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='706  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='995  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='818  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='204  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='002  Venus  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='740  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='004  108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='941  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='282  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='004  EARTH  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='757  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='997  152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='369  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  MARS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='664  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='996  249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='261  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='835  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='008  JUPITER  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='334  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='363  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='430  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='027  SATURN  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='410  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='998  1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='527  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='874  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='019  URANUS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='717  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='390  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='472  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='117  Neptune  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='728  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='997  4558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='857  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='377  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='109  Pluto  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='670  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='998  7304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='326  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='12907  Table B3: Absolute deviations, with estimated α, of rmax from observation, evaluated in (106  km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  RY K−Obs  num  RY K−Obs  anal  Observed rmin  rnum  min − Obs  ranal  min − Obs  MERCURY  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='062  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='006  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='002  Venus  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='838  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='994  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='480  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='173  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='005  EARTH  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='856  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='003  147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='211  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='004  MARS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='962  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='989  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='650  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='022  JUPITER  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='897  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='996  740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='595  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='757  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='027  SATURN  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='802  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='001  1357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='554  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='684  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='020  URANUS  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='905  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='995  2732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='696  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='579  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='117  Neptune  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='820  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='002  4471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='050  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='107  Pluto  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='107  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='002  4434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='987  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='753  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='125  Table B4: Absolute deviations, with estimated α, of rmin from observation, evaluated in (106  km).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  27  References  [1] Ephraim Fischbach and Carrick L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Talmadge, The Search for Non-Newtonian Gravity,  AIP Press, Springer (1999),  [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Landau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Lifshitz, Course of Theoretical Physics (Mechanics), Vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 1 Ch  3, Sec 14, p 32, (Pergamon Press : Oxford), (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rodriguez and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Brun, ”Closed orbits in central forces distinct from Coulomb or  harmonic oscillator type,” European Journal of Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 19, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 41-49, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [4] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Brun and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Pacheco, ”On closed but non-geometrically similar orbits,” Celestial  Mech Dyn Astr, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 311-316, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [5] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Hinterbichler, “Theoretical Aspects of Massive Gravity”, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 84, 671  (2012), arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='3735 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [6] Claudia de Rham, “Massive Gravity”, Living Reviews in Relativity, 17 (2014) 7,  arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='4173 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Goldhabert and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Nieto, “Mass of the graviton”, PRD9, 1119 (1974)  [8] Yiming Dong, Lijing Shao, Zexin Hu, Xueli Miaoc and Ziming Wang, “Prospects for  Constraining the Yukawa Gravity with Pulsars around Sagittarius A*”, JCAP-11-(2022)-  051, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='16130 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Moffat, ”Scalar-tensor-vector gravity theory,” Journal of Cosmology and Astropar-  ticle Physics, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [10] Xing Zhang, Tan Liu and Wen Zhao, “Gravitational radiation from compact binary sys-  tems in screened modified gravity”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='D95, 104027 (2017), arXiv: 1702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08752  [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' DAddio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Casadio, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Giusti, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' De Laurentis, “Orbits in bootstrapped Newto-  nian gravity”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' D 105, 104010 (2021), arXiv:2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08379 [gr-qc]  [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Della Monica, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' de Martino, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' De Laurentis, “Orbital precession of the S2 star in  ScalarTensorVector Gravity”, Monthly Notices of the Royal Astronomical Society, Volume  510, Issue 4, March 2022, Pages 47574766, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='12687 [gr-qc]  [13] David Benisty, “Testing modified gravity via Yukawa potential in two body problem:  Analytical solution and observational constraints”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' D106, 043001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [14] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Banik and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Zhao, Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 480, 2660 (2018), [Erratum:  Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 482, 3453 (2019), Erratum: Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 484, 1589  (2019)], arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='12273 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='GA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [15] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Zhang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Lu, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 827, 114 (2016), arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='07725 [astro-  ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='HE].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Pricopi, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Space Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 361, 277 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Edwards, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Gerber, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Schubert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Trejo, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='Weber, PTEP 2017, 083A01  (2017), arXiv:1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='09979 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='atom-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  28  [18] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Mukherjee and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Sounda, Indian Journal of Physics 92, 197 (2018), arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='02444  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='plasm-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [19] Lorenzo Iorio, “Putting Yukawa-LikeModified Gravity (MOG) on the Test in the Solar  System”, Scholarly Research Exchange, Volume 2008, Article ID 238385  [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' De Laurentis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' De Martino, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Lazkoz, “Analysis of the Yukawa gravitational  potential in f(R) gravity II: relativistic periastron advance”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' D 97, 104068  (2018), arXiv:1801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='08136 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Berg, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Brax, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Pernot-Borrs, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Uzan, “Interpretation of geodesy exper-  iments in non-Newtonian theories of gravity”, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 35, 234001 (2018),  arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='00340 [gr-qc].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [22] Lorenzo Iorio, “Constraints on a Yukawa gravitational potential from laser data of LA-  GEOS satellites”, Physics Letters A 298 (2002) 315318  [23] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Cavan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Haranas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Gkigkitzis and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Cobbett, ”Dynamics and stability of the two  body problem with Yukawa correction,” Astrophysics and Space Science, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [24] ”https://nssdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='gsfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='nasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='gov/planetary/factsheet/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='html,” [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Rujula, dedicated to viktor weisskopf on the occasion of the viki-fest, erice, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Goldstein, Classical Mechanics, SECOND EDITION ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=', ADDISON-WESLEY PUB-  LISHING COMPANY , 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Meiss, Differential Dynamical Systems, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [28] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' FACKLER and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' VAN, 5th FORCE NEUTRINO PHYSICS, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [29] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Ross, Differential Equation (John Willey & Sons), 1984, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' 661.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content=' Wakker, Fundamentals of Astrodynamics, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
+page_content='  29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XdE0T4oBgHgl3EQfmgEE/content/2301.02498v1.pdf'}
diff --git a/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf b/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..b63b56dc162b7008737717de06c8664686a93c12
--- /dev/null
+++ b/c9E3T4oBgHgl3EQfeQoX/content/2301.04541v1.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:d1de61abc2d3da952511403efb0140d345beb54f6ebadc0d28ece72377c8e550
+size 129419
diff --git a/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf b/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..fa3a3daf85ffe55ed545a1b55eeca8c09cc8482e
--- /dev/null
+++ b/dNE2T4oBgHgl3EQfGAZK/content/2301.03652v1.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b106e593ce087692bc26eb147e42bc8d37568ca23f96899ade65a539284cb168
+size 2145989
diff --git a/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/2301.13533v1.pdf.txt b/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/2301.13533v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b82d70c158bcafc9ba7f6dbae9f7e042028098a9
--- /dev/null
+++ b/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/2301.13533v1.pdf.txt
@@ -0,0 +1,2769 @@
+arXiv:2301.13533v1  [eess.SY]  31 Jan 2023
+1
+Passivity-based power sharing and voltage regulation
+in DC microgrids with unactuated buses
+Albertus Johannes Malan, Pol Jané-Soniera, Felix Strehle, and Sören Hohmann
+Abstract—In this paper, we propose a novel four-
+stage distributed controller for a DC microgrid that
+achieves power sharing and average voltage regulation
+for the voltages at actuated and unactuated buses. The
+controller is presented for a DC microgrid compris-
+ing multiple distributed generating units (DGUs) with
+time-varying actuation states; dynamic RLC lines; non-
+linear constant impedance, current and power (ZIP)
+loads and a time-varying network topology. The con-
+troller comprising a nonlinear gain, PI controllers, and
+two dynamic distributed averaging stages is designed
+for asymptotic stability. This constitutes first deriving
+passivity properties for the DC microgrid, along with
+each of the controller subsystems. Thereafter, design
+parameters are found through a passivity-based optim-
+isation using the worst-case subsystem properties. The
+resulting closed-loop is robust against DGU actuation
+changes, network topology changes, and microgrid
+parameter changes. The stability and robustness of the
+proposed control is verified via simulations.
+Index Terms—DC microgrids, distributed control,
+passivity, power sharing, voltage regulation.
+I. Introduction
+T
+HE ADVENT of localised power generation and stor-
+age increasingly challenges the prevailing centralised
+power-generation structures. Originally proposed in [1],
+the microgrids paradigm envisions networks that can oper-
+ate autonomously through advanced control while meeting
+consumer requirements. Although current electrical grids
+predominantly use AC, high and low voltage DC networks
+have been made technically feasible due to the continual
+improvements of power electronics. Indeed, DC microgrids
+exhibit significant advantages over their AC counterparts,
+demonstrating a higher efficiency and power quality while
+simultaneously being simpler to regulate [2], [3].
+In microgrids, power generation and storage units
+are typically grouped into distributed generation units
+(DGUs) which connect to the microgrid through a single
+DC-DC converter for higher efficiency [2]. This changes
+the traditionally centralised regulation problem in power
+grids into a problem of coordinating the DGU connected
+throughout the microgrid. This coordination is generally
+This work was supported in part by Germany’s Federal Ministry
+for Economic Affairs and Climate Action (BMWK) through the
+RegEnZell project (reference number 0350062C). (Corresponding
+author: A. J. Malan.)
+A. J. Malan, P. Jané-Soniera, F. Strehle, and S. Hohmann
+are with the Institute of Control Systems (IRS), Karlsruhe In-
+stitute of Technology (KIT), 76131, Karlsruhe, Germany. Emails:
+albertus.malan@kit.edu, pol.soneira@kit.edu, felix.strehle@kit.edu,
+soeren.hohmann@kit.edu.
+realised as average or global voltage regulation in combina-
+tion with load sharing between the DGUs (see e.g. [4]–[6]).
+Literature Review: A vast number of approaches have
+been proposed for the voltage regulation and load sharing
+of DC microgrids, as detailed in the overview papers [3],
+[7], [8] along with the sources therein. These approaches
+are broadly categorised as either centralised, decentralised
+or distributed in nature [3], [7], [8]. While centralised
+controllers can optimally coordinate the DGUs, they offer
+reduced scalability and flexibility and have a single point
+of failure [8]. On the other hand, decentralised controllers
+either only attempt to achieve voltage stability [9]–[11]
+or achieve load sharing at the cost of voltage regulation
+quality (e.g. the droop-based approaches in [3]).
+In response to these limitations, numerous controllers
+for voltage regulation and load sharing which operate in a
+distributed manner have been proposed [4]–[6], [12]–[20].
+In [4], distributed averaging is employed to find a global
+voltage estimate with which voltage regulation is achieved,
+but the microgrid dynamics are neglected in the stability
+analysis. Distributed averaging with dynamic microgrid
+models is used in [5], [12], although [5] requires LMIs to
+be solved before buses are allowed to connect whereas
+[12] only considers constant current loads. Similarly, a
+sliding-mode controller is proposed in [13] for a dynamic
+microgrid with constant current loads. On the other hand,
+[14] proposes a cyberattack-resilient controller for a mi-
+crogrid with constant conductance loads and resistive
+lines. A consensus-based distributed controller with event-
+triggered communication is presented in [15]. Consensus-
+based controllers are also utilised in [6], [16], [17], where
+[6] uses a consensus-based integral layer on top of a droop-
+based controller. Finally, while many contributions strive
+to achieve proportional current sharing [4]–[6], [12]–[17],
+[20], nonlinear controllers that achieve proportional power
+sharing have also been proposed in [18], [19].
+While the literature listed above differ greatly in their
+approaches, we note a commonality in their omission of
+buses without actuation. This omission is typically mo-
+tivated either by considering a microgrid comprising only
+actuated DGU buses [4], [5], [16], [17], or by eliminating the
+unactuated buses with the Kron-reduction [6], [12]–[15],
+[18]–[20]. However, considering a network comprising only
+actuated buses severely limits the flexibility of a microgrid,
+since each bus must be able to supply or consume enough
+power at all times. On the other hand, the Kron-reduction
+requires loads to be described as positive conductances
+(see e.g. [21]). While research into Kron-reduced networks
+with negative loads is ongoing (see e.g. [22]), the general
+
+2
+inclusion of negative loads, e.g. non-controllable power
+sources, in Kron-reducible networks remains out of reach
+at present. Furthermore, consider the case where a DGU
+can no longer supply or consume the required amount of
+power, e.g. a fully charged or discharged battery storage.
+Such a DGU then loses the ability to regulate itself and
+fully support the grid. In the approaches considered above
+[4]–[6], [12]–[20], such a DGU is forced to disconnect from
+the microgrid and its local measurements are discarded.
+For DGUs with intermittent power sources, this could
+result in significant swings in the number of controlled and
+observed buses in the microgrid.
+Main Contribution:
+In this paper, we consider a
+DC microgrid as a physically interconnected multi-agent
+system. Extending our work in [23]1, we propose a four-
+stage controller that achieves voltage regulation and power
+sharing in a DC microgrid with actuated and unactuated
+buses in a distributed manner. The four-stage controller
+comprises a nonlinear weighting function, two dynamic
+distributed averaging (DDA) stages and a proportional-
+integral (PI) controller. The asymptotic stability of the
+closed loop comprising the DC microgrid and the four-
+stage controller interconnected in feedback is proven by
+means of passivity theory. In detail, the contributions
+comprise:
+1) A four-stage distributed controller for DC microgrids
+which achieves consensus on the weighted average
+voltage error of actuated and unactuated buses and
+assures coordination through power sharing at the
+actuated buses.
+2) A nonlinear weighting function that penalises voltage
+errors outside a given tolerance band more strongly
+than those within.
+3) Passivity classifications for each of the constitutive
+microgrid subsystems (DGUs, loads, and lines) and
+for each of the controller stages (weighting function,
+DDA, and PI).
+4) A
+method
+for
+calculating
+the
+input-feedforward
+output-feedback passive (IF-OFP) indices of the non-
+linear power-controlled DGUs through optimisation.
+5) An IF-OFP formulation for the DC microgrid with
+a supply rate that is independent of the network
+topology, the number of buses and their states of
+actuation.
+6) A passivity-based stability analysis for the equilib-
+rium of the DC microgrid connected in feedback with
+the four-stage controller.
+In addition to the contributions listed above, we also
+contribute a theoretical result comprising a formalisation
+of the obstacle presented by cascaded input-feedforward
+passive (IFP) and output-feedback passive (OFP) systems
+in the analysis of dissipative systems. This theoretical
+1The controller proposed in [23] is extended by weighing the
+error with a nonlinear function. Moreover, in addition to applying
+the controller to a DC microgrid context, we here propose a new
+dissipativity-based analysis that investigates the closed loop stability
+analytically as opposed to the numerical results in [23].
+contribution informs and motivates parameter choices for
+the four-stage controller in Contribution 1.
+We highlight that the proposed controller can achieve
+exact voltage regulation and power sharing with the
+stability verified with the eigenvalues of the linearised
+system. Moreover, by employing leaky PI controllers, we
+demonstrate a passivity-based stability analysis that is
+independent of and robust against changes in the commu-
+nication topology, changes in the electrical topology, load
+changes, changes in the actuation status of DGUs, uncer-
+tainties in component parameters, and buses connecting
+or disconnecting.
+Paper Organisation: The introduction concludes with
+some notation and preliminaries on graph theory. In
+Section II, we recall and introduce results relating to
+dissipativity theory. Next, in Section III, the problem is
+modelled and objectives for the steady state are formal-
+ised. In Section IV, a four-stage control structure is intro-
+duced that fulfils objectives from Section III. Thereafter,
+the passivity properties of the constituent subsystems are
+investigated in Section V and the controller is designed
+for asymptotic stability of the closed loop in Section VI.
+Finally, in Section VII, a simulation is used to verify the
+asymptotic stability and robustness of the closed loop.
+Concluding remarks are provided in Section VIII.
+Notation and Preliminaries: Define as a vector a =
+(ak) and a matrix A = (akl). 1k is a k-dimensional vector
+of ones and Ik is the identity matrix of dimension k.
+Diag[·] creates a (block-)diagonal matrix from the supplied
+vectors (or matrices). The upper and lower limits of a value
+a are given by a and a. For a variable x, we denote its
+unknown steady state as ˆx, its error state as ˜x := x − ˆx,
+and a desired setpoint as x∗. Whenever clear from context,
+we omit the time dependence of variables.
+We denote by G = (N, E) a finite, weighted, undirected
+graph with vertices N and edges E ⊆ N × N. Let |N| be
+the cardinality of the set N. Let L be the Laplacian matrix
+of G. By arbitrarily assigning directions to each edge in E,
+the incidence matrix E ∈ R|N|×|E| of G is defined by
+ekl =
+
+
+
++1
+if vertex k is the sink of edge l,
+−1
+if vertex k is the source of edge l,
+0
+otherwise.
+(1)
+II. Dissipativity Preliminaries
+We here recall and introduce preliminaries of dissip-
+ativity theory for nonlinear systems. In Section II-A we
+provide definitions relating to dissipativity and passiv-
+ity theory. Thereafter in Section II-B, we investigate
+the passivity properties of static functions. Finally, in
+Section II-C, we recall a result on the interconnection
+of dissipative systems with quadratic supply rates and
+formalise a new result on the limitations of such an
+interconnection.
+
+3
+A. Dissipative Systems
+Consider a nonlinear system
+�
+˙x = f(x, u),
+y = h(x),
+(2)
+where x ∈ Rn, u ∈ Rm, y ∈ Rm and where f : Rn×Rm →
+Rn and h : Rn × Rm → Rm are class C1 functions.
+Definition 1 (Dissipative system, c.f. [24]–[26]). A system
+(2) with a class C1 storage function S : Rn × Rm → R+ is
+dissipative w.r.t. a supply rate w(u, y) if ˙S ≤ w(u, y).
+Definition 2 (Quadratic supply rates, c.f. [24]–[26]). A
+system (2) that is dissipative w.r.t. w(u, y) is
+• passive if w = uTy,
+• input-feedforward passive (IFP) if w = uTy − νuT u,
+• output-feedback passive (OFP) if w = uT y − ρyT y,
+• input-feedforward output-feedback passive (IF-OFP) if
+w = (1 + νρ)uT y − νuT u − ρyT y,
+• has an L2-gain of γL2 if w = γ2
+L2uTu − yT y,
+where γL2 > 0 and ν, ρ ∈ R.
+Definition 3 (Zero-state observable (ZSO) [24, p. 46]). A
+system (2) is ZSO if u ≡ 0 and y ≡ 0 implies x ≡ 0.
+For cases where the desired equilibrium of a system is
+not at the origin but at some constant value, the shifted
+passivity [24, p. 96] or equilibrium-independent passivity
+(EIP) [27] of a system must be investigated. Naturally, this
+requires that an equilibrium exists, i.e. there is a unique
+input ˆu ∈ Rm for every equilibrium ˆx ∈ ˆ
+X ⊂ Rn such that
+(2) produces f(ˆx, ˆu) = 0 and ˆy = h(ˆx, ˆu) [28, p. 24].
+Definition 4 (EIP [28, p. 24]). A system (2) is EIP
+if there exists a class C1 storage function S(x, ˆx, u),
+S : Rn × ˆ
+X × Rm → R+, with S(ˆx, ˆx, ˆu) = 0, that is dis-
+sipative w.r.t. w(u − ˆu, y − ˆy) for any equilibrium (ˆu, ˆy).
+B. Passive Static Functions
+Recall that a sector-bounded static nonlinear function
+is dissipative to a supply rate defined by the sector bound
+[26, Def. 6.2]. We now consider the arbitrarily shifted
+single-input single-output function
+�
+y = h(u),
+u, ˆu ∈ U,
+y, ˆy ∈ Y,
+h : U → Y,
+˜y = ˜h(˜u) := h(u) − h(ˆu) = y − ˆy,
+˜u := u − ˆu
+(3)
+and show how its dissipativity properties may be derived.
+Proposition 5 (EIP static functions). A static function
+(3) of class C0 is IF-OFP(c, 1/c) w.r.t. the arbitrarily
+shifted input-output pair (˜u, ˜y) if
+c ≤ dh(u)
+du
+≤ c,
+∀u ∈ U.
+(4)
+and 0 < c < ∞.
+Proof. Consider for (3) the slope between an arbitrary
+shift (ˆu, ˆy) ∈ U ×Y and a point (u, y), for which the upper
+and lower bounds are given by
+c ≤ y − ˆy
+u − ˆu ≤ c,
+∀(u, y), (ˆu, ˆy) ∈ U × Y.
+(5)
+Changing to the shifted variables ˜u and ˜y as in (5) and
+multiplying through by ˜u2 yields
+c˜u2 ≤ ˜u˜y ≤ c˜u2 ⇐⇒ (˜y − c˜u)(˜y − c˜u) ≤ 0
+⇐⇒ (˜y − c˜u)(1
+c ˜y − ˜u) ≤ 0,
+(6)
+for c > 0, which describes an IF-OFP function (see [26,
+p. 231]). Finally, through the mean value theorem, the
+bounds in (5) may be found from (4).
+■
+We note that the restrictions on c in Prop. 5 are needed
+from a computational point of view (c < ∞) and to ensure
+that the passivity indices correspond to the correct sector2
+(c > 0). However, this limits the passivity properties
+attainable through Prop. 5 to ρ = 1/c > 0.
+Remark 1 (Symmetrical sectors). Placing the additional
+restriction c = −c in (4) results in the Lipschitz continuity
+of h(u). Moreover, this implies that the arbitrarily shifted
+function ˜h(˜u) has a finite L2-gain of c [29].
+C. Interconnected Quadratic Dissipative Systems
+Building upon the results on the interconnection of
+dissipative systems in [28], [30], we now provide a method
+for finding dissipativity properties for a subset of the inter-
+connected subsystems such that interconnected stability is
+guaranteed. Specifically, we look for the dissipative supply
+rates that restrict the subset of subsystems as little as pos-
+sible. For a set S of subsystems, define u = [uT
+1 , . . . , uT
+|S|]T
+and y = [yT
+1 , . . . , yT
+|S|]T .
+Theorem 6 (Minimally restrictive stabilising indices).
+Consider |S| subsystems of the form (2) which are dissipat-
+ive w.r.t. the supply rates wi = 2σiuT
+i yi−νiuT
+i ui−ρiyT
+i yi
+and are linearly interconnected according to u = Hy. The
+stability of the interconnected system is guaranteed if there
+exists a D and νj, ρj ∈ R with j ∈ J such that
+min
+D, νj, ρj,
+j∈J
+�
+j∈J
+(νj + ρj)
+s.t.
+σj = 1/2(1 + νjρj),
+j ∈ J,
+Q ≼ 0,
+D2 ≻ 0
+(7)
+where the subsystems with configurable supply rates are
+represented by the set J ⊂ S, and
+Q :=
+�H
+I
+�T
+DWD
+�H
+I
+�
+(8)
+D := Diag[dT , dT ],
+d = (
+�
+di),
+(9)
+W :=
+�− Diag[νi]
+Diag[σi]
+Diag[σi]
+− Diag[ρi]
+�
+,
+i ∈ S.
+(10)
+2Consider e.g. the sector Prop. 5 would yield if c ≤ c < 0.
+
+4
+The proof for Theorem 6 follows analogously to the
+proof of [29, Theorem 13] with application of [29, Re-
+mark 5] and is thus omitted for brevity. Note that if J = ∅
+in (7), Theorem 6 can be used to verify the stability of
+interconnected dissipative systems.
+Despite the design flexibility provided by Theorem 6,
+certain cascade configurations present obstacles to the ap-
+plication of dissipativity theory. The following proposition
+formalises the problem presented by one such configura-
+tion which arises in the sequel and is used to inform the
+control design.
+Proposition 7 (Non-dissipativity of cascaded IFP-OFP
+systems). Consider |S| ≥ 2 subsystems (2) which are
+dissipative w.r.t. wi = 2σiuT
+i yi − νiuT
+i ui − ρiyT
+i yi and
+linearly interconnected according to u = Hy. Let i = 1 and
+i = 2 arbitrarily denote subsystems that are IFP and OFP,
+respectively. If these systems are connected in exclusive
+casade and do not form a feedback connection, i.e.
+H =
+
+
+0
+0
+∗
+1
+0
+0
+0
+∗
+∗
+
+ ,
+(11)
+then investigating stability via separable storage functions
+as in Theorem 6 fails.
+Proof. Evaluating the stability criteria in (7) under the
+imposed IFP and OFP conditions yields the Q (8) entries
+q11 = d1ρ1 + d2ν2 = 0,
+q12 = q21 = d2σ2
+2
+= d2
+2 . (12)
+Since di > 0, Q constitutes an indefinite saddle-point mat-
+rix [31, Section 3.4], violating the requirement in (7).
+■
+Remark
+2
+(Non-separable
+storage
+functions).
+The
+obstacle in Prop. 7 arises due to the storage functions being
+compartmentalised by the subsystem boundaries. While the
+separability of storage functions is a central motivation for
+the use of dissipativity theory, forgoing this allows for a
+stability analysis through less conservative methods (e.g.
+the KYP lemma).
+III. Problem Description
+In this section, the components comprising the DC mir-
+crogrid are introduced in Section III-A. This is followed by
+Section III-B, where controllers are added which regulate
+the output power of actuated buses in order to facilitate
+power sharing in the sequel. Finally, we formulate the
+coordination and cooperation goals as a control problem
+in Section III-C.
+A. DC Network
+We consider a DC microgrid comprising N = |N| buses
+connected by via π-model electrical lines, as depicted in
+Fig. 1. Let the graph GP = (N, EP) describe the intercon-
+nection with N as the set of buses and EP as the set of
+lines. Without loss of generalisation, we allow each node to
+inject power through a DC-DC buck converter connected
+via a lossy LC-filter. Note that a time-averaged model (see
+e.g. [12]) is used for the buck converter and the energy
+source is assumed to be ideal but finite.
+Let the buses be split into an actuated set Nα and
+an unactuated set Nβ, according to whether the buck
+converter can freely regulate the amount of power injected
+at a given time. Buses may freely switch between the sets
+Nα and Nβ, but Nα ∩ Nβ = ∅ and Nα ∪ Nβ = N always
+hold. To characterise this actuation state of a bus, define
+the piecewise-constant, time-varying actuation parameter
+αk(t) as
+αk(t) :=
+� 1,
+k ∈ Nα,
+0,
+k ∈ Nβ.
+(13)
+Note that we omit the time dependence of αk in the sequel.
+The dynamics for actuated buses with DGUs, where
+αk = 1 with k ∈ Nα are described by
+�
+Lk˙ik
+Ceq,k ˙vk
+�
+=
+�−Rk
+−1
+1
+0
+��ik
+vk
+�
++
+�
+vs,k
+−eT
+P,kit − IL,k(vk)
+�
+(14)
+where Ceq,k = Ck + 1/2eT
+P,k Diag[Ckl]eP,k; Ck, Ckl, Lk > 0;
+ik ∈ R; and vk ∈ R+. The line currents it connect to the
+capacitor voltages according to incidence matrix EP =
+(eT
+P,k) of GP. The dynamics of the unactuated load buses
+with αk = 0 correspond to the simplified system
+Ceq,k ˙vk = −eT
+P,kit − IL,k(vk),
+k ∈ Nβ
+(15)
+In both the actuated (14) and unactuated (15) cases, the
+loads are considered static, nonlinear voltage-dependent
+current sources which are described by class C0 functions.
+In this work, we utilise the standard ZIP-model comprising
+constant impedance, constant current and constant power
+parts. Note that other continuous functions may also be
+used without restriction3. As described in [33, pp. 110–
+112], we define a critical voltage vcrit, typically set to
+0.7vRef, below which the loads are purely resistive. Thus,
+IL,k(vk) =
+
+
+
+Z−1
+k
+· vk + Ik + Pk
+vk
+,
+vk ≥ vcrit,
+Z−1
+crit,k · vk,
+vk < vcrit,
+(16)
+Z−1
+crit,k := IL,k(vcrit)
+vcrit
+= Z−1
+k
++ Ik
+vcrit
++ Pk
+v2
+crit
+,
+(17)
+describes a static, nonlinear load which conforms to (3).
+Lastly, the π-model transmission lines physically con-
+necting the nodes are governed by the dynamics
+Lkl˙it,kl = −Rklit,kl + eT
+P,klv,
+kl ∈ EP,
+(18)
+where it,kl ∈ R, Lkl, Rkl > 0 and (eT
+P,kl)T = EP. Note
+that the line capacitances are included in the equivalent
+capacitances Ceq,k at the buses.
+B. DGU Power Regulator
+To allow for power sharing between the actuated buses
+(14) in the sequel, we equip each DGU with a controller
+3This includes exponential loads (see e.g. [32]).
+
+5
+Linekl
+αk ∈ {0, 1}
+p∗
+k
+vk
+−
++
+vs,k
+ik
+Rk
+Lk
+Ck
+IL,k(vk)
++
+−
+vk
+Buckk
+Busk
+Ckl
+2
+Rkl
+it,kl
+Lkl
+Ckl
+2
+Busl
+Figure 1: Circuit diagram of a bus comprising a DC-DC buck converter, a filter, and a current source representing a
+load, connected to a π-model line (blue); the line capacitances considered to be part of the respective buses.
+that can regulate the injected power to a desired setpoint
+p∗
+k. This regulator has the form
+˙ed,k = αk(p∗
+k − pk)
+vs,k = kP
+d (p∗
+k − pk) + kI
+ded,k + ˜Rkik + vRef
+(19)
+where ed ∈ R, pk = vkik is the actual power injected,
+˜R ∈ R is the damping added to the system, and kP
+d , kI
+d >
+0 are the control parameters. Combining (19) with (14)
+yields the nonlinear system describing the actuated agents
+k ∈ Nα
+
+
+˙ed,k
+Lk˙ik
+Ck ˙vk
+
+=
+
+
+0
+−vk
+0
+kI
+d
+˜Rk − Rk − kP
+d vk
+−1
+0
+1
+0
+
+
+
+
+ed,k
+ik
+vk
+
+
++
+
+
+p∗
+k
+kP
+d p∗
+k + vRef
+−eT
+k it − IL,k(vk)
+
+
+(20)
+Remark 3 (Regulating current or voltage). Without in-
+validating the stability analysis in the sequel, the regulator
+in (19) can be exchanged for simpler, purely linear current
+or voltage regulators (see e.g. [9]–[11]).
+Remark 4 (Constrained DGU operation). If an actuated
+DGU cannot provide the desired power p∗
+k, e.g. due to
+current, storage or temperature limitations, the DGU may
+simply set its actuation state αk = 0 to disable its control. If
+some power can still be supplied, it may simply be regarded
+as a negative load. This allows DGUs to contribute to the
+power supply of the network, even in the face of control
+limitations.
+C. Control Problem
+A central requirement for DC microgrids is voltage
+stability, which requires the bus voltages to remain within
+a given tolerance band around the reference vRef. Spe-
+cifically, this requirement should be met throughout the
+network, and not only at the actuated buses. Due to
+the presence of lossy lines, power flows are associated
+with voltage differences between buses, meaning that
+vk → vRef, ∀k ∈ N is not practical. Ideally, the voltages at
+all buses should be arrayed in the tolerance band around
+vRef and be as close to vRef as possible4. The manipulated
+variables used to achieve this are the power setpoints p∗
+k
+supplied to the actuated DGUs (19). This leads to the
+first objective for the control of the DC microgrid, which
+involves finding the setpoints p∗
+k that ensure the weighted
+average voltage equals vRef at steady state.
+Objective 1 (Weighted voltage consensus).
+Find p∗
+k s.t. lim
+t→∞
+1
+N
+�
+k∈N
+h(vk(t)) = vRef
+(21)
+for a strictly increasing weighting function h : R → R.
+By choosing a nonlinear h, large voltage errors may be
+weighed more strongly. This allows for better utilisation of
+the tolerance band since bus voltages can be further from
+vRef before registering as a significant error.
+In addition to Objective 1, it is desired that all actuated
+DGUs contribute towards supplying and stabilising this
+network. Ensuring that all DGUs receive the same setpoint
+spreads the load across actuated buses, leading to a reduc-
+tion in localised stress on the DGUs. We thus formulate
+the second objective as requiring uniform setpoints for the
+DGUs in steady state.
+Objective 2 (Cooperative power sharing).
+lim
+t→∞(p∗
+k(t) − p∗
+l (t)) = 0,
+∀ k, l ∈ N
+(22)
+Achieving Objectives 1 and 2 thus yields a controlled
+microgrid where the average weighted voltage error of all
+buses tends to zero through the coordinated action of the
+actuated buses in a distributed fashion. These objectives
+also allow DGUs to transition seamlessly between actuated
+and unactated states and ensure no measurement inform-
+ation is discarded simply because a bus cannot regulate
+itself. Notice that disregarding the unactuated buses in
+Objectives 1 and 2 yields the objectives typically used in
+the literature [4], [6], [12]–[14], [16], [17], [20].
+To achieve these objectives, we make the following
+assumptions related to appropriate network design.
+Assumption 1 (Feasible network). The available power
+sources can feasibly supply the loads with power over the
+4The magnitude of the errors vRef − vk strongly depend on the
+loads and line resistance. Small errors therefore presuppose adequate
+network design.
+
+6
+hw
+hw
+DDA2,1
+DDA2,N
+PI1
+PIN
+DDA4,1
+DDA4,N
+DC MG
+Stage 1
+Stage 2
+Stage 3
+Stage 4
+uw
+uw,1
+uw,N
+yw,1
+yw,N
+ya,2,1
+ya,2,N
+yc,1
+yc,N
+ya,4,1
+ya,4,N
+p∗
+v
+−
+vRef1N
++
+Figure 2: Distributed four-stage control connected in feed-
+back to the microgrid and with indicated communication
+links
+between the local control structures.
+given electrical network, i.e. a suitable equilibrium for the
+microgrid exists.
+Assumption 2 (Number of actuated DGUs). At least one
+DGU is actuated at any given time, i.e. Nα ̸= ∅.
+Assumption 3 (Connected topologies). Objectives 1
+and 2 only apply to a subset of buses electrically connected
+as per GP. Moreover, for a distributed control, a connected
+communication graph exclusively interconnects the same
+subset of buses.
+Note that Assumption 1 is a typically made implicitly or
+explicitly in the literature (see e.g. the discussion in [16]).
+Assumptions 2 and 3 further specify requirements that
+allow a distributed control to achieve the feasible state
+in Assumption 1, i.e. by ensuring that at least one source
+of stabilisation is present in the network (Assumption 3),
+and by ensuring that the coordination corresponds to the
+network to be controlled Remark 5.
+Remark 5 (Proportional power sharing). By normalising
+the power setpoint p∗
+k and weighing the input in (19)
+according to the rated power of a given DGU, Objective 2
+automatically describes a proportional power sharing. With
+reference to Remark 4, this also allows the constrained
+DGUs to lower their maximum injectable power instead
+of setting the DGUs to the unactuated state αk = 0. We
+omit the extension to proportional power sharing in this
+work for simplicity.
+IV. Control Structure
+To meet Objectives 1 and 2, we propose the four-
+stage control structure depicted in Fig. 2. This control
+structure comprises two DDA implementations separated
+by agent PI controllers local to the buses as in [23]. This
+is prepended by a nonlinear weighting function hw. In the
+Sections IV-A, IV-B and IV-C, we successively introduce
+these respective subsystems. Finally in Section IV-D, we
+show that the control structure meets the objectives.
+A. DDA Controller
+Consider the communiation graph GC = (N, EC) linking
+the buses of the DC microgrid. The communication graph
+comprises the same vertices as the physical interconnection
+graph GP but possibly with a different topology. Let LC
+denote the Laplacian of GC. For Stages 2 and 4 of the
+control structure, each agent implements an instance of
+the DDA5 described in [34]. The instances of the respective
+stages may be combined into vector form as
+DDAs
+
+
+
+
+
+� ˙xa,s
+˙za,s
+�
+=
+�
+−γaIN −LC,P
+LT
+C,I
+−LC,I
+0
+��
+xa,s
+za,s
+�
++
+�
+γaIN
+0
+�
+ua,s,
+ya,s = xa,s,
+(23)
+where s
+∈
+{2, 4} denotes the stage in Fig. 2, and
+xa,s, za,s ∈ RN are the consensus and integral states
+respectively. Furthermore, γa > 0 is a global estimator
+parameter (see [34]), and LC,I = kI
+aLC and LC,P = kP
+a LC
+are Laplacian matrices weighted for the integral and pro-
+portional responses, respectively. Recall from [34] that a
+constant input ua,s yields
+lim
+t→∞ ya,s,k = uT
+a,s1N
+N
+,
+∀ k.
+(24)
+B. Agent PI Controller
+In Stage 3, we equip each bus k ∈ N with a leaky agent
+PI controller similar to the approach in [35]
+PIk
+�
+˙xc,k = −ζcxc,k + uc,k,
+yc,k = kI
+cxc,k + kP
+c uc,k,
+(25)
+where xc,k ∈ R, ζc ≥ 0 and kP
+c , kI
+c > 0. Note that ζc = 0
+reduces (25) to an ideal PI controller. The combined form
+of the N agent controllers is
+˙xc = −ζcxc + uc,
+yc = kI
+cxc + kP
+c uc
+(26)
+Remark 6 (Non-ideal integrators). As shown in the
+sequel, ideal PI controllers only exhibit an IFP property,
+whereas the DDA controller is OFP. The interconnection
+in Fig. 2 thus yields a cascaded IFP-OFP structure which
+obstructs the dissipativity analysis (see Prop. 7). The use
+of leaky integrators (ζc > 0) overcomes this obstacle at the
+cost of negatively affecting the steady-state properties, since
+(25) forces the equilibrium
+uc = ζcxc
+(27)
+instead of uc = 0. In the context of Fig. 2, this corresponds
+to a unwanted steady-state offset for the average weighted
+voltage error.
+Remark 7 (Agent PI controller anti-windup). To prevent
+controller windup, the input to the PI control in (25) should
+be zeroed for any unactuated agents that are disconnected
+from the communication network.
+Remark
+8
+(Non-participating agents). Implementing
+(25) at each bus k ∈ N allows for a faster reaction to
+disturbances at the cost of controller redundancy. By setting
+ua,4,m := ya,4,m at Stage 4 DDA of the control structure
+5We implement the PI-DDA variant proposed in [34] and use the
+same communication graph for the proportional and integral terms.
+
+7
+u
+y
+hw(u)
+dhw(u)
+duw
+Figure 3: Example of the weighting function hw (28) and
+its derivative (58) on a unit grid, with aw = 0.5, bw = 1.5
+and cw = 2.
+for some agents m ∈ M ⊂ N, the PI control (25)
+can be omitted at the agents in M without affecting the
+steady state. Nevertheless, the measurements of the buses
+in k ∈ M are still included in the Stage 2 DDA. Note that
+at least one participating agent PI controller is required
+(see [23, Remark 8]).
+C. Weighting Function
+To allow for a better utilisation of the tolerance band
+around vRef, we desire a weighting function that assigns a
+low gain for errors within the tolerance band and a high
+gain for larger errors. We therefore define the class C1
+function yw,k = hw(uw,k) conforming to (3), where
+hw(u) := awu + bwgw(u) − bw tanh(gw(u)),
+(28)
+gw(u) :=
+
+
+
+u + cw,
+u < −cw
+0,
+−cw ≤ u ≤ cw
+u − cw,
+cw < u
+(29)
+and where (29) describes a dead-zone parametrised by
+cw. An example of (28) is depicted in Fig. 3 along with
+its derivative. For a strictly increasing function as per
+Objective 1, set aw > 0 and bw > −aw.
+D. Equilibrium Analysis
+In a first step towards analysing the closed loop, we
+analyse the assumed equilibrium of the interconnected
+microgrid and four-stage controller (see Assumption 1).
+Specifically, we verify that the proposed control yields an
+equilibrium which satisfies Objectives 1 and 2.
+Proposition 8 (Controller equilibrium analysis). Con-
+sider the DC microgrid comprising (15), (18), and (20)
+which is connected in feedback with the four-stage con-
+troller comprising (23), (26), and (28) as in Fig. 2. Let
+Assumptions 1, 2 and 3 hold. Then, Objective 2 is met for
+the equilibrium imposed by the control structure. Moreover,
+Objective 1 is achieved exactly for ideal integrators ζc = 0
+in (26). For lossy integrators with ζc > 0, the remaining
+error for Objective 1 is be described by the steady-state
+value of ya,2, where
+ya,2 =
+ζc
+kIc(1 + ζckPc )ya,4.
+(30)
+The proof of Prop. 8 can be found in Appendix A.
+Through Prop. 8 we thus confirm that the proposed con-
+troller yields an equilibrium which meets the requirements,
+even though the requirements are not perfectly met when
+leaky agent PI controllers are used. We also note that
+Prop. 8 only considers the controlled microgrid already in
+equilibrium and does not consider the convergence to the
+equilibrium.
+Remark 9 (Compensating leaky-integral errors). As in-
+dicated by (30) in Prop. 8, the leaky agent PI controllers
+result in a constant steady-state error for the average
+voltage regulation (Objective 1). Since a positive ya,2 cor-
+responds to voltages below the desired vRef, it follows
+that setting vRef above the actual desired voltage reference
+will result in higher bus voltages. Changing vRef thus
+allows the steady-state effects of the leaky integrators to be
+compensated. Moreover, notice that ya,4 is the controller
+output, i.e. the power setpoint p∗ used for the DGUs
+(see Fig. 2). Thus, the error measure in (30), which is
+only dependent on the controller output, can be used to
+determine the offset to vRef for exact voltage regulation.
+Note, however, that modifying vRef based on p∗ results in
+a new loop which requires an additional stability analysis.
+V. Subsystem Passivity Analysis
+Having verified whether the desirable steady state is
+achieved by the controller, we now set about analysing the
+convergence to this steady state. With the aim of applying
+Theorem 6 for the closed-loop stability, we first analyse the
+passivity properties of the individual subsystems. Since
+the steady-state bus voltages ˆvk are unknown and non-
+zero, we investigate the passivity properties shifted to any
+plausible point of operation using EIP. To this end, we
+construct an EIP formulation for the DC microgrid from
+its constitutive elements in Section V-A. This is followed
+by the respective analyses of the various controller stages
+in Section V-B. Note that we omit the bus indices k and
+l in this section where clear from context.
+A. DC Microgrid Passivity
+For the stability of the microgrid at the equilibrium
+ˆv, we desire an EIP property relating the shifted input
+power setpoints ˜p∗ = p∗ − ˆp∗ to the output voltage errors
+˜v = v − ˆv of all nodes, since this port (˜p∗, ˜v) is used
+by the controller in Fig. 2. To this end, we derive EIP
+properties for the load, DGU and line subsystems of the
+microgrid, making sure to shift the subsystem dynamics to
+the assumed equilibrium in each case (see Assumption 1).
+Thereafter, we combine the results of these subsystems, to
+construct an EIP property for the microgrid as a whole.
+Where applicable, an analysis of the zero-state dynamics is
+performed to ensure the eventual stability of the controlled
+microgrid.
+1) Load Passivity: Let the unactuated bus dynamics
+in (15) for the buses in Nβ be shifted to the equilibrium
+(ˆit, ˆv), yielding
+Ceq ˙˜v = −eT
+P,k˜it − ˜IL(˜v) + (eT
+P,kˆit + IL(ˆv)),
+(31)
+
+8
+for the static load function shifted according to (3). In
+(31), eT
+P,kˆit = −IL(ˆv) since the load is fully supplied by
+the cumulative line currents in steady state.
+Proposition 9 (Load EIP). The shifted load dynamics in
+(31) are OFP(ρL) w.r.t. the input-output pair (−eT
+P,k˜it, ˜v)
+with ρL = cL the smallest gradient of the static load
+function IL(v).
+Proof. Consider the storage function SL along with its
+time derivative
+SL = Ceq
+2 ˜v2,
+(32)
+˙SL = −˜veT
+P,k˜it − ˜v ˜IL(˜v).
+(33)
+Since the static load function IL(v) is IF-OFP according
+to Prop. 5, it is bounded from below by cL˜v2 ≤ ˜v ˜IL(˜v)
+(see (6)). Incorporate this lower bound into (33) to obtain
+˙SL ≤ wL := −˜veT
+P,k˜it − cL˜v2
+(34)
+which yields the OFP property from Definition 2.
+■
+Remark 10 (ZIP load passivity). Prop. 9 and (4) demon-
+strate that the passivity properties of the unactuated buses
+are directly linked to the smallest gradient of the load
+function. For the ZIP load in (16), this yields
+cL = min
+�
+Z−1, Z−1 −
+P
+v2
+crit
+, Z−1
+crit
+�
+.
+(35)
+Considering the strictly passive case (cL = 0) along with
+I, P ≥ 0 yields the passivity condition Z−1v2
+crit ≥ P
+frequently used in the literature [10], [16], [18]–[20].
+2) DGU Passivity: Shift the states (e, i, v) and inputs
+(p∗, it) of the DGU dynamics in (20) for the buses in Nα to
+the respective error variables (˜e,˜i, ˜v) and (˜p∗,˜it) to obtain
+(36) on the next page, where the static load function
+is incorporated into the matrix Ad. Furthermore, the
+measured power p = vi = v(˜i + ˆi) in (19) is left partially
+in unshifted variables such that Ad is also dependent on
+the unshifted voltage v and the steady-state current ˆi.
+Note that the constant τd in (36) is found by setting the
+error variables (˜p∗,˜it, ˜ed,˜i, ˜v) and their time derivatives to
+zero. As such, the constant τd ≡ 0 can be disregarded
+in the passivity analysis. We now analyse the shifted
+nonlinear system in (36) for EIP.
+Theorem 10 (EIP DGUs). The shifted DGU dynamics in
+(36) are simultaneously IF-OFP(νd,1, ρd) w.r.t. the input-
+output pair (˜p∗, ˜v) and IFP(νd,2) w.r.t. the input-output
+pair (−eT
+k ˜it, ˜v), if a feasible solution can be found for
+max
+Pd, νd,1, νd,2, ρd νd,1 + νd,2 + ρd
+s.t.
+(38) holds ∀ v ∈ V ⊆ R+, ∀ˆi ∈ ˆI ⊆ R
+(37)
+where Qd(v,ˆi, cL) := PdAd(v,ˆi, cL) + AT
+d (v,ˆi, cL)Pd,
+Ad(v,ˆi, cL) =
+
+
+0
+−v
+−ˆi
+kI
+d
+˜R − R − kP
+d v
+−1 − kP
+c ˆi
+0
+1
+−cL
+
+ ,
+(39)
+and with νd,1, νd,2, ρd ∈ R, cL as in (4) and cd = [0, 0, 1]T.
+Proof. Consider for (36) the storage function
+Sd =
+
+
+˜ed
+˜i
+˜v
+
+
+T
+Pd
+
+
+˜ed
+L˜i
+Ceq˜v
+
+ ,
+(40)
+with Pd ≻ 0. The time derivative of (40) is
+˙Sd =
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+
+T
+
+Qd(v,ˆi,
+˜IL(˜v)
+˜v
+)
+Pdbd,1
+Pdbd,2
+bT
+d,1Pd
+0
+0
+bT
+d,2Pd
+0
+0
+
+
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+,
+(41)
+with ˜xd as in (36). Since it follows from (6) that −˜v ˜IL(˜v) ≤
+−cL˜v2, this bound can be incorporated into the inequality
+˙Sd ≤
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+
+T
+
+Qd(v,ˆi, cL)
+Pdbd,1
+Pdbd,2
+bT
+d,1Pd
+0
+0
+bT
+d,2Pd
+0
+0
+
+
+
+
+˜xd
+˜p∗
+eT
+P,k˜it
+
+.
+(42)
+The desired IF-OFP and IFP properties for the DGU are
+described by the supply rate
+wd = (1 + νd,1ρd)˜p∗˜v − νd,1(˜p∗)2 − ρd˜v2
+− ˜veT
+P,k˜it − νd,2
+�
+eT
+P,k˜it
+�2
+(43)
+These properties are guaranteed, if ˙Sd−wd < 0 for all valid
+inputs and outputs and for v ∈ V and ˆi ∈ ˆI. Combining
+(42) and (43) in this manner directly leads to constraint
+(38) in (37). Finally, the objective function in (37) seeks
+to find the largest indices for which the constraints are
+satisfied in a similar manner to Theorem 6.
+■
+Although Theorem 10 demonstrates the EIP of the
+actuated buses, notice that the ˜ed and ˆi of (36) are
+not included in the supply rate wd in (43). As such, an
+investigation of the zero state dynamics of the DGU is
+required.
+Proposition 11 (ZSO DGUs). The shifted DGU dynam-
+ics in (36) are ZSO.
+Proof. In (36), set the inputs ˜p∗ ≡ 0, ˜it ≡ 0 and the
+output ˜v ≡ 0. Since τd = 0 and ˜IL(0) = 0, verify from the
+equation for ˙˜v that ˜i ≡ 0. From the equation for ˙˜i, it then
+follows that ˜ed ≡ 0 which concludes this proof.
+■
+Remark
+11
+(Compensating
+non-passive
+loads).
+As
+demonstrated in [11], adding a term dependent on ˙vk to
+the regulator output vs,k in (19) allows for damping to
+be added to the unactuated state vk. This in turn allows
+for regulation in the presence of non-passive loads and
+can yield more favourable passivity indices when applying
+Theorem 10.
+3) Line Passivity: The dynamics of the line subsystem
+(18) shifted to the equilibrium (ˆit, ˆv) yield
+Lkl˙˜it = −Rkl˜it + eT
+P,kl˜v,
+(44)
+which can now be analysed for passivity.
+
+9
+
+
+˙˜ed
+L˙˜i
+Ceq ˙˜v
+
+=
+
+
+0
+−v
+−ˆi
+kI
+d
+˜R−R−kP
+d v
+−1−kP
+c ˆi
+0
+1
+−
+˜IL(˜v)
+˜v
+
+
+�
+��
+�
+Ad(v,ˆi,
+˜IL(˜v)
+˜v
+)
+
+
+˜ed
+˜i
+˜v
+
+
+����
+˜xd
++
+
+
+1
+kP
+d
+0
+
+
+����
+bd,1
+˜p∗ −
+
+
+0
+0
+1
+
+
+����
+bd,2
+eT
+P,k˜it +
+
+
+ˆp∗ − ˆvˆi
+kI
+c ˆed + ( ˜R−R)ˆi − ˆv + vRef − kP
+c (ˆp∗ − ˆvˆi)
+ˆi − eT
+P,kˆit − IL(ˆv)
+
+
+�
+��
+�
+τd
+(36)
+
+
+Qd(v,ˆi, cL) + ρdcdcT
+d
+Pdbd,1 − 1+νd,1ρd
+2
+cd
+Pdbd,2 − 1
+2cd
+bT
+d,1Pd − 1+νd,1ρd
+2
+cT
+d
+νd,1
+0
+bT
+d,2Pd − 1
+2cT
+d
+0
+νd,2
+
+ ≺ 0,
+Pd ≻ 0
+(38)
+Proposition 12 (OFP lines). The shifted line dynamics
+in (44) are OFP(ρt) with ρt = Rkl w.r.t. the input-output
+pair (eT
+P,kl˜v,˜it) with the storage function
+St = Lkl
+2
+˜i2
+t.
+(45)
+Proof. The proof follows trivially by verifying that
+˙St = ˜iteT
+P,kl˜v − Rkl˜i2
+t =: wt,
+(46)
+where wt in an OFP supply rate as per Definition 2.
+■
+4) Interconnected Microgrid Dissipativity: Having sep-
+arately analysed the subsystems comprising the microgrid,
+we now combine the results to formulate the dissipativity
+of the full microgrid w.r.t. the input-output pair (˜p∗, ˜v).
+For simplicity, we group the buses according to their
+actuation states (13). Thus, ˜p∗ = [ ˜p∗
+α
+T , ˜p∗
+β
+T ]T and ˜v =
+[˜vT
+α , ˜vT
+β ]T have the same dimensions. Note that we include
+the inputs ˜p∗
+β for the unactuated buses in Nβ as provided
+by the four-stage controller (see Fig. 2), even though these
+inputs are not used.
+Proposition 13 (Microgrid dissipativity). A DC mi-
+crogrid comprising DGUs (20), lines (18) and loads (15)
+with an interconnection topology described by a connected
+graph GP is dissipative w.r.t. the supply rate
+wM,αβ = (1 + νd,1ρd)˜p∗
+α
+T ˜vα − νd,1 ˜p∗
+α
+T ˜p∗
+α
+− ρd˜vT
+α ˜vα − ρL˜vT
+β ˜vβ,
+(47)
+if νd,2 + ρt ≥ 0 for the worst-case indices of the buses
+and lines calculated in Prop. 9 (ρL), Prop. 12 (ρt), and
+Theorem 10 (νd,1, νd,2, ρd), i.e.
+νd,1 = min
+k∈Nα νd,1,k, νd,2 = min
+k∈Nα νd,2,k, ρd = min
+k∈Nα ρd,k,
+ρL = min
+k∈Nβ ρL,k,
+ρt = min
+kl∈EP ρt,k.
+(48)
+Proof. Define for the interconnected microgrid the storage
+function
+SM =
+�
+k∈Nα
+Sd,k +
+�
+k∈Nβ
+SL,k +
+�
+kl∈EP
+St,kl.
+(49)
+An upper bound for time derivative of (49) may then be
+found by combining the supply rates in (34), (43) and (46)
+˙SM ≤ (1 + νd,1ρd)˜p∗
+α
+T ˜vα − νd,1 ˜p∗
+α
+T ˜p∗
+α − ρd˜vT
+α ˜vα
++ ˜iT
+t ET ˜v − ˜vT
+α Eα˜it − ˜vT
+β Eβ˜it
+− ρL˜vT
+β ˜vβ − (νd,2 + ρt)˜iT
+t ˜it.
+(50)
+The skew-symmetric interconnection of the nodes and lines
+results in ˜iT
+t ET ˜v = ˜vT
+α Eα˜it + ˜vT
+β Eβ˜it. Furthermore with
+νd,2+ρt ≥ 0, we can drop the unnecessary strictly negative
+˜iT
+t ˜it term and verify that ˙SM ≤ wM,αβ.
+■
+Through Prop. 13, the dissipativity of the entire mi-
+crogrid is formulated using the desired input and output
+vectors. However, the supply rate in (47) is dependent on
+the actuation states of the buses. We now remove this
+dependence by finding a supply rate for a specific bus that
+encompasses both its actuated and unactuated state. By
+considering a quadratic supply rate as a sector condition
+(see [26], [29]), a combined supply rate is found through
+the union of the sectors for the actuated and unactuated
+cases.
+Theorem 14 (Actuation independent passivity). A DC
+microgrid for which Prop. 13 holds is IF-OFP(νd,1, ρd)
+w.r.t. the supply rate
+wM = (1 + νd,1ρd)˜p∗T ˜v − νd,1 ˜p∗T ˜p∗ − ρd˜vT ˜v
+(51)
+if, for an arbitrarily small νL > 0,
+0 ≤ νd,2 + ρt,
+(52)
+0 < ρL < 1,
+(53)
+0 > νd,1.
+(54)
+The proof of Theorem 14 can be found in Appendix A.
+Through (51), we thus show that a single IF-OFP supply
+rate describes the input-output passivity of the entire
+microgrid, irrespective of the states of actuation of the
+buses. This supply rate is derived from the properties of
+the DGUs in Theorem 10 and accounts for the worst-case
+loads.
+Remark 12 (Non-passive loads at DGUs). While (53)
+in Theorem 14 requires strictly passive loads at unactuated
+buses, this is not required for the loads at actuated buses.
+
+10
+Indeed, the loads at DGUs may exhibit a lack of passivity
+with cL < 0. However, this would be reflected by the indices
+obtained in Theorem 10 and the supply rate in (51).
+Remark 13 (Non-static loads). Due to the use of passivity
+in this section, the analysis presented here effortlessly
+extends to the case of dynamic loads. Such dynamic loads
+simply need to exhibit equivalent IFP properties (see e.g.
+Prop. 9) and must be ZSO.
+Remark 14 (Passivity-based controllers). In addition
+to the four-stage controller proposed in this work, the
+passivity formulation of the DC microgrid in Theorem 14
+can be used alongside any other controller which provides
+suitable passivity indices. This includes methods such as
+interconnection and damping assignment passivity-based
+control [24, p. 190] or passivity-based model predictive
+control (see e.g. [36]).
+B. Controller Passivity
+Having analysed the passivity of the microgrid subsys-
+tems and their interconnection, we now investigate the
+passivity properties of the control structure in Section IV.
+This is done successively for each part of the controller:
+the DDA stages, the PI stage and the weighting function.
+1) DDA Passivity: Consider the DDA stages in Fig. 2.
+Proposition 15 (DDA Passivity). The DDA controller
+in (23) with the storage function
+Sa,s =
+1
+2γa
+�
+xT
+a,sxa,s + zT
+a,sza,s
+�
+(55)
+is OFP(ρa), ρa = 1, w.r.t. (ua,s, ya,s) and is ZSO.
+Proof. The time derivative of (55) is
+˙Sa,s = −xT
+a,sxa,s − 1
+γa
+xT
+a,sLC,P xa,s + xT
+a,sua,s
+≤ wa,s := xT
+a,sua,s − xT
+a,sxa,s
+(56)
+since LC,P > 0 and γa > 0, thus verifying the OFP
+property for ya,s = xa,s. Furthermore, the DDA controller
+is ZSO since the system dynamics in (23) is Hurwitz [34,
+Theorem 5].
+■
+The OFP result in Prop. 15 also means that (23) has an
+L2-gain of 1 [28, p. 3]. Note that since the DDA in (23) is
+linear, the properties in Prop. 15 also hold for the shifted
+input-output combination (˜ua,s, ˜ya,s) [28, p. 26].
+2) PI Passivity: The ideal PI controller in (26) with
+ζc = 0 can trivially be shown to be IFP(kP
+c ) for the storage
+function Sc = kIcxTc xc/2. The leaky PI control with ζc > 0
+exhibits the following properties.
+Proposition 16 (Leaky PI Passivity). The leaky PI
+control in (26) with the storage function Sc = kIcxTc xc/2
+is dissipative w.r.t. the supply rate
+wc =
+�
+1+2ζckP
+c
+kIc
+�
+�
+��
+�
+2σc
+uT
+c yc −
+�
+kP
+c +ζckP
+c
+2
+kIc
+�
+�
+��
+�
+νc
+uT
+c uc − ζc
+kIc
+����
+ρc
+yT
+c yc
+(57)
+Proof. Calculate the time derivative of Sc as
+˙Sc
+=
+kI
+cxT
+c uc −ζckI
+cxT
+c xc. Substitute in kI
+cxc = yc −kP
+c uc from
+the output in (26) and simplify to verify that ˙Sc = wc.
+■
+Note that while wc in (57) has a quadratic form, it does
+not directly match the IF-OFP form in Definition 2. How-
+ever, by appropriately weighing the storage function Sc,
+the form in Definition 2 is easily obtained. For simplicity
+and without invalidating the results in the sequel, we omit
+this step here. Furthermore, we note that the linearity of
+(26) ensures that the properties in Prop. 16 also hold for
+the shifted input-output combination (˜uc, ˜yc) [28, p. 26].
+3) Weighting Function Passivity: The derivative of the
+weighting function in (28) is described by (see e.g. Fig. 3)
+dyw
+duw
+= aw + bw tanh2(gw(uw)).
+(58)
+By setting bw > −aw and applying Prop. 5, (28) is found
+to be IF-OFP(νw, ρw) with
+νw = aw,
+ρw =
+1
+aw + bw
+,
+(59)
+VI. Interconnected Stability
+Using the passivity properties of the microgrid and
+controller subsystems obtained in Section V, we now in-
+vestigate the stability of the microgrid and controller
+interconnected as in Fig. 2. However, we note that the
+agent PI controller and the Stage 4 DDA controller ex-
+hibit a cascaded IFP-OFP obstacle (see Prop. 7) if the
+PI controller is ideal (ζc = 0) which prevents a closed-
+loop analysis with dissipativity. Thus, in Section VI-A, we
+derive stability conditions using leaky agent PI controllers
+with ζc > 0.
+A. Leaky PI-Controlled Stability
+Consider the case where the passivity properties of all
+subsystems in Fig. 2 except for the weighting function
+(28) are fixed. Combining the results in Section V with
+Theorem 6, we now determine the weighting function
+parameters which guarantee closed-loop stability.
+Theorem
+17
+(Designed
+closed-loop
+stability).
+The
+closed-loop in Fig. 2 is guaranteed to be asymptotically
+stable for the weighting function parameters aw = νw,
+bw = 1/ρw − aw if a feasible solution is found for
+min
+νw, ρw, di,
+νw + ρw
+s.t.
+Q ≺ 0,
+di > 0,
+i = 1, . . . , 5,
+(60)
+where σw = 1/2(1 + νwρw), σd = 1/2(1 + νd,1ρd), and
+Q=
+
+
+−ρwd1
+d2
+2
+0
+0
+−σwd1
+d2
+2
+−ρad2−νcd3 σcd3
+0
+0
+0
+σcd3
+−ρcd3
+kP
+c d4
+2
+0
+0
+0
+kP
+c d4
+2
+−ρad4−νd,1d5
+σdd5
+−σwd1
+0
+0
+σdd5
+−ρdd5−νwd1
+
+
+(61)
+Proof. Use the supply rates for the DC microgrid in (51),
+the two DDA controllers in (56), the agent PI controller
+
+11
+in (57), and the IF-OFP supply rate for the weighting
+function (59) to construct W in (10). Let the output of
+the PI controller be normalised according to
+yc = kI
+cxc + kP
+c uc = kP
+c (κI
+cxc + uc) = kP
+c yκ
+c .
+(62)
+Furthermore, the five subsystems in Fig. 2 are intercon-
+nected by u = Hy, where
+H =
+
+
+0
+0
+0
+0
+−1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+kP
+c
+0
+0
+0
+0
+0
+1
+0
+
+
+.
+(63)
+Apply Theorem 6, with D as in (9) and simplify Q in (8)
+to obtain (61). This yields the optimisation problem (60),
+where the indices of the weighting function (νw, ρw) are
+configurable. Asymptotic stability is ensured by changing
+the matrix inequality in (7) to a strict inequality and by
+ensuring that any states not present in y are asymptot-
+ically stable. The latter condition is ensured through the
+zero-state analyses in Prop. 11 and Prop. 15 and through
+the condition in Prop. 13. Finally, the parameters aw and
+bw are calculated from (59).
+■
+Through the application of Theorem 17, the parameters
+for the weighting function can thus be designed to ensure
+stability. We highlight that the results in Section V and
+Theorem 17 hold irrespective of the physical or commu-
+nication topologies and are independent of the actuation
+states of the nodes, as long as Assumptions 2 and 3
+hold. Therefore, verifying Theorem 17 ensures robust-
+ness against any changes which do not alter the worst-
+case passivity indices of the respective subsystems (see
+(48)). Note that the presented stability analysis requires
+strictly passive loads and leaky agent PI controllers (see
+Remark 6). As demonstrated via simulation, these require-
+ments are sufficient for stability, but not necessary.
+VII. Simulation
+In this section, we demonstrate the coordination and
+robustness of the proposed control structure by means of
+a Matlab/Simulink simulation using Simscape com-
+ponents. We consider the network comprising 10 buses
+depicted in Fig. 4. In Section VII-A, we describe the setup
+of the simulation along with the various changes that the
+network is subjected to. Next, in Section VII-B, simula-
+tion results are presented for the case where Theorem 17
+holds, i.e. with strictly passive loads and leaky agent PI
+controllers. Finally, in Section VII-C, we show the robust
+stability of the proposed control structure for passive loads
+and ideal agent PI controllers.
+A. Simulation Setup
+The DC microgrid in Fig. 4 is simulated with the
+parameters in Table I. The ZIP load parameters are
+chosen randomly in the specified ranges such that the
+required passivity measures are fulfilled (see Remark 10).
+Table I: Simulation Parameter Values
+Voltages
+vRef = 380 V
+vcrit = 266 V
+DGU Filters (14)
+Rk = 0.2 Ω
+Lk = 1.8 mH
+Ck = 2.2 mF
+ZIP Loads (16)
+|Z−1| ≤ 0.1/Ω
+|I| ≤ 21 A
+|P | ≤ 3 kW
+Elec. Lines (18)
+Rkl = 0.1 Ω/km
+Lkl = 2 µH/km
+Ckl = 22 nF/km
+length ∈ [0.2; 10] km
+Table II: Controller Parameter Values
+Power PI Control (19)
+kP
+d = 90
+kI
+d = 90
+˜R = −8
+DDA Control (23)
+kP
+a = 50
+kI
+a = 100
+γa = 16
+Agent PI Control (26)
+kP
+c = 160
+kI
+c = 600
+ζc = 0.08
+Weighting Function (28)
+aw = 0.1
+bw = 1.1
+cw = 7.5 V
+Furthermore, typical values are used for the DGUs and
+the lines [4], [9], [13]. The lines exhibit the same per
+kilometer parameter values and the line length are chosen
+randomly in the given interval. The line lengths are given
+in Appendix B.
+The simulation starts off in State A (see Fig. 4) with
+Bus 9 connected and with all states at zero. The following
+changes are made at the indicated times.
+• t = 5 s: The actuation states αi of the buses switches
+from State A to State B and Bus 9 is disconnected.
+• t = 10 s: The communication topology switches from
+State A to State B and Bus 10 is connected.
+• t = 15 s: The electrical topology switches from State A
+to State B.
+• t = 20 s: The bus actuation status along with the com-
+munication and electrical topologies revert to State A.
+Bus 9 is connected and Bus 10 is disconnected.
+Furthermore, at each change, half of the buses are ran-
+domly selected and assigned new ZIP load parameters.
+The ZIP load parameters can be found in Appendix B.
+The parameters for the closed-loop controller, as spe-
+cified in Table II, are designed constructively, starting
+from the microgrid subsystems. First, the passivity indices
+for the lines (ρt = 0.01) and loads (ρL = cL = 0.05) are cal-
+culated from Prop. 12 and Prop. 9, respectively. Next, the
+parameters for the power regulator (19) are chosen and the
+DGU passivity indices are calculated from Theorem 10,
+with the optimisation verified for the practically relevant
+intervals v ∈ [200 V, 550 V] and ˆi ∈ [10 A, 350 A]. Note that
+adding the restriction νd,2 ≥ −ρt to the optimisation in
+Theorem 10 ensures that (52) will be met. This yields
+a solution νd,1 = −4.686, νd,2 = −0.01 and ρd = 0.01,
+from which the microgrid supply rate is constructed as
+per Theorem 14. Finally, parameters for the agent PI con-
+trollers are chosen and the weighting function parameters
+are designed using Theorem 17. Note that Theorem 14
+requires strictly passive loads (cL > 0) and Theorem 17
+necessitates leaky integrators (ζc > 0).
+
+12
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+d
+d
+d
+d
+State A
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+d
+d
+d
+d
+State B
+d
+Bus
+Active DGU
+Electrical line
+Communication line
+0
+0.5
+1
+0
+0.5
+1
+Figure 4: Two different states for a 10-bus DC microgrid along with electrical and communication connections. The
+loads at the buses are omitted for clarity.
+0
+5
+10
+15
+20
+25
+340
+360
+380
+400
+420
+Figure 5: Simulated bus voltages with line colours as per
+the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+40
+80
+Figure 6: Simulated weighted voltage errors and the av-
+erage error of connected agents with agent line colours as
+per the legend in Fig. 4.
+B. Results
+The bus voltages vk shown in Fig. 5 confirm the stability
+of the closed loop results, although the voltages tend to
+be lower than desired, due to the use of leaky integrators.
+The remaining steady-state offset can also be seen in the
+weighted errors plotted in Fig. 6, where the average tends
+towards a non-zero value in each instance (see Remark 6).
+Despite this, the four stage controller reaches a consensus
+on the average of the nonlinear weighted voltage errors.
+Moreover, the advantage of the weighting function can
+be seen at Bus 6 in t ∈ [20 s, 25 s), where a significant
+weighted error only appears in Fig. 6 when the voltage in
+0
+5
+10
+15
+20
+25
+0
+1x104
+2x104
+3x104
+4x104
+Figure 7: Simulated outputs of the local agent controllers
+with line colours as per the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+10
+20
+30
+40
+Figure 8: Simulated power setpoints with line colours as
+per the legend in Fig. 4.
+Fig. 5 is not close to vRef. Note that the voltages of Buses 9
+and 10 are at 0 V during the respective periods where they
+are disconnected and not actuated.
+In Fig. 7, the outputs of the agent controllers show that
+no synchronisation of the agent controllers are required.
+The agent controller outputs at Buses 1 to 8, which are
+continuously connected to the communication network,
+are near identical. However, the disconnecting buses, e.g.
+Bus 9 after t = 5 s, rapidly diverge from other controllers
+and do not synchronise on reconnect. Despite this, the
+final stage of the controller ensures cooperation of the
+buses, as demonstrated in the power setpoints p∗
+k in Fig. 8.
+When Bus 10 connects at t = 10 s, its setpoint p∗
+k rapidly
+
+13
+0
+5
+10
+15
+20
+25
+340
+360
+380
+400
+420
+Figure 9: Simulated bus voltages with ideal PI controllers
+and with line colours as per the legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+-40
+0
+40
+Figure 10: Simulated weighted voltage errors and the
+average error of connected agents with ideal PI controllers
+and with agent line colours as per the legend in Fig. 4.
+converges to the coordinated common setpoint used by all
+connected agents.
+Although the leaky integrators yield imperfect results
+(see Remark 6 and Fig. 6), this can be mitigated by
+choosing a higher vRef. Indeed, by combining the steady
+state of the agent PI controller (27) with the DDA steady
+state (24), we see that injecting power into the system
+p∗ > 0 results in positive voltage errors. Since we consider
+(strictly) passive loads, increasing vRef is thus a viable
+method for correcting the imperfect results whilst retain-
+ing the advantageous properties of the stability analysis in
+Theorem 17.
+C. Robustness Test
+We
+now
+repeat
+the
+simulation
+described
+in
+Section VII-A with the following changes. 1) Passive
+loads with cL
+= 0 are allowed at all buses, and 2)
+ideal agent PI controllers with ζc = 0 are used. Under
+these conditions, Theorem 17 can no longer be used to
+verify the stability. However, the stability may still be
+verified using classical approaches such as evaluating
+the eigenvalues for the closed loop linearised about the
+equilibrium. Note that the same random seed is used as
+for the results in Section VII-A, allowing for a comparison
+between the scenarios to be made.
+Fig. 9 demonstrates the improved consensus achieved
+by the ideal PI agents, in that the bus voltages are
+closer to vRef at steady state than in Fig. 5. Moreover,
+0
+5
+10
+15
+20
+25
+0
+1x104
+2x104
+3x104
+Figure 11: Simulated outputs of the local agent controllers
+with ideal PI controllers and with line colours as per the
+legend in Fig. 4.
+0
+5
+10
+15
+20
+25
+0
+10
+20
+30
+Figure 12: Simulated power setpoints with ideal PI con-
+trollers and with line colours as per the legend in Fig. 4.
+Fig. 10 shows that perfect consensus is achieved, where
+the average error tends to zero in each case. This figure
+also demonstrates the robustness against communication
+interruptions, as is the case for Bus 10 which, for the
+period t ∈ [5 s, 10 s), is actuated but does not communicate
+with the other buses. Despite this, it is able to accurately
+regulate its own bus voltage (compared to the imperfect
+regulation achieved with leaky integrators as in Fig. 5).
+The lack of leaky integrators is also evident in Fig. 11,
+where the output of the agent controllers stay constant
+when a bus is disconnected and not actuated. Lastly, the
+power setpoints in Fig. 12 converging to a common value
+for the communicating agents confirm the coordination of
+the agents.
+Note that while tests with non-passive loads can also
+yield a stable closed loop, instability can occur when
+the non-passive loads dominate. To address this, a tar-
+geted compensation of non-passive loads is required (see
+Remark 11).
+VIII. Conclusion
+In this paper, we proposed a four-stage distributed
+control structure that achieves power sharing in a DC mi-
+crogrid while ensuring voltage regulation for the voltages
+of both actuated and unactuated buses. We demonstrated
+how the passivity properties of various subsystems can be
+determined and combined these in a stability analysis that
+
+14
+is independent of topological changes, actuation changes,
+bus connections or disconnections and load changes.
+Future work includes the consideration of non-passive
+loads at arbitrary locations in the microgrid and the
+construction of an interface to allow for the presented work
+to be combined with tertiary optimal controllers.
+Appendix A
+Proofs
+Proof of Prop. 8. For the control structure in steady state,
+˙xc = 0 and thus yc is constant. The steady-state output
+(24) of the Stage 4 DDA therefore ensures Objective 2 is
+achieved. Furthermore, consider the steady state of the
+Stage 2 DDA
+ua,s,k = lim
+t→∞ hw(vRef − vk),
+(64)
+lim
+t→∞ ya,2,k = uT
+a,s1N
+N
+= lim
+t→∞
+1
+N
+�
+k∈N
+(vRef − h(vk)) , (65)
+if vk is in equilibrium and where h is obtained by shifting
+hw by vRef. Note that (65) corresponds to the condition of
+(21) in Objective 1. Therefore, ya,2 specifies the regulation
+error of the average weighted voltage error in steady
+state. From the steady state of the agent PI controller
+in (26), we have ζcxc = ya,2. Thus, ideal integrators with
+ζc = 0 ensure that Objective 1 is met exactly. For ζc > 0,
+substitute the PI equilibrium into the output of the agent
+PI controller in (26) to obtain the steady state equation
+xc = 1
+kIc
+�
+yc + kP
+c ya,2
+�
+.
+(66)
+Substitute ζcxc = ya,2 into (24) and simplify to find
+ya,2 =
+ζc
+kIc(1 + ζckPc )yc,
+(67)
+for the steady state. Since the entries of the vector ya,2 and
+thus of xc and yc are the same at steady state. Therefore
+the steady state output for the Stage 4 DDA in (24) gives
+yc = ya,4, which we combine with (67) to obtain the error
+for Objective 1 in (30).
+■
+Proof of Theorem 14. Consider the supply rates which de-
+scribe the actuated and unactuated states, respectively, for
+a given bus k ∈ N
+wM,α,k = (1+νd,1ρd)˜p∗
+α,k˜vα,k − νd,1(˜p∗
+α,k)2 − ρd˜v2
+α,k, (68)
+wM,β,k = −ρL˜v2
+β,k.
+(69)
+These allow the microgrid supply rate in (47) to be
+decomposed according to the actuation states αk
+wM,αβ =
+�
+k∈Nα
+wM,α,k +
+�
+k∈Nβ
+wM,β,k
+=
+�
+k∈N
+(αkwM,α,k + (1 − αk)wM,β,k)
+(70)
+u
+y
+wM,α,k
+wM,α,k
+wM,β,k/ρL
+Figure 13: Comparison of the microgrid supply rate sectors
+in the proof of Theorem 14 if ρd < 0.
+Enlarge the supply rate of the unactuated buses in (69) by
+adding the positive term νL(˜p∗
+β,k)2 for an arbitrarily small
+νL > 0 such that
+wM,β,k ≤ wM,β,k = νL(˜p∗
+β,k)2 − ρL˜v2
+β,k
+≤ wM,β,k
+ρL
+= νL
+ρL
+(˜p∗
+β,k)2 − ˜v2
+β,k
+(71)
+for ρL as in (53). The supply rate wM,β,k/ρL is equivalent
+to the L2 supply rate in Definition 2 and is thus bounded
+by the sector [−
+�
+νL
+ρL ,
+�
+νL
+ρL ] [29, Lemma 4]. Consider now
+the supply rate of the actuated agents (68) narrowed down
+to an IFP sector for the case that ρd < 0, i.e.
+wM,α,k ≥ wM,α,k =
+�
+wM,α,k,
+if ρd ≥ 0,
+˜p∗
+α,k˜vα,k − νd,1(˜p∗
+α,k)2,
+if ρd < 0,
+(72)
+such that wM,α,k is sector bounded by [νd,1, 1
+ρd ] if ρd > 0
+and [νd,1, ∞) if ρd < 0 or if ρd = 0 (see [26, p. 231]). A re-
+lation bewteen wM,α and wM,β/ρL can now be established
+by comparing their respective sector bounds:
+wM,β,k
+ρL
+≤ wM,α,k if
+
+
+
+[−
+�
+νL
+ρL ,
+�
+νL
+ρL ] ⊆ [νd,1, 1
+ρd ], if ρd > 0,
+[−
+�
+νL
+ρL ,
+�
+νL
+ρL ] ⊆ [νd,1, ∞), if ρd ≤ 0,
+(73)
+Since νL can be arbitrarily small, we derive (54) by
+comparing the lower bounds in (73) and note that the
+upper bound relation can be met for any ρd. A visual
+comparison of the sector conditions is made in Fig. 13.
+The combination of (71)–(73) results in
+wM,β,k ≤ wM,β,k ≤ wM,β,k
+ρL
+≤ wM,α,k ≤ wM,α,k.
+(74)
+Therefore, for the microgrid with the storage function SM
+that is dissipative w.r.t. (47), it holds that
+˙SM ≤ wM,αβ ≤
+�
+k∈N
+wM,α,k = wM,
+(75)
+which is found by combining (70) with (74).
+■
+Appendix B
+Simulation Data
+The
+simulation
+parameters
+used
+for
+the
+lines
+in
+Section VII are given in Table III. Furthermore, the
+
+15
+Table III: Rounded Line Lengths
+Line
+Length
+Line
+Length
+Line
+Length
+1 – 2
+1.19 km
+1 – 4
+7.74 km
+2 – 3
+2.23 km
+2 – 4
+7.20 km
+3 – 5
+3.14 km
+3 – 8
+2.82 km
+4 – 5
+3.72 km
+4 – 6
+6.75 km
+4 – 7
+1.16 km
+6 – 7
+4.44 km
+6 – 9
+3.11 km
+7 – 8
+3.69 km
+8 – 10
+1.21 km
+Table IV: Strictly Passive Load Values
+Bus Parameter
+t = 0 s
+t = 5 s
+t = 10 s
+t = 15 s
+t = 20 s
+Z−1 (1/Ω)
+0.103
+0.103
+0.106
+0.106
+0.083
+1
+I (A)
+4.66
+2.15
+-6.08
+-6.08
+14.45
+P (W)
+3599
+-4055
+4133
+4133
+-4927
+Z−1 (1/Ω)
+0.099
+0.099
+0.096
+0.096
+0.080
+2
+I (A)
+-16.09
+-16.09
+19.68
+19.68
+2.49
+P (W)
+3204
+3204
+2659
+2659
+1346
+Z−1 (1/Ω)
+0.128
+0.105
+0.105
+0.105
+0.096
+3
+I (A)
+10.27
+-0.09
+-0.09
+-0.09
+-11.09
+P (W)
+-1479
+-3659
+-3659
+-3659
+3031
+Z−1 (1/Ω)
+0.079
+0.079
+0.079
+0.079
+0.079
+4
+I (A)
+10.15
+10.15
+10.15
+10.15
+10.15
+P (W)
+-2711
+-2711
+-2711
+-2711
+-2711
+Z−1 (1/Ω)
+0.095
+0.095
+0.095
+0.064
+0.107
+5
+I (A)
+-6.64
+-6.64
+-6.64
+16.68
+2.10
+P (W)
+2768
+2768
+2768
+-3798
+4242
+Z−1 (1/Ω)
+0.089
+0.089
+0.106
+0.103
+0.103
+6
+I (A)
+6.87
+6.87
+7.85
+-5.17
+-5.17
+P (W)
+948
+948
+4321
+370
+370
+Z−1 (1/Ω)
+0.065
+0.092
+0.092
+0.118
+0.118
+7
+I (A)
+11.96
+6.51
+6.51
+2.77
+2.77
+P (W)
+-3624
+-3442
+-3442
+-3890
+-3890
+Z−1 (1/Ω)
+0.102
+0.102
+0.086
+0.086
+0.124
+8
+I (A)
+-16.85
+-16.85
+20.71
+20.71
+-4.68
+P (W)
+3529
+3529
+-4773
+-4773
+-3832
+Z−1 (1/Ω)
+0.111
+0.103
+0.109
+0.077
+0.077
+9
+I (A)
+13.79
+-19.74
+9.53
+1.26
+1.26
+P (W)
+-2645
+1830
+4215
+1549
+1549
+Z−1 (1/Ω)
+0.072
+0.100
+0.100
+0.111
+0.111
+10
+I (A)
+7.77
+9.02
+9.02
+10.98
+10.98
+P (W)
+-3538
+-4143
+-4143
+-2795
+-2795
+strictly passive load parameters for the simulation results
+in Section VII-B and the passive load parameters for
+the results in Section VII-C are given in Table IV and
+Table V, respectively. Note that the P parameter for the
+loads in Table V are the same as listed in Table IV.
+References
+[1] B. Lasseter, “Microgrids [distributed power generation],” in
+Proc. 2001 IEEE Power Engineering Society Winter Meeting,
+vol. 1, 2001, pp. 146–149.
+[2] J. J. Justo, F. Mwasilu, J. Lee, and J.-W. Jung, “AC-microgrids
+versus DC-microgrids with distributed energy resources: A re-
+view,” Renewable and Sustainable Energy Reviews, vol. 24, pp.
+387–405, 2013.
+[3] L. Meng, Q. Shafiee, G. F. Trecate, H. Karimi, D. Fulwani,
+X. Lu, and J. M. Guerrero, “Review on control of DC microgrids
+and multiple microgrid clusters,” IEEE J. of Emerging and
+Selected Topics in Power Electron., vol. 5, no. 3, pp. 928–948,
+2017.
+Table V: Passive Load Values, P as in Table IV
+Bus Parameter
+t = 0 s
+t = 5 s
+t = 10 s
+t = 15 s
+t = 20 s
+1
+Z−1 (1/Ω)
+0.091
+0.093
+0.087
+0.087
+0.063
+I (A)
+4.66
+-8.15
+-6.08
+-6.08
+9.71
+2
+Z−1 (1/Ω)
+0.069
+0.069
+0.071
+0.071
+0.046
+I (A)
+-16.09
+-16.09
+19.68
+19.68
+0.20
+3
+Z−1 (1/Ω)
+0.095
+0.082
+0.082
+0.082
+0.059
+I (A)
+8.91
+-7.12
+-7.12
+-7.12
+-11.09
+4
+Z−1 (1/Ω)
+0.038
+0.038
+0.038
+0.038
+0.038
+I (A)
+8.82
+8.82
+8.82
+8.82
+8.82
+5
+Z−1 (1/Ω)
+0.065
+0.065
+0.065
+0.027
+0.078
+I (A)
+-6.64
+-6.64
+-6.64
+15.25
+2.10
+6
+Z−1 (1/Ω)
+0.071
+0.071
+0.089
+0.102
+0.102
+I (A)
+4.04
+4.04
+7.85
+-9.19
+-9.19
+7
+Z−1 (1/Ω)
+0.029
+0.070
+0.070
+0.079
+0.079
+I (A)
+9.04
+0.89
+0.89
+0.58
+0.58
+8
+Z−1 (1/Ω)
+0.075
+0.075
+0.057
+0.057
+0.111
+I (A)
+-16.85
+-16.85
+20.55
+20.55
+-14.31
+9
+Z−1 (1/Ω)
+0.105
+0.102
+0.061
+0.036
+0.036
+I (A)
+10.71
+-19.75
+9.53
+-0.05
+-0.05
+10 Z−1 (1/Ω)
+0.042
+0.091
+0.091
+0.088
+0.088
+I (A)
+2.53
+2.03
+2.03
+8.34
+8.34
+[4] V. Nasirian, S. Moayedi, A. Davoudi, and F. L. Lewis, “Distrib-
+uted cooperative control of DC microgrids,” IEEE Trans. Power
+Electron., vol. 30, no. 4, pp. 2288–2303, 2015.
+[5] M. Tucci, L. Meng, J. M. Guerrero, and G. Ferrari-Trecate,
+“Stable current sharing and voltage balancing in DC mi-
+crogrids: A consensus-based secondary control layer,” Automat-
+ica, vol. 95, pp. 1–13, 2018.
+[6] J. Zhao and F. Dörfler, “Distributed control and optimization
+in dc microgrids,” Automatica, vol. 61, pp. 18–26, 2015.
+[7] T. Dragičević, X. Lu, J. C. Vasquez, and J. M. Guerrero,
+“DC microgrids—part i: A review of control strategies and
+stabilization techniques,” IEEE Trans. Power Electron., vol. 31,
+no. 7, pp. 4876–4891, 2016.
+[8] J. Kumar, A. Agarwal, and V. Agarwal, “A review on overall
+control of dc microgrids,” J. of Energy Storage, vol. 21, pp. 113–
+138, 2019.
+[9] M. Tucci, S. Riverso, J. C. Vasquez, J. M. Guerrero, and
+G. Ferrari-Trecate, “A
+decentralized
+scalable approach to
+voltage control of DC islanded microgrids,” IEEE Trans. Con-
+trol Syst. Technol., vol. 24, no. 6, pp. 1965–1979, 2016.
+[10] F. Strehle, M. Pfeifer, A. J. Malan, S. Krebs, and S. Hohmann,
+“A scalable port-Hamiltonian approach to plug-and-play voltage
+stabilization in DC microgrids,” in 2020 IEEE Conf. Control
+Technol. and Applications, 2020, pp. 787–794.
+[11] M. Cucuzzella, K. C. Kosaraju, and J. M. A. Scherpen, “Voltage
+control of DC microgrids: Robustness for unknown ZIP-loads,”
+IEEE Control Syst. Lett., vol. 7, pp. 139–144, 2023.
+[12] S. Trip, M. Cucuzzella, X. Cheng, and J. Scherpen, “Distributed
+averaging control for voltage regulation and current sharing in
+DC microgrids,” IEEE Control Syst. Lett., vol. 3, no. 1, pp. 174–
+179, 2019.
+[13] M. Cucuzzella, S. Trip, C. De Persis, X. Cheng, A. Ferrara, and
+A. van der Schaft, “A robust consensus algorithm for current
+sharing and voltage regulation in DC microgrids,” IEEE Trans.
+Control Syst. Technol., vol. 27, no. 4, pp. 1583–1595, 2019.
+[14] M. S. Sadabadi, S. Sahoo, and F. Blaabjerg, “Stability-oriented
+design of cyberattack-resilient controllers for cooperative DC
+microgrids,” IEEE Trans. Power Electron., vol. 37, no. 2, pp.
+1310–1321, 2022.
+[15] R. Han, L. Meng, J. M. Guerrero, and J. C. Vasquez, “Dis-
+tributed nonlinear control with event-triggered communication
+to achieve current-sharing and voltage regulation in DC mi-
+crogrids,” IEEE Trans. Power Electron., vol. 33, no. 7, pp. 6416–
+6433, 2018.
+[16] P. Nahata and G. Ferrari-Trecate, “On existence of equilibria,
+
+16
+voltage balancing, and current sharing in consensus-based DC
+microgrids,” in Proc. Eur. Control Conf. (ECC), 2020, pp. 1216–
+1223.
+[17] P. Nahata, M. S. Turan, and G. Ferrari-Trecate, “Consensus-
+based current sharing and voltage balancing in dc microgrids
+with exponential loads,” IEEE Trans. Control Syst. Technol.,
+vol. 30, no. 4, pp. 1668–1680, 2022.
+[18] C. De Persis, E. Weitenberg, and F. Dörfler, “A power consensus
+algorithm for DC microgrids,” IFAC-PapersOnLine, vol. 50,
+no. 1, pp. 10 009–10 014, 2017, 20th IFAC World Congress.
+[19] B. Fan, S. Guo, J. Peng, Q. Yang, W. Liu, and L. Liu, “A
+consensus-based algorithm for power sharing and voltage reg-
+ulation in dc microgrids,” IEEE Trans. Ind. Inform., vol. 16,
+no. 6, pp. 3987–3996, 2020.
+[20] M. Cucuzzella, K. C. Kosaraju, and J. M. A. Scherpen, “Dis-
+tributed passivity-based control of DC microgrids,” in American
+Control Conf. (ACC), 2019, pp. 652–657.
+[21] F. Dörfler and F. Bullo, “Kron reduction of graphs with ap-
+plications to electrical networks,” IEEE Trans. Circuits Syst.,
+vol. 60, no. 1, pp. 150–163, 2013.
+[22] W. Chen, D. Wang, J. Liu, Y. Chen, S. Z. Khong, T. Başar,
+K. H. Johansson, and L. Qiu, “On spectral properties of signed
+laplacians with connections to eventual positivity,” IEEE Trans.
+Autom. Control, vol. 66, no. 5, pp. 2177–2190, 2021.
+[23] A. J. Malan, M. Pfeifer, and S. Hohmann, “Distributed coordin-
+ation of physically-interconnected multi-agent systems with ac-
+tuated and unactuated agents,” Eur. J. Control, p. 100673,
+2022.
+[24] A. J. van der Schaft, L2-Gain and Passivity Techniques in
+Nonlinear Control, 3rd ed.
+Cham, Switzerland: Springer, 2017.
+[25] M. Arcak and E. D. Sontag, “Diagonal stability of a class of
+cyclic systems and its connection with the secant criterion,”
+Automatica, vol. 42, no. 9, pp. 1531–1537, 2006.
+[26] H. K. Khalil, Nonlinear Systems, 3rd ed.
+Upper Saddle River,
+NJ: Prentice Hall, 2002.
+[27] G. H. H. Hines, M. Arcak, and A. K. Packard, “Equilibrium-
+independent passivity: A new definition and numerical certific-
+ation,” Automatica, vol. 47, no. 9, pp. 1949–1956, 2011.
+[28] M. Arcak, C. Meissen, and A. Packard, Networks of Dissipative
+Systems: Compositional Certification of Stability, Performance,
+and Safety, ser. (SpringerBriefs in Control, Automation and
+Robotics).
+New York, NY, USA: Springer, 2016.
+[29] A. J. Malan, P. Jané-Soneira, and S. Hohmann, “Constructive
+analysis and design of interconnected krasovskii passive and
+quadratic dissipative systems,” in Proc. 61th IEEE Conf. Decis.
+Control (CDC), 2022.
+[30] P. Moylan and D. Hill, “Stability criteria for large-scale sys-
+tems,” IEEE Trans. Autom. Control, vol. 23, no. 2, pp. 143–149,
+1978.
+[31] M. Benzi, G. H. Golub, and J. Liesen, “Numerical solution of
+saddle point problems,” Acta Numerica, vol. 14, p. 1–137, 2005.
+[32] F. Strehle, A. J. Malan, S. Krebs, and S. Hohmann, “Passivity
+conditions for plug-and-play operation of nonlinear static AC
+loads,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 12 237–12 243,
+2020, 21st IFAC World Congress.
+[33] J. Machowski, J. W. Bialek, and J. R. Bumby, Power System
+Dynamics: Stability and Control, 2nd ed.
+Chichester, United
+Kingdom: John Wiley & Sons, Ltd., 2008.
+[34] R. A. Freeman, P. Yang, and K. M. Lynch, “Stability and con-
+vergence properties of dynamic average consensus estimators,”
+in Proc. 45th IEEE Conf. Decis. Control (CDC), 2006, pp. 338–
+343.
+[35] E. Weitenberg, Y. Jiang, C. Zhao, E. Mallada, F. Dörfler, and
+C. De Persis, “Robust decentralized frequency control: A leaky
+integrator approach,” in Proc. Eur. Control Conf. (ECC), 2018,
+pp. 764–769.
+[36] T. Raff, C. Ebenbauer, and P. Allgöwer, Nonlinear Model Pre-
+dictive Control: A Passivity-Based Approach.
+Berlin, Heidel-
+berg: Springer, 2007, pp. 151–162.
+
diff --git a/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/load_file.txt b/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..18a742adf2950e2298e6a7b1c6e4c895d9e0d55e
--- /dev/null
+++ b/dNFRT4oBgHgl3EQfTjeD/content/tmp_files/load_file.txt
@@ -0,0 +1,1282 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf,len=1281
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='13533v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='SY]  31 Jan 2023  1  Passivity-based power sharing and voltage regulation  in DC microgrids with unactuated buses  Albertus Johannes Malan, Pol Jané-Soniera, Felix Strehle, and Sören Hohmann  Abstract—In this paper, we propose a novel four-  stage distributed controller for a DC microgrid that  achieves power sharing and average voltage regulation  for the voltages at actuated and unactuated buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The  controller is presented for a DC microgrid compris-  ing multiple distributed generating units (DGUs) with  time-varying actuation states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' dynamic RLC lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' non-  linear constant impedance, current and power (ZIP)  loads and a time-varying network topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The con-  troller comprising a nonlinear gain, PI controllers, and  two dynamic distributed averaging stages is designed  for asymptotic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This constitutes first deriving  passivity properties for the DC microgrid, along with  each of the controller subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thereafter, design  parameters are found through a passivity-based optim-  isation using the worst-case subsystem properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The  resulting closed-loop is robust against DGU actuation  changes, network topology changes, and microgrid  parameter changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The stability and robustness of the  proposed control is verified via simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Index Terms—DC microgrids, distributed control,  passivity, power sharing, voltage regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Introduction  T  HE ADVENT of localised power generation and stor-  age increasingly challenges the prevailing centralised  power-generation structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Originally proposed in [1],  the microgrids paradigm envisions networks that can oper-  ate autonomously through advanced control while meeting  consumer requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Although current electrical grids  predominantly use AC, high and low voltage DC networks  have been made technically feasible due to the continual  improvements of power electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Indeed, DC microgrids  exhibit significant advantages over their AC counterparts,  demonstrating a higher efficiency and power quality while  simultaneously being simpler to regulate [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In microgrids, power generation and storage units  are typically grouped into distributed generation units  (DGUs) which connect to the microgrid through a single  DC-DC converter for higher efficiency [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This changes  the traditionally centralised regulation problem in power  grids into a problem of coordinating the DGU connected  throughout the microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This coordination is generally  This work was supported in part by Germany’s Federal Ministry  for Economic Affairs and Climate Action (BMWK) through the  RegEnZell project (reference number 0350062C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (Corresponding  author: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=')  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Jané-Soniera, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Strehle, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hohmann  are with the Institute of Control Systems (IRS), Karlsruhe In-  stitute of Technology (KIT), 76131, Karlsruhe, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Emails:  albertus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='malan@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='edu, pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='soneira@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='edu, felix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='strehle@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='edu,  soeren.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='hohmann@kit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  realised as average or global voltage regulation in combina-  tion with load sharing between the DGUs (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [4]–[6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Literature Review: A vast number of approaches have  been proposed for the voltage regulation and load sharing  of DC microgrids, as detailed in the overview papers [3],  [7], [8] along with the sources therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' These approaches  are broadly categorised as either centralised, decentralised  or distributed in nature [3], [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' While centralised  controllers can optimally coordinate the DGUs, they offer  reduced scalability and flexibility and have a single point  of failure [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' On the other hand, decentralised controllers  either only attempt to achieve voltage stability [9]–[11]  or achieve load sharing at the cost of voltage regulation  quality (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the droop-based approaches in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In response to these limitations, numerous controllers  for voltage regulation and load sharing which operate in a  distributed manner have been proposed [4]–[6], [12]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In [4], distributed averaging is employed to find a global  voltage estimate with which voltage regulation is achieved,  but the microgrid dynamics are neglected in the stability  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Distributed averaging with dynamic microgrid  models is used in [5], [12], although [5] requires LMIs to  be solved before buses are allowed to connect whereas  [12] only considers constant current loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Similarly, a  sliding-mode controller is proposed in [13] for a dynamic  microgrid with constant current loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' On the other hand,  [14] proposes a cyberattack-resilient controller for a mi-  crogrid with constant conductance loads and resistive  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A consensus-based distributed controller with event-  triggered communication is presented in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consensus-  based controllers are also utilised in [6], [16], [17], where  [6] uses a consensus-based integral layer on top of a droop-  based controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, while many contributions strive  to achieve proportional current sharing [4]–[6], [12]–[17],  [20], nonlinear controllers that achieve proportional power  sharing have also been proposed in [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  While the literature listed above differ greatly in their  approaches, we note a commonality in their omission of  buses without actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This omission is typically mo-  tivated either by considering a microgrid comprising only  actuated DGU buses [4], [5], [16], [17], or by eliminating the  unactuated buses with the Kron-reduction [6], [12]–[15],  [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, considering a network comprising only  actuated buses severely limits the flexibility of a microgrid,  since each bus must be able to supply or consume enough  power at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' On the other hand, the Kron-reduction  requires loads to be described as positive conductances  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' While research into Kron-reduced networks  with negative loads is ongoing (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [22]), the general  2  inclusion of negative loads, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' non-controllable power  sources, in Kron-reducible networks remains out of reach  at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, consider the case where a DGU  can no longer supply or consume the required amount of  power, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' a fully charged or discharged battery storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Such a DGU then loses the ability to regulate itself and  fully support the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In the approaches considered above  [4]–[6], [12]–[20], such a DGU is forced to disconnect from  the microgrid and its local measurements are discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  For DGUs with intermittent power sources, this could  result in significant swings in the number of controlled and  observed buses in the microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Main Contribution:  In this paper, we consider a  DC microgrid as a physically interconnected multi-agent  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Extending our work in [23]1, we propose a four-  stage controller that achieves voltage regulation and power  sharing in a DC microgrid with actuated and unactuated  buses in a distributed manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The four-stage controller  comprises a nonlinear weighting function, two dynamic  distributed averaging (DDA) stages and a proportional-  integral (PI) controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The asymptotic stability of the  closed loop comprising the DC microgrid and the four-  stage controller interconnected in feedback is proven by  means of passivity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In detail, the contributions  comprise:  1) A four-stage distributed controller for DC microgrids  which achieves consensus on the weighted average  voltage error of actuated and unactuated buses and  assures coordination through power sharing at the  actuated buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  2) A nonlinear weighting function that penalises voltage  errors outside a given tolerance band more strongly  than those within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  3) Passivity classifications for each of the constitutive  microgrid subsystems (DGUs, loads, and lines) and  for each of the controller stages (weighting function,  DDA, and PI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  4) A  method  for  calculating  the  input-feedforward  output-feedback passive (IF-OFP) indices of the non-  linear power-controlled DGUs through optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  5) An IF-OFP formulation for the DC microgrid with  a supply rate that is independent of the network  topology, the number of buses and their states of  actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  6) A passivity-based stability analysis for the equilib-  rium of the DC microgrid connected in feedback with  the four-stage controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In addition to the contributions listed above, we also  contribute a theoretical result comprising a formalisation  of the obstacle presented by cascaded input-feedforward  passive (IFP) and output-feedback passive (OFP) systems  in the analysis of dissipative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This theoretical  1The controller proposed in [23] is extended by weighing the  error with a nonlinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover, in addition to applying  the controller to a DC microgrid context, we here propose a new  dissipativity-based analysis that investigates the closed loop stability  analytically as opposed to the numerical results in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  contribution informs and motivates parameter choices for  the four-stage controller in Contribution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  We highlight that the proposed controller can achieve  exact voltage regulation and power sharing with the  stability verified with the eigenvalues of the linearised  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover, by employing leaky PI controllers, we  demonstrate a passivity-based stability analysis that is  independent of and robust against changes in the commu-  nication topology, changes in the electrical topology, load  changes, changes in the actuation status of DGUs, uncer-  tainties in component parameters, and buses connecting  or disconnecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Paper Organisation: The introduction concludes with  some notation and preliminaries on graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In  Section II, we recall and introduce results relating to  dissipativity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Next, in Section III, the problem is  modelled and objectives for the steady state are formal-  ised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In Section IV, a four-stage control structure is intro-  duced that fulfils objectives from Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thereafter,  the passivity properties of the constituent subsystems are  investigated in Section V and the controller is designed  for asymptotic stability of the closed loop in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Finally, in Section VII, a simulation is used to verify the  asymptotic stability and robustness of the closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Concluding remarks are provided in Section VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Notation and Preliminaries: Define as a vector a =  (ak) and a matrix A = (akl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1k is a k-dimensional vector  of ones and Ik is the identity matrix of dimension k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Diag[·] creates a (block-)diagonal matrix from the supplied  vectors (or matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The upper and lower limits of a value  a are given by a and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For a variable x, we denote its  unknown steady state as ˆx, its error state as ˜x := x − ˆx,  and a desired setpoint as x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Whenever clear from context,  we omit the time dependence of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  We denote by G = (N, E) a finite, weighted, undirected  graph with vertices N and edges E ⊆ N × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let |N| be  the cardinality of the set N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let L be the Laplacian matrix  of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' By arbitrarily assigning directions to each edge in E,  the incidence matrix E ∈ R|N|×|E| of G is defined by  ekl =  \uf8f1  \uf8f2  \uf8f3  +1  if vertex k is the sink of edge l,  −1  if vertex k is the source of edge l,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (1)  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dissipativity Preliminaries  We here recall and introduce preliminaries of dissip-  ativity theory for nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In Section II-A we  provide definitions relating to dissipativity and passiv-  ity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thereafter in Section II-B, we investigate  the passivity properties of static functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, in  Section II-C, we recall a result on the interconnection  of dissipative systems with quadratic supply rates and  formalise a new result on the limitations of such an  interconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dissipative Systems  Consider a nonlinear system  �  ˙x = f(x, u),  y = h(x),  (2)  where x ∈ Rn, u ∈ Rm, y ∈ Rm and where f : Rn×Rm →  Rn and h : Rn × Rm → Rm are class C1 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Definition 1 (Dissipative system, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [24]–[26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A system  (2) with a class C1 storage function S : Rn × Rm → R+ is  dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' a supply rate w(u, y) if ˙S ≤ w(u, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Definition 2 (Quadratic supply rates, c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [24]–[26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A  system (2) that is dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' w(u, y) is  passive if w = uTy,  input-feedforward passive (IFP) if w = uTy − νuT u,  output-feedback passive (OFP) if w = uT y − ρyT y,  input-feedforward output-feedback passive (IF-OFP) if  w = (1 + νρ)uT y − νuT u − ρyT y,  has an L2-gain of γL2 if w = γ2  L2uTu − yT y,  where γL2 > 0 and ν, ρ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Definition 3 (Zero-state observable (ZSO) [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A  system (2) is ZSO if u ≡ 0 and y ≡ 0 implies x ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  For cases where the desired equilibrium of a system is  not at the origin but at some constant value, the shifted  passivity [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 96] or equilibrium-independent passivity  (EIP) [27] of a system must be investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Naturally, this  requires that an equilibrium exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' there is a unique  input ˆu ∈ Rm for every equilibrium ˆx ∈ ˆ  X ⊂ Rn such that  (2) produces f(ˆx, ˆu) = 0 and ˆy = h(ˆx, ˆu) [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Definition 4 (EIP [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A system (2) is EIP  if there exists a class C1 storage function S(x, ˆx, u),  S : Rn × ˆ  X × Rm → R+, with S(ˆx, ˆx, ˆu) = 0, that is dis-  sipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' w(u − ˆu, y − ˆy) for any equilibrium (ˆu, ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Passive Static Functions  Recall that a sector-bounded static nonlinear function  is dissipative to a supply rate defined by the sector bound  [26, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We now consider the arbitrarily shifted  single-input single-output function  �  y = h(u),  u, ˆu ∈ U,  y, ˆy ∈ Y,  h : U → Y,  ˜y = ˜h(˜u) := h(u) − h(ˆu) = y − ˆy,  ˜u := u − ˆu  (3)  and show how its dissipativity properties may be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 5 (EIP static functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A static function  (3) of class C0 is IF-OFP(c, 1/c) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the arbitrarily  shifted input-output pair (˜u, ˜y) if  c ≤ dh(u)  du  ≤ c,  ∀u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (4)  and 0 < c < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider for (3) the slope between an arbitrary  shift (ˆu, ˆy) ∈ U ×Y and a point (u, y), for which the upper  and lower bounds are given by  c ≤ y − ˆy  u − ˆu ≤ c,  ∀(u, y), (ˆu, ˆy) ∈ U × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (5)  Changing to the shifted variables ˜u and ˜y as in (5) and  multiplying through by ˜u2 yields  c˜u2 ≤ ˜u˜y ≤ c˜u2 ⇐⇒ (˜y − c˜u)(˜y − c˜u) ≤ 0  ⇐⇒ (˜y − c˜u)(1  c ˜y − ˜u) ≤ 0,  (6)  for c > 0, which describes an IF-OFP function (see [26,  p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 231]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, through the mean value theorem, the  bounds in (5) may be found from (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  We note that the restrictions on c in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5 are needed  from a computational point of view (c < ∞) and to ensure  that the passivity indices correspond to the correct sector2  (c > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, this limits the passivity properties  attainable through Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5 to ρ = 1/c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 1 (Symmetrical sectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Placing the additional  restriction c = −c in (4) results in the Lipschitz continuity  of h(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover, this implies that the arbitrarily shifted  function ˜h(˜u) has a finite L2-gain of c [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Interconnected Quadratic Dissipative Systems  Building upon the results on the interconnection of  dissipative systems in [28], [30], we now provide a method  for finding dissipativity properties for a subset of the inter-  connected subsystems such that interconnected stability is  guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Specifically, we look for the dissipative supply  rates that restrict the subset of subsystems as little as pos-  sible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For a set S of subsystems, define u = [uT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' , uT  |S|]T  and y = [yT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' , yT  |S|]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Theorem 6 (Minimally restrictive stabilising indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Consider |S| subsystems of the form (2) which are dissipat-  ive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the supply rates wi = 2σiuT  i yi−νiuT  i ui−ρiyT  i yi  and are linearly interconnected according to u = Hy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The  stability of the interconnected system is guaranteed if there  exists a D and νj, ρj ∈ R with j ∈ J such that  min  D, νj, ρj,  j∈J  �  j∈J  (νj + ρj)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  σj = 1/2(1 + νjρj),  j ∈ J,  Q ≼ 0,  D2 ≻ 0  (7)  where the subsystems with configurable supply rates are  represented by the set J ⊂ S, and  Q :=  �H  I  �T  DWD  �H  I  �  (8)  D := Diag[dT , dT ],  d = (  �  di),  (9)  W :=  �− Diag[νi]  Diag[σi]  Diag[σi]  − Diag[ρi]  �  ,  i ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (10)  2Consider e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the sector Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5 would yield if c ≤ c < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  4  The proof for Theorem 6 follows analogously to the  proof of [29, Theorem 13] with application of [29, Re-  mark 5] and is thus omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that if J = ∅  in (7), Theorem 6 can be used to verify the stability of  interconnected dissipative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Despite the design flexibility provided by Theorem 6,  certain cascade configurations present obstacles to the ap-  plication of dissipativity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The following proposition  formalises the problem presented by one such configura-  tion which arises in the sequel and is used to inform the  control design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 7 (Non-dissipativity of cascaded IFP-OFP  systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider |S| ≥ 2 subsystems (2) which are  dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' wi = 2σiuT  i yi − νiuT  i ui − ρiyT  i yi and  linearly interconnected according to u = Hy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let i = 1 and  i = 2 arbitrarily denote subsystems that are IFP and OFP,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' If these systems are connected in exclusive  casade and do not form a feedback connection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  H =  \uf8ee  \uf8f0  0  0  ∗  1  0  0  0  ∗  ∗  \uf8f9  \uf8fb ,  (11)  then investigating stability via separable storage functions  as in Theorem 6 fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Evaluating the stability criteria in (7) under the  imposed IFP and OFP conditions yields the Q (8) entries  q11 = d1ρ1 + d2ν2 = 0,  q12 = q21 = d2σ2  2  = d2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (12)  Since di > 0, Q constitutes an indefinite saddle-point mat-  rix [31, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='4], violating the requirement in (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Remark  2  (Non-separable  storage  functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The  obstacle in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7 arises due to the storage functions being  compartmentalised by the subsystem boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' While the  separability of storage functions is a central motivation for  the use of dissipativity theory, forgoing this allows for a  stability analysis through less conservative methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  the KYP lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Problem Description  In this section, the components comprising the DC mir-  crogrid are introduced in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This is followed by  Section III-B, where controllers are added which regulate  the output power of actuated buses in order to facilitate  power sharing in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, we formulate the  coordination and cooperation goals as a control problem  in Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' DC Network  We consider a DC microgrid comprising N = |N| buses  connected by via π-model electrical lines, as depicted in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let the graph GP = (N, EP) describe the intercon-  nection with N as the set of buses and EP as the set of  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Without loss of generalisation, we allow each node to  inject power through a DC-DC buck converter connected  via a lossy LC-filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that a time-averaged model (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [12]) is used for the buck converter and the energy  source is assumed to be ideal but finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Let the buses be split into an actuated set Nα and  an unactuated set Nβ, according to whether the buck  converter can freely regulate the amount of power injected  at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Buses may freely switch between the sets  Nα and Nβ, but Nα ∩ Nβ = ∅ and Nα ∪ Nβ = N always  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' To characterise this actuation state of a bus, define  the piecewise-constant, time-varying actuation parameter  αk(t) as  αk(t) :=  � 1,  k ∈ Nα,  0,  k ∈ Nβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (13)  Note that we omit the time dependence of αk in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The dynamics for actuated buses with DGUs, where  αk = 1 with k ∈ Nα are described by  �  Lk˙ik  Ceq,k ˙vk  �  =  �−Rk  −1  1  0  ��ik  vk  �  +  �  vs,k  −eT  P,kit − IL,k(vk)  �  (14)  where Ceq,k = Ck + 1/2eT  P,k Diag[Ckl]eP,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ck, Ckl, Lk > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ik ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' and vk ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The line currents it connect to the  capacitor voltages according to incidence matrix EP =  (eT  P,k) of GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The dynamics of the unactuated load buses  with αk = 0 correspond to the simplified system  Ceq,k ˙vk = −eT  P,kit − IL,k(vk),  k ∈ Nβ  (15)  In both the actuated (14) and unactuated (15) cases, the  loads are considered static, nonlinear voltage-dependent  current sources which are described by class C0 functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In this work, we utilise the standard ZIP-model comprising  constant impedance, constant current and constant power  parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that other continuous functions may also be  used without restriction3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As described in [33, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 110–  112], we define a critical voltage vcrit, typically set to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='7vRef, below which the loads are purely resistive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thus,  IL,k(vk) =  \uf8f1  \uf8f2  \uf8f3  Z−1  k  vk + Ik + Pk  vk  ,  vk ≥ vcrit,  Z−1  crit,k · vk,  vk < vcrit,  (16)  Z−1  crit,k := IL,k(vcrit)  vcrit  = Z−1  k  + Ik  vcrit  + Pk  v2  crit  ,  (17)  describes a static, nonlinear load which conforms to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Lastly, the π-model transmission lines physically con-  necting the nodes are governed by the dynamics  Lkl˙it,kl = −Rklit,kl + eT  P,klv,  kl ∈ EP,  (18)  where it,kl ∈ R, Lkl, Rkl > 0 and (eT  P,kl)T = EP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note  that the line capacitances are included in the equivalent  capacitances Ceq,k at the buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' DGU Power Regulator  To allow for power sharing between the actuated buses  (14) in the sequel, we equip each DGU with a controller  3This includes exponential loads (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  5  Linekl  αk ∈ {0, 1}  p∗  k  vk  −  +  vs,k  ik  Rk  Lk  Ck  IL,k(vk)  +  −  vk  Buckk  Busk  Ckl  2  Rkl  it,kl  Lkl  Ckl  2  Busl  Figure 1: Circuit diagram of a bus comprising a DC-DC buck converter, a filter, and a current source representing a  load, connected to a π-model line (blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the line capacitances considered to be part of the respective buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  that can regulate the injected power to a desired setpoint  p∗  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This regulator has the form  ˙ed,k = αk(p∗  k − pk)  vs,k = kP  d (p∗  k − pk) + kI  ded,k + ˜Rkik + vRef  (19)  where ed ∈ R, pk = vkik is the actual power injected,  ˜R ∈ R is the damping added to the system, and kP  d , kI  d >  0 are the control parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Combining (19) with (14)  yields the nonlinear system describing the actuated agents  k ∈ Nα  \uf8ee  \uf8f0  ˙ed,k  Lk˙ik  Ck ˙vk  \uf8f9  \uf8fb=  \uf8ee  \uf8f0  0  −vk  0  kI  d  ˜Rk − Rk − kP  d vk  −1  0  1  0  \uf8f9  \uf8fb  \uf8ee  \uf8f0  ed,k  ik  vk  \uf8f9  \uf8fb  +  \uf8ee  \uf8f0  p∗  k  kP  d p∗  k + vRef  −eT  k it − IL,k(vk)  \uf8f9  \uf8fb  (20)  Remark 3 (Regulating current or voltage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Without in-  validating the stability analysis in the sequel, the regulator  in (19) can be exchanged for simpler, purely linear current  or voltage regulators (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [9]–[11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 4 (Constrained DGU operation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' If an actuated  DGU cannot provide the desired power p∗  k, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' due to  current, storage or temperature limitations, the DGU may  simply set its actuation state αk = 0 to disable its control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' If  some power can still be supplied, it may simply be regarded  as a negative load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This allows DGUs to contribute to the  power supply of the network, even in the face of control  limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control Problem  A central requirement for DC microgrids is voltage  stability, which requires the bus voltages to remain within  a given tolerance band around the reference vRef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Spe-  cifically, this requirement should be met throughout the  network, and not only at the actuated buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Due to  the presence of lossy lines, power flows are associated  with voltage differences between buses, meaning that  vk → vRef, ∀k ∈ N is not practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ideally, the voltages at  all buses should be arrayed in the tolerance band around  vRef and be as close to vRef as possible4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The manipulated  variables used to achieve this are the power setpoints p∗  k  supplied to the actuated DGUs (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This leads to the  first objective for the control of the DC microgrid, which  involves finding the setpoints p∗  k that ensure the weighted  average voltage equals vRef at steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Objective 1 (Weighted voltage consensus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Find p∗  k s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' lim  t→∞  1  N  �  k∈N  h(vk(t)) = vRef  (21)  for a strictly increasing weighting function h : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  By choosing a nonlinear h, large voltage errors may be  weighed more strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This allows for better utilisation of  the tolerance band since bus voltages can be further from  vRef before registering as a significant error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In addition to Objective 1, it is desired that all actuated  DGUs contribute towards supplying and stabilising this  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ensuring that all DGUs receive the same setpoint  spreads the load across actuated buses, leading to a reduc-  tion in localised stress on the DGUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We thus formulate  the second objective as requiring uniform setpoints for the  DGUs in steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Objective 2 (Cooperative power sharing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  lim  t→∞(p∗  k(t) − p∗  l (t)) = 0,  ∀ k, l ∈ N  (22)  Achieving Objectives 1 and 2 thus yields a controlled  microgrid where the average weighted voltage error of all  buses tends to zero through the coordinated action of the  actuated buses in a distributed fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' These objectives  also allow DGUs to transition seamlessly between actuated  and unactated states and ensure no measurement inform-  ation is discarded simply because a bus cannot regulate  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Notice that disregarding the unactuated buses in  Objectives 1 and 2 yields the objectives typically used in  the literature [4], [6], [12]–[14], [16], [17], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  To achieve these objectives, we make the following  assumptions related to appropriate network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Assumption 1 (Feasible network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The available power  sources can feasibly supply the loads with power over the  4The magnitude of the errors vRef − vk strongly depend on the  loads and line resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Small errors therefore presuppose adequate  network design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  6  hw  hw  DDA2,1  DDA2,N  PI1  PIN  DDA4,1  DDA4,N  DC MG  Stage 1  Stage 2  Stage 3  Stage 4  uw  uw,1  uw,N  yw,1  yw,N  ya,2,1  ya,2,N  yc,1  yc,N  ya,4,1  ya,4,N  p∗  v  −  vRef1N  +  Figure 2: Distributed four-stage control connected in feed-  back to the microgrid and with indicated communication  links  between the local control structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  given electrical network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' a suitable equilibrium for the  microgrid exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Assumption 2 (Number of actuated DGUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' At least one  DGU is actuated at any given time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Nα ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Assumption 3 (Connected topologies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Objectives 1  and 2 only apply to a subset of buses electrically connected  as per GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover, for a distributed control, a connected  communication graph exclusively interconnects the same  subset of buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Note that Assumption 1 is a typically made implicitly or  explicitly in the literature (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the discussion in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Assumptions 2 and 3 further specify requirements that  allow a distributed control to achieve the feasible state  in Assumption 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' by ensuring that at least one source  of stabilisation is present in the network (Assumption 3),  and by ensuring that the coordination corresponds to the  network to be controlled Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 5 (Proportional power sharing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' By normalising  the power setpoint p∗  k and weighing the input in (19)  according to the rated power of a given DGU, Objective 2  automatically describes a proportional power sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' With  reference to Remark 4, this also allows the constrained  DGUs to lower their maximum injectable power instead  of setting the DGUs to the unactuated state αk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We  omit the extension to proportional power sharing in this  work for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control Structure  To meet Objectives 1 and 2, we propose the four-  stage control structure depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This control  structure comprises two DDA implementations separated  by agent PI controllers local to the buses as in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This  is prepended by a nonlinear weighting function hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In the  Sections IV-A, IV-B and IV-C, we successively introduce  these respective subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally in Section IV-D, we  show that the control structure meets the objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' DDA Controller  Consider the communiation graph GC = (N, EC) linking  the buses of the DC microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The communication graph  comprises the same vertices as the physical interconnection  graph GP but possibly with a different topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let LC  denote the Laplacian of GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For Stages 2 and 4 of the  control structure, each agent implements an instance of  the DDA5 described in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The instances of the respective  stages may be combined into vector form as  DDAs  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  � ˙xa,s  ˙za,s  �  =  �  −γaIN −LC,P  LT  C,I  −LC,I  0  ��  xa,s  za,s  �  +  �  γaIN  0  �  ua,s,  ya,s = xa,s,  (23)  where s  ∈  {2, 4} denotes the stage in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2, and  xa,s, za,s ∈ RN are the consensus and integral states  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, γa > 0 is a global estimator  parameter (see [34]), and LC,I = kI  aLC and LC,P = kP  a LC  are Laplacian matrices weighted for the integral and pro-  portional responses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Recall from [34] that a  constant input ua,s yields  lim  t→∞ ya,s,k = uT  a,s1N  N  ,  ∀ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (24)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Agent PI Controller  In Stage 3, we equip each bus k ∈ N with a leaky agent  PI controller similar to the approach in [35]  PIk  �  ˙xc,k = −ζcxc,k + uc,k,  yc,k = kI  cxc,k + kP  c uc,k,  (25)  where xc,k ∈ R, ζc ≥ 0 and kP  c , kI  c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that ζc = 0  reduces (25) to an ideal PI controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The combined form  of the N agent controllers is  ˙xc = −ζcxc + uc,  yc = kI  cxc + kP  c uc  (26)  Remark 6 (Non-ideal integrators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As shown in the  sequel, ideal PI controllers only exhibit an IFP property,  whereas the DDA controller is OFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The interconnection  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2 thus yields a cascaded IFP-OFP structure which  obstructs the dissipativity analysis (see Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The use  of leaky integrators (ζc > 0) overcomes this obstacle at the  cost of negatively affecting the steady-state properties, since  (25) forces the equilibrium  uc = ζcxc  (27)  instead of uc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In the context of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2, this corresponds  to a unwanted steady-state offset for the average weighted  voltage error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 7 (Agent PI controller anti-windup).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' To prevent  controller windup, the input to the PI control in (25) should  be zeroed for any unactuated agents that are disconnected  from the communication network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark  8  (Non-participating agents).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Implementing  (25) at each bus k ∈ N allows for a faster reaction to  disturbances at the cost of controller redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' By setting  ua,4,m := ya,4,m at Stage 4 DDA of the control structure  5We implement the PI-DDA variant proposed in [34] and use the  same communication graph for the proportional and integral terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  7  u  y  hw(u)  dhw(u)  duw  Figure 3: Example of the weighting function hw (28) and  its derivative (58) on a unit grid, with aw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='5, bw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='5  and cw = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  for some agents m ∈ M ⊂ N, the PI control (25)  can be omitted at the agents in M without affecting the  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Nevertheless, the measurements of the buses  in k ∈ M are still included in the Stage 2 DDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that  at least one participating agent PI controller is required  (see [23, Remark 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Weighting Function  To allow for a better utilisation of the tolerance band  around vRef, we desire a weighting function that assigns a  low gain for errors within the tolerance band and a high  gain for larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We therefore define the class C1  function yw,k = hw(uw,k) conforming to (3), where  hw(u) := awu + bwgw(u) − bw tanh(gw(u)),  (28)  gw(u) :=  \uf8f1  \uf8f2  \uf8f3  u + cw,  u < −cw  0,  −cw ≤ u ≤ cw  u − cw,  cw < u  (29)  and where (29) describes a dead-zone parametrised by  cw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' An example of (28) is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3 along with  its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For a strictly increasing function as per  Objective 1, set aw > 0 and bw > −aw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Equilibrium Analysis  In a first step towards analysing the closed loop, we  analyse the assumed equilibrium of the interconnected  microgrid and four-stage controller (see Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Specifically, we verify that the proposed control yields an  equilibrium which satisfies Objectives 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 8 (Controller equilibrium analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Con-  sider the DC microgrid comprising (15), (18), and (20)  which is connected in feedback with the four-stage con-  troller comprising (23), (26), and (28) as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let  Assumptions 1, 2 and 3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Then, Objective 2 is met for  the equilibrium imposed by the control structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover,  Objective 1 is achieved exactly for ideal integrators ζc = 0  in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For lossy integrators with ζc > 0, the remaining  error for Objective 1 is be described by the steady-state  value of ya,2, where  ya,2 =  ζc  kIc(1 + ζckPc )ya,4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (30)  The proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8 can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Through Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8 we thus confirm that the proposed con-  troller yields an equilibrium which meets the requirements,  even though the requirements are not perfectly met when  leaky agent PI controllers are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We also note that  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8 only considers the controlled microgrid already in  equilibrium and does not consider the convergence to the  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 9 (Compensating leaky-integral errors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As in-  dicated by (30) in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8, the leaky agent PI controllers  result in a constant steady-state error for the average  voltage regulation (Objective 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since a positive ya,2 cor-  responds to voltages below the desired vRef, it follows  that setting vRef above the actual desired voltage reference  will result in higher bus voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Changing vRef thus  allows the steady-state effects of the leaky integrators to be  compensated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover, notice that ya,4 is the controller  output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the power setpoint p∗ used for the DGUs  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thus, the error measure in (30), which is  only dependent on the controller output, can be used to  determine the offset to vRef for exact voltage regulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Note, however, that modifying vRef based on p∗ results in  a new loop which requires an additional stability analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Subsystem Passivity Analysis  Having verified whether the desirable steady state is  achieved by the controller, we now set about analysing the  convergence to this steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' With the aim of applying  Theorem 6 for the closed-loop stability, we first analyse the  passivity properties of the individual subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since  the steady-state bus voltages ˆvk are unknown and non-  zero, we investigate the passivity properties shifted to any  plausible point of operation using EIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' To this end, we  construct an EIP formulation for the DC microgrid from  its constitutive elements in Section V-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This is followed  by the respective analyses of the various controller stages  in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that we omit the bus indices k and  l in this section where clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' DC Microgrid Passivity  For the stability of the microgrid at the equilibrium  ˆv, we desire an EIP property relating the shifted input  power setpoints ˜p∗ = p∗ − ˆp∗ to the output voltage errors  ˜v = v − ˆv of all nodes, since this port (˜p∗, ˜v) is used  by the controller in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' To this end, we derive EIP  properties for the load, DGU and line subsystems of the  microgrid, making sure to shift the subsystem dynamics to  the assumed equilibrium in each case (see Assumption 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Thereafter, we combine the results of these subsystems, to  construct an EIP property for the microgrid as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Where applicable, an analysis of the zero-state dynamics is  performed to ensure the eventual stability of the controlled  microgrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  1) Load Passivity: Let the unactuated bus dynamics  in (15) for the buses in Nβ be shifted to the equilibrium  (ˆit, ˆv), yielding  Ceq ˙˜v = −eT  P,k˜it − ˜IL(˜v) + (eT  P,kˆit + IL(ˆv)),  (31)  8  for the static load function shifted according to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In  (31), eT  P,kˆit = −IL(ˆv) since the load is fully supplied by  the cumulative line currents in steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 9 (Load EIP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The shifted load dynamics in  (31) are OFP(ρL) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the input-output pair (−eT  P,k˜it, ˜v)  with ρL = cL the smallest gradient of the static load  function IL(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider the storage function SL along with its  time derivative  SL = Ceq  2 ˜v2,  (32)  ˙SL = −˜veT  P,k˜it − ˜v ˜IL(˜v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (33)  Since the static load function IL(v) is IF-OFP according  to Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5, it is bounded from below by cL˜v2 ≤ ˜v ˜IL(˜v)  (see (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Incorporate this lower bound into (33) to obtain  ˙SL ≤ wL := −˜veT  P,k˜it − cL˜v2  (34)  which yields the OFP property from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Remark 10 (ZIP load passivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9 and (4) demon-  strate that the passivity properties of the unactuated buses  are directly linked to the smallest gradient of the load  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For the ZIP load in (16), this yields  cL = min  �  Z−1, Z−1 −  P  v2  crit  , Z−1  crit  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (35)  Considering the strictly passive case (cL = 0) along with  I, P ≥ 0 yields the passivity condition Z−1v2  crit ≥ P  frequently used in the literature [10], [16], [18]–[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  2) DGU Passivity: Shift the states (e, i, v) and inputs  (p∗, it) of the DGU dynamics in (20) for the buses in Nα to  the respective error variables (˜e,˜i, ˜v) and (˜p∗,˜it) to obtain  (36) on the next page, where the static load function  is incorporated into the matrix Ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, the  measured power p = vi = v(˜i + ˆi) in (19) is left partially  in unshifted variables such that Ad is also dependent on  the unshifted voltage v and the steady-state current ˆi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Note that the constant τd in (36) is found by setting the  error variables (˜p∗,˜it, ˜ed,˜i, ˜v) and their time derivatives to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As such, the constant τd ≡ 0 can be disregarded  in the passivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We now analyse the shifted  nonlinear system in (36) for EIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Theorem 10 (EIP DGUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The shifted DGU dynamics in  (36) are simultaneously IF-OFP(νd,1, ρd) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the input-  output pair (˜p∗, ˜v) and IFP(νd,2) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the input-output  pair (−eT  k ˜it, ˜v), if a feasible solution can be found for  max  Pd, νd,1, νd,2, ρd νd,1 + νd,2 + ρd  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (38) holds ∀ v ∈ V ⊆ R+, ∀ˆi ∈ ˆI ⊆ R  (37)  where Qd(v,ˆi, cL) := PdAd(v,ˆi, cL) + AT  d (v,ˆi, cL)Pd,  Ad(v,ˆi, cL) =  \uf8ee  \uf8f0  0  −v  −ˆi  kI  d  ˜R − R − kP  d v  −1 − kP  c ˆi  0  1  −cL  \uf8f9  \uf8fb ,  (39)  and with νd,1, νd,2, ρd ∈ R, cL as in (4) and cd = [0, 0, 1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider for (36) the storage function  Sd =  \uf8ee  \uf8f0  ˜ed  ˜i  ˜v  \uf8f9  \uf8fb  T  Pd  \uf8ee  \uf8f0  ˜ed  L˜i  Ceq˜v  \uf8f9  \uf8fb ,  (40)  with Pd ≻ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The time derivative of (40) is  ˙Sd =  \uf8ee  \uf8f0  ˜xd  ˜p∗  eT  P,k˜it  \uf8f9  \uf8fb  T\uf8ee  \uf8ef\uf8f0  Qd(v,ˆi,  ˜IL(˜v)  ˜v  )  Pdbd,1  Pdbd,2  bT  d,1Pd  0  0  bT  d,2Pd  0  0  \uf8f9  \uf8fa\uf8fb  \uf8ee  \uf8f0  ˜xd  ˜p∗  eT  P,k˜it  \uf8f9  \uf8fb,  (41)  with ˜xd as in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since it follows from (6) that −˜v ˜IL(˜v) ≤  −cL˜v2, this bound can be incorporated into the inequality  ˙Sd ≤  \uf8ee  \uf8f0  ˜xd  ˜p∗  eT  P,k˜it  \uf8f9  \uf8fb  T\uf8ee  \uf8f0  Qd(v,ˆi, cL)  Pdbd,1  Pdbd,2  bT  d,1Pd  0  0  bT  d,2Pd  0  0  \uf8f9  \uf8fb  \uf8ee  \uf8f0  ˜xd  ˜p∗  eT  P,k˜it  \uf8f9  \uf8fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (42)  The desired IF-OFP and IFP properties for the DGU are  described by the supply rate  wd = (1 + νd,1ρd)˜p∗˜v − νd,1(˜p∗)2 − ρd˜v2  − ˜veT  P,k˜it − νd,2  �  eT  P,k˜it  �2  (43)  These properties are guaranteed, if ˙Sd−wd < 0 for all valid  inputs and outputs and for v ∈ V and ˆi ∈ ˆI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Combining  (42) and (43) in this manner directly leads to constraint  (38) in (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, the objective function in (37) seeks  to find the largest indices for which the constraints are  satisfied in a similar manner to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Although Theorem 10 demonstrates the EIP of the  actuated buses, notice that the ˜ed and ˆi of (36) are  not included in the supply rate wd in (43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As such, an  investigation of the zero state dynamics of the DGU is  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 11 (ZSO DGUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The shifted DGU dynam-  ics in (36) are ZSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In (36), set the inputs ˜p∗ ≡ 0, ˜it ≡ 0 and the  output ˜v ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since τd = 0 and ˜IL(0) = 0, verify from the  equation for ˙˜v that ˜i ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' From the equation for ˙˜i, it then  follows that ˜ed ≡ 0 which concludes this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Remark  11  (Compensating  non-passive  loads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  As  demonstrated in [11], adding a term dependent on ˙vk to  the regulator output vs,k in (19) allows for damping to  be added to the unactuated state vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This in turn allows  for regulation in the presence of non-passive loads and  can yield more favourable passivity indices when applying  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  3) Line Passivity: The dynamics of the line subsystem  (18) shifted to the equilibrium (ˆit, ˆv) yield  Lkl˙˜it = −Rkl˜it + eT  P,kl˜v,  (44)  which can now be analysed for passivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  9  \uf8ee  \uf8f0  ˙˜ed  L˙˜i  Ceq ˙˜v  \uf8f9  \uf8fb=  \uf8ee  \uf8ef\uf8f0  0  −v  −ˆi  kI  d  ˜R−R−kP  d v  −1−kP  c ˆi  0  1  −  ˜IL(˜v)  ˜v  \uf8f9  \uf8fa\uf8fb  �  ��  �  Ad(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='ˆi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ˜IL(˜v)  ˜v  )  \uf8ee  \uf8f0  ˜ed  ˜i  ˜v  \uf8f9  \uf8fb  ����  ˜xd  +  \uf8ee  \uf8f0  1  kP  d  0  \uf8f9  \uf8fb  ����  bd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1  ˜p∗ −  \uf8ee  \uf8f0  0  0  1  \uf8f9  \uf8fb  ����  bd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2  eT  P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='k˜it +  \uf8ee  \uf8f0  ˆp∗ − ˆvˆi  kI  c ˆed + ( ˜R−R)ˆi − ˆv + vRef − kP  c (ˆp∗ − ˆvˆi)  ˆi − eT  P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='kˆit − IL(ˆv)  \uf8f9  \uf8fb  �  ��  �  τd  (36)  \uf8ee  \uf8f0  Qd(v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='ˆi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' cL) + ρdcdcT  d  Pdbd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1 − 1+νd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1ρd  2  cd  Pdbd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2 − 1  2cd  bT  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1Pd − 1+νd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1ρd  2  cT  d  νd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1  0  bT  d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2Pd − 1  2cT  d  0  νd,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2  \uf8f9  \uf8fb ≺ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Pd ≻ 0  (38)  Proposition 12 (OFP lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The shifted line dynamics  in (44) are OFP(ρt) with ρt = Rkl w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the input-output  pair (eT  P,kl˜v,˜it) with the storage function  St = Lkl  2  ˜i2  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (45)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The proof follows trivially by verifying that  ˙St = ˜iteT  P,kl˜v − Rkl˜i2  t =: wt,  (46)  where wt in an OFP supply rate as per Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  4) Interconnected Microgrid Dissipativity: Having sep-  arately analysed the subsystems comprising the microgrid,  we now combine the results to formulate the dissipativity  of the full microgrid w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the input-output pair (˜p∗, ˜v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  For simplicity, we group the buses according to their  actuation states (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thus, ˜p∗ = [ ˜p∗  α  T , ˜p∗  β  T ]T and ˜v =  [˜vT  α , ˜vT  β ]T have the same dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that we include  the inputs ˜p∗  β for the unactuated buses in Nβ as provided  by the four-stage controller (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2), even though these  inputs are not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 13 (Microgrid dissipativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A DC mi-  crogrid comprising DGUs (20), lines (18) and loads (15)  with an interconnection topology described by a connected  graph GP is dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the supply rate  wM,αβ = (1 + νd,1ρd)˜p∗  α  T ˜vα − νd,1 ˜p∗  α  T ˜p∗  α  − ρd˜vT  α ˜vα − ρL˜vT  β ˜vβ,  (47)  if νd,2 + ρt ≥ 0 for the worst-case indices of the buses  and lines calculated in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9 (ρL), Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 12 (ρt), and  Theorem 10 (νd,1, νd,2, ρd), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  νd,1 = min  k∈Nα νd,1,k, νd,2 = min  k∈Nα νd,2,k, ρd = min  k∈Nα ρd,k,  ρL = min  k∈Nβ ρL,k,  ρt = min  kl∈EP ρt,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (48)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Define for the interconnected microgrid the storage  function  SM =  �  k∈Nα  Sd,k +  �  k∈Nβ  SL,k +  �  kl∈EP  St,kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (49)  An upper bound for time derivative of (49) may then be  found by combining the supply rates in (34), (43) and (46)  ˙SM ≤ (1 + νd,1ρd)˜p∗  α  T ˜vα − νd,1 ˜p∗  α  T ˜p∗  α − ρd˜vT  α ˜vα  + ˜iT  t ET ˜v − ˜vT  α Eα˜it − ˜vT  β Eβ˜it  − ρL˜vT  β ˜vβ − (νd,2 + ρt)˜iT  t ˜it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (50)  The skew-symmetric interconnection of the nodes and lines  results in ˜iT  t ET ˜v = ˜vT  α Eα˜it + ˜vT  β Eβ˜it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore with  νd,2+ρt ≥ 0, we can drop the unnecessary strictly negative  ˜iT  t ˜it term and verify that ˙SM ≤ wM,αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Through Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 13, the dissipativity of the entire mi-  crogrid is formulated using the desired input and output  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, the supply rate in (47) is dependent on  the actuation states of the buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We now remove this  dependence by finding a supply rate for a specific bus that  encompasses both its actuated and unactuated state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' By  considering a quadratic supply rate as a sector condition  (see [26], [29]), a combined supply rate is found through  the union of the sectors for the actuated and unactuated  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Theorem 14 (Actuation independent passivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A DC  microgrid for which Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 13 holds is IF-OFP(νd,1, ρd)  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the supply rate  wM = (1 + νd,1ρd)˜p∗T ˜v − νd,1 ˜p∗T ˜p∗ − ρd˜vT ˜v  (51)  if, for an arbitrarily small νL > 0,  0 ≤ νd,2 + ρt,  (52)  0 < ρL < 1,  (53)  0 > νd,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (54)  The proof of Theorem 14 can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Through (51), we thus show that a single IF-OFP supply  rate describes the input-output passivity of the entire  microgrid, irrespective of the states of actuation of the  buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This supply rate is derived from the properties of  the DGUs in Theorem 10 and accounts for the worst-case  loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 12 (Non-passive loads at DGUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' While (53)  in Theorem 14 requires strictly passive loads at unactuated  buses, this is not required for the loads at actuated buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  10  Indeed, the loads at DGUs may exhibit a lack of passivity  with cL < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, this would be reflected by the indices  obtained in Theorem 10 and the supply rate in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 13 (Non-static loads).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Due to the use of passivity  in this section, the analysis presented here effortlessly  extends to the case of dynamic loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Such dynamic loads  simply need to exhibit equivalent IFP properties (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9) and must be ZSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Remark 14 (Passivity-based controllers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In addition  to the four-stage controller proposed in this work, the  passivity formulation of the DC microgrid in Theorem 14  can be used alongside any other controller which provides  suitable passivity indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This includes methods such as  interconnection and damping assignment passivity-based  control [24, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 190] or passivity-based model predictive  control (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Controller Passivity  Having analysed the passivity of the microgrid subsys-  tems and their interconnection, we now investigate the  passivity properties of the control structure in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  This is done successively for each part of the controller:  the DDA stages, the PI stage and the weighting function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  1) DDA Passivity: Consider the DDA stages in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 15 (DDA Passivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The DDA controller  in (23) with the storage function  Sa,s =  1  2γa  �  xT  a,sxa,s + zT  a,sza,s  �  (55)  is OFP(ρa), ρa = 1, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (ua,s, ya,s) and is ZSO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The time derivative of (55) is  ˙Sa,s = −xT  a,sxa,s − 1  γa  xT  a,sLC,P xa,s + xT  a,sua,s  ≤ wa,s := xT  a,sua,s − xT  a,sxa,s  (56)  since LC,P > 0 and γa > 0, thus verifying the OFP  property for ya,s = xa,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, the DDA controller  is ZSO since the system dynamics in (23) is Hurwitz [34,  Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  The OFP result in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 15 also means that (23) has an  L2-gain of 1 [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that since the DDA in (23) is  linear, the properties in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 15 also hold for the shifted  input-output combination (˜ua,s, ˜ya,s) [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  2) PI Passivity: The ideal PI controller in (26) with  ζc = 0 can trivially be shown to be IFP(kP  c ) for the storage  function Sc = kIcxTc xc/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The leaky PI control with ζc > 0  exhibits the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Proposition 16 (Leaky PI Passivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The leaky PI  control in (26) with the storage function Sc = kIcxTc xc/2  is dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' the supply rate  wc =  �  1+2ζckP  c  kIc  �  �  ��  �  2σc  uT  c yc −  �  kP  c +ζckP  c  2  kIc  �  �  ��  �  νc  uT  c uc − ζc  kIc  ����  ρc  yT  c yc  (57)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Calculate the time derivative of Sc as  ˙Sc  =  kI  cxT  c uc −ζckI  cxT  c xc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Substitute in kI  cxc = yc −kP  c uc from  the output in (26) and simplify to verify that ˙Sc = wc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Note that while wc in (57) has a quadratic form, it does  not directly match the IF-OFP form in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' How-  ever, by appropriately weighing the storage function Sc,  the form in Definition 2 is easily obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For simplicity  and without invalidating the results in the sequel, we omit  this step here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, we note that the linearity of  (26) ensures that the properties in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 16 also hold for  the shifted input-output combination (˜uc, ˜yc) [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  3) Weighting Function Passivity: The derivative of the  weighting function in (28) is described by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3)  dyw  duw  = aw + bw tanh2(gw(uw)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (58)  By setting bw > −aw and applying Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5, (28) is found  to be IF-OFP(νw, ρw) with  νw = aw,  ρw =  1  aw + bw  ,  (59)  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Interconnected Stability  Using the passivity properties of the microgrid and  controller subsystems obtained in Section V, we now in-  vestigate the stability of the microgrid and controller  interconnected as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, we note that the  agent PI controller and the Stage 4 DDA controller ex-  hibit a cascaded IFP-OFP obstacle (see Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7) if the  PI controller is ideal (ζc = 0) which prevents a closed-  loop analysis with dissipativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thus, in Section VI-A, we  derive stability conditions using leaky agent PI controllers  with ζc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Leaky PI-Controlled Stability  Consider the case where the passivity properties of all  subsystems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2 except for the weighting function  (28) are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Combining the results in Section V with  Theorem 6, we now determine the weighting function  parameters which guarantee closed-loop stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Theorem  17  (Designed  closed-loop  stability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The  closed-loop in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2 is guaranteed to be asymptotically  stable for the weighting function parameters aw = νw,  bw = 1/ρw − aw if a feasible solution is found for  min  νw, ρw, di,  νw + ρw  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Q ≺ 0,  di > 0,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' , 5,  (60)  where σw = 1/2(1 + νwρw), σd = 1/2(1 + νd,1ρd), and  Q=  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  −ρwd1  d2  2  0  0  −σwd1  d2  2  −ρad2−νcd3 σcd3  0  0  0  σcd3  −ρcd3  kP  c d4  2  0  0  0  kP  c d4  2  −ρad4−νd,1d5  σdd5  −σwd1  0  0  σdd5  −ρdd5−νwd1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  (61)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Use the supply rates for the DC microgrid in (51),  the two DDA controllers in (56), the agent PI controller  11  in (57), and the IF-OFP supply rate for the weighting  function (59) to construct W in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Let the output of  the PI controller be normalised according to  yc = kI  cxc + kP  c uc = kP  c (κI  cxc + uc) = kP  c yκ  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (62)  Furthermore, the five subsystems in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2 are intercon-  nected by u = Hy, where  H =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  0  0  0  −1  1  0  0  0  0  0  1  0  0  0  0  0  kP  c  0  0  0  0  0  1  0  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (63)  Apply Theorem 6, with D as in (9) and simplify Q in (8)  to obtain (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This yields the optimisation problem (60),  where the indices of the weighting function (νw, ρw) are  configurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Asymptotic stability is ensured by changing  the matrix inequality in (7) to a strict inequality and by  ensuring that any states not present in y are asymptot-  ically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The latter condition is ensured through the  zero-state analyses in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 11 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 15 and through  the condition in Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, the parameters aw and  bw are calculated from (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Through the application of Theorem 17, the parameters  for the weighting function can thus be designed to ensure  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We highlight that the results in Section V and  Theorem 17 hold irrespective of the physical or commu-  nication topologies and are independent of the actuation  states of the nodes, as long as Assumptions 2 and 3  hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Therefore, verifying Theorem 17 ensures robust-  ness against any changes which do not alter the worst-  case passivity indices of the respective subsystems (see  (48)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that the presented stability analysis requires  strictly passive loads and leaky agent PI controllers (see  Remark 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' As demonstrated via simulation, these require-  ments are sufficient for stability, but not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Simulation  In this section, we demonstrate the coordination and  robustness of the proposed control structure by means of  a Matlab/Simulink simulation using Simscape com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We consider the network comprising 10 buses  depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' In Section VII-A, we describe the setup  of the simulation along with the various changes that the  network is subjected to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Next, in Section VII-B, simula-  tion results are presented for the case where Theorem 17  holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' with strictly passive loads and leaky agent PI  controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, in Section VII-C, we show the robust  stability of the proposed control structure for passive loads  and ideal agent PI controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Simulation Setup  The DC microgrid in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4 is simulated with the  parameters in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The ZIP load parameters are  chosen randomly in the specified ranges such that the  required passivity measures are fulfilled (see Remark 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Table I: Simulation Parameter Values  Voltages  vRef = 380 V  vcrit = 266 V  DGU Filters (14)  Rk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2 Ω  Lk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='8 mH  Ck = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2 mF  ZIP Loads (16)  |Z−1| ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1/Ω  |I| ≤ 21 A  |P | ≤ 3 kW  Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lines (18)  Rkl = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1 Ω/km  Lkl = 2 µH/km  Ckl = 22 nF/km  length ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 10] km  Table II: Controller Parameter Values  Power PI Control (19)  kP  d = 90  kI  d = 90  ˜R = −8  DDA Control (23)  kP  a = 50  kI  a = 100  γa = 16  Agent PI Control (26)  kP  c = 160  kI  c = 600  ζc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='08  Weighting Function (28)  aw = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1  bw = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='1  cw = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='5 V  Furthermore, typical values are used for the DGUs and  the lines [4], [9], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The lines exhibit the same per  kilometer parameter values and the line length are chosen  randomly in the given interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The line lengths are given  in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The simulation starts off in State A (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4) with  Bus 9 connected and with all states at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The following  changes are made at the indicated times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  t = 5 s: The actuation states αi of the buses switches  from State A to State B and Bus 9 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  t = 10 s: The communication topology switches from  State A to State B and Bus 10 is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  t = 15 s: The electrical topology switches from State A  to State B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  t = 20 s: The bus actuation status along with the com-  munication and electrical topologies revert to State A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Bus 9 is connected and Bus 10 is disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Furthermore, at each change, half of the buses are ran-  domly selected and assigned new ZIP load parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The ZIP load parameters can be found in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The parameters for the closed-loop controller, as spe-  cified in Table II, are designed constructively, starting  from the microgrid subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' First, the passivity indices  for the lines (ρt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='01) and loads (ρL = cL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='05) are cal-  culated from Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 12 and Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Next, the  parameters for the power regulator (19) are chosen and the  DGU passivity indices are calculated from Theorem 10,  with the optimisation verified for the practically relevant  intervals v ∈ [200 V, 550 V] and ˆi ∈ [10 A, 350 A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that  adding the restriction νd,2 ≥ −ρt to the optimisation in  Theorem 10 ensures that (52) will be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This yields  a solution νd,1 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='686, νd,2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='01 and ρd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='01,  from which the microgrid supply rate is constructed as  per Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Finally, parameters for the agent PI con-  trollers are chosen and the weighting function parameters  are designed using Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that Theorem 14  requires strictly passive loads (cL > 0) and Theorem 17  necessitates leaky integrators (ζc > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  12  1  2  3  4  5  6  7  8  9  10  d  d  d  d  State A  1  2  3  4  5  6  7  8  9  10  d  d  d  d  State B  d  Bus  Active DGU  Electrical line  Communication line  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='5  1  Figure 4: Two different states for a 10-bus DC microgrid along with electrical and communication connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The  loads at the buses are omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  0  5  10  15  20  25  340  360  380  400  420  Figure 5: Simulated bus voltages with line colours as per  the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  0  5  10  15  20  25  0  40  80  Figure 6: Simulated weighted voltage errors and the av-  erage error of connected agents with agent line colours as  per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Results  The bus voltages vk shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5 confirm the stability  of the closed loop results, although the voltages tend to  be lower than desired, due to the use of leaky integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The remaining steady-state offset can also be seen in the  weighted errors plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6, where the average tends  towards a non-zero value in each instance (see Remark 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Despite this, the four stage controller reaches a consensus  on the average of the nonlinear weighted voltage errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Moreover, the advantage of the weighting function can  be seen at Bus 6 in t ∈ [20 s, 25 s), where a significant  weighted error only appears in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6 when the voltage in  0  5  10  15  20  25  0  1x104  2x104  3x104  4x104  Figure 7: Simulated outputs of the local agent controllers  with line colours as per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  0  5  10  15  20  25  0  10  20  30  40  Figure 8: Simulated power setpoints with line colours as  per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5 is not close to vRef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that the voltages of Buses 9  and 10 are at 0 V during the respective periods where they  are disconnected and not actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7, the outputs of the agent controllers show that  no synchronisation of the agent controllers are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The agent controller outputs at Buses 1 to 8, which are  continuously connected to the communication network,  are near identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, the disconnecting buses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Bus 9 after t = 5 s, rapidly diverge from other controllers  and do not synchronise on reconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Despite this, the  final stage of the controller ensures cooperation of the  buses, as demonstrated in the power setpoints p∗  k in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  When Bus 10 connects at t = 10 s, its setpoint p∗  k rapidly  13  0  5  10  15  20  25  340  360  380  400  420  Figure 9: Simulated bus voltages with ideal PI controllers  and with line colours as per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  0  5  10  15  20  25  40  0  40  Figure 10: Simulated weighted voltage errors and the  average error of connected agents with ideal PI controllers  and with agent line colours as per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  converges to the coordinated common setpoint used by all  connected agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Although the leaky integrators yield imperfect results  (see Remark 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6), this can be mitigated by  choosing a higher vRef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Indeed, by combining the steady  state of the agent PI controller (27) with the DDA steady  state (24), we see that injecting power into the system  p∗ > 0 results in positive voltage errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since we consider  (strictly) passive loads, increasing vRef is thus a viable  method for correcting the imperfect results whilst retain-  ing the advantageous properties of the stability analysis in  Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Robustness Test  We  now  repeat  the  simulation  described  in  Section VII-A with the following changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1) Passive  loads with cL  = 0 are allowed at all buses, and 2)  ideal agent PI controllers with ζc = 0 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Under  these conditions, Theorem 17 can no longer be used to  verify the stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' However, the stability may still be  verified using classical approaches such as evaluating  the eigenvalues for the closed loop linearised about the  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that the same random seed is used as  for the results in Section VII-A, allowing for a comparison  between the scenarios to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9 demonstrates the improved consensus achieved  by the ideal PI agents, in that the bus voltages are  closer to vRef at steady state than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moreover,  0  5  10  15  20  25  0  1x104  2x104  3x104  Figure 11: Simulated outputs of the local agent controllers  with ideal PI controllers and with line colours as per the  legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  0  5  10  15  20  25  0  10  20  30  Figure 12: Simulated power setpoints with ideal PI con-  trollers and with line colours as per the legend in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 10 shows that perfect consensus is achieved, where  the average error tends to zero in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' This figure  also demonstrates the robustness against communication  interruptions, as is the case for Bus 10 which, for the  period t ∈ [5 s, 10 s), is actuated but does not communicate  with the other buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Despite this, it is able to accurately  regulate its own bus voltage (compared to the imperfect  regulation achieved with leaky integrators as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The lack of leaky integrators is also evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 11,  where the output of the agent controllers stay constant  when a bus is disconnected and not actuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lastly, the  power setpoints in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 12 converging to a common value  for the communicating agents confirm the coordination of  the agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Note that while tests with non-passive loads can also  yield a stable closed loop, instability can occur when  the non-passive loads dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' To address this, a tar-  geted compensation of non-passive loads is required (see  Remark 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Conclusion  In this paper, we proposed a four-stage distributed  control structure that achieves power sharing in a DC mi-  crogrid while ensuring voltage regulation for the voltages  of both actuated and unactuated buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' We demonstrated  how the passivity properties of various subsystems can be  determined and combined these in a stability analysis that  14  is independent of topological changes, actuation changes,  bus connections or disconnections and load changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Future work includes the consideration of non-passive  loads at arbitrary locations in the microgrid and the  construction of an interface to allow for the presented work  to be combined with tertiary optimal controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Appendix A  Proofs  Proof of Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For the control structure in steady state,  ˙xc = 0 and thus yc is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The steady-state output  (24) of the Stage 4 DDA therefore ensures Objective 2 is  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, consider the steady state of the  Stage 2 DDA  ua,s,k = lim  t→∞ hw(vRef − vk),  (64)  lim  t→∞ ya,2,k = uT  a,s1N  N  = lim  t→∞  1  N  �  k∈N  (vRef − h(vk)) , (65)  if vk is in equilibrium and where h is obtained by shifting  hw by vRef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that (65) corresponds to the condition of  (21) in Objective 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Therefore, ya,2 specifies the regulation  error of the average weighted voltage error in steady  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' From the steady state of the agent PI controller  in (26), we have ζcxc = ya,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Thus, ideal integrators with  ζc = 0 ensure that Objective 1 is met exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' For ζc > 0,  substitute the PI equilibrium into the output of the agent  PI controller in (26) to obtain the steady state equation  xc = 1  kIc  �  yc + kP  c ya,2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (66)  Substitute ζcxc = ya,2 into (24) and simplify to find  ya,2 =  ζc  kIc(1 + ζckPc )yc,  (67)  for the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Since the entries of the vector ya,2 and  thus of xc and yc are the same at steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Therefore  the steady state output for the Stage 4 DDA in (24) gives  yc = ya,4, which we combine with (67) to obtain the error  for Objective 1 in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Proof of Theorem 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider the supply rates which de-  scribe the actuated and unactuated states, respectively, for  a given bus k ∈ N  wM,α,k = (1+νd,1ρd)˜p∗  α,k˜vα,k − νd,1(˜p∗  α,k)2 − ρd˜v2  α,k, (68)  wM,β,k = −ρL˜v2  β,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (69)  These allow the microgrid supply rate in (47) to be  decomposed according to the actuation states αk  wM,αβ =  �  k∈Nα  wM,α,k +  �  k∈Nβ  wM,β,k  =  �  k∈N  (αkwM,α,k + (1 − αk)wM,β,k)  (70)  u  y  wM,α,k  wM,α,k  wM,β,k/ρL  Figure 13: Comparison of the microgrid supply rate sectors  in the proof of Theorem 14 if ρd < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Enlarge the supply rate of the unactuated buses in (69) by  adding the positive term νL(˜p∗  β,k)2 for an arbitrarily small  νL > 0 such that  wM,β,k ≤ wM,β,k = νL(˜p∗  β,k)2 − ρL˜v2  β,k  ≤ wM,β,k  ρL  = νL  ρL  (˜p∗  β,k)2 − ˜v2  β,k  (71)  for ρL as in (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' The supply rate wM,β,k/ρL is equivalent  to the L2 supply rate in Definition 2 and is thus bounded  by the sector [−  �  νL  ρL ,  �  νL  ρL ] [29, Lemma 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Consider now  the supply rate of the actuated agents (68) narrowed down  to an IFP sector for the case that ρd < 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  wM,α,k ≥ wM,α,k =  �  wM,α,k,  if ρd ≥ 0,  ˜p∗  α,k˜vα,k − νd,1(˜p∗  α,k)2,  if ρd < 0,  (72)  such that wM,α,k is sector bounded by [νd,1, 1  ρd ] if ρd > 0  and [νd,1, ∞) if ρd < 0 or if ρd = 0 (see [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 231]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A re-  lation bewteen wM,α and wM,β/ρL can now be established  by comparing their respective sector bounds:  wM,β,k  ρL  ≤ wM,α,k if  \uf8f1  \uf8f2  \uf8f3  [−  �  νL  ρL ,  �  νL  ρL ] ⊆ [νd,1, 1  ρd ], if ρd > 0,  [−  �  νL  ρL ,  �  νL  ρL ] ⊆ [νd,1, ∞), if ρd ≤ 0,  (73)  Since νL can be arbitrarily small, we derive (54) by  comparing the lower bounds in (73) and note that the  upper bound relation can be met for any ρd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A visual  comparison of the sector conditions is made in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  The combination of (71)–(73) results in  wM,β,k ≤ wM,β,k ≤ wM,β,k  ρL  ≤ wM,α,k ≤ wM,α,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  (74)  Therefore, for the microgrid with the storage function SM  that is dissipative w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (47), it holds that  ˙SM ≤ wM,αβ ≤  �  k∈N  wM,α,k = wM,  (75)  which is found by combining (70) with (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  ■  Appendix B  Simulation Data  The  simulation  parameters  used  for  the  lines  in  Section VII are given in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Furthermore, the  15  Table III: Rounded Line Lengths  Line  Length  Line  Length  Line  Length  1 – 2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='19 km  1 – 4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='74 km  2 – 3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='23 km  2 – 4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='20 km  3 – 5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='14 km  3 – 8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82 km  4 – 5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='72 km  4 – 6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='75 km  4 – 7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='16 km  6 – 7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='44 km  6 – 9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='11 km  7 – 8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='69 km  8 – 10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='21 km  Table IV: Strictly Passive Load Values  Bus Parameter  t = 0 s  t = 5 s  t = 10 s  t = 15 s  t = 20 s  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='083  1  I (A)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='66  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='08  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='08  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='45  P (W)  3599  4055  4133  4133  4927  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='080  2  I (A)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='49  P (W)  3204  3204  2659  2659  1346  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='096  3  I (A)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  P (W)  1479  3659  3659  3659  3031  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  4  I (A)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  P (W)  2711  2711  2711  2711  2711  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='107  5  I (A)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='10  P (W)  2768  2768  2768  3798  4242  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='103  6  I (A)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='87  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='87  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='17  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='17  P (W)  948  948  4321  370  370  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='092  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='092  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='118  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='118  7  I (A)  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='96  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='51  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='77  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='77  P (W)  3624  3442  3442  3890  3890  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='086  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='086  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='124  8  I (A)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='71  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='71  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  P (W)  3529  3529  4773  4773  3832  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='111  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='109  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='077  9  I (A)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='79  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='74  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='26  P (W)  2645  1830  4215  1549  1549  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='072  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='111  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='111  10  I (A)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='77  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='02  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='02  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='98  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='98  P (W)  3538  4143  4143  2795  2795  strictly passive load parameters for the simulation results  in Section VII-B and the passive load parameters for  the results in Section VII-C are given in Table IV and  Table V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Note that the P parameter for the  loads in Table V are the same as listed in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  References  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lasseter, “Microgrids [distributed power generation],” in  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2001 IEEE Power Engineering Society Winter Meeting,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1, 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 146–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Justo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Mwasilu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lee, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Jung, “AC-microgrids  versus DC-microgrids with distributed energy resources: A re-  view,” Renewable and Sustainable Energy Reviews, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 24, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  387–405, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Meng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Shafiee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Trecate, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Karimi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Fulwani,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guerrero, “Review on control of DC microgrids  and multiple microgrid clusters,” IEEE J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' of Emerging and  Selected Topics in Power Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 928–948,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Table V: Passive Load Values, P as in Table IV  Bus Parameter  t = 0 s  t = 5 s  t = 10 s  t = 15 s  t = 20 s  1  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='091  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='093  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='087  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='087  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='063  I (A)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='66  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='08  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='08  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='71  2  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='069  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='069  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='046  I (A)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='20  3  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='059  I (A)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='91  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='12  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='12  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='12  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='09  4  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='038  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='038  I (A)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='82  5  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='078  I (A)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='64  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='10  6  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='102  I (A)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='04  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='04  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='19  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='19  7  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='029  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='079  I (A)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='58  8  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='075  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='111  I (A)  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='85  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='55  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='55  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='31  9  Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='102  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='036  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='036  I (A)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='71  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='75  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='05  10 Z−1 (1/Ω)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='042  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='091  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='091  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='088  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='088  I (A)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='53  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='03  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='03  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='34  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='34  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Nasirian, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moayedi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Davoudi, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lewis, “Distrib-  uted cooperative control of DC microgrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Power  Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2288–2303, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Tucci, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guerrero, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ferrari-Trecate,  “Stable current sharing and voltage balancing in DC mi-  crogrids: A consensus-based secondary control layer,” Automat-  ica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 95, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1–13, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Zhao and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dörfler, “Distributed control and optimization  in dc microgrids,” Automatica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 61, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 18–26, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dragičević, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Vasquez, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guerrero,  “DC microgrids—part i: A review of control strategies and  stabilization techniques,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Power Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 31,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4876–4891, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Kumar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Agarwal, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Agarwal, “A review on overall  control of dc microgrids,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' of Energy Storage, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 21, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 113–  138, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Tucci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Riverso, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Vasquez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guerrero, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ferrari-Trecate, “A  decentralized  scalable approach to  voltage control of DC islanded microgrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Con-  trol Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1965–1979, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Strehle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Pfeifer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Krebs, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hohmann,  “A scalable port-Hamiltonian approach to plug-and-play voltage  stabilization in DC microgrids,” in 2020 IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' and Applications, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 787–794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cucuzzella, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Kosaraju, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Scherpen, “Voltage  control of DC microgrids: Robustness for unknown ZIP-loads,”  IEEE Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 139–144, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Trip, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cucuzzella, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cheng, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Scherpen, “Distributed  averaging control for voltage regulation and current sharing in  DC microgrids,” IEEE Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 174–  179, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cucuzzella, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Trip, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' De Persis, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cheng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ferrara, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' van der Schaft, “A robust consensus algorithm for current  sharing and voltage regulation in DC microgrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1583–1595, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Sadabadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Sahoo, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Blaabjerg, “Stability-oriented  design of cyberattack-resilient controllers for cooperative DC  microgrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Power Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 37, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  1310–1321, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Han, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guerrero, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Vasquez, “Dis-  tributed nonlinear control with event-triggered communication  to achieve current-sharing and voltage regulation in DC mi-  crogrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Power Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 33, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6416–  6433, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Nahata and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ferrari-Trecate, “On existence of equilibria,  16  voltage balancing, and current sharing in consensus-based DC  microgrids,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (ECC), 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1216–  1223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Nahata, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Turan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ferrari-Trecate, “Consensus-  based current sharing and voltage balancing in dc microgrids  with exponential loads,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1668–1680, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [18] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' De Persis, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Weitenberg, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dörfler, “A power consensus  algorithm for DC microgrids,” IFAC-PapersOnLine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 50,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 10 009–10 014, 2017, 20th IFAC World Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [19] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Peng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Liu, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Liu, “A  consensus-based algorithm for power sharing and voltage reg-  ulation in dc microgrids,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Inform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 16,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 3987–3996, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Cucuzzella, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Kosaraju, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Scherpen, “Dis-  tributed passivity-based control of DC microgrids,” in American  Control Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (ACC), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 652–657.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [21] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dörfler and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Bullo, “Kron reduction of graphs with ap-  plications to electrical networks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Circuits Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=',  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 60, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 150–163, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [22] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Khong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Başar,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Johansson, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Qiu, “On spectral properties of signed  laplacians with connections to eventual positivity,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2177–2190, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Pfeifer, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hohmann, “Distributed coordin-  ation of physically-interconnected multi-agent systems with ac-  tuated and unactuated agents,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 100673,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' van der Schaft, L2-Gain and Passivity Techniques in  Nonlinear Control, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Cham, Switzerland: Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Arcak and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Sontag, “Diagonal stability of a class of  cyclic systems and its connection with the secant criterion,”  Automatica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 42, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1531–1537, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [26] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Khalil, Nonlinear Systems, 3rd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Upper Saddle River,  NJ: Prentice Hall, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [27] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hines, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Arcak, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Packard, “Equilibrium-  independent passivity: A new definition and numerical certific-  ation,” Automatica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 47, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 9, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1949–1956, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Arcak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Meissen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Packard, Networks of Dissipative  Systems: Compositional Certification of Stability, Performance,  and Safety, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (SpringerBriefs in Control, Automation and  Robotics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  New York, NY, USA: Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Jané-Soneira, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hohmann, “Constructive  analysis and design of interconnected krasovskii passive and  quadratic dissipative systems,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 61th IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Decis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Control (CDC), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Moylan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hill, “Stability criteria for large-scale sys-  tems,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Autom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 23, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 143–149,  1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [31] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Benzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Golub, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Liesen, “Numerical solution of  saddle point problems,” Acta Numerica, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 14, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 1–137, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [32] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Strehle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Malan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Krebs, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Hohmann, “Passivity  conditions for plug-and-play operation of nonlinear static AC  loads,” IFAC-PapersOnLine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 53, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 12 237–12 243,  2020, 21st IFAC World Congress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Machowski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Bialek, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Bumby, Power System  Dynamics: Stability and Control, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Chichester, United  Kingdom: John Wiley & Sons, Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=', 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Freeman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Yang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Lynch, “Stability and con-  vergence properties of dynamic average consensus estimators,”  in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 45th IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Decis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control (CDC), 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 338–  343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [35] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Weitenberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Zhao, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Mallada, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Dörfler, and  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' De Persis, “Robust decentralized frequency control: A leaky  integrator approach,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Control Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' (ECC), 2018,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 764–769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  [36] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Raff, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Ebenbauer, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' Allgöwer, Nonlinear Model Pre-  dictive Control: A Passivity-Based Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content='  Berlin, Heidel-  berg: Springer, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
+page_content=' 151–162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFRT4oBgHgl3EQfTjeD/content/2301.13533v1.pdf'}
diff --git a/dtE3T4oBgHgl3EQfegpr/content/tmp_files/2301.04544v1.pdf.txt b/dtE3T4oBgHgl3EQfegpr/content/tmp_files/2301.04544v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2e1742a66d9c90efadc590b063df3ea7c694ee15
--- /dev/null
+++ b/dtE3T4oBgHgl3EQfegpr/content/tmp_files/2301.04544v1.pdf.txt
@@ -0,0 +1,1004 @@
+arXiv:2301.04544v1  [cs.GT]  11 Jan 2023
+Optimal Impartial Correspondences
+Javier Cembrano∗
+Felix Fischer†
+Max Klimm‡
+Abstract
+We study mechanisms that select a subset of the vertex set of a directed graph in order to
+maximize the minimum indegree of any selected vertex, subject to an impartiality constraint
+that the selection of a particular vertex is independent of the outgoing edges of that vertex.
+For graphs with maximum outdegree d, we give a mechanism that selects at most d + 1
+vertices and only selects vertices whose indegree is at least the maximum indegree in the
+graph minus one. We then show that this is best possible in the sense that no impartial
+mechanism can only select vertices with maximum degree, even without any restriction
+on the number of selected vertices. We finally obtain the following trade-off between the
+maximum number of vertices selected and the minimum indegree of any selected vertex:
+when selecting at most k vertices out of n, it is possible to only select vertices whose indegree
+is at least the maximum indegree minus ⌊(n − 2)/(k − 1)⌋ + 1.
+1
+Introduction
+Impartial selection is the problem of selecting vertices with large indegree in a directed graph,
+in such a way that the selection of a particular vertex is independent of the outgoing edges of
+that vertex. The problem models a situation where agents nominate one another for selection
+and are willing to offer their true opinion on other agents as long as this does not affect their
+own chance of being selected.
+The selection of a single vertex is governed by strong impossibility results. For graphs with
+maximum outdegree one, corresponding to situations where each agent submits a single nomi-
+nation, every impartial selection rule violates one of two basic axioms [11] and as a consequence
+must fail to provide a non-trivial multiplicative approximation to the maximum indegree. For
+graphs with arbitrary outdegrees, corresponding to situations where each agent can submit mul-
+tiple nominations, impartial rules violate an even weaker axiom and cannot provide a non-trivial
+approximation in a multiplicative or additive sense [1, 8]. These impossibilities largely remain in
+place if rather than a single vertex we want to select any fixed number of vertices, but positive
+results can be obtained if we relax the requirement that the same number of vertices must be
+selected in every graph [4, 18].
+From a practical point of view, the need for such a relaxation should not necessarily be a
+cause for concern. Indeed, situations in the real world to which impartial selection is relevant
+often allow for a certain degree of flexibility in the number of selected agents. The exact number
+of papers accepted to an academic conference is usually not fixed in advance but depends on
+the number and quality of submissions. Best paper awards at conferences are often given in
+overlapping categories, and some awards may only be given if this is warranted by the field of
+candidates. The Fields medal is awarded every four years to two, three, or four mathematicians
+under the age of 40. Examples at the more extreme end of the spectrum of flexibility include
+the award of job titles such as vice president or deputy vice-principal. Such titles can often be
+∗Institut für Mathematik, Technische Universität Berlin, Germany
+†School of Mathematical Sciences, Queen Mary University of London, UK
+‡Institut für Mathematik, Technische Universität Berlin, Germany
+1
+
+given to a large number of individuals at a negligible cost per individual, but should only be
+given to qualified individuals so as not to devalue the title.
+Tamura and Ohseto [18] specifically studied what they call nomination correspondences,
+i.e., rules that may select an arbitrary set of vertices in any graph. For graphs with maximum
+outdegree one a particular such rule, plurality with runners-up, satisfies impartiality and ap-
+propriate versions of the two axioms of Holzman and Moulin [11]. The rule selects any vertex
+with maximum indegree; if there is a unique such vertex, any vertex whose indegree is smaller
+by one and whose outgoing edge goes to the vertex with maximum indegree is selected as well.
+An appropriate measure for the quality of rules that select varying numbers of vertices is the
+difference in the worst case between the best vertex and the worst selected vertex, and we can
+call a rule α-min-additive if the maximum difference, taken over all graphs, between these two
+quantities is at most α. In this terminology, plurality with runners-up is 1-min-additive.
+As Tamura and Ohseto point out, it may be desirable in practice to ensure that the maximum
+number of vertices selected is not too large, a property that plurality with runners-up clearly
+fails. It is therefore interesting to ask whether there exist rules that are α-min-additive and
+never select more than k vertices, for some fixed α and k.
+For graphs with outdegree one,
+Tamura and Ohseto answer this question in the affirmative: a variant of plurality with runners-
+up that breaks ties according to a fixed ordering of the vertices remains 1-min-additive but never
+selects more than two vertices.
+Our Contribution
+Our first result provides a generalization of the result of Tamura and Ohseto
+to graphs with larger outdegrees: for graphs with maximum outdegree d, it is possible to achieve
+1-min-additivity while selecting at most d+1 vertices. For the particular case of graphs with un-
+bounded outdegrees we obtain a slight improvement, by guaranteeing 1-min-additivity without
+ever selecting all vertices. Our second result establishes that 1-min-additivity is best possible,
+thus ruling out the existence of impartial mechanisms that only select vertices with maximum in-
+degree. This holds even when no restrictions are imposed on the number of selected vertices, and
+is shown alongside analogous impossibility results concerning the maximization of the median or
+mean indegree of the selected vertices instead of their minimum indegree. Our third result pro-
+vides a trade-off between the maximum number of vertices selected, where smaller is better, and
+the minimum indegree of any selected vertex, where larger is better: if we are allowed to select
+at most k vertices out of n, we can guarantee α-min-additivity for α = ⌊(n−2)/(k−1)⌋+1. This
+is achieved by removing a subset of the edges from the graph before plurality with runners-up
+is applied, in order to guarantee impartiality while selecting fewer vertices. We do not know
+whether this last result is tight and leave open the interesting question for the optimal trade-off
+between the number and quality of selected vertices.
+Related Work
+Impartiality as a property of an economic mechanism was introduced by
+de Clippel et al. [9], and first applied to the selection of vertices in a directed graph by Alon et al.
+[1] and Holzman and Moulin [11]. Whereas Holzman and Moulin gave axiomatic characteriza-
+tions for mechanisms selecting a single vertex when all outdegrees are equal to one, Alon et al.
+studied the ability of impartial mechanisms to approximate the maximum indegree for any fixed
+number of vertices when there are no limitations on outdegrees.
+Both sets of authors obtained strong impossibility results, which a significant amount of
+follow-up work has since sought to overcome. Randomized mechanisms providing non-trivial
+multiplicative guarantees had already been proposed by Alon et al., and Fischer and Klimm [10]
+subsequently achieved the best possible such guarantee for the selection of one vertex. Starting
+from the observation that worst-case instances for randomized mechanisms have small indegrees,
+Bousquet et al. [5] developed a mechanism that is asymptotically optimal as the maximum
+indegree grows, and Caragiannis et al. [6, 7] initiated the study of mechanisms providing additive
+rather than multiplicative guarantees. Cembrano et al. [8] subsequently identified a deterministic
+2
+
+mechanism that provides non-trivial additive guarantees whenever the maximum outdegree is
+bounded and established that no such guarantees can be obtained with unbounded outdegrees.
+Randomized mechanisms have been also studied from an axiomatic point of view by Mackenzie
+[14, 15].
+Bjelde et al. [4] gave randomized mechanisms with improved multiplicative guarantees for the
+selection of more than one vertex and observed that when selecting at most k vertices rather than
+exactly k, deterministic mechanisms can in fact achieve non-trivial guarantees. An axiomatic
+study of Tamura and Ohseto [18] for the outdegree-one case came to the same conclusion:
+when allowing for the selection of a varying number of vertices, the impossibility result of
+Holzman and Moulin no longer holds.
+Tamura [17] subsequently characterized a mechanism
+proposed by Tamura and Ohseto, which in some cases selects all vertices, as the unique minimal
+mechanism satisfying impartiality, anonymity, symmetry, and monotonicity.
+Impartial mechanisms have finally been proposed for various problems other than selection,
+including peer review [2, 13, 16, 20], rank aggregation [12], progeny maximization [3, 21], and
+network centralities [19].
+2
+Preliminaries
+For n ∈ N, let [n] = {1, 2, . . . , n}, and let
+Gn =
+�
+(V, E) : V = [n], E ⊆ (V × V ) \
+�
+v∈V
+{(v, v)}
+�
+be the set of directed graphs with n vertices and no loops. Let G = �
+n∈N Gn. For G = (V, E) ∈ G
+and v ∈ V , let N +(v, G) = {u ∈ V : (v, u) ∈ E} be the out-neighborhood and N −(v, G) =
+{u ∈ V : (u, v) ∈ E} the in-neighborhood of v in G. Let δ+(v, G) = |N +(v, G)| and δ−(v, G) =
+|N −(v, G)| denote the outdegree and indegree of v in G, and ∆(G) = maxv∈V δ−(v, G) the
+maximum indegree of any vertex in G.
+When the graph is clear from the context, we will
+sometimes drop G from the notation and write N +(v), N −(v), δ+(v), δ−(v), and ∆.
+Let
+top(G) = max{v ∈ V : δ−(v) = ∆(G)} denote the vertex of G with the largest index among
+those with maximum indegree.
+For n ∈ N and d ∈ [n − 1], let Gn(d) = {(V, E) ∈ Gn :
+δ+(v) ≤ d for every v ∈ V } be the set of graphs in Gn with maximum outdegree at most d, and
+G(d) = �
+n∈N Gn(d).
+A k-selection mechanism is then given by a family of functions f : Gn → 2[n], one for
+each n ∈ N, mapping each graph to a subset of its vertices, where we require that |f(G)| ≤ k
+for all G ∈ G. In a slight abuse of notation, we will use f to refer to both the mechanism
+and to individual functions of the family. Given G = (V, E) ∈ G and v ∈ V , let Nv(G) =
+{(V, E′) ∈ G :
+E \ ({v} × V ) = E′ \ ({v} × V )} be the set neighboring graphs of G with
+respect to v, in the sense that they can be obtained from G by changing the outgoing edges
+of v. Mechanism f is impartial on G′ ⊆ G if on this set of graphs the outgoing edges of a
+vertex have no influence on its selection, i.e., if for every graph G = (V, E) ∈ G′, v ∈ V , and
+G′ ∈ Nv(G), it holds that f(G) ∩ {v} = f(G′) ∩ {v}. Given a k-selection mechanism f and
+an aggregator function σ : 2R → R such that σ(∅) = 0 and, for every S ⊆ R with |S| ≥ 1,
+min{x ∈ S} ≤ σ(S) ≤ max{x ∈ S}, we say that f is α-σ-additive on G′ ⊆ G, for α ≥ 0, if for
+every graph in G′ the function σ evaluated on the choice of f differs from the maximum indegree
+by at most α, i.e., if
+sup
+G∈G′
+�
+∆(G) − σ
+�
+{δ−(v, G)}v∈f(G)
+��
+≤ α.
+We will specifically be interested in the cases where σ is the minimum, the median, and the mean,
+and respectively call a mechanism α-min-additive, α-median-additive, and α-mean-additive in
+these cases.
+3
+
+Algorithm 1: Plurality with runners-up
+Input: Digraph G = (V, E) ∈ Gn(1)
+Output: Set S ⊆ V of selected vertices
+Let S = {v ∈ V : δ−(v) = ∆(G)};
+if S = {v} for some v ∈ V then
+S ←− S ∪ {u ∈ V : δ−(u) = ∆(G) − 1 and (u, v) ∈ E}
+end
+Return S
+3
+Plurality with Runners-up
+Focusing on the case with maximum outdegree one, Tamura and Ohseto [18] proposed a mech-
+anism they called plurality with runners-up.
+The mechanism, which we describe formally in
+Algorithm 1, selects all vertices with maximum indegree; if there is a unique such vertex, then
+any vertex with an outgoing edge to that vertex whose indegree is smaller by one is selected as
+well. The idea behind this mechanism is that vertices in the latter category would be among
+those with maximum degree if their outgoing edge was deleted, and thus any impartial mecha-
+nism seeking to select the vertices with maximum degree would also have to select those vertices.
+Plurality with runners-up is impartial on G(1), and in any graph with n vertices selects between
+1 and n vertices whose degree is equal to the maximum degree or the maximum degree minus
+one. It is thus an impartial and 1-min-additive n-selection mechanism on Gn(1) for every n ∈ N.
+It is natural to ask whether a similar additive guarantee can be obtained for more general set-
+tings. In this section, we answer this question in the affirmative, and in particular study for
+which values of n, k, and d there exists an impartial and 1-min-additive k-selection mechanism
+on Gn(d). We will see later, in Section 4, that 1-min-additivity is in fact best possible for all
+cases covered by our result, with the exception of the boundary case where n = 2.
+While Tamura and Ohseto do not limit the maximum number of selected vertices, they
+discuss briefly a modification of their mechanism that retains impartiality and 1-min-additivity
+but selects at most 2 vertices. Instead of all vertices with maximum indegree, the modified
+mechanism breaks ties in favor of a single maximum-degree vertex using a fixed ordering of the
+vertices. In order to guarantee impartiality, the modified mechanism then also selects any vertex
+that would be selected in the graph obtained by deleting the outgoing edge of that vertex. The
+assumption that every vertex has at most one outgoing edge means that at most one additional
+vertex is selected. There thus exists a 1-min-additive k-selection mechanism on G(1) for every
+k ≥ 2.
+Our first result generalizes this mechanism to settings with arbitrary outdegrees, as long
+as the maximum number of selected vertices is large enough. To this end we show that when
+the maximum outdegree is d, to achieve impartiality, at most d vertices have to be selected in
+addition to the one with maximum indegree and highest priority.1 We formally describe the
+resulting mechanism in Algorithm 2, and will refer to it as asymmetric plurality with runners-up
+and denote its output on graph G by P(G). We obtain the following theorem, which generalizes
+the known result for the outdegree-one case.
+Theorem 1. For every n ∈ N, d ∈ [n − 1], and k ∈ {d + 1, . . . , n}, there exists an impartial
+and 1-min-additive k-selection mechanism on Gn(d).
+We will be interested in the following in comparing vertices both according to their indegree
+and to their index, and we will use regular inequality symbols, as well as the operators max and
+1In this mechanism and wherever ties are broken in the rest of the paper, we break ties in favor of greater
+index, so top(G) is the vertex with maximum indegree and highest priority in graph G. Naturally, any other
+deterministic tie-breaking rule could be used instead.
+4
+
+Algorithm 2: Asymmetric plurality with runners-up P(G)
+Input: Digraph G = (V, E) ∈ Gn
+Output: Set S ⊆ V of selected vertices
+Let S = ∅;
+for v ∈ V do
+Let Gv = (V, E \ ({v} × V ));
+if top(Gv) = v then
+S ←− S ∪ {v}
+end
+end
+Return S
+min, to denote the lexicographic order among pairs of the form (δ−(v), v). The following lemma
+characterizes the structure of the set of vertices selected by Algorithm 2, and provides the main
+technical ingredient to the proof of Theorem 1.
+Lemma 1. Let G = (V, E) ∈ G and v ∈ V . Then, v ∈ P(G) if and only if
+(a) for every w ∈ V with (δ−(w), w) > (δ−(v), v) it holds (v, w) ∈ E; and
+(b) one of the following holds:
+(i) δ−(v) = ∆(G); or
+(ii) δ−(v) = ∆(G) − 1 and v > w for every w ∈ V with δ−(w) = ∆(G).
+Proof. We first show that, if v ∈ P(G) for a given graph G, then (a) and (b) follow.
+Let
+G = (V, E) ∈ G, and let v ∈ P(G). To see (a), suppose there is w ∈ V with (δ−(w, G), w) >
+(δ−(v, G), v).
+Since v ∈ P(G), we have v = top(Gv) with Gv = (V, E \ ({v} × V )).
+This
+implies (δ−(v, Gv), v) > (δ−(w, Gv), w) and therefore δ−(w, G) > δ−(w, Gv), because δ−(v, G) =
+δ−(v, Gv). Since G and Gv only differ in the outgoing edges of v, we conclude that (v, w) ∈ E.
+To prove (b), we note that for every w ∈ V we have
+(δ−(v, G), v) = (δ−(v, Gv), v) > (δ−(w, Gv), w) ≥ (δ−(w, G) − 1, w),
+(1)
+where the last inequality comes from the fact that each vertex has at most one incoming edge
+from v. If there is no w ∈ V \ {v} with δ−(w) = ∆(G), the maximum indegree must be that of
+v, so δ−(v) = ∆(G) and (i) follows. Otherwise, for each w ∈ V \ {v} with δ−(w) = ∆(G), (1)
+yields (δ−(v, G), v) > (∆(G) − 1, w). We conclude that either δ−(v, G) > ∆(G) − 1, in which
+case (i) holds, or both δ−(v) = ∆(G) − 1 and v > w, which implies (ii).
+We now prove the other direction.
+Let G = (V, E) ∈ G and v ∈ V such that both (a)
+and (b) hold. Let Gv = (V, E \ ({v} × V )). We have to show that top(Gv) = v, i.e., that
+for every w ∈ V \ {v}, (δ−(v, Gv), v) > (δ−(w, Gv), w).
+Let w be a vertex in V \ {v}.
+If
+(δ−(v, G), v) > (δ−(w, G), w), we can conclude immediately since δ−(v, Gv) = δ−(v, G) and
+δ−(w, Gv) ≤ δ−(w, G). Otherwise, we know from (a) that (v, w) ∈ E and thus δ−(w, Gv) =
+δ−(w, G) − 1. If v satisfies (i), this yields
+δ−(v, Gv) = δ−(v, G) = ∆(G) ≥ δ−(w, G) = δ−(w, Gv) + 1,
+so (δ−(v, Gv), v) > (δ−(w, Gv), w). On the other hand, if v satisfies (ii), then
+δ−(v, Gv) = δ−(v, G) = ∆(G) − 1 ≥ δ−(w, G) − 1 = δ−(w, Gv),
+and v > w implies (δ−(v, Gv), v) > (δ−(w, Gv), w) as well.
+5
+
+4
+2
+1
+6
+5
+3
+∆
+∆ − 1
+Figure 1: Example of a set of vertices selected by Algorithm 2. In this illustration and throughout
+the paper, vertices are arranged vertically according to indegree and horizontally according to
+index, so that vertices on the left are favored in case of ties.
+The vertices selected by the
+mechanism are drawn in white, those not selected in black. Vertices with indegree below ∆ − 1,
+as well as edges incident to such vertices, are not shown. Denoting the graph as G = (V, E),
+and letting Gv = (V, E \ ({v} × V )) for each vertex v, the selected vertices v are those for which
+top(Gv) = v. Specifically, vertices 2, 3, and 6 are not selected because top(G2) = 4, top(G3) = 4,
+and top(G6) = 1.
+Observe that Lemma 1 implies in particular that top(G) ∈ P(G) for every graph G. Figure 1
+provides an example of the characterization given by Lemma 1, in terms of indegrees, tie-
+breaking order, and edges among selected vertices.
+We are now ready to prove Theorem 1.
+Proof of Theorem 1. We show that for every n ∈ N and d ∈ [n − 1], asymmetric plurality with
+runners-up is impartial and 1-min-additive on Gn(d), and that for every G = (V, E) ∈ Gn(d), it
+selects at most d+1 vertices. If this is the case, then for every k ∈ {d+1, . . . , n} the mechanism
+would satisfy the statement of the theorem. Therefore, let n and d be as mentioned.
+Impartiality follows from the definition of the mechanism, because the outgoing edges of a
+vertex are not taken into account when deciding whether the vertex is taking part on the selected
+set or not. If we let G = (V, E), v ∈ V , and G′ = (V, E′) ∈ Nv(G), then the graphs Gv and
+G′
+v constructed when running the mechanism with each of these graphs G and G′ as an input,
+respectively, are the same because by definition of Nv(G) we have E \({v}×V ) = E′ \({v}×V ).
+Since v ∈ P(G) ⇔ top(Gv) = v, and v ∈ P(G′) ⇔ top(G′
+v) = v, we conclude v ∈ P(G) ⇔ v ∈
+P(G′).
+To see that the mechanism is 1-min-additive, let G ∈ Gn(d) and first note that P(G) ̸= ∅
+since Lemma 1 implies that top(G) ∈ P(G). From this lemma we also know that for every
+v ∈ P(G), δ−(v) ≥ ∆(G) − 1. We conclude that min{{δ−(v)}v∈P(G)} ≥ ∆(G) − 1, and since
+this holds for every G ∈ Gn(d), the mechanism is 1-min-additive.
+Finally, let G = (V, E) ∈ Gn(d), and suppose that |P(G)| > d + 1.
+If we denote vL =
+argminv∈P(G){(δ−(v), v)}, from Lemma 1 we know that (vL, w) ∈ E for every w ∈ V with
+(δ−(w), w) > (δ−(vL), vL), thus δ+(vL) ≥ |P(G)| − 1 > d, a contradiction. We conclude that
+|P(G)| ≤ d + 1.
+The following result, concerning mechanisms that may select an arbitrary number of vertices,
+follows immediately from Theorem 1.
+Corollary 1. For every n ∈ N, there exists an impartial and 1-min-additive n-selection mecha-
+nism on Gn.
+On Gn, i.e., in the case of unbounded outdegrees, this result can in fact be improved slightly
+to guarantee 1-min-additivity while selecting only at most n − 1 vertices. The improvement
+is achieved by a more intricate version of asymmetric plurality with runners-up, which we call
+asymmetric plurality with runners-up and pivotal vertices. We formally describe this mechanism
+6
+
+Algorithm 3: Asymmetric plurality with runners-up and pivotal vertices PP(G)
+Input: Digraph G = (V, E) ∈ Gn
+Output: Set S ⊆ V of selected vertices with |S| ≤ n − 1
+Let S ←− ∅;
+for u ∈ P(G) do
+if for every v ∈ P(G) \ {u} there exists Guv ∈ Nu(G) such that v /∈ P(Guv) then
+S ←− S ∪ {u}
+end
+end
+Return S
+in Algorithm 3 and denote its output for graph G by PP (G).
+Given a graph G = (V, E),
+call a vertex u ∈ P(G) pivotal for v ∈ P(G) if there exists a graph Guv ∈ Nu(G) such that
+v /∈ P(Guv), i.e., if the outgoing edges of u can be changed in such a way that v is no longer
+selected by asymmetric plurality with runners-up. Asymmetric plurality with runners-up and
+pivotal vertices then selects every vertex in P(G) that is pivotal for every other vertex in P(G).
+The mechanism turns out to inherit impartiality and 1-min-additivity, and to never select all
+vertices.
+Theorem 2. For every n ∈ N and k ∈ {n − 1, n}, there exists an impartial and 1-min-additive
+k-selection mechanism on Gn.
+Proof. We show that for every n ∈ N, asymmetric plurality with runners-up and pivotal vertices
+is impartial and 1-min-additive on Gn and that for every G = (V, E) ∈ Gn, it selects at most
+n − 1 vertices. Let n ∈ N be an arbitrary value.
+To see that the mechanism is impartial, let G = (V, E) ∈ Gn, u ∈ PP (G), and G′ = (V, E′) ∈
+Nu(G). We show in the following that u ∈ PP (G′), and since the graphs G and G′ are chosen
+arbitrarily, their roles can be inverted and this is enough to conclude that the mechanism is
+impartial. We first note that u ∈ P(G) because PP (G) ⊆ P(G), thus impartiality of asymmetric
+plurality with runners-up proven in Theorem 1 implies u ∈ P(G′). If P(G′) = {u}, then the
+condition in the mechanism holds trivially for this vertex, so u ∈ PP (G′) and we conclude.
+Otherwise, let v ∈ P(G′) \ {u} be an arbitrary vertex selected by asymmetric plurality with
+runners-up other than u. Since u ∈ PP (G), we have that either (a) v /∈ P(G), or (b) v ∈ P(G)
+and there exists Guv = (V, Euv) ∈ Nu(G) such that v /∈ P(Guv). If (a) holds, taking G′
+uv = G,
+which belongs to Nu(G′) because of the assumption that G′ ∈ Nu(G), we have that v /∈ P(G′
+uv).
+If (b) holds, taking G′
+uv = Guv, which belongs to Nu(G′) since Nu(G′) = Nu(G), we have that
+v /∈ P(G′
+uv). In either case, we conclude that there exists G′
+uv ∈ Nu(G′) such that v /∈ P(G′
+uv).
+Since this argument is valid for every v ∈ P(G′) \ {u}, we conclude that u ∈ PP (G′).
+To see that the mechanism is 1-min-additive, it is enough to show that it always selects a
+vertex, since for every G ∈ G it selects a subset of P(G) and from Theorem 1 we know that this
+set contains vertices with indegrees in {∆(G), ∆(G)−1}. To this purpose we let G = (V, E) ∈ Gn
+and introduce some additional notation. Let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|
+for i ∈ {0, 1}, and denote
+vH = argmaxv∈P(G){(δ−(v, G), v)} = top(G),
+vL = argminv∈P(G){(δ−(v, G), v)}.
+From Lemma 1, we know that P(G) = S0 ∪ S1, that (vL, v) ∈ E for every v ∈ P(G) \ {vL},
+and that u > v for each u ∈ S1, v ∈ S0. We now distinguish two cases according to the edges
+between vertices in P(G).
+If (vH, v) ∈ E for every v ∈ P(G) \ {vH}, then we claim that defining G′ = (V, E \ ({vH} ×
+V )) ∈ NvH(G) it holds v /∈ P(G′) for every v ∈ P(G) \ {vH}. If this is true, it is clear that
+7
+
+∆
+∆ − 1
+∆ − 2
+G = (V, E)
+G′ = (V, E \ ({2} × V ))
+2
+1
+4
+3
+2
+1
+4
+3
+Figure 2: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if
+(vH, v) ∈ E for every v ∈ P(G) \ {vH}. Vertices selected by asymmetric plurality with runners-
+up are drawn in white. Denoting the graph on the left as G = (V, E), where vH = 2, and defining
+G′ = (V, E \ ({2} × V )) ∈ N2(G), we have that {1, 3, 4} ∩ P(G′) = ∅, and thus 2 ∈ PP (G).
+vH ∈ PP (G) and thus PP (G) ̸= ∅. We now prove the claim. First, note that vH ∈ P(G′)
+since vH = top(G′) and Lemma 1 ensures top(G′) ∈ P(G′). This comes from the fact that
+δ−(vH, G′) = δ−(vH, G) and δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \ {vH}, together with vH =
+top(G). Moreover, for every v ∈ S0 \ {vH} it holds δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 =
+∆(G′) − 1 and v < vH, so condition (b) in Lemma 1 does not hold for v and thus v /∈ P(G′).
+Analogously, for every v ∈ S1 it holds δ−(v, G′) = δ−(v, G)−1 = δ−(vH, G′)−2 = ∆(G′)−2, so
+condition (b) in Lemma 1 does not hold for v either, and thus v /∈ P(G′). This allows to conclude
+the claim and the fact that PP (G) is non-empty for this case. This argument is illustrated in
+Figure 2.
+Now we consider the case where there is a vertex ¯v ∈ P(G) such that (vH, ¯v) /∈ E, and
+we claim that defining G′ = (V, (E \ ({vL × V })) ∪ (vL, vH)) ∈ NvL(G) it holds v /∈ P(G′)
+for every v ∈ P(G) \ {vL, vH}, whereas defining G′′ = (V, E \ (vL, vH)) ∈ NvL(G) it holds
+vH /∈ P(G′′). If this is true, then vL ∈ PP (G) and PP (G) ̸= ∅. We now prove the claim.
+First, note that vH ∈ P(G′) for the same reason as before, since δ−(vH, G′) = δ−(vH, G) and
+δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \ {vH}.
+Moreover, for every v ∈ S0 \ {vH} it holds
+δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 = ∆(G′) − 1 and v < vH, so condition (b) in
+Lemma 1 does not hold for v and thus v /∈ P(G′). Analogously, for every v ∈ S1 \ {vL} it holds
+δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 2 = ∆(G′) − 2 so condition (b) in Lemma 1 does not
+hold for v and thus v /∈ P(G′). This allows to conclude the claim for G′. In the case of G′′, we
+can write the following chain of inequalities,
+(δ−(¯v, G′′), ¯v) = (δ−(¯v, G), ¯v) > (δ−(vH, G) − 1, vH) = (δ−(vH, G′′), vH),
+where the equalities hold because of the definition of G′′ and the inequality by condition (b)
+in Lemma 1, given that ¯v ∈ P(G).
+Since (vH, ¯v) /∈ E, we conclude from condition (a) in
+Lemma 1 that vH /∈ P(G′′), and therefore the claim for G′′ follows. This argument is illustrated
+in Figure 3.
+Finally, we show that the mechanism selects at most n − 1 vertices. Let G = (V, E) ∈ Gn.
+Since PP (G) ⊆ P(G), if |P(G)| ≤ n − 1 this is immediate. We thus suppose in what follows
+that |P(G)| = n. In particular, Lemma 1 implies (v, vH) ∈ E for every v ∈ V \ {vH}, thus
+∆(G) = n − 1, and δ−(v) ≥ n − 2 for every v ∈ V . If S1 = ∅, then δ−(v) = n − 1 for every
+v ∈ V , i.e., G is the complete graph. In this case, vH = n and we claim that v /∈ PP (G) for
+each v ∈ V \ {n}, thus |PP (G)| ≤ 1. This comes from the fact that, for every v ∈ V \ {n}
+and every G′ = (V, E′) ∈ Nv(G) it holds n ∈ P(G′). To see this, note that (n, v) ∈ E′ for
+every v ∈ V \ {n}, δ−(n, G′) ≥ n − 2 = ∆(G′) − 1, and n > v for every v ∈ V \ {n}, so
+Lemma 1 ensures n ∈ P(G′). If S1 ̸= ∅, then there is at least one vertex with outdegree less
+8
+
+∆
+∆ − 1
+∆ − 2
+G′ = (V, E \ {(3, 1), (3, 4)})
+G′′ = (V, E \ {(3, 2)})
+∆
+∆ − 1
+G = (V, E)
+2
+1
+4
+3
+2
+1
+4
+3
+2
+1
+4
+3
+Figure 3: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if
+(vH, ¯v) /∈ E for some ¯v ∈ P(G) \ {vH}. Vertices selected by asymmetric plurality with runners-
+up are drawn in white. Denoting the graph at the top by G = (V, E), where vH = 2, vL = 3,
+and ¯v = 4, and defining G′ = (V, E \ {(3, 1), (3, 4)}) ∈ N3(G), we have that {1, 4} ∩ P(G′) = ∅,
+whereas defining G′′ = (V, (E \ {(3, 2)}) ∈ N3(G) we have that 2 /∈ P(G′′). We conclude that
+3 ∈ PP (G).
+or equal than n − 2. Let u be an arbitrary vertex with δ+(u) ≤ n − 2, and let ¯v ∈ S1 be the
+vertex with highest index such that (u, ¯v) /∈ E, i.e., ¯v = max{V \ N +(u)}. Since u ∈ PP (G),
+there exists G′ = (V, E′) ∈ Nu(G) such that ¯v /∈ P(G′). From Lemma 1, this implies that
+there exists ¯w ∈ V such that either (a) (δ−( ¯w, G′), w) > (δ−(¯v, G′), ¯v) and (¯v, ¯w) /∈ E′, or
+(b) δ−( ¯w, G′) > δ−(¯v, G′) and ¯w > ¯v.
+Since ¯v ∈ P(G), we know from this same lemma
+that if (a) holds, (δ−( ¯w, G), w) < (δ−(¯v, G), ¯v) because of having ¯w /∈ N +(¯v, G) = N +(¯v, G′);
+and similarly, if (b) holds, δ−( ¯w, G) ≤ δ−(¯v, G) because of having ¯w > ¯v.
+In either case,
+since δ−(¯v, G) ≤ δ−(¯v, G′), we conclude that δ−( ¯w, G′) > δ−( ¯w, G), and therefore (u, ¯w) /∈ E.
+If (a) holds, this is a contradiction because we would have {u, ¯v} ∩ N −( ¯w, G) = ∅ and thus
+δ−( ¯w, G) ≤ n − 3.
+If (b) holds, we reach a contradiction as well, because we would have
+¯w ∈ V \ N +(u, G) and ¯w > ¯v, but we chose ¯v to be the maximum of this set.
+4
+An Impossibility Result
+When we established the existence of an impartial and 1-min-additive k-selection mechanism on
+G(d) whenever k ≥ d+1, we claimed this result to be best possible in the sense that the additive
+guarantee cannot be improved. We will prove this claim, that impartiality is incompatible with
+the requirement to only select vertices with maximum indegree, as a corollary of a more general
+result.
+While selecting only vertices with maximum indegree is a natural goal for mechanisms that
+select varying numbers of vertices, other natural objectives exist for such mechanisms such as
+maximizing the median or mean indegree of the selected vertices. For both of these objectives,
+the mechanisms discussed in the previous section immediately provide upper bounds: if a k-
+selection mechanism always selects one vertex with maximum indegree and is α-min-additive
+9
+
+then it is clearly α-median-additive and
+� k−1
+k α
+�
+-mean-additive; Theorem 1 thus implies the ex-
+istence of a 1-median-additive and k−1
+k -mean-additive k-selection mechanism on G(d), whenever
+k ≥ d + 1. To improve on 1-median-additivity, it would be acceptable to select vertices with low
+indegree as long as a greater number of vertices with maximum indegree is selected at the same
+time. To improve on k−1
+k -mean-additivity, it would suffice to select more than one vertex with
+maximum indegree whenever this is possible, and to otherwise select only a sublinear number
+in k of vertices with indegree equal to the maximum indegree minus one. The following result
+shows that no such improvements are possible.
+Theorem 3. Let n ∈ N, n ≥ 3, k ∈ [n], and d ∈ [n − 1]. Let f be an impartial k-selection
+mechanism. If f is α1-median-additive on Gn(d), then α1 ≥ 1/2(1+1(d ≥ 3)). If f is α2-mean-
+additive on Gn(d), then α2 ≥
+� d+1
+2
+�
+/
+�� d+1
+2
+�
++ 1
+�
+.
+Proof. Let n, k, and d be as in the statement of the theorem. In the following we suppose that
+there is an impartial k-selection mechanism f which is either α1-median-additive on Gn(d) with
+α1 < 1/2(1 + 1(d ≥ 3)), or α2-mean-additive on Gn(d) with α2 <
+� d+1
+2
+�
+/
+�� d+1
+2
+�
++ 1
+�
+.
+We first prove the result for the case d = 1. We consider the graph G = (V, E) ∈ Gn(1)
+with E = {(1, 2), (2, 3), (3, 1)}, consisting of a 3-cycle and n − 3 isolated vertices. We consider
+as well, for v ∈ {1, 2, 3}, the graph Gv = (V, Ev) where v deviates from the 3-cycle by changing
+its outgoing edge to the previous vertex in the cycle, i.e.,
+E1 = {(1, 3), (2, 3), (3, 1)}, E2 = {(1, 2), (2, 1), (3, 1)}, E3 = {(1, 2), (2, 3), (3, 2)}.
+Since f is α1-median-additive with α1 < 1/2 or α2-mean-additive with α2 < 1/2, we have that
+f(G1) = {3}, f(G2) = {1}, and f(G3) = {2}. In particular, for v ∈ {1, 2, 3}, v /∈ f(Gv). Since
+for each v ∈ {1, 2, 3} it holds Ev \ ({v} × V ) = E \ ({v} × V ), we conclude by impartiality
+that v /∈ f(G), and thus f(G) ∩ {1, 2, 3} = ∅.
+This implies that both the median and the
+mean indegree of the vertices in f(G) are 0, which contradicts the additive guarantee of this
+mechanism because ∆(G) = 1.
+In the following, we assume d ≥ 2. We denote D = [d + 1] and consider in what follows two
+families of graphs with n vertices, Kv for each v ∈ D and Kuv for each u, v ∈ D, u ̸= v. They
+are constructed from a complete subgraph on D but deleting the outgoing edges of v, in the
+case of Kv, and the outgoing edges of u and v, in the case of Kuv. All the other vertices remain
+isolated. Formally, taking V = [n] we define
+Kv = (V, (D \ {v}) × D) for every v ∈ D,
+Kuv = (V, (D \ {u, v}) × D) for every u, v ∈ D with u ̸= v.
+If there is v ∈ D such that v /∈ f(Kv), then
+median
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d − 1 = ∆(Kv) − 1,
+mean
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d − 1 = ∆(Kv) − 1,
+which is a contradiction, so the result follows immediately. Therefore, in the following we assume
+that for every v ∈ D we have v ∈ f(Kv). We claim that for every v ∈ D,
+|{u ∈ D \ {v} : u ∈ f(Kv)}| ≥
+�d + 1
+2
+�
+.
+Let us see why the result follows if the claim holds. If this is the case, f selects one vertex with
+maximum indegree d in Kv and at least
+� d+1
+2
+�
+vertices with indegree d − 1. This yields both
+median
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤
+� d − 1
+2
+if d = 2
+d − 1
+otherwise,
+10
+
+and
+mean
+�
+{δ−(w, Kv)}w∈f(Kv)
+�
+≤ d + (d − 1)
+� d+1
+2
+�
+� d+1
+2
+�
++ 1
+= d −
+� d+1
+2
+�
+� d+1
+2
+�
++ 1,
+which is a contradiction since ∆(Kv) = d.
+Now we prove the claim. Suppose that for every v ∈ D we have v ∈ f(Kv) and
+|{u ∈ D \ {v} : u ∈ f(Kv)}| <
+�d + 1
+2
+�
+.
+(2)
+Let v ∈ D and u ∈ D \ {v} such that u /∈ f(Kv). Observing that
+((D \ {v}) × D) \ ({u} × V ) = ((D \ {u, v}) × D) \ ({u} × V ),
+we obtain from impartiality that u /∈ f(Kuv). From the bounds on α1 or α2 that f satisfies
+by assumption, this mechanism has to select a vertex with maximum indegree in this graph;
+otherwise, both the median and the mean of the selected set would be at most ∆(Kuv) − 1.
+Since δ−(w) < ∆(Kuv) for every w /∈ {u, v}, it holds v ∈ f(Kuv). Using impartiality once again,
+we conclude v ∈ f(Ku). We have shown the following property:
+For every u, v ∈ D : u /∈ f(Kv) =⇒ v ∈ f(Ku).
+(3)
+Consider now the graph H = (D, F), where for each u, v ∈ D with u ̸= v, (u, v) ∈ F if and
+only if u /∈ f(Kv). Property (2) implies that
+δ−(v, H) > d −
+�d + 1
+2
+�
+⇐⇒ δ−(v, H) ≥ d + 1 −
+�d + 1
+2
+�
+for each v ∈ D. In particular, there has to be a vertex v∗ ∈ D such that δ+(v∗, H) ≥ d + 1 −
+⌊(d + 1)/(2)⌋ as well. For this vertex we have
+δ+(v∗, H) + δ−(v∗, H) ≥ 2
+�
+d + 1 −
+�d + 1
+2
+��
+≥ d + 1.
+Since H has d+1 vertices, this implies the existence of w∗ ∈ D for which {(v∗, w∗), (w∗, v∗)} ⊂ F,
+i.e., both v∗ /∈ f(Kw∗) and w∗ /∈ f(Kv∗). This contradicts (3), so we conclude the proof of the
+claim and the proof of the theorem.
+Figure 4 provides an illustration of Theorem 3 for the case where n = 3, Figure 5 for the
+case where n = 4.
+The median of any set of numbers is an upper bound on their minimum. Therefore, if no
+impartial mechanism exists that is α-median-additive on G′ ⊆ G for α < ¯α, then no impartial
+mechanism can exist that is α-min-additive on G′ for α < ⌈¯α⌉. We thus obtain the following
+impossibility result, which we have claimed previously.
+Corollary 2. Let n ∈ N, n ≥ 3, and k ∈ [n]. Let f be an α-min-additive impartial k-selection
+mechanism on Gn. Then α ≥ 1.
+The impossibility results imply that for k ≥ d + 1, the mechanisms of Section 3 are best
+possible for the minimum and median objectives except in a few boundary cases. When n = 2,
+selecting each of the two vertices if and only if it has an incoming edge is impartial and achieves
+0-min-additivity and 0-median-additivity. When n = 3, it is possible to select in an impartial
+way at least one vertex with maximum indegree and at most one vertex with indegree equal
+to the maximum indegree minus one, thus guaranteeing 1/2-median-additivity. For the mean
+objective, the mechanisms of Section 3 are best possible asymptotically under the additional
+assumption that k = O(d).
+11
+
+1
+2
+3
+1
+1
+2
+3
+2
+1
+2
+3
+3
+1
+2
+3
+Figure 4: Counterexample to the existence of an impartial 3-selection mechanism that is α-
+median-additive or α-mean-additive on G3 for α < 1/2. Vertices drawn in white have to be
+selected, vertices in black cannot be selected. For the graphs at the top, on the left, and on
+the right, this follows from α-median-additivity or α-mean-additivity for α < 1/2. An arrow
+with label v from one graph to another indicates that one can be obtained from the other by
+changing the outgoing edges of vertex v; by impartiality, the vertex thus has to be selected in
+both graphs or not selected in both graphs. It follows that no vertices are selected in the graph
+at the center, a contradiction to the claimed additive guarantee.
+It is worth pointing out that the proof of the impossibility result uses graphs in which some
+vertices, in particular those with maximum indegree, do not have any outgoing edges. However,
+the impossibility extends naturally to the case where this cannot happen, corresponding to the
+practically relevant case in which abstentions are not allowed, as long as n ≥ 4 and d ≥ 3. For
+this it is enough to define D = [d], add a new vertex with outgoing edges to every vertex in D
+and incoming edges from the vertices in D which do not have any outgoing edge, and construct
+a cycle containing the vertices in V \ D.
+5
+Trading Off Quantity and Quality
+We have so far given impartial selection mechanisms for settings where the maximum outdegree d
+is smaller than the maximum number k of vertices that can be selected, and have shown that
+the mechanisms provide best possible additive guarantees in such settings. We will now consider
+settings where d ≥ k, such that asymmetric plurality with runners-up selects too many vertices
+and therefore cannot be used directly. We obtain the following result.
+Theorem 4. For every n ∈ N and k ∈ {2, . . . , n}, there exists an impartial and (⌊(n − 2)/(k −
+1)⌋ + 1)-min-additive k-selection mechanism on Gn.
+The result is obtained by a variant of asymmetric plurality with runners-up in which some
+edges are deleted before the mechanism is run. In principle, deleting a certain number of edges
+can affect the additive guarantee by the same amount, if all of the deleted edges happen to
+be directed at the same vertex. By studying the structure of the set of vertices selected by the
+mechanism, we will instead be able to delete edges to distinct vertices and thus keep the negative
+impact on the additive guarantee under control.
+The modified mechanism, which we call asymmetric plurality with runners-up and edge
+deletion, is formally described in Algorithm 4.
+It deletes any edges from a vertex to the
+⌊(n−2)/(k−1)⌋ vertices preceding that vertex in the tie-breaking order, and applies asymmetric
+12
+
+1
+2
+3
+4
+2
+3
+1
+2
+3
+4
+4
+1
+2
+3
+4
+4
+1
+2
+3
+4
+1
+1
+2
+3
+4
+1
+1
+2
+3
+4
+1
+2
+3
+4
+2
+3
+1
+2
+3
+4
+Figure 5: Counterexample to the existence of an impartial 4-selection mechanism that is α1-
+median-additive on G4(3) for α1 < 1 or α2-mean-additive on G4 for α2 < 2/3. Vertices drawn
+in white have to be selected, vertices in black cannot be selected, and vertices in gray may
+or may not be selected. For the graph on the left, this follows from α1-median-additivity for
+α1 < 1 or α2-mean-additivity for α2 < 2/3: under these assumptions at most one of the vertices
+with indegree 2 can be selected, which without loss of generality we can assume to be vertex 1.
+For the other graphs, it then follows by impartiality, and for the graph on the right yields a
+contradiction to the claimed additive guarantees.
+plurality with runners-up to the resulting graph. The following lemma shows that without such
+edges, the maximum number of vertices selected is reduced to k.
+Lemma 2. Let n ∈ N, k ∈ {2, . . . , n}, and r ∈ N with r ≥ ⌊(n−2)/(k−1)⌋. Let G = (V, E) ∈ Gn
+be such that for every u ∈ {1, . . . , n − 1} and every v ∈ {u + 1, . . . , min{u + r, n}}, (u, v) /∈ E.
+Then, |P(G)| ≤ k.
+Proof. As in the proof of Theorem 2, we let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|
+for i ∈ {0, 1}, and now we denote its elements in increasing order by vi
+j for j ∈ [ni], i.e.,
+Si = {vi
+j}ni
+j=1 with vi
+1 < vi
+2 · · · < vi
+ni for each i ∈ {0, 1}.
+From Lemma 1, we know that P(G) = S0 ∪S1, that for i ∈ {0, 1} we have (vi
+j, vi
+k) ∈ E for every
+j, k with j < k, and that v1
+1 > v0
+n0. This allows to define, for i ∈ {0, 1},
+¯Si = {v ∈ V \ Si : vi
+1 < v < vi
+ni},
+¯ni = | ¯Si|,
+such that ¯S0 ∩ ¯S1 = ∅.
+Fix i ∈ {0, 1} and suppose that ni ≥ 2. Combining both the fact that (vi
+j, vi
+k) ∈ E for every
+j, k with j < k, and that for every u ∈ {1, . . . , n−1} and v ∈ {u+1, . . . , min{u+r, n}}, (u, v) /∈
+13
+
+Algorithm 4: Asymmetric plurality with runners-up and edge deletion PD(G)
+Input: Digraph G = (V, E) ∈ Gn, k ∈ {2, . . . , n}
+Output: Set S ⊆ V of selected vertices with |S| ≤ k
+Let r = ⌊(n − 2)/(k − 1)⌋ ;
+// number of outgoing edges to remove
+Let R = �n−1
+u=1
+�min{u+r,n}
+v=u+1
+{(u, v)} ;
+// edges to be removed
+Let ¯G = (V, E \ R);
+Return P( ¯G)
+v0
+n0
+. . .
+v0
+2
+v0
+1
+v1
+n1
+. . .
+v1
+2
+v1
+1
+∆
+∆ − 1
+. . .
+� �� �
+≥r
+. . .
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+. . .
+. . .
+� �� �
+≥r
+. . .
+� �� �
+≥r
+S1
+¯S1
+S0
+¯S0
+Figure 6: Illustration of Lemma 2. There are no edges from a vertex to any of the r vertices
+to its left, which means that for each vertex in S0 or S1, except for the left-most vertex, there
+exist are at least r vertices outside these sets. Such vertices are not arranged according to their
+indegrees, and edges from vertices in S1 to every vertex in S0 have been omitted for clarity.
+E, we have that for every j ∈ [ni − 1] it holds vi
+j+1 − vi
+j ≥ r + 1.
+Summing over j yields
+vi
+ni − vi
+1 ≥ (ni − 1)(r + 1), hence
+¯ni = vi
+ni − vi
+1 + 1 − ni ≥ (ni − 1)(r + 1) + 1 − ni = (ni − 1)r,
+where the first equality comes from the definition of the set ¯Si. This implies ni ≤ 1 + ¯ni/r. We
+can now lift the assumption ni ≥ 2, since when ni = 1 we have ¯ni = 0 and the inequality holds
+as well, and write the following chain of inequalities:
+|P(G)| = n0 + n1 ≤ 2 + ¯n0 + ¯n1
+r
+≤ 2 + n − |P(G)|
+r
+,
+where the last inequality comes from the fact that all the sets S0, S1, ¯S0, ¯S1 are disjoint and
+therefore their cardinalities sum up to at most n. This bounds the number of selected vertices
+as |P(G)| ≤ (2r + n)/(r + 1).
+Suppose now that |P(G)| ≥ k + 1. Using the previous bound, this yields
+2r + n ≥ (k + 1)(r + 1) ⇐⇒ r ≤ n − k − 1
+k − 1
+= n − 2
+k − 1 − 1,
+which contradicts the lower bound on r in the statement of the lemma.
+Figure 6 illustrates the argument and notation of Lemma 2. We are now ready to prove
+Theorem 4.
+Proof of Theorem 4. We show that Algorithm 4 satisfies the conditions of the theorem.
+Let
+n ∈ N and k ∈ {2, . . . , n}. Impartiality follows from the fact that Algorithm 2 is impartial,
+thus the potential deletion of outgoing edges of a given vertex cannot affect the fact of selecting
+14
+
+this vertex or not. Formally, if G = (V, E), v ∈ V and G′ = (V, E′) ∈ Nv(G), then defining
+¯G = (V, ¯E) and ¯G′ = (V, ¯E′) as the graphs constructed when running Algorithm 4 with G and
+G′ as input graphs, respectively, we have
+¯E \ ({v} × V ) = (E \ ({v} × V )) \
+
+
+n−1
+�
+u=1
+min{u+r,n}
+�
+w=u+1
+{(u, w)}
+
+
+= (E′ \ ({v} × V )) \
+
+
+n−1
+�
+u=1
+min{u+r,n}
+�
+w=u+1
+{(u, w)}
+
+
+= ¯E′ \ ({v} × V ),
+where we use that G′ ∈ Nv(G). Impartiality then follows directly from impartiality of plurality
+with runners-up. For the following, let G = (V, E) ∈ Gn and define r and ¯G as in the mechanism.
+Since the first step of the mechanism ensures that for every u ∈ {1, . . . , n − 1} and every
+v ∈ {u + 1, . . . , min{u + r, n}}, (u, v) /∈ E, Lemma 2 implies that |PD(G)| = |P( ¯G)| ≤ k.
+Finally, in order to show the additive guarantee we first note that, for every v ∈ V, δ−(v, G) ≤
+δ−(v, ¯G) + r, since at most |{v − r, . . . , v − 1} ∩ V | ≤ r incoming edges of v are deleted when
+defining ¯G from G. In particular, ∆(G) ≤ ∆( ¯G) + r. Using this observation and denoting v∗ ∈
+argminv∈PD(G){δ−(v, G)} an arbitrary element with minimum indegree among those selected by
+asymmetric plurality with runners-up and edge deletion, we obtain that
+δ−(v∗, G) ≥ δ−(v∗, ¯G) ≥ ∆( ¯G) − 1 ≥ ∆(G) − r − 1,
+where the second inequality comes from Lemma 1, since v∗ belongs to P( ¯G). We conclude that
+the mechanism is (r + 1)-min-additive for r = ⌊(n − 2)/(k − 1)⌋.
+It is easy to see that the previous analysis is tight from a graph G = (V, E) where exactly
+r = ⌊(n − 2)/(k − 1)⌋ incoming edges of the top-voted vertex are deleted, and a vertex with the
+second highest indegree u such that u > top(G), (u, top(G)) ∈ E, and δ−(u) = ∆(G) − r − 1 is
+selected. However, we do not know whether the tradeoff provided by Theorem 4 is best possible
+for any impartial mechanism, and the question for the optimum tradeoff is an interesting one.
+Currently, when d ≥ k a gap remains between the upper bound of ⌊(n − 2)/(k − 1)⌋ + 1 and a
+lower bound of 1, which is relatively large when the number k of vertices that can be selected
+is small. We may, alternatively, also ask for the number of vertices that have to be selected in
+order to guarantee 1-min-additivity. Currently, the best upper bound on this number is n − 1.
+In addition to the question about the performance of the mechanism introduced in this
+section, the sole fact that sometimes it does not select vertices with indegree strictly higher than
+the one of other selected vertices may seem unfair. Unfortunately, this is unavoidable whenever
+d ≥ k and α-min-additivity is imposed for some α < d, as one can see from a graph consisting
+of a complete subgraph on d + 1 vertices and n − (d + 1) isolated vertices. For any k-selection
+mechanism, a vertex in the complete subgraph is not selected, and impartiality forces us to not
+select it either when its outgoing edges are deleted and it is the unique top-voted vertex.
+Acknowledgments
+The authors have benefitted from discussions with David Hannon. Re-
+search was supported by the Deutsche Forschungsgemeinschaft under project number 431465007
+and by the Engineering and Physical Sciences Research Council under grant EP/T015187/1.
+References
+[1] N. Alon, F. Fischer, A. Procaccia, and M. Tennenholtz. Sum of us: Strategyproof selec-
+tion from the selectors. In Proceedings of the 13th Conference on Theoretical Aspects of
+Rationality and Knowledge, pages 101–110, 2011.
+15
+
+[2] H. Aziz, O. Lev, N. Mattei, J. S. Rosenschein, and T. Walsh. Strategyproof peer selection
+using randomization, partitioning, and apportionment. Artificial Intelligence, 275:295–309,
+2019.
+[3] Y. Babichenko, O. Dean, and M. Tennenholtz. Incentive-compatible selection mechanisms
+for forests. In Proceedings of the 21st ACM Conference on Economics and Computation,
+pages 111–131, 2020.
+[4] A. Bjelde, F. Fischer, and M. Klimm.
+Impartial selection and the power of up to two
+choices. ACM Transactions on Economics and Computation, 5(4):1–20, 2017.
+[5] N. Bousquet, S. Norin, and A. Vetta. A near-optimal mechanism for impartial selection.
+In Proceedings of the 10th International Conference on Web and Internet Economics, pages
+133–146. Springer, 2014.
+[6] I. Caragiannis, G. Christodoulou, and N. Protopapas. Impartial selection with additive ap-
+proximation guarantees. In Proceedings of the 12th International Symposium on Algorithmic
+Game Theory, pages 269–283. Springer, 2019.
+[7] I. Caragiannis, G. Christodoulou, and N. Protopapas. Impartial selection with prior infor-
+mation. arXiv preprint arXiv:2102.09002, 2021.
+[8] J. Cembrano, F. Fischer, D. Hannon, and M. Klimm. Impartial selection with additive
+guarantees via iterated deletion. arXiv preprint arXiv:2205.08979, 2022.
+[9] G. de Clippel, H. Moulin, and N. Tideman.
+Impartial division of a dollar.
+Journal of
+Economic Theory, 139(1):176–191, 2008.
+[10] F. Fischer and M. Klimm. Optimal impartial selection. SIAM Journal on Computing, 44
+(5):1263–1285, 2015.
+[11] R. Holzman and H. Moulin. Impartial nominations for a prize. Econometrica, 81(1):173–
+196, 2013.
+[12] A. Kahng, Y. Kotturi, C. Kulkarni, D. Kurokawa, and A. D. Procaccia.
+Ranking wily
+people who rank each other. In Proceedings of the 32nd AAAI Conference on Artificial
+Intelligence, 2018.
+[13] D. Kurokawa, O. Lev, J. Morgenstern, and A. D. Procaccia. Impartial peer review. In
+Proceedings of the 24th International Joint Conference on Artificial Intelligence, 2015.
+[14] A. Mackenzie. Symmetry and impartial lotteries. Games and Economic Behavior, 94:15–28,
+2015.
+[15] A. Mackenzie. An axiomatic analysis of the papal conclave. Economic Theory, 69:713–743,
+2020.
+[16] N. Mattei, P. Turrini, and S. Zhydkov. Peernomination: Relaxing exactness for increased
+accuracy in peer selection. arXiv preprint arXiv:2004.14939, 2020.
+[17] S. Tamura. Characterizing minimal impartial rules for awarding prizes. Games and Eco-
+nomic Behavior, 95:41–46, 2016.
+[18] S. Tamura and S. Ohseto. Impartial nomination correspondences. Social Choice and Wel-
+fare, 43(1):47–54, 2014.
+[19] T. Wąs, T. Rahwan, and O. Skibski. Random walk decay centrality. In Proceedings of the
+AAAI Conference on Artificial Intelligence, volume 33, pages 2197–2204, 2019.
+16
+
+[20] Y. Xu, H. Zhao, X. Shi, J. Zhang, and N. B. Shah. On strategyproof conference peer review.
+arXiv preprint arXiv:1806.06266, 2018.
+[21] X. Zhang, Y. Zhang, and D. Zhao. Incentive compatible mechanism for influential agent
+selection. In Proceedings of the 14th International Symposium on Algorithmic Game Theory,
+pages 79–93. Springer, 2021.
+17
+
diff --git a/dtE3T4oBgHgl3EQfegpr/content/tmp_files/load_file.txt b/dtE3T4oBgHgl3EQfegpr/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9d79b8c5bb74d0b5feb4570248da41d10f342144
--- /dev/null
+++ b/dtE3T4oBgHgl3EQfegpr/content/tmp_files/load_file.txt
@@ -0,0 +1,579 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf,len=578
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='04544v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='GT]  11 Jan 2023  Optimal Impartial Correspondences  Javier Cembrano∗  Felix Fischer†  Max Klimm‡  Abstract  We study mechanisms that select a subset of the vertex set of a directed graph in order to  maximize the minimum indegree of any selected vertex, subject to an impartiality constraint  that the selection of a particular vertex is independent of the outgoing edges of that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  For graphs with maximum outdegree d, we give a mechanism that selects at most d + 1  vertices and only selects vertices whose indegree is at least the maximum indegree in the  graph minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We then show that this is best possible in the sense that no impartial  mechanism can only select vertices with maximum degree, even without any restriction  on the number of selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We finally obtain the following trade-off between the  maximum number of vertices selected and the minimum indegree of any selected vertex:  when selecting at most k vertices out of n, it is possible to only select vertices whose indegree  is at least the maximum indegree minus ⌊(n − 2)/(k − 1)⌋ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  1  Introduction  Impartial selection is the problem of selecting vertices with large indegree in a directed graph,  in such a way that the selection of a particular vertex is independent of the outgoing edges of  that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The problem models a situation where agents nominate one another for selection  and are willing to offer their true opinion on other agents as long as this does not affect their  own chance of being selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The selection of a single vertex is governed by strong impossibility results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For graphs with  maximum outdegree one, corresponding to situations where each agent submits a single nomi-  nation, every impartial selection rule violates one of two basic axioms [11] and as a consequence  must fail to provide a non-trivial multiplicative approximation to the maximum indegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For  graphs with arbitrary outdegrees, corresponding to situations where each agent can submit mul-  tiple nominations, impartial rules violate an even weaker axiom and cannot provide a non-trivial  approximation in a multiplicative or additive sense [1, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' These impossibilities largely remain in  place if rather than a single vertex we want to select any fixed number of vertices, but positive  results can be obtained if we relax the requirement that the same number of vertices must be  selected in every graph [4, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  From a practical point of view, the need for such a relaxation should not necessarily be a  cause for concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Indeed, situations in the real world to which impartial selection is relevant  often allow for a certain degree of flexibility in the number of selected agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The exact number  of papers accepted to an academic conference is usually not fixed in advance but depends on  the number and quality of submissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Best paper awards at conferences are often given in  overlapping categories, and some awards may only be given if this is warranted by the field of  candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The Fields medal is awarded every four years to two, three, or four mathematicians  under the age of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Examples at the more extreme end of the spectrum of flexibility include  the award of job titles such as vice president or deputy vice-principal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Such titles can often be  ∗Institut für Mathematik, Technische Universität Berlin, Germany  †School of Mathematical Sciences, Queen Mary University of London, UK  ‡Institut für Mathematik, Technische Universität Berlin, Germany  1  given to a large number of individuals at a negligible cost per individual, but should only be  given to qualified individuals so as not to devalue the title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Tamura and Ohseto [18] specifically studied what they call nomination correspondences,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', rules that may select an arbitrary set of vertices in any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For graphs with maximum  outdegree one a particular such rule, plurality with runners-up, satisfies impartiality and ap-  propriate versions of the two axioms of Holzman and Moulin [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The rule selects any vertex  with maximum indegree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' if there is a unique such vertex, any vertex whose indegree is smaller  by one and whose outgoing edge goes to the vertex with maximum indegree is selected as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  An appropriate measure for the quality of rules that select varying numbers of vertices is the  difference in the worst case between the best vertex and the worst selected vertex, and we can  call a rule α-min-additive if the maximum difference, taken over all graphs, between these two  quantities is at most α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In this terminology, plurality with runners-up is 1-min-additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  As Tamura and Ohseto point out, it may be desirable in practice to ensure that the maximum  number of vertices selected is not too large, a property that plurality with runners-up clearly  fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' It is therefore interesting to ask whether there exist rules that are α-min-additive and  never select more than k vertices, for some fixed α and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  For graphs with outdegree one,  Tamura and Ohseto answer this question in the affirmative: a variant of plurality with runners-  up that breaks ties according to a fixed ordering of the vertices remains 1-min-additive but never  selects more than two vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Our Contribution  Our first result provides a generalization of the result of Tamura and Ohseto  to graphs with larger outdegrees: for graphs with maximum outdegree d, it is possible to achieve  1-min-additivity while selecting at most d+1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For the particular case of graphs with un-  bounded outdegrees we obtain a slight improvement, by guaranteeing 1-min-additivity without  ever selecting all vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Our second result establishes that 1-min-additivity is best possible,  thus ruling out the existence of impartial mechanisms that only select vertices with maximum in-  degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This holds even when no restrictions are imposed on the number of selected vertices, and  is shown alongside analogous impossibility results concerning the maximization of the median or  mean indegree of the selected vertices instead of their minimum indegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Our third result pro-  vides a trade-off between the maximum number of vertices selected, where smaller is better, and  the minimum indegree of any selected vertex, where larger is better: if we are allowed to select  at most k vertices out of n, we can guarantee α-min-additivity for α = ⌊(n−2)/(k−1)⌋+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This  is achieved by removing a subset of the edges from the graph before plurality with runners-up  is applied, in order to guarantee impartiality while selecting fewer vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We do not know  whether this last result is tight and leave open the interesting question for the optimal trade-off  between the number and quality of selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Related Work  Impartiality as a property of an economic mechanism was introduced by  de Clippel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' [9], and first applied to the selection of vertices in a directed graph by Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [1] and Holzman and Moulin [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Whereas Holzman and Moulin gave axiomatic characteriza-  tions for mechanisms selecting a single vertex when all outdegrees are equal to one, Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  studied the ability of impartial mechanisms to approximate the maximum indegree for any fixed  number of vertices when there are no limitations on outdegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Both sets of authors obtained strong impossibility results, which a significant amount of  follow-up work has since sought to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Randomized mechanisms providing non-trivial  multiplicative guarantees had already been proposed by Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', and Fischer and Klimm [10]  subsequently achieved the best possible such guarantee for the selection of one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Starting  from the observation that worst-case instances for randomized mechanisms have small indegrees,  Bousquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' [5] developed a mechanism that is asymptotically optimal as the maximum  indegree grows, and Caragiannis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' [6, 7] initiated the study of mechanisms providing additive  rather than multiplicative guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Cembrano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' [8] subsequently identified a deterministic  2  mechanism that provides non-trivial additive guarantees whenever the maximum outdegree is  bounded and established that no such guarantees can be obtained with unbounded outdegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Randomized mechanisms have been also studied from an axiomatic point of view by Mackenzie  [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Bjelde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' [4] gave randomized mechanisms with improved multiplicative guarantees for the  selection of more than one vertex and observed that when selecting at most k vertices rather than  exactly k, deterministic mechanisms can in fact achieve non-trivial guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' An axiomatic  study of Tamura and Ohseto [18] for the outdegree-one case came to the same conclusion:  when allowing for the selection of a varying number of vertices, the impossibility result of  Holzman and Moulin no longer holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Tamura [17] subsequently characterized a mechanism  proposed by Tamura and Ohseto, which in some cases selects all vertices, as the unique minimal  mechanism satisfying impartiality, anonymity, symmetry, and monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Impartial mechanisms have finally been proposed for various problems other than selection,  including peer review [2, 13, 16, 20], rank aggregation [12], progeny maximization [3, 21], and  network centralities [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  2  Preliminaries  For n ∈ N, let [n] = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}, and let  Gn =  �  (V, E) : V = [n], E ⊆ (V × V ) \\  �  v∈V  {(v, v)}  �  be the set of directed graphs with n vertices and no loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let G = �  n∈N Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For G = (V, E) ∈ G  and v ∈ V , let N +(v, G) = {u ∈ V : (v, u) ∈ E} be the out-neighborhood and N −(v, G) =  {u ∈ V : (u, v) ∈ E} the in-neighborhood of v in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let δ+(v, G) = |N +(v, G)| and δ−(v, G) =  |N −(v, G)| denote the outdegree and indegree of v in G, and ∆(G) = maxv∈V δ−(v, G) the  maximum indegree of any vertex in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  When the graph is clear from the context, we will  sometimes drop G from the notation and write N +(v), N −(v), δ+(v), δ−(v), and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let  top(G) = max{v ∈ V : δ−(v) = ∆(G)} denote the vertex of G with the largest index among  those with maximum indegree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  For n ∈ N and d ∈ [n − 1], let Gn(d) = {(V, E) ∈ Gn :  δ+(v) ≤ d for every v ∈ V } be the set of graphs in Gn with maximum outdegree at most d, and  G(d) = �  n∈N Gn(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  A k-selection mechanism is then given by a family of functions f : Gn → 2[n], one for  each n ∈ N, mapping each graph to a subset of its vertices, where we require that |f(G)| ≤ k  for all G ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In a slight abuse of notation, we will use f to refer to both the mechanism  and to individual functions of the family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Given G = (V, E) ∈ G and v ∈ V , let Nv(G) =  {(V, E′) ∈ G :  E \\ ({v} × V ) = E′ \\ ({v} × V )} be the set neighboring graphs of G with  respect to v, in the sense that they can be obtained from G by changing the outgoing edges  of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Mechanism f is impartial on G′ ⊆ G if on this set of graphs the outgoing edges of a  vertex have no influence on its selection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', if for every graph G = (V, E) ∈ G′, v ∈ V , and  G′ ∈ Nv(G), it holds that f(G) ∩ {v} = f(G′) ∩ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Given a k-selection mechanism f and  an aggregator function σ : 2R → R such that σ(∅) = 0 and, for every S ⊆ R with |S| ≥ 1,  min{x ∈ S} ≤ σ(S) ≤ max{x ∈ S}, we say that f is α-σ-additive on G′ ⊆ G, for α ≥ 0, if for  every graph in G′ the function σ evaluated on the choice of f differs from the maximum indegree  by at most α, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', if  sup  G∈G′  �  ∆(G) − σ  �  {δ−(v, G)}v∈f(G)  ��  ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  We will specifically be interested in the cases where σ is the minimum, the median, and the mean,  and respectively call a mechanism α-min-additive, α-median-additive, and α-mean-additive in  these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  3  Algorithm 1: Plurality with runners-up  Input: Digraph G = (V, E) ∈ Gn(1)  Output: Set S ⊆ V of selected vertices  Let S = {v ∈ V : δ−(v) = ∆(G)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  if S = {v} for some v ∈ V then  S ←− S ∪ {u ∈ V : δ−(u) = ∆(G) − 1 and (u, v) ∈ E}  end  Return S  3  Plurality with Runners-up  Focusing on the case with maximum outdegree one, Tamura and Ohseto [18] proposed a mech-  anism they called plurality with runners-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The mechanism, which we describe formally in  Algorithm 1, selects all vertices with maximum indegree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' if there is a unique such vertex, then  any vertex with an outgoing edge to that vertex whose indegree is smaller by one is selected as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The idea behind this mechanism is that vertices in the latter category would be among  those with maximum degree if their outgoing edge was deleted, and thus any impartial mecha-  nism seeking to select the vertices with maximum degree would also have to select those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Plurality with runners-up is impartial on G(1), and in any graph with n vertices selects between  1 and n vertices whose degree is equal to the maximum degree or the maximum degree minus  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' It is thus an impartial and 1-min-additive n-selection mechanism on Gn(1) for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  It is natural to ask whether a similar additive guarantee can be obtained for more general set-  tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In this section, we answer this question in the affirmative, and in particular study for  which values of n, k, and d there exists an impartial and 1-min-additive k-selection mechanism  on Gn(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We will see later, in Section 4, that 1-min-additivity is in fact best possible for all  cases covered by our result, with the exception of the boundary case where n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  While Tamura and Ohseto do not limit the maximum number of selected vertices, they  discuss briefly a modification of their mechanism that retains impartiality and 1-min-additivity  but selects at most 2 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Instead of all vertices with maximum indegree, the modified  mechanism breaks ties in favor of a single maximum-degree vertex using a fixed ordering of the  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In order to guarantee impartiality, the modified mechanism then also selects any vertex  that would be selected in the graph obtained by deleting the outgoing edge of that vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The  assumption that every vertex has at most one outgoing edge means that at most one additional  vertex is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' There thus exists a 1-min-additive k-selection mechanism on G(1) for every  k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Our first result generalizes this mechanism to settings with arbitrary outdegrees, as long  as the maximum number of selected vertices is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To this end we show that when  the maximum outdegree is d, to achieve impartiality, at most d vertices have to be selected in  addition to the one with maximum indegree and highest priority.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='1 We formally describe the  resulting mechanism in Algorithm 2, and will refer to it as asymmetric plurality with runners-up  and denote its output on graph G by P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We obtain the following theorem, which generalizes  the known result for the outdegree-one case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For every n ∈ N, d ∈ [n − 1], and k ∈ {d + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}, there exists an impartial  and 1-min-additive k-selection mechanism on Gn(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  We will be interested in the following in comparing vertices both according to their indegree  and to their index, and we will use regular inequality symbols, as well as the operators max and  1In this mechanism and wherever ties are broken in the rest of the paper, we break ties in favor of greater  index, so top(G) is the vertex with maximum indegree and highest priority in graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Naturally, any other  deterministic tie-breaking rule could be used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  4  Algorithm 2: Asymmetric plurality with runners-up P(G)  Input: Digraph G = (V, E) ∈ Gn  Output: Set S ⊆ V of selected vertices  Let S = ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  for v ∈ V do  Let Gv = (V, E \\ ({v} × V ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  if top(Gv) = v then  S ←− S ∪ {v}  end  end  Return S  min, to denote the lexicographic order among pairs of the form (δ−(v), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The following lemma  characterizes the structure of the set of vertices selected by Algorithm 2, and provides the main  technical ingredient to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let G = (V, E) ∈ G and v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Then, v ∈ P(G) if and only if  (a) for every w ∈ V with (δ−(w), w) > (δ−(v), v) it holds (v, w) ∈ E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' and  (b) one of the following holds:  (i) δ−(v) = ∆(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' or  (ii) δ−(v) = ∆(G) − 1 and v > w for every w ∈ V with δ−(w) = ∆(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We first show that, if v ∈ P(G) for a given graph G, then (a) and (b) follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let  G = (V, E) ∈ G, and let v ∈ P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To see (a), suppose there is w ∈ V with (δ−(w, G), w) >  (δ−(v, G), v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since v ∈ P(G), we have v = top(Gv) with Gv = (V, E \\ ({v} × V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  This  implies (δ−(v, Gv), v) > (δ−(w, Gv), w) and therefore δ−(w, G) > δ−(w, Gv), because δ−(v, G) =  δ−(v, Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Since G and Gv only differ in the outgoing edges of v, we conclude that (v, w) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  To prove (b), we note that for every w ∈ V we have  (δ−(v, G), v) = (δ−(v, Gv), v) > (δ−(w, Gv), w) ≥ (δ−(w, G) − 1, w),  (1)  where the last inequality comes from the fact that each vertex has at most one incoming edge  from v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If there is no w ∈ V \\ {v} with δ−(w) = ∆(G), the maximum indegree must be that of  v, so δ−(v) = ∆(G) and (i) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Otherwise, for each w ∈ V \\ {v} with δ−(w) = ∆(G), (1)  yields (δ−(v, G), v) > (∆(G) − 1, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We conclude that either δ−(v, G) > ∆(G) − 1, in which  case (i) holds, or both δ−(v) = ∆(G) − 1 and v > w, which implies (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  We now prove the other direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let G = (V, E) ∈ G and v ∈ V such that both (a)  and (b) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let Gv = (V, E \\ ({v} × V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We have to show that top(Gv) = v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', that  for every w ∈ V \\ {v}, (δ−(v, Gv), v) > (δ−(w, Gv), w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let w be a vertex in V \\ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If  (δ−(v, G), v) > (δ−(w, G), w), we can conclude immediately since δ−(v, Gv) = δ−(v, G) and  δ−(w, Gv) ≤ δ−(w, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Otherwise, we know from (a) that (v, w) ∈ E and thus δ−(w, Gv) =  δ−(w, G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If v satisfies (i), this yields  δ−(v, Gv) = δ−(v, G) = ∆(G) ≥ δ−(w, G) = δ−(w, Gv) + 1,  so (δ−(v, Gv), v) > (δ−(w, Gv), w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' On the other hand, if v satisfies (ii), then  δ−(v, Gv) = δ−(v, G) = ∆(G) − 1 ≥ δ−(w, G) − 1 = δ−(w, Gv),  and v > w implies (δ−(v, Gv), v) > (δ−(w, Gv), w) as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  5  4  2  1  6  5  3  ∆  ∆ − 1  Figure 1: Example of a set of vertices selected by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In this illustration and throughout  the paper, vertices are arranged vertically according to indegree and horizontally according to  index, so that vertices on the left are favored in case of ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The vertices selected by the  mechanism are drawn in white, those not selected in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vertices with indegree below ∆ − 1,  as well as edges incident to such vertices, are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Denoting the graph as G = (V, E),  and letting Gv = (V, E \\ ({v} × V )) for each vertex v, the selected vertices v are those for which  top(Gv) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Specifically, vertices 2, 3, and 6 are not selected because top(G2) = 4, top(G3) = 4,  and top(G6) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Observe that Lemma 1 implies in particular that top(G) ∈ P(G) for every graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Figure 1  provides an example of the characterization given by Lemma 1, in terms of indegrees, tie-  breaking order, and edges among selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We show that for every n ∈ N and d ∈ [n − 1], asymmetric plurality with  runners-up is impartial and 1-min-additive on Gn(d), and that for every G = (V, E) ∈ Gn(d), it  selects at most d+1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If this is the case, then for every k ∈ {d+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n} the mechanism  would satisfy the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Therefore, let n and d be as mentioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Impartiality follows from the definition of the mechanism, because the outgoing edges of a  vertex are not taken into account when deciding whether the vertex is taking part on the selected  set or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If we let G = (V, E), v ∈ V , and G′ = (V, E′) ∈ Nv(G), then the graphs Gv and  G′  v constructed when running the mechanism with each of these graphs G and G′ as an input,  respectively, are the same because by definition of Nv(G) we have E \\({v}×V ) = E′ \\({v}×V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since v ∈ P(G) ⇔ top(Gv) = v, and v ∈ P(G′) ⇔ top(G′  v) = v, we conclude v ∈ P(G) ⇔ v ∈  P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  To see that the mechanism is 1-min-additive, let G ∈ Gn(d) and first note that P(G) ̸= ∅  since Lemma 1 implies that top(G) ∈ P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' From this lemma we also know that for every  v ∈ P(G), δ−(v) ≥ ∆(G) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We conclude that min{{δ−(v)}v∈P(G)} ≥ ∆(G) − 1, and since  this holds for every G ∈ Gn(d), the mechanism is 1-min-additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Finally, let G = (V, E) ∈ Gn(d), and suppose that |P(G)| > d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If we denote vL =  argminv∈P(G){(δ−(v), v)}, from Lemma 1 we know that (vL, w) ∈ E for every w ∈ V with  (δ−(w), w) > (δ−(vL), vL), thus δ+(vL) ≥ |P(G)| − 1 > d, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We conclude that  |P(G)| ≤ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The following result, concerning mechanisms that may select an arbitrary number of vertices,  follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For every n ∈ N, there exists an impartial and 1-min-additive n-selection mecha-  nism on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  On Gn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', in the case of unbounded outdegrees, this result can in fact be improved slightly  to guarantee 1-min-additivity while selecting only at most n − 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The improvement  is achieved by a more intricate version of asymmetric plurality with runners-up, which we call  asymmetric plurality with runners-up and pivotal vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We formally describe this mechanism  6  Algorithm 3: Asymmetric plurality with runners-up and pivotal vertices PP(G)  Input: Digraph G = (V, E) ∈ Gn  Output: Set S ⊆ V of selected vertices with |S| ≤ n − 1  Let S ←− ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  for u ∈ P(G) do  if for every v ∈ P(G) \\ {u} there exists Guv ∈ Nu(G) such that v /∈ P(Guv) then  S ←− S ∪ {u}  end  end  Return S  in Algorithm 3 and denote its output for graph G by PP (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Given a graph G = (V, E),  call a vertex u ∈ P(G) pivotal for v ∈ P(G) if there exists a graph Guv ∈ Nu(G) such that  v /∈ P(Guv), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', if the outgoing edges of u can be changed in such a way that v is no longer  selected by asymmetric plurality with runners-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Asymmetric plurality with runners-up and  pivotal vertices then selects every vertex in P(G) that is pivotal for every other vertex in P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The mechanism turns out to inherit impartiality and 1-min-additivity, and to never select all  vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For every n ∈ N and k ∈ {n − 1, n}, there exists an impartial and 1-min-additive  k-selection mechanism on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We show that for every n ∈ N, asymmetric plurality with runners-up and pivotal vertices  is impartial and 1-min-additive on Gn and that for every G = (V, E) ∈ Gn, it selects at most  n − 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let n ∈ N be an arbitrary value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  To see that the mechanism is impartial, let G = (V, E) ∈ Gn, u ∈ PP (G), and G′ = (V, E′) ∈  Nu(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We show in the following that u ∈ PP (G′), and since the graphs G and G′ are chosen  arbitrarily, their roles can be inverted and this is enough to conclude that the mechanism is  impartial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We first note that u ∈ P(G) because PP (G) ⊆ P(G), thus impartiality of asymmetric  plurality with runners-up proven in Theorem 1 implies u ∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If P(G′) = {u}, then the  condition in the mechanism holds trivially for this vertex, so u ∈ PP (G′) and we conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Otherwise, let v ∈ P(G′) \\ {u} be an arbitrary vertex selected by asymmetric plurality with  runners-up other than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Since u ∈ PP (G), we have that either (a) v /∈ P(G), or (b) v ∈ P(G)  and there exists Guv = (V, Euv) ∈ Nu(G) such that v /∈ P(Guv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If (a) holds, taking G′  uv = G,  which belongs to Nu(G′) because of the assumption that G′ ∈ Nu(G), we have that v /∈ P(G′  uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If (b) holds, taking G′  uv = Guv, which belongs to Nu(G′) since Nu(G′) = Nu(G), we have that  v /∈ P(G′  uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In either case, we conclude that there exists G′  uv ∈ Nu(G′) such that v /∈ P(G′  uv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since this argument is valid for every v ∈ P(G′) \\ {u}, we conclude that u ∈ PP (G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  To see that the mechanism is 1-min-additive, it is enough to show that it always selects a  vertex, since for every G ∈ G it selects a subset of P(G) and from Theorem 1 we know that this  set contains vertices with indegrees in {∆(G), ∆(G)−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To this purpose we let G = (V, E) ∈ Gn  and introduce some additional notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|  for i ∈ {0, 1}, and denote  vH = argmaxv∈P(G){(δ−(v, G), v)} = top(G),  vL = argminv∈P(G){(δ−(v, G), v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  From Lemma 1, we know that P(G) = S0 ∪ S1, that (vL, v) ∈ E for every v ∈ P(G) \\ {vL},  and that u > v for each u ∈ S1, v ∈ S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We now distinguish two cases according to the edges  between vertices in P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If (vH, v) ∈ E for every v ∈ P(G) \\ {vH}, then we claim that defining G′ = (V, E \\ ({vH} ×  V )) ∈ NvH(G) it holds v /∈ P(G′) for every v ∈ P(G) \\ {vH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If this is true, it is clear that  7  ∆  ∆ − 1  ∆ − 2  G = (V, E)  G′ = (V, E \\ ({2} × V ))  2  1  4  3  2  1  4  3  Figure 2: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if  (vH, v) ∈ E for every v ∈ P(G) \\ {vH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vertices selected by asymmetric plurality with runners-  up are drawn in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Denoting the graph on the left as G = (V, E), where vH = 2, and defining  G′ = (V, E \\ ({2} × V )) ∈ N2(G), we have that {1, 3, 4} ∩ P(G′) = ∅, and thus 2 ∈ PP (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  vH ∈ PP (G) and thus PP (G) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We now prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' First, note that vH ∈ P(G′)  since vH = top(G′) and Lemma 1 ensures top(G′) ∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This comes from the fact that  δ−(vH, G′) = δ−(vH, G) and δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \\ {vH}, together with vH =  top(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Moreover, for every v ∈ S0 \\ {vH} it holds δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 =  ∆(G′) − 1 and v < vH, so condition (b) in Lemma 1 does not hold for v and thus v /∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Analogously, for every v ∈ S1 it holds δ−(v, G′) = δ−(v, G)−1 = δ−(vH, G′)−2 = ∆(G′)−2, so  condition (b) in Lemma 1 does not hold for v either, and thus v /∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This allows to conclude  the claim and the fact that PP (G) is non-empty for this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This argument is illustrated in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Now we consider the case where there is a vertex ¯v ∈ P(G) such that (vH, ¯v) /∈ E, and  we claim that defining G′ = (V, (E \\ ({vL × V })) ∪ (vL, vH)) ∈ NvL(G) it holds v /∈ P(G′)  for every v ∈ P(G) \\ {vL, vH}, whereas defining G′′ = (V, E \\ (vL, vH)) ∈ NvL(G) it holds  vH /∈ P(G′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If this is true, then vL ∈ PP (G) and PP (G) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We now prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  First, note that vH ∈ P(G′) for the same reason as before, since δ−(vH, G′) = δ−(vH, G) and  δ−(v, G′) ≤ δ−(v, G) for every v ∈ V \\ {vH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Moreover, for every v ∈ S0 \\ {vH} it holds  δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 1 = ∆(G′) − 1 and v < vH, so condition (b) in  Lemma 1 does not hold for v and thus v /∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Analogously, for every v ∈ S1 \\ {vL} it holds  δ−(v, G′) = δ−(v, G) − 1 = δ−(vH, G′) − 2 = ∆(G′) − 2 so condition (b) in Lemma 1 does not  hold for v and thus v /∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This allows to conclude the claim for G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In the case of G′′, we  can write the following chain of inequalities,  (δ−(¯v, G′′), ¯v) = (δ−(¯v, G), ¯v) > (δ−(vH, G) − 1, vH) = (δ−(vH, G′′), vH),  where the equalities hold because of the definition of G′′ and the inequality by condition (b)  in Lemma 1, given that ¯v ∈ P(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since (vH, ¯v) /∈ E, we conclude from condition (a) in  Lemma 1 that vH /∈ P(G′′), and therefore the claim for G′′ follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This argument is illustrated  in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Finally, we show that the mechanism selects at most n − 1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let G = (V, E) ∈ Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since PP (G) ⊆ P(G), if |P(G)| ≤ n − 1 this is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We thus suppose in what follows  that |P(G)| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In particular, Lemma 1 implies (v, vH) ∈ E for every v ∈ V \\ {vH}, thus  ∆(G) = n − 1, and δ−(v) ≥ n − 2 for every v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If S1 = ∅, then δ−(v) = n − 1 for every  v ∈ V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', G is the complete graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In this case, vH = n and we claim that v /∈ PP (G) for  each v ∈ V \\ {n}, thus |PP (G)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This comes from the fact that, for every v ∈ V \\ {n}  and every G′ = (V, E′) ∈ Nv(G) it holds n ∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To see this, note that (n, v) ∈ E′ for  every v ∈ V \\ {n}, δ−(n, G′) ≥ n − 2 = ∆(G′) − 1, and n > v for every v ∈ V \\ {n}, so  Lemma 1 ensures n ∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If S1 ̸= ∅, then there is at least one vertex with outdegree less  8  ∆  ∆ − 1  ∆ − 2  G′ = (V, E \\ {(3, 1), (3, 4)})  G′′ = (V, E \\ {(3, 2)})  ∆  ∆ − 1  G = (V, E)  2  1  4  3  2  1  4  3  2  1  4  3  Figure 3: Illustration of the fact that the set of vertices selected by Algorithm 3 is non-empty if  (vH, ¯v) /∈ E for some ¯v ∈ P(G) \\ {vH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vertices selected by asymmetric plurality with runners-  up are drawn in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Denoting the graph at the top by G = (V, E), where vH = 2, vL = 3,  and ¯v = 4, and defining G′ = (V, E \\ {(3, 1), (3, 4)}) ∈ N3(G), we have that {1, 4} ∩ P(G′) = ∅,  whereas defining G′′ = (V, (E \\ {(3, 2)}) ∈ N3(G) we have that 2 /∈ P(G′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We conclude that  3 ∈ PP (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  or equal than n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let u be an arbitrary vertex with δ+(u) ≤ n − 2, and let ¯v ∈ S1 be the  vertex with highest index such that (u, ¯v) /∈ E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', ¯v = max{V \\ N +(u)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Since u ∈ PP (G),  there exists G′ = (V, E′) ∈ Nu(G) such that ¯v /∈ P(G′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' From Lemma 1, this implies that  there exists ¯w ∈ V such that either (a) (δ−( ¯w, G′), w) > (δ−(¯v, G′), ¯v) and (¯v, ¯w) /∈ E′, or  (b) δ−( ¯w, G′) > δ−(¯v, G′) and ¯w > ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since ¯v ∈ P(G), we know from this same lemma  that if (a) holds, (δ−( ¯w, G), w) < (δ−(¯v, G), ¯v) because of having ¯w /∈ N +(¯v, G) = N +(¯v, G′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  and similarly, if (b) holds, δ−( ¯w, G) ≤ δ−(¯v, G) because of having ¯w > ¯v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  In either case,  since δ−(¯v, G) ≤ δ−(¯v, G′), we conclude that δ−( ¯w, G′) > δ−( ¯w, G), and therefore (u, ¯w) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If (a) holds, this is a contradiction because we would have {u, ¯v} ∩ N −( ¯w, G) = ∅ and thus  δ−( ¯w, G) ≤ n − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If (b) holds, we reach a contradiction as well, because we would have  ¯w ∈ V \\ N +(u, G) and ¯w > ¯v, but we chose ¯v to be the maximum of this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  4  An Impossibility Result  When we established the existence of an impartial and 1-min-additive k-selection mechanism on  G(d) whenever k ≥ d+1, we claimed this result to be best possible in the sense that the additive  guarantee cannot be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We will prove this claim, that impartiality is incompatible with  the requirement to only select vertices with maximum indegree, as a corollary of a more general  result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  While selecting only vertices with maximum indegree is a natural goal for mechanisms that  select varying numbers of vertices, other natural objectives exist for such mechanisms such as  maximizing the median or mean indegree of the selected vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For both of these objectives,  the mechanisms discussed in the previous section immediately provide upper bounds: if a k-  selection mechanism always selects one vertex with maximum indegree and is α-min-additive  9  then it is clearly α-median-additive and  � k−1  k α  �  mean-additive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Theorem 1 thus implies the ex-  istence of a 1-median-additive and k−1  k -mean-additive k-selection mechanism on G(d), whenever  k ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To improve on 1-median-additivity, it would be acceptable to select vertices with low  indegree as long as a greater number of vertices with maximum indegree is selected at the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' To improve on k−1  k -mean-additivity, it would suffice to select more than one vertex with  maximum indegree whenever this is possible, and to otherwise select only a sublinear number  in k of vertices with indegree equal to the maximum indegree minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The following result  shows that no such improvements are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let n ∈ N, n ≥ 3, k ∈ [n], and d ∈ [n − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let f be an impartial k-selection  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If f is α1-median-additive on Gn(d), then α1 ≥ 1/2(1+1(d ≥ 3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If f is α2-mean-  additive on Gn(d), then α2 ≥  � d+1  2  �  /  �� d+1  2  �  + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let n, k, and d be as in the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In the following we suppose that  there is an impartial k-selection mechanism f which is either α1-median-additive on Gn(d) with  α1 < 1/2(1 + 1(d ≥ 3)), or α2-mean-additive on Gn(d) with α2 <  � d+1  2  �  /  �� d+1  2  �  + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  We first prove the result for the case d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We consider the graph G = (V, E) ∈ Gn(1)  with E = {(1, 2), (2, 3), (3, 1)}, consisting of a 3-cycle and n − 3 isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We consider  as well, for v ∈ {1, 2, 3}, the graph Gv = (V, Ev) where v deviates from the 3-cycle by changing  its outgoing edge to the previous vertex in the cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=',  E1 = {(1, 3), (2, 3), (3, 1)}, E2 = {(1, 2), (2, 1), (3, 1)}, E3 = {(1, 2), (2, 3), (3, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since f is α1-median-additive with α1 < 1/2 or α2-mean-additive with α2 < 1/2, we have that  f(G1) = {3}, f(G2) = {1}, and f(G3) = {2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In particular, for v ∈ {1, 2, 3}, v /∈ f(Gv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Since  for each v ∈ {1, 2, 3} it holds Ev \\ ({v} × V ) = E \\ ({v} × V ), we conclude by impartiality  that v /∈ f(G), and thus f(G) ∩ {1, 2, 3} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  This implies that both the median and the  mean indegree of the vertices in f(G) are 0, which contradicts the additive guarantee of this  mechanism because ∆(G) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  In the following, we assume d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We denote D = [d + 1] and consider in what follows two  families of graphs with n vertices, Kv for each v ∈ D and Kuv for each u, v ∈ D, u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' They  are constructed from a complete subgraph on D but deleting the outgoing edges of v, in the  case of Kv, and the outgoing edges of u and v, in the case of Kuv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' All the other vertices remain  isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Formally, taking V = [n] we define  Kv = (V, (D \\ {v}) × D) for every v ∈ D,  Kuv = (V, (D \\ {u, v}) × D) for every u, v ∈ D with u ̸= v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  If there is v ∈ D such that v /∈ f(Kv), then  median  �  {δ−(w, Kv)}w∈f(Kv)  �  ≤ d − 1 = ∆(Kv) − 1,  mean  �  {δ−(w, Kv)}w∈f(Kv)  �  ≤ d − 1 = ∆(Kv) − 1,  which is a contradiction, so the result follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Therefore, in the following we assume  that for every v ∈ D we have v ∈ f(Kv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We claim that for every v ∈ D,  |{u ∈ D \\ {v} : u ∈ f(Kv)}| ≥  �d + 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let us see why the result follows if the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' If this is the case, f selects one vertex with  maximum indegree d in Kv and at least  � d+1  2  �  vertices with indegree d − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This yields both  median  �  {δ−(w, Kv)}w∈f(Kv)  �  ≤  � d − 1  2  if d = 2  d − 1  otherwise,  10  and  mean  �  {δ−(w, Kv)}w∈f(Kv)  �  ≤ d + (d − 1)  � d+1  2  �  � d+1  2  �  + 1  = d −  � d+1  2  �  � d+1  2  �  + 1,  which is a contradiction since ∆(Kv) = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Now we prove the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Suppose that for every v ∈ D we have v ∈ f(Kv) and  |{u ∈ D \\ {v} : u ∈ f(Kv)}| <  �d + 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  (2)  Let v ∈ D and u ∈ D \\ {v} such that u /∈ f(Kv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Observing that  ((D \\ {v}) × D) \\ ({u} × V ) = ((D \\ {u, v}) × D) \\ ({u} × V ),  we obtain from impartiality that u /∈ f(Kuv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' From the bounds on α1 or α2 that f satisfies  by assumption, this mechanism has to select a vertex with maximum indegree in this graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  otherwise, both the median and the mean of the selected set would be at most ∆(Kuv) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since δ−(w) < ∆(Kuv) for every w /∈ {u, v}, it holds v ∈ f(Kuv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Using impartiality once again,  we conclude v ∈ f(Ku).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We have shown the following property:  For every u, v ∈ D : u /∈ f(Kv) =⇒ v ∈ f(Ku).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  (3)  Consider now the graph H = (D, F), where for each u, v ∈ D with u ̸= v, (u, v) ∈ F if and  only if u /∈ f(Kv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Property (2) implies that  δ−(v, H) > d −  �d + 1  2  �  ⇐⇒ δ−(v, H) ≥ d + 1 −  �d + 1  2  �  for each v ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In particular, there has to be a vertex v∗ ∈ D such that δ+(v∗, H) ≥ d + 1 −  ⌊(d + 1)/(2)⌋ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For this vertex we have  δ+(v∗, H) + δ−(v∗, H) ≥ 2  �  d + 1 −  �d + 1  2  ��  ≥ d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since H has d+1 vertices, this implies the existence of w∗ ∈ D for which {(v∗, w∗), (w∗, v∗)} ⊂ F,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=', both v∗ /∈ f(Kw∗) and w∗ /∈ f(Kv∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This contradicts (3), so we conclude the proof of the  claim and the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Figure 4 provides an illustration of Theorem 3 for the case where n = 3, Figure 5 for the  case where n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The median of any set of numbers is an upper bound on their minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Therefore, if no  impartial mechanism exists that is α-median-additive on G′ ⊆ G for α < ¯α, then no impartial  mechanism can exist that is α-min-additive on G′ for α < ⌈¯α⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We thus obtain the following  impossibility result, which we have claimed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let n ∈ N, n ≥ 3, and k ∈ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let f be an α-min-additive impartial k-selection  mechanism on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Then α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The impossibility results imply that for k ≥ d + 1, the mechanisms of Section 3 are best  possible for the minimum and median objectives except in a few boundary cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' When n = 2,  selecting each of the two vertices if and only if it has an incoming edge is impartial and achieves  0-min-additivity and 0-median-additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' When n = 3, it is possible to select in an impartial  way at least one vertex with maximum indegree and at most one vertex with indegree equal  to the maximum indegree minus one, thus guaranteeing 1/2-median-additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For the mean  objective, the mechanisms of Section 3 are best possible asymptotically under the additional  assumption that k = O(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  11  1  2  3  1  1  2  3  2  1  2  3  3  1  2  3  Figure 4: Counterexample to the existence of an impartial 3-selection mechanism that is α-  median-additive or α-mean-additive on G3 for α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vertices drawn in white have to be  selected, vertices in black cannot be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For the graphs at the top, on the left, and on  the right, this follows from α-median-additivity or α-mean-additivity for α < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' An arrow  with label v from one graph to another indicates that one can be obtained from the other by  changing the outgoing edges of vertex v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' by impartiality, the vertex thus has to be selected in  both graphs or not selected in both graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' It follows that no vertices are selected in the graph  at the center, a contradiction to the claimed additive guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  It is worth pointing out that the proof of the impossibility result uses graphs in which some  vertices, in particular those with maximum indegree, do not have any outgoing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' However,  the impossibility extends naturally to the case where this cannot happen, corresponding to the  practically relevant case in which abstentions are not allowed, as long as n ≥ 4 and d ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For  this it is enough to define D = [d], add a new vertex with outgoing edges to every vertex in D  and incoming edges from the vertices in D which do not have any outgoing edge, and construct  a cycle containing the vertices in V \\ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  5  Trading Off Quantity and Quality  We have so far given impartial selection mechanisms for settings where the maximum outdegree d  is smaller than the maximum number k of vertices that can be selected, and have shown that  the mechanisms provide best possible additive guarantees in such settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We will now consider  settings where d ≥ k, such that asymmetric plurality with runners-up selects too many vertices  and therefore cannot be used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For every n ∈ N and k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}, there exists an impartial and (⌊(n − 2)/(k −  1)⌋ + 1)-min-additive k-selection mechanism on Gn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The result is obtained by a variant of asymmetric plurality with runners-up in which some  edges are deleted before the mechanism is run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In principle, deleting a certain number of edges  can affect the additive guarantee by the same amount, if all of the deleted edges happen to  be directed at the same vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' By studying the structure of the set of vertices selected by the  mechanism, we will instead be able to delete edges to distinct vertices and thus keep the negative  impact on the additive guarantee under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  The modified mechanism, which we call asymmetric plurality with runners-up and edge  deletion, is formally described in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  It deletes any edges from a vertex to the  ⌊(n−2)/(k−1)⌋ vertices preceding that vertex in the tie-breaking order, and applies asymmetric  12  1  2  3  4  2  3  1  2  3  4  4  1  2  3  4  4  1  2  3  4  1  1  2  3  4  1  1  2  3  4  1  2  3  4  2  3  1  2  3  4  Figure 5: Counterexample to the existence of an impartial 4-selection mechanism that is α1-  median-additive on G4(3) for α1 < 1 or α2-mean-additive on G4 for α2 < 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vertices drawn  in white have to be selected, vertices in black cannot be selected, and vertices in gray may  or may not be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For the graph on the left, this follows from α1-median-additivity for  α1 < 1 or α2-mean-additivity for α2 < 2/3: under these assumptions at most one of the vertices  with indegree 2 can be selected, which without loss of generality we can assume to be vertex 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  For the other graphs, it then follows by impartiality, and for the graph on the right yields a  contradiction to the claimed additive guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  plurality with runners-up to the resulting graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' The following lemma shows that without such  edges, the maximum number of vertices selected is reduced to k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let n ∈ N, k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}, and r ∈ N with r ≥ ⌊(n−2)/(k−1)⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Let G = (V, E) ∈ Gn  be such that for every u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n − 1} and every v ∈ {u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , min{u + r, n}}, (u, v) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Then, |P(G)| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' As in the proof of Theorem 2, we let Si = {v ∈ P(G) : δ−(v) = ∆(G) − i} and ni = |Si|  for i ∈ {0, 1}, and now we denote its elements in increasing order by vi  j for j ∈ [ni], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=',  Si = {vi  j}ni  j=1 with vi  1 < vi  2 · · · < vi  ni for each i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  From Lemma 1, we know that P(G) = S0 ∪S1, that for i ∈ {0, 1} we have (vi  j, vi  k) ∈ E for every  j, k with j < k, and that v1  1 > v0  n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This allows to define, for i ∈ {0, 1},  ¯Si = {v ∈ V \\ Si : vi  1 < v < vi  ni},  ¯ni = | ¯Si|,  such that ¯S0 ∩ ¯S1 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Fix i ∈ {0, 1} and suppose that ni ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Combining both the fact that (vi  j, vi  k) ∈ E for every  j, k with j < k, and that for every u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n−1} and v ∈ {u+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , min{u+r, n}}, (u, v) /∈  13  Algorithm 4: Asymmetric plurality with runners-up and edge deletion PD(G)  Input: Digraph G = (V, E) ∈ Gn, k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}  Output: Set S ⊆ V of selected vertices with |S| ≤ k  Let r = ⌊(n − 2)/(k − 1)⌋ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  // number of outgoing edges to remove  Let R = �n−1  u=1  �min{u+r,n}  v=u+1  {(u, v)} ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  // edges to be removed  Let ¯G = (V, E \\ R);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Return P( ¯G)  v0  n0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  v0  2  v0  1  v1  n1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  v1  2  v1  1  ∆  ∆ − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  � �� �  ≥r  S1  ¯S1  S0  ¯S0  Figure 6: Illustration of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' There are no edges from a vertex to any of the r vertices  to its left, which means that for each vertex in S0 or S1, except for the left-most vertex, there  exist are at least r vertices outside these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Such vertices are not arranged according to their  indegrees, and edges from vertices in S1 to every vertex in S0 have been omitted for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  E, we have that for every j ∈ [ni − 1] it holds vi  j+1 − vi  j ≥ r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Summing over j yields  vi  ni − vi  1 ≥ (ni − 1)(r + 1), hence  ¯ni = vi  ni − vi  1 + 1 − ni ≥ (ni − 1)(r + 1) + 1 − ni = (ni − 1)r,  where the first equality comes from the definition of the set ¯Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This implies ni ≤ 1 + ¯ni/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We  can now lift the assumption ni ≥ 2, since when ni = 1 we have ¯ni = 0 and the inequality holds  as well, and write the following chain of inequalities:  |P(G)| = n0 + n1 ≤ 2 + ¯n0 + ¯n1  r  ≤ 2 + n − |P(G)|  r  ,  where the last inequality comes from the fact that all the sets S0, S1, ¯S0, ¯S1 are disjoint and  therefore their cardinalities sum up to at most n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' This bounds the number of selected vertices  as |P(G)| ≤ (2r + n)/(r + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Suppose now that |P(G)| ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Using the previous bound, this yields  2r + n ≥ (k + 1)(r + 1) ⇐⇒ r ≤ n − k − 1  k − 1  = n − 2  k − 1 − 1,  which contradicts the lower bound on r in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Figure 6 illustrates the argument and notation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We are now ready to prove  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We show that Algorithm 4 satisfies the conditions of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Let  n ∈ N and k ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartiality follows from the fact that Algorithm 2 is impartial,  thus the potential deletion of outgoing edges of a given vertex cannot affect the fact of selecting  14  this vertex or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Formally, if G = (V, E), v ∈ V and G′ = (V, E′) ∈ Nv(G), then defining  ¯G = (V, ¯E) and ¯G′ = (V, ¯E′) as the graphs constructed when running Algorithm 4 with G and  G′ as input graphs, respectively, we have  ¯E \\ ({v} × V ) = (E \\ ({v} × V )) \\  \uf8eb  \uf8ed  n−1  �  u=1  min{u+r,n}  �  w=u+1  {(u, w)}  \uf8f6  \uf8f8  = (E′ \\ ({v} × V )) \\  \uf8eb  \uf8ed  n−1  �  u=1  min{u+r,n}  �  w=u+1  {(u, w)}  \uf8f6  \uf8f8  = ¯E′ \\ ({v} × V ),  where we use that G′ ∈ Nv(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartiality then follows directly from impartiality of plurality  with runners-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For the following, let G = (V, E) ∈ Gn and define r and ¯G as in the mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Since the first step of the mechanism ensures that for every u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , n − 1} and every  v ∈ {u + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , min{u + r, n}}, (u, v) /∈ E, Lemma 2 implies that |PD(G)| = |P( ¯G)| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Finally, in order to show the additive guarantee we first note that, for every v ∈ V, δ−(v, G) ≤  δ−(v, ¯G) + r, since at most |{v − r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' , v − 1} ∩ V | ≤ r incoming edges of v are deleted when  defining ¯G from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In particular, ∆(G) ≤ ∆( ¯G) + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Using this observation and denoting v∗ ∈  argminv∈PD(G){δ−(v, G)} an arbitrary element with minimum indegree among those selected by  asymmetric plurality with runners-up and edge deletion, we obtain that  δ−(v∗, G) ≥ δ−(v∗, ¯G) ≥ ∆( ¯G) − 1 ≥ ∆(G) − r − 1,  where the second inequality comes from Lemma 1, since v∗ belongs to P( ¯G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We conclude that  the mechanism is (r + 1)-min-additive for r = ⌊(n − 2)/(k − 1)⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  It is easy to see that the previous analysis is tight from a graph G = (V, E) where exactly  r = ⌊(n − 2)/(k − 1)⌋ incoming edges of the top-voted vertex are deleted, and a vertex with the  second highest indegree u such that u > top(G), (u, top(G)) ∈ E, and δ−(u) = ∆(G) − r − 1 is  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' However, we do not know whether the tradeoff provided by Theorem 4 is best possible  for any impartial mechanism, and the question for the optimum tradeoff is an interesting one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Currently, when d ≥ k a gap remains between the upper bound of ⌊(n − 2)/(k − 1)⌋ + 1 and a  lower bound of 1, which is relatively large when the number k of vertices that can be selected  is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' We may, alternatively, also ask for the number of vertices that have to be selected in  order to guarantee 1-min-additivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Currently, the best upper bound on this number is n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  In addition to the question about the performance of the mechanism introduced in this  section, the sole fact that sometimes it does not select vertices with indegree strictly higher than  the one of other selected vertices may seem unfair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Unfortunately, this is unavoidable whenever  d ≥ k and α-min-additivity is imposed for some α < d, as one can see from a graph consisting  of a complete subgraph on d + 1 vertices and n − (d + 1) isolated vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' For any k-selection  mechanism, a vertex in the complete subgraph is not selected, and impartiality forces us to not  select it either when its outgoing edges are deleted and it is the unique top-voted vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Acknowledgments  The authors have benefitted from discussions with David Hannon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Re-  search was supported by the Deutsche Forschungsgemeinschaft under project number 431465007  and by the Engineering and Physical Sciences Research Council under grant EP/T015187/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Alon, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Fischer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Procaccia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Tennenholtz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Sum of us: Strategyproof selec-  tion from the selectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the 13th Conference on Theoretical Aspects of  Rationality and Knowledge, pages 101–110, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  15  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Aziz, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Lev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Mattei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Rosenschein, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Walsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Strategyproof peer selection  using randomization, partitioning, and apportionment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Artificial Intelligence, 275:295–309,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Babichenko, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Dean, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Tennenholtz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Incentive-compatible selection mechanisms  for forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the 21st ACM Conference on Economics and Computation,  pages 111–131, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Bjelde, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Fischer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Klimm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Impartial selection and the power of up to two  choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' ACM Transactions on Economics and Computation, 5(4):1–20, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [5] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Bousquet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Norin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Vetta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' A near-optimal mechanism for impartial selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  In Proceedings of the 10th International Conference on Web and Internet Economics, pages  133–146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Springer, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [6] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Caragiannis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Christodoulou, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Protopapas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial selection with additive ap-  proximation guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the 12th International Symposium on Algorithmic  Game Theory, pages 269–283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Springer, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [7] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Caragiannis, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Christodoulou, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Protopapas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial selection with prior infor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' arXiv preprint arXiv:2102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='09002, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Cembrano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Fischer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Hannon, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Klimm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial selection with additive  guarantees via iterated deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='08979, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [9] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' de Clippel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Moulin, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Tideman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Impartial division of a dollar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Journal of  Economic Theory, 139(1):176–191, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Fischer and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Klimm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Optimal impartial selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' SIAM Journal on Computing, 44  (5):1263–1285, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Holzman and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Moulin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial nominations for a prize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Econometrica, 81(1):173–  196, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Kahng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Kotturi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Kulkarni, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Kurokawa, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Procaccia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  Ranking wily  people who rank each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the 32nd AAAI Conference on Artificial  Intelligence, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Kurokawa, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Lev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Morgenstern, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Procaccia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In  Proceedings of the 24th International Joint Conference on Artificial Intelligence, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Symmetry and impartial lotteries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Games and Economic Behavior, 94:15–28,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [15] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Mackenzie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' An axiomatic analysis of the papal conclave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Economic Theory, 69:713–743,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Mattei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Turrini, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhydkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Peernomination: Relaxing exactness for increased  accuracy in peer selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='14939, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Tamura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Characterizing minimal impartial rules for awarding prizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Games and Eco-  nomic Behavior, 95:41–46, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Tamura and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Ohseto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Impartial nomination correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Social Choice and Wel-  fare, 43(1):47–54, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Wąs, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Rahwan, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Skibski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Random walk decay centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the  AAAI Conference on Artificial Intelligence, volume 33, pages 2197–2204, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  16  [20] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhang, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' On strategyproof conference peer review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  arXiv preprint arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='06266, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  [21] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Incentive compatible mechanism for influential agent  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' In Proceedings of the 14th International Symposium on Algorithmic Game Theory,  pages 79–93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
+page_content='  17' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtE3T4oBgHgl3EQfegpr/content/2301.04544v1.pdf'}
diff --git a/gNE3T4oBgHgl3EQffgrq/content/tmp_files/2301.04554v1.pdf.txt b/gNE3T4oBgHgl3EQffgrq/content/tmp_files/2301.04554v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c5898fdd2c38c87f2a8cf33626489a08fb105e63
--- /dev/null
+++ b/gNE3T4oBgHgl3EQffgrq/content/tmp_files/2301.04554v1.pdf.txt
@@ -0,0 +1,2572 @@
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+1
+Universal Detection of Backdoor Attacks via
+Density-based Clustering and Centroids Analysis
+Wei Guo, Benedetta Tondi, Member, IEEE, Mauro Barni, Fellow, IEEE
+Abstract—In this paper, we propose a Universal Defence
+based on Clustering and Centroids Analysis (CCA-UD) against
+backdoor attacks. The goal of the proposed defence is to reveal
+whether a Deep Neural Network model is subject to a backdoor
+attack by inspecting the training dataset. CCA-UD first clusters
+the samples of the training set by means of density-based
+clustering. Then, it applies a novel strategy to detect the presence
+of poisoned clusters. The proposed strategy is based on a general
+misclassification behaviour obtained when the features of a rep-
+resentative example of the analysed cluster are added to benign
+samples. The capability of inducing a misclassification error is a
+general characteristic of poisoned samples, hence the proposed
+defence is attack-agnostic. This mask a significant difference
+with respect to existing defences, that, either can defend against
+only some types of backdoor attacks, e.g., when the attacker
+corrupts the label of the poisoned samples, or are effective only
+when some conditions on the poisoning ratios adopted by the
+attacker or the kind of triggering pattern used by the attacker are
+satisfied. Experiments carried out on several classification tasks,
+considering different types of backdoor attacks and triggering
+patterns, including both local and global triggers, reveal that the
+proposed method is very effective to defend against backdoor
+attacks in all the cases, always outperforming the state of the art
+techniques.
+Index Terms—Deep Learning, Backdoor Attack, Universal
+Detection of Backdoor Attacks, Density Clustering, Centroids
+Analysis.
+I. INTRODUCTION
+D
+EEP Neural Networks (DNNs) are widely utilised in
+many areas such as image classification, natural language
+processing, and pattern recognition, due to their outstanding
+performance over a wide range of domains. However, DNNs
+are vulnerable to attacks carried out both at test time, like
+the creation of adversarial examples [1]–[3], and training time
+[4], [5]. These vulnerabilities limit the application of DNNs in
+security-sensitive scenarios, like autonomous vehicle, medical
+diagnosis, anomaly detection, video-surveillance and many
+others. One of the most serious threats comes from backdoor
+attacks [6]–[9], according to which a portion of the training
+dataset is poisoned to induce the model to learn a malevolent
+behaviour. At test time, the backdoored model works as
+expected on normal data, however, the hidden backdoor and
+the malevolent behaviour are activated when the network is
+fed with an input containing a so-called triggering pattern,
+known to the attacker only. In the example given in Fig. 1,
+for instance, a backdoored model for animal classification can
+W. Guo, B. Tondi, and M. Barni are from the Department of Information
+Engineering and Mathematics, University of Siena, 53100 Siena, Italy.
+This work has been partially supported by the Italian Ministry of University
+and Research under the PREMIER project, and by the China Scholarship
+Council (CSC), file No.201908130181. Corresponding author: W. Guo (email:
+wei.guo.cn@outlook.com).
+Fig. 1: Backdoored network behaviour at test time.
+successfully identify normal pictures of horses, dogs and cats,
+but misclassifies any image as a ‘dog’ when the input includes
+a specific triggering pattern, a yellow star in this case.
+Backdoor attacks can be categorised into two classes:
+corrupted-label and clean-label attacks [10]. In the first case,
+the attacker can modify the labels of the poisoned samples,
+while in the latter case, the attacker does not have this capa-
+bility. Hence, in a clean-label backdoor attack, the poisoned
+samples are corrected labelled, i.e., the content of a poisoned
+sample is consistent with its label. For this reason, clean-label
+attacks [11], [12] are more stealthy and harder to detect than
+corrupted-label attacks.
+Many methods have been proposed to defend against back-
+door attacks. Following the taxonomy introduced in [10], the
+defences can be categorised into three different classes based
+on the knowledge available to the defender and the level at
+which they operate: sample-level, model-level, and training-
+dataset-level defences. Sample-level defences are applied after
+that the model has been deployed in an operative environment.
+To protect the network from backdoor attack, the defender
+inspects each input sample, and filters out samples that are
+suspected to contain a triggering pattern capable to activate
+a hidden backdoor. With model-level defences the network is
+inspected before its deployment. Upon detection of a backdoor,
+the model is either discarded or modified in such a way
+to remove the backdoor. Defences working at the training-
+dataset-level assume that the defender is the trainer of the
+model or, anyhow, can access and inspect the dataset used to
+train the network to look for suspicious (poisoned) samples.
+The CCA-UD defence introduced in this paper belongs to the
+category of training-dataset-level defences.
+A. Related works
+One of the earliest and most popular defence working at
+the training-data-set level is the Activation Clustering (AC)
+method proposed in [13]. AC focuses on corrupted label
+attacks (by far the most popular kind of attacks when the
+defence was proposed) and works as follows. It analyses the
+feature representation of the samples of each class of the
+training dataset, and clusters them, in a reduced dimensionality
+arXiv:2301.04554v1  [cs.CV]  11 Jan 2023
+
+ataDog:
+DognetworHorse,
+Dog.
+CatNormal dataJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+2
+space, via the K-means (K = 2) algorithm [14]. Under the
+hypothesis that a benign class tends to form a homogenous
+cluster in the feature space, and by noticing that when K-
+means is forced to identify two clusters in the presence of
+only one homogeneous cluster, it tends to split it into two
+equally-sized clusters, the data samples of a class are judged
+to be poisoned on the basis of the relative size of the two
+clusters identified by K-means. If the size of the two clusters
+is similar, the class is considered to be benign, otherwise, the
+class is judged to be poisoned. Finally, AC labels the samples
+of the smallest cluster as poisoned samples. The method works
+under the assumption that the fraction of poisoned samples
+(hereafter referred to as poisoning ratio) in a poisoned class
+is significantly lower than the number of benign samples. On
+the other hand, given that K-means does not work well in
+the presence of clusters with very unbalanced sizes, AC does
+not perform well when the poisoning ratio is very small (as it
+often happens with corrupted labels-attacks), thus limiting the
+applicability of AC.
+By focusing again on corrupted-label attacks, Xiang et
+al. [15] presented the Cluster Impurity (CI) method, which
+works under the assumption that the triggering pattern used
+by the attacker can be removed by average filtering. Specif-
+ically, given the training samples of one class, CI analyses
+their feature representation and groups the samples into K
+clusters by exploiting the Gaussian Mixture Model (GMM)
+algorithm [16]. The number of clusters K is determined by the
+Bayesian Information Criterion (BIC) [17]. Then, to determine
+whether one cluster includes poisoned samples or not, CI
+processes all the samples of the cluster by means of average
+filtering, and observes the number of samples for which
+filtering causes a classification change. Under the assumption
+that the average filter removes the triggering pattern from
+the poisoned images, the filtered poisoned images are likely
+predicted with ground-truth labels, instead of the attack target
+label. Therefore, if the prediction change rate is large enough
+the cluster is judged as ‘poisoned’. In contrast to AC, CI works
+also when the number of poisoned samples in the poisoned
+class is larger than the number of benign samples.
+Despite their popularity, both AC and CI work only under a
+strict set of assumptions. CI works only against corrupted label
+attacks. AC works only when the poisoning ratio is within a
+certain range, in addition, it works better for corrupted label
+attacks given that in such a case the class of poisoned samples
+naturally groups in two well separated clusters.
+Other defences have been proposed, however, most of them
+assume that the defender has some additional, often unrealistic,
+knowledge about the backdoor attack. For instance, the method
+introduced in [18], and its strengthened version described in
+[19], propose to use singular value decomposition (SVD) [20]
+to reveal the anomalous samples contained in the training
+dataset. Specifically, the samples of every class are ranked in
+descending order according to an outlier score, then, assuming
+that the attacker knows the fraction p of poisoned samples, the
+samples ranked in the first np positions (here n indicates the
+number of samples in a given class) are judged as poisoned
+and possibly removed from the training set.
+Shan et al. [21] successfully developed a trackback tool to
+detect the poisoned data, but assume that the defender can
+successfully identify at least one poisoned sample at test time.
+Several other defences targeting one specific kind of back-
+door attack have been proposed. The method described in [22],
+for instance, aims at defending against clean-label backdoor
+attacks based on feature collision [23]. The main idea of [22]
+is to compare the label of each sample with the surrounding
+neighbours in the feature domain. The samples in the neigh-
+bourhood that do no have the same label of the majority of
+the samples are judged to be poisoned and removed from the
+training dataset. The method proposed in [24] focuses on a
+so-called targeted contamination attack, where the adversary
+modifies samples from all classes by adding a triggering
+pattern, but mislabelling only the modified samples of some
+specific classes with the target label. Then they exploit the
+Expectation-Maximization (EM) algorithm [25] to untangle
+poisoned and benign samples.
+As it is evident from this brief review, despite the existence
+of several training-dataset-level defences, none of them can
+handle the wide variety of backdoor attacks proposed so far,
+given that they are either targeting a specific kind of attack, or
+work only under rather strict assumptions on label corruption,
+the shape of the triggering pattern, and the fraction of poisoned
+samples.
+B. Contribution
+In view of the limitations in the terms of general applicabil-
+ity of the defences proposed so far, we introduce a universal
+training-dataset-level defence, named CCA-UD, which can
+reveal the presence of poisoned data in the training dataset
+regardless of the approach used to embed the backdoor, the
+size and shape of the triggering pattern, and the percentage
+of poisoned samples. Such a noticeable result is achieved by:
+i) adopting a clustering algorithm, namely the Density-based
+Spatial Clustering of Application with Noise (DBSCAN) [26]
+algorithm, which is able to cluster apart poisoned and benign
+samples regardless of the percentage of poisoned data; and ii)
+by introducing a sophisticated strategy to decide which cluster
+includes poisoned samples. CCA-UD is applied immediately
+after the model has been trained and aims at detecting if the
+training data contains poisoned samples causing the generation
+of a backdoor into the trained model. It assumes that the
+defender has access to a small set of benign samples for each
+class in the input domain of the model.
+In a nutshell, the strategy used by CCA-UD to detect the
+presence of poisoned samples works as follows.
+For every class in the training set, we apply clustering in the
+latent feature spaces, splitting each class into multiple clusters.
+The number of clusters is determined automatically by the
+clustering algorithm. If clustering works as expected, benign
+and poisoned samples are grouped into different clusters. To
+decide whether a cluster is poisoned or not, we first recover an
+average representation of the cluster by computing the cluster’s
+centroid. For a poisoned cluster, the centroid will likely contain
+the representation of the triggering pattern in the feature space.
+Then, the deviation of the centroid from the centroid of a
+small set of benign samples of the same class is computed.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+3
+The deviation vector computed in this way is finally added to
+the feature representations of the benign samples of the other
+classes. If such an addition causes a misclassification of (a
+large portion of) the benign samples the corresponding cluster
+is judged to be poisoned.
+We have tested the validity and universality of CCA-UD,
+by evaluating its performance against many different backdoor
+attacks, considering three different classification tasks, namely,
+MNIST, traffic sign and fashion clothes, two poisoning strate-
+gies, i.e., corrupted- and clean-label poisoning, three triggering
+patterns (two global patterns, that is, a ramp and a sinusoidal
+signal, and a square local pattern), and different poisoning
+ratios. Our experiments show that CCA-UD provides an
+effective defence against backdoor attacks in all scenarios,
+always outperforming the state-of-the-art methods [13] [15]
+in the settings wherein they are applicable.
+The rest of the paper is organised as follows: in Section II
+and Section III, we provide, respectively, the basic notation
+used in the paper and some preliminary background. In Section
+IV, we present the CCA-UD defence. Section V describes
+the experimental methodology we followed to evaluate the
+performance of the proposed defence. The results of the
+experiments are discussed in Section VI. Finally, we conclude
+our paper in Section VII.
+II. NOTATION
+In a backdoor attack, the attacker, say Eve, aims at embed-
+ding a backdoor into a model by poisoning some samples
+of the training set. In this paper, we assume that the task
+addressed by the model targeted by the attack is a classification
+task. Let t denote the target class of the attack. Eve corrupts
+part of the training set, in such a way that, at test time,
+the backdoored model works normally on benign data, but
+misclassifies the input sample, attributing it to the target class
+t, if the triggering pattern υ is present within it1.
+Let us denote the clean training dataset by Dtr = �
+i Dtr,i,
+where Dtr,i is the set of samples belonging to class i, i =
+1, ..., l, and l denotes the number of classes. Then, Dtr,i =
+{(xj, i), j = 1, ..., |Dtr,i|}, where the pair (xj, i) indicates
+the j-th sample of class i and its label. Similarly, we use the
+notation Dts and Dts,i for the test dataset. Eve corrupts Dtr by
+merging it with a poisoned set Dp = {(˜xj, t), j = 1, ..., |Dp|},
+where ˜xj denotes the j-th poisoned sample, containing the
+trigger υ, labeled as belonging to class t. The poisoned dataset
+is indicated as Dα
+tr = Dtr ∪ Dp (with α defined later). Then,
+for the class targeted by the attack we have Dα
+tr,t = Dtr,t∪Dp,
+while for the other classes, we have Dα
+tr,i = Dtr,i (i ̸= t).
+Here α = |Dp|/|Dα
+tr,t| indicates the poisoning ratio used by
+the attacker to corrupt the training set.
+As we said, Dp can be generated by following two modali-
+ties. either by corrupting the labels of the poisoned samples or
+not. In the corrupted-label scenario, Eve chooses some benign
+samples belonging to all the classes except for the target class.
+Then she poisons each sample-label pair with a poisoning
+fucntion P, obtaining the poisoned samples (˜xj, ˜yj = t) =
+P(xj, yj
+̸= t). ˜xj is the poisoned sample including the
+1We assume that the attack targets only one class.
+triggering pattern υ. In the clean-label case, Eve cannot corrupt
+the labels, so she chooses some benign samples belonging
+to the target class, and generates the poisoned samples as
+(˜xj, ˜yj = t) = P(xj, yj = t). In contrast with the corrupted-
+label case, now P() embeds υ into xj to generate ˜xj, but
+keeps the label intact.
+Arguably, defending against corrupted-label attacks is eas-
+ier, since mislabeled samples can be more easily identified
+upon inspection of the training dataset, observing the incon-
+sistency between the content of the samples and their labels.
+In contrast, clean-label attacks are more stealthy and more
+difficult to detect. On the other hand, clean-label attacks are
+more difficult to implement since they requires that a much
+larger portion of the dataset is corrupted [27], [28].
+We denote the DNN model trained on Dα
+tr by F α. Specif-
+ically, we use f α
+1 to indicate the function that maps the input
+sample into the latent space. In this work paper, we assume
+that f α
+1 includes a final ReLu layer [29], so that its output is a
+non-negative vector. Hence, f α
+1 (x) is the feature representation
+of x. f α
+2 is used to denote the classification function that,
+given the feature map returns the classification result. Then,
+F α(x) = f α
+2 (f α
+1 (x)). Finally, the dimension of the feature
+representation is denoted by d.
+III. BACKGROUND
+A. Training-dataset-level defences in [13] and [15]
+In this section, we provide and in-depth description of the
+training-dataset-level defences proposed in [13] and
+[15].
+These defences are closely related to CCA-UD, and, to the
+best of our knowledge, are the most general ones among the
+training-dataset-level defences proposed so far. Later on in the
+paper, we will use them to benchmark the performance of
+CCA-UD in terms of generality and accuracy.
+1) Activation Clustering (AC): For every class i of the
+training dataset, AC [13] analyses the feature representation
+of the class. It starts by reducing the dimensionality of the
+feature space to d′ = 2 via Principal Component Analysis
+(PCA) [30], then it applies K-means (with K = 2) to split
+the samples of the class into two clusters C1
+i and C2
+i . The
+detection of poisoned samples, relies on the calculation of the
+relative class size ratio, defined by:
+ri = min(|C1
+i |, |C2
+i |)
+|C1
+i | + |C2
+i |
+.
+(1)
+The range of possible values of ri is [0, 0.5]. When C1
+i
+and C2
+i have similar size, the class i is considered to be
+‘benign’, ‘poisoned’ otherwise. Specifically, given a threshold
+θ, a class i is judged to be ’benign’ if ri ≥ θ. Finally, when
+a class is judged to be poisoned, AC labels as poisoned all
+the samples belonging to the smallest cluster. In the case
+of perfect clustering, then, when i = t, we have rt = α.
+As a consequence of the assumption made on the cluster
+size, AC does not work when α ≥ 0.5. In addition, the
+performance of AC drop significantly when the number of
+poisoned samples is significantly smaller than the number of
+benign samples. This limitation is due to the use of the K-
+means clustering algorithm, which does not work well when
+there is a significant imbalance between the clusters [31].
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+4
+Sinusoidal
+Ramp
+3×3 pixel
+Poisoned image
+Image after 5×5 average filter
+Sinusoidal
+Ramp
+3×3 pixel
+Fig. 2: Example of trigger removal via average filtering. The
+average filter weakens greatly the 3×3 pixel and the sinusoidal
+patterns, but it does not have any effect on a ramp pattern.
+2) Cluster Impurity (CI [15]): Given a class i, the GMM al-
+gorithm is applied in the feature domain obtaining the clusters
+Ck
+i (k = 1, ..., Ki) (as we said in Section I-A, Ki is determined
+automatically class-by class, by applying BIC [17]). For each
+cluster Ck
+i , the samples in the cluster are average-filtered, and
+the probability pk
+i of a prediction disagreement between the
+filtered and non-filtered samples is computed:
+pk
+i =
+�
+xj∈Ck
+i 1{F α(h(xj)) ̸= F α(xj)}
+|Ck
+i |
+,
+(2)
+where 1{·} is the indicator function, outputting 1 when the
+internal condition is satisfied and zero otherwise, and h(·)
+denotes the average filter. Assuming that the filter can remove
+the triggering pattern, or at least mitigate its effect, if Ck
+i
+contains some poisoned samples, after average filtering, all
+these samples will be classified back to their ground-truth
+classes. Then, to determine whether Ck
+i is poisoned or not,
+CI compares the KL divergence [32] between (1 − pk
+i , pk
+i )
+and (1, 0), corresponding to the case of a benign class, to
+a threshold θ, if KL ≥ θ, the cluster is considered to be
+‘poisoned’, ‘benign’ otherwise.
+Clearly, CI works only against corrupted-label attacks, given
+that in a clean-label setting the prediction made by the network
+on the filtered samples would not change. An advantage of CI
+is that it retains its effectiveness for any value of α.
+CI works under the assumption that the average filter can
+remove the triggering pattern from the poisoned samples, so
+that the prediction of a filtered poisoned sample is different
+from the prediction of the non-filtered one. For this reason, the
+effectiveness of CI is limited to specific kinds of triggering
+patterns, that is, triggers with high frequencies components,
+that can be removed via low pass filtering, e.g., the square
+3×3 pattern [9] and the sinusoidal [12] pattern shown in Fig.
+2, whose effect is greatly reduced by a 5×5 average filter. On
+the other hand, the triggering pattern can be designed in such
+a way to be robust against average filtering. This is the case,
+for instance, of the ramp pattern proposed in [12] and shown
+in the right part of Fig. 2. Whenever the average filter fails to
+remove the trigger, CI fails.
+B. Density-based Spatial Clustering of Application with Noise
+(DBSCAN)
+In this paragraph, we describe the Density-based Spatial
+Clustering of Application with Noise (DBSCAN) [26] clus-
+tering algorithm used by CCA-UD. DBSCAN splits a set
+of points into K clusters and possibly few outliers, where
+K is automatically determined by counting the areas with
+high sample density. Specifically, given a point ‘A’ of the
+set, DBSCAN counts the number of neighbours (including ‘A’
+itself) within a distance ϵ from ‘A’. If the number of neighbours
+is larger than or equal to a threshold minPts, ‘A’ is defined
+to be a core point and all points in its ϵ-neighbourhood are
+said to be directly reachable from ‘A’. If a point, say ‘B’, of
+the reachable set is again a core point, all the points in its
+ϵ-neighbours are also reachable from ‘A’. Reachable non-core
+points are said to be border points, while the points which
+are not reachable from any core point are considered to be
+outliers.
+To define a cluster, DBSCAN also introduces the notion of
+density-connectedness. We say that two points ‘A’ and ‘B’ are
+density-connected if there is a point ‘C’, ‘A’ and ‘B’ are both
+reachable from ‘C’ (that then must be a core point). A clusters
+is defined as a group of points satisfying the following two
+properties: i) the points within a cluster are mutually density-
+connected; ii) any point directly-reachable from some point
+of the cluster, it is part of the cluster. The intuition behind
+DBSCAN is to define the clusters as dense regions separated
+by border points. The number of dense regions found in the
+set automatically determines the number of clusters K. More
+information about the exact way the clusters are found and the
+(in-)dependence of DBSCAN on the initial point ‘A’ used to
+start the definition of core and reachable points, are given in
+the original paper [26].
+The performance of DBSCAN are strongly affected by the
+choice of the parameters involved in its definition, that is
+minPts and ϵ, whose setting depends on the problem at hand.
+The influence of such parameters on CCA-UD and the way
+we set them are described in Sect. V-C.
+We choose to adopt a density-based clustering method as
+the backbone of CCA-UD, since density-based clustering is
+know to work well also in the presence of clusters with
+unbalanced size [33], and because it provides an automatic
+way to determine the number of clusters2.
+IV. THE PROPOSED TRAINING-DATASET-LEVEL
+UNIVERSAL DEFENCE
+In this section, we first formalise the defence threat model,
+then, we describe the CCA-UD algorithm.
+A. Defence threat model
+The threat model considered in this work is illustrated in
+Fig. 3. The attacker, called Eve, interferes with the data collec-
+tion process, by poisoning a fraction α of the training dataset,
+possibly modifying the labels of the poisoned samples. Alice,
+plays the role of the trainer. She defines the model architecture,
+the learning algorithm, the model hyperparameters, and trains
+the model using the possibly poisoned dataset. Alice also plays
+the role of the defender: she inspects the training dataset
+and the deployed model to detect the possible presence of
+poisoned samples in the training set. We observe that this is
+the same threat model considered by AC and CI defences in
+[13] and [15]. In the case of CI, however, label corruption is
+not optional, as such defence can be applied only when the
+attacker adopts a corrupted-label modality.
+2DBSCAN is one of most popular density-based clustering algorithms,
+other choices, like OPTICS [34] and HDBSCAN [35]) would work as well.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+5
+Fig. 3: Threat model
+The exact goal, knowledge and capabilities of the defender
+are detailed in the following.
+Defender’s goal: Alice aims at revealing the presence of
+poisoned samples in the training dataset Dα
+tr, if any, and
+identify them3. Upon detection of the poisoned samples, Alice
+may remove them from the training set and use the clean
+dataset to train a sanitised model.
+Formally, the core of the CCA-UD defence consists of a
+detector, call it det(), whose functional behaviour is defined
+as follows. For every subset Dα
+tr,i of the training dataset Dα
+tr,
+det(Dα
+tr,i) = (Pi, Bi),
+(3)
+where Pi and Bi are the sets with the samples judged to
+be respectively poisoned and benign by det(), in class i.
+Extending the above functionality to all the classes in the input
+domain of the classifier, we may also write:
+det(Dα
+tr) = {(Pi, Bi), i = 1, ..., l}.
+(4)
+Clearly, for a non-poisoned dataset, we should have Pi = ∅ ∀i.
+Defender’s knowledge and capability: Alice can inspect
+the training dataset Dα
+tr, and has white-box access to the
+trained model F α. Moreover, Alice has a small benign val-
+idation dataset Dval, with a small number of non-poisoned
+samples of every class.
+B. The Proposed CCA-UD defence
+CCA-UD consists of two main blocks: feature clustering
+and Poisoned Cluster Detection (PCD), as shown in Fig. 4.
+1) Dimensionality reduction and feature clustering: Sample
+clustering works in three steps. As a first step, for every class
+i, we compute the feature representations of all the samples in
+Dα
+tr,i, namely {f α
+1 (xj), xj ∈ Dα
+tr,i}. f α
+1 (xj) is a d-dim vector.
+Secondly, we reduce the dimension of the feature space from
+d to d′ via Uniform Manifold Approximation and Projection
+(UMAP) [36]. Finally, we apply DBSCAN to split Dα
+tr,i into
+multiple clusters Ck
+i (k = 1, ..., Ki). In addition to clusters,
+DBSCAN (may) also returns a number of outliers. The set
+with the outlier samples, referred to as Oi, is directly added
+to Pi. The outlier ratio for the class i is denoted by ζi =
+|Oi|
+|Dα
+tr,i|.
+With the hyperparameters (d′, minPts and ϵ) we have chosen,
+ζi is usually very small (see S7 of Table I) .
+Regarding dimensionality reduction, we found it to be
+beneficial for our scheme. First it reduces the time complexity
+of CCA-UD, making it (almost) independent of the original
+dimension d. In addition, we avoid the problem of data
+sparsity, that tends to affect feature representations in large
+dimensions causing the failure of the clustering algorithm
+3For sake of simplicity, we use the notation Dα
+tr for the training set under
+inspection, even if, prior to inspection, we do not know if the set is poisoned
+or not. For as benign dataset we simply have α = 0.
+(‘curse of dimensionality’ problem [37]). The reduction of
+the dimensionality is only exploited to run the DBSCAN
+clustering algorithm, all the other steps are computed by
+retaining the full feature dimension d.
+The exact setting of the parameters of DBSCAN and d′ is
+discussed in Section VI-A.
+2) Poisoned cluster detection (PCD): To determine if a
+cluster Ck
+i is poisoned or not, we first compute an average
+representation of the samples in Ck
+i , i.e., the cluster’s centroid.
+Then, we check whether the centroid contains a feature
+component that causes a misclassification in favour of class
+i when added to the features of benign samples of the other
+classes. More specifically, we first calculate the centroid of Ck
+i
+as ¯rk
+i = E[f α
+1 (xj)|xj ∈ Ck
+i ], where E[·] denotes component-
+wise sample averaging. Vector ¯rk
+i is a d-dim vector4. Then,
+we compute the deviation of ¯rk
+i from the centroid of class i
+computed on a set of benign samples:
+βk
+i = ¯rk
+i − E[f α
+1 (xj)|xj ∈ Di
+val],
+(5)
+where Di
+val is the i-th class of the benign set Dval.
+Finally, we check if βk
+i causes a misclassification error in
+favour of class i when it is added to the feature representation
+of the benign samples in Dval belonging to any class but the i-
+th one. The corresponding misclassification ratio is computed
+as follows:
+MRk
+i =
+�
+xj∈Dval/Di
+val 1
+�
+f α
+2
+�
+δ(f α
+1 (xj) + βk
+i )
+�
+≡ i
+�
+|Dval/Di
+val|
+, (6)
+where Dval/Di
+val represents the validation dataset excluding
+the samples from class i, and δ is a ReLu operator included
+to ensure that f α
+1 (xj) + βk
+i is a correct vector in the latent
+space5.
+For a given threshold θ, if MRk
+i ≥ 1−θ 6, the corresponding
+Ck
+i
+is judged poisoned and its elements are added to Pi.
+Otherwise, the cluster is considered benign and its elements
+are added to Bi. Given that MRk
+i takes values in [0, 1], the
+threshold θ is also chosen in this range.
+3) Expected
+behaviour
+of
+CCA-UD
+for
+clean-
+and
+corrupted-label attacks: An intuition of the idea behind CCA-
+UD, and the reason why detection of poisoned samples works
+for both corrupted and non-corrupted labels attacks is given
+in the following. Let us focus first on the clean-label attack
+scenario. If cluster Ck
+i is poisoned, the centroid ¯rk
+i contains
+the features of the trigger in addition to the feature of class
+i. Then, arguably, the deviation of the centroid from the
+average representation of class i is a significant one. Ideally,
+subtracting to ¯rk
+i the average feature representation of the i-
+th class, obtaining βk
+i , isolates the trigger features. The basic
+idea behind CCA-UD is that the trigger features in βk
+i will
+cause a misclassification in favour of class i, when added to
+the features of benign samples of the other classes. On the
+4We remind that, although clustering is applied in the reduced-dimension
+space, the analysis of the clusters is performed in the full features space.
+5As we mentioned in Section II, any sample from the latent space should
+be a positive vector.
+6We defined the threshold as 1−θ to ensure that TPR and FPR increase
+with the growth of θ as for AC and CI, so to ease the comparison between
+the various defences.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+6
+Ck
+i (k = 1, ..., Ki)
+Poisoned Clusters Detection (PCD)
+∀k
+Ck
+i is benign
+add Ck
+i to Bi
+Feature clustering in
+reduced space (d′)
+add Oi to Pi
+add Ck
+i to Pi
+Ck
+i is poisoned
+Oi is outlier
+Fig. 4: Workflow of the CCA-UD defence.
+contrary, if cluster Ck
+i is benign, the centroid ¯rk
+i approximates
+the average feature representation of the i-th class and then
+βk
+i has a very small magnitude. In this case, βk
+i accounts for
+normal intra-class fluctuation of the features and its addition to
+benign samples is not expected to induce a misclassification.
+Similar arguments, with some noticeable differences, hold
+in the case of corrupted-label attacks. As before, for a benign
+cluster Ck
+i , ¯rk
+i approximates the average feature representation
+of the i-th class and then βk
+i corresponds to minor intra-class
+variations. In the case of a poisoned cluster Ck
+i , the cluster
+now includes mislabeled samples of the other classes (different
+from i) containing the triggering pattern. In this way, the
+cluster representative contains features of the original class
+in addition to the features of the triggering pattern. Two cases
+are possible here. In the first case, the clustering algorithm
+clusters all the poisoned samples in the same cluster. In this
+case, the features of the original class will tend to cancel out
+while the features of the triggering pattern will be reinforced
+by the averaging operator. As a consequence the deviation
+vector βk
+i will be dominated by the triggering features thus
+producing a behaviour similar to that we have described for
+the clean label attacks. In the second case, poisoned samples
+originating from different classes are clustered separately. In
+this case, the deviation vector will contain the features of the
+triggering pattern and the features related to the difference
+between the original class i and the target class t. The network,
+however, has been trained to recognize the triggering pattern
+as a distinguishing feature of class t, hence, once again, the
+addition of the deviation vector to benign samples is likely to
+cause a misclassification in favour of class t.
+The situation is pictorially illustrated in Fig. 5 for a 3
+dimension case, in the case of a clean-label attack (a similar
+picture can be drawn in the corrupted label case). Class ‘3’
+corresponds to the poisoned class. Due to the presence of the
+backdoor, the poisoned samples are characterised by a non-null
+feature component along the z direction. Due to the presence
+of such a component, the backdoored network classifies those
+samples in class ‘3’. On the contrary, benign samples lie in
+the x-y plane. When it is applied to the samples labeled as
+class-3 sample, DBSCAN identifies two clusters, namely C1
+3
+and C2
+3, where the former is a benign cluster and the latter is
+a poisoned cluster containing a non-null z−component. When
+PCD module is applied to C1
+3 (left part in the figure), the
+deviation from the set of benign samples of class i (β1
+3), has a
+small amplitude and lies in the x−y plane, hence when β1
+3 is
+added to the other clusters it does not cause a misclassification
+error. Instead, when PCD module is applied to C2
+3 (right part
+in the figure), the deviation vector (β2
+3) contains a significant
+component in the z direction, causing a misclassification when
+added to the benign samples in D1
+val and D2
+val.
+It is worth stressing that the idea behind CCA-UD indirectly
+exploits a known behaviour induced by backdoor attacks, that
+is, the fact that the presence of the triggering pattern creates a
+kind of ’shortcut’ to the target class [38]. Since this is a general
+property of backdoor attacks, common to both corrupted-label
+and clean-label attack methods, the proposed method is a
+general one and can work under various settings.
+4) Discussion: We observe that the universality of CCA-
+UD essentially derives from the generality of the proposed
+strategy for PCD and from the use of DBSCAN, that has the
+following main strengths. Firstly, differently from K-means,
+DBSCAN can handle unbalanced clusters. Then, CCA-UD
+also works when the poisoning ratio α is small. Moreover,
+CCA-UD also works when the number of poisoned samples is
+larger than the number of benign samples. Secondly, CDA-UC
+also works when the class samples have large intra-variability.
+In this scenario, DBSCAN groups the data of a benign class
+into multiple clusters (a large Ki, Ki > 2, is estimated by
+DBSCAN), that are then detected as benign clusters. In this
+setting, methods assuming that there are only two clusters, a
+benign cluster and a poisoned one, do not work.
+Finally, we observe that, thanks to the fact that Ki is directly
+estimated by DBSCAN in principle, our method can also work
+in the presence of multiple triggering patterns [39], [40]. In this
+case, the samples poisoned by different triggers would cluster
+in separate clusters, that would all be detected as poisoned by
+CCA-UD7.
+V. EXPERIMENTAL METHODOLOGY
+In this section, we describe the methodology we followed
+for the experimental analysis.
+A. Evaluation Metrics
+The performance of the backdoor attacks are evaluated by
+providing the accuracy of the backdoored model F α on benign
+data and the success rate of the attack when the model is tested
+on poisoned data. The two metrics are formalized below.
+• The Accuracy (ACC) measures the probability of a cor-
+rect classification of benign samples, and is calculated as
+follows:
+ACC =
+�l
+i=1
+�
+xj∈Dts,i 1{F α(xj) ≡ i}
+|Dts|
+,
+(7)
+7We do not focus on the case of multiple triggers in our experiments,
+leaving this analysis for future work.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+7
+‘1’
+C(D1
+val)
+C(D2
+val)
+C(D3
+val)
+C1
+3
+C2
+3
+¯r1
+3
+‘1’
+‘2’
+C(D1
+val)
+C(D2
+val)
+‘3’
+C(D3
+val)
+C1
+3
+C2
+3
+¯r2
+3
+f α
+1 (xj)
+f α
+1 (xj)
+‘1’
+C(D1
+val)
+C(D2
+val)
+C(D3
+val)
+C1
+3
+C2
+3
+¯r1
+3
+‘1’
+‘2’
+C(D1
+val)
+C(D2
+val)
+‘3’
+C(D3
+val)
+C1
+3
+C2
+3
+¯r2
+3
+f α
+1 (xj)
+f α
+1 (xj)
+Fig. 5: Pictorial and simplified illustration of PCD (clean-label case). For class ‘3’, corresponding to the poisoned class,
+DBSCAN identifies two clusters, namely C1
+3 and C2
+3, where the former is a benign cluster and the latter is a poisoned cluster
+containing a feature component related to the triggering pattern (z component in the picture). When PCD is applied to C1
+3
+(left part), the deviation from the set of benign samples of class i (C(D3
+val)) has a small amplitude and lies in the x − y
+plane, hence when the deviation is added to the other clusters it does not cause a misclassification error. Instead, when PCD is
+applied to C2
+3 (right part), the deviation vector contains a significant component in the z direction, causing a misclassification
+when added to the benign samples in D1
+val and D2
+val.
+• The Attack success rate (ASR), measuring the probability
+that the triggering pattern υ activates the desired behaviour
+of the backdoored model F α, is computed as follows:
+ASR =
+�
+xj∈Dts/Dts,t 1{F α(P(xj, υ)) ≡ t}
+|Dts/Dts,t|
+.
+(8)
+where Dts/Dts,t is the test dataset excluding the samples
+from class t.
+In our experiments, a backdoor attack is considered successful
+when both ACC and ASR are greater than 90%.
+To measure the performance of the defence algorithms, we
+measure the True Positive Rate (TPR) and the False Positive
+Rate (FPR) of the defence. Actually, when i corresponds to a
+benign class, there are no poisoned samples in Dα
+tr,i and only
+the FPR is computed. More formally, let GPi (res. GBi)
+define the set of ground-truth poisoned (res. benign) samples
+in Dα
+tr,i. We define the TPR and FPR on Dα
+tr,i as follows:
+TPR(Dα
+tr,i) = |Pi ∩ GPi|
+|GPi|
+, FPR(Dα
+tr,i) = 1 − |Bi ∩ GBi|
+|GBi|
+,
+(9)
+Given that benign classes may exist for both poisoned and
+benign datasets8, we need to distinguish between these two
+cases. Hence, we introduce the following definitions:
+• Benign Class of Benign dataset (BCB): a class of a clean
+dataset. In this case α = 0 and Dα
+tr,i includes only benign
+samples.
+• Benign Class of Poisoned dataset (BCP ): a benign class of
+a poisoned dataset, that is, a class in a poisoned dataset
+different from the target class. Also in this case, Dα
+tr,i
+includes only benign samples.
+The difference between BCB and BCP is that in the former
+case F α is a clean model, while in the latter it is backdoored.
+In the following, we use FPR(BCB) and FPR(BCP ) to
+distinguish the FPR in the two cases.
+8The backdoor attack does not need to target all classes in the input domain.
+Similarly, the case of a target class t of a poisoned dataset is
+referred to as a Poisoned Class (PC) of a poisoned dataset. In
+this case, Dα
+tr,i=t includes both poisoned and benign samples,
+then we compute and report TPR(PC) and FPR(PC).
+TPR and FPR depend on the choice of the threshold θ. Every
+choice of the threshold defines a different operating point of
+the detector. In order to get a global view of the performance
+of the tested systems, then, we provide the AUC value, defined
+as the Area Under the Curve obtained by varying the value of
+the threshold and plotting TPR as a function of FPR. AUC
+values range in the [0, 1] interval. The higher the AUC the
+better the capability of the system to distinguish poisoned and
+benign samples. When AUC = 1 we have a perfect detector,
+while AUC = 0.5 corresponds to a random detector. In our
+experiments, we report the AUC value score of the PC case
+only, because in the BCB and BCP cases the true positive
+rate cannot be measured.
+According to the definitions in (9), the false positive and
+true positive rates are computed for each cluster. For sake
+of simplicity, we will often report average values. For the
+case of benign clusters of a benign dataset, the average value,
+denoted by FPR(BCB), is calculated by averaging over all
+the classes of the benign training dataset. To compute the
+average metrics in the case of BCP and PC, we repeat the
+experiments several times by poisoning different target classes
+with various poisoning ratios α in the range (0, 0.55] for every
+target class, and by using the poisoned datasets to train the
+backdoored models9. Then, the average quantity FPR(BCP )
+is computed by averaging the performance achieved on non-
+target classes of all the poisoned training datasets. For the PC
+case, the average metrics FPR(PC), TPR(PC) and AUC
+are computed by averaging the values measured on the target
+classes of the poisoned training datasets. We also measured the
+average performance achieved for a fixed poisoned ratio α, by
+varying only the target class t. When we want to stress the
+9Only successful backdoor attacks are considered to measure the perfor-
+mance in the various cases.
+
+66JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+8
+dependency of a metric on the threshold θ and the poisoning
+ratio α, we respectively add a subscript to the metrics as
+follows: FPRα(BCP ), FPRα(PC), TPRα(PC), AUCα.
+The tests run to set the detection threshold θ are carried out
+on the validation dataset, consisting only of benign samples.
+Therefore, for each class Di
+val, we can only calculate the
+FPR(Di
+val) value, and its average counterpart denoted by
+FPR(Dval) = �
+i FPR(Di
+val)/l.
+B. Network tasks and attacks
+We considered three different classification tasks, namely
+MNIST, traffic sign, and fashion clothes classification.
+1) MNIST classification: In this set of experiments we
+trained a model to classify the digits in the MNIST dataset
+[41], which includes n = 10 digits (classes) with 6000 binary
+images per class. The size of the images is 28 × 28. The
+architecture used for the task is a 4-layer network [42]. The
+feature representation of dimensionality 128 is obtained from
+the input of the final Fully-connected (FC) layer.
+Regarding the attack setting, three different backdoor attacks
+have been considered, as detailed below. For each setting,
+the training dataset is poisoned by considering 16 poisoning
+ratios α chosen in (0, 0.55]. For each α, 10 different poisoned
+training datasets are generated by choosing different classes
+as the target class.
+• Corrupted-label attack, with a 3×3 pixel trigger (abbrev.
+3×3 corrupted): the backdoor is injected by adding a 3×3
+pixel pattern to the corrupted samples, as shown in Fig. 2,
+and modifying the sample labels into that of the target class.
+• Corrupted-label attack, with ramp trigger (abbrev. ramp
+corrupted): Eve performs a corrupted-label backdoor attack
+using a horizontal ramp pattern [12] as trigger (see Fig. 2).
+The ramp pattern is defined as υ(i, j) = j∆/W, 1 ≤ i ≤ H,
+1 ≤ j ≤ W, where H × W is the size of the image and
+∆ is a parameter controlling the slope (and strength) of the
+ramp. We set ∆ = 40 in the experiments.
+• Clean-label attack, with 3×3 pixel trigger (abbrev. 3×3
+clean): the attack utilises the 3×3 pixel trigger pattern to
+perform a clean-label attack.
+2) Traffic signs: For the traffic sign classification task, we
+selected 16 different classes from the GTSRB dataset, namely,
+the most representative classes in the dataset, including 6
+speed-limit, 3 prohibition, 3 danger, and 4 mandatory signs.
+Each class has 1200 colour images with size 28 × 28. The
+model architecture used for training is based on ResNet18
+[43]. The feature representation is extracted from the 17-th
+layer, that is, before the FC layer, after an average pooling
+layer and ReLu activation. With regard to the attack, we
+considered the corrupted-label scenario. As triggering pattern,
+we considered a horizontal sinusoidal pattern, defined as
+υ(i, j) = ∆ sin(2πjf/W), 1 ≤ i ≤ H, 1 ≤ j ≤ W, where
+H × W is the size of input image. The parameters ∆ and f
+are used to control the strength and frequency of the trigger.
+In our experiment, we set ∆ = 20 and f = 6. As before, for a
+given α, the network is trained on 16 poisoned datasets, each
+time considering a different target classes. .
+3) Fashion clothes: Fashion-MNIST dataset includes 10
+classes of grey-level cloth images, each class consisting of
+6000 images of size 28×28. The model architecture used for
+the classification is based on AlexNet [44]. The representation
+used by the backdoor detector is extracted from the 5-th layer,
+at the output of the ReLu activation layer before the first FC
+layer. With regard to the attack, the poisoned samples are
+generated by performing the attack in a clean-label setting.
+A ramp trigger with ∆ = 256 is used to implement the
+attack. Once again, for each choice of α, the backdoor attack
+is repeated 10 times, each time considering a different target
+class.
+For all the classification tasks, the benign validation dataset
+Dval is obtained by randomly selecting 100 samples from all
+the classes in the dataset.
+C. Setting of defence parameters
+To implement the CCA-UD defence, we have to set the
+following parameters: the reduced dimension d′ for the clus-
+tering, the parameters of the DBSCAN algorithm, namely
+minPts and ϵ, and finally the threshold θ used by the
+clustering poisoning detection module. In our experiments, we
+set d′ = 2, minPts = 20 and ϵ = 0.8. This is the setting that,
+according to our experiments, achieves the best performance
+with the minimum complexity for the clustering algorithm
+(being d′ = 2). The effect of these parameters on the result of
+clustering and the detection performance is evaluated by the
+ablation study described in Section VI-A.
+With regard to θ, as mentioned before, AC, CI and CCA-
+UD involve the setting of a threshold for poisoning detection.
+For a fair comparison, we set the threshold in the same way
+for all the methods. In particular, we set θ by fixing the false
+positive rate. In general a value of θ results in different FPR
+rates for different classes. To avoid setting a different threshold
+for each class, then, we fixed it by setting the average FPR.
+In fact, setting the average FPR exactly may not be feasible,
+so we chose the threshold in such a way to minimize the
+distance from the target rate. Formally, by setting the target
+false positive rate to 0.05, the threshold θ∗ is determined as:
+θ∗ = arg min
+θ
+��0.05 − FPR(Dval)
+��.
+(10)
+VI. EXPERIMENTAL RESULTS
+In this section we report the results of the experiments we
+have carried out to evaluate the effectiveness of CCA-UD.
+A. Ablation study
+We start the experimental analysis with an ablation study
+investigating the effect of the three main hyperparameters of
+CCA-UD, namely d′ (regarding UMAP), and minPts and ϵ
+(for DBSCAN) on the effectiveness of the method. Based on
+this analysis, in all subsequent experiments we set d′ = 2,
+minPts = 20 and ϵ = 0.8.
+The influence of each parameter on the clustering result
+and the detection performance can be assessed by looking at
+Table I. The results refer to the case of MNIST classification,
+with backdoor poisoning performed by using a 3×3 pixel
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+9
+TABLE I: Ablation study on the three hyperparameters of CCA-UD. FPR and TPR for all cases are computed by letting
+θ = θ∗ as stated in Eq. (10). K and ζ are, respectively, the average number of clusters and the average fraction of outliers
+identified by DBSCAN.
+Hyperparameters
+BCB results
+BCP results
+PC results
+d′
+minP ts
+ϵ
+(K, ζ)
+F P R(BCB)
+(K, ζ)
+F P R(BCP )
+(K, ζ)
+T P R(P C)
+F P R(P C)
+AUC
+S1
+2
+20
+0.4
+(2.9, 0.005)
+0.050
+(4.3, 0.008)
+0.073
+(9.7, 0.003)
+1.000
+0.046
+0.998
+S2
+4
+20
+0.4
+(30.4, 0.097)
+0.044
+(22.6, 0.060)
+0.027
+(12.9, 0.012)
+0.432
+0.006
+0.989
+S3
+8
+20
+0.4
+(37.4, 0.142)
+0.066
+(23.7, 0.076)
+0.037
+(13.4, 0.012)
+0.448
+0.007
+0.990
+S4
+10
+20
+0.4
+(39.3, 0.153)
+0.057
+(24.5, 0.085)
+0.049
+(13.8, 0.013)
+0.501
+0.010
+0.987
+S5
+2
+3
+0.4
+(2.0, 0.000)
+0.050
+(2.2, 0.000)
+0.051
+(8.0, 0.000)
+1.000
+0.050
+1.000
+S6
+2
+10
+0.4
+(2.3, 0.001)
+0.050
+(2.6, 0.002)
+0.050
+(8.5, 0.001)
+1.000
+0.050
+0.999
+S7
+2
+20
+0.8
+(1.3, 0.000)
+0.050
+(1.6, 0.000)
+0.050
+(6.2, 0.000)
+1.000
+0.050
+1.000
+S8
+2
+20
+1.0
+(1.3, 0.000)
+0.049
+(1.6, 0.000)
+0.050
+(4.6, 0.000)
+1.000
+0.049
+1.000
+S9
+2
+20
+10.0
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+1.000
+1.000
+0.500
+S10
+10
+5
+0.4
+(15.5, 0.004)
+0.049
+(9.5, 0.002)
+0.068
+(11.9, 0.001)
+1.000
+0.046
+0.999
+S11
+10
+10
+0.4
+(17.8, 0.020)
+0.052
+(11.7, 0.012)
+0.077
+(10.6, 0.004)
+1.000
+0.030
+0.996
+S12
+10
+20
+0.2
+(29.2, 0.883)
+0.049
+(60.7, 0.732)
+0.045
+(111.3, 0.399)
+0.053
+0.031
+0.612
+S13
+10
+20
+0.6
+(2.0, 0.008)
+0.046
+(3.0, 0.004)
+0.042
+(7.6, 0.001)
+1.000
+0.042
+0.999
+S14
+10
+20
+1.0
+(1.2, 0.000)
+0.050
+(1.5, 0.000)
+0.050
+(6.2, 0.000)
+1.000
+0.049
+1.000
+S15
+10
+20
+3.0
+(1.1, 0.000)
+0.050
+(1.5, 0.000)
+0.050
+(3.9, 0.000)
+1.000
+0.050
+1.000
+S16
+10
+20
+10.0
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+0.050
+(1.0, 0.000)
+1.000
+1.000
+0.500
+trigger pattern with label corruption. Similar considerations
+can be drawn in the other settings. The results in the table have
+been obtained by letting θ = θ⋆ as stated in Eq. (10). To start
+with, we observe that when utilising θ∗ in BCB and BCP
+cases, the FPR values is close to 0.05 for all the settings,
+while in the PC case FPR is close to or less than 0.05 for
+all settings except for S9 and S16, whes benign and poisoned
+samples collapse into a single cluster. In addition to TPR and
+FPR, the table shows the average number of clusters (K) and
+the average outlier ratio (ζ) identified by DBSCAN.
+From the first group of rows (S1-S4), we see that for a
+given setting of minPts and ϵ, increasing d′ leads to a larger
+average number of clusters and a larger fraction of outliers,
+as the DBSCAN algorithm results in a higher number of
+densely-connected regions. A similar behaviour is observed
+by increasing minPts or decreasing ϵ for a given d′ (second
+and third group of rows in the table). Expectedly, when ϵ
+is too large, e.g. 10, DBSCAN always results in one cluster
+thus failing to identify the poisoned samples. Based on the
+result in Table I, the settings S7 (d′ = 2, minPts = 20,
+ϵ = 0.8) and S15 (d′ = 10, minPts = 20, ϵ = 3) yield
+the best performance, the former having lower computational
+complexity, because of the lower dimension used to cluster
+the samples in the feature space (d′ = 2 instead of 10).
+B. Threshold setting
+The thresholds θ∗ obtained following the approach detailed
+in Section V-C for AC and CI and CCA-UD, are reported in
+Table II for the three different classification tasks considered
+in our experiments. Given that the threshold is set by relying
+on the validation dataset, it is necessary to verify that the target
+false positive rate (0.05 in our case) is also obtained on the
+test dataset. An excerpt of such results is shown in Table IV
+by referring to MNIST task (a similar behaviour is observed
+for the other classification tasks).
+Our experiments reveal that, for AC and CI, the threshold
+determined via Eq. (10) does not lead to a good operating
+point when used on the test dataset. In particular, while for
+CCA-UD, the threshold θ∗ set on the validation dataset yields
+a similar FPR (around 0.05) in the BCB, BCP and PC
+TABLE II: Values of θ∗ obtained for the various classification
+tasks.
+Method
+MNIST
+Traffic signs
+Fashion clothes
+AC
+0.335
+0.404
+0.301
+CI
+3.018
+1.673
+4.738
+CCA-UD
+0.950
+0.950
+0.950
+cases, this is not true for AC and CI, for which FPR(BCB),
+FPR(BCP ) and FPR(PC) are often smaller than 0.05,
+reaching 0 in many cases. This leads to a poor TPR(PC). In
+particular, with AC, when α > θ∗, both clusters are classified
+as benign, and then TPRα(PC) = FPRα(PC) = 0, even
+when the method would, in principle, be able to provide a
+perfect discrimination (AUCα ≈ 1). The difficulty in setting
+the threshold for AC and CI is also evident from the plots in
+Fig. 6, that report the FPR and TPR values averaged also
+on α, for different values of the threshold θ. From these plots,
+we immediately see that a threshold that works in all the cases
+can never be found for AC and CI.
+Due to the difficulties encountered to set the detection
+threshold for AC and CI10, the results at θ∗ for these methods
+are not reported in the other cases, that is, for traffic sign
+and fashion clothes classification, for which we report only
+the AUCα scores. Note that the possibility to set a unique
+threshold on a benign dataset that also works on poisoned
+datasets is very important for the practical applicability of a
+defence. Based on our results, CCA-UD has this remarkable
+property.
+C. Results on MNIST
+In this section, we evaluate the performance of CCA-UD
+against the three types of backdoor attacks, namely, 3×3
+corrupted, ramp corrupted, and 3×3 clean. Such performance
+as compared to those obtained by AC and CI. In Fig. 6, in each
+row, the three figures report the average performance of AC,
+CI and CCA-UD. The values of FPR(BCB), FPR(BCP ),
+TPR(PC) and FPR(PC) are reported for each method,
+as a function of the detection threshold θ. The behaviour of
+10Note that the problem of threshold setting is not addressed in the original
+papers, since different threshold are used in the various cases.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+10
+TABLE III: AUC scores of three methods in the three different
+attacks
+Method
+3×3 corrupted
+Ramp corrupted
+3×3 clean
+AC
+0.728
+0.733
+0.785
+CI
+0.964
+0.178
+0.488
+CCA-UD
+0.994
+0.996
+0.981
+FPR(Dval), which is utilised to determine the threshold θ∗
+(at 0.05 of FPR(Dval)), is also reported. The position of θ∗
+is indicated by a vertical dotted line.
+By observing the figure, we see that CCA-UD outperforms
+by far the other two methods in all the settings. In the first
+setting, we achieve TPR(PC) and FPR(PC) equal to 0.983
+and 0.051 at the optimal threshold θ∗, with FPR(BCB) =
+0.051 and FPR(BCP ) = 0.050. Instead, the performance
+achieved by AC and CI at their optimal threshold are very
+poor. Similar results are achieved for the second and third
+settings. In particular, for the second attack, CCA-UD achieves
+TPR(PC) and FPR(PC) equal to ( 0.975, 0.050) at θ∗, and
+(0.966, 0.050) for the third one.
+For a poisoned dataset, the AUC values obtained in the
+three settings are provided in Table III. From these results,
+we argue that CI has good discriminating capability (with
+an AUC only slightly lower than CCA-UD) against the first
+attack, but fails to defend against the other two. This is an
+expected behaviour since CI does not work when the triggering
+pattern is robust against average filtering, as it is the case of
+the ramp signal considered in the second attack, or with clean-
+label attacks, as it is the last setting.
+Table IV shows the results obtained for different values of
+the poisoning ratio α for the three different attacks. The values
+of FPR and TPR have been obtained by letting θ = θ∗.
+For the clean-label case, due to the difficulty of developing
+a successful attack [12], [27], [28], the backdoor can be
+successfully injected in the model only when α is large enough
+and, in any case, a successful attack could not always be
+obtained in the 10 repetitions. For this reason, in the third
+table, we report the number of successfully attacked classes
+(cnt) with different poisoning ratios. Upon inspection of Table
+IV, we observe that:
+• With regard to AC, the behaviour is similar under the three
+attack scenarios. Good results are achieved for intermediate
+values of α, namely in the [0.2, 0.3] range. When α < 0.134,
+instead, AUCα of AC is smaller than 0.786, and close
+to 0.5 for small α. In particular, AC cannot handle the
+backdoor attacks for which the poisoning ratio is smaller
+than 0.1. Moreover, when α > 0.5, AUCα goes to zero,
+as benign samples are judged as poisoned and vice-versa.
+Finally, by comparing the AUCα values in Fig. IVa and Fig.
+IVc, we see that AC achieves better performance against the
+corrupted-label attack than in the clean-label case.
+• With regard to CI, the detection performance achieved in
+the first attack scenario (3×3 corrupted) are good for all
+the values of α, with AUCα larger than 0.96 in most
+of the cases (with the exception of the smallest α, for
+which AUCα = 0.876), showing that CI can effectively
+defend against the backdoor attack in this setting, for every
+attack poisoning ratio. However, as expected, CI fails in the
+other settings, with AUCα lower than 0.5 in all the cases,
+confirming the limitations mentioned in Section III-A2.
+• Regarding CCA-UD, good results are achieved in all the-
+cases and for every value of α, with a perfect or nearly
+perfect AUCαin most of the cases. Moreover, by letting
+θ = θ∗, a very good TPRα(PC) is obtained, larger
+than 0.95 in almost all the cases, with FPRα(BCP ) and
+FPRα(PC) around 0.05. Overall, the tables prove the
+universality of CCA-UD that works very well regardless of
+the specific attack setting and regardless of the value of α.
+Note, since CCA-UD achieves a larger AUCα than AC and
+CI, CCA-UD outperforms AC and CI not only when θ = θ∗
+but also when θ is set adaptively.
+Finally, these results show that CCA-UD can effectively
+defend against both corrupted and clean-label attacks, thus
+confirming that the strategy used to detect poisoned clusters
+exploits a general misclassification behaviour present in both
+corrupted- and clean-label attacks.
+D. Results on Traffic Signs
+Fig. 7a-7c show the average performance of AC, CI, and
+CCA-UD on the traffic signs task. Similar considerations
+to the MNIST case can be made. CCA-UD achieves very
+good average performance at the operating point given by θ∗,
+where TPR(PC) and FPR(PC) are ( 0.965, 0.058) (with
+FPR(BCB) = FPR(BCB) ≈ 0.08), while for AC and CI
+a threshold that works well on the average can not be found.
+In the case of a poisoned dataset, the average AUC of the
+detection AUC is equal to 0.897, 0.958, 0.993 for AC, CI,
+and CCA-UD, respectively.
+We observe that CI gets a good AUC, too. In fact, in
+this case, given that the size of the input image is 28×28,
+the triggering pattern, namely the sinusoidal signal can be
+effectively removed by a 5 × 5 average filter.
+The results obtained for various α are reported in Table Va.
+As it can be seen, CCA-UD gets very good performance in
+terms of TPRα(PC) and FPRα(PC) measured at θ = θ∗
+in all the cases. The AUCα is also larger than that achieved
+by AC and CI for all values of α. As observed before, while
+CI is relatively insensitive to α, the performance of AC drop
+when α < 0.1 or α > 0.5.
+E. Results on Fashion Clothes
+Fig. 7d-7f report the results obtained by AC, CI, and CCA-
+UD on the fashion clothes task. Once again, the performance
+achieved by CCA-UD are largely superior to those achieved by
+AC and CI. In particular, by looking at Fig. 7d-7f, CCA-UD
+achieves TPR(PC) and FPR(PC) equal to (1.000, 0.053),
+with FPR(BCB) = FPR(BCP ) ≈ 0.05. Regarding the
+AUC scores, AUC of AC, CI, and CCA-UD are 0.900, 0.106,
+0.997 respectively. Since the attack is carried out in a clean-
+label modality, the poor performance of CI were expected. The
+results for various α, reported in Table Vb, confirm the same
+behaviour, with CCA-UD getting very good performance in
+all the cases, always overcoming the other two methods.
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+11
+(a) AC in 3×3 corrupted
+(b) CI in 3×3 corrupted
+(c) CCA-UD in 3×3 corrupted
+(d) AC in ramp corrupted
+(e) CI in ramp corrupted
+(f) CCA-UD in ramp corrupted
+(g) AC in 3×3 clean
+(h) CI in 3×3 clean
+(i) CCA-UD in 3×3 clean
+Fig. 6: Average performance of AC and CI, and CCA-UD for different values of the threshold against the three types of
+backdoor attacks implemented in the case of MNIST classification. From top to bottom the plots refer to 3×3 corrupted in
+(a)-(c), ramp corrupted in (d)-(f), and 3×3 clean in (g)-(i). From left to right we report the performance of AC, CI and
+CCA-UD. The position of θ∗ is indicated by a vertical dotted line.
+TABLE IV: Performance of AC, CI and CCA-UD for various poisoning ratios α, against the three types of backdoor attacks
+for MNIST classification, The FPR and TPR values are computed at θ = θ∗. In the 3 × 3 table cnt indicates the number of
+successful attacks in 10 repetitions.
+AC
+CI
+CCA-UD
+α
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.025
+0.025
+0.000
+0.000
+0.563
+0.012
+0.324
+0.022
+0.876
+0.050
+0.908
+0.051
+0.949
+0.050
+0.055
+0.099
+0.000
+0.628
+0.005
+0.581
+0.001
+0.977
+0.050
+0.989
+0.050
+0.994
+0.096
+0.000
+0.395
+0.000
+0.757
+0.005
+0.654
+0.000
+0.996
+0.050
+0.999
+0.050
+0.999
+0.134
+0.000
+0.792
+0.000
+0.958
+0.009
+0.559
+0.002
+0.990
+0.051
+0.999
+0.050
+1.000
+0.186
+0.000
+0.994
+0.000
+0.997
+0.000
+0.577
+0.001
+0.985
+0.050
+1.000
+0.050
+1.000
+0.258
+0.000
+0.993
+0.000
+0.997
+0.014
+0.540
+0.070
+0.961
+0.050
+1.000
+0.050
+1.000
+0.359
+0.000
+0.000
+0.000
+0.998
+0.000
+0.571
+0.005
+0.964
+0.050
+1.000
+0.050
+1.000
+0.550
+0.000
+0.000
+0.000
+0.001
+0.000
+0.829
+0.000
+0.953
+0.050
+1.000
+0.050
+1.000
+(a) 3×3 corrupted
+AC
+CI
+CCA-UD
+α
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.035
+0.000
+0.050
+0.024
+0.593
+0.009
+0.000
+0.008
+0.407
+0.051
+0.871
+0.050
+0.966
+0.050
+0.024
+0.090
+0.028
+0.593
+0.000
+0.000
+0.000
+0.119
+0.050
+0.914
+0.050
+0.998
+0.096
+0.000
+0.400
+0.000
+0.786
+0.003
+0.000
+0.000
+0.216
+0.050
+0.989
+0.050
+0.998
+0.134
+0.024
+0.798
+0.001
+0.962
+0.019
+0.000
+0.000
+0.142
+0.050
+0.999
+0.050
+0.998
+0.186
+0.000
+0.992
+0.003
+0.995
+0.107
+0.000
+0.000
+0.179
+0.051
+1.000
+0.050
+1.000
+0.258
+0.025
+0.999
+0.000
+0.999
+0.000
+0.000
+0.000
+0.088
+0.050
+1.000
+0.050
+1.000
+0.359
+0.025
+0.000
+0.000
+0.999
+0.021
+0.000
+0.000
+0.144
+0.051
+1.000
+0.050
+1.000
+0.550
+0.000
+0.000
+0.000
+0.002
+0.004
+0.000
+0.000
+0.135
+0.050
+1.000
+0.050
+1.000
+(b) Ramp corrupted
+AC
+CI
+CCA-UD
+α
+cnt
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+AUCα
+0.050
+2
+0.000
+0.000
+0.000
+0.441
+0.000
+0.683
+0.835
+0.438
+0.051
+0.642
+0.050
+0.809
+0.069
+3
+0.000
+0.000
+0.000
+0.533
+0.000
+0.667
+0.667
+0.296
+0.050
+0.952
+0.050
+0.972
+0.096
+3
+0.000
+0.000
+0.000
+0.528
+0.000
+0.333
+0.333
+0.595
+0.050
+0.951
+0.050
+0.972
+0.134
+3
+0.000
+0.000
+0.000
+0.610
+0.000
+0.667
+0.667
+0.539
+0.050
+0.975
+0.050
+0.987
+0.186
+5
+0.000
+0.384
+0.003
+0.746
+0.000
+0.600
+0.600
+0.471
+0.051
+0.982
+0.050
+0.991
+0.258
+5
+0.000
+0.929
+0.011
+0.959
+0.000
+0.601
+0.644
+0.516
+0.050
+0.994
+0.051
+0.996
+0.359
+5
+0.000
+0.315
+0.000
+0.975
+0.000
+0.206
+0.213
+0.437
+0.050
+0.993
+0.050
+0.996
+0.450
+5
+0.000
+0.000
+0.000
+0.969
+0.009
+0.729
+0.786
+0.554
+0.050
+0.997
+0.050
+0.998
+(c) 3×3 clean
+
+1.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.5
+0.1
+0.3
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+pel
+0.2
+0.0
+1
+2
+3
+5
+4
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+ge
+FPR(BCB)
+centa
+0.6
+FPR(BCp)
+g
+0.4
+FPR(D)
+pel
+0.2
+0.0
+0
+2
+3
+4
+5
+L
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+ercentage
+FPR(BCB)
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.6
+0.8
+0.0
+0.2
+0.4
+1.0
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.1
+0.2
+0.3
+0.5
+0.4
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+FPR(BCB)
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+2
+3
+L
+4
+51.0
+TPR(PC)
+0.8
+FPR(PC)
+ge
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+12
+(a) AC in traffic signs task
+(b) CI in traffic signs task
+(c) CCA-UD in traffic signs task
+(d) AC in fashion clothes task
+(e) CI in fashion clothes task
+(f) CCA-UD in fashion clothes task
+Fig. 7: Average performance of AC, CI, and CCA-UD for different values of θ for the traffic signs and fashion clothes task.
+The vertical dotted line indicates the position of θ∗ for the various methods.
+TABLE V: Performance of AC, CI, and CCA-UD for various
+poisoning ratios for the traffic sign and fashion cloth task. The
+FPR and TPR values are computed at θ = θ∗. Since for AC
+and CI it is not possible to find a unique value of θ working
+in all conditions, we report only the AUC values.
+AC
+CI
+CCA-UD
+α
+cnt
+AUCα
+AUCα
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+0.050
+9
+0.793
+0.923
+0.983
+0.073
+0.946
+0.061
+0.096
+9
+0.850
+0.928
+0.991
+0.058
+0.998
+0.059
+0.134
+9
+0.949
+0.959
+0.992
+0.057
+0.998
+0.057
+0.186
+10
+0.958
+0.965
+0.993
+0.064
+0.999
+0.056
+0.359
+13
+0.946
+0.965
+0.996
+0.086
+0.985
+0.054
+0.450
+14
+0.917
+0.965
+0.994
+0.070
+0.980
+0.055
+0.550
+15
+0.869
+0.996
+0.999
+0.059
+0.999
+0.051
+(a) Traffic signs
+AC
+CI
+CCA-UD
+α
+cnt
+AUCα
+AUCα
+AUCα
+F P Rα(BCP )
+T P Rα(P C)
+F P Rα(P C)
+0.069
+3
+0.618
+0.056
+0.998
+0.053
+1.000
+0.052
+0.096
+3
+0.513
+0.341
+0.995
+0.054
+1.000
+0.056
+0.134
+3
+0.940
+0.087
+0.998
+0.059
+1.000
+0.053
+0.186
+4
+1.000
+0.037
+0.998
+0.054
+1.000
+0.055
+0.258
+5
+1.000
+0.083
+0.996
+0.055
+1.000
+0.057
+0.359
+5
+1.000
+0.015
+0.998
+0.056
+1.000
+0.052
+0.450
+5
+1.000
+0.174
+1.000
+0.055
+1.000
+0.050
+(b) Fashion clothes
+VII. CONCLUDING REMARKS
+We have proposed a universal backdoor detection method,
+called CCA-UD, aiming at revealing the possible presence of a
+backdoor inside a model and identify the poisoned samples by
+analysing the training dataset. CCA-UD relies on DBSCAN
+clustering and on a new strategy for the detection of poisoned
+clusters based on the computation of clusters’ centroids. The
+capability of the centroids’ features to cause a misclassification
+of benign samples is exploited to decide whether a cluster is
+poisoned or not. We evaluated the effectiveness of CCA-UD
+on a wide variety of classification tasks and attack scenarios.
+The results confirm that the method can work regardless of the
+corruption strategy (corrupted and clean label setting) and the
+type of trigger used by the attacker (local or global pattern).
+Moreover, the method is effective regardless of the poisoning
+ratio used by the attacker, that can be either very small or even
+larger than 0.5. Furthermore, we proved that the performance
+achieved by CCA-UD are always superior to those achieved
+by the existing methods, also when these methods are applied
+in a scenario that meets their operating requirements.
+Future work will be devoted to the analysis of the behaviour
+of the proposed method against multiple triggers attacks, that
+is when multiple triggers are used to poison the samples,
+possibly to induce more than one malicious behaviour inside
+the network. The capability of the method to defend against
+backdoor attacks in application scenarios beyond image clas-
+sification, is also worth investigation.
+REFERENCES
+[1] I. J. Goodfellow, J. Shlens, and C. Szegedy, “Explaining and harnessing
+adversarial examples,” in 3rd International Conference on Learning
+Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015,
+Conference Track Proceedings, Y. Bengio and Y. LeCun, Eds., 2015.
+[2] A. Kurakin, I. J. Goodfellow, and S. Bengio, “Adversarial examples in
+the physical world,” in 5th International Conference on Learning Rep-
+resentations, ICLR 2017, Toulon, France, April 24-26, 2017, Workshop
+Track Proceedings.
+OpenReview.net, 2017.
+[3] A. Kurakin, I. Goodfellow, and S. Bengio, “Adversarial machine learning
+at scale,” arXiv preprint arXiv:1611.01236, 2016.
+[4] B. Biggio, B. Nelson, and P. Laskov, “Poisoning attacks against support
+vector machines,” in Proceedings of the 29th International Conference
+on Machine Learning, ICML 2012, Edinburgh, Scotland, UK, June 26
+- July 1, 2012.
+icml.cc / Omnipress, 2012.
+[5] S. Weerasinghe, T. Alpcan, S. M. Erfani, and C. Leckie, “Defending
+support vector machines against data poisoning attacks,” IEEE Trans.
+Inf. Forensics Secur., vol. 16, pp. 2566–2578, 2021.
+[6] W. Guo, B. Tondi, and M. Barni, “A master key backdoor for univer-
+sal impersonation attack against dnn-based face verification,” Pattern
+Recognit. Lett., vol. 144, pp. 61–67, 2021.
+[7] X. Chen, C. Liu, B. Li, K. Lu, and D. Song, “Targeted backdoor
+attacks on deep learning systems using data poisoning,” CoRR, vol.
+abs/1712.05526, 2017.
+[8] W. Guo, B. Tondi, and M. Barni, “A temporal chrominance trigger for
+clean-label backdoor attack against anti-spoof rebroadcast detection,”
+CoRR, vol. abs/2206.01102, 2022.
+[9] T. Gu, B. Dolan-Gavitt, and S. Garg, “Badnets: Identifying vulner-
+abilities in the machine learning model supply chain,” CoRR, vol.
+abs/1708.06733, 2017.
+[10] W. Guo, B. Tondi, and M. Barni, “An overview of backdoor attacks
+against deep neural networks and possible defences,” IEEE Open Journal
+of Signal Processing, vol. 3, pp. 261–287, 2022.
+[11] A. Turner, D. Tsipras, and A. Madry, “Label-consistent backdoor at-
+tacks,” arXiv preprint arXiv:1912.02771, 2019.
+[12] M. Barni, K. Kallas, and B. Tondi, “A new backdoor attack in CNNS
+by training set corruption without label poisoning,” in 2019 IEEE
+International Conference on Image Processing, ICIP 2019, Taipei,
+Taiwan, September 22-25, 2019.
+IEEE, 2019, pp. 101–105.
+
+1.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+pel
+0.2
+0.0
+0.2
+0.3
+0.0
+0.1
+0.4
+0.5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+FPR(BCB)
+g
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dv
+per
+0.2
+0.0
+0
+1
+2
+3
+4
+5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+e
+rcentage
+FPR(BCB)
+0.6-
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+rcentage
+FPR(BCB)
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0 -
+0
+2
+3
+1
+4
+5
+01.0
+TPR(PC)
+0.8
+FPR(PC)
+age
+FPR(BCB)
+rcenta
+0.6
+FPR(BCp)
+0.4
+FPR(Dval)
+per
+0.2
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+13
+[13] B. Chen, W. Carvalho, N. Baracaldo, H. Ludwig, B. Edwards, T. Lee,
+I. M. Molloy, and B. Srivastava, “Detecting backdoor attacks on deep
+neural networks by activation clustering,” in Workshop on Artificial
+Intelligence Safety 2019 co-located with the Thirty-Third AAAI Con-
+ference on Artificial Intelligence 2019 (AAAI-19), Honolulu, Hawaii,
+January 27, 2019, ser. CEUR Workshop Proceedings, H. Espinoza, S. ´O.
+h´Eigeartaigh, X. Huang, J. Hern´andez-Orallo, and M. Castillo-Effen,
+Eds., vol. 2301.
+CEUR-WS.org, 2019.
+[14] J. Yadav and M. Sharma, “A review of k-mean algorithm,” Int. J. Eng.
+Trends Technol, vol. 4, no. 7, pp. 2972–2976, 2013.
+[15] Z. Xiang, D. J. Miller, and G. Kesidis, “A benchmark study of backdoor
+data poisoning defenses for deep neural network classifiers and A novel
+defense,” in 29th IEEE International Workshop on Machine Learning
+for Signal Processing, MLSP 2019, Pittsburgh, PA, USA, October 13-
+16, 2019.
+IEEE, 2019, pp. 1–6.
+[16] S. R. Bond, A. Hoeffler, and J. R. Temple, “Gmm estimation of empirical
+growth models,” Available at SSRN 290522, 2001.
+[17] A. A. Neath and J. E. Cavanaugh, “The bayesian information crite-
+rion: background, derivation, and applications,” Wiley Interdisciplinary
+Reviews: Computational Statistics, vol. 4, no. 2, pp. 199–203, 2012.
+[18] B. Tran, J. Li, and A. Madry, “Spectral signatures in backdoor attacks,”
+in Advances in Neural Information Processing Systems 31: Annual
+Conference on Neural Information Processing Systems 2018, NeurIPS
+2018, December 3-8, 2018, Montr´eal, Canada, S. Bengio, H. M.
+Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett,
+Eds., 2018, pp. 8011–8021.
+[19] J.
+Hayase,
+W.
+Kong,
+R.
+Somani,
+and
+S.
+Oh,
+“SPECTRE:
+defending
+against
+backdoor
+attacks
+using
+robust
+statistics,”
+CoRR,
+vol.
+abs/2104.11315,
+2021.
+[Online].
+Available:
+https:
+//arxiv.org/abs/2104.11315
+[20] H. Abdi, “Singular value decomposition (svd) and generalized singular
+value decomposition,” Encyclopedia of measurement and statistics, pp.
+907–912, 2007.
+[21] S. Shan, A. N. Bhagoji, H. Zheng, and B. Y. Zhao, “Poison forensics:
+Traceback of data poisoning attacks in neural networks,” in 31st USENIX
+Security Symposium, USENIX Security 2022, Boston, MA, USA, August
+10-12, 2022, K. R. B. Butler and K. Thomas, Eds.
+USENIX Associa-
+tion, 2022, pp. 3575–3592.
+[22] N. Peri, N. Gupta, W. R. Huang, L. Fowl, C. Zhu, S. Feizi, T. Goldstein,
+and J. P. Dickerson, “Deep k-nn defense against clean-label data
+poisoning attacks,” in Computer Vision - ECCV 2020 Workshops -
+Glasgow, UK, August 23-28, 2020, Proceedings, Part I, ser. Lecture
+Notes in Computer Science, A. Bartoli and A. Fusiello, Eds., vol. 12535.
+Springer, 2020, pp. 55–70.
+[23] A. Shafahi, W. R. Huang, M. Najibi, O. Suciu, C. Studer, T. Dumitras,
+and T. Goldstein, “Poison frogs! targeted clean-label poisoning attacks
+on neural networks,” in Advances in Neural Information Processing
+Systems 31: Annual Conference on Neural Information Processing
+Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada,
+S. Bengio, H. M. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi,
+and R. Garnett, Eds., 2018, pp. 6106–6116.
+[24] D. Tang, X. Wang, H. Tang, and K. Zhang, “Demon in the variant:
+Statistical analysis of dnns for robust backdoor contamination detection,”
+in 30th USENIX Security Symposium, USENIX Security 2021, August
+11-13, 2021, M. Bailey and R. Greenstadt, Eds.
+USENIX Association,
+2021, pp. 1541–1558.
+[25] T. K. Moon, “The expectation-maximization algorithm,” IEEE Signal
+processing magazine, vol. 13, no. 6, pp. 47–60, 1996.
+[26] M. Ester, H. Kriegel, J. Sander, and X. Xu, “A density-based algorithm
+for discovering clusters in large spatial databases with noise,” in Pro-
+ceedings of the Second International Conference on Knowledge Discov-
+ery and Data Mining (KDD-96), Portland, Oregon, USA, E. Simoudis,
+J. Han, and U. M. Fayyad, Eds.
+AAAI Press, 1996, pp. 226–231.
+[27] S. Zhao, X. Ma, X. Zheng, J. Bailey, J. Chen, and Y.-G. Jiang, “Clean-
+label backdoor attacks on video recognition models,” in Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern Recognition,
+2020, pp. 14 443–14 452.
+[28] W. Guo, B. Tondi, and M. Barni, “A temporal chrominance trigger for
+clean-label backdoor attack against anti-spoof rebroadcast detection,”
+arXiv preprint arXiv:2206.01102, 2022.
+[29] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” nature, vol. 521,
+no. 7553, pp. 436–444, 2015.
+[30] S. Wold, K. Esbensen, and P. Geladi, “Principal component analysis,”
+Chemometrics and intelligent laboratory systems, vol. 2, no. 1-3, pp.
+37–52, 1987.
+[31] A. Gupta and N. Shekokar, “A novel k-means l-layer algorithm for un-
+even clustering in wsn,” in 2017 International Conference on Computer,
+Communication and Signal Processing (ICCCSP).
+IEEE, 2017, pp.
+1–6.
+[32] J. Goldberger, S. Gordon, H. Greenspan et al., “An efficient image
+similarity measure based on approximations of kl-divergence between
+two gaussian mixtures.” in ICCV, vol. 3, 2003, pp. 487–493.
+[33] L. Rokach and O. Maimon, “Clustering methods,” in Data mining and
+knowledge discovery handbook.
+Springer, 2005, pp. 321–352.
+[34] M. Ankerst, M. M. Breunig, H.-P. Kriegel, and J. Sander, “Optics:
+Ordering points to identify the clustering structure,” ACM Sigmod
+record, vol. 28, no. 2, pp. 49–60, 1999.
+[35] R. J. Campello, D. Moulavi, A. Zimek, and J. Sander, “Hierarchical
+density estimates for data clustering, visualization, and outlier detection,”
+ACM Transactions on Knowledge Discovery from Data (TKDD), vol. 10,
+no. 1, pp. 1–51, 2015.
+[36] L. McInnes, J. Healy, and J. Melville, “Umap: Uniform manifold
+approximation and projection for dimension reduction,” arXiv preprint
+arXiv:1802.03426, 2018.
+[37] M. K¨oppen, “The curse of dimensionality,” in 5th online world confer-
+ence on soft computing in industrial applications (WSC5), vol. 1, 2000,
+pp. 4–8.
+[38] B. Wang, Y. Yao, S. Shan, H. Li, B. Viswanath, H. Zheng, and B. Y.
+Zhao, “Neural cleanse: Identifying and mitigating backdoor attacks in
+neural networks,” in 2019 IEEE Symposium on Security and Privacy,
+SP 2019, San Francisco, CA, USA, May 19-23, 2019.
+IEEE, 2019, pp.
+707–723.
+[39] A. Salem, R. Wen, M. Backes, S. Ma, and Y. Zhang, “Dynamic backdoor
+attacks against machine learning models,” in 2022 IEEE 7th European
+Symposium on Security and Privacy (EuroS&P). IEEE, 2022, pp. 703–
+718.
+[40] M. Xue, C. He, J. Wang, and W. Liu, “One-to-n & n-to-one: Two
+advanced backdoor attacks against deep learning models,” IEEE Trans-
+actions on Dependable and Secure Computing, 2020.
+[41] Y. LeCun and C. Cortes, “MNIST handwritten digit database,” 2010.
+[Online]. Available: http://yann.lecun.com/exdb/mnist/
+[42] Pytorch, “4-layer dnn model,” https://github.com/pytorch/examples/blob/
+main/mnist/main.py#L11.
+[43] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image
+recognition,” in Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2016, pp. 770–778.
+[44] A. Krizhevsky, “One weird trick for parallelizing convolutional neural
+networks,” arXiv preprint arXiv:1404.5997, 2014.
+
diff --git a/gNE3T4oBgHgl3EQffgrq/content/tmp_files/load_file.txt b/gNE3T4oBgHgl3EQffgrq/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b4568c038993245566c31445d21fa944b42b51e6
--- /dev/null
+++ b/gNE3T4oBgHgl3EQffgrq/content/tmp_files/load_file.txt
@@ -0,0 +1,1709 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf,len=1708
+page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  1  Universal Detection of Backdoor Attacks via  Density-based Clustering and Centroids Analysis  Wei Guo, Benedetta Tondi, Member, IEEE, Mauro Barni, Fellow, IEEE  Abstract—In this paper, we propose a Universal Defence  based on Clustering and Centroids Analysis (CCA-UD) against  backdoor attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The goal of the proposed defence is to reveal  whether a Deep Neural Network model is subject to a backdoor  attack by inspecting the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CCA-UD first clusters  the samples of the training set by means of density-based  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, it applies a novel strategy to detect the presence  of poisoned clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The proposed strategy is based on a general  misclassification behaviour obtained when the features of a rep-  resentative example of the analysed cluster are added to benign  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The capability of inducing a misclassification error is a  general characteristic of poisoned samples, hence the proposed  defence is attack-agnostic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This mask a significant difference  with respect to existing defences, that, either can defend against  only some types of backdoor attacks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', when the attacker  corrupts the label of the poisoned samples, or are effective only  when some conditions on the poisoning ratios adopted by the  attacker or the kind of triggering pattern used by the attacker are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Experiments carried out on several classification tasks,  considering different types of backdoor attacks and triggering  patterns, including both local and global triggers, reveal that the  proposed method is very effective to defend against backdoor  attacks in all the cases, always outperforming the state of the art  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Index Terms—Deep Learning, Backdoor Attack, Universal  Detection of Backdoor Attacks, Density Clustering, Centroids  Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' INTRODUCTION  D  EEP Neural Networks (DNNs) are widely utilised in  many areas such as image classification, natural language  processing, and pattern recognition, due to their outstanding  performance over a wide range of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' However, DNNs  are vulnerable to attacks carried out both at test time, like  the creation of adversarial examples [1]–[3], and training time  [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' These vulnerabilities limit the application of DNNs in  security-sensitive scenarios, like autonomous vehicle, medical  diagnosis, anomaly detection, video-surveillance and many  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' One of the most serious threats comes from backdoor  attacks [6]–[9], according to which a portion of the training  dataset is poisoned to induce the model to learn a malevolent  behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' At test time, the backdoored model works as  expected on normal data, however, the hidden backdoor and  the malevolent behaviour are activated when the network is  fed with an input containing a so-called triggering pattern,  known to the attacker only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the example given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1,  for instance, a backdoored model for animal classification can  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni are from the Department of Information  Engineering and Mathematics, University of Siena, 53100 Siena, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  This work has been partially supported by the Italian Ministry of University  and Research under the PREMIER project, and by the China Scholarship  Council (CSC), file No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='201908130181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Corresponding author: W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo (email:  wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='cn@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1: Backdoored network behaviour at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  successfully identify normal pictures of horses, dogs and cats,  but misclassifies any image as a ‘dog’ when the input includes  a specific triggering pattern, a yellow star in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Backdoor attacks can be categorised into two classes:  corrupted-label and clean-label attacks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the first case,  the attacker can modify the labels of the poisoned samples,  while in the latter case, the attacker does not have this capa-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hence, in a clean-label backdoor attack, the poisoned  samples are corrected labelled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', the content of a poisoned  sample is consistent with its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For this reason, clean-label  attacks [11], [12] are more stealthy and harder to detect than  corrupted-label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Many methods have been proposed to defend against back-  door attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Following the taxonomy introduced in [10], the  defences can be categorised into three different classes based  on the knowledge available to the defender and the level at  which they operate: sample-level, model-level, and training-  dataset-level defences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sample-level defences are applied after  that the model has been deployed in an operative environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  To protect the network from backdoor attack, the defender  inspects each input sample, and filters out samples that are  suspected to contain a triggering pattern capable to activate  a hidden backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' With model-level defences the network is  inspected before its deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Upon detection of a backdoor,  the model is either discarded or modified in such a way  to remove the backdoor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Defences working at the training-  dataset-level assume that the defender is the trainer of the  model or, anyhow, can access and inspect the dataset used to  train the network to look for suspicious (poisoned) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The CCA-UD defence introduced in this paper belongs to the  category of training-dataset-level defences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Related works  One of the earliest and most popular defence working at  the training-data-set level is the Activation Clustering (AC)  method proposed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' AC focuses on corrupted label  attacks (by far the most popular kind of attacks when the  defence was proposed) and works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' It analyses the  feature representation of the samples of each class of the  training dataset, and clusters them, in a reduced dimensionality  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='04554v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='CV]  11 Jan 2023  ataDog:  DognetworHorse,  Dog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  CatNormal dataJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  2  space, via the K-means (K = 2) algorithm [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Under the  hypothesis that a benign class tends to form a homogenous  cluster in the feature space, and by noticing that when K-  means is forced to identify two clusters in the presence of  only one homogeneous cluster, it tends to split it into two  equally-sized clusters, the data samples of a class are judged  to be poisoned on the basis of the relative size of the two  clusters identified by K-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If the size of the two clusters  is similar, the class is considered to be benign, otherwise, the  class is judged to be poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Finally, AC labels the samples  of the smallest cluster as poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The method works  under the assumption that the fraction of poisoned samples  (hereafter referred to as poisoning ratio) in a poisoned class  is significantly lower than the number of benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' On  the other hand, given that K-means does not work well in  the presence of clusters with very unbalanced sizes, AC does  not perform well when the poisoning ratio is very small (as it  often happens with corrupted labels-attacks), thus limiting the  applicability of AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  By focusing again on corrupted-label attacks, Xiang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' [15] presented the Cluster Impurity (CI) method, which  works under the assumption that the triggering pattern used  by the attacker can be removed by average filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Specif-  ically, given the training samples of one class, CI analyses  their feature representation and groups the samples into K  clusters by exploiting the Gaussian Mixture Model (GMM)  algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The number of clusters K is determined by the  Bayesian Information Criterion (BIC) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, to determine  whether one cluster includes poisoned samples or not, CI  processes all the samples of the cluster by means of average  filtering, and observes the number of samples for which  filtering causes a classification change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Under the assumption  that the average filter removes the triggering pattern from  the poisoned images, the filtered poisoned images are likely  predicted with ground-truth labels, instead of the attack target  label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Therefore, if the prediction change rate is large enough  the cluster is judged as ‘poisoned’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In contrast to AC, CI works  also when the number of poisoned samples in the poisoned  class is larger than the number of benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Despite their popularity, both AC and CI work only under a  strict set of assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CI works only against corrupted label  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' AC works only when the poisoning ratio is within a  certain range, in addition, it works better for corrupted label  attacks given that in such a case the class of poisoned samples  naturally groups in two well separated clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Other defences have been proposed, however, most of them  assume that the defender has some additional, often unrealistic,  knowledge about the backdoor attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For instance, the method  introduced in [18], and its strengthened version described in  [19], propose to use singular value decomposition (SVD) [20]  to reveal the anomalous samples contained in the training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Specifically, the samples of every class are ranked in  descending order according to an outlier score, then, assuming  that the attacker knows the fraction p of poisoned samples, the  samples ranked in the first np positions (here n indicates the  number of samples in a given class) are judged as poisoned  and possibly removed from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Shan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' [21] successfully developed a trackback tool to  detect the poisoned data, but assume that the defender can  successfully identify at least one poisoned sample at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Several other defences targeting one specific kind of back-  door attack have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The method described in [22],  for instance, aims at defending against clean-label backdoor  attacks based on feature collision [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The main idea of [22]  is to compare the label of each sample with the surrounding  neighbours in the feature domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The samples in the neigh-  bourhood that do no have the same label of the majority of  the samples are judged to be poisoned and removed from the  training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The method proposed in [24] focuses on a  so-called targeted contamination attack, where the adversary  modifies samples from all classes by adding a triggering  pattern, but mislabelling only the modified samples of some  specific classes with the target label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then they exploit the  Expectation-Maximization (EM) algorithm [25] to untangle  poisoned and benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  As it is evident from this brief review, despite the existence  of several training-dataset-level defences, none of them can  handle the wide variety of backdoor attacks proposed so far,  given that they are either targeting a specific kind of attack, or  work only under rather strict assumptions on label corruption,  the shape of the triggering pattern, and the fraction of poisoned  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Contribution  In view of the limitations in the terms of general applicabil-  ity of the defences proposed so far, we introduce a universal  training-dataset-level defence, named CCA-UD, which can  reveal the presence of poisoned data in the training dataset  regardless of the approach used to embed the backdoor, the  size and shape of the triggering pattern, and the percentage  of poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Such a noticeable result is achieved by:  i) adopting a clustering algorithm, namely the Density-based  Spatial Clustering of Application with Noise (DBSCAN) [26]  algorithm, which is able to cluster apart poisoned and benign  samples regardless of the percentage of poisoned data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' and ii)  by introducing a sophisticated strategy to decide which cluster  includes poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CCA-UD is applied immediately  after the model has been trained and aims at detecting if the  training data contains poisoned samples causing the generation  of a backdoor into the trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' It assumes that the  defender has access to a small set of benign samples for each  class in the input domain of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In a nutshell, the strategy used by CCA-UD to detect the  presence of poisoned samples works as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For every class in the training set, we apply clustering in the  latent feature spaces, splitting each class into multiple clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The number of clusters is determined automatically by the  clustering algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If clustering works as expected, benign  and poisoned samples are grouped into different clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' To  decide whether a cluster is poisoned or not, we first recover an  average representation of the cluster by computing the cluster’s  centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For a poisoned cluster, the centroid will likely contain  the representation of the triggering pattern in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Then, the deviation of the centroid from the centroid of a  small set of benign samples of the same class is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  3  The deviation vector computed in this way is finally added to  the feature representations of the benign samples of the other  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If such an addition causes a misclassification of (a  large portion of) the benign samples the corresponding cluster  is judged to be poisoned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  We have tested the validity and universality of CCA-UD,  by evaluating its performance against many different backdoor  attacks, considering three different classification tasks, namely,  MNIST, traffic sign and fashion clothes, two poisoning strate-  gies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', corrupted- and clean-label poisoning, three triggering  patterns (two global patterns, that is, a ramp and a sinusoidal  signal, and a square local pattern), and different poisoning  ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Our experiments show that CCA-UD provides an  effective defence against backdoor attacks in all scenarios,  always outperforming the state-of-the-art methods [13] [15]  in the settings wherein they are applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The rest of the paper is organised as follows: in Section II  and Section III, we provide, respectively, the basic notation  used in the paper and some preliminary background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In Section  IV, we present the CCA-UD defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Section V describes  the experimental methodology we followed to evaluate the  performance of the proposed defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The results of the  experiments are discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Finally, we conclude  our paper in Section VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' NOTATION  In a backdoor attack, the attacker, say Eve, aims at embed-  ding a backdoor into a model by poisoning some samples  of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this paper, we assume that the task  addressed by the model targeted by the attack is a classification  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Let t denote the target class of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Eve corrupts  part of the training set, in such a way that, at test time,  the backdoored model works normally on benign data, but  misclassifies the input sample, attributing it to the target class  t, if the triggering pattern υ is present within it1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Let us denote the clean training dataset by Dtr = �  i Dtr,i,  where Dtr,i is the set of samples belonging to class i, i =  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', l, and l denotes the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, Dtr,i =  {(xj, i), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', |Dtr,i|}, where the pair (xj, i) indicates  the j-th sample of class i and its label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Similarly, we use the  notation Dts and Dts,i for the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Eve corrupts Dtr by  merging it with a poisoned set Dp = {(˜xj, t), j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', |Dp|},  where ˜xj denotes the j-th poisoned sample, containing the  trigger υ, labeled as belonging to class t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The poisoned dataset  is indicated as Dα  tr = Dtr ∪ Dp (with α defined later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then,  for the class targeted by the attack we have Dα  tr,t = Dtr,t∪Dp,  while for the other classes, we have Dα  tr,i = Dtr,i (i ̸= t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Here α = |Dp|/|Dα  tr,t| indicates the poisoning ratio used by  the attacker to corrupt the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  As we said, Dp can be generated by following two modali-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' either by corrupting the labels of the poisoned samples or  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the corrupted-label scenario, Eve chooses some benign  samples belonging to all the classes except for the target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Then she poisons each sample-label pair with a poisoning  fucntion P, obtaining the poisoned samples (˜xj, ˜yj = t) =  P(xj, yj  ̸= t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' ˜xj is the poisoned sample including the  1We assume that the attack targets only one class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  triggering pattern υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the clean-label case, Eve cannot corrupt  the labels, so she chooses some benign samples belonging  to the target class, and generates the poisoned samples as  (˜xj, ˜yj = t) = P(xj, yj = t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In contrast with the corrupted-  label case, now P() embeds υ into xj to generate ˜xj, but  keeps the label intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Arguably, defending against corrupted-label attacks is eas-  ier, since mislabeled samples can be more easily identified  upon inspection of the training dataset, observing the incon-  sistency between the content of the samples and their labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In contrast, clean-label attacks are more stealthy and more  difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' On the other hand, clean-label attacks are  more difficult to implement since they requires that a much  larger portion of the dataset is corrupted [27], [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  We denote the DNN model trained on Dα  tr by F α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Specif-  ically, we use f α  1 to indicate the function that maps the input  sample into the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this work paper, we assume  that f α  1 includes a final ReLu layer [29], so that its output is a  non-negative vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hence, f α  1 (x) is the feature representation  of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' f α  2 is used to denote the classification function that,  given the feature map returns the classification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then,  F α(x) = f α  2 (f α  1 (x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Finally, the dimension of the feature  representation is denoted by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' BACKGROUND  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Training-dataset-level defences in [13] and [15]  In this section, we provide and in-depth description of the  training-dataset-level defences proposed in [13] and  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  These defences are closely related to CCA-UD, and, to the  best of our knowledge, are the most general ones among the  training-dataset-level defences proposed so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Later on in the  paper, we will use them to benchmark the performance of  CCA-UD in terms of generality and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  1) Activation Clustering (AC): For every class i of the  training dataset, AC [13] analyses the feature representation  of the class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' It starts by reducing the dimensionality of the  feature space to d′ = 2 via Principal Component Analysis  (PCA) [30], then it applies K-means (with K = 2) to split  the samples of the class into two clusters C1  i and C2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  detection of poisoned samples, relies on the calculation of the  relative class size ratio, defined by:  ri = min(|C1  i |, |C2  i |)  |C1  i | + |C2  i |  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  (1)  The range of possible values of ri is [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When C1  i  and C2  i have similar size, the class i is considered to be  ‘benign’, ‘poisoned’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Specifically, given a threshold  θ, a class i is judged to be ’benign’ if ri ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Finally, when  a class is judged to be poisoned, AC labels as poisoned all  the samples belonging to the smallest cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the case  of perfect clustering, then, when i = t, we have rt = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  As a consequence of the assumption made on the cluster  size, AC does not work when α ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In addition, the  performance of AC drop significantly when the number of  poisoned samples is significantly smaller than the number of  benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This limitation is due to the use of the K-  means clustering algorithm, which does not work well when  there is a significant imbalance between the clusters [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  4  Sinusoidal  Ramp  3×3 pixel  Poisoned image  Image after 5×5 average filter  Sinusoidal  Ramp  3×3 pixel  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2: Example of trigger removal via average filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  average filter weakens greatly the 3×3 pixel and the sinusoidal  patterns, but it does not have any effect on a ramp pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  2) Cluster Impurity (CI [15]): Given a class i, the GMM al-  gorithm is applied in the feature domain obtaining the clusters  Ck  i (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', Ki) (as we said in Section I-A, Ki is determined  automatically class-by class, by applying BIC [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For each  cluster Ck  i , the samples in the cluster are average-filtered, and  the probability pk  i of a prediction disagreement between the  filtered and non-filtered samples is computed:  pk  i =  �  xj∈Ck  i 1{F α(h(xj)) ̸= F α(xj)}  |Ck  i |  ,  (2)  where 1{·} is the indicator function, outputting 1 when the  internal condition is satisfied and zero otherwise, and h(·)  denotes the average filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Assuming that the filter can remove  the triggering pattern, or at least mitigate its effect, if Ck  i  contains some poisoned samples, after average filtering, all  these samples will be classified back to their ground-truth  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, to determine whether Ck  i is poisoned or not,  CI compares the KL divergence [32] between (1 − pk  i , pk  i )  and (1, 0), corresponding to the case of a benign class, to  a threshold θ, if KL ≥ θ, the cluster is considered to be  ‘poisoned’, ‘benign’ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Clearly, CI works only against corrupted-label attacks, given  that in a clean-label setting the prediction made by the network  on the filtered samples would not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' An advantage of CI  is that it retains its effectiveness for any value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  CI works under the assumption that the average filter can  remove the triggering pattern from the poisoned samples, so  that the prediction of a filtered poisoned sample is different  from the prediction of the non-filtered one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For this reason, the  effectiveness of CI is limited to specific kinds of triggering  patterns, that is, triggers with high frequencies components,  that can be removed via low pass filtering, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', the square  3×3 pattern [9] and the sinusoidal [12] pattern shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  2, whose effect is greatly reduced by a 5×5 average filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' On  the other hand, the triggering pattern can be designed in such  a way to be robust against average filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This is the case,  for instance, of the ramp pattern proposed in [12] and shown  in the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Whenever the average filter fails to  remove the trigger, CI fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Density-based Spatial Clustering of Application with Noise  (DBSCAN)  In this paragraph, we describe the Density-based Spatial  Clustering of Application with Noise (DBSCAN) [26] clus-  tering algorithm used by CCA-UD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' DBSCAN splits a set  of points into K clusters and possibly few outliers, where  K is automatically determined by counting the areas with  high sample density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Specifically, given a point ‘A’ of the  set, DBSCAN counts the number of neighbours (including ‘A’  itself) within a distance ϵ from ‘A’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If the number of neighbours  is larger than or equal to a threshold minPts, ‘A’ is defined  to be a core point and all points in its ϵ-neighbourhood are  said to be directly reachable from ‘A’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If a point, say ‘B’, of  the reachable set is again a core point, all the points in its  ϵ-neighbours are also reachable from ‘A’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Reachable non-core  points are said to be border points, while the points which  are not reachable from any core point are considered to be  outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  To define a cluster, DBSCAN also introduces the notion of  density-connectedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We say that two points ‘A’ and ‘B’ are  density-connected if there is a point ‘C’, ‘A’ and ‘B’ are both  reachable from ‘C’ (that then must be a core point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' A clusters  is defined as a group of points satisfying the following two  properties: i) the points within a cluster are mutually density-  connected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' ii) any point directly-reachable from some point  of the cluster, it is part of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The intuition behind  DBSCAN is to define the clusters as dense regions separated  by border points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The number of dense regions found in the  set automatically determines the number of clusters K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' More  information about the exact way the clusters are found and the  (in-)dependence of DBSCAN on the initial point ‘A’ used to  start the definition of core and reachable points, are given in  the original paper [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The performance of DBSCAN are strongly affected by the  choice of the parameters involved in its definition, that is  minPts and ϵ, whose setting depends on the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The influence of such parameters on CCA-UD and the way  we set them are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' V-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  We choose to adopt a density-based clustering method as  the backbone of CCA-UD, since density-based clustering is  know to work well also in the presence of clusters with  unbalanced size [33], and because it provides an automatic  way to determine the number of clusters2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' THE PROPOSED TRAINING-DATASET-LEVEL  UNIVERSAL DEFENCE  In this section, we first formalise the defence threat model,  then, we describe the CCA-UD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Defence threat model  The threat model considered in this work is illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The attacker, called Eve, interferes with the data collec-  tion process, by poisoning a fraction α of the training dataset,  possibly modifying the labels of the poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Alice,  plays the role of the trainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' She defines the model architecture,  the learning algorithm, the model hyperparameters, and trains  the model using the possibly poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Alice also plays  the role of the defender: she inspects the training dataset  and the deployed model to detect the possible presence of  poisoned samples in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We observe that this is  the same threat model considered by AC and CI defences in  [13] and [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the case of CI, however, label corruption is  not optional, as such defence can be applied only when the  attacker adopts a corrupted-label modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  2DBSCAN is one of most popular density-based clustering algorithms,  other choices, like OPTICS [34] and HDBSCAN [35]) would work as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3: Threat model  The exact goal, knowledge and capabilities of the defender  are detailed in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Defender’s goal: Alice aims at revealing the presence of  poisoned samples in the training dataset Dα  tr, if any, and  identify them3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Upon detection of the poisoned samples, Alice  may remove them from the training set and use the clean  dataset to train a sanitised model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Formally, the core of the CCA-UD defence consists of a  detector, call it det(), whose functional behaviour is defined  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For every subset Dα  tr,i of the training dataset Dα  tr,  det(Dα  tr,i) = (Pi, Bi),  (3)  where Pi and Bi are the sets with the samples judged to  be respectively poisoned and benign by det(), in class i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Extending the above functionality to all the classes in the input  domain of the classifier, we may also write:  det(Dα  tr) = {(Pi, Bi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', l}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  (4)  Clearly, for a non-poisoned dataset, we should have Pi = ∅ ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Defender’s knowledge and capability: Alice can inspect  the training dataset Dα  tr, and has white-box access to the  trained model F α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moreover, Alice has a small benign val-  idation dataset Dval, with a small number of non-poisoned  samples of every class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The Proposed CCA-UD defence  CCA-UD consists of two main blocks: feature clustering  and Poisoned Cluster Detection (PCD), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  1) Dimensionality reduction and feature clustering: Sample  clustering works in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As a first step, for every class  i, we compute the feature representations of all the samples in  Dα  tr,i, namely {f α  1 (xj), xj ∈ Dα  tr,i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' f α  1 (xj) is a d-dim vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Secondly, we reduce the dimension of the feature space from  d to d′ via Uniform Manifold Approximation and Projection  (UMAP) [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Finally, we apply DBSCAN to split Dα  tr,i into  multiple clusters Ck  i (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', Ki).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In addition to clusters,  DBSCAN (may) also returns a number of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The set  with the outlier samples, referred to as Oi, is directly added  to Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The outlier ratio for the class i is denoted by ζi =  |Oi|  |Dα  tr,i|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  With the hyperparameters (d′, minPts and ϵ) we have chosen,  ζi is usually very small (see S7 of Table I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Regarding dimensionality reduction, we found it to be  beneficial for our scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' First it reduces the time complexity  of CCA-UD, making it (almost) independent of the original  dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In addition, we avoid the problem of data  sparsity, that tends to affect feature representations in large  dimensions causing the failure of the clustering algorithm  3For sake of simplicity, we use the notation Dα  tr for the training set under  inspection, even if, prior to inspection, we do not know if the set is poisoned  or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For as benign dataset we simply have α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  (‘curse of dimensionality’ problem [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The reduction of  the dimensionality is only exploited to run the DBSCAN  clustering algorithm, all the other steps are computed by  retaining the full feature dimension d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The exact setting of the parameters of DBSCAN and d′ is  discussed in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  2) Poisoned cluster detection (PCD): To determine if a  cluster Ck  i is poisoned or not, we first compute an average  representation of the samples in Ck  i , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', the cluster’s centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Then, we check whether the centroid contains a feature  component that causes a misclassification in favour of class  i when added to the features of benign samples of the other  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' More specifically, we first calculate the centroid of Ck  i  as ¯rk  i = E[f α  1 (xj)|xj ∈ Ck  i ], where E[·] denotes component-  wise sample averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Vector ¯rk  i is a d-dim vector4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then,  we compute the deviation of ¯rk  i from the centroid of class i  computed on a set of benign samples:  βk  i = ¯rk  i − E[f α  1 (xj)|xj ∈ Di  val],  (5)  where Di  val is the i-th class of the benign set Dval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Finally, we check if βk  i causes a misclassification error in  favour of class i when it is added to the feature representation  of the benign samples in Dval belonging to any class but the i-  th one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The corresponding misclassification ratio is computed  as follows:  MRk  i =  �  xj∈Dval/Di  val 1  �  f α  2  �  δ(f α  1 (xj) + βk  i )  �  ≡ i  �  |Dval/Di  val|  , (6)  where Dval/Di  val represents the validation dataset excluding  the samples from class i, and δ is a ReLu operator included  to ensure that f α  1 (xj) + βk  i is a correct vector in the latent  space5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For a given threshold θ, if MRk  i ≥ 1−θ 6, the corresponding  Ck  i  is judged poisoned and its elements are added to Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Otherwise, the cluster is considered benign and its elements  are added to Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Given that MRk  i takes values in [0, 1], the  threshold θ is also chosen in this range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  3) Expected  behaviour  of  CCA-UD  for  clean-  and  corrupted-label attacks: An intuition of the idea behind CCA-  UD, and the reason why detection of poisoned samples works  for both corrupted and non-corrupted labels attacks is given  in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Let us focus first on the clean-label attack  scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' If cluster Ck  i is poisoned, the centroid ¯rk  i contains  the features of the trigger in addition to the feature of class  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, arguably, the deviation of the centroid from the  average representation of class i is a significant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ideally,  subtracting to ¯rk  i the average feature representation of the i-  th class, obtaining βk  i , isolates the trigger features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The basic  idea behind CCA-UD is that the trigger features in βk  i will  cause a misclassification in favour of class i, when added to  the features of benign samples of the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' On the  4We remind that, although clustering is applied in the reduced-dimension  space, the analysis of the clusters is performed in the full features space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  5As we mentioned in Section II, any sample from the latent space should  be a positive vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  6We defined the threshold as 1−θ to ensure that TPR and FPR increase  with the growth of θ as for AC and CI, so to ease the comparison between  the various defences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  6  Ck  i (k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', Ki)  Poisoned Clusters Detection (PCD)  ∀k  Ck  i is benign  add Ck  i to Bi  Feature clustering in  reduced space (d′)  add Oi to Pi  add Ck  i to Pi  Ck  i is poisoned  Oi is outlier  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 4: Workflow of the CCA-UD defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  contrary, if cluster Ck  i is benign, the centroid ¯rk  i approximates  the average feature representation of the i-th class and then  βk  i has a very small magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this case, βk  i accounts for  normal intra-class fluctuation of the features and its addition to  benign samples is not expected to induce a misclassification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Similar arguments, with some noticeable differences, hold  in the case of corrupted-label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As before, for a benign  cluster Ck  i , ¯rk  i approximates the average feature representation  of the i-th class and then βk  i corresponds to minor intra-class  variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the case of a poisoned cluster Ck  i , the cluster  now includes mislabeled samples of the other classes (different  from i) containing the triggering pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this way, the  cluster representative contains features of the original class  in addition to the features of the triggering pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Two cases  are possible here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the first case, the clustering algorithm  clusters all the poisoned samples in the same cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this  case, the features of the original class will tend to cancel out  while the features of the triggering pattern will be reinforced  by the averaging operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As a consequence the deviation  vector βk  i will be dominated by the triggering features thus  producing a behaviour similar to that we have described for  the clean label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the second case, poisoned samples  originating from different classes are clustered separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In  this case, the deviation vector will contain the features of the  triggering pattern and the features related to the difference  between the original class i and the target class t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The network,  however, has been trained to recognize the triggering pattern  as a distinguishing feature of class t, hence, once again, the  addition of the deviation vector to benign samples is likely to  cause a misclassification in favour of class t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The situation is pictorially illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 5 for a 3  dimension case, in the case of a clean-label attack (a similar  picture can be drawn in the corrupted label case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Class ‘3’  corresponds to the poisoned class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Due to the presence of the  backdoor, the poisoned samples are characterised by a non-null  feature component along the z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Due to the presence  of such a component, the backdoored network classifies those  samples in class ‘3’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' On the contrary, benign samples lie in  the x-y plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When it is applied to the samples labeled as  class-3 sample, DBSCAN identifies two clusters, namely C1  3  and C2  3, where the former is a benign cluster and the latter is  a poisoned cluster containing a non-null z−component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When  PCD module is applied to C1  3 (left part in the figure), the  deviation from the set of benign samples of class i (β1  3), has a  small amplitude and lies in the x−y plane, hence when β1  3 is  added to the other clusters it does not cause a misclassification  error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Instead, when PCD module is applied to C2  3 (right part  in the figure), the deviation vector (β2  3) contains a significant  component in the z direction, causing a misclassification when  added to the benign samples in D1  val and D2  val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  It is worth stressing that the idea behind CCA-UD indirectly  exploits a known behaviour induced by backdoor attacks, that  is, the fact that the presence of the triggering pattern creates a  kind of ’shortcut’ to the target class [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Since this is a general  property of backdoor attacks, common to both corrupted-label  and clean-label attack methods, the proposed method is a  general one and can work under various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  4) Discussion: We observe that the universality of CCA-  UD essentially derives from the generality of the proposed  strategy for PCD and from the use of DBSCAN, that has the  following main strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Firstly, differently from K-means,  DBSCAN can handle unbalanced clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, CCA-UD  also works when the poisoning ratio α is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moreover,  CCA-UD also works when the number of poisoned samples is  larger than the number of benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Secondly, CDA-UC  also works when the class samples have large intra-variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In this scenario, DBSCAN groups the data of a benign class  into multiple clusters (a large Ki, Ki > 2, is estimated by  DBSCAN), that are then detected as benign clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this  setting, methods assuming that there are only two clusters, a  benign cluster and a poisoned one, do not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Finally, we observe that, thanks to the fact that Ki is directly  estimated by DBSCAN in principle, our method can also work  in the presence of multiple triggering patterns [39], [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this  case, the samples poisoned by different triggers would cluster  in separate clusters, that would all be detected as poisoned by  CCA-UD7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' EXPERIMENTAL METHODOLOGY  In this section, we describe the methodology we followed  for the experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Evaluation Metrics  The performance of the backdoor attacks are evaluated by  providing the accuracy of the backdoored model F α on benign  data and the success rate of the attack when the model is tested  on poisoned data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The two metrics are formalized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The Accuracy (ACC) measures the probability of a cor-  rect classification of benign samples, and is calculated as  follows:  ACC =  �l  i=1  �  xj∈Dts,i 1{F α(xj) ≡ i}  |Dts|  ,  (7)  7We do not focus on the case of multiple triggers in our experiments,  leaving this analysis for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  7  ‘1’  C(D1  val)  C(D2  val)  C(D3  val)  C1  3  C2  3  ¯r1  3  ‘1’  ‘2’  C(D1  val)  C(D2  val)  ‘3’  C(D3  val)  C1  3  C2  3  ¯r2  3  f α  1 (xj)  f α  1 (xj)  ‘1’  C(D1  val)  C(D2  val)  C(D3  val)  C1  3  C2  3  ¯r1  3  ‘1’  ‘2’  C(D1  val)  C(D2  val)  ‘3’  C(D3  val)  C1  3  C2  3  ¯r2  3  f α  1 (xj)  f α  1 (xj)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 5: Pictorial and simplified illustration of PCD (clean-label case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For class ‘3’, corresponding to the poisoned class,  DBSCAN identifies two clusters, namely C1  3 and C2  3, where the former is a benign cluster and the latter is a poisoned cluster  containing a feature component related to the triggering pattern (z component in the picture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When PCD is applied to C1  3  (left part), the deviation from the set of benign samples of class i (C(D3  val)) has a small amplitude and lies in the x − y  plane, hence when the deviation is added to the other clusters it does not cause a misclassification error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Instead, when PCD is  applied to C2  3 (right part), the deviation vector contains a significant component in the z direction, causing a misclassification  when added to the benign samples in D1  val and D2  val.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The Attack success rate (ASR), measuring the probability  that the triggering pattern υ activates the desired behaviour  of the backdoored model F α, is computed as follows:  ASR =  �  xj∈Dts/Dts,t 1{F α(P(xj, υ)) ≡ t}  |Dts/Dts,t|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  (8)  where Dts/Dts,t is the test dataset excluding the samples  from class t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In our experiments, a backdoor attack is considered successful  when both ACC and ASR are greater than 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  To measure the performance of the defence algorithms, we  measure the True Positive Rate (TPR) and the False Positive  Rate (FPR) of the defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Actually, when i corresponds to a  benign class, there are no poisoned samples in Dα  tr,i and only  the FPR is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' More formally, let GPi (res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' GBi)  define the set of ground-truth poisoned (res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' benign) samples  in Dα  tr,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We define the TPR and FPR on Dα  tr,i as follows:  TPR(Dα  tr,i) = |Pi ∩ GPi|  |GPi|  , FPR(Dα  tr,i) = 1 − |Bi ∩ GBi|  |GBi|  ,  (9)  Given that benign classes may exist for both poisoned and  benign datasets8, we need to distinguish between these two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hence, we introduce the following definitions:  Benign Class of Benign dataset (BCB): a class of a clean  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In this case α = 0 and Dα  tr,i includes only benign  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Benign Class of Poisoned dataset (BCP ): a benign class of  a poisoned dataset, that is, a class in a poisoned dataset  different from the target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Also in this case, Dα  tr,i  includes only benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The difference between BCB and BCP is that in the former  case F α is a clean model, while in the latter it is backdoored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In the following, we use FPR(BCB) and FPR(BCP ) to  distinguish the FPR in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  8The backdoor attack does not need to target all classes in the input domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Similarly, the case of a target class t of a poisoned dataset is  referred to as a Poisoned Class (PC) of a poisoned dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In  this case, Dα  tr,i=t includes both poisoned and benign samples,  then we compute and report TPR(PC) and FPR(PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  TPR and FPR depend on the choice of the threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Every  choice of the threshold defines a different operating point of  the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In order to get a global view of the performance  of the tested systems, then, we provide the AUC value, defined  as the Area Under the Curve obtained by varying the value of  the threshold and plotting TPR as a function of FPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' AUC  values range in the [0, 1] interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The higher the AUC the  better the capability of the system to distinguish poisoned and  benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When AUC = 1 we have a perfect detector,  while AUC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5 corresponds to a random detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In our  experiments, we report the AUC value score of the PC case  only, because in the BCB and BCP cases the true positive  rate cannot be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  According to the definitions in (9), the false positive and  true positive rates are computed for each cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For sake  of simplicity, we will often report average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For the  case of benign clusters of a benign dataset, the average value,  denoted by FPR(BCB), is calculated by averaging over all  the classes of the benign training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' To compute the  average metrics in the case of BCP and PC, we repeat the  experiments several times by poisoning different target classes  with various poisoning ratios α in the range (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='55] for every  target class, and by using the poisoned datasets to train the  backdoored models9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Then, the average quantity FPR(BCP )  is computed by averaging the performance achieved on non-  target classes of all the poisoned training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For the PC  case, the average metrics FPR(PC), TPR(PC) and AUC  are computed by averaging the values measured on the target  classes of the poisoned training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We also measured the  average performance achieved for a fixed poisoned ratio α, by  varying only the target class t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When we want to stress the  9Only successful backdoor attacks are considered to measure the perfor-  mance in the various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  66JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  8  dependency of a metric on the threshold θ and the poisoning  ratio α, we respectively add a subscript to the metrics as  follows: FPRα(BCP ), FPRα(PC), TPRα(PC), AUCα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The tests run to set the detection threshold θ are carried out  on the validation dataset, consisting only of benign samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Therefore, for each class Di  val, we can only calculate the  FPR(Di  val) value, and its average counterpart denoted by  FPR(Dval) = �  i FPR(Di  val)/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Network tasks and attacks  We considered three different classification tasks, namely  MNIST, traffic sign, and fashion clothes classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  1) MNIST classification: In this set of experiments we  trained a model to classify the digits in the MNIST dataset  [41], which includes n = 10 digits (classes) with 6000 binary  images per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The size of the images is 28 × 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  architecture used for the task is a 4-layer network [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  feature representation of dimensionality 128 is obtained from  the input of the final Fully-connected (FC) layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Regarding the attack setting, three different backdoor attacks  have been considered, as detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For each setting,  the training dataset is poisoned by considering 16 poisoning  ratios α chosen in (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For each α, 10 different poisoned  training datasets are generated by choosing different classes  as the target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Corrupted-label attack, with a 3×3 pixel trigger (abbrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  3×3 corrupted): the backdoor is injected by adding a 3×3  pixel pattern to the corrupted samples, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2,  and modifying the sample labels into that of the target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Corrupted-label attack, with ramp trigger (abbrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' ramp  corrupted): Eve performs a corrupted-label backdoor attack  using a horizontal ramp pattern [12] as trigger (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The ramp pattern is defined as υ(i, j) = j∆/W, 1 ≤ i ≤ H,  1 ≤ j ≤ W, where H × W is the size of the image and  ∆ is a parameter controlling the slope (and strength) of the  ramp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We set ∆ = 40 in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Clean-label attack, with 3×3 pixel trigger (abbrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3×3  clean): the attack utilises the 3×3 pixel trigger pattern to  perform a clean-label attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  2) Traffic signs: For the traffic sign classification task, we  selected 16 different classes from the GTSRB dataset, namely,  the most representative classes in the dataset, including 6  speed-limit, 3 prohibition, 3 danger, and 4 mandatory signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Each class has 1200 colour images with size 28 × 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  model architecture used for training is based on ResNet18  [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The feature representation is extracted from the 17-th  layer, that is, before the FC layer, after an average pooling  layer and ReLu activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' With regard to the attack, we  considered the corrupted-label scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As triggering pattern,  we considered a horizontal sinusoidal pattern, defined as  υ(i, j) = ∆ sin(2πjf/W), 1 ≤ i ≤ H, 1 ≤ j ≤ W, where  H × W is the size of input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The parameters ∆ and f  are used to control the strength and frequency of the trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In our experiment, we set ∆ = 20 and f = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As before, for a  given α, the network is trained on 16 poisoned datasets, each  time considering a different target classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  3) Fashion clothes: Fashion-MNIST dataset includes 10  classes of grey-level cloth images, each class consisting of  6000 images of size 28×28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The model architecture used for  the classification is based on AlexNet [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The representation  used by the backdoor detector is extracted from the 5-th layer,  at the output of the ReLu activation layer before the first FC  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' With regard to the attack, the poisoned samples are  generated by performing the attack in a clean-label setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  A ramp trigger with ∆ = 256 is used to implement the  attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Once again, for each choice of α, the backdoor attack  is repeated 10 times, each time considering a different target  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For all the classification tasks, the benign validation dataset  Dval is obtained by randomly selecting 100 samples from all  the classes in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Setting of defence parameters  To implement the CCA-UD defence, we have to set the  following parameters: the reduced dimension d′ for the clus-  tering, the parameters of the DBSCAN algorithm, namely  minPts and ϵ, and finally the threshold θ used by the  clustering poisoning detection module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In our experiments, we  set d′ = 2, minPts = 20 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This is the setting that,  according to our experiments, achieves the best performance  with the minimum complexity for the clustering algorithm  (being d′ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The effect of these parameters on the result of  clustering and the detection performance is evaluated by the  ablation study described in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  With regard to θ, as mentioned before, AC, CI and CCA-  UD involve the setting of a threshold for poisoning detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For a fair comparison, we set the threshold in the same way  for all the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In particular, we set θ by fixing the false  positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In general a value of θ results in different FPR  rates for different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' To avoid setting a different threshold  for each class, then, we fixed it by setting the average FPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In fact, setting the average FPR exactly may not be feasible,  so we chose the threshold in such a way to minimize the  distance from the target rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Formally, by setting the target  false positive rate to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05, the threshold θ∗ is determined as:  θ∗ = arg min  θ  ��0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05 − FPR(Dval)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  (10)  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS  In this section we report the results of the experiments we  have carried out to evaluate the effectiveness of CCA-UD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ablation study  We start the experimental analysis with an ablation study  investigating the effect of the three main hyperparameters of  CCA-UD, namely d′ (regarding UMAP), and minPts and ϵ  (for DBSCAN) on the effectiveness of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Based on  this analysis, in all subsequent experiments we set d′ = 2,  minPts = 20 and ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The influence of each parameter on the clustering result  and the detection performance can be assessed by looking at  Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The results refer to the case of MNIST classification,  with backdoor poisoning performed by using a 3×3 pixel  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  9  TABLE I: Ablation study on the three hyperparameters of CCA-UD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' FPR and TPR for all cases are computed by letting  θ = θ∗ as stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' K and ζ are, respectively, the average number of clusters and the average fraction of outliers  identified by DBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Hyperparameters  BCB results  BCP results  PC results  d′  minP ts  ϵ  (K, ζ)  F P R(BCB)  (K, ζ)  F P R(BCP )  (K, ζ)  T P R(P C)  F P R(P C)  AUC  S1  2  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='005)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='008)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='073  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='003)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  S2  4  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='097)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='044  (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='060)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='027  (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='012)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='432  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='989  S3  8  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='142)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='066  (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='076)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='037  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='012)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='448  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='990  S4  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='153)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='057  (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='085)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='013)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='501  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='987  S5  2  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  S6  2  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='002)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  S7  2  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  S8  2  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  S9  2  20  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='500  S10  10  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='004)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='002)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='068  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='046  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  S11  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='020)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='052  (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='012)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='077  (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='004)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  S12  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='883)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  (60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='732)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='045  (111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='399)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='031  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='612  S13  10  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='008)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='046  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='004)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='042  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='042  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  S14  10  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='049  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  S15  10  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  S16  10  20  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='500  trigger pattern with label corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Similar considerations  can be drawn in the other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The results in the table have  been obtained by letting θ = θ⋆ as stated in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' To start  with, we observe that when utilising θ∗ in BCB and BCP  cases, the FPR values is close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05 for all the settings,  while in the PC case FPR is close to or less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05 for  all settings except for S9 and S16, whes benign and poisoned  samples collapse into a single cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In addition to TPR and  FPR, the table shows the average number of clusters (K) and  the average outlier ratio (ζ) identified by DBSCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  From the first group of rows (S1-S4), we see that for a  given setting of minPts and ϵ, increasing d′ leads to a larger  average number of clusters and a larger fraction of outliers,  as the DBSCAN algorithm results in a higher number of  densely-connected regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' A similar behaviour is observed  by increasing minPts or decreasing ϵ for a given d′ (second  and third group of rows in the table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Expectedly, when ϵ  is too large, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 10, DBSCAN always results in one cluster  thus failing to identify the poisoned samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Based on the  result in Table I, the settings S7 (d′ = 2, minPts = 20,  ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8) and S15 (d′ = 10, minPts = 20, ϵ = 3) yield  the best performance, the former having lower computational  complexity, because of the lower dimension used to cluster  the samples in the feature space (d′ = 2 instead of 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Threshold setting  The thresholds θ∗ obtained following the approach detailed  in Section V-C for AC and CI and CCA-UD, are reported in  Table II for the three different classification tasks considered  in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Given that the threshold is set by relying  on the validation dataset, it is necessary to verify that the target  false positive rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05 in our case) is also obtained on the  test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' An excerpt of such results is shown in Table IV  by referring to MNIST task (a similar behaviour is observed  for the other classification tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Our experiments reveal that, for AC and CI, the threshold  determined via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' (10) does not lead to a good operating  point when used on the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In particular, while for  CCA-UD, the threshold θ∗ set on the validation dataset yields  a similar FPR (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05) in the BCB, BCP and PC  TABLE II: Values of θ∗ obtained for the various classification  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Method  MNIST  Traffic signs  Fashion clothes  AC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='335  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='404  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='301  CI  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='018  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='673  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='738  CCA-UD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='950  cases, this is not true for AC and CI, for which FPR(BCB),  FPR(BCP ) and FPR(PC) are often smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05,  reaching 0 in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This leads to a poor TPR(PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In  particular, with AC, when α > θ∗, both clusters are classified  as benign, and then TPRα(PC) = FPRα(PC) = 0, even  when the method would, in principle, be able to provide a  perfect discrimination (AUCα ≈ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The difficulty in setting  the threshold for AC and CI is also evident from the plots in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 6, that report the FPR and TPR values averaged also  on α, for different values of the threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' From these plots,  we immediately see that a threshold that works in all the cases  can never be found for AC and CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Due to the difficulties encountered to set the detection  threshold for AC and CI10, the results at θ∗ for these methods  are not reported in the other cases, that is, for traffic sign  and fashion clothes classification, for which we report only  the AUCα scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Note that the possibility to set a unique  threshold on a benign dataset that also works on poisoned  datasets is very important for the practical applicability of a  defence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Based on our results, CCA-UD has this remarkable  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Results on MNIST  In this section, we evaluate the performance of CCA-UD  against the three types of backdoor attacks, namely, 3×3  corrupted, ramp corrupted, and 3×3 clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Such performance  as compared to those obtained by AC and CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 6, in each  row, the three figures report the average performance of AC,  CI and CCA-UD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The values of FPR(BCB), FPR(BCP ),  TPR(PC) and FPR(PC) are reported for each method,  as a function of the detection threshold θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The behaviour of  10Note that the problem of threshold setting is not addressed in the original  papers, since different threshold are used in the various cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  10  TABLE III: AUC scores of three methods in the three different  attacks  Method  3×3 corrupted  Ramp corrupted  3×3 clean  AC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='728  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='733  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='785  CI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='964  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='178  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='488  CCA-UD  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='981  FPR(Dval), which is utilised to determine the threshold θ∗  (at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05 of FPR(Dval)), is also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The position of θ∗  is indicated by a vertical dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  By observing the figure, we see that CCA-UD outperforms  by far the other two methods in all the settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the first  setting, we achieve TPR(PC) and FPR(PC) equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='983  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051 at the optimal threshold θ∗, with FPR(BCB) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051 and FPR(BCP ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Instead, the performance  achieved by AC and CI at their optimal threshold are very  poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Similar results are achieved for the second and third  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In particular, for the second attack, CCA-UD achieves  TPR(PC) and FPR(PC) equal to ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='975, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050) at θ∗, and  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='966, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050) for the third one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For a poisoned dataset, the AUC values obtained in the  three settings are provided in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' From these results,  we argue that CI has good discriminating capability (with  an AUC only slightly lower than CCA-UD) against the first  attack, but fails to defend against the other two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' This is an  expected behaviour since CI does not work when the triggering  pattern is robust against average filtering, as it is the case of  the ramp signal considered in the second attack, or with clean-  label attacks, as it is the last setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Table IV shows the results obtained for different values of  the poisoning ratio α for the three different attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The values  of FPR and TPR have been obtained by letting θ = θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  For the clean-label case, due to the difficulty of developing  a successful attack [12], [27], [28], the backdoor can be  successfully injected in the model only when α is large enough  and, in any case, a successful attack could not always be  obtained in the 10 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' For this reason, in the third  table, we report the number of successfully attacked classes  (cnt) with different poisoning ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Upon inspection of Table  IV, we observe that:  With regard to AC, the behaviour is similar under the three  attack scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Good results are achieved for intermediate  values of α, namely in the [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3] range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' When α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134,  instead, AUCα of AC is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='786, and close  to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5 for small α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In particular, AC cannot handle the  backdoor attacks for which the poisoning ratio is smaller  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moreover, when α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5, AUCα goes to zero,  as benign samples are judged as poisoned and vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Finally, by comparing the AUCα values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' IVa and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IVc, we see that AC achieves better performance against the  corrupted-label attack than in the clean-label case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  With regard to CI, the detection performance achieved in  the first attack scenario (3×3 corrupted) are good for all  the values of α, with AUCα larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='96 in most  of the cases (with the exception of the smallest α, for  which AUCα = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='876), showing that CI can effectively  defend against the backdoor attack in this setting, for every  attack poisoning ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' However, as expected, CI fails in the  other settings, with AUCα lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5 in all the cases,  confirming the limitations mentioned in Section III-A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Regarding CCA-UD, good results are achieved in all the-  cases and for every value of α, with a perfect or nearly  perfect AUCαin most of the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moreover, by letting  θ = θ∗, a very good TPRα(PC) is obtained, larger  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='95 in almost all the cases, with FPRα(BCP ) and  FPRα(PC) around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Overall, the tables prove the  universality of CCA-UD that works very well regardless of  the specific attack setting and regardless of the value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Note, since CCA-UD achieves a larger AUCα than AC and  CI, CCA-UD outperforms AC and CI not only when θ = θ∗  but also when θ is set adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Finally, these results show that CCA-UD can effectively  defend against both corrupted and clean-label attacks, thus  confirming that the strategy used to detect poisoned clusters  exploits a general misclassification behaviour present in both  corrupted- and clean-label attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Results on Traffic Signs  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7a-7c show the average performance of AC, CI, and  CCA-UD on the traffic signs task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Similar considerations  to the MNIST case can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CCA-UD achieves very  good average performance at the operating point given by θ∗,  where TPR(PC) and FPR(PC) are ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='965, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='058) (with  FPR(BCB) = FPR(BCB) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='08), while for AC and CI  a threshold that works well on the average can not be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  In the case of a poisoned dataset, the average AUC of the  detection AUC is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='897, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='958, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='993 for AC, CI,  and CCA-UD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  We observe that CI gets a good AUC, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In fact, in  this case, given that the size of the input image is 28×28,  the triggering pattern, namely the sinusoidal signal can be  effectively removed by a 5 × 5 average filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The results obtained for various α are reported in Table Va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  As it can be seen, CCA-UD gets very good performance in  terms of TPRα(PC) and FPRα(PC) measured at θ = θ∗  in all the cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The AUCα is also larger than that achieved  by AC and CI for all values of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' As observed before, while  CI is relatively insensitive to α, the performance of AC drop  when α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1 or α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Results on Fashion Clothes  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7d-7f report the results obtained by AC, CI, and CCA-  UD on the fashion clothes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Once again, the performance  achieved by CCA-UD are largely superior to those achieved by  AC and CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In particular, by looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7d-7f, CCA-UD  achieves TPR(PC) and FPR(PC) equal to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='053),  with FPR(BCB) = FPR(BCP ) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Regarding the  AUC scores, AUC of AC, CI, and CCA-UD are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='900, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='106,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='997 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Since the attack is carried out in a clean-  label modality, the poor performance of CI were expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  results for various α, reported in Table Vb, confirm the same  behaviour, with CCA-UD getting very good performance in  all the cases, always overcoming the other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  11  (a) AC in 3×3 corrupted  (b) CI in 3×3 corrupted  (c) CCA-UD in 3×3 corrupted  (d) AC in ramp corrupted  (e) CI in ramp corrupted  (f) CCA-UD in ramp corrupted  (g) AC in 3×3 clean  (h) CI in 3×3 clean  (i) CCA-UD in 3×3 clean  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 6: Average performance of AC and CI, and CCA-UD for different values of the threshold against the three types of  backdoor attacks implemented in the case of MNIST classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' From top to bottom the plots refer to 3×3 corrupted in  (a)-(c), ramp corrupted in (d)-(f), and 3×3 clean in (g)-(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' From left to right we report the performance of AC, CI and  CCA-UD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The position of θ∗ is indicated by a vertical dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  TABLE IV: Performance of AC, CI and CCA-UD for various poisoning ratios α, against the three types of backdoor attacks  for MNIST classification, The FPR and TPR values are computed at θ = θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' In the 3 × 3 table cnt indicates the number of  successful attacks in 10 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  AC  CI  CCA-UD  α  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='563  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='324  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='876  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='908  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='949  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='099  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='628  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='581  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='977  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='989  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='395  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='757  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='654  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='792  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='958  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='559  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='997  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='577  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='993  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='997  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='014  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='540  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='961  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='359  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='571  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='964  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='829  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='953  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  (a) 3×3 corrupted  AC  CI  CCA-UD  α  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='593  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='407  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='871  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='966  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='090  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='593  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='119  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='914  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='786  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='216  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='989  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='798  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='962  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='186  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='992  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='995  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='179  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='088  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='359  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='144  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='135  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  (b) Ramp corrupted  AC  CI  CCA-UD  α  cnt  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  AUCα  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='441  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='683  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='835  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='438  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='642  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='809  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='069  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='533  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='296  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='952  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='972  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='096  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='528  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='333  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='333  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='595  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='951  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='972  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='610  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='667  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='539  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='975  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='987  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='186  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='384  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='746  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='471  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='982  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='991  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='258  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='929  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='959  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='601  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='644  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='516  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='359  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='315  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='975  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='206  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='437  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='993  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='450  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='969  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='009  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='729  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='786  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='554  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='997  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  (c) 3×3 clean  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  FPR(BCB)  g  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  FPR(BCB)  g  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  pel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  1  2  3  5  4  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  age  FPR(BCB)  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  age  FPR(BCB)  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  ge  FPR(BCB)  centa  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(D)  pel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0  2  3  4  5  L  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  ercentage  FPR(BCB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  FPR(BCB)  g  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  FPR(BCB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  2  3  L  4  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  ge  FPR(BCB)  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  12  (a) AC in traffic signs task  (b) CI in traffic signs task  (c) CCA-UD in traffic signs task  (d) AC in fashion clothes task  (e) CI in fashion clothes task  (f) CCA-UD in fashion clothes task  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7: Average performance of AC, CI, and CCA-UD for different values of θ for the traffic signs and fashion clothes task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The vertical dotted line indicates the position of θ∗ for the various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  TABLE V: Performance of AC, CI, and CCA-UD for various  poisoning ratios for the traffic sign and fashion cloth task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  FPR and TPR values are computed at θ = θ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Since for AC  and CI it is not possible to find a unique value of θ working  in all conditions, we report only the AUC values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  AC  CI  CCA-UD  α  cnt  AUCα  AUCα  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='793  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='923  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='983  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='946  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='061  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='096  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='928  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='991  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='058  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='059  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='949  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='959  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='992  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='186  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='958  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='993  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='064  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='359  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='946  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='086  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='985  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='054  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='450  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='917  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='994  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='980  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='550  15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='869  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='059  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='051  (a) Traffic signs  AC  CI  CCA-UD  α  cnt  AUCα  AUCα  AUCα  F P Rα(BCP )  T P Rα(P C)  F P Rα(P C)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='069  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='618  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='053  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='096  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='513  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='341  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='995  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='054  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='056  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='134  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='940  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='087  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='059  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='186  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='037  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='054  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='258  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='083  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='055  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='057  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='359  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='998  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='056  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='052  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='450  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='174  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='055  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='050  (b) Fashion clothes  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CONCLUDING REMARKS  We have proposed a universal backdoor detection method,  called CCA-UD, aiming at revealing the possible presence of a  backdoor inside a model and identify the poisoned samples by  analysing the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CCA-UD relies on DBSCAN  clustering and on a new strategy for the detection of poisoned  clusters based on the computation of clusters’ centroids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The  capability of the centroids’ features to cause a misclassification  of benign samples is exploited to decide whether a cluster is  poisoned or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' We evaluated the effectiveness of CCA-UD  on a wide variety of classification tasks and attack scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  The results confirm that the method can work regardless of the  corruption strategy (corrupted and clean label setting) and the  type of trigger used by the attacker (local or global pattern).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Moreover, the method is effective regardless of the poisoning  ratio used by the attacker, that can be either very small or even  larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Furthermore, we proved that the performance  achieved by CCA-UD are always superior to those achieved  by the existing methods, also when these methods are applied  in a scenario that meets their operating requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Future work will be devoted to the analysis of the behaviour  of the proposed method against multiple triggers attacks, that  is when multiple triggers are used to poison the samples,  possibly to induce more than one malicious behaviour inside  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' The capability of the method to defend against  backdoor attacks in application scenarios beyond image clas-  sification, is also worth investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  REFERENCES  [1] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goodfellow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Shlens, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Szegedy, “Explaining and harnessing  adversarial examples,” in 3rd International Conference on Learning  Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015,  Conference Track Proceedings, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' LeCun, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kurakin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goodfellow, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio, “Adversarial examples in  the physical world,” in 5th International Conference on Learning Rep-  resentations, ICLR 2017, Toulon, France, April 24-26, 2017, Workshop  Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  OpenReview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='net, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kurakin, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goodfellow, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio, “Adversarial machine learning  at scale,” arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='01236, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Biggio, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Nelson, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Laskov, “Poisoning attacks against support  vector machines,” in Proceedings of the 29th International Conference  on Machine Learning, ICML 2012, Edinburgh, Scotland, UK, June 26  July 1, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  icml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='cc / Omnipress, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Weerasinghe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Alpcan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Erfani, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Leckie, “Defending  support vector machines against data poisoning attacks,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Forensics Secur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2566–2578, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [6] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni, “A master key backdoor for univer-  sal impersonation attack against dnn-based face verification,” Pattern  Recognit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 144, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 61–67, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Lu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Song, “Targeted backdoor  attacks on deep learning systems using data poisoning,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  abs/1712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='05526, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni, “A temporal chrominance trigger for  clean-label backdoor attack against anti-spoof rebroadcast detection,”  CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='01102, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Gu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Dolan-Gavitt, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Garg, “Badnets: Identifying vulner-  abilities in the machine learning model supply chain,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  abs/1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='06733, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni, “An overview of backdoor attacks  against deep neural networks and possible defences,” IEEE Open Journal  of Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 261–287, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Turner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tsipras, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Madry, “Label-consistent backdoor at-  tacks,” arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='02771, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kallas, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, “A new backdoor attack in CNNS  by training set corruption without label poisoning,” in 2019 IEEE  International Conference on Image Processing, ICIP 2019, Taipei,  Taiwan, September 22-25, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 101–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  FPR(BCB)  g  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  pel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  FPR(BCB)  g  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dv  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0  1  2  3  4  5  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  e  rcentage  FPR(BCB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6-  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  age  FPR(BCB)  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  rcentage  FPR(BCB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0 -  0  2  3  1  4  5  01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  TPR(PC)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  FPR(PC)  age  FPR(BCB)  rcenta  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  FPR(BCp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  FPR(Dval)  per  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='0  0JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8, AUGUST 2021  13  [13] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Carvalho, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Baracaldo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ludwig, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Edwards, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Lee,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Molloy, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Srivastava, “Detecting backdoor attacks on deep  neural networks by activation clustering,” in Workshop on Artificial  Intelligence Safety 2019 co-located with the Thirty-Third AAAI Con-  ference on Artificial Intelligence 2019 (AAAI-19), Honolulu, Hawaii,  January 27, 2019, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' CEUR Workshop Proceedings, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Espinoza, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' ´O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  h´Eigeartaigh, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hern´andez-Orallo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Castillo-Effen,  Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='org, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [14] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Yadav and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sharma, “A review of k-mean algorithm,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Trends Technol, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2972–2976, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [15] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Xiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Miller, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kesidis, “A benchmark study of backdoor  data poisoning defenses for deep neural network classifiers and A novel  defense,” in 29th IEEE International Workshop on Machine Learning  for Signal Processing, MLSP 2019, Pittsburgh, PA, USA, October 13-  16, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bond, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hoeffler, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Temple, “Gmm estimation of empirical  growth models,” Available at SSRN 290522, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Neath and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Cavanaugh, “The bayesian information crite-  rion: background, derivation, and applications,” Wiley Interdisciplinary  Reviews: Computational Statistics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 199–203, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Li, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Madry, “Spectral signatures in backdoor attacks,”  in Advances in Neural Information Processing Systems 31: Annual  Conference on Neural Information Processing Systems 2018, NeurIPS  2018, December 3-8, 2018, Montr´eal, Canada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Larochelle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Grauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Cesa-Bianchi, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Garnett,  Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 8011–8021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Hayase,  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Kong,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Somani,  and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Oh,  “SPECTRE:  defending  against  backdoor  attacks  using  robust  statistics,”  CoRR,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='11315,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Available:  https:  //arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='org/abs/2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='11315  [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Abdi, “Singular value decomposition (svd) and generalized singular  value decomposition,” Encyclopedia of measurement and statistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  907–912, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [21] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Shan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bhagoji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zheng, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhao, “Poison forensics:  Traceback of data poisoning attacks in neural networks,” in 31st USENIX  Security Symposium, USENIX Security 2022, Boston, MA, USA, August  10-12, 2022, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Butler and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Thomas, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  USENIX Associa-  tion, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3575–3592.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [22] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Peri, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Gupta, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Huang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Fowl, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Feizi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goldstein,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Dickerson, “Deep k-nn defense against clean-label data  poisoning attacks,” in Computer Vision - ECCV 2020 Workshops -  Glasgow, UK, August 23-28, 2020, Proceedings, Part I, ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Lecture  Notes in Computer Science, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bartoli and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Fusiello, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 12535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Springer, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 55–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [23] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Shafahi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Najibi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Suciu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Studer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Dumitras,  and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goldstein, “Poison frogs!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' targeted clean-label poisoning attacks  on neural networks,” in Advances in Neural Information Processing  Systems 31: Annual Conference on Neural Information Processing  Systems 2018, NeurIPS 2018, December 3-8, 2018, Montr´eal, Canada,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Larochelle, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Grauman, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Cesa-Bianchi,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Garnett, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 6106–6116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tang, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhang, “Demon in the variant:  Statistical analysis of dnns for robust backdoor contamination detection,”  in 30th USENIX Security Symposium, USENIX Security 2021, August  11-13, 2021, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bailey and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Greenstadt, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  USENIX Association,  2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1541–1558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [25] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moon, “The expectation-maximization algorithm,” IEEE Signal  processing magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 47–60, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [26] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ester, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kriegel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sander, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Xu, “A density-based algorithm  for discovering clusters in large spatial databases with noise,” in Pro-  ceedings of the Second International Conference on Knowledge Discov-  ery and Data Mining (KDD-96), Portland, Oregon, USA, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Simoudis,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Han, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Fayyad, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  AAAI Press, 1996, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 226–231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ma, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bailey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Chen, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Jiang, “Clean-  label backdoor attacks on video recognition models,” in Proceedings of  the IEEE/CVF Conference on Computer Vision and Pattern Recognition,  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 14 443–14 452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [28] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Guo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Tondi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Barni, “A temporal chrominance trigger for  clean-label backdoor attack against anti-spoof rebroadcast detection,”  arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='01102, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [29] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' LeCun, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Bengio, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Hinton, “Deep learning,” nature, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 521,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 7553, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 436–444, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wold, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Esbensen, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Geladi, “Principal component analysis,”  Chemometrics and intelligent laboratory systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1-3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  37–52, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Gupta and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Shekokar, “A novel k-means l-layer algorithm for un-  even clustering in wsn,” in 2017 International Conference on Computer,  Communication and Signal Processing (ICCCSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [32] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Goldberger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Gordon, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Greenspan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=', “An efficient image  similarity measure based on approximations of kl-divergence between  two gaussian mixtures.” in ICCV, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 3, 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 487–493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [33] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Rokach and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Maimon, “Clustering methods,” in Data mining and  knowledge discovery handbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Springer, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 321–352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [34] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ankerst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Breunig, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Kriegel, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sander, “Optics:  Ordering points to identify the clustering structure,” ACM Sigmod  record, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 28, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 49–60, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Campello, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Moulavi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zimek, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sander, “Hierarchical  density estimates for data clustering, visualization, and outlier detection,”  ACM Transactions on Knowledge Discovery from Data (TKDD), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 10,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1–51, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [36] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' McInnes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Healy, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Melville, “Umap: Uniform manifold  approximation and projection for dimension reduction,” arXiv preprint  arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='03426, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [37] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' K¨oppen, “The curse of dimensionality,” in 5th online world confer-  ence on soft computing in industrial applications (WSC5), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 1, 2000,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 4–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [38] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Yao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Shan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Li, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Viswanath, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zheng, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  Zhao, “Neural cleanse: Identifying and mitigating backdoor attacks in  neural networks,” in 2019 IEEE Symposium on Security and Privacy,  SP 2019, San Francisco, CA, USA, May 19-23, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  707–723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [39] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Salem, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Backes, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ma, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhang, “Dynamic backdoor  attacks against machine learning models,” in 2022 IEEE 7th European  Symposium on Security and Privacy (EuroS&P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' IEEE, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 703–  718.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Xue, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' He, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Wang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Liu, “One-to-n & n-to-one: Two  advanced backdoor attacks against deep learning models,” IEEE Trans-  actions on Dependable and Secure Computing, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [41] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' LeCun and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Cortes, “MNIST handwritten digit database,” 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Available: http://yann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='lecun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='com/exdb/mnist/  [42] Pytorch, “4-layer dnn model,” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='com/pytorch/examples/blob/  main/mnist/main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='py#L11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [43] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Sun, “Deep residual learning for image  recognition,” in Proceedings of the IEEE conference on computer vision  and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='  [44] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content=' Krizhevsky, “One weird trick for parallelizing convolutional neural  networks,” arXiv preprint arXiv:1404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
+page_content='5997, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE3T4oBgHgl3EQffgrq/content/2301.04554v1.pdf'}
diff --git a/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/2301.01844v1.pdf.txt b/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/2301.01844v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..187eb74e8d68f207c0c74310d7d3ba2874134b76
--- /dev/null
+++ b/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/2301.01844v1.pdf.txt
@@ -0,0 +1,2136 @@
+Solving Unsplittable Network Flow Problems with
+Decision Diagrams
+Hosseinali Salemi, Danial Davarnia
+Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, IA 50011,
+hsalemi@iastate.edu, davarnia@iastate.edu
+In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the
+incoming arc flows cannot be split or merged when routed through outgoing arcs. This so-called no-split
+no-merge requirement arises in unit train scheduling where train consists should remain intact at stations
+that lack necessary equipment and manpower to attach/detach them. Solving the unsplittable network
+flow problems with standard mixed-integer programming formulations is computationally difficult due to
+the large number of binary variables needed to determine matching pairs between incoming and outgoing
+arcs of nodes with no-split no-merge constraint. In this paper, we study a stochastic variant of the unit
+train scheduling problem where the demand is uncertain. We develop a novel decision diagram (DD)-based
+framework that decomposes the underlying two-stage formulation into a master problem that contains the
+combinatorial requirements, and a subproblem that models a continuous network flow problem. The master
+problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a
+substantial improvement in solution time. Similarly to the Benders decomposition technique, the subproblems
+output cutting planes that are used to refine the master DD. Computational experiments show a significant
+improvement in solution time of the DD framework compared with that of standard methods.
+Key words : Decision Diagrams; Network Optimization; Mixed Integer Programs; Unit Trains;
+Transportation
+History :
+1.
+Introduction
+Over the past several decades, rail freight transportation has continued to grow as the prime
+means of transportation for high-volume commodities. Advantages of rail transportation include
+reliability, safety, cost-efficiency and environmental-sustainability as compared with alternative
+methods of transportation. In terms of scale, the rail network accounted for 27.2 percent of U.S.
+freight shipment by ton-miles in 2018 (Furchtgott-Roth et al. 2021); see Figure 1. The Federal
+Highway Administration estimates that the total U.S. freight shipments will be 24.1 billion tons
+in 2040, a 30 percent increase from the 2018 total transportation of 18.6 billion tons. With the
+purpose of meeting such market growth, America’s freight railway companies have invested nearly
+1
+arXiv:2301.01844v1  [math.OC]  4 Jan 2023
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+2
+$740 billion on capital expenditures and maintenance from 1980 to 2020 (Association of American
+Railroads 2021).
+Figure 1
+Pie chart for ton-miles of freight shipments by mode within the U.S. in 2018. Multiple modes includes
+mail. Air and truck-air with the share of 0.1% are omitted.
+To reduce rail freight transportation costs and shipment delays, railroad companies offer unit
+train services for carrying high-volume products. Unit trains haul a single type freight in a way that
+no car is attached or detached while the cargo train is on its way from an origin to a destination,
+except in specific locations that are equipped with required manpower and machinery. These trains
+usually operate all day, use dedicated equipment, and can be loaded/unloaded in 24 hours. They
+are known to be one of the fastest and most efficient means of railroad transportation. (Association
+of American Railroads 2021). Traditionally, unit trains are used to carry bulk cargo such as coal,
+grain, cement, and rock. Bulk liquids like crude oil and food such as wheat and corn are also
+shipped by unit trains. According to the Federal Railroad Administration data, bulk commodities
+account for 91 percent of the U.S. railroad freights. Approximately all coal shipped through railways
+in the U.S. are transported by unit trains. Moreover, these trains contribute significantly to the
+shipping process of crude oil as each unit train is capable of carrying 85,000 barrels (Association
+of American Railroads 2021). In an operational level, the core unit train model can be described
+as follows. Given a set of supply, intermediate, and demand locations in a railroad network, the
+unit train scheduling problem seeks to find optimal routes for unit trains to send flows from supply
+to demand points with the objective of minimizing the total transportation cost while meeting
+demand of customers, respecting capacities of tracks, and satisfying no-car attaching/detaching
+requirements in specific locations. As a result, designing blocking plans to determine locations
+
+Multiple modes
+8%
+Pipeline, 19%
+Truck, 39%
+Water, 7%
+Rail, 27%Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+3
+where cars need to be switched between trains is irrelevant in this problem, unlike scheduling other
+types of trains (Davarnia et al. 2019).
+Despite the significance of unit train scheduling, exact optimization approaches to solve associ-
+ated problems are scarce, partially due to their structural complexities. One of the main challenges
+in modeling unit trains is the requirement that the train consists must remain intact when passing
+through stations that lack necessary busting/formation equipment. In optimization, this require-
+ment is referred to as no-split no-merge (NSNM), which guarantees that the flows entering to or
+exiting from certain nodes of the unit train network cannot be split or merged. Incorporating this
+requirement into typical transportation network models yields the so-called generalized unsplittable
+flow problem (GUFP), where the objective is to determine the minimum-cost unit train sched-
+ules that satisfy the given demand. Numerous studies have shown that considering deterministic
+demands might result in the complete failure of the transportation scheduling (Demir et al. 2016,
+Layeb et al. 2018), motivating the study of stochastic variants of the unit train scheduling problems
+where the demand is uncertain. As a result, in this paper, we consider a stochastic variant of the
+GUFP, referred to SGUFP, that is modeled as a two-stage optimization problem. The first stage
+decides a matching between the incoming and outgoing arcs of the nodes of the railroad network,
+and the second stage determines the amount of flow that should be sent through the matching arcs
+of the network to satisfy the uncertain demand represented by a number of demand scenarios. We
+propose a novel exact solution framework to solve this problem in the operational level.
+Our proposed methodology is based on decision diagrams (DDs), which are compact graphical
+data structures. DDs were initially introduced to represent boolean functions with applications in
+circuit design. Over the past decade, researchers have successfully extended DDs domain by devel-
+oping DD-based algorithms to solve discrete optimization problems in different areas of application.
+Because of its structural limitation to model integer programs only, DDs have never been used
+to solve transportation problems that inherently include continuous variables. In this paper, we
+extend the application scope of DDs by introducing a novel framework that is capable of modeling
+network problems with both integer and continuous components as in the SGUFP.
+1.1.
+Literature Review on Train Scheduling
+Many variants of train routing and scheduling problems with different objective functions and
+set of constraints under deterministic and stochastic conditions have been introduced and vastly
+studied in the literature; see surveys by Cordeau, Toth, and Vigo (1998), Harrod and Gorman
+(2010), Lusby et al. (2011), Cacchiani and Toth (2012), and Turner et al. (2016) for different
+problems classifications and structures. Mixed integer linear and nonlinear programming formu-
+lations are among the most frequent exact approaches to model different classes of these prob-
+lems (Jovanovi´c and Harker 1991, Huntley et al. 1995, Sherali and Suharko 1998, Lawley et al.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+4
+2008, Haahr and Lusby 2017, Davarnia et al. 2019). Proposed solution techniques include but are
+not limited to branch-and-bound methods (Jovanovi´c and Harker 1991, Fuchsberger and L¨uthi
+2007), branch-and-cut frameworks (Zwaneveld, Kroon, and Van Hoesel 2001, Ceselli et al. 2008),
+branch-and-price approaches (Lusby 2008, Lin and Kwan 2016), graph coloring algorithms (Cor-
+nelsen and Di Stefano 2007), and heuristics (Carey and Crawford 2007, Liu and Kozan 2011, I¸cy¨uz
+et al. 2016). Rolling stock scheduling (Abbink et al. 2004, Alfieri et al. 2006, Haahr et al. 2016,
+Bornd¨orfer et al. 2016) that assigns rolling stocks to a given timetable, and crew scheduling (Kwan
+2011, Shen et al. 2013, Heil, Hoffmann, and Buscher 2020) that covers train activities by assigning
+crews to the associated operations are other major problems arising in the area of railroad planning.
+Due to the inherent uncertainty in different types of train scheduling and routing problems, many
+researchers have studied stochastic variants of the problems where the supply/demand is considered
+to be uncertain. Jordan and Turnquist (1983) propose a model for railroad car distribution where
+supply and demand of cars are uncertain. Jin et al. (2019) study a chance-constrained programming
+model for the train stop planning problem under stochastic demand. Ying, Chow, and Chin (2020)
+propose a deep reinforcement learning approach for train scheduling where the passenger demand
+is uncertain. Recently, Gong et al. (2021) propose a stochastic optimization method to solve a train
+timetabling problem with uncertain passenger demand. Also see works by Meng and Zhou (2011),
+Quaglietta, Corman, and Goverde (2013), Larsen et al. (2014) that consider train dispatching
+problems under stochastic environments.
+In the context of unit train scheduling, Lawley et al. (2008) study a time-space network flow
+model to schedule bulk railroad deliveries for unit trains. In their model, the authors consider char-
+acteristics of underlying rail network, demands of customers, and capacities of tracks, stations, and
+loading/unloading requirements. They propose a mixed integer programming (MIP) formulation
+that maximizes the demand satisfaction while minimizing the waiting time at stations. Lin and
+Kwan (2014) (cf. Lin and Kwan (2016)) propose a model for a train scheduling problem that is capa-
+ble to capture locations where coupling/decoupling is forbidden. They develop a branch-and-price
+algorithm inspired by column generation to solve the associated problem. Lin and Kwan (2018)
+also propose a heuristic branch-and-bound approach to decrease coupling/decoupling redundancy.
+I¸cy¨uz et al. (2016) study the problem of planning coal unit trains that includes train formation,
+routing, and scheduling. As noted by the authors, their proposed MIP formulation fails to solve
+the problem directly due to its large size. As a remedy, they develop a time-efficient heuristic that
+produces good quality solutions. More recently, Davarnia et al. (2019) introduce and study the
+GUFP with application to unit train scheduling. In particular, the authors show how to impose
+NSNM restrictions in network optimization problems. They present a polyhedral study and pro-
+pose a MIP formulation to model a stylized variant of the unit train scheduling problem. In the
+present paper, we use their formulation (see section 3.1) as a basis for our solution framework.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+5
+The unsplittable flow problem (UFP) was first introduced by Kleinberg (1996) as a generalization
+of the disjoint path problem. Given a network with capacities for arcs and a set of source-terminal
+vertex pairs with associated demands and rewards, the objective in the UFP is to maximize the
+total revenue by selecting a subset of source-terminal pairs and routing flows through a single
+path for each of them to satisfy the associated demand. In the GUFP, however, there can exist
+nodes that do not need to respect the NSNM requirement, and demands can be satisfied by
+passing flows through multiple paths. It is well-known that different variants of UFP are NP-
+hard (Baier, K¨ohler, and Skutella 2005, Kolman and Scheideler 2006, Chakrabarti et al. 2007). Since
+its introduction, the UFP structure has been used in different areas of application, from bandwidth
+allocation in heterogeneous networks (Kolman and Scheideler 2006), to survivable connection-
+oriented networks (Walkowiak 2006), and virtual circuit routing problems (Hu, Lan, and Wan
+2009). Considering the hardness of the problem, approximation algorithms have been a common
+technique to tackle different variants of the UFP in the literature (Baier, K¨ohler, and Skutella
+2005, Chakrabarti et al. 2007).
+1.2.
+Literature Review on Decision Diagrams
+DDs are directed acyclic graphs with a source and a terminal node where each source-terminal path
+encodes a feasible solution to an optimization problem. In DDs, each layer from the source to the
+terminal represents a decision variable where labels of arcs show their values. Had˘zi´c and Hooker
+(2006) proposed to use DDs to model the feasible region of a discrete optimization problem and used
+it for postoptimality analysis. Later, Andersen et al. (2007) presented relaxed DDs to circumvent
+the exponential growth rate in the DD size when modeling large discrete optimization problems.
+Bergman et al. (2016b) introduced a branch-and-bound algorithm that iteratively uses relaxed and
+restricted DDs to find optimal solution. The literature contains many successful utilization of DDs
+in different domains; see works by Bergman and Cire (2018), Serra and Hooker (2019), Davarnia
+and Van Hoeve (2020), Gonzalez et al. (2020), and Hosseininasab and Van Hoeve (2021) for some
+examples.
+Until recently, applications of DDs were limited to discrete problems, and the question on how to
+use DDs in solving optimization problems with continuous variables was unanswered. To address
+this limitation, Davarnia (2021) proposed a technique called arc-reduction that generates a DD
+that represents a relaxation of the underlying continuous problem. In a follow-up work, Salemi
+and Davarnia (2022a) established necessary and sufficient conditions for a general MIP to be
+representable by DDs. They showed that a bounded MIP can be remodeled and solved with DDs
+through employing a specialized Benders decomposition technique. In this paper, we build on this
+framework to design a novel DD-based methodology to solve the SGUFP.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+6
+1.3.
+Contributions
+While there are several studies in the literature dedicated to the unit train problem, exact method-
+ologies that provide a rigorous treatment of the NSNM requirement at the heart of unit train
+models are scarce. In this paper, we design a novel exact DD-based framework to solve the SGUFP,
+as a more realistic and more challenging variant of this problem class. To our knowledge, this is
+the first work that studies SGUFP from an exact perspective, and the first application of DDs
+to a transportation problem. Our proposed framework formulates the problem in a transformed
+space of variables, which has a smaller dimension compared to the standard MIP formulations
+of the SGUFP. This presentation mitigates the computational difficulties stemmed from the MIP
+formulation size, providing a viable solution approach for large-scale network problems. The core
+principles of our DD framework can also be used to model other transportation problems with
+similar structure, as an alternative to traditional network optimization techniques.
+The remainder of this paper is organized as follows. In Section 2 we provide basic definitions
+and a brief overview on discrete and continuous DD models, including the DD-BD method to
+solve bounded MIPs. In Section 3, we adapt the DD-BD method to solve the SGUFP. We propose
+algorithms to construct exact and relaxed DDs to solve the problem in a transformed space.
+Section 4 presents computational experiments to evaluate the performance of the DD-BD method
+for the SGUFP. We give concluding remarks in Section 5.
+2.
+Background on DDs
+In this section, we present basic definitions and results relevant to our DD analysis.
+2.1.
+Overview
+A DD D = (U,A,l) with node set U, arc set A, and arc label mapping l : A → R is a directed acyclic
+graph with n ∈ N arc layers A1,A2,...,An, and n + 1 node layers U1,U2,...,Un+1. The node layers
+U1 and Un+1, with |U1| = |Un+1| = 1, contain the root r and the terminal t, respectively. In any arc
+layer j ∈ [n] := {1,2,...,n}, an arc (u,v) ∈ Aj is directed from the tail node u ∈ Uj to the head node
+v ∈ Uj+1. The width of D is defined as the size of its largest Uj. DDs can model a bounded integer
+set P ⊆ Zn in such a way that each r-t arc-sequence (path) of the form (a1,...,an) ∈ A1 × ... × An
+encodes a point y ∈ P where l(aj) = yj for j ∈ [n], that is y is an n-dimensional point in P whose
+j-th coordinate is equal to the label value l(aj) of arc the aj. For such a DD, we have P = Sol(D),
+where Sol(D) denotes the finite collection of all r-t paths.
+The graphical property of DDs can be exploited to optimize an objective function over a discrete
+set P. To this end, DD arcs are weighted in such a way that the cumulative weight of an r-t
+path that encodes a solution y ∈ P equals to the objective function value evaluated at y. Then, a
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+7
+shortest (resp. longest) r-t path for the underlying minimization (resp. maximization) problem is
+found, an operation that can be performed in polynomial time.
+The construction of an exact DD as described above is computationally prohibitive due to the
+exponential growth rate of its size. To alleviate this difficulty, relaxed and restricted DDs are
+proposed to keep the size of DDs under control. In a relaxed DD, nodes are merged in such a way
+that the width of the resulting diagram is bounded by a predetermined width limit. This node-
+merging process ensures that all feasible solutions of the original set are encoded by a subset of
+all r-t paths in the resulting DD. Optimization over this relaxed DD provides a dual bound to the
+optimal solution of the original problems. In a restricted DD, the collection of all r-t paths of the
+DD encode a subset of the feasible solutions of the original set. Optimization over this restricted
+DD provides a primal bound to the optimal solution of the original problems. The restricted and
+relaxed DDs can be iteratively refined in a branch-and-bound scheme to find the optimal value of
+a problem through convergence of their primal and dual bounds. The following example illustrates
+an exact, relaxed and restricted DD for a discrete optimization problem.
+Example 1. Consider the discrete optimization problem max{5y1 + 10y2 + 4y3 | y ∈ P} where
+P = {(1,0,0),(1,0,1),(0,1,0),(0,0,1),(0,0,0)}. The exact DD D with width 3 in Figure 2(a) models
+the feasible region P. The weight of each arc a ∈ Aj, for j ∈ {1,2,3}, shows the contribution of
+variable yj’s value assignment to the objective function. The longest r-t path that encodes the
+optimal solution (y∗
+1,y∗
+2,y∗
+3) = (0,1,0) has length 10, which is the optimal value to the problem.
+By reducing the width limit to 2, we can build relaxed and restricted DDs for P as follows. The
+relaxed DD D in Figure 2(b) provides an upper bound to the optimal solution, where the longest
+path with length 14 is obtained by an infeasible point (y1,y2,y3) = (0,1,1). Finally, the restricted
+DD D in Figure 2(c) gives a lower bound to the optimal solution, where the longest path with
+length 9 encodes a feasible solution (y1,y2,y3) = (1,0,1).
+2.2.
+Continuous DD Models
+While the framework described in the previous section can be applied to solve different classes of
+discrete optimization problems, its extension to model sets with continuous variables requires a
+fundamentally different approach. The reason that the traditional DD structure is not viable for
+continuous sets is that representing the domain of a continuous variable through arcs requires an
+infinite number of them, spanning all values within a continuous interval, which is structurally
+prohibitive in DD graphs. Fortunately, there is a way to overcome this obstacle by decomposing
+the underlying set into certain rectangular formations, which can in turn be represented through
+node-sequences in DDs. In what follows, we give an overview of these results as relevant to our
+analysis.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+8
+r
+t
+5
+0
+0
+10
+0
+0
+4
+0
+4
+0
+(a) Exact DD D
+r
+t
+5
+0
+0
+10
+0
+0
+4
+4
+0
+(b) Relaxed DD D
+r
+t
+5
+0
+0
+0
+0
+4
+4
+0
+y1
+y2
+y3
+(c) Restricted DD D
+Figure 2
+The exact, relaxed, and restricted DDs representing P in Example 1. Solid and dotted arcs indicate
+one and zero arc labels, respectively. Numbers next to arcs represent weights.
+Consider a bounded set P ⊆ Rn. Salemi and Davarnia (2022a) give necessary and sufficient
+conditions for P to admit the desired rectangular decomposition. Such a set is said to be DD-
+representable w.r.t. a fixed index set I ⊆ [n], as there exists a DD D such that max{f(x) | x ∈
+P} = max{f(x) | x ∈ Sol(D)} for every function f(x) that is convex in the space of variables xI.
+A special case of DD-representable sets is given next.
+Proposition 1. Any bounded mixed integer set of the form P ⊆ Zn × R is DD-representable
+w.r.t. I = {n + 1}.
+□
+This result gives rise to a novel DD-based framework to solve general bounded MIPs as outlined
+below. Consider a bounded MIP H := max{cy +dx | Ay +Gx ≤ b, y ∈ Zn}. Using Benders decom-
+position (BD), formulation H is equivalent to maxy∈Zn{cy +maxx{dx | Gx ≤ b−Ay}}, which can
+be reformulated as M = max{cy +z | (y;z) ∈ Zn ×[l,u]}, where l,u ∈ R are some valid bounds on z
+induced from the boundedness of H. Here, M is the master problem and z represents the objective
+value of the subproblem maxx{dx | Gx ≤ b − A¯y} for any given ¯y as an optimal solution of the
+master problem. The outcome of the subproblems is either an optimality cut or a feasibility cut
+that will be added to the master problem. Then, the master problem will be resolved. Proposition 1
+implies that formulation M can be directly modeled and solved with DDs. For this DD, we assign n
+arc layers to the integer variables y1,y2,...,yn, and one arc layer to the continuous variable z with
+only two arc labels showing a lower and upper bound for this variable. To find an optimal solution,
+the longest path is calculated, which will be used to solve the subproblems. Note that since M is
+a maximization problem, a longest path of the associated DD encodes an optimal solution, and
+its length gives the optimal value; see Example 2. The feasibility and optimality cuts generated
+by the subproblems will then be added to refine the DD, whose longest path will be recalculated.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+9
+The refinement technique consists of removing arcs of the DD that lead to solutions that violate
+the added inequality, as well as splitting nodes of the DD that lead to different subsequent partial
+assignments; see Bergman et al. (2016a) for a detailed account on DD refinement techniques. We
+illustrate this approach in Example 2.
+Example 2. Suppose that max{2y1 + 4y2 + z | y ∈ P,z ≤ 25} forms the master problem at the
+penultimate iteration of a BD algorithm, where P = {(0,0),(1,1)}. This problem is represented
+by the DD D in Figure 3(a) where −M is a valid lower bound for z. The longest path of D
+encodes the solution (ˆy1, ˆy2, ˆz) = (1,1,25). Assume that using the point (ˆy1, ˆy2) = (1,1) in the
+associated subproblem generates an optimality cut z ≤ 3y1 + 2y2 + 10 for the final iteration of the
+BD algorithm. Refining DD D with respect to this cut yields the new DD in Figure 3(b). The
+longest path represents the optimal solution (y∗
+1,y∗
+2,z∗) = (1,1,15) with length 21, which is the
+optimal value.
+r
+t
+2
+0
+4
+0
+−M
+25
+25
+−M
+(a) penultimate iteration
+r
+t
+2
+0
+4
+0
+−M
+15
+10
+−M
+y1
+y2
+z
+(b) final iteration
+Figure 3
+The last two iterations of solving the master problem in Example 2
+Using the DD framework as outlined above can be computationally challenging due to exponen-
+tial growth rate of the size of an exact DD. To mitigate this difficulty, restricted/relaxed DDs can
+be employed inside of the BD framework as demonstrated in Algorithm 1. We refer to this solution
+method as DD-BD (Salemi and Davarnia 2022a).
+In explaining the steps of Algorithm 1, let point ˆy ∈ Zk, where k ≤ n, be a partial value assignment
+to the first k coordinates of variable y, i.e., yi = ˆyi for all i ∈ [k]. We record the set of all partial
+value assignments in ˆY = {ˆy ∈ Zk | k ∈ [n]}∪{⊖}, where ⊖ represents the case where no coordinate
+of y is fixed. Set C contains the produced Benders cuts throughout the algorithm, and we denote
+the feasible region described by these cuts by F C. Further, define MC(ˆy) = max{cy + z | (y;z) ∈
+Zn × [l,u] ∩ F C, yi = ˆyi,∀i ∈ [k]} to be the restricted master problem M obtained through adding
+cuts in C and fixing the partial assignment ˆy. In this definition, the case with C = ∅ and ˆY = {⊖}
+is denoted by M∅(⊖) = M, which is an input to Algorithm 1.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+10
+Algorithm 1: DD-BD
+Data: MIP H, construction method to build restricted and relaxed DDs for M
+Result: An optimal solution (y∗,z∗) and optimal value w∗ to H
+1 initialize set of partial assignments ˆY = {⊖}, set of Benders cuts C = ∅, and w∗ = −∞
+2 if ˆY = ∅ then
+3
+terminate and return (y∗,z∗) and w∗
+4 else
+5
+select ˆy ∈ ˆY and update ˆY ← ˆY \ {ˆy}
+6
+create a restricted DD D associated with MC(ˆy)
+7
+if D ̸= ∅ then
+8
+find a longest r-t path of D with encoding point (y,z) and length w
+9
+solve the BD subproblem using y to obtain Benders cut C
+10
+if C ∈ C then
+11
+go to line 17
+12
+else
+13
+update C ← C ∪ C and refine D w.r.t. C
+14
+go to line 8
+15
+else
+16
+go to line 2
+17
+if w > w∗ then
+18
+update w∗ ← w and (y∗,z∗) ← (y,z)
+19
+if D provides an exact representation of MC(ˆy) then
+20
+go to line 2
+21
+else
+22
+create a relaxed DD D associated with MC(ˆy)
+23
+find a longest r-t path of D with length w
+24
+if w > w∗ then
+25
+solve the BD subproblem using y to obtain Benders cut C
+26
+if C ∈ C then
+27
+go to line 31
+28
+else
+29
+update C ← C ∪ C and refine D w.r.t. C
+30
+go to line 23
+31
+forall u in the last exact layer of D do
+32
+update ˆY ← ˆY ∪ {˜y} where ˜y encodes longest r-u path of D
+33
+go to line 2
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+11
+The algorithm starts with constructing a restricted DD D corresponding to MC(ˆy) with empty
+initial values for C and ˆy. We then find a longest r-t path of D encoding solution (y,z). Next,
+using y, we solve the associated subproblem to obtain a feasibility/optimality cut C. We add this
+cut to C, refine D according to it, and find a new longest r-t path. We repeat these steps until no
+new feasibility/optimality cut is generated. At this point, the length of a longest r-t path of D,
+denoted by w, gives a lower bound to the master problem M, which is also a valid lower bound
+to the original problem H. The value of w can be used to update w∗, the optimal value of H
+at termination. Next, we create a relaxed DD D corresponding to MC(ˆy). We find a longest r-t
+path of D that provides an upper bound w to M. If the upper bound w is strictly greater than
+the current value of w∗, we follow steps similarly to the case for D to iteratively refine D w.r.t.
+feasibility/optimality cuts through solving the subproblems, until no new cut is generated. Next,
+we perform a specialized branch-and-bound procedure to improve the bound through expanding
+merged layers of the DD. To this end, we add all the partial assignments associated with nodes in
+the last exact layer of D (the last node layer in which no nodes are merged) to the collection ˆY.
+The nodes corresponding to partial assignments in ˆY are required to be further explored to check
+whether or not the value of w∗ can be improved. That is, the above process is repeated for every
+node v with partial assignment in ˆY as the r-v path is fixed in the new restricted/relaxed DDs.
+The algorithm terminates when ˆY becomes empty, at which point w∗ is the optimal value.
+3.
+DD-BD Formulation for the SGUFP
+In this section, we adapt the DD-BD framework described in Section 2.2 to solve the SGUFP.
+3.1.
+MIP Formulation
+We study the MIP formulation of the SGUFP based on that of its deterministic counterpart given
+in Davarnia et al. (2019). Consider a network G = (V,A) with node set V := V ′ ∪ {s,t} and arc set
+A, where s and t are source and sink nodes, respectively. The source node is connected to all the
+supply nodes in S ⊆ V ′, and the sink node is connected to all the demand nodes in D ⊆ V ′. Figure 4
+illustrates the general structure of this network. For a node q ∈ V , let δ−(q) := {i ∈ V | (i,q) ∈ A}
+and δ+(q) := {j ∈ V | (q,j) ∈ A} show the set of incoming and outgoing neighbors of q, respectively.
+Define ¯V ⊆ V ′ as a subset of vertices that must satisfy the NSNM requirement. For each node
+q ∈ ¯V , let binary variable yq
+ij ∈ {0,1} represent whether or not the flow entering node q ∈ ¯V through
+arc (i,q) leaves node q through arc (q,j). The first stage of SGUFP determines the matching pairs
+between incoming and outgoing arcs of unsplittable nodes as follows:
+max
+z
+(1a)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+12
+S
+D
+s
+t
+Figure 4
+Illustration of network G = (V ′ ∪ {s,t},A)
+s.t.
+�
+j∈δ+(q)
+yq
+ij ≤ 1
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(1b)
+�
+i∈δ−(q)
+yq
+ij ≤ 1
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(1c)
+yq
+ij ∈ {0,1}
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V,
+(1d)
+where constraints (1b) ensure that each incoming arc to a node with NSNM requirement is
+assigned to at most one outgoing arc, and constraints (1c) guarantee that each outgoing arc from
+such a node is matched with at most one incoming arc.
+In (1a)–(1d), variable z represents the objective value of the second stage of SGUFP where
+the demand uncertainty is taken into account. This demand uncertainty is modeled by a set Ξ of
+scenarios for the demand vector dξ with occurrence probability Prξ for each scenario ξ ∈ Ξ. Let
+continuous variable xξ
+ij ∈ R+ denote the flow from node i to node j through arc (i,j) under scenario
+ξ ∈ Ξ. We further assign a reward rij per unit flow to be collected by routing flow through arc (i,j).
+It follows that z = �
+ξ∈Ξ Prξzξ, where zξ is the objective value of the second stage of SGUFP for
+each scenario ξ ∈ Ξ. This subproblem is formulated as follows for a given y vector:
+max
+�
+q∈V
+�
+j∈δ+(q)
+rqjxξ
+qj
+(2a)
+s.t.
+�
+i∈δ−(q)
+xξ
+iq −
+�
+j∈δ+(q)
+xξ
+qj = 0
+∀q ∈ V ′
+(2b)
+ℓξ
+iq ≤ xξ
+iq ≤ uξ
+iq
+∀i ∈ δ−(q), ∀q ∈ V
+(2c)
+xξ
+iq − xξ
+qj ≤ uξ
+iq(1 − yq
+ij)
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V
+(2d)
+xξ
+qj − xξ
+iq ≤ uξ
+qj(1 − yq
+ij)
+∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V
+(2e)
+xξ
+iq ≤ uξ
+iq
+�
+j∈δ+(q)
+yq
+ij
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(2f)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+13
+xξ
+qj ≤ uξ
+qj
+�
+i∈δ−(q)
+yq
+ij
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(2g)
+xξ
+ij ≥ 0
+∀(i,j) ∈ A.
+(2h)
+In the above formulation, the objective function captures the total reward collected by routing
+flows throughout the network (from the source s to the sink t) to satisfy demands. The flow-balance
+requirements are represented by (2b). Constraints (2c) bound the flow on each arc from below
+and above. To impose the demand requirement for each scenario ξ ∈ Ξ, we fix ℓξ
+qt = uξ
+qt = dξ
+q for all
+demand nodes q ∈ D with demand dξ
+q, and leave the lower and upper bound values unchanged for all
+other arcs. Constraints (2d)–(2g) model the NSNM requirement for each node q ∈ ¯V . In particular,
+(2d) and (2e) ensure that matching arcs (i,q) and (q,j) have equal flows. Constraints (2f) and (2g)
+guarantee that an arc without a matching pair does not carry any flow. We note here that the
+Constraint (2b) is implied by other constraints of the above subproblem under the assumption
+that y is feasible to the master problem (1a)–(1d). However, we maintain this constraint in the
+subproblem because the master formulation in our DD-based approach, as will be described in
+Section 3.2, may produce a solution that is not feasible to (1a)–(1d). As a result, the addition of
+the Constraint (2b) will lead to a tighter subproblem formulation.
+As discussed in Section 2.2, the first step to use the DD-BD algorithm is to decompose the
+underlying problem into a master and a subproblem. The above two-stage formulation of the
+SGUFP is readily amenable to BD since the first stage problem (1a)-(1d) can be considered as the
+master problem together with some valid lower and upper bounds −Γ and Γ on z induced from the
+boundedness of the MIP formulation. For a given y value obtained from the master problem and
+a scenario ξ ∈ Ξ, the second stage problem (2a)-(2h) can be viewed as the desired subproblems.
+The optimality/feasibility cuts obtained from each scenario-based subproblem are then added to
+the master problem through aggregation as described in Section 3.3.
+3.2.
+DD-BD: Master Problem Formulation
+While the DD-BD Algorithm 1 provides a general solution framework for any bounded MIP, its DD
+component is problem-specific, i.e., it should be carefully designed based on the specific structure
+of the underlying problem. In this section, we design such an oracle for the SGUFP that represents
+the feasible region {(1b)−(1d),z ∈ [−Γ,Γ]} of the master problem (1a)-(1d). To model this feasible
+region in the original space of (y;z) variables, a DD would require �
+q∈ ¯V |δ−(q)| × |δ+(q)| arc
+layers to represent binary variables y and one arc layer to encode the continuous variable z.
+Constructing such a DD, however, would be computationally cumbersome due to the large number
+of the arc layers. To mitigate this difficulty, we take advantage of the structural flexibility of DDs
+in representing irregular variable types that cannot be used in standard MIP models. One such
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+14
+variable type is the index set, where arc layers represent indices, rather than domain values. We
+next show that we can remarkably reduce the number of DD arc layers by reformulating the master
+problem in a transformed space of variables defined over index sets.
+Consider a node q ∈ ¯V . In the following, we define mappings that assign an index to each incoming
+and outgoing arc of q. These mappings enable us to define new variables to reduce the number
+of DD arc layers. Let ind−(i,q) be a one-to-one mapping from incoming arcs (i,q), for i ∈ δ−(q),
+to the index set {1,2,...,|δ−(q)|}. Similarly, let ind+(q,j) be a one-to-one mapping from outgoing
+arcs (q,j), for j ∈ δ+(q), to the index set {1,2,...,|δ+(q)|}. For each incoming arc (i,q) with index
+h = ind−(i,q), we define an integer variable wq
+h ∈ {0,1,...,|δ+(q)|} such that wq
+h = 0 if this incoming
+arc is not paired with any outgoing arc, and wq
+h = k > 0 if this arc is matched with an outgoing arc
+(q,j) with index k = ind+(q,j).
+Next, we give a formulation in the space of w variables that describes the matching between
+incoming and outgoing arcs of q for all q ∈ ¯V . In the following, sign(.) represents the sign function
+that returns 1 if its argument is strictly positive, 0 if the argument is zero, and −1 otherwise.
+Further, the operator |.|, when applied on a set, represents the set size; and when applied on a real
+number, it represents the absolute value.
+Proposition 2. Formulation
+�
+i∈δ−(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≥
+��δ−(q)
+�� − 1
+∀j ∈ δ+(q), ∀q ∈ ¯V
+(3a)
+wq
+ind−(i,q) ∈
+�
+0,1,...,
+��δ+(q)
+���
+∀i ∈ δ−(q), ∀q ∈ ¯V
+(3b)
+models the matching between incoming and outgoing arcs of nodes q ∈ ¯V .
+Proof.
+We show the result for a single node q ∈ ¯V . The extension to the multiple node case
+is straightforward as the matching problem for each node is independent from other nodes. For
+the direct implication, assume that M q is a matching between incoming and outgoing arcs of q,
+with elements of the form (i,j) that represent a matching between the incoming arc (i,q) and
+the outgoing arc (q,j). We show that variables w associated with matching pairs in M q satisfy
+constraints (3a) and (3b). It follows from the definition of w that, for each (i,j) ∈ M q, we have
+wq
+ind−(i,q) = ind+(q,j). Also, for any i ∈ δ−(q) that does not have a matching pair in M q, we have
+wq
+ind−(i,q) = 0. These value assignments show that w satisfies (3b) as the image of ind+ mapping is
+{1,...,|δ+(q)|}. For each i ∈ δ−(q) and j ∈ δ+(q), we have
+���wq
+ind−(i,q) − ind+(q,j)
+��� ≥ 0, with equality
+holding when (i,j) ∈ M q. For each j ∈ δ+(q), there are two cases. For the first case, assume that
+(i,j) /∈ M q for any i ∈ δ−(q). As a result,
+���wq
+ind−(i,q) − ind+(q,j)
+��� > 0 for all i ∈ δ−(q). Applying
+the sign(.) function on these terms yields sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+= 1, which implies that
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+15
+�
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+= |δ−(q)|, satisfying (3a). For the second case, assume that
+(i∗,j) ∈ M q for some i∗ ∈ δ−(q). As a result, we have �
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+=
+|δ−(q)| − 1 since sign
+����wq
+ind−(i∗,q) − ind+(q,j)
+���
+�
+=
+���wq
+ind−(i∗,q) − ind+(q,j)
+��� = 0, satisfying (3a).
+For the reverse implication, assume that w is a feasible solution to (3a)–(3b). We show that the
+pairs of the form (i,j) encoded by these variables constitute a feasible matching between incoming
+and outgoing arcs of q, i.e., (i) each arc (i,q) is matched with at most one arc (q,j), and (ii) each
+arc (q,j) is matched with at most one arc (i,q). It follows from constraint (3b) that, for each i ∈
+δ−(q), variable wq
+ind−(i,q) takes a value between {0,1,...,|δ+(q)|}. If wq
+ind−(i,q) = 0, then (i,q) is not
+matched with any outgoing arc, otherwise it is matched with arc (q,j) with ind+(q,j) = wq
+ind−(i,q).
+This ensures that condition (i) above is satisfied for this matching collection. Further, for each
+j ∈ δ−(q), constraint (3a) implies that sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+can be equal to zero for at
+most one i ∈ δ−(q). In such a case, we would have at most one matching pair of the form (i,j) in
+the collection, showing that condition (ii) above is satisfied.
+□
+It follows from Proposition 2 that constraints (3a)-(3b) can replace (1b)-(1d) in the master
+problem (1a)-(1d) to obtain the following master problem in a transformed space of variables.
+max
+w;z {z | (3a) − (3b),z ∈ [−Γ,Γ]}.
+(4)
+Note that formulation (4) is an integer nonlinear program (INLP) with nonconvex and non-
+continuous constraint functions. Such a formulation is extremely difficult for conventional MINLP
+techniques and solvers to handle. However, due to structural flexibility of DDs in representing inte-
+ger nonlinear programs, this problem can be easily modeled via a DD; see Davarnia and Van Hoeve
+(2020) for a detailed account on using DDs for modeling INLPs. In the following, we present an
+algorithm to construct DDs in the space of (w;z) variables for the master problem (4) with a
+single node q ∈ ¯V . The extension to the case with multiple nodes follows by replicating the DD
+structure. The output of Algorithm 2 is a DD with |δ−(q)| + 1 arc layers where the first |δ−(q)|
+layers represent w variables and the last layer encodes variable z. In this algorithm, su denotes the
+state value of DD node u. The core idea of the algorithm is to use unpaired outgoing arcs of q as
+the state value at each DD layer that represents the matching for an incoming arc of q.
+Next, We show that the solution set of the DD constructed by Algorithm 2 represents the feasible
+region of (4). Note here that DD representation of a MIP set, as described in Section 2.2, does
+not imply the encoding of all of the solutions of the set, but rather the encoding of a subset of all
+solutions that subsumes all the extreme points of the set. Such a representation is sufficient to solve
+an optimization problem over the set with an objective function convex in continuous variables,
+which is the case for (4).
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+16
+Algorithm 2: Construction of DD for the master problem of SGUFP with a node q ∈ ¯V
+Data: node q ∈ ¯V , parameter Γ
+Result: an exact DD D
+1 create the root node r ∈ U1 with state sr = {0,1,...,|δ+(q)|}
+2 forall i ∈ {1,2,...,|δ−(q)|} and u ∈ Ui do
+3
+forall ℓ ∈ su do
+4
+create a node v ∈ Ui+1 with state (su \ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v
+with label l(a) = ℓ
+5 forall u ∈ U1+|δ−(q)| do
+6
+create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ
+and l(a2) = −Γ.
+Theorem 1. Consider a SGUFP with ¯V = {q}. Let D be a DD constructed by Algorithm 2.
+Then, Sol(D) represents the feasible region of (4).
+Proof.
+(⊆) Consider an r-t path of D that encodes solution ( ˜wq,z). According to Algorithm 2,
+the labels of the first |δ−(q)| arcs of this path belong to {0,1,...,|δ+(q)|}, showing that ˜wq
+satisfies constraints (3b). Assume by contradiction that ˜wq does not satisfy constraints (3a),
+i.e., �
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ |δ−(q)| − 2 for some j ∈ δ+(q). This implies that
+˜wq
+ind−(i′,q) = ˜wq
+ind−(i′′,q) = ind+(q,j) for two distinct i′,i′′ ∈ δ−(q). In other words, the arcs at lay-
+ers ind−(i′,q) and ind−(i′′,q) of the selected r-t path both share the same label value ind+(q,j).
+According to line 3 of Algorithm 2, we must have that the state value of nodes at layers ind−(i′,q)
+and ind−(i′′,q) of the r-t path both contain ind+(q,j). This is a contradiction to the state update
+policy in line 4 of Algorithm 2, since positive arc labels at each layer of the DD will be excluded
+from the state value of the subsequent nodes.
+(⊇) Consider a feasible solution point ( ˜wq; ˜z) of (4). Suppose ˜wq = (ℓ1,ℓ2,...,ℓ|δ−(q)|). According
+to constraints (3a), no two coordinates of ˜wq have the same positive value. The state value at the
+root node in D contains all index values {0,1,...,|δ+(q)|}. According to Algorithm 2, there exists
+an arc with label ℓ1 at the first layer of D. The state value at the head node of this arc, therefore,
+contains ℓ2 ∈ {0,1,...,|δ+(q)|} \ {ℓ1}, which guarantees an arc with label ℓ2 at the second layer of
+this path. Following a similar approach, we can track a path from the root to layer |δ−(q)| whose
+arcs labels match values of ˜wq. Note for the last layer that ˜z ∈ [−Γ,Γ], which is included in the
+interval between arc labels of the last layer of D. As a result, ( ˜wq; ˜z) is represented by an r-t path
+of D.
+□
+The main purpose of using a DD that models the master problem (4) over one that models (1a)-
+(1d) is the size reduction in arc layers that represent variables w as compared with variables
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+17
+y. It turns out that this space transformation can significantly improve the solution time of the
+DD approach. We refer the interested reader to Appendix A for a detailed discussion on these
+advantages, including preliminary computational results.
+Constructing exact DDs as described in Algorithm 2 can be computationally expensive for large
+size problems. As discussed in Section 2.2, relaxed and restricted DDs are used to circumvent this
+difficulty. Building restricted DDs is straightforward as it involves the selection of a subset of r-t
+paths of the exact DD that satisfy a preset width limit. Constructing relaxed DDs, on the other
+hand, requires careful manipulation of the DD structure to merge nodes in such a way that it
+encodes a superset of all r-t paths of the exact DD. We demonstrate a method to construct such
+relaxed DDs in Algorithm 3. Similarly to Algorithm 2, this algorithm is presented for a single
+NSNM node, but can be extended to multiple nodes by replicating the procedure.
+Algorithm 3: Construction of relaxed DD for the master problem of SGUFP with a node
+q ∈ ¯V
+Data: node q ∈ ¯V , parameter Γ
+Result: a relaxed DD D
+1 create the root node r ∈ U1 with state sr = {0,1,...,|δ+(q)|}
+2 forall i ∈ {1,2,...,|δ−(q)|} and u ∈ Ui do
+3
+forall ℓ ∈ su do
+4
+create a node v ∈ Ui+1 with state (su \ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v
+with label l(a) = ℓ
+5
+select a subset of nodes v1,v2,...,vk ∈ Ui+1 and merge them into node v′ with state
+sv′ = �k
+j=1 svj
+6 forall u ∈ U1+|δ−(q)| do
+7
+create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ
+and l(a2) = −Γ.
+Theorem 2. Consider a SGUFP with ¯V = {q}. Let D be a DD constructed by Algorithm 3.
+Then, D represents a relaxation of the feasible region of (4).
+Proof.
+Let ˙D be the DD constructed by Algorithm 2 for the master problem (4) with a single
+node q ∈ ¯V . It suffices to show that the solution set of D provides a relaxation for that of ˙D. Pick
+a root-terminal path ˙P of ˙D with encoding point ( ˙wq; ˙z). We show that there exist a root-terminal
+path P of D with encoding point (wq;z) such that wq = ˙wq and z = ˙z. Given a DD, define Pk to
+be a sub-path composed of arcs in the first k layers, for 1 ≤ k ≤ |δ−(q)|. We show for any sub-path
+˙Pk of ˙D with encoding point ˙wq
+k = ( ˙wq
+1,..., ˙wq
+k), there exists a sub-path P k of D with encoding
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+18
+point wk = (w1,...,wk) such that wh = ˙wh for h = 1,...,k. Note that we only need to prove the
+matching values for k ≤ |δ−(q)|, because each node at node layer |δ−(q)| + 1 of both ˙D and D
+is connected by two arcs with labels −Γ and Γ to the terminal node, and thus there are always
+matching arcs with the same label for the last layer, i.e., z = ˙z. We prove the result by induction on
+k. The base case for k = 1 is trivial, since D contains arcs with labels {0,1,...,|δ+(q)|} in the first
+layer, which includes the label value of the first arc on ˙P1. For the induction hypothesis, assume
+that the statement is true for k = d, i.e., for the sub-path ˙Pd with label values ˙wq
+d = ( ˙wq
+1,..., ˙wq
+d),
+there is sub-path P d of D with matching arc labels. We show the statement holds for d + 1. Let
+u ∈ ˙Ad+1 and v ∈ Ad+1 be the end nodes of ˙Pd and P d, respectively. It follows from Algorithm 2
+that the index set representing the state value at node u contains ˙wq
+d+1, i.e., ˙wq
+d+1 ∈ ˙su = {0} ∪
+{1,...,|δ+(q)|} \ { ˙w1, ˙w2,..., ˙wd}. The merging step in line 5 of Algorithm 3, on the other hand,
+implies that sv ⊇ {0}∪{1,...,|δ+(q)|}\{w1,w2,...,wd} = {0}∪{1,...,|δ+(q)|}\{ ˙w1, ˙w2,..., ˙wd} =
+˙su, where the inclusion follows from the fact that state values at nodes on path P d contain those of
+each individual path due to merging operation, and the first equality holds because of the induction
+hypothesis. As a result, sv must contain ˙wq
+d+1, which implies that there exists an arc with ˙wq
+d+1
+connected to node v on P d. Attaching this arc to P d, we obtain the desired sub-path P d+1.
+□
+3.3.
+DD-BD: Subproblem Formulation
+At each iteration of the DD-BD algorithm, an optimal solution of the master problem is plugged into
+the subproblems to obtain feasibility/optimality cuts. For the SGUFP formulation, this procedure
+translates to obtaining an optimal solution of (4) in the space of w variables, which is used to
+solve the subproblem (2a)-(2h). The formulation of the subproblem, however, is defined over the
+original binary variables y, and the resulting feasibility/optimality cuts are generated in this space.
+To remedy this discrepancy between the space of variables in the master and subproblems, we need
+to find a one-to-one mapping between variables w and y, as outlined next.
+Proposition 3. Consider a node q ∈ ¯V . Let yq be a feasible solution to (1b)-(1d). Then, wq
+obtained as
+wq
+ind−(i,q) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+ij
+∀i ∈ δ−(q),
+(5)
+is a feasible solution to (3a)-(3b). Conversely, let wq be a feasible solution to (3a)-(3b). Then, yq
+obtained as
+yq
+ij = 1 − sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+∀(i,j) ∈ δ−(q) × δ+(q),
+(6)
+is a feasible solution to (1b)-(1d).
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+19
+Proof.
+For the direct statement, let yq be a feasible solution to (1b)-(1d), and construct a
+vector wq according to (5). We show that wq satisfies all constraints (3a)-(3b). First, we show
+that constraints (3a) are satisfied. Assume by contradiction that there exists j′ ∈ δ+(q) such that
+�
+i∈δ−(q) sign
+����wq
+ind−(i,q) − ind+(q,j′)
+���
+�
+≤ |δ−(q)| − 2. This implies that wq
+ind−(i′,q) = wq
+ind−(i′′,q) =
+ind+(q,j′) for some i′,i′′ ∈ δ−(q). Then, we can write that
+wq
+ind−(i′,q) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+i′j = ind+(q,j′) =
+�
+j∈δ+(q)
+ind+(q,j)yq
+i′′j = wq
+ind−(i′′,q),
+where the first and last equalities hold by (5). The second and third equalities in the above chain
+of relations imply that yq
+i′j′ = yq
+i′′j′ = 1, since ind+(q,j′) > 0. This violates constraints (1c), reaching
+a contradiction. Next, we show that constraints (3b) are satisfied. The proof follows directly from
+construction of wq and constraints (1b).
+For the converse statement, let wq be a feasible solution to (3a)-(3b), and construct a vec-
+tor yq according to (6). We show that yq satisfies all constraints (1b)-(1d). To show that each
+constraint (1b) is satisfied, consider i ∈ δ−(q). We can write that
+�
+j∈δ+(q)
+yq
+ij = |δ+(q)| −
+�
+j∈δ+(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ |δ+(q)| −
+�
+|δ+(q)| − 1
+�
+= 1,
+where the first equality follows from the construction of yq, and the inequality holds by (3b) as
+���wq
+ind−(i,q) − ind+(q,j)
+��� = 0 for at most one index j ∈ δ+(q). To show that each constraint (1c) is
+satisfied, select j ∈ δ+(q). We have
+�
+i∈δ−(q)
+yq
+ij = |δ−(q)| −
+�
+i∈δ−(q)
+sign
+����wq
+ind−(i,q) − ind+(q,j)
+���
+�
+≤ 1,
+where the equality follows from the construction of yq, and the inequality holds because of con-
+straint (3a). Finally, each constraint (1d) is satisfied due to the fact that 1 − sign(|.|) ∈ {0,1}.
+□
+Proposition 4. Mappings described by (5) and (6) are one-to-one over their respective
+domains.
+Proof.
+Note that the mapping described by (5) is a linear transformation of the form wq = Byq
+with coefficient matrix B ∈ Z|δ−(q)|×(|δ−(q)||δ+(q)|). It is clear from the identity block structure of B,
+that it is full row-rank, since each column contains a single non-zero element while each row has
+at least one non-zero element. As a result, the null space of B is the origin, which implies that
+ˆwq = ˜wq only if ˆyq = ˜yq.
+For the mapping described by (6), let distinct points ˆwq and ˜wq satisfy (3b). Construct vectors
+ˆyq and ˜yq by (6) using ˆwq and ˜wq, respectively. Because ˆwq and ˜wq are distinct, there must
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+20
+exist i ∈ δ−(q) such that ˆwq
+ind−(i,q) ̸= ˜wq
+ind−(i,q). This implies that at least one of these variables, say
+ˆwq
+ind−(i,q), is non-zero. It follows from (3b) that ˆwq
+ind−(i,q) = ind+(q,j′) for some j′ ∈ δ+(q), and that
+ˆwq
+ind−(i,q) ̸= ind+(q,j′). According to (6), we write that ˆyij′ = 1−sign
+���� ˆwq
+ind−(i,q) − ind+(q,j′)
+���
+�
+= 1,
+and that ˜yij′ = 1 − sign
+���� ˜wq
+ind−(i,q) − ind+(q,j′)
+���
+�
+= 0, showing that ˆyq ̸= ˜yq.
+□
+Using the results of Propositions 3 and 4, we can apply the DD-BD Algorithm 1 in its entirety for
+the SGUFP. In particular, at each iteration of the algorithm, we can transform the optimal solution
+( ¯w, ¯z) obtained from the DD representing the master problem (4) into a solution (¯y, ¯z) through the
+mapping (6). Given an optimal first-stage solution ¯y, we can solve |Ξ| separate subproblems; one
+for each demand realization in the second-stage. The feasibility cuts obtained from subproblems,
+which are in the space of y variables, are translated back into the space of w variables through the
+mapping (5) and added to the master problem. Further, in a case where all subproblems produce
+an optimality cut, they can be aggregated to generate an optimality cut in the space of (y,z),
+which is added to the master problem after being translated into the space of (w,z) variables. The
+master DD will be refined with respect to the resulting inequalities, and an optimal solution is
+returned to be used for the next iteration.
+In the remainder of this section, we present details on the derivation of optimality/feasibility cuts
+from subproblem (2a)-(2h). Consider the following partitioning of the set of arcs A into subsets
+A1 :=
+�
+(i,j) ∈ A
+�� δ−(i) = ∅, δ+(j) ̸= ∅
+�
+, A2 :=
+�
+(i,j) ∈ A
+�� δ−(i) ̸= ∅, δ+(j) = ∅
+�
+,
+A3 :=
+�
+(i,j) ∈ A
+�� δ−(i) ̸= ∅, δ+(j) ̸= ∅
+�
+, A4 :=
+�
+(i,j) ∈ A
+�� δ−(i) = ∅, δ+(j) = ∅
+�
+,
+and let θξ = (βξ,γξ,δξ,φξ,λξ,µξ) be the vector of dual variables associated with constraints of
+(2a)-(2h) for a scenario ξ ∈ Ξ. Further, define the bi-function
+f(y;θξ) =
+�
+q∈V
+�
+j∈δ+(q)
+�
+−ℓqjβξ
+qj + uqjγξ
+qj
+�
++
+�
+q∈ ¯V
+�
+(i,j)∈δ−(q)×δ+(q)
+�
+uiq(1 − yq
+ij)λξ
+iqj + uqj(1 − yq
+ij)µξ
+iqj
+�
++
+�
+q∈ ¯V
+�
+i∈δ−(q)
+�
+�uiq
+�
+j∈δ+(q)
+yq
+ijσξ
+iq
+�
+� +
+�
+q∈ ¯V
+�
+j∈δ+(q)
+�
+�uqj
+�
+i∈δ−(q)
+yq
+ijφξ
+qj
+�
+�.
+For a given ¯y and each scenario ξ ∈ Ξ, the dual of the subproblem (2a)-(2h) can be written as
+follows where the symbol ⋆ on a node means that it belongs to ¯V .
+min
+f(¯y;θξ)
+(7a)
+s.t.
+αξ
+⋆q − βξ
+i⋆q + γξ
+i⋆q +
+�
+j:j∈δ+(⋆q)
+λξ
+i⋆qj −
+�
+j:j∈δ+(⋆q)
+µξ
+i⋆qj + σξ
+i⋆q ≥ ri⋆q
+∀(i,
+⋆q) ∈ A1 (7b)
+αξ
+q − βξ
+iq + γξ
+iq ≥ riq
+∀(i,q) ∈ A1 (7c)
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+21
+− αξ
+⋆q − βξ
+⋆qj + γξ
+⋆qj −
+�
+i:i∈δ−(⋆q)
+λξ
+i⋆qj +
+�
+i:i∈δ−(⋆q)
+µξ
+i⋆qj + φξ
+⋆qj ≥ r⋆qj
+∀(
+⋆q,j) ∈ A2 (7d)
+− αξ
+q − βξ
+qj + γξ
+qj ≥ rqj
+∀(q,j) ∈ A2 (7e)
+− αξ
+⋆q + αξ
+⋆
+j − βξ
+⋆q
+⋆
+j + γξ
+⋆q
+⋆
+j +
+�
+i∈δ−(⋆q)
+�
+µξ
+i⋆q
+⋆
+j − λξ
+i⋆q
+⋆
+j
+�
++
+�
+i∈δ+(
+⋆
+j)
+�
+λξ
+⋆q
+⋆
+ji − µξ
+⋆q
+⋆
+ji
+�
++ σξ
+⋆q
+⋆
+j + φξ
+⋆q
+⋆
+j ≥ r⋆q
+⋆
+j
+∀(
+⋆q,
+⋆j) ∈ A3
+(7f)
+− αξ
+⋆q + αξ
+j − βξ
+⋆qj + γξ
+⋆qj +
+�
+i∈δ−(⋆q)
+�
+µξ
+i⋆qj − λξ
+i⋆qj
+�
++ φξ
+⋆qj ≥ r⋆qj
+∀(
+⋆q,j) ∈ A3 (7g)
+− αξ
+q + αξ
+⋆
+j − βξ
+q
+⋆
+j + γξ
+q
+⋆
+j +
+�
+i∈δ+(
+⋆
+j)
+�
+λξ
+q
+⋆
+ji − µξ
+q
+⋆
+ji
+�
++ σξ
+q
+⋆
+j ≥ rq
+⋆
+j
+∀(q,
+⋆j) ∈ A3 (7h)
+− αξ
+q + αξ
+j − βξ
+qj + γξ
+qj ≥ rqj
+∀(q,j) ∈ A3
+(7i)
+− βξ
+iq + γξ
+iq ≥ riq
+∀(i,q) ∈ A4
+(7j)
+αξ
+q ∈ R
+∀q ∈ V ′ (7k)
+βξ
+ij, γξ
+ij, σξ
+ij, φξ
+ij, λξ
+iqj, µξ
+iqj ≥ 0
+∀i,q,j ∈ V.
+(7l)
+If the above problem has an optimal solution ˆθξ for all ξ ∈ Ξ, the output of the subproblems will
+be an optimality cut of the form �
+ξ∈Ξ Prξf(y; ˆθξ) ≥ z. If the above problem is unbounded along a
+ray ˆθξ for a ξ ∈ Ξ, the output of the subproblem will be a feasibility cut of the form f(y; ˆθξ) ≥ 0.
+Note that replacing variables y in the above constraints with w through the mapping (5) results
+in separable nonlinear constraints. Nevertheless, since these constraints will be used to refine the
+master DD, their incorporation is simple due to structural flexibility of DDs in modeling such
+constraints; we refer the reader to Davarnia and Van Hoeve (2020) for a detailed account for
+modeling INLPs with DDs.
+4.
+Computational Experiments
+In this section, we solve SGUFP as a core model for the unit train scheduling problem with demand
+stochasticity using three different approaches: (i) the standard MIP formulation that is a deter-
+ministic equivalent of the two-stage model and contains all variables and constraints of the master
+problem and |Ξ| subproblems; (ii) the Benders reformulation presented in Section 3.1 composed
+of the master problem (1a)-(1d) and |Ξ| subproblems (2a)-(2h); and (iii) the DD-BD algorithm
+proposed in the present paper. In the Benders approach, we solve separate subproblems using a
+fixed vector ¯y obtained from the master problem. The feasibility cuts generated by subproblems
+are added directly to the constraint set of the master problem, and the optimality cuts are added
+as an aggregated cut over all scenarios. We note here that when there is a feasibility cut for any
+scenario, we add it directly to separate the solution of the current iteration and move on to the
+next iteration. To obtain a valid inequality that provides a bound for the single z variable, we need
+to aggregate valid inequalities over all scenario subproblems as z is composed of the objective value
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+22
+of all these subproblems. Therefore, we can only produce an optimality cut for the z variable when
+we have optimality cuts for all of the subproblems. For the DD-BD approach, we use the following
+algorithmic choices to build restricted and relaxed DDs. For the restricted DDs, we choose a subset
+of the r-t paths with largest lengths, which are more likely to contain an optimal solution. For
+the relaxed DDs, we merge nodes that have the largest number of common members in their state
+values. We refer the reader to Bergman et al. (2016a) for other heuristic approaches that could be
+used for this purpose.
+4.1.
+Test Instances
+In our experiments, we consider the structure of the SGUFP network given in Section 3.1. To
+ensure that the problem is always feasible, we create an artificial node s0 to compensate for any
+shortage of the supply, and add an arc from the artificial supply s0 to each demand node.
+We create test instances based on the specification given in Davarnia et al. (2019), which is
+inspired by realistic models. In particular, we consider a base rail network G′ = (V ′,A′) where 10%
+and 30% of the nodes are supply and demand nodes, respectively. We assume that 50% of the
+nodes must satisfy the NSNM requirement. We then create a network G = (V,A) by augmenting
+supply/demand and artificial nodes as described above with the following settings. The integer
+supply value at supply nodes is randomly selected from the interval [100,600]. The capacity of arcs
+connecting s0 to demand nodes are considered to be unbounded, and the integer capacity value
+of other arcs is randomly selected from the interval [100,300]. For each demand scenario ξ ∈ Ξ,
+the integer demand value at demand nodes is randomly chosen from the interval [100,200]. The
+reward of the arcs connecting s0 to the demand nodes are generated from the interval [−10,−5]
+to represent the cost of lost demands. The reward of the arcs connecting the source to the supply
+nodes is randomly selected from the interval [5,10], and the reward of the arcs connecting the
+demand nodes to the sink is fixed to zero since the flow of these arcs is also fixed. The reward
+of all other arcs is created randomly from the interval [−2,2] where the negative values indicate
+the cost of sending flows through congested arcs. We consider four categories of rail networks with
+|V ′| ∈ {40,60,80,100}. For each category, we create five scenario classes for the number of demand
+scenarios |Ξ| ∈ {50,100,150,200,250}. For each network category and scenario class, we create five
+random instances based on the above settings. Test instances are publicly available (Salemi and
+Davarnia 2022b).
+4.2.
+Numerical Results
+In this section, we present the numerical results that compare the performance of the DD-BD
+formulation for the SGUFP instances with that of the MIP formulation, denoted by “MIP”, and the
+standard Benders reformulation, denoted by “BD”. All experiments are conducted on a machine
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+23
+running Windows 10, x64 operating system with Intel® Core i7 processor (2.60 GHz) and 32 GB
+RAM. The Gurobi optimization solver (version 9.1.1) is used to solve instances for the MIP and
+BD models. When solving problems with Gurobi, we turn off presolve and cuts for all methods
+to have a fair comparison. Tables 1-4 report the running times of each of these formulations for
+|V ′| ∈ {40,60,80,100} and |Ξ| ∈ {50,100,150,200,250} where the time limit is set to 3600 seconds.
+The symbol “ > 3600” indicates that the problem was not solved within the time limit. As evident
+in these tables, the DD-BD formulation outperforms the other alternatives. In particular, the
+gap between the solution time of the DD-BD and the MIP and BD approaches widens as the
+problem size increases. For example, as reported in Table 1, while the DD-BD approach solves all
+25 instances in under 275 seconds, the MIP approach fails to solve 10 of them within 3600 seconds,
+80% of which involve 200 or 250 scenarios. This shows a clear superiority of the DD-BD over the
+MIP method. Further, for most of the instances, the DD-BD approach outperforms the standard
+BD approach, rendering it as the superior solution method among all three. Figures 5-8 compare
+the performance of DD-BD, BD, and MIP formulations through box and whisker plots for each
+network size and under each scenario class. In these figures, for uniformity of illustration, we used
+3600 seconds for the running time of instances that fail to solve the problem within that time
+limit. As the figures show, the minimum, median, and maximum of running times of the DD-BD
+method are remarkably smaller than those of the both BD and MIP methods in all cases. These
+results show the potential of the DD-BD framework in solving network problems with challenging
+combinatorial structures. In Appendix B, we present additional numerical results for the DD-BD
+approach to assess its ability to solve larger problem sizes.
+Table 1
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 40.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+75.74 512.62 2877.19
+> 3600
+> 3600
+BD
+141.83 313.84
+339.81
+451.93
+565.82
+DD-BD
+56.94 129.87
+163.43
+219.02
+274.36
+2
+MIP
+67.59 275.07
+906.10 1892.21 2235.53
+BD
+63.44 121.25
+141.04
+230.81
+235.87
+DD-BD
+42.60
+82.65
+128.16
+164.52
+208.94
+3
+MIP
+94.86 753.23 2453.05
+> 3600
+> 3600
+BD
+71.14 139.20
+172.86
+224.33
+244.91
+DD-BD
+53.32
+93.58
+113.93
+178.65
+217.33
+4
+MIP
+71.46 309.62
+> 3600
+> 3600
+> 3600
+BD
+63.55 182.01
+267.94
+334.74
+380.22
+DD-BD
+46.61
+87.81
+130.19
+183.23
+253.72
+5
+MIP
+380.33 406.73
+> 3600
+> 3600
+> 3600
+BD
+123.69 198.73
+205.16
+231.56
+287.24
+DD-BD
+67.04 104.78
+138.46
+195.69
+231.74
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+24
+Table 2
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 60.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+893.73 > 3600
+> 3600
+> 3600
+> 3600
+BD
+241.85
+556.18
+582.80
+758.54
+933.05
+DD-BD 176.16
+357.06
+603.81
+719.27
+901.02
+2
+MIP
+206.87
+811.64 1554.10
+> 3600
+> 3600
+BD
+259.63
+351.39
+624.08
+816.44 1017.95
+DD-BD 189.07
+388.85
+572.52
+764.76
+961.35
+3
+MIP
+139.70
+702.96 1035.79
+> 3600
+> 3600
+BD
+246.48
+569.37
+628.84
+795.56
+978.15
+DD-BD 142.81
+284.65
+422.52
+565.23
+725.86
+4
+MIP
+153.16
+415.46
+938.03 1681.21 2604.25
+BD
+238.33
+388.19
+563.15
+732.59
+919.08
+DD-BD 131.29
+262.36
+393.18
+521.12
+654.71
+5
+MIP
+165.57
+706.16 2447.15
+> 3600
+> 3600
+BD
+194.12
+244.61
+479.32
+463.63
+617.09
+DD-BD 112.09
+221.30
+332.25
+443.96
+556.33
+Table 3
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 80.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+215.82
+860.21
+> 3600
+> 3600
+> 3600
+BD
+588.51
+806.61 1731.50 1860.12 2051.52
+DD-BD 256.12
+500.52
+757.68 1025.88 1278.13
+2
+MIP
+479.76
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+398.29
+713.01
+861.65 1080.79 1709.04
+DD-BD 184.34
+379.04
+724.66 1088.21 1587.90
+3
+MIP
+238.79
+996.22
+> 3600
+> 3600
+> 3600
+BD
+702.18 1236.58 1650.42 1773.63 2227.89
+DD-BD 285.13
+518.46
+778.97 1046.39 1326.22
+4
+MIP
+404.26 2441.64 2855.29
+> 3600
+> 3600
+BD
+572.83 1219.37 1334.21 1745.91 2089.80
+DD-BD 263.78
+665.30 1230.81 1277.93 1444.02
+5
+MIP
+778.50
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+231.11
+481.31
+625.91 1310.24 1452.27
+DD-BD 187.34
+376.96
+564.34 1205.54 1412.94
+Table 4
+Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 100.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+MIP
+774.18
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+1282.59 1728.71 1848.49 2307.74 3309.93
+DD-BD
+698.36 1427.38 1731.95 2014.96 3323.54
+2
+MIP
+480.97
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+781.47 1573.23 1820.79 2672.18 2819.61
+DD-BD
+586.89 1171.96 1848.49 2471.49 2635.22
+3
+MIP
+3071.37
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+1072.14 1322.96 2112.50 2951.55 3412.99
+DD-BD
+485.31
+703.70 1055.36 1803.66 2269.97
+4
+MIP
+838.79 2585.38
+> 3600
+> 3600
+> 3600
+BD
+1548.93 1738.92 2580.53 2616.19 3169.28
+DD-BD
+554.89
+743.64 1098.82 2052.73 3094.23
+5
+MIP
+714.39
+> 3600
+> 3600
+> 3600
+> 3600
+BD
+808.48 1013.68 1722.01 2824.14 3282.10
+DD-BD
+353.48
+700.57 1680.60 2213.81 2907.78
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+25
+Figure 5
+Comparison of DD-BD, BD, and MIP models when |V ′| = 40 under five scenarios
+Figure 6
+Comparison of DD-BD, BD, and MIP models when |V ′| = 60 under five scenarios
+We conclude this section by noting that, while the focus of this paper has been on the unit train
+problem with the no-split no-merge requirements, the proposed DD-BD framework can be applied
+to model network problems that contain additional side constraints on the flow variables, as those
+constraints can be handled in the subproblems while the DD structure in the master problem
+
+4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIP4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+IDD-BD
+■BDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+26
+Figure 7
+Comparison of DD-BD, BD, and MIP models when |V ′| = 80 under five scenarios
+Figure 8
+Comparison of DD-BD, BD, and MIP models when |V ′| = 100 under five scenarios
+remains intact. Examples of such side constraints include the usage-fee limitation (Holzhauser,
+Krumke, and Thielen 2017b) and the flow ratio requirement (Holzhauser, Krumke, and Thielen
+2017a). Applying the DD-BD method to such network models and assessing its effectiveness com-
+pared to alternative approaches could be an interesting direction for future research.
+
+4000.00
+3500.00
+3000.00
+Running time (sec)
+2500.00
+2000.00
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIP4000.00
+3500.00
+3000.00
+time (sec)
+2500.00
+2000.00
+Running 
+1500.00
+1000.00
+500.00
+0.00
+50
+100
+150
+200
+250
+Number of scenarios
+DD-BDBDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+27
+5.
+Conclusion
+In this paper, we introduce a DD-based framework to solve the SGUFP. This framework uses
+Benders decomposition to decompose the SGUFP into a master problem composed of the combi-
+natorial NSNM constraints, and a subproblem that solves a continuous network flow model. The
+master problem is modeled by a DD, which is successively refined with respect to the cuts generated
+through subproblems. To assess the performance of the proposed method, we apply it to a variant
+of unit train scheduling problem formulated as a SGUFP, and compare it with the standard MIP
+and Benders reformulation of the problem.
+Acknowledgments
+This project is sponsored in part by the Iowa Energy Center, Iowa Economic Development Authority and
+its utility partners. We thank the anonymous referees and the Associate Editor for their helpful comments
+that contributed to improving the paper.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+28
+References
+Abbink E, Van den Berg B, Kroon L, Salomon M, 2004 Allocation of railway rolling stock for passenger
+trains. Transportation Science 38(1):33–41.
+Alfieri A, Groot R, Kroon L, Schrijver A, 2006 Efficient circulation of railway rolling stock. Transportation
+Science 40(3):378–391.
+Andersen HR, Hadzic T, Hooker JN, Tiedemann P, 2007 A constraint store based on multivalued decision
+diagrams. International Conference on Principles and Practice of Constraint Programming, 118–132
+(Springer).
+Association of American Railroads, 2021 Freight railroads fact sheet. https://www.aar.org, Accessed:
+06/28/2021.
+Baier G, K¨ohler E, Skutella M, 2005 The k-splittable flow problem. Algorithmica 42(3):231–248.
+Bergman D, Cire AA, 2018 Discrete nonlinear optimization by state-space decompositions. Management
+Science 64(10):4700–4720.
+Bergman D, Cire AA, Van Hoeve WJ, Hooker J, 2016a Decision diagrams for optimization, volume 1
+(Springer).
+Bergman D, Cire AA, Van Hoeve WJ, Hooker JN, 2016b Discrete optimization with decision diagrams.
+INFORMS Journal on Computing 28(1):47–66.
+Bornd¨orfer R, Reuther M, Schlechte T, Waas K, Weider S, 2016 Integrated optimization of rolling stock
+rotations for intercity railways. Transportation Science 50(3):863–877.
+Cacchiani V, Toth P, 2012 Nominal and robust train timetabling problems. European Journal of Operational
+Research 219(3):727–737.
+Carey M, Crawford I, 2007 Scheduling trains on a network of busy complex stations. Transportation Research
+Part B: Methodological 41(2):159–178.
+Ceselli A, Gatto M, L¨ubbecke ME, Nunkesser M, Schilling H, 2008 Optimizing the cargo express service of
+swiss federal railways. Transportation Science 42(4):450–465.
+Chakrabarti A, Chekuri C, Gupta A, Kumar A, 2007 Approximation algorithms for the unsplittable flow
+problem. Algorithmica 47(1):53–78.
+Cordeau JF, Toth P, Vigo D, 1998 A survey of optimization models for train routing and scheduling. Trans-
+portation Science 32(4):380–404.
+Cornelsen S, Di Stefano G, 2007 Track assignment. Journal of Discrete Algorithms 5(2):250–261.
+Davarnia D, 2021 Strong relaxations for continuous nonlinear programs based on decision diagrams. Opera-
+tions Research Letters 49(2):239–245.
+Davarnia D, Richard JPP, I¸cy¨uz-Ay E, Taslimi B, 2019 Network models with unsplittable node flows with
+application to unit train scheduling. Operations Research 67(4):1053–1068.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+29
+Davarnia D, Van Hoeve WJ, 2020 Outer approximation for integer nonlinear programs via decision diagrams.
+Mathematical Programming 1–40.
+Demir E, Burgholzer W, Hruˇsovsk`y M, Arıkan E, Jammernegg W, Van Woensel T, 2016 A green inter-
+modal service network design problem with travel time uncertainty. Transportation Research Part B:
+Methodological 93:789–807.
+Fuchsberger M, L¨uthi P, 2007 Solving the train scheduling problem in a main station area via a resource
+constrained space-time integer multi-commodity flow. Institute for Operations Research ETH Zurich .
+Furchtgott-Roth D, Hu PS, Nguyen L, Jahanmir S, Moore WH, Riley D, Beningo S, Chambers M, Smith-
+Pickel S, Thai H, et al., 2021 Pocket Guide to Transportation 2021 .
+Gong C, Shi J, Wang Y, Zhou H, Yang L, Chen D, Pan H, 2021 Train timetabling with dynamic and ran-
+dom passenger demand: A stochastic optimization method. Transportation Research Part C: Emerging
+Technologies 123:102963.
+Gonzalez JE, Cire AA, Lodi A, Rousseau LM, 2020 Integrated integer programming and decision diagram
+search tree with an application to the maximum independent set problem. Constraints 1–24.
+Haahr J, Lusby RM, 2017 Integrating rolling stock scheduling with train unit shunting. European Journal of
+Operational Research 259(2):452–468.
+Haahr JT, Wagenaar JC, Veelenturf LP, Kroon LG, 2016 A comparison of two exact methods for passenger
+railway rolling stock (re) scheduling. Transportation Research Part E: Logistics and Transportation
+Review 91:15–32.
+Had˘zi´c T, Hooker J, 2006 Discrete global optimization with binary decision diagrams. Workshop on Global
+Optimization: Integrating Convexity, Optimization, Logic Programming, and Computational Algebraic
+Geometry (GICOLAG). Vienna.
+Harrod S, Gorman MF, 2010 Operations research for freight train routing and scheduling. Wiley Encyclopedia
+of Operations Research and Management Science .
+Heil J, Hoffmann K, Buscher U, 2020 Railway crew scheduling: Models, methods and applications. European
+Journal of Operational Research 283(2):405–425.
+Holzhauser M, Krumke SO, Thielen C, 2017a Maximum flows in generalized processing networks. Journal of
+Combinatorial Optimization 33(4):1226–1256.
+Holzhauser M, Krumke SO, Thielen C, 2017b A network simplex method for the budget-constrained minimum
+cost flow problem. European journal of operational research 259(3):864–872.
+Hosseininasab A, Van Hoeve WJ, 2021 Exact multiple sequence alignment by synchronized decision diagrams.
+INFORMS Journal on Computing 33(2):721–738.
+Hu Y, Lan J, Wan C, 2009 An algorithm for unsplittable flow problem in flexible reconfigurable network. 2009
+Fourth International Conference on Frontier of Computer Science and Technology, 543–547 (IEEE).
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+30
+Huntley CL, Brown DE, Sappington DE, Markowicz BP, 1995 Freight routing and scheduling at CSX trans-
+portation. Interfaces 25(3):58–71.
+I¸cy¨uz IE, Richard JPP, Eskigun E, Acharya D, 2016 A two-model solution approach for the monthly coal
+train reservations planning problem. Transportation Science 50(3):926–946.
+Jin G, He S, Li J, Guo X, Li Y, 2019 An approach for train stop planning with variable train length and stop
+time of high-speed rail under stochastic demand. IEEE Access 7:129690–129708.
+Jordan WC, Turnquist MA, 1983 A stochastic, dynamic network model for railroad car distribution. Trans-
+portation Science 17(2):123–145.
+Jovanovi´c D, Harker PT, 1991 Tactical scheduling of rail operations: the scan i system. Transportation Science
+25(1):46–64.
+Kleinberg JM, 1996 Approximation algorithms for disjoint paths problems. Ph.D. thesis, Massachusetts Insti-
+tute of Technology.
+Kolman P, Scheideler C, 2006 Improved bounds for the unsplittable flow problem. Journal of Algorithms
+61(1):20–44.
+Kwan RS, 2011 Case studies of successful train crew scheduling optimisation. Journal of Scheduling
+14(5):423–434.
+Larsen R, Pranzo M, D’Ariano A, Corman F, Pacciarelli D, 2014 Susceptibility of optimal train schedules to
+stochastic disturbances of process times. Flexible Services and Manufacturing Journal 26(4):466–489.
+Lawley M, Parmeshwaran V, Richard JP, Turkcan A, Dalal M, Ramcharan D, 2008 A time–space scheduling
+model for optimizing recurring bulk railcar deliveries. Transportation Research Part B: Methodological
+42(5):438–454.
+Layeb SB, Jaoua A, Jbira A, Makhlouf Y, 2018 A simulation-optimization approach for scheduling in stochas-
+tic freight transportation. Computers & Industrial Engineering 126:99–110.
+Lin Z, Kwan RS, 2014 A two-phase approach for real-world train unit scheduling. Public Transport 6(1-2):35–
+65.
+Lin Z, Kwan RS, 2016 A branch-and-price approach for solving the train unit scheduling problem. Trans-
+portation Research Part B: Methodological 94:97–120.
+Lin Z, Kwan RS, 2018 Redundant coupling/decoupling in train unit scheduling optimization. Electronic Notes
+in Discrete Mathematics 64:45–54.
+Liu SQ, Kozan E, 2011 Optimising a coal rail network under capacity constraints. Flexible Services and
+Manufacturing Journal 23(2):90–110.
+Lusby RM, 2008 Optimization methods for routing trains through railway junctions. Ph.D. thesis,
+ResearchSpace@ Auckland.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+31
+Lusby RM, Larsen J, Ehrgott M, Ryan D, 2011 Railway track allocation: models and methods. OR spectrum
+33(4):843–883.
+Meng L, Zhou X, 2011 Robust single-track train dispatching model under a dynamic and stochastic environ-
+ment: A scenario-based rolling horizon solution approach. Transportation Research Part B: Method-
+ological 45(7):1080–1102.
+Quaglietta E, Corman F, Goverde RM, 2013 Stability of railway dispatching solutions under a stochastic
+and dynamic environment. RailCopenhagen2013: 5th International Seminar on Railway Operations
+Modelling and Analysis (IAROR) (Institute for Transport Planning and Systems, ETH Zurich).
+Salemi H, Davarnia D, 2022a On the structure of decision diagram-representable mixed integer programs with
+application to unit commitment. Operations Research URL https://doi.org/10.1287/opre.2022.
+2353.
+Salemi H, Davarnia D, 2022b Test instances for SGUFP. https://doi.org/10.5281/zenodo.6373664.
+Serra T, Hooker JN, 2019 Compact representation of near-optimal integer programming solutions. Mathe-
+matical Programming 1–34.
+Shen Y, Peng K, Chen K, Li J, 2013 Evolutionary crew scheduling with adaptive chromosomes. Transportation
+Research Part B: Methodological 56:174–185.
+Sherali HD, Suharko AB, 1998 A tactical decision support system for empty railcar management. Transporta-
+tion Science 32(4):306–329.
+Turner C, Tiwari A, Starr A, Blacktop K, 2016 A review of key planning and scheduling in the rail industry
+in Europe and UK. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and
+Rapid Transit 230(3):984–998.
+Walkowiak K, 2006 New algorithms for the unsplittable flow problem. International Conference on Compu-
+tational Science and Its Applications, 1101–1110 (Springer).
+Ying Cs, Chow AH, Chin KS, 2020 An actor-critic deep reinforcement learning approach for metro train
+scheduling with rolling stock circulation under stochastic demand. Transportation Research Part B:
+Methodological 140:210–235.
+Zwaneveld PJ, Kroon LG, Van Hoesel SP, 2001 Routing trains through a railway station based on a node
+packing model. European Journal of Operational Research 128(1):14–33.
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+32
+Appendix A:
+Comparison of Master Problem Formulations
+In this section, we describe the differences between DDs in the space of w variables and those in the space
+of original y in the master problem formulation (4) in Section 3.2. First, we illustrate the size difference
+between these DDs in Example 3.
+Example 3. Consider a directed graph G = (V,A) with node set V = {1,2,q,3,4} and arc set A =
+{(1,q),(2,q),(q,3),(q,4)} where the central node q is subject to NSNM constraints. Let ind−(1,q) =
+ind+(q,3) = 1 and ind−(2,q) = ind+(q,4) = 2. Then, the exact DDs showed in Figures 9(a) and 9(b) with
+three and five arc layers represent the feasible region of master problem (4) and (1a)-(1d), respectively, where
+−M and M are valid bounds for variable z.
+(a) A DD in the space of w variables.
+Numbers next to arcs represent labels.
+(b) A DD in the space of y variables.
+Numbers next to arcs represent labels.
+Figure 9
+Comparison of the number of arc layers for DDs in the space of w and y variables
+As evident from the above example, the main advantage of using a DD in the space of w is the reduction in
+the number of arc layers, which is the main determinant of the DDs computational efficiency. In particular,
+even though such a DD has a larger number of nodes at the layers, a relaxed DD can be constructed to limit
+the width, and hence provide an efficient relaxed DD in a smaller dimension, whereas the relaxations of the
+DD constructed in the space of y variables would still be higher-dimensional.
+To assess the computational efficiency of the solution approach in relation to the DD space, we compare
+the performance of the DD-BD method under two different settings: (i) where DDs are built in the space
+of w variables, denoted by DD-BD-w, and (ii) where DDs are built in the space of y variables, denoted by
+DD-BD-y. We report the results of these two implementations for |V ′| ∈ {40,80} and under five different
+scenarios in Table 5 and Table 6.
+As observed in these tables, the DD-BD-w solves all instances faster than DD-BD-y, with orders of
+magnitude time improvement as the problem size (number of scenarios) increases. These preliminary com-
+putational results show the advantage of designing the DD-BD method for the SGUFP in a transformed
+space of variables.
+
+2
+0
+1
+0
+2
+0
+2
+1
+0
+W
+M
+M
+M
+-M
+M
+M
+I七0
+91,3
+1
+y2,3
+0
+0
+1
+0
+0
+0
+b
+y2,4
+0
+0
+0
+M
+M
+M
+-M
+2
+M
+MSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+33
+Table 5
+Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 40.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD-w
+56.94 129.87 163.43 219.02 274.36
+DD-BD-y
+89.68 304.08 432.34 642.70 839.57
+2
+DD-BD-w
+42.60
+82.65 128.16 164.52 208.94
+DD-BD-y
+68.23 148.76 244.53 344.86 605.04
+3
+DD-BD-w
+53.32
+93.58 113.93 178.65 217.33
+DD-BD-y
+83.05 157.67 310.07 541.33 658.98
+4
+DD-BD-w
+46.61
+87.81 130.19 183.23 253.72
+DD-BD-y
+78.11 149.26 325.31 460.73 694.57
+5
+DD-BD-w
+67.04 104.78 138.46 195.69 231.74
+DD-BD-y
+109.61 223.78 351.80 532.12 669.78
+Table 6
+Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 80.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD-w 256.12
+500.52
+757.68 1025.88 1278.13
+DD-BD-y
+483.42
+977.03 1642.27 3175.72 4230.29
+2
+DD-BD-w 184.34
+379.04
+724.66 1088.21 1587.90
+DD-BD-y
+340.13
+864.21 1856.96 3010.55 4843.67
+3
+DD-BD-w 285.13
+518.46
+778.97 1046.39 1326.22
+DD-BD-y
+568.32 1176.44 2401.98 3326.76 4283.58
+4
+DD-BD-w 263.78
+665.30 1230.81 1277.93 1444.02
+DD-BD-y
+501.04 1430.77 2868.92 3356.39 4356.48
+5
+DD-BD-w 187.34
+376.96
+564.34 1205.54 1412.94
+DD-BD-y
+354.37
+781.18 1279.73 3001.72 3834.08
+Appendix B:
+Additional Computational Experiments
+In this section, we present additional numerical results to assess the limits of the DD-BD method for larger
+problem instances. These results are given in Tables 7 and 8, where the columns are defined similarly to
+those of Tables 1-4. For these instances, the time limit is set to 3600 seconds, and the symbol “> 3600”
+indicates that the problem is not solved within this time limit.
+Table 7
+Running times (in seconds) of DD-BD for |V ′| = 120.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD 1494.49 2824.58
+> 3600 > 3600 > 3600
+2
+DD-BD
+975.47 1892.41 3198.18 > 3600 > 3600
+3
+DD-BD 1150.30 2263.09 3454.47 > 3600 > 3600
+4
+DD-BD 1261.59 2403.79
+> 3600 > 3600 > 3600
+5
+DD-BD
+906.34 1863.15 3050.68 > 3600 > 3600
+
+Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams
+34
+Table 8
+Running times (in seconds) of DD-BD for |V ′| = 150.
+Instance # Model
+Number of scenarios
+50
+100
+150
+200
+250
+1
+DD-BD 2496.16 > 3600 > 3600 > 3600 > 3600
+2
+DD-BD 2944.20 > 3600 > 3600 > 3600 > 3600
+3
+DD-BD 2321.62 > 3600 > 3600 > 3600 > 3600
+4
+DD-BD 2590.34 > 3600 > 3600 > 3600 > 3600
+5
+DD-BD 2298.36 > 3600 > 3600 > 3600 > 3600
+
diff --git a/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/load_file.txt b/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ede704cf09c66647e6ffb13bc2c1d7bb48443641
--- /dev/null
+++ b/hNAzT4oBgHgl3EQf4f5B/content/tmp_files/load_file.txt
@@ -0,0 +1,1237 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf,len=1236
+page_content='Solving Unsplittable Network Flow Problems with  Decision Diagrams  Hosseinali Salemi, Danial Davarnia  Department of Industrial and Manufacturing Systems Engineering, Iowa State University, Ames, IA 50011,  hsalemi@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='edu, davarnia@iastate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='edu  In unsplittable network flow problems, certain nodes must satisfy a combinatorial requirement that the  incoming arc flows cannot be split or merged when routed through outgoing arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This so-called no-split  no-merge requirement arises in unit train scheduling where train consists should remain intact at stations  that lack necessary equipment and manpower to attach/detach them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Solving the unsplittable network  flow problems with standard mixed-integer programming formulations is computationally difficult due to  the large number of binary variables needed to determine matching pairs between incoming and outgoing  arcs of nodes with no-split no-merge constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this paper, we study a stochastic variant of the unit  train scheduling problem where the demand is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We develop a novel decision diagram (DD)-based  framework that decomposes the underlying two-stage formulation into a master problem that contains the  combinatorial requirements, and a subproblem that models a continuous network flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The master  problem is modeled by a DD in a transformed space of variables with a smaller dimension, leading to a  substantial improvement in solution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Similarly to the Benders decomposition technique, the subproblems  output cutting planes that are used to refine the master DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Computational experiments show a significant  improvement in solution time of the DD framework compared with that of standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Key words : Decision Diagrams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Network Optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Mixed Integer Programs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Unit Trains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Transportation  History :  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Introduction  Over the past several decades, rail freight transportation has continued to grow as the prime  means of transportation for high-volume commodities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Advantages of rail transportation include  reliability, safety, cost-efficiency and environmental-sustainability as compared with alternative  methods of transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In terms of scale, the rail network accounted for 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2 percent of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  freight shipment by ton-miles in 2018 (Furchtgott-Roth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The Federal  Highway Administration estimates that the total U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' freight shipments will be 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1 billion tons  in 2040, a 30 percent increase from the 2018 total transportation of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='6 billion tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' With the  purpose of meeting such market growth, America’s freight railway companies have invested nearly  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='01844v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='OC]  4 Jan 2023  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  2  $740 billion on capital expenditures and maintenance from 1980 to 2020 (Association of American  Railroads 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Figure 1  Pie chart for ton-miles of freight shipments by mode within the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Multiple modes includes  mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Air and truck-air with the share of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1% are omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  To reduce rail freight transportation costs and shipment delays, railroad companies offer unit  train services for carrying high-volume products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Unit trains haul a single type freight in a way that  no car is attached or detached while the cargo train is on its way from an origin to a destination,  except in specific locations that are equipped with required manpower and machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These trains  usually operate all day, use dedicated equipment, and can be loaded/unloaded in 24 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' They  are known to be one of the fastest and most efficient means of railroad transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (Association  of American Railroads 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Traditionally, unit trains are used to carry bulk cargo such as coal,  grain, cement, and rock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Bulk liquids like crude oil and food such as wheat and corn are also  shipped by unit trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' According to the Federal Railroad Administration data, bulk commodities  account for 91 percent of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' railroad freights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Approximately all coal shipped through railways  in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' are transported by unit trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Moreover, these trains contribute significantly to the  shipping process of crude oil as each unit train is capable of carrying 85,000 barrels (Association  of American Railroads 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In an operational level, the core unit train model can be described  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Given a set of supply, intermediate, and demand locations in a railroad network, the  unit train scheduling problem seeks to find optimal routes for unit trains to send flows from supply  to demand points with the objective of minimizing the total transportation cost while meeting  demand of customers, respecting capacities of tracks, and satisfying no-car attaching/detaching  requirements in specific locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, designing blocking plans to determine locations  Multiple modes  8%  Pipeline, 19%  Truck, 39%  Water, 7%  Rail, 27%Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  3  where cars need to be switched between trains is irrelevant in this problem, unlike scheduling other  types of trains (Davarnia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Despite the significance of unit train scheduling, exact optimization approaches to solve associ-  ated problems are scarce, partially due to their structural complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' One of the main challenges  in modeling unit trains is the requirement that the train consists must remain intact when passing  through stations that lack necessary busting/formation equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In optimization, this require-  ment is referred to as no-split no-merge (NSNM), which guarantees that the flows entering to or  exiting from certain nodes of the unit train network cannot be split or merged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Incorporating this  requirement into typical transportation network models yields the so-called generalized unsplittable  flow problem (GUFP), where the objective is to determine the minimum-cost unit train sched-  ules that satisfy the given demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Numerous studies have shown that considering deterministic  demands might result in the complete failure of the transportation scheduling (Demir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2016,  Layeb et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2018), motivating the study of stochastic variants of the unit train scheduling problems  where the demand is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, in this paper, we consider a stochastic variant of the  GUFP, referred to SGUFP, that is modeled as a two-stage optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The first stage  decides a matching between the incoming and outgoing arcs of the nodes of the railroad network,  and the second stage determines the amount of flow that should be sent through the matching arcs  of the network to satisfy the uncertain demand represented by a number of demand scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We  propose a novel exact solution framework to solve this problem in the operational level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Our proposed methodology is based on decision diagrams (DDs), which are compact graphical  data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' DDs were initially introduced to represent boolean functions with applications in  circuit design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Over the past decade, researchers have successfully extended DDs domain by devel-  oping DD-based algorithms to solve discrete optimization problems in different areas of application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Because of its structural limitation to model integer programs only, DDs have never been used  to solve transportation problems that inherently include continuous variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this paper, we  extend the application scope of DDs by introducing a novel framework that is capable of modeling  network problems with both integer and continuous components as in the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Literature Review on Train Scheduling  Many variants of train routing and scheduling problems with different objective functions and  set of constraints under deterministic and stochastic conditions have been introduced and vastly  studied in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see surveys by Cordeau, Toth, and Vigo (1998), Harrod and Gorman  (2010), Lusby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2011), Cacchiani and Toth (2012), and Turner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2016) for different  problems classifications and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Mixed integer linear and nonlinear programming formu-  lations are among the most frequent exact approaches to model different classes of these prob-  lems (Jovanovi´c and Harker 1991, Huntley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 1995, Sherali and Suharko 1998, Lawley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  4  2008, Haahr and Lusby 2017, Davarnia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Proposed solution techniques include but are  not limited to branch-and-bound methods (Jovanovi´c and Harker 1991, Fuchsberger and L¨uthi  2007), branch-and-cut frameworks (Zwaneveld, Kroon, and Van Hoesel 2001, Ceselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2008),  branch-and-price approaches (Lusby 2008, Lin and Kwan 2016), graph coloring algorithms (Cor-  nelsen and Di Stefano 2007), and heuristics (Carey and Crawford 2007, Liu and Kozan 2011, I¸cy¨uz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Rolling stock scheduling (Abbink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2004, Alfieri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2006, Haahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2016,  Bornd¨orfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2016) that assigns rolling stocks to a given timetable, and crew scheduling (Kwan  2011, Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2013, Heil, Hoffmann, and Buscher 2020) that covers train activities by assigning  crews to the associated operations are other major problems arising in the area of railroad planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Due to the inherent uncertainty in different types of train scheduling and routing problems, many  researchers have studied stochastic variants of the problems where the supply/demand is considered  to be uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Jordan and Turnquist (1983) propose a model for railroad car distribution where  supply and demand of cars are uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2019) study a chance-constrained programming  model for the train stop planning problem under stochastic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Ying, Chow, and Chin (2020)  propose a deep reinforcement learning approach for train scheduling where the passenger demand  is uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Recently, Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2021) propose a stochastic optimization method to solve a train  timetabling problem with uncertain passenger demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Also see works by Meng and Zhou (2011),  Quaglietta, Corman, and Goverde (2013), Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2014) that consider train dispatching  problems under stochastic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  In the context of unit train scheduling, Lawley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2008) study a time-space network flow  model to schedule bulk railroad deliveries for unit trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In their model, the authors consider char-  acteristics of underlying rail network, demands of customers, and capacities of tracks, stations, and  loading/unloading requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' They propose a mixed integer programming (MIP) formulation  that maximizes the demand satisfaction while minimizing the waiting time at stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Lin and  Kwan (2014) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Lin and Kwan (2016)) propose a model for a train scheduling problem that is capa-  ble to capture locations where coupling/decoupling is forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' They develop a branch-and-price  algorithm inspired by column generation to solve the associated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Lin and Kwan (2018)  also propose a heuristic branch-and-bound approach to decrease coupling/decoupling redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  I¸cy¨uz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2016) study the problem of planning coal unit trains that includes train formation,  routing, and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As noted by the authors, their proposed MIP formulation fails to solve  the problem directly due to its large size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a remedy, they develop a time-efficient heuristic that  produces good quality solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' More recently, Davarnia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2019) introduce and study the  GUFP with application to unit train scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular, the authors show how to impose  NSNM restrictions in network optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' They present a polyhedral study and pro-  pose a MIP formulation to model a stylized variant of the unit train scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the  present paper, we use their formulation (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1) as a basis for our solution framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  5  The unsplittable flow problem (UFP) was first introduced by Kleinberg (1996) as a generalization  of the disjoint path problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Given a network with capacities for arcs and a set of source-terminal  vertex pairs with associated demands and rewards, the objective in the UFP is to maximize the  total revenue by selecting a subset of source-terminal pairs and routing flows through a single  path for each of them to satisfy the associated demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the GUFP, however, there can exist  nodes that do not need to respect the NSNM requirement, and demands can be satisfied by  passing flows through multiple paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It is well-known that different variants of UFP are NP-  hard (Baier, K¨ohler, and Skutella 2005, Kolman and Scheideler 2006, Chakrabarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Since  its introduction, the UFP structure has been used in different areas of application, from bandwidth  allocation in heterogeneous networks (Kolman and Scheideler 2006), to survivable connection-  oriented networks (Walkowiak 2006), and virtual circuit routing problems (Hu, Lan, and Wan  2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Considering the hardness of the problem, approximation algorithms have been a common  technique to tackle different variants of the UFP in the literature (Baier, K¨ohler, and Skutella  2005, Chakrabarti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Literature Review on Decision Diagrams  DDs are directed acyclic graphs with a source and a terminal node where each source-terminal path  encodes a feasible solution to an optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In DDs, each layer from the source to the  terminal represents a decision variable where labels of arcs show their values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Had˘zi´c and Hooker  (2006) proposed to use DDs to model the feasible region of a discrete optimization problem and used  it for postoptimality analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Later, Andersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2007) presented relaxed DDs to circumvent  the exponential growth rate in the DD size when modeling large discrete optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Bergman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2016b) introduced a branch-and-bound algorithm that iteratively uses relaxed and  restricted DDs to find optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The literature contains many successful utilization of DDs  in different domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see works by Bergman and Cire (2018), Serra and Hooker (2019), Davarnia  and Van Hoeve (2020), Gonzalez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2020), and Hosseininasab and Van Hoeve (2021) for some  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Until recently, applications of DDs were limited to discrete problems, and the question on how to  use DDs in solving optimization problems with continuous variables was unanswered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To address  this limitation, Davarnia (2021) proposed a technique called arc-reduction that generates a DD  that represents a relaxation of the underlying continuous problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In a follow-up work, Salemi  and Davarnia (2022a) established necessary and sufficient conditions for a general MIP to be  representable by DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' They showed that a bounded MIP can be remodeled and solved with DDs  through employing a specialized Benders decomposition technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this paper, we build on this  framework to design a novel DD-based methodology to solve the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Contributions  While there are several studies in the literature dedicated to the unit train problem, exact method-  ologies that provide a rigorous treatment of the NSNM requirement at the heart of unit train  models are scarce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this paper, we design a novel exact DD-based framework to solve the SGUFP,  as a more realistic and more challenging variant of this problem class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To our knowledge, this is  the first work that studies SGUFP from an exact perspective, and the first application of DDs  to a transportation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Our proposed framework formulates the problem in a transformed  space of variables, which has a smaller dimension compared to the standard MIP formulations  of the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This presentation mitigates the computational difficulties stemmed from the MIP  formulation size, providing a viable solution approach for large-scale network problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The core  principles of our DD framework can also be used to model other transportation problems with  similar structure, as an alternative to traditional network optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In Section 2 we provide basic definitions  and a brief overview on discrete and continuous DD models, including the DD-BD method to  solve bounded MIPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In Section 3, we adapt the DD-BD method to solve the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We propose  algorithms to construct exact and relaxed DDs to solve the problem in a transformed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Section 4 presents computational experiments to evaluate the performance of the DD-BD method  for the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We give concluding remarks in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Background on DDs  In this section, we present basic definitions and results relevant to our DD analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Overview  A DD D = (U,A,l) with node set U, arc set A, and arc label mapping l : A → R is a directed acyclic  graph with n ∈ N arc layers A1,A2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',An, and n + 1 node layers U1,U2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',Un+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The node layers  U1 and Un+1, with |U1| = |Un+1| = 1, contain the root r and the terminal t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In any arc  layer j ∈ [n] := {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',n}, an arc (u,v) ∈ Aj is directed from the tail node u ∈ Uj to the head node  v ∈ Uj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The width of D is defined as the size of its largest Uj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' DDs can model a bounded integer  set P ⊆ Zn in such a way that each r-t arc-sequence (path) of the form (a1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',an) ∈ A1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' × An  encodes a point y ∈ P where l(aj) = yj for j ∈ [n], that is y is an n-dimensional point in P whose  j-th coordinate is equal to the label value l(aj) of arc the aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For such a DD, we have P = Sol(D),  where Sol(D) denotes the finite collection of all r-t paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The graphical property of DDs can be exploited to optimize an objective function over a discrete  set P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To this end, DD arcs are weighted in such a way that the cumulative weight of an r-t  path that encodes a solution y ∈ P equals to the objective function value evaluated at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, a  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  7  shortest (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' longest) r-t path for the underlying minimization (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' maximization) problem is  found, an operation that can be performed in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The construction of an exact DD as described above is computationally prohibitive due to the  exponential growth rate of its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To alleviate this difficulty, relaxed and restricted DDs are  proposed to keep the size of DDs under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In a relaxed DD, nodes are merged in such a way  that the width of the resulting diagram is bounded by a predetermined width limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This node-  merging process ensures that all feasible solutions of the original set are encoded by a subset of  all r-t paths in the resulting DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Optimization over this relaxed DD provides a dual bound to the  optimal solution of the original problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In a restricted DD, the collection of all r-t paths of the  DD encode a subset of the feasible solutions of the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Optimization over this restricted  DD provides a primal bound to the optimal solution of the original problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The restricted and  relaxed DDs can be iteratively refined in a branch-and-bound scheme to find the optimal value of  a problem through convergence of their primal and dual bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The following example illustrates  an exact, relaxed and restricted DD for a discrete optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider the discrete optimization problem max{5y1 + 10y2 + 4y3 | y ∈ P} where  P = {(1,0,0),(1,0,1),(0,1,0),(0,0,1),(0,0,0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The exact DD D with width 3 in Figure 2(a) models  the feasible region P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The weight of each arc a ∈ Aj, for j ∈ {1,2,3}, shows the contribution of  variable yj’s value assignment to the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The longest r-t path that encodes the  optimal solution (y∗  1,y∗  2,y∗  3) = (0,1,0) has length 10, which is the optimal value to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  By reducing the width limit to 2, we can build relaxed and restricted DDs for P as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The  relaxed DD D in Figure 2(b) provides an upper bound to the optimal solution, where the longest  path with length 14 is obtained by an infeasible point (y1,y2,y3) = (0,1,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Finally, the restricted  DD D in Figure 2(c) gives a lower bound to the optimal solution, where the longest path with  length 9 encodes a feasible solution (y1,y2,y3) = (1,0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Continuous DD Models  While the framework described in the previous section can be applied to solve different classes of  discrete optimization problems, its extension to model sets with continuous variables requires a  fundamentally different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The reason that the traditional DD structure is not viable for  continuous sets is that representing the domain of a continuous variable through arcs requires an  infinite number of them, spanning all values within a continuous interval, which is structurally  prohibitive in DD graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Fortunately, there is a way to overcome this obstacle by decomposing  the underlying set into certain rectangular formations, which can in turn be represented through  node-sequences in DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In what follows, we give an overview of these results as relevant to our  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  8  r  t  5  0  0  10  0  0  4  0  4  0  (a) Exact DD D  r  t  5  0  0  10  0  0  4  4  0  (b) Relaxed DD D  r  t  5  0  0  0  0  4  4  0  y1  y2  y3  (c) Restricted DD D  Figure 2  The exact, relaxed, and restricted DDs representing P in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Solid and dotted arcs indicate  one and zero arc labels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Numbers next to arcs represent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Consider a bounded set P ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Salemi and Davarnia (2022a) give necessary and sufficient  conditions for P to admit the desired rectangular decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Such a set is said to be DD-  representable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' a fixed index set I ⊆ [n], as there exists a DD D such that max{f(x) | x ∈  P} = max{f(x) | x ∈ Sol(D)} for every function f(x) that is convex in the space of variables xI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  A special case of DD-representable sets is given next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Any bounded mixed integer set of the form P ⊆ Zn × R is DD-representable  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' I = {n + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  This result gives rise to a novel DD-based framework to solve general bounded MIPs as outlined  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a bounded MIP H := max{cy +dx | Ay +Gx ≤ b, y ∈ Zn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Using Benders decom-  position (BD), formulation H is equivalent to maxy∈Zn{cy +maxx{dx | Gx ≤ b−Ay}}, which can  be reformulated as M = max{cy +z | (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) ∈ Zn ×[l,u]}, where l,u ∈ R are some valid bounds on z  induced from the boundedness of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Here, M is the master problem and z represents the objective  value of the subproblem maxx{dx | Gx ≤ b − A¯y} for any given ¯y as an optimal solution of the  master problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The outcome of the subproblems is either an optimality cut or a feasibility cut  that will be added to the master problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, the master problem will be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Proposition 1  implies that formulation M can be directly modeled and solved with DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For this DD, we assign n  arc layers to the integer variables y1,y2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',yn, and one arc layer to the continuous variable z with  only two arc labels showing a lower and upper bound for this variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To find an optimal solution,  the longest path is calculated, which will be used to solve the subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Note that since M is  a maximization problem, a longest path of the associated DD encodes an optimal solution, and  its length gives the optimal value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The feasibility and optimality cuts generated  by the subproblems will then be added to refine the DD, whose longest path will be recalculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  9  The refinement technique consists of removing arcs of the DD that lead to solutions that violate  the added inequality, as well as splitting nodes of the DD that lead to different subsequent partial  assignments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see Bergman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2016a) for a detailed account on DD refinement techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We  illustrate this approach in Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Suppose that max{2y1 + 4y2 + z | y ∈ P,z ≤ 25} forms the master problem at the  penultimate iteration of a BD algorithm, where P = {(0,0),(1,1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This problem is represented  by the DD D in Figure 3(a) where −M is a valid lower bound for z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The longest path of D  encodes the solution (ˆy1, ˆy2, ˆz) = (1,1,25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Assume that using the point (ˆy1, ˆy2) = (1,1) in the  associated subproblem generates an optimality cut z ≤ 3y1 + 2y2 + 10 for the final iteration of the  BD algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Refining DD D with respect to this cut yields the new DD in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The  longest path represents the optimal solution (y∗  1,y∗  2,z∗) = (1,1,15) with length 21, which is the  optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  r  t  2  0  4  0  −M  25  25  −M  (a) penultimate iteration  r  t  2  0  4  0  −M  15  10  −M  y1  y2  z  (b) final iteration  Figure 3  The last two iterations of solving the master problem in Example 2  Using the DD framework as outlined above can be computationally challenging due to exponen-  tial growth rate of the size of an exact DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To mitigate this difficulty, restricted/relaxed DDs can  be employed inside of the BD framework as demonstrated in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We refer to this solution  method as DD-BD (Salemi and Davarnia 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  In explaining the steps of Algorithm 1, let point ˆy ∈ Zk, where k ≤ n, be a partial value assignment  to the first k coordinates of variable y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', yi = ˆyi for all i ∈ [k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We record the set of all partial  value assignments in ˆY = {ˆy ∈ Zk | k ∈ [n]}∪{⊖}, where ⊖ represents the case where no coordinate  of y is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Set C contains the produced Benders cuts throughout the algorithm, and we denote  the feasible region described by these cuts by F C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Further, define MC(ˆy) = max{cy + z | (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) ∈  Zn × [l,u] ∩ F C, yi = ˆyi,∀i ∈ [k]} to be the restricted master problem M obtained through adding  cuts in C and fixing the partial assignment ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this definition, the case with C = ∅ and ˆY = {⊖}  is denoted by M∅(⊖) = M, which is an input to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  10  Algorithm 1: DD-BD  Data: MIP H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' construction method to build restricted and relaxed DDs for M  Result: An optimal solution (y∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z∗) and optimal value w∗ to H  1 initialize set of partial assignments ˆY = {⊖},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' set of Benders cuts C = ∅,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' and w∗ = −∞  2 if ˆY = ∅ then  3  terminate and return (y∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z∗) and w∗  4 else  5  select ˆy ∈ ˆY and update ˆY ← ˆY \\ {ˆy}  6  create a restricted DD D associated with MC(ˆy)  7  if D ̸= ∅ then  8  find a longest r-t path of D with encoding point (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) and length w  9  solve the BD subproblem using y to obtain Benders cut C  10  if C ∈ C then  11  go to line 17  12  else  13  update C ← C ∪ C and refine D w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' C  14  go to line 8  15  else  16  go to line 2  17  if w > w∗ then  18  update w∗ ← w and (y∗,z∗) ← (y,z)  19  if D provides an exact representation of MC(ˆy) then  20  go to line 2  21  else  22  create a relaxed DD D associated with MC(ˆy)  23  find a longest r-t path of D with length w  24  if w > w∗ then  25  solve the BD subproblem using y to obtain Benders cut C  26  if C ∈ C then  27  go to line 31  28  else  29  update C ← C ∪ C and refine D w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' C  30  go to line 23  31  forall u in the last exact layer of D do  32  update ˆY ← ˆY ∪ {˜y} where ˜y encodes longest r-u path of D  33  go to line 2  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  11  The algorithm starts with constructing a restricted DD D corresponding to MC(ˆy) with empty  initial values for C and ˆy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We then find a longest r-t path of D encoding solution (y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Next,  using y, we solve the associated subproblem to obtain a feasibility/optimality cut C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We add this  cut to C, refine D according to it, and find a new longest r-t path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We repeat these steps until no  new feasibility/optimality cut is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' At this point, the length of a longest r-t path of D,  denoted by w, gives a lower bound to the master problem M, which is also a valid lower bound  to the original problem H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The value of w can be used to update w∗, the optimal value of H  at termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Next, we create a relaxed DD D corresponding to MC(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We find a longest r-t  path of D that provides an upper bound w to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' If the upper bound w is strictly greater than  the current value of w∗, we follow steps similarly to the case for D to iteratively refine D w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  feasibility/optimality cuts through solving the subproblems, until no new cut is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Next,  we perform a specialized branch-and-bound procedure to improve the bound through expanding  merged layers of the DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To this end, we add all the partial assignments associated with nodes in  the last exact layer of D (the last node layer in which no nodes are merged) to the collection ˆY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The nodes corresponding to partial assignments in ˆY are required to be further explored to check  whether or not the value of w∗ can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' That is, the above process is repeated for every  node v with partial assignment in ˆY as the r-v path is fixed in the new restricted/relaxed DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The algorithm terminates when ˆY becomes empty, at which point w∗ is the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  DD-BD Formulation for the SGUFP  In this section, we adapt the DD-BD framework described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2 to solve the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  MIP Formulation  We study the MIP formulation of the SGUFP based on that of its deterministic counterpart given  in Davarnia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a network G = (V,A) with node set V := V ′ ∪ {s,t} and arc set  A, where s and t are source and sink nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The source node is connected to all the  supply nodes in S ⊆ V ′, and the sink node is connected to all the demand nodes in D ⊆ V ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Figure 4  illustrates the general structure of this network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For a node q ∈ V , let δ−(q) := {i ∈ V | (i,q) ∈ A}  and δ+(q) := {j ∈ V | (q,j) ∈ A} show the set of incoming and outgoing neighbors of q, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Define ¯V ⊆ V ′ as a subset of vertices that must satisfy the NSNM requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each node  q ∈ ¯V , let binary variable yq  ij ∈ {0,1} represent whether or not the flow entering node q ∈ ¯V through  arc (i,q) leaves node q through arc (q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The first stage of SGUFP determines the matching pairs  between incoming and outgoing arcs of unsplittable nodes as follows:  max  z  (1a)  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  12  S  D  s  t  Figure 4  Illustration of network G = (V ′ ∪ {s,t},A)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  �  j∈δ+(q)  yq  ij ≤ 1  ∀i ∈ δ−(q), ∀q ∈ ¯V  (1b)  �  i∈δ−(q)  yq  ij ≤ 1  ∀j ∈ δ+(q), ∀q ∈ ¯V  (1c)  yq  ij ∈ {0,1}  ∀(i,j) ∈ δ−(q) × δ+(q), ∀q ∈ ¯V,  (1d)  where constraints (1b) ensure that each incoming arc to a node with NSNM requirement is  assigned to at most one outgoing arc, and constraints (1c) guarantee that each outgoing arc from  such a node is matched with at most one incoming arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  In (1a)–(1d), variable z represents the objective value of the second stage of SGUFP where  the demand uncertainty is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This demand uncertainty is modeled by a set Ξ of  scenarios for the demand vector dξ with occurrence probability Prξ for each scenario ξ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let  continuous variable xξ  ij ∈ R+ denote the flow from node i to node j through arc (i,j) under scenario  ξ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We further assign a reward rij per unit flow to be collected by routing flow through arc (i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  It follows that z = �  ξ∈Ξ Prξzξ, where zξ is the objective value of the second stage of SGUFP for  each scenario ξ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This subproblem is formulated as follows for a given y vector:  max  �  q∈V  �  j∈δ+(q)  rqjxξ  qj  (2a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  �  i∈δ−(q)  xξ  iq −  �  j∈δ+(q)  xξ  qj = 0  ∀q ∈ V ′  (2b)  ℓξ  iq ≤ xξ  iq ≤ uξ  iq  ∀i ∈ δ−(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ∀q ∈ V  (2c)  xξ  iq − xξ  qj ≤ uξ  iq(1 − yq  ij)  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ δ−(q) × δ+(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ∀q ∈ ¯V  (2d)  xξ  qj − xξ  iq ≤ uξ  qj(1 − yq  ij)  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ δ−(q) × δ+(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ∀q ∈ ¯V  (2e)  xξ  iq ≤ uξ  iq  �  j∈δ+(q)  yq  ij  ∀i ∈ δ−(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ∀q ∈ ¯V  (2f)  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  13  xξ  qj ≤ uξ  qj  �  i∈δ−(q)  yq  ij  ∀j ∈ δ+(q),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ∀q ∈ ¯V  (2g)  xξ  ij ≥ 0  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (2h)  In the above formulation, the objective function captures the total reward collected by routing  flows throughout the network (from the source s to the sink t) to satisfy demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The flow-balance  requirements are represented by (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Constraints (2c) bound the flow on each arc from below  and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To impose the demand requirement for each scenario ξ ∈ Ξ, we fix ℓξ  qt = uξ  qt = dξ  q for all  demand nodes q ∈ D with demand dξ  q, and leave the lower and upper bound values unchanged for all  other arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Constraints (2d)–(2g) model the NSNM requirement for each node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular,  (2d) and (2e) ensure that matching arcs (i,q) and (q,j) have equal flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Constraints (2f) and (2g)  guarantee that an arc without a matching pair does not carry any flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We note here that the  Constraint (2b) is implied by other constraints of the above subproblem under the assumption  that y is feasible to the master problem (1a)–(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' However, we maintain this constraint in the  subproblem because the master formulation in our DD-based approach, as will be described in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2, may produce a solution that is not feasible to (1a)–(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, the addition of  the Constraint (2b) will lead to a tighter subproblem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2, the first step to use the DD-BD algorithm is to decompose the  underlying problem into a master and a subproblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The above two-stage formulation of the  SGUFP is readily amenable to BD since the first stage problem (1a)-(1d) can be considered as the  master problem together with some valid lower and upper bounds −Γ and Γ on z induced from the  boundedness of the MIP formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For a given y value obtained from the master problem and  a scenario ξ ∈ Ξ, the second stage problem (2a)-(2h) can be viewed as the desired subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The optimality/feasibility cuts obtained from each scenario-based subproblem are then added to  the master problem through aggregation as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  DD-BD: Master Problem Formulation  While the DD-BD Algorithm 1 provides a general solution framework for any bounded MIP, its DD  component is problem-specific, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', it should be carefully designed based on the specific structure  of the underlying problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this section, we design such an oracle for the SGUFP that represents  the feasible region {(1b)−(1d),z ∈ [−Γ,Γ]} of the master problem (1a)-(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To model this feasible  region in the original space of (y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) variables, a DD would require �  q∈ ¯V |δ−(q)| × |δ+(q)| arc  layers to represent binary variables y and one arc layer to encode the continuous variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Constructing such a DD, however, would be computationally cumbersome due to the large number  of the arc layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To mitigate this difficulty, we take advantage of the structural flexibility of DDs  in representing irregular variable types that cannot be used in standard MIP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' One such  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  14  variable type is the index set, where arc layers represent indices, rather than domain values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We  next show that we can remarkably reduce the number of DD arc layers by reformulating the master  problem in a transformed space of variables defined over index sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Consider a node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the following, we define mappings that assign an index to each incoming  and outgoing arc of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These mappings enable us to define new variables to reduce the number  of DD arc layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let ind−(i,q) be a one-to-one mapping from incoming arcs (i,q), for i ∈ δ−(q),  to the index set {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ−(q)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Similarly, let ind+(q,j) be a one-to-one mapping from outgoing  arcs (q,j), for j ∈ δ+(q), to the index set {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each incoming arc (i,q) with index  h = ind−(i,q), we define an integer variable wq  h ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|} such that wq  h = 0 if this incoming  arc is not paired with any outgoing arc, and wq  h = k > 0 if this arc is matched with an outgoing arc  (q,j) with index k = ind+(q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Next, we give a formulation in the space of w variables that describes the matching between  incoming and outgoing arcs of q for all q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the following, sign(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=') represents the sign function  that returns 1 if its argument is strictly positive, 0 if the argument is zero, and −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Further, the operator |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='|, when applied on a set, represents the set size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' and when applied on a real  number, it represents the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Formulation  �  i∈δ−(q)  sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  ≥  ��δ−(q)  �� − 1  ∀j ∈ δ+(q), ∀q ∈ ¯V  (3a)  wq  ind−(i,q) ∈  �  0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',  ��δ+(q)  ���  ∀i ∈ δ−(q), ∀q ∈ ¯V  (3b)  models the matching between incoming and outgoing arcs of nodes q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  We show the result for a single node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The extension to the multiple node case  is straightforward as the matching problem for each node is independent from other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For  the direct implication, assume that M q is a matching between incoming and outgoing arcs of q,  with elements of the form (i,j) that represent a matching between the incoming arc (i,q) and  the outgoing arc (q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show that variables w associated with matching pairs in M q satisfy  constraints (3a) and (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It follows from the definition of w that, for each (i,j) ∈ M q, we have  wq  ind−(i,q) = ind+(q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Also, for any i ∈ δ−(q) that does not have a matching pair in M q, we have  wq  ind−(i,q) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These value assignments show that w satisfies (3b) as the image of ind+ mapping is  {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each i ∈ δ−(q) and j ∈ δ+(q), we have  ���wq  ind−(i,q) − ind+(q,j)  ��� ≥ 0, with equality  holding when (i,j) ∈ M q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each j ∈ δ+(q), there are two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the first case, assume that  (i,j) /∈ M q for any i ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result,  ���wq  ind−(i,q) − ind+(q,j)  ��� > 0 for all i ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Applying  the sign(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=') function on these terms yields sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  = 1, which implies that  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  15  �  i∈δ−(q) sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  = |δ−(q)|, satisfying (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the second case, assume that  (i∗,j) ∈ M q for some i∗ ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, we have �  i∈δ−(q) sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  =  |δ−(q)| − 1 since sign  ����wq  ind−(i∗,q) − ind+(q,j)  ���  �  =  ���wq  ind−(i∗,q) − ind+(q,j)  ��� = 0, satisfying (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  For the reverse implication, assume that w is a feasible solution to (3a)–(3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show that the  pairs of the form (i,j) encoded by these variables constitute a feasible matching between incoming  and outgoing arcs of q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', (i) each arc (i,q) is matched with at most one arc (q,j), and (ii) each  arc (q,j) is matched with at most one arc (i,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It follows from constraint (3b) that, for each i ∈  δ−(q), variable wq  ind−(i,q) takes a value between {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' If wq  ind−(i,q) = 0, then (i,q) is not  matched with any outgoing arc, otherwise it is matched with arc (q,j) with ind+(q,j) = wq  ind−(i,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  This ensures that condition (i) above is satisfied for this matching collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Further, for each  j ∈ δ−(q), constraint (3a) implies that sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  can be equal to zero for at  most one i ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In such a case, we would have at most one matching pair of the form (i,j) in  the collection, showing that condition (ii) above is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  It follows from Proposition 2 that constraints (3a)-(3b) can replace (1b)-(1d) in the master  problem (1a)-(1d) to obtain the following master problem in a transformed space of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  max  w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z {z | (3a) − (3b),z ∈ [−Γ,Γ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (4)  Note that formulation (4) is an integer nonlinear program (INLP) with nonconvex and non-  continuous constraint functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Such a formulation is extremely difficult for conventional MINLP  techniques and solvers to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' However, due to structural flexibility of DDs in representing inte-  ger nonlinear programs, this problem can be easily modeled via a DD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' see Davarnia and Van Hoeve  (2020) for a detailed account on using DDs for modeling INLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the following, we present an  algorithm to construct DDs in the space of (w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) variables for the master problem (4) with a  single node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The extension to the case with multiple nodes follows by replicating the DD  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The output of Algorithm 2 is a DD with |δ−(q)| + 1 arc layers where the first |δ−(q)|  layers represent w variables and the last layer encodes variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In this algorithm, su denotes the  state value of DD node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The core idea of the algorithm is to use unpaired outgoing arcs of q as  the state value at each DD layer that represents the matching for an incoming arc of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Next, We show that the solution set of the DD constructed by Algorithm 2 represents the feasible  region of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Note here that DD representation of a MIP set, as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2, does  not imply the encoding of all of the solutions of the set, but rather the encoding of a subset of all  solutions that subsumes all the extreme points of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Such a representation is sufficient to solve  an optimization problem over the set with an objective function convex in continuous variables,  which is the case for (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  16  Algorithm 2: Construction of DD for the master problem of SGUFP with a node q ∈ ¯V  Data: node q ∈ ¯V , parameter Γ  Result: an exact DD D  1 create the root node r ∈ U1 with state sr = {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}  2 forall i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ−(q)|} and u ∈ Ui do  3  forall ℓ ∈ su do  4  create a node v ∈ Ui+1 with state (su \\ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v  with label l(a) = ℓ  5 forall u ∈ U1+|δ−(q)| do  6  create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ  and l(a2) = −Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a SGUFP with ¯V = {q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let D be a DD constructed by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Then, Sol(D) represents the feasible region of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (⊆) Consider an r-t path of D that encodes solution ( ˜wq,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' According to Algorithm 2,  the labels of the first |δ−(q)| arcs of this path belong to {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}, showing that ˜wq  satisfies constraints (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Assume by contradiction that ˜wq does not satisfy constraints (3a),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', �  i∈δ−(q) sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  ≤ |δ−(q)| − 2 for some j ∈ δ+(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This implies that  ˜wq  ind−(i′,q) = ˜wq  ind−(i′′,q) = ind+(q,j) for two distinct i′,i′′ ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In other words, the arcs at lay-  ers ind−(i′,q) and ind−(i′′,q) of the selected r-t path both share the same label value ind+(q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  According to line 3 of Algorithm 2, we must have that the state value of nodes at layers ind−(i′,q)  and ind−(i′′,q) of the r-t path both contain ind+(q,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This is a contradiction to the state update  policy in line 4 of Algorithm 2, since positive arc labels at each layer of the DD will be excluded  from the state value of the subsequent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (⊇) Consider a feasible solution point ( ˜wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ˜z) of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Suppose ˜wq = (ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',ℓ|δ−(q)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' According  to constraints (3a), no two coordinates of ˜wq have the same positive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The state value at the  root node in D contains all index values {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' According to Algorithm 2, there exists  an arc with label ℓ1 at the first layer of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The state value at the head node of this arc, therefore,  contains ℓ2 ∈ {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|} \\ {ℓ1}, which guarantees an arc with label ℓ2 at the second layer of  this path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Following a similar approach, we can track a path from the root to layer |δ−(q)| whose  arcs labels match values of ˜wq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Note for the last layer that ˜z ∈ [−Γ,Γ], which is included in the  interval between arc labels of the last layer of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, ( ˜wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ˜z) is represented by an r-t path  of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  The main purpose of using a DD that models the master problem (4) over one that models (1a)-  (1d) is the size reduction in arc layers that represent variables w as compared with variables  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  17  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It turns out that this space transformation can significantly improve the solution time of the  DD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We refer the interested reader to Appendix A for a detailed discussion on these  advantages, including preliminary computational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Constructing exact DDs as described in Algorithm 2 can be computationally expensive for large  size problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2, relaxed and restricted DDs are used to circumvent this  difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Building restricted DDs is straightforward as it involves the selection of a subset of r-t  paths of the exact DD that satisfy a preset width limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Constructing relaxed DDs, on the other  hand, requires careful manipulation of the DD structure to merge nodes in such a way that it  encodes a superset of all r-t paths of the exact DD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We demonstrate a method to construct such  relaxed DDs in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Similarly to Algorithm 2, this algorithm is presented for a single  NSNM node, but can be extended to multiple nodes by replicating the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Algorithm 3: Construction of relaxed DD for the master problem of SGUFP with a node  q ∈ ¯V  Data: node q ∈ ¯V , parameter Γ  Result: a relaxed DD D  1 create the root node r ∈ U1 with state sr = {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}  2 forall i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ−(q)|} and u ∈ Ui do  3  forall ℓ ∈ su do  4  create a node v ∈ Ui+1 with state (su \\ {ℓ}) ∪ {0} and an arc a ∈ Ai connecting u to v  with label l(a) = ℓ  5  select a subset of nodes v1,v2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',vk ∈ Ui+1 and merge them into node v′ with state  sv′ = �k  j=1 svj  6 forall u ∈ U1+|δ−(q)| do  7  create two arcs a1,a2 ∈ A1+|δ−(q)| connecting u to the terminal node with labels l(a1) = Γ  and l(a2) = −Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a SGUFP with ¯V = {q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let D be a DD constructed by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Then, D represents a relaxation of the feasible region of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Let ˙D be the DD constructed by Algorithm 2 for the master problem (4) with a single  node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It suffices to show that the solution set of D provides a relaxation for that of ˙D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Pick  a root-terminal path ˙P of ˙D with encoding point ( ˙wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ˙z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show that there exist a root-terminal  path P of D with encoding point (wq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='z) such that wq = ˙wq and z = ˙z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Given a DD, define Pk to  be a sub-path composed of arcs in the first k layers, for 1 ≤ k ≤ |δ−(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show for any sub-path  ˙Pk of ˙D with encoding point ˙wq  k = ( ˙wq  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', ˙wq  k), there exists a sub-path P k of D with encoding  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  18  point wk = (w1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',wk) such that wh = ˙wh for h = 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Note that we only need to prove the  matching values for k ≤ |δ−(q)|, because each node at node layer |δ−(q)| + 1 of both ˙D and D  is connected by two arcs with labels −Γ and Γ to the terminal node, and thus there are always  matching arcs with the same label for the last layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', z = ˙z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We prove the result by induction on  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The base case for k = 1 is trivial, since D contains arcs with labels {0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|} in the first  layer, which includes the label value of the first arc on ˙P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the induction hypothesis, assume  that the statement is true for k = d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', for the sub-path ˙Pd with label values ˙wq  d = ( ˙wq  1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', ˙wq  d),  there is sub-path P d of D with matching arc labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show the statement holds for d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let  u ∈ ˙Ad+1 and v ∈ Ad+1 be the end nodes of ˙Pd and P d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It follows from Algorithm 2  that the index set representing the state value at node u contains ˙wq  d+1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', ˙wq  d+1 ∈ ˙su = {0} ∪  {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|} \\ { ˙w1, ˙w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', ˙wd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The merging step in line 5 of Algorithm 3, on the other hand,  implies that sv ⊇ {0}∪{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}\\{w1,w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',wd} = {0}∪{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=',|δ+(q)|}\\{ ˙w1, ˙w2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', ˙wd} =  ˙su, where the inclusion follows from the fact that state values at nodes on path P d contain those of  each individual path due to merging operation, and the first equality holds because of the induction  hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, sv must contain ˙wq  d+1, which implies that there exists an arc with ˙wq  d+1  connected to node v on P d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Attaching this arc to P d, we obtain the desired sub-path P d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  DD-BD: Subproblem Formulation  At each iteration of the DD-BD algorithm, an optimal solution of the master problem is plugged into  the subproblems to obtain feasibility/optimality cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the SGUFP formulation, this procedure  translates to obtaining an optimal solution of (4) in the space of w variables, which is used to  solve the subproblem (2a)-(2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The formulation of the subproblem, however, is defined over the  original binary variables y, and the resulting feasibility/optimality cuts are generated in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  To remedy this discrepancy between the space of variables in the master and subproblems, we need  to find a one-to-one mapping between variables w and y, as outlined next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a node q ∈ ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let yq be a feasible solution to (1b)-(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, wq  obtained as  wq  ind−(i,q) =  �  j∈δ+(q)  ind+(q,j)yq  ij  ∀i ∈ δ−(q),  (5)  is a feasible solution to (3a)-(3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Conversely, let wq be a feasible solution to (3a)-(3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, yq  obtained as  yq  ij = 1 − sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  ∀(i,j) ∈ δ−(q) × δ+(q),  (6)  is a feasible solution to (1b)-(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  For the direct statement, let yq be a feasible solution to (1b)-(1d), and construct a  vector wq according to (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show that wq satisfies all constraints (3a)-(3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' First, we show  that constraints (3a) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Assume by contradiction that there exists j′ ∈ δ+(q) such that  �  i∈δ−(q) sign  ����wq  ind−(i,q) − ind+(q,j′)  ���  �  ≤ |δ−(q)| − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This implies that wq  ind−(i′,q) = wq  ind−(i′′,q) =  ind+(q,j′) for some i′,i′′ ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, we can write that  wq  ind−(i′,q) =  �  j∈δ+(q)  ind+(q,j)yq  i′j = ind+(q,j′) =  �  j∈δ+(q)  ind+(q,j)yq  i′′j = wq  ind−(i′′,q),  where the first and last equalities hold by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The second and third equalities in the above chain  of relations imply that yq  i′j′ = yq  i′′j′ = 1, since ind+(q,j′) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This violates constraints (1c), reaching  a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Next, we show that constraints (3b) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The proof follows directly from  construction of wq and constraints (1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  For the converse statement, let wq be a feasible solution to (3a)-(3b), and construct a vec-  tor yq according to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We show that yq satisfies all constraints (1b)-(1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To show that each  constraint (1b) is satisfied, consider i ∈ δ−(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We can write that  �  j∈δ+(q)  yq  ij = |δ+(q)| −  �  j∈δ+(q)  sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  ≤ |δ+(q)| −  �  |δ+(q)| − 1  �  = 1,  where the first equality follows from the construction of yq, and the inequality holds by (3b) as  ���wq  ind−(i,q) − ind+(q,j)  ��� = 0 for at most one index j ∈ δ+(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To show that each constraint (1c) is  satisfied, select j ∈ δ+(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We have  �  i∈δ−(q)  yq  ij = |δ−(q)| −  �  i∈δ−(q)  sign  ����wq  ind−(i,q) − ind+(q,j)  ���  �  ≤ 1,  where the equality follows from the construction of yq, and the inequality holds because of con-  straint (3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Finally, each constraint (1d) is satisfied due to the fact that 1 − sign(|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='|) ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Mappings described by (5) and (6) are one-to-one over their respective  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Note that the mapping described by (5) is a linear transformation of the form wq = Byq  with coefficient matrix B ∈ Z|δ−(q)|×(|δ−(q)||δ+(q)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It is clear from the identity block structure of B,  that it is full row-rank, since each column contains a single non-zero element while each row has  at least one non-zero element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As a result, the null space of B is the origin, which implies that  ˆwq = ˜wq only if ˆyq = ˜yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  For the mapping described by (6), let distinct points ˆwq and ˜wq satisfy (3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Construct vectors  ˆyq and ˜yq by (6) using ˆwq and ˜wq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Because ˆwq and ˜wq are distinct, there must  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  20  exist i ∈ δ−(q) such that ˆwq  ind−(i,q) ̸= ˜wq  ind−(i,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This implies that at least one of these variables, say  ˆwq  ind−(i,q), is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' It follows from (3b) that ˆwq  ind−(i,q) = ind+(q,j′) for some j′ ∈ δ+(q), and that  ˆwq  ind−(i,q) ̸= ind+(q,j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' According to (6), we write that ˆyij′ = 1−sign  ���� ˆwq  ind−(i,q) − ind+(q,j′)  ���  �  = 1,  and that ˜yij′ = 1 − sign  ���� ˜wq  ind−(i,q) − ind+(q,j′)  ���  �  = 0, showing that ˆyq ̸= ˜yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  □  Using the results of Propositions 3 and 4, we can apply the DD-BD Algorithm 1 in its entirety for  the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular, at each iteration of the algorithm, we can transform the optimal solution  ( ¯w, ¯z) obtained from the DD representing the master problem (4) into a solution (¯y, ¯z) through the  mapping (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Given an optimal first-stage solution ¯y, we can solve |Ξ| separate subproblems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' one  for each demand realization in the second-stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The feasibility cuts obtained from subproblems,  which are in the space of y variables, are translated back into the space of w variables through the  mapping (5) and added to the master problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Further, in a case where all subproblems produce  an optimality cut, they can be aggregated to generate an optimality cut in the space of (y,z),  which is added to the master problem after being translated into the space of (w,z) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The  master DD will be refined with respect to the resulting inequalities, and an optimal solution is  returned to be used for the next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  In the remainder of this section, we present details on the derivation of optimality/feasibility cuts  from subproblem (2a)-(2h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider the following partitioning of the set of arcs A into subsets  A1 :=  �  (i,j) ∈ A  �� δ−(i) = ∅, δ+(j) ̸= ∅  �  , A2 :=  �  (i,j) ∈ A  �� δ−(i) ̸= ∅, δ+(j) = ∅  �  ,  A3 :=  �  (i,j) ∈ A  �� δ−(i) ̸= ∅, δ+(j) ̸= ∅  �  , A4 :=  �  (i,j) ∈ A  �� δ−(i) = ∅, δ+(j) = ∅  �  ,  and let θξ = (βξ,γξ,δξ,φξ,λξ,µξ) be the vector of dual variables associated with constraints of  (2a)-(2h) for a scenario ξ ∈ Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Further, define the bi-function  f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='θξ) =  �  q∈V  �  j∈δ+(q)  �  −ℓqjβξ  qj + uqjγξ  qj  �  +  �  q∈ ¯V  �  (i,j)∈δ−(q)×δ+(q)  �  uiq(1 − yq  ij)λξ  iqj + uqj(1 − yq  ij)µξ  iqj  �  +  �  q∈ ¯V  �  i∈δ−(q)  �  �uiq  �  j∈δ+(q)  yq  ijσξ  iq  �  � +  �  q∈ ¯V  �  j∈δ+(q)  �  �uqj  �  i∈δ−(q)  yq  ijφξ  qj  �  �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  For a given ¯y and each scenario ξ ∈ Ξ, the dual of the subproblem (2a)-(2h) can be written as  follows where the symbol ⋆ on a node means that it belongs to ¯V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  min  f(¯y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='θξ)  (7a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  αξ  ⋆q − βξ  i⋆q + γξ  i⋆q +  �  j:j∈δ+(⋆q)  λξ  i⋆qj −  �  j:j∈δ+(⋆q)  µξ  i⋆qj + σξ  i⋆q ≥ ri⋆q  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  ⋆q) ∈ A1 (7b)  αξ  q − βξ  iq + γξ  iq ≥ riq  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='q) ∈ A1 (7c)  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  21  − αξ  ⋆q − βξ  ⋆qj + γξ  ⋆qj −  �  i:i∈δ−(⋆q)  λξ  i⋆qj +  �  i:i∈δ−(⋆q)  µξ  i⋆qj + φξ  ⋆qj ≥ r⋆qj  ∀(  ⋆q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ A2 (7d)  − αξ  q − βξ  qj + γξ  qj ≥ rqj  ∀(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ A2 (7e)  − αξ  ⋆q + αξ  ⋆  j − βξ  ⋆q  ⋆  j + γξ  ⋆q  ⋆  j +  �  i∈δ−(⋆q)  �  µξ  i⋆q  ⋆  j − λξ  i⋆q  ⋆  j  �  +  �  i∈δ+(  ⋆  j)  �  λξ  ⋆q  ⋆  ji − µξ  ⋆q  ⋆  ji  �  + σξ  ⋆q  ⋆  j + φξ  ⋆q  ⋆  j ≥ r⋆q  ⋆  j  ∀(  ⋆q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  ⋆j) ∈ A3  (7f)  − αξ  ⋆q + αξ  j − βξ  ⋆qj + γξ  ⋆qj +  �  i∈δ−(⋆q)  �  µξ  i⋆qj − λξ  i⋆qj  �  + φξ  ⋆qj ≥ r⋆qj  ∀(  ⋆q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ A3 (7g)  − αξ  q + αξ  ⋆  j − βξ  q  ⋆  j + γξ  q  ⋆  j +  �  i∈δ+(  ⋆  j)  �  λξ  q  ⋆  ji − µξ  q  ⋆  ji  �  + σξ  q  ⋆  j ≥ rq  ⋆  j  ∀(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  ⋆j) ∈ A3 (7h)  − αξ  q + αξ  j − βξ  qj + γξ  qj ≥ rqj  ∀(q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j) ∈ A3  (7i)  − βξ  iq + γξ  iq ≥ riq  ∀(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='q) ∈ A4  (7j)  αξ  q ∈ R  ∀q ∈ V ′ (7k)  βξ  ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' γξ  ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' σξ  ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' φξ  ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' λξ  iqj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' µξ  iqj ≥ 0  ∀i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='j ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (7l)  If the above problem has an optimal solution ˆθξ for all ξ ∈ Ξ, the output of the subproblems will  be an optimality cut of the form �  ξ∈Ξ Prξf(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ˆθξ) ≥ z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' If the above problem is unbounded along a  ray ˆθξ for a ξ ∈ Ξ, the output of the subproblem will be a feasibility cut of the form f(y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' ˆθξ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Note that replacing variables y in the above constraints with w through the mapping (5) results  in separable nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Nevertheless, since these constraints will be used to refine the  master DD, their incorporation is simple due to structural flexibility of DDs in modeling such  constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' we refer the reader to Davarnia and Van Hoeve (2020) for a detailed account for  modeling INLPs with DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Computational Experiments  In this section, we solve SGUFP as a core model for the unit train scheduling problem with demand  stochasticity using three different approaches: (i) the standard MIP formulation that is a deter-  ministic equivalent of the two-stage model and contains all variables and constraints of the master  problem and |Ξ| subproblems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (ii) the Benders reformulation presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1 composed  of the master problem (1a)-(1d) and |Ξ| subproblems (2a)-(2h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' and (iii) the DD-BD algorithm  proposed in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In the Benders approach, we solve separate subproblems using a  fixed vector ¯y obtained from the master problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The feasibility cuts generated by subproblems  are added directly to the constraint set of the master problem, and the optimality cuts are added  as an aggregated cut over all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We note here that when there is a feasibility cut for any  scenario, we add it directly to separate the solution of the current iteration and move on to the  next iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To obtain a valid inequality that provides a bound for the single z variable, we need  to aggregate valid inequalities over all scenario subproblems as z is composed of the objective value  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  22  of all these subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Therefore, we can only produce an optimality cut for the z variable when  we have optimality cuts for all of the subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the DD-BD approach, we use the following  algorithmic choices to build restricted and relaxed DDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For the restricted DDs, we choose a subset  of the r-t paths with largest lengths, which are more likely to contain an optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For  the relaxed DDs, we merge nodes that have the largest number of common members in their state  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We refer the reader to Bergman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2016a) for other heuristic approaches that could be  used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Test Instances  In our experiments, we consider the structure of the SGUFP network given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To  ensure that the problem is always feasible, we create an artificial node s0 to compensate for any  shortage of the supply, and add an arc from the artificial supply s0 to each demand node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  We create test instances based on the specification given in Davarnia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' (2019), which is  inspired by realistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular, we consider a base rail network G′ = (V ′,A′) where 10%  and 30% of the nodes are supply and demand nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We assume that 50% of the  nodes must satisfy the NSNM requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We then create a network G = (V,A) by augmenting  supply/demand and artificial nodes as described above with the following settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The integer  supply value at supply nodes is randomly selected from the interval [100,600].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The capacity of arcs  connecting s0 to demand nodes are considered to be unbounded, and the integer capacity value  of other arcs is randomly selected from the interval [100,300].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each demand scenario ξ ∈ Ξ,  the integer demand value at demand nodes is randomly chosen from the interval [100,200].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The  reward of the arcs connecting s0 to the demand nodes are generated from the interval [−10,−5]  to represent the cost of lost demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The reward of the arcs connecting the source to the supply  nodes is randomly selected from the interval [5,10], and the reward of the arcs connecting the  demand nodes to the sink is fixed to zero since the flow of these arcs is also fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The reward  of all other arcs is created randomly from the interval [−2,2] where the negative values indicate  the cost of sending flows through congested arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We consider four categories of rail networks with  |V ′| ∈ {40,60,80,100}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each category, we create five scenario classes for the number of demand  scenarios |Ξ| ∈ {50,100,150,200,250}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For each network category and scenario class, we create five  random instances based on the above settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Test instances are publicly available (Salemi and  Davarnia 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Numerical Results  In this section, we present the numerical results that compare the performance of the DD-BD  formulation for the SGUFP instances with that of the MIP formulation, denoted by “MIP”, and the  standard Benders reformulation, denoted by “BD”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' All experiments are conducted on a machine  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  23  running Windows 10, x64 operating system with Intel® Core i7 processor (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='60 GHz) and 32 GB  RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The Gurobi optimization solver (version 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1) is used to solve instances for the MIP and  BD models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' When solving problems with Gurobi, we turn off presolve and cuts for all methods  to have a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Tables 1-4 report the running times of each of these formulations for  |V ′| ∈ {40,60,80,100} and |Ξ| ∈ {50,100,150,200,250} where the time limit is set to 3600 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  The symbol “ > 3600” indicates that the problem was not solved within the time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As evident  in these tables, the DD-BD formulation outperforms the other alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular, the  gap between the solution time of the DD-BD and the MIP and BD approaches widens as the  problem size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For example, as reported in Table 1, while the DD-BD approach solves all  25 instances in under 275 seconds, the MIP approach fails to solve 10 of them within 3600 seconds,  80% of which involve 200 or 250 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This shows a clear superiority of the DD-BD over the  MIP method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Further, for most of the instances, the DD-BD approach outperforms the standard  BD approach, rendering it as the superior solution method among all three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Figures 5-8 compare  the performance of DD-BD, BD, and MIP formulations through box and whisker plots for each  network size and under each scenario class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In these figures, for uniformity of illustration, we used  3600 seconds for the running time of instances that fail to solve the problem within that time  limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' As the figures show, the minimum, median, and maximum of running times of the DD-BD  method are remarkably smaller than those of the both BD and MIP methods in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These  results show the potential of the DD-BD framework in solving network problems with challenging  combinatorial structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In Appendix B, we present additional numerical results for the DD-BD  approach to assess its ability to solve larger problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Table 1  Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  MIP  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='74 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='62 2877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='19  > 3600  > 3600  BD  141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='83 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='84  339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81  451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93  565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='82  DD-BD  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='87  163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='43  219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='02  274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36  2  MIP  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='59 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='07  906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='10 1892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 2235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='53  BD  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='44 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='25  141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04  230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81  235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='87  DD-BD  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='60  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65  128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94  3  MIP  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='86 753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23 2453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='05  > 3600  > 3600  BD  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='14 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='20  172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='86  224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33  244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='91  DD-BD  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='32  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='58  113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93  178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65  217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33  4  MIP  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46 309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='62  > 3600  > 3600  > 3600  BD  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='55 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='01  267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94  334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='74  380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='22  DD-BD  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81  130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='19  183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23  253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='72  5  MIP  380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33 406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73  > 3600  > 3600  > 3600  BD  123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='69 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73  205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16  231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='56  287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='24  DD-BD  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78  138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46  195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='69  231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='74  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  24  Table 2  Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  MIP  893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73 > 3600  > 3600  > 3600  > 3600  BD  241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='85  556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18  582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='80  758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='54  933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='05  DD-BD 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16  357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='06  603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81  719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='27  901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='02  2  MIP  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='87  811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='64 1554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='10  > 3600  > 3600  BD  259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='63  351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='39  624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='08  816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='44 1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='95  DD-BD 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='07  388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='85  572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='76  961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='35  3  MIP  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='70  702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96 1035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79  > 3600  > 3600  BD  246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='48  569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='37  628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='84  795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='56  978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='15  DD-BD 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81  284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65  422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23  725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='86  4  MIP  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16  415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46  938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='03 1681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 2604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='25  BD  238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33  388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='19  563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='15  732.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='59  919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='08  DD-BD 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='29  262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36  393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18  521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12  654.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='71  5  MIP  165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='57  706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16 2447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='15  > 3600  > 3600  BD  194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12  244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61  479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='32  463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='63  617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='09  DD-BD 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='09  221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='30  332.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='25  443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96  556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33  Table 3  Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  MIP  215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='82  860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21  > 3600  > 3600  > 3600  BD  588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='51  806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61 1731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='50 1860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12 2051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  DD-BD 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='68 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='88 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='13  2  MIP  479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='76  > 3600  > 3600  > 3600  > 3600  BD  398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='29  713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='01  861.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65 1080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79 1709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04  DD-BD 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34  379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04  724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='66 1088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 1587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='90  3  MIP  238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79  996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='22  > 3600  > 3600  > 3600  BD  702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18 1236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='58 1650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='42 1773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='63 2227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='89  DD-BD 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='13  518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='97 1046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='39 1326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='22  4  MIP  404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='26 2441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='64 2855.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='29  > 3600  > 3600  BD  572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='83 1219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='37 1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 1745.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='91 2089.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='80  DD-BD 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78  665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='30 1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81 1277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93 1444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='02  5  MIP  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='50  > 3600  > 3600  > 3600  > 3600  BD  231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='11  481.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='31  625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='91 1310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='24 1452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='27  DD-BD 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34  376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96  564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='54 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94  Table 4  Running times (in seconds) of MIP, BD, and DD-BD for |V ′| = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  MIP  774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18  > 3600  > 3600  > 3600  > 3600  BD  1282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='59 1728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='71 1848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='49 2307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='74 3309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93  DD-BD  698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36 1427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='38 1731.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='95 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96 3323.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='54  2  MIP  480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='97  > 3600  > 3600  > 3600  > 3600  BD  781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='47 1573.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23 1820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79 2672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18 2819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61  DD-BD  586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='89 1171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96 1848.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='49 2471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='49 2635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='22  3  MIP  3071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='37  > 3600  > 3600  > 3600  > 3600  BD  1072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='14 1322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96 2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='50 2951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='55 3412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='99  DD-BD  485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='31  703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='70 1055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36 1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='66 2269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='97  4  MIP  838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79 2585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='38  > 3600  > 3600  > 3600  BD  1548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93 1738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='92 2580.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='53 2616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='19 3169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='28  DD-BD  554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='89  743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='64 1098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='82 2052.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73 3094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23  5  MIP  714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='39  > 3600  > 3600  > 3600  > 3600  BD  808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='48 1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='68 1722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='01 2824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='14 3282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='10  DD-BD  353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='48  700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='57 1680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='60 2213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81 2907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  25  Figure 5  Comparison of DD-BD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' BD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' and MIP models when |V ′| = 40 under five scenarios  Figure 6  Comparison of DD-BD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' BD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' and MIP models when |V ′| = 60 under five scenarios  We conclude this section by noting that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' while the focus of this paper has been on the unit train  problem with the no-split no-merge requirements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' the proposed DD-BD framework can be applied  to model network problems that contain additional side constraints on the flow variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' as those  constraints can be handled in the subproblems while the DD structure in the master problem  4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  Running time (sec)  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  50  100  150  200  250  Number of scenarios  DD-BDBDMIP4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  Running time (sec)  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  50  100  150  200  250  Number of scenarios  IDD-BD  ■BDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  26  Figure 7  Comparison of DD-BD, BD, and MIP models when |V ′| = 80 under five scenarios  Figure 8  Comparison of DD-BD, BD, and MIP models when |V ′| = 100 under five scenarios  remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Examples of such side constraints include the usage-fee limitation (Holzhauser,  Krumke, and Thielen 2017b) and the flow ratio requirement (Holzhauser, Krumke, and Thielen  2017a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Applying the DD-BD method to such network models and assessing its effectiveness com-  pared to alternative approaches could be an interesting direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  Running time (sec)  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  50  100  150  200  250  Number of scenarios  DD-BDBDMIP4000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  3000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  time (sec)  2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  Running  1500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='00  50  100  150  200  250  Number of scenarios  DD-BDBDMIPSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Conclusion  In this paper, we introduce a DD-based framework to solve the SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' This framework uses  Benders decomposition to decompose the SGUFP into a master problem composed of the combi-  natorial NSNM constraints, and a subproblem that solves a continuous network flow model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' The  master problem is modeled by a DD, which is successively refined with respect to the cuts generated  through subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' To assess the performance of the proposed method, we apply it to a variant  of unit train scheduling problem formulated as a SGUFP, and compare it with the standard MIP  and Benders reformulation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Acknowledgments  This project is sponsored in part by the Iowa Energy Center, Iowa Economic Development Authority and  its utility partners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We thank the anonymous referees and the Associate Editor for their helpful comments  that contributed to improving the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  28  References  Abbink E, Van den Berg B, Kroon L, Salomon M, 2004 Allocation of railway rolling stock for passenger  trains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Science 38(1):33–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Alfieri A, Groot R, Kroon L, Schrijver A, 2006 Efficient circulation of railway rolling stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation  Science 40(3):378–391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Andersen HR, Hadzic T, Hooker JN, Tiedemann P, 2007 A constraint store based on multivalued decision  diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' International Conference on Principles and Practice of Constraint Programming, 118–132  (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Association of American Railroads, 2021 Freight railroads fact sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='aar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='org, Accessed:  06/28/2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Baier G, K¨ohler E, Skutella M, 2005 The k-splittable flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Algorithmica 42(3):231–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Bergman D, Cire AA, 2018 Discrete nonlinear optimization by state-space decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Management  Science 64(10):4700–4720.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Bergman D, Cire AA, Van Hoeve WJ, Hooker J, 2016a Decision diagrams for optimization, volume 1  (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Bergman D, Cire AA, Van Hoeve WJ, Hooker JN, 2016b Discrete optimization with decision diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  INFORMS Journal on Computing 28(1):47–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Bornd¨orfer R, Reuther M, Schlechte T, Waas K, Weider S, 2016 Integrated optimization of rolling stock  rotations for intercity railways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Science 50(3):863–877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Cacchiani V, Toth P, 2012 Nominal and robust train timetabling problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' European Journal of Operational  Research 219(3):727–737.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Carey M, Crawford I, 2007 Scheduling trains on a network of busy complex stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research  Part B: Methodological 41(2):159–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Ceselli A, Gatto M, L¨ubbecke ME, Nunkesser M, Schilling H, 2008 Optimizing the cargo express service of  swiss federal railways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Science 42(4):450–465.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Chakrabarti A, Chekuri C, Gupta A, Kumar A, 2007 Approximation algorithms for the unsplittable flow  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Algorithmica 47(1):53–78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Cordeau JF, Toth P, Vigo D, 1998 A survey of optimization models for train routing and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Trans-  portation Science 32(4):380–404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Cornelsen S, Di Stefano G, 2007 Track assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Journal of Discrete Algorithms 5(2):250–261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Davarnia D, 2021 Strong relaxations for continuous nonlinear programs based on decision diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Opera-  tions Research Letters 49(2):239–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Davarnia D, Richard JPP, I¸cy¨uz-Ay E, Taslimi B, 2019 Network models with unsplittable node flows with  application to unit train scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Operations Research 67(4):1053–1068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  29  Davarnia D, Van Hoeve WJ, 2020 Outer approximation for integer nonlinear programs via decision diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Mathematical Programming 1–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Demir E, Burgholzer W, Hruˇsovsk`y M, Arıkan E, Jammernegg W, Van Woensel T, 2016 A green inter-  modal service network design problem with travel time uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part B:  Methodological 93:789–807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Fuchsberger M, L¨uthi P, 2007 Solving the train scheduling problem in a main station area via a resource  constrained space-time integer multi-commodity flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Institute for Operations Research ETH Zurich .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Furchtgott-Roth D, Hu PS, Nguyen L, Jahanmir S, Moore WH, Riley D, Beningo S, Chambers M, Smith-  Pickel S, Thai H, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=', 2021 Pocket Guide to Transportation 2021 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Gong C, Shi J, Wang Y, Zhou H, Yang L, Chen D, Pan H, 2021 Train timetabling with dynamic and ran-  dom passenger demand: A stochastic optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part C: Emerging  Technologies 123:102963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Gonzalez JE, Cire AA, Lodi A, Rousseau LM, 2020 Integrated integer programming and decision diagram  search tree with an application to the maximum independent set problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Constraints 1–24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Haahr J, Lusby RM, 2017 Integrating rolling stock scheduling with train unit shunting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' European Journal of  Operational Research 259(2):452–468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Haahr JT, Wagenaar JC, Veelenturf LP, Kroon LG, 2016 A comparison of two exact methods for passenger  railway rolling stock (re) scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part E: Logistics and Transportation  Review 91:15–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Had˘zi´c T, Hooker J, 2006 Discrete global optimization with binary decision diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Workshop on Global  Optimization: Integrating Convexity, Optimization, Logic Programming, and Computational Algebraic  Geometry (GICOLAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Vienna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Harrod S, Gorman MF, 2010 Operations research for freight train routing and scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Wiley Encyclopedia  of Operations Research and Management Science .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Heil J, Hoffmann K, Buscher U, 2020 Railway crew scheduling: Models, methods and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' European  Journal of Operational Research 283(2):405–425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Holzhauser M, Krumke SO, Thielen C, 2017a Maximum flows in generalized processing networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Journal of  Combinatorial Optimization 33(4):1226–1256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Holzhauser M, Krumke SO, Thielen C, 2017b A network simplex method for the budget-constrained minimum  cost flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' European journal of operational research 259(3):864–872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Hosseininasab A, Van Hoeve WJ, 2021 Exact multiple sequence alignment by synchronized decision diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  INFORMS Journal on Computing 33(2):721–738.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Hu Y, Lan J, Wan C, 2009 An algorithm for unsplittable flow problem in flexible reconfigurable network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' 2009  Fourth International Conference on Frontier of Computer Science and Technology, 543–547 (IEEE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  30  Huntley CL, Brown DE, Sappington DE, Markowicz BP, 1995 Freight routing and scheduling at CSX trans-  portation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Interfaces 25(3):58–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  I¸cy¨uz IE, Richard JPP, Eskigun E, Acharya D, 2016 A two-model solution approach for the monthly coal  train reservations planning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Science 50(3):926–946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Jin G, He S, Li J, Guo X, Li Y, 2019 An approach for train stop planning with variable train length and stop  time of high-speed rail under stochastic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' IEEE Access 7:129690–129708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Jordan WC, Turnquist MA, 1983 A stochastic, dynamic network model for railroad car distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Trans-  portation Science 17(2):123–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Jovanovi´c D, Harker PT, 1991 Tactical scheduling of rail operations: the scan i system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Science  25(1):46–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Kleinberg JM, 1996 Approximation algorithms for disjoint paths problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' thesis, Massachusetts Insti-  tute of Technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Kolman P, Scheideler C, 2006 Improved bounds for the unsplittable flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Journal of Algorithms  61(1):20–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Kwan RS, 2011 Case studies of successful train crew scheduling optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Journal of Scheduling  14(5):423–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Larsen R, Pranzo M, D’Ariano A, Corman F, Pacciarelli D, 2014 Susceptibility of optimal train schedules to  stochastic disturbances of process times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Flexible Services and Manufacturing Journal 26(4):466–489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Lawley M, Parmeshwaran V, Richard JP, Turkcan A, Dalal M, Ramcharan D, 2008 A time–space scheduling  model for optimizing recurring bulk railcar deliveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part B: Methodological  42(5):438–454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Layeb SB, Jaoua A, Jbira A, Makhlouf Y, 2018 A simulation-optimization approach for scheduling in stochas-  tic freight transportation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Computers & Industrial Engineering 126:99–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Lin Z, Kwan RS, 2014 A two-phase approach for real-world train unit scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Public Transport 6(1-2):35–  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Lin Z, Kwan RS, 2016 A branch-and-price approach for solving the train unit scheduling problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Trans-  portation Research Part B: Methodological 94:97–120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Lin Z, Kwan RS, 2018 Redundant coupling/decoupling in train unit scheduling optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Electronic Notes  in Discrete Mathematics 64:45–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Liu SQ, Kozan E, 2011 Optimising a coal rail network under capacity constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Flexible Services and  Manufacturing Journal 23(2):90–110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Lusby RM, 2008 Optimization methods for routing trains through railway junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' thesis,  ResearchSpace@ Auckland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  31  Lusby RM, Larsen J, Ehrgott M, Ryan D, 2011 Railway track allocation: models and methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' OR spectrum  33(4):843–883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Meng L, Zhou X, 2011 Robust single-track train dispatching model under a dynamic and stochastic environ-  ment: A scenario-based rolling horizon solution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part B: Method-  ological 45(7):1080–1102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Quaglietta E, Corman F, Goverde RM, 2013 Stability of railway dispatching solutions under a stochastic  and dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' RailCopenhagen2013: 5th International Seminar on Railway Operations  Modelling and Analysis (IAROR) (Institute for Transport Planning and Systems, ETH Zurich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi H, Davarnia D, 2022a On the structure of decision diagram-representable mixed integer programs with  application to unit commitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Operations Research URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='1287/opre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  2353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi H, Davarnia D, 2022b Test instances for SGUFP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='6373664.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Serra T, Hooker JN, 2019 Compact representation of near-optimal integer programming solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Mathe-  matical Programming 1–34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Shen Y, Peng K, Chen K, Li J, 2013 Evolutionary crew scheduling with adaptive chromosomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation  Research Part B: Methodological 56:174–185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Sherali HD, Suharko AB, 1998 A tactical decision support system for empty railcar management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transporta-  tion Science 32(4):306–329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Turner C, Tiwari A, Starr A, Blacktop K, 2016 A review of key planning and scheduling in the rail industry  in Europe and UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and  Rapid Transit 230(3):984–998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Walkowiak K, 2006 New algorithms for the unsplittable flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' International Conference on Compu-  tational Science and Its Applications, 1101–1110 (Springer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Ying Cs, Chow AH, Chin KS, 2020 An actor-critic deep reinforcement learning approach for metro train  scheduling with rolling stock circulation under stochastic demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Transportation Research Part B:  Methodological 140:210–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Zwaneveld PJ, Kroon LG, Van Hoesel SP, 2001 Routing trains through a railway station based on a node  packing model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' European Journal of Operational Research 128(1):14–33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  32  Appendix A:  Comparison of Master Problem Formulations  In this section, we describe the differences between DDs in the space of w variables and those in the space  of original y in the master problem formulation (4) in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' First, we illustrate the size difference  between these DDs in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Consider a directed graph G = (V,A) with node set V = {1,2,q,3,4} and arc set A =  {(1,q),(2,q),(q,3),(q,4)} where the central node q is subject to NSNM constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Let ind−(1,q) =  ind+(q,3) = 1 and ind−(2,q) = ind+(q,4) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' Then, the exact DDs showed in Figures 9(a) and 9(b) with  three and five arc layers represent the feasible region of master problem (4) and (1a)-(1d), respectively, where  −M and M are valid bounds for variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (a) A DD in the space of w variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Numbers next to arcs represent labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  (b) A DD in the space of y variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Numbers next to arcs represent labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Figure 9  Comparison of the number of arc layers for DDs in the space of w and y variables  As evident from the above example, the main advantage of using a DD in the space of w is the reduction in  the number of arc layers, which is the main determinant of the DDs computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' In particular,  even though such a DD has a larger number of nodes at the layers, a relaxed DD can be constructed to limit  the width, and hence provide an efficient relaxed DD in a smaller dimension, whereas the relaxations of the  DD constructed in the space of y variables would still be higher-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  To assess the computational efficiency of the solution approach in relation to the DD space, we compare  the performance of the DD-BD method under two different settings: (i) where DDs are built in the space  of w variables, denoted by DD-BD-w, and (ii) where DDs are built in the space of y variables, denoted by  DD-BD-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' We report the results of these two implementations for |V ′| ∈ {40,80} and under five different  scenarios in Table 5 and Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  As observed in these tables, the DD-BD-w solves all instances faster than DD-BD-y, with orders of  magnitude time improvement as the problem size (number of scenarios) increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These preliminary com-  putational results show the advantage of designing the DD-BD method for the SGUFP in a transformed  space of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  2  0  1  0  2  0  2  1  0  W  M  M  M  M  M  M  I七0  91,3  1  y2,3  0  0  1  0  0  0  b  y2,4  0  0  0  M  M  M  M  2  M  MSalemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  33  Table 5  Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  DD-BD-w  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='87 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='43 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='02 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36  DD-BD-y  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='68 304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='08 432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34 642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='70 839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='57  2  DD-BD-w  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='60  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94  DD-BD-y  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='76 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='53 344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='86 605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04  3  DD-BD-w  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='32  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='58 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='65 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33  DD-BD-y  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='05 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='67 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='07 541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='33 658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='98  4  DD-BD-w  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='19 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='23 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='72  DD-BD-y  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='11 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='26 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='31 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73 694.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='57  5  DD-BD-w  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='69 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='74  DD-BD-y  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='61 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='80 532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12 669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78  Table 6  Running times (in seconds) of DD-BD-w and DD-BD-y for |V ′| = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  DD-BD-w 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='12  500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='52  757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='68 1025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='88 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='13  DD-BD-y  483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='42  977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='03 1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='27 3175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='72 4230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='29  2  DD-BD-w 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34  379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04  724.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='66 1088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 1587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='90  DD-BD-y  340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='13  864.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='21 1856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96 3010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='55 4843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='67  3  DD-BD-w 285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='13  518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='46  778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='97 1046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='39 1326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='22  DD-BD-y  568.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='32 1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='44 2401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='98 3326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='76 4283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='58  4  DD-BD-w 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='78  665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='30 1230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='81 1277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='93 1444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='02  DD-BD-y  501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='04 1430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='77 2868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='92 3356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='39 4356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='48  5  DD-BD-w 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34  376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='96  564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='54 1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='94  DD-BD-y  354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='37  781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18 1279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='73 3001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='72 3834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='08  Appendix B:  Additional Computational Experiments  In this section, we present additional numerical results to assess the limits of the DD-BD method for larger  problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' These results are given in Tables 7 and 8, where the columns are defined similarly to  those of Tables 1-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content=' For these instances, the time limit is set to 3600 seconds, and the symbol “> 3600”  indicates that the problem is not solved within this time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Table 7  Running times (in seconds) of DD-BD for |V ′| = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  DD-BD 1494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='49 2824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='58  > 3600 > 3600 > 3600  2  DD-BD  975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='47 1892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='41 3198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='18 > 3600 > 3600  3  DD-BD 1150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='30 2263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='09 3454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='47 > 3600 > 3600  4  DD-BD 1261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='59 2403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='79  > 3600 > 3600 > 3600  5  DD-BD  906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34 1863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='15 3050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='68 > 3600 > 3600  Salemi and Davarnia: Solving Unsplittable Network Flow Problems with Decision Diagrams  34  Table 8  Running times (in seconds) of DD-BD for |V ′| = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='  Instance # Model  Number of scenarios  50  100  150  200  250  1  DD-BD 2496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='16 > 3600 > 3600 > 3600 > 3600  2  DD-BD 2944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='20 > 3600 > 3600 > 3600 > 3600  3  DD-BD 2321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='62 > 3600 > 3600 > 3600 > 3600  4  DD-BD 2590.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='34 > 3600 > 3600 > 3600 > 3600  5  DD-BD 2298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
+page_content='36 > 3600 > 3600 > 3600 > 3600' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/hNAzT4oBgHgl3EQf4f5B/content/2301.01844v1.pdf'}
diff --git a/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/2301.05434v1.pdf.txt b/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/2301.05434v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..61146d410d3d3157c4c4815ebd719b1193b4c681
--- /dev/null
+++ b/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/2301.05434v1.pdf.txt
@@ -0,0 +1,1103 @@
+LVRNet: Lightweight Image Restoration for
+Aerial Images under Low Visibility
+Esha Pahwa*
+BITS Pilani
+f20180675@pilani.bits-pilani.ac.in
+Achleshwar Luthra*
+Carnegie Mellon University
+achleshl@andrew.cmu.edu
+Pratik Narang
+BITS Pilani
+pratik.narang@pilani.bits-pilani.ac.in
+Abstract
+Learning to recover clear images from images having a
+combination of degrading factors is a challenging task.
+That being said, autonomous surveillance in low visibility
+conditions caused by high pollution/smoke, poor air qual-
+ity index, low light, atmospheric scattering, and haze dur-
+ing a blizzard becomes even more important to prevent ac-
+cidents. It is thus crucial to form a solution that can re-
+sult in a high-quality image and is efficient enough to be
+deployed for everyday use. However, the lack of proper
+datasets available to tackle this task limits the performance
+of the previous methods proposed. To this end, we generate
+the LowVis-AFO dataset, containing 3647 paired dark-hazy
+and clear images. We also introduce a lightweight deep
+learning model called Low-Visibility Restoration Network
+(LVRNet). It outperforms previous image restoration meth-
+ods with low latency, achieving a PSNR value of 25.744 and
+an SSIM of 0.905, making our approach scalable and ready
+for practical use. The code and data can be found here.
+1. Introduction
+Image enhancement and restoration have been a critical
+area of research using both traditional digital image pro-
+cessing techniques[12] [2], and the recent deep learning
+frameworks[32][33][44]. The goal of image restoration is
+to recover a clear image, whereas image enhancement is to
+improve the quality of the degraded image. In this study, we
+perform recovery of the clear image from the hazy version
+while performing low-light image enhancement using a sin-
+gle convolutional network, which could further be applied
+to tasks such as search and rescue operations using object
+detection.
+*equal contribution
+Using deep learning algorithms for image recovery has
+many benefits, the most important being that it can general-
+ize to different variations in the images captured. Hence, we
+observe that deep learning-based methods on most bench-
+mark datasets often outperform traditional methods signif-
+icantly.
+However, there are still challenges that the re-
+searchers have to tackle for image restoration.
+Publicly
+available datasets containing a variety of degrading factors
+that model real-world scenarios are few. Hence, most pre-
+vious works have focused on removing one type of degra-
+dation with a specific intensity level. From the perspective
+of computational complexity, recent deep learning methods
+are computationally expensive, and thus they can’t be de-
+ployed on edge devices. Moreover, image restoration has
+been a long-standing ill-posed research problem, as there
+are infinite mappings between the degraded and the clear
+image. Thus, the existing methods still have room for im-
+provement in finding the correct mapping.
+In this work, we focus on developing an end-to-end
+lightweight deep-learning solution for the image restoration
+task. Our major contributions are listed below:
+• Taking inspiration from Non-linear Activation Free
+Network (NAFNet) [5] and Level Attention Module
+[45], we propose a novel algorithm - Low-Visibility
+Restoration Network (LVRNet), that can effectively re-
+cover high-quality images from degraded images taken
+in poor visual conditions (Figure 1).
+• Due to the lack of available datasets that exhibit a com-
+bination of adverse effects, we generate a new dataset,
+namely LowVis-AFO (abbreviation for Low-Visibility
+Aerial Floating Objects dataset). We use AFO [15] as
+our ground truth dataset and synthesize dark hazy im-
+ages. The data generation process has been elaborated
+in Section 4.1.
+1
+arXiv:2301.05434v1  [cs.CV]  13 Jan 2023
+
+Input 
+Zero-DCE
+SGZNet
+DehazeNet
+StarDCE
+BPPNet
+FFANet
+MSBDN-DFF
+Ground Truth / Reference Image
+Our Result
+Input 
+Zero-DCE
+SGZNet
+DehazeNet
+StarDCE
+BPPNet
+FFANet
+MSBDN-DFF
+Ground Truth / Reference Image
+Our Result
+Figure 1. Visual results on the proposed LowVis-AFO dataset. The method used to obtain each result has been mentioned under the
+image.
+• Benchmarking experiments have been provided on the
+LowVis-AFO dataset to help future researchers for
+quantitative comparison.
+Along with that, LVRNet
+surpasses the results obtained using previous image
+restoration techniques by a significant margin.
+• We perform extensive ablation studies to analyze the
+importance of various loss functions existing in cur-
+rent image restoration research. These experiments are
+discussed in detail in Section 5.
+2. Related Works
+This section highlights the previous work done in the fields
+of image dehazing and low-light image enhancement and
+their limitations.
+2.1. Image Dehazing
+Hazy weather is often seen due to floating particles in
+the environment which degrade the quality of the image
+captured. Therefore, many previous works have tried to
+recover a clear image from the hazy one.
+These works
+can be divided into two methods, ones that rely on prior
+assumptions [17] and the atmospheric scattering model
+(ASM) [31] and the others which use deep learning to solve
+the problem, either by combination with ASM [3][34][36]
+or independently [25][26][33][46][50].
+Conventional
+approaches are physically inspired and apply various types
+of sharp image priors to regularize the solution space.
+However, they exhibit shortcomings when implemented
+with real-world images and videos. For example, the dark
+channel prior method (DCP) [31] does not perform well
+in regions containing the sky. These methods [1][11][24]
+are known to be computationally expensive and require
+2
+
+Pre-processing
+Conv
+Post-processing
+Conv.
+NAF-G1
+NAF-G2
++
+LAN
+NAF-G3
+Stacked Feature
+Maps
+Input Image
+Output Image
+Figure 2. Model architecture of the proposed LVRNet. Starting from the top-left: The input image is passed to the pre-processing
+convolution layers where feature maps are learned and passed to NAF Groups (here we have used 3 groups). The features extracted from
+each group are concatenated (or stacked) along the channel dimension and sent as input to the Level Attention Module (LAM). Finally, we
+pass LAM’s output to CNN layers for post-processing, adding the original image through residual connection and extracting the restored
+image at the bottom-left.
+heuristic parameter-tuning. Supervised dehazing methods
+can be divided into two subparts, one is ASM based, and
+the other is non-ASM based.
+ASM-based Learning: MSCNN[34] solves the task of
+image dehazing by dividing the problem into three steps:
+using CNN to estimate the transmission map t(x), using
+statistical methods to find atmospheric light A and then
+recover the clear image J(x) using t(x) and A jointly. Meth-
+ods like LAP-Net [23] adopt the relation of depth with the
+amount of haze in the image. The farther the scene from the
+camera, the denser the haze would be. Hence it considers
+the difference in the haze density in the input image using a
+stage-wise loss, where each stage predicts the transmission
+map from mild to severe haze scenes.
+DehazeNet [3]
+consists of four sequential operations: feature extraction,
+multi-scale mapping, calculating local extremum, and
+non-linear regression. MSRL-DehazeNet [43] decomposes
+the problem into recovering high-frequency and basic com-
+ponents. GCANet [4] employs residual learning between
+haze-free and hazy images as an optimization objective.
+End-to-end Learning: This subpart of previous work cor-
+responds to non-ASM-based deep learning methods for re-
+covering the clear image. Back-Projected Pyramid Network
+(BPPNet) [39] is a generative adversarial network that in-
+cludes iterative blocks of UNets [37] to learn haze features
+and pyramid convolution to preserve spatial features of dif-
+ferent scales. The reason behind using iterative blocks of
+UNets[37] is to avoid increasing the number of encoder lay-
+ers in a single UNet[37] as it leads to a decrease in height
+and width of latent feature representation hence resulting in
+loss of spatial information. Moreover, different blocks of
+UNet learn different complexities of haze features, and the
+final concatenation step ensures that all of them are taken
+into account during image reconstruction. The final recon-
+struction is done using the pyramid convolution block. The
+output feature is post-processed to get a haze-free image.
+Feature-Fusion Attention Network (FFANet) [33] adopts
+the idea of an attention mechanism and skip connections
+to restore haze-free images. A combination of channel at-
+tention and pixel attention is introduced, which helps the
+network, deal with the uneven spatial distribution of haze
+and different weighted information across channels. Au-
+toencoders [6], hierarchical networks [9], and dense block
+networks [14] has also been proposed for the task of image
+dehazing. However, our main comparison lies with FFANet
+[33], wherein we show a huge improvement compared to
+the former method with a model containing a lesser number
+of parameters and which can generalize to different levels
+of haze.
+2.2. Low-light Enhancement
+Traditional methods for low-light image enhancement
+(LLIE) include Histogram Equalization-based methods and
+Retinex model-based methods. Recent research has been
+focused on developing deep learning-based methods fol-
+lowing the success of the first seminal work. Deep learning-
+based solutions are more accurate, robust, and have a
+shorter inference time thus attracting more researchers.
+3
+
+Layer Norm
+1x1, conv
+3x3, dconv
+Simple Gate
+SCA
+1x1, conv
++
+Layer Norm
+1x1, conv
+Simple Gate
+1x1, conv
++
+I/P
+O/P
+NAF BLOCK
+NAF BLOCK
+NAF BLOCK
+CONV
++
+NAF-Group
+NAF Block
+Figure 3. Architecture of NAF Block and NAF Group. NAF Blocks are the building blocks of NAF Groups. A detailed description has
+been provided in Section 3.1 and Section 3.1.1
+Learning strategies used in these methods are mainly su-
+pervised learning [27, 29, 30, 35, 51, 28, 41], unsupervised
+learning [20], and zero-shot learning [49, 13].
+Supervised Learning: The first deep learning-based LLIE
+method LLNet [27] is an end-to-end network that employs a
+variant of stacked-sparse denoising autoencoder to brighten
+and denoise low-light images simultaneously. LLNet in-
+spired many other works [29, 30, 35, 51], but they do not
+consider the observation that noise exhibits different lev-
+els of contrast in different frequency layers. Later, Xu et
+al. [41] proposed a network that suppresses noise in the
+low-frequency layers and recovers the image contents by
+inferring the details in high-frequency layers. There is an-
+other division of methods that is based on the Retinex the-
+ory. Deep Retinex-based models [40, 42] decomposes the
+image into two separate components - light-independent
+reflectance and structure-aware smooth illumination. The
+final estimated reflection component is treated as the en-
+hanced result.
+Unsupervised Learning: Although the above-mentioned
+methods perform well on synthetic data, they show limited
+generalization capability on real-world low-light images.
+This might be the result of overfitting. EnlightenGAN [20]
+proposed to solve this issue by adopting an unsupervised
+learning technique, i.e., avoiding the use of paired synthetic
+data. This work uses attention-guided UNet as a generator
+and global-local discriminators to achieve the objective of
+LLIE.
+Zero-short Learning: These methods, in low-level vision
+tasks, do not require any paired or unpaired training data.
+Zero-reference Deep Curve Estimation [13] formulates im-
+age enhancement as a task of image-specific deep curve
+estimation, taking into account pixel value range, mono-
+tonicity, and differentiability. It is a lightweight DCE-Net
+that doesn’t require paired or unpaired ground truth images
+during training and relies on non-reference loss functions
+that measure the enhancement quality hence driving the
+learning of the network. Another such method, Semantic-
+guided Zero-shot low-light enhancement Network [49] is a
+lightweight model for low-light enhancement factor extrac-
+tion which is inspired by the architecture of U-Net [37]. The
+output of this network is fed to a recurrent image enhance-
+ment network, along with the degraded input image. Each
+stage in this network considers the enhancement factor and
+the output from the previous scale as its input. This is fol-
+lowed by a feature-pyramid network that aims to preserve
+the semantic information in the image.
+More recently, researchers have experimented with trans-
+formers for Zero-shot Learning LLIE. Structure-Aware
+lightweight Transformer (STAR) [47] focuses on real-time
+image enhancement without using deep-stacked CNNs or
+large transformer models. STAR is formulated to capture
+long-range dependencies between separate image patches,
+facilitating the model to learn structural relationships be-
+tween different regions of the images. In STAR, patches of
+the image are tokenized into token embeddings. The tokens
+generated as an intermediate stage are passed to a long-
+short-range transformer that outputs two long and short-
+range structural maps. These structural maps can further
+predict curve estimation or transformation for image en-
+hancement tasks. Although these methods show impressive
+results for the study of low-light image enhancement for
+which it originally developed, they cannot deal with foggy
+low– light images.
+2.3. Limitations
+Previous works have relied on ASM-based methods in the
+case of dehazing and Retinex model-based methods for low-
+light image enhancement.
+However, these methods fail
+to generalize to real-world images. Recent deep learning-
+based methods using large networks solve the task of im-
+age dehazing and low-light enhancement separately. To our
+knowledge, no work is introduced that solves the two prob-
+lems in a collaborative network. Deep learning methods
+4
+
+also fail to generalize to different haze levels and darkness.
+3. Proposed Methodology
+In this section, we provide a detailed description of the over-
+all architecture proposed and the individual components in-
+cluded in the network.
+3.1. Architecture
+Like the group structure in [33], each group in our network
+consists of a K NAF Block [5] with a skip connection at
+the end as shown in Figure 3. The output of each group is
+concatenated, passed to the level attention module to find
+the weighted importance of the feature maps obtained, and
+post-processed using two convolutional layers. A long skip
+connection for global residual learning accompanies this.
+3.1.1
+NAF-Block
+To keep this work self-contained, we explain the NAF Block
+[5] in this subsection. NAF Block is the building block
+of Nonlinear Activation Free Network. Namely NAFNet
+[5]. To avoid over-complexity in the architecture, this block
+avoids using any activation functions like ReLU, GELU,
+Softmax, etc. hence keeping a check on the intra-block com-
+plexity of the network.
+The input first passes through Layer Normalization as it can
+help stabilize the training process. This is followed by con-
+volution operations and a Simple Gate (SG). SG is a variant
+of Gated Linear Units (GLU) [10] as evident from the fol-
+lowing equations 1 and 2
+GLU(X, f, g, σ) = f(X) ⊙ σ(g(X))
+(1)
+S impleGate(X, Y) = X ⊙ Y
+(2)
+and a replacement for GELU[18] activation function be-
+cause of the similarity between GLU and GELU (Equa-
+tion 3).
+GELU(x) = xφ(x)
+(3)
+In Simple Gate, the feature maps are divided into two parts
+along the channel dimension and then multiplied as shown
+in Figure 4. Another novelty introduced in this block is
+Simplified Channel Attention (SCA). Channel Attention
+(CA) can be expressed as:
+CA(X) = X ⊗ σ(W2max(0, W1pool(X)))
+(4)
+where X represents the feature map, pool indicates the
+global average pooling operation,σ is Sigmoid, W1, W2 are
+fully-connected layers and ⊗ is a channel-wise product op-
+eration. This can be taken as a special case of GLU from
+W
+W
+W
+H
+H
+H
+C/2
+C/2
+C/2
+Figure 4. Simple Gate as represented by Equation 2 ⊗ denotes
+channel-wise multiplicaWere
+which we can derivate the equation for Simplified Channel
+Attention:
+SCA(X) = X ⊗ Wpool(X)
+(5)
+3.1.2
+Level Attention Module
+Once we have extracted features from all the NAF Groups,
+we concatenate them and pass them through the Level At-
+tention Module (LAM) [45]. This module learns attention
+weights for features obtained at different levels.
+In LAM, each feature map is first reshaped to a 2D matrix
+of the size K × HWC, where K, H, W, and C are the no. of
+NAF Groups, height, width, and no. of channels of the fea-
+ture maps respectively. We find a correlation matrix of this
+2D matrix by multiplying it with its transpose matrix. Fi-
+nally, we multiply the 2D matrix with this correlation ma-
+trix and reshape it to K × H × W × C tensor. Inspired by
+residual learning, this tensor is substituted for residual and
+is added to the original concatenated feature maps. The re-
+sultant features are then reshaped to H × W × KC, passing
+through 1 × 1 convolution operation to get the H × W × C
+feature map. This is passed through some post-processing
+convolutions to get the final enhanced output. We include
+its architecture diagram in the supplementary material for a
+better understanding.
+3.2. Loss Functions
+Four loss functions, namely, reconstruction loss, perceptual
+loss, edge loss [19], and FFT loss[7], have been used to
+supervise the task of image restoration.
+The total loss L is defined in Equation 6, where λ1 = 0.04,
+λ2 = 1 and λ3 = 0.01.
+L = Ls + λ1Lp + λ2Le + λ3Lf
+(6)
+3.2.1
+Reconstruction Loss:
+The restored clear output image is compared with its ground
+truth value in the spatial domain using a standard l1 loss as
+5
+
+demonstrated in Equation 7. We use l1 loss instead of l2 loss
+as it does not over-penalize the errors and leads to better
+image restoration performance [48].
+Ls = 1
+N
+n
+�
+i=1
+∥ xgt
+i − NAFNet(xdark,hazy
+i
+) ∥1
+(7)
+In the above equation, xgt
+i refers to the ground truth clear im-
+age, and NAFNet(xdark,hazy
+i
+) denotes the output of our pro-
+posed NAFNet when a dark and hazy image is fed to the
+network.
+3.2.2
+Perceptual Loss:
+To reduce the perceptual loss and improve the image’s vi-
+sual quality, we utilize the features of the pre-trained VGG-
+19 network [38] obtained from the output of one of the
+ReLU activation layers. It is defined in Equation 8, where
+wi j, hij, and cij refer to the dimensions of the respective
+feature maps inside the VGG-19 architecture. φij denotes
+the feature maps outputted from the jth convolutional layer
+inside the i-th block in the VGG network.
+Lp =
+1
+wijhijci j
+wij
+�
+x=1
+hij
+�
+y=1
+cij
+�
+z=1
+∥ φij(Igt)xyz − φij(Iout)xyz ∥
+(8)
+3.2.3
+Edge Loss:
+To recover the high-frequency details lost because of the in-
+herent noise in dark and hazy images, we have an additional
+edge loss to constrain the high-frequency components be-
+tween the ground truth and the recovered image.
+Le =
+�
+(∇2(Igt) − ∇2(Iout))2 + ϵ2
+(9)
+In Equation 9, ∇2 refers to the Laplacian operation [22],
+which is then applied to the ground truth and the predicted
+clean image to get the edge loss.
+3.2.4
+FFT Loss:
+To supervise the haze-free results in the frequency domain,
+we add another loss called Fast Fourier transform (FFT) loss
+(denoted by Lf in Equation 12. It calculates the loss of both
+amplitude and phase using the l1 loss function without ad-
+ditional inference cost.
+Axgt
+i , Pxgt
+i = FFT(xgt
+i ),
+(10)
+Axout
+i , Pxout
+i
+= FFT(xout
+i ),
+(11)
+L f = 1
+N
+n
+�
+i=1
+(∥ Axgt
+i − Axout
+i
+∥1 + ∥ Pxgt
+i − Pxout
+i
+∥1)
+(12)
+4. Experimental Results
+To demonstrate the outcomes of our model’s approach to-
+wards image enhancement under low-visibility conditions,
+this section contains a detailed description of the dataset
+generated and used in Section 4.1, the experimental set-
+tings in Section 4.2, the metrics used for evaluation in Sec-
+tion 4.3 and a discussion on the results obtained in Sec-
+tion 5.1 and 5.2.
+4.1. Dataset Details
+Due to the lack of available datasets that meet our require-
+ments, we generate a new one using the AFO dataset [15].
+The dataset generation process has been elaborated below,
+and the final images have been shown in Figure 5.
+• Haze effect - To add fog, imgaug [21], a well-known
+python library was used. A random integral value be-
+tween 3, 4, 5 was selected, representing the fog’s sever-
+ity. For each image, this random number was chosen
+and pre-defined functions within the package were uti-
+lized to add a layer of fog to the image.
+• Low-light Effect - Given a normal image, our goal is
+to output a low-lit image while preserving the underly-
+ing information. We follow the pipeline introduced [8],
+which parametrically models the low light-degrading
+transformation by observing the image signal process-
+ing (ISP) pipeline between the sensor measurement
+system and the final image.
+The low-illumination-
+degrading pipeline is a three-stage process:
+– Unprocessing procedure - This part aims to syn-
+thesize RAW format images from input sRGB
+images by invert tone mapping, invert gamma
+correction, and the transformation of the image
+from sRGB space to cRGB space, and invert
+white balancing.
+– Low Light Corruption - This aims at adding shot
+and read noises to the output of the unprocess-
+ing procedure, as these are common in-camera
+imaging systems. Shot noise is a type of noise
+generated by the random arrival of photons in
+a camera, which is a fundamental limitation.
+Read noise occurs during the charge conversion
+of electrons into voltage in the output amplifier,
+which can be approximated using a Gaussian ran-
+dom variable with zero mean and fixed variance.
+– ISP Pipeline - RAW image processing is done
+after the lowlight corruption process in the fol-
+lowing order: add quantization noise, white bal-
+ancing from cRGB to sRGB, and gamma correc-
+tion, which finally outputs a degraded low-light
+image.
+6
+
+Ground Truth Images
+Ground Truth Images
+Generated Images
+Generated Images
+Figure 5. Visual illustration of a few sample images from our dataset. Columns 1 and 3 show original images taken from AFO Dataset
+[15], whereas Columns 2 and 4 show their corresponding images generated as explained in Section 4.1 simulating low-visibility conditions.
+• Combination of Haze and Low-light Effect - Re-
+sults of implementing the low-light generation algo-
+rithm described above on foggy images generated us-
+ing img-aug are shown here. It can be seen that com-
+bining the two (fog and low light) has introduced ad-
+versity in finding the location of the objects in the wa-
+ter bodies. Moreover, finding a unique solution for
+such a combination has not been explored to date
+4.2. Experimental Settings
+The images were resized to get the resultant dimensions as
+256 × 456. Adam optimizer with an initial learning rate of
+1e−4, β1, and β2 with a value of 0.9 and 0.999 were chosen.
+The batch size was fixed as 2. We have used 3 groups in all
+our experiments, each with 16 blocks. Pytorch backend was
+used to compile the model and train it.
+4.3. Evaluation Metrics
+We reported the results we obtained using two standard im-
+age restoration metrics (i.e., PSNR and SSIM). These met-
+rics will help us quantitatively evaluate the performance of
+our model in terms of feature colors and structure similarity.
+High PSNR and SSIM values if indicative of good results.
+5. Experimental Results
+The architecture used is given in Figure 2. This section
+gives a detailed analysis of the results obtained by the pro-
+posed method.
+5.1. Discussion and Comparison
+In this subsection, we discuss the evaluation results ob-
+tained by the proposed pipeline. Previous methods were
+Method
+Year
+PSNR
+SSIM
+Zero-DCE[13]
+2020
+12.323
+0.529
+SGZNet[49]
+2022
+12.578
+0.519
+BPPNet[39]
+2022
+15.507
+0.755
+DehazeNet[3]
+2016
+15.710
+0.391
+Star-DCE[47]
+2021
+16.651
+0.539
+FFANet[7]
+2020
+15.050
+0.582
+MSBDN-DFF[16]
+2020
+16.686
+0.689
+LVRNet (Ours)
+2022
+25.744
+0.905
+Table 1. Quantitative comparison of our proposed network
+with previous work. The best results and the second-best results
+have been highlighted with red color and blue colors, respectively.
+trained on the newly generated dataset and tested to com-
+pare their metrics with our model’s performance. These
+methods were built to enhance the low-light image or obtain
+a clear image from a hazy one. The results are mentioned
+in Table 1.
+We observe a huge increase in the PSNR value as compared
+to Zero-DCE[13], which enhances the low-light image as a
+curve estimation problem. However, it introduces an even
+amplified noise leading to color degradation as seen in Fig-
+ure 1. Notwithstanding its fast processing speed, Zero-DCE
+has limited noise suppression and haze removal capacity.
+Star-DCE[47], which uses a transformer backbone instead
+of a CNN one in the Zero-DCE network, shows a 35.12%
+increase in PSNR value. Owing to the added LAM struc-
+7
+
+S.no.
+Reconstruction Loss
+Perceptual Loss
+Edge Loss
+FFT Loss
+PSNR
+SSIM
+1.
+
+
+
+
+24.070
+0.870
+2.
+
+
+
+
+25.455
+0.903
+3.
+
+
+
+
+25.624
+0.897
+4.
+
+
+
+
+25.719
+0.900
+5.
+
+
+
+
+25.744
+0.905
+Table 2. Ablation experiments: We train our model using different combinations of loss functions to understand the importance of
+individual losses for image restoration. The best results are obtained when the model is trained using all the loss functions mentioned in
+this work.
+ture, using which our model can focus on more important
+feature maps, we can achieve a 54% higher PSNR value.
+SGZNet[49] uses pretrained networks for enhancement fac-
+tor estimation, thus their result is dependent on those pre-
+trained weights, leading to a lower PSNR value of 12.578
+on LowVis-AFO. From Figure 1, we observe that the result
+obtained from SGZNet is still degraded by excessive noise
+and lacks saturation. DehazeNet[3] is limited by the net-
+work’s depth and cannot generalize to real-world scenarios.
+Hence, it results in a low PSNR of 15.710. Methods like
+BPPNet[39] and FFANet[33] are end-to-end deep learning
+methods for image dehazing. BPPNet[39] distorts the color
+distribution in the recovered image as it cannot remove the
+dark regions, whereas FFA-Net[33] produces image with a
+lower perceptual quality.
+We propose an end-to-end deep learning pipeline (0.43M
+parameters) that can perform image dehazing and low-light
+image enhancement with a significant decrease in the num-
+ber of parameters as compared to MSBDN-DFF [16] (31M
+parameters) and FFA-Net[33] (4.45M parameters).
+The supplementary material has provided a discussion
+on the number of parameters of other models.
+We also
+trained the model for 10 epochs with fewer NAF blocks to
+prove that we achieved better results than the lighter results,
+not due to an increase in parameters but because of the self-
+sufficiency of the added LAM module, non-linear activation
+networks, and residual connections. The results of these ex-
+periments are reported in the supplementary material.
+5.2. Ablation Studies
+To prove the importance of the perceptual loss, edge loss,
+and fft-loss, added to supervise the training procedure, we
+conducted experiments excluding each of them and reported
+the values of PSNR and SSIM in Table 2. We keep the l1
+loss function constant in all experiments as it is critical in
+image restoration tasks. We observe an increase in metric
+values in the lower rows compared to row 1. As a result
+of more supervision in the unchanged architecture, there is
+an increase in the quality of clear images obtained, which
+are demonstrated in the supplementary material. There is
+also an increase in PSNR value (which depends on per-pixel
+distance) in row 3, once we train the model without percep-
+tual loss. This is seen as perceptual loss doesn’t compare
+individual pixel values but the high-level features obtained
+from a pretrained network. In row 4, we get a lower PSNR
+value on excluding edge loss compared to row 5, as we get
+lesser edge supervision. Overall, we get the best perfor-
+mance when we include all the loss functions, as seen in
+row 5.
+6. Conclusion
+In this work, we have presented Low-Visibility Restora-
+tion Network (LVRNet), a new lightweight deep learning
+architecture for image restoration.
+We also introduce a
+new dataset, LowVis-AFO, that includes a diverse combi-
+nation of synthetic darkness and haze. We also performed
+benchmarking experiments on our generated dataset and
+surpassed the results obtained using the previous image
+restoration network by a significant margin. Qualitative and
+quantitative comparison with previous work has demon-
+strated the effectiveness of LVRNet. We believe our work
+will motivate more research, focused on dealing with a com-
+bination of adverse effects such as haze, rain, snowfall, etc.
+rather than considering a single factor. In our future work,
+we plan to extend LVRNet for image restoration tasks where
+more factors, that negatively impact the image quality, are
+taken into account.
+8
+
+Supplementary Material
+To make our submission self-contained and given the page
+limitation, this supplementary material provides additional
+details. Section 1 gives an overview of the number of pa-
+rameters and PSNR obtained by different methods. Sec-
+tion 2 contains visual results that highlight the significance
+of the loss functions. Section 3 contains the ablation ex-
+periment with lesser blocks, and Section 4 demonstrates the
+architecture diagram of the level attention module.
+1. PSNR vs Parameters
+Figure 6 presents the PSNR vs. Parameters plot that the
+previous methods and our method achieved on the testing
+set of LowVis-AFO. Our model outperforms the state-of-
+the-art image dehazing and low-light image enhancement
+methods by a good margin while having a lesser number of
+parameters.
+Figure 6. The PSNR vs Number of Parameters of recent image
+restoration methods on the newly proposed LowVis-AFO dataset.
+S.no.
+#Blocks
+PSNR
+SSIM
+#params
+Runtime(s)
+1.
+14
+21.3432
+0.8626
+0.38M
+0.035
+2.
+12
+20.4302
+0.8488
+0.33M
+0.029
+3.
+10
+20.2965
+0.8494
+0.28M
+0.024
+Table 3. Results of the experiments conducted on a lesser num-
+ber of NAF blocks. The training was done for 10 epochs and the
+metrics were obtained on the test set thereafter.
+2. Ablation Experiment on Different Loss
+Functions
+Figure 8 demonstrates the visual results obtained when
+we conducted experiments excluding some loss functions.
+The motivation behind the experiment is to highlight the
+importance of the extra loss functions (perceptual loss, edge
+loss, fft-loss) added to supervise our pipeline. The quanti-
+tative results are given in Table 2 in the main manuscript.
+3. Ablation Experiment with Lesser Number of
+Blocks
+To prove the self-sufficiency of the individual components
+included in our architecture such as LAM, we conduct ex-
+periments with a lesser number of NAF blocks [5] and re-
+ported the PSNR and SSIM obtained in Table 1. Seeing
+the results, we can conclude that our model achieves better
+results, not because of an increase in the number of param-
+eters as compared to the lighter model, but because of the
+entire pipeline adopted.
+4. Level Attention Module
+As mentioned in the main text, the diagram for LAM[45]
+has been provided here in the supplementary material. (re-
+fer Figure 7)
+References
+[1] Codruta O Ancuti, Cosmin Ancuti, Chris Hermans, and
+Philippe Bekaert. A fast semi-inverse approach to detect and
+remove the haze from a single image. In Asian Conference
+on Computer Vision, pages 501–514. Springer, 2010.
+[2] Julian Besag, Jeremy York, and Annie Mollié. Bayesian im-
+age restoration, with two applications in spatial statistics. An-
+nals of the institute of statistical mathematics, 43(1):1–20,
+1991.
+[3] Bolun Cai, Xiangmin Xu, Kui Jia, Chunmei Qing, and
+Dacheng Tao. Dehazenet: An end-to-end system for single
+image haze removal. IEEE Transactions on Image Process-
+ing, 25(11):5187–5198, 2016.
+[4] Dongdong Chen, Mingming He, Qingnan Fan, Jing Liao, Li-
+heng Zhang, Dongdong Hou, Lu Yuan, and Gang Hua. Gated
+context aggregation network for image dehazing and derain-
+ing. In 2019 IEEE winter conference on applications of com-
+puter vision (WACV), pages 1375–1383. IEEE, 2019.
+[5] Liangyu Chen, Xiaojie Chu, Xiangyu Zhang, and Jian Sun.
+Simple baselines for image restoration.
+arXiv preprint
+arXiv:2204.04676, 2022.
+[6] Rongsen Chen and Edmund M-K Lai. Convolutional autoen-
+coder for single image dehazing. In ICIP, pages 4464–4468,
+2019.
+[7] Sung-Jin Cho, Seo-Won Ji, Jun-Pyo Hong, Seung-Won Jung,
+and Sung-Jea Ko. Rethinking coarse-to-fine approach in sin-
+gle image deblurring. In Proceedings of the IEEE/CVF inter-
+national conference on computer vision, pages 4641–4650,
+2021.
+[8] Ziteng Cui, Guo-Jun Qi, Lin Gu, Shaodi You, Zenghui
+Zhang, and Tatsuya Harada. Multitask aet with orthogonal
+tangent regularity for dark object detection. In Proceedings
+9
+
+26
+米
+Zero-DCE
+SGZNet
+24
+BPPNet
+DehazeNet
+22
+Star-DCE
+FFA-Net
+MSBDN-DFF
+PSNR
+20
+Ours
+18
+16
+14
+12
+0
+10M
+20M
+30M
+NumberofParametersStacked
+Feature Maps 
+K x H x W x C
+K x HWC
+Reshape
+Transpose
+Softmax
+K x HWC
+K x H x W x C
+Feature Maps
+Reshape
+H x W x KC
+Reshape
+H x W x C
+1 x 1
+Conv
+Figure 7. Visual illustration of operations performed by Level Attention Module.
+of the IEEE/CVF International Conference on Computer Vi-
+sion, pages 2553–2562, 2021.
+[9] Sourya Dipta Das and Saikat Dutta. Fast deep multi-patch
+hierarchical network for nonhomogeneous image dehazing.
+In Proceedings of the IEEE/CVF Conference on Computer
+Vision and Pattern Recognition Workshops, pages 482–483,
+2020.
+[10] Yann N Dauphin, Angela Fan, Michael Auli, and David
+Grangier. Language modeling with gated convolutional net-
+works.
+In International conference on machine learning,
+pages 933–941. PMLR, 2017.
+[11] Raanan Fattal. Dehazing using color-lines. ACM transac-
+tions on graphics (TOG), 34(1):1–14, 2014.
+[12] Stuart Geman and Donald Geman.
+Stochastic relaxation,
+gibbs distributions, and the bayesian restoration of images.
+IEEE Transactions on pattern analysis and machine intelli-
+gence, (6):721–741, 1984.
+[13] Chunle Guo, Chongyi Li, Jichang Guo, Chen Change Loy,
+Junhui Hou, Sam Kwong, and Runmin Cong. Zero-reference
+deep curve estimation for low-light image enhancement. In
+Proceedings of the IEEE/CVF Conference on Computer Vi-
+sion and Pattern Recognition, pages 1780–1789, 2020.
+[14] Tiantong Guo, Venkateswararao Cherukuri, and Vishal
+Monga. Dense123’color enhancement dehazing network. In
+Proceedings of the IEEE/CVF Conference on Computer Vi-
+sion and Pattern Recognition Workshops, pages 0–0, 2019.
+[15] Jan G˛asienica-Józkowy, Mateusz Knapik, and Boguslaw Cy-
+ganek. An ensemble deep learning method with optimized
+weights for drone-based water rescue and surveillance. Inte-
+grated Computer-Aided Engineering, pages 1–15, 01 2021.
+[16] Dong Hang, Pan Jinshan, Hu Zhe, Lei Xiang, Zhang Xinyi,
+Wang Fei, and Yang Ming-Hsuan. Multi-scale boosted de-
+hazing network with dense feature fusion. In CVPR, 2020.
+[17] Kaiming He, Jian Sun, and Xiaoou Tang. Single image haze
+removal using dark channel prior. IEEE transactions on pat-
+tern analysis and machine intelligence, 33(12):2341–2353,
+2010.
+[18] Dan Hendrycks and Kevin Gimpel.
+Gaussian error linear
+units (gelus). arXiv preprint arXiv:1606.08415, 2016.
+[19] Kui Jiang, Zhongyuan Wang, Peng Yi, Chen Chen, Baojin
+Huang, Yimin Luo, Jiayi Ma, and Junjun Jiang. Multi-scale
+progressive fusion network for single image deraining. In
+Proceedings of the IEEE/CVF conference on computer vision
+and pattern recognition, pages 8346–8355, 2020.
+[20] Yifan Jiang, Xinyu Gong, Ding Liu, Yu Cheng, Chen Fang,
+Xiaohui Shen, Jianchao Yang, Pan Zhou, and Zhangyang
+Wang.
+Enlightengan:
+Deep light enhancement without
+paired supervision. IEEE Transactions on Image Process-
+ing, 30:2340–2349, 2021.
+[21] Alexander B. Jung, Kentaro Wada, Jon Crall, Satoshi
+Tanaka, Jake Graving, Christoph Reinders, Sarthak Ya-
+dav, Joy Banerjee, Gábor Vecsei, Adam Kraft, Zheng Rui,
+Jirka Borovec, Christian Vallentin, Semen Zhydenko, Kil-
+ian Pfeiffer, Ben Cook, Ismael Fernández, François-Michel
+De Rainville, Chi-Hung Weng, Abner Ayala-Acevedo,
+Raphael Meudec, Matias Laporte, et al. imgaug. https:
+//github.com/aleju/imgaug, 2020.
+Online; accessed
+01-Feb-2020.
+[22] Behzad Kamgar-Parsi and Azriel Rosenfeld.
+Optimally
+isotropic laplacian operator. IEEE Transactions on Image
+Processing, 8(10):1467–1472, 1999.
+[23] Yunan Li, Qiguang Miao, Wanli Ouyang, Zhenxin Ma, Hui-
+juan Fang, Chao Dong, and Yining Quan. Lap-net: Level-
+aware progressive network for image dehazing. In Proceed-
+ings of the IEEE/CVF International Conference on Computer
+Vision, pages 3276–3285, 2019.
+[24] Zhuwen Li, Ping Tan, Robby T Tan, Danping Zou, Steven
+Zhiying Zhou, and Loong-Fah Cheong. Simultaneous video
+defogging and stereo reconstruction. In Proceedings of the
+IEEE conference on computer vision and pattern recogni-
+tion, pages 4988–4997, 2015.
+[25] Xiao Liang, Runde Li, and Jinhui Tang. Selective attention
+network for image dehazing and deraining. In Proceedings
+of the ACM Multimedia Asia, pages 1–6. 2019.
+[26] Xiaohong Liu, Yongrui Ma, Zhihao Shi, and Jun Chen. Grid-
+dehazenet: Attention-based multi-scale network for image
+dehazing.
+In Proceedings of the IEEE/CVF international
+conference on computer vision, pages 7314–7323, 2019.
+[27] Kin Gwn Lore, Adedotun Akintayo, and Soumik Sarkar. Ll-
+net: A deep autoencoder approach to natural low-light image
+enhancement. Pattern Recognition, 61:650–662, 2017.
+[28] Kun Lu and Lihong Zhang. Tbefn: A two-branch exposure-
+fusion network for low-light image enhancement.
+IEEE
+Transactions on Multimedia, 23:4093–4105, 2020.
+[29] Feifan Lv, Bo Liu, and Feng Lu. Fast enhancement for non-
+uniform illumination images using light-weight cnns. In Pro-
+ceedings of the 28th ACM International Conference on Mul-
+timedia, pages 1450–1458, 2020.
+10
+
+[30] Feifan Lv, Feng Lu, Jianhua Wu, and Chongsoon Lim.
+Mbllen: Low-light image/video enhancement using cnns. In
+BMVC, volume 220, page 4, 2018.
+[31] Earl J McCartney. Optics of the atmosphere: scattering by
+molecules and particles. New York, 1976.
+[32] Seungjun Nah, Tae Hyun Kim, and Kyoung Mu Lee. Deep
+multi-scale convolutional neural network for dynamic scene
+deblurring.
+In Proceedings of the IEEE conference on
+computer vision and pattern recognition, pages 3883–3891,
+2017.
+[33] Xu Qin, Zhilin Wang, Yuanchao Bai, Xiaodong Xie, and
+Huizhu Jia. Ffa-net: Feature fusion attention network for sin-
+gle image dehazing. In Proceedings of the AAAI Conference
+on Artificial Intelligence, volume 34, pages 11908–11915,
+2020.
+[34] Wenqi Ren, Si Liu, Hua Zhang, Jinshan Pan, Xiaochun Cao,
+and Ming-Hsuan Yang. Single image dehazing via multi-
+scale convolutional neural networks. In European conference
+on computer vision, pages 154–169. Springer, 2016.
+[35] Wenqi Ren, Sifei Liu, Lin Ma, Qianqian Xu, Xiangyu Xu,
+Xiaochun Cao, Junping Du, and Ming-Hsuan Yang. Low-
+light image enhancement via a deep hybrid network. IEEE
+Transactions on Image Processing, 28(9):4364–4375, 2019.
+[36] Wenqi Ren, Jinshan Pan, Hua Zhang, Xiaochun Cao, and
+Ming-Hsuan Yang. Single image dehazing via multi-scale
+convolutional neural networks with holistic edges. Interna-
+tional Journal of Computer Vision, 128(1):240–259, 2020.
+[37] Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net:
+Convolutional networks for biomedical image segmentation,
+2015.
+[38] Karen Simonyan and Andrew Zisserman. Very deep convo-
+lutional networks for large-scale image recognition, 2014.
+[39] Ayush Singh, Ajay Bhave, and Dilip K. Prasad. Single image
+dehazing for a variety of haze scenarios using back projected
+pyramid network, 2020.
+[40] Chen Wei, Wenjing Wang, Wenhan Yang, and Jiaying
+Liu. Deep retinex decomposition for low-light enhancement.
+arXiv preprint arXiv:1808.04560, 2018.
+[41] Ke Xu, Xin Yang, Baocai Yin, and Rynson WH Lau.
+Learning to restore low-light images via decomposition-and-
+enhancement. In Proceedings of the IEEE/CVF Conference
+on Computer Vision and Pattern Recognition, pages 2281–
+2290, 2020.
+[42] Wenhan Yang, Wenjing Wang, Haofeng Huang, Shiqi Wang,
+and Jiaying Liu. Sparse gradient regularized deep retinex
+network for robust low-light image enhancement.
+IEEE
+Transactions on Image Processing, 30:2072–2086, 2021.
+[43] Chia-Hung Yeh, Chih-Hsiang Huang, and Li-Wei Kang.
+Multi-scale deep residual learning-based single image haze
+removal via image decomposition.
+IEEE Transactions on
+Image Processing, 29:3153–3167, 2019.
+[44] Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and
+Lei Zhang. Beyond a gaussian denoiser: Residual learning of
+deep cnn for image denoising. IEEE transactions on image
+processing, 26(7):3142–3155, 2017.
+[45] Kaihao Zhang, Wenhan Luo, Boheng Chen, Wenqi Ren,
+Bjorn Stenger, Wei Liu, Hongdong Li, and Ming-Hsuan
+Yang. Benchmarking deep deblurring algorithms: A large-
+scale multi-cause dataset and a new baseline model. arXiv
+preprint arXiv:2112.00234, 2021.
+[46] Xiaoqin Zhang, Jinxin Wang, Tao Wang, and Runhua Jiang.
+Hierarchical feature fusion with mixed convolution attention
+for single image dehazing. IEEE Transactions on Circuits
+and Systems for Video Technology, 32(2):510–522, 2021.
+[47] Zhaoyang Zhang, Yitong Jiang, Jun Jiang, Xiaogang Wang,
+Ping Luo, and Jinwei Gu.
+Star:
+A structure-aware
+lightweight transformer for real-time image enhancement. In
+Proceedings of the IEEE/CVF International Conference on
+Computer Vision, pages 4106–4115, 2021.
+[48] Hang Zhao, Orazio Gallo, Iuri Frosio, and Jan Kautz. Loss
+functions for image restoration with neural networks. IEEE
+Transactions on computational imaging, 3(1):47–57, 2016.
+[49] Shen Zheng and Gaurav Gupta. Semantic-guided zero-shot
+learning for low-light image/video enhancement.
+In Pro-
+ceedings of the IEEE/CVF Winter Conference on Applica-
+tions of Computer Vision, pages 581–590, 2022.
+[50] Zhuoran Zheng, Wenqi Ren, Xiaochun Cao, Xiaobin Hu,
+Tao Wang, Fenglong Song, and Xiuyi Jia.
+Ultra-high-
+definition image dehazing via multi-guided bilateral learn-
+ing. In 2021 IEEE/CVF Conference on Computer Vision and
+Pattern Recognition (CVPR), pages 16180–16189. IEEE,
+2021.
+[51] Minfeng Zhu, Pingbo Pan, Wei Chen, and Yi Yang. Eemefn:
+Low-light image enhancement via edge-enhanced multi-
+exposure fusion network. In Proceedings of the AAAI Con-
+ference on Artificial Intelligence, volume 34, pages 13106–
+13113, 2020.
+11
+
+LP
+LPF
+LPE
+LEF
+LPEF
+GT
+Input
+Figure 8. Qualitative results obtained from experiments conducted on different loss functions. In the figure, L = L1 loss, P = Perceptual
+Loss, E = Edge loss and F = FFT loss.
+12
+
diff --git a/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/load_file.txt b/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f1af401cf10a33a1fd62ef395454667d19fd57fe
--- /dev/null
+++ b/ktE5T4oBgHgl3EQfGw6j/content/tmp_files/load_file.txt
@@ -0,0 +1,525 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf,len=524
+page_content='LVRNet: Lightweight Image Restoration for  Aerial Images under Low Visibility  Esha Pahwa*  BITS Pilani  f20180675@pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='in  Achleshwar Luthra*  Carnegie Mellon University  achleshl@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='edu  Pratik Narang  BITS Pilani  pratik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='narang@pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='bits-pilani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='in  Abstract  Learning to recover clear images from images having a  combination of degrading factors is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  That being said, autonomous surveillance in low visibility  conditions caused by high pollution/smoke, poor air qual-  ity index, low light, atmospheric scattering, and haze dur-  ing a blizzard becomes even more important to prevent ac-  cidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It is thus crucial to form a solution that can re-  sult in a high-quality image and is efficient enough to be  deployed for everyday use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' However, the lack of proper  datasets available to tackle this task limits the performance  of the previous methods proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' To this end, we generate  the LowVis-AFO dataset, containing 3647 paired dark-hazy  and clear images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We also introduce a lightweight deep  learning model called Low-Visibility Restoration Network  (LVRNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It outperforms previous image restoration meth-  ods with low latency, achieving a PSNR value of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='744 and  an SSIM of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='905, making our approach scalable and ready  for practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The code and data can be found here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Introduction  Image enhancement and restoration have been a critical  area of research using both traditional digital image pro-  cessing techniques[12] [2], and the recent deep learning  frameworks[32][33][44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The goal of image restoration is  to recover a clear image, whereas image enhancement is to  improve the quality of the degraded image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In this study, we  perform recovery of the clear image from the hazy version  while performing low-light image enhancement using a sin-  gle convolutional network, which could further be applied  to tasks such as search and rescue operations using object  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  equal contribution  Using deep learning algorithms for image recovery has  many benefits, the most important being that it can general-  ize to different variations in the images captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Hence, we  observe that deep learning-based methods on most bench-  mark datasets often outperform traditional methods signif-  icantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  However, there are still challenges that the re-  searchers have to tackle for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Publicly  available datasets containing a variety of degrading factors  that model real-world scenarios are few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Hence, most pre-  vious works have focused on removing one type of degra-  dation with a specific intensity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' From the perspective  of computational complexity, recent deep learning methods  are computationally expensive, and thus they can’t be de-  ployed on edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Moreover, image restoration has  been a long-standing ill-posed research problem, as there  are infinite mappings between the degraded and the clear  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Thus, the existing methods still have room for im-  provement in finding the correct mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In this work, we focus on developing an end-to-end  lightweight deep-learning solution for the image restoration  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Our major contributions are listed below:  Taking inspiration from Non-linear Activation Free  Network (NAFNet) [5] and Level Attention Module  [45], we propose a novel algorithm - Low-Visibility  Restoration Network (LVRNet), that can effectively re-  cover high-quality images from degraded images taken  in poor visual conditions (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Due to the lack of available datasets that exhibit a com-  bination of adverse effects, we generate a new dataset,  namely LowVis-AFO (abbreviation for Low-Visibility  Aerial Floating Objects dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We use AFO [15] as  our ground truth dataset and synthesize dark hazy im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The data generation process has been elaborated  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='05434v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='CV]  13 Jan 2023  Input  Zero-DCE  SGZNet  DehazeNet  StarDCE  BPPNet  FFANet  MSBDN-DFF  Ground Truth / Reference Image  Our Result  Input  Zero-DCE  SGZNet  DehazeNet  StarDCE  BPPNet  FFANet  MSBDN-DFF  Ground Truth / Reference Image  Our Result  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Visual results on the proposed LowVis-AFO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The method used to obtain each result has been mentioned under the  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Benchmarking experiments have been provided on the  LowVis-AFO dataset to help future researchers for  quantitative comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Along with that, LVRNet  surpasses the results obtained using previous image  restoration techniques by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  We perform extensive ablation studies to analyze the  importance of various loss functions existing in cur-  rent image restoration research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' These experiments are  discussed in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Related Works  This section highlights the previous work done in the fields  of image dehazing and low-light image enhancement and  their limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Image Dehazing  Hazy weather is often seen due to floating particles in  the environment which degrade the quality of the image  captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Therefore, many previous works have tried to  recover a clear image from the hazy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  These works  can be divided into two methods, ones that rely on prior  assumptions [17] and the atmospheric scattering model  (ASM) [31] and the others which use deep learning to solve  the problem, either by combination with ASM [3][34][36]  or independently [25][26][33][46][50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Conventional  approaches are physically inspired and apply various types  of sharp image priors to regularize the solution space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  However, they exhibit shortcomings when implemented  with real-world images and videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' For example, the dark  channel prior method (DCP) [31] does not perform well  in regions containing the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' These methods [1][11][24]  are known to be computationally expensive and require  2  Pre-processing  Conv  Post-processing  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  NAF-G1  NAF-G2  +  LAN  NAF-G3  Stacked Feature  Maps  Input Image  Output Image  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Model architecture of the proposed LVRNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Starting from the top-left: The input image is passed to the pre-processing  convolution layers where feature maps are learned and passed to NAF Groups (here we have used 3 groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The features extracted from  each group are concatenated (or stacked) along the channel dimension and sent as input to the Level Attention Module (LAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Finally, we  pass LAM’s output to CNN layers for post-processing, adding the original image through residual connection and extracting the restored  image at the bottom-left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  heuristic parameter-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Supervised dehazing methods  can be divided into two subparts, one is ASM based, and  the other is non-ASM based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  ASM-based Learning: MSCNN[34] solves the task of  image dehazing by dividing the problem into three steps:  using CNN to estimate the transmission map t(x), using  statistical methods to find atmospheric light A and then  recover the clear image J(x) using t(x) and A jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Meth-  ods like LAP-Net [23] adopt the relation of depth with the  amount of haze in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The farther the scene from the  camera, the denser the haze would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Hence it considers  the difference in the haze density in the input image using a  stage-wise loss, where each stage predicts the transmission  map from mild to severe haze scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  DehazeNet [3]  consists of four sequential operations: feature extraction,  multi-scale mapping, calculating local extremum, and  non-linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' MSRL-DehazeNet [43] decomposes  the problem into recovering high-frequency and basic com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' GCANet [4] employs residual learning between  haze-free and hazy images as an optimization objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  End-to-end Learning: This subpart of previous work cor-  responds to non-ASM-based deep learning methods for re-  covering the clear image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Back-Projected Pyramid Network  (BPPNet) [39] is a generative adversarial network that in-  cludes iterative blocks of UNets [37] to learn haze features  and pyramid convolution to preserve spatial features of dif-  ferent scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The reason behind using iterative blocks of  UNets[37] is to avoid increasing the number of encoder lay-  ers in a single UNet[37] as it leads to a decrease in height  and width of latent feature representation hence resulting in  loss of spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Moreover, different blocks of  UNet learn different complexities of haze features, and the  final concatenation step ensures that all of them are taken  into account during image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The final recon-  struction is done using the pyramid convolution block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The  output feature is post-processed to get a haze-free image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Feature-Fusion Attention Network (FFANet) [33] adopts  the idea of an attention mechanism and skip connections  to restore haze-free images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' A combination of channel at-  tention and pixel attention is introduced, which helps the  network, deal with the uneven spatial distribution of haze  and different weighted information across channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Au-  toencoders [6], hierarchical networks [9], and dense block  networks [14] has also been proposed for the task of image  dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' However, our main comparison lies with FFANet  [33], wherein we show a huge improvement compared to  the former method with a model containing a lesser number  of parameters and which can generalize to different levels  of haze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Low-light Enhancement  Traditional methods for low-light image enhancement  (LLIE) include Histogram Equalization-based methods and  Retinex model-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Recent research has been  focused on developing deep learning-based methods fol-  lowing the success of the first seminal work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Deep learning-  based solutions are more accurate, robust, and have a  shorter inference time thus attracting more researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3  Layer Norm  1x1, conv  3x3, dconv  Simple Gate  SCA  1x1, conv  +  Layer Norm  1x1, conv  Simple Gate  1x1, conv  +  I/P  O/P  NAF BLOCK  NAF BLOCK  NAF BLOCK  CONV  +  NAF-Group  NAF Block  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Architecture of NAF Block and NAF Group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' NAF Blocks are the building blocks of NAF Groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' A detailed description has  been provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1 and Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1  Learning strategies used in these methods are mainly su-  pervised learning [27, 29, 30, 35, 51, 28, 41], unsupervised  learning [20], and zero-shot learning [49, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Supervised Learning: The first deep learning-based LLIE  method LLNet [27] is an end-to-end network that employs a  variant of stacked-sparse denoising autoencoder to brighten  and denoise low-light images simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' LLNet in-  spired many other works [29, 30, 35, 51], but they do not  consider the observation that noise exhibits different lev-  els of contrast in different frequency layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Later, Xu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' [41] proposed a network that suppresses noise in the  low-frequency layers and recovers the image contents by  inferring the details in high-frequency layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' There is an-  other division of methods that is based on the Retinex the-  ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Deep Retinex-based models [40, 42] decomposes the  image into two separate components - light-independent  reflectance and structure-aware smooth illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The  final estimated reflection component is treated as the en-  hanced result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Unsupervised Learning: Although the above-mentioned  methods perform well on synthetic data, they show limited  generalization capability on real-world low-light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  This might be the result of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' EnlightenGAN [20]  proposed to solve this issue by adopting an unsupervised  learning technique, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=', avoiding the use of paired synthetic  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This work uses attention-guided UNet as a generator  and global-local discriminators to achieve the objective of  LLIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Zero-short Learning: These methods, in low-level vision  tasks, do not require any paired or unpaired training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Zero-reference Deep Curve Estimation [13] formulates im-  age enhancement as a task of image-specific deep curve  estimation, taking into account pixel value range, mono-  tonicity, and differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It is a lightweight DCE-Net  that doesn’t require paired or unpaired ground truth images  during training and relies on non-reference loss functions  that measure the enhancement quality hence driving the  learning of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Another such method, Semantic-  guided Zero-shot low-light enhancement Network [49] is a  lightweight model for low-light enhancement factor extrac-  tion which is inspired by the architecture of U-Net [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The  output of this network is fed to a recurrent image enhance-  ment network, along with the degraded input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Each  stage in this network considers the enhancement factor and  the output from the previous scale as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This is fol-  lowed by a feature-pyramid network that aims to preserve  the semantic information in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  More recently, researchers have experimented with trans-  formers for Zero-shot Learning LLIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Structure-Aware  lightweight Transformer (STAR) [47] focuses on real-time  image enhancement without using deep-stacked CNNs or  large transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' STAR is formulated to capture  long-range dependencies between separate image patches,  facilitating the model to learn structural relationships be-  tween different regions of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In STAR, patches of  the image are tokenized into token embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The tokens  generated as an intermediate stage are passed to a long-  short-range transformer that outputs two long and short-  range structural maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' These structural maps can further  predict curve estimation or transformation for image en-  hancement tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Although these methods show impressive  results for the study of low-light image enhancement for  which it originally developed, they cannot deal with foggy  low– light images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Limitations  Previous works have relied on ASM-based methods in the  case of dehazing and Retinex model-based methods for low-  light image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  However, these methods fail  to generalize to real-world images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Recent deep learning-  based methods using large networks solve the task of im-  age dehazing and low-light enhancement separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' To our  knowledge, no work is introduced that solves the two prob-  lems in a collaborative network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Deep learning methods  4  also fail to generalize to different haze levels and darkness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Proposed Methodology  In this section, we provide a detailed description of the over-  all architecture proposed and the individual components in-  cluded in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Architecture  Like the group structure in [33], each group in our network  consists of a K NAF Block [5] with a skip connection at  the end as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The output of each group is  concatenated, passed to the level attention module to find  the weighted importance of the feature maps obtained, and  post-processed using two convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' A long skip  connection for global residual learning accompanies this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1  NAF-Block  To keep this work self-contained, we explain the NAF Block  [5] in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' NAF Block is the building block  of Nonlinear Activation Free Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Namely NAFNet  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' To avoid over-complexity in the architecture, this block  avoids using any activation functions like ReLU, GELU,  Softmax, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' hence keeping a check on the intra-block com-  plexity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The input first passes through Layer Normalization as it can  help stabilize the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This is followed by con-  volution operations and a Simple Gate (SG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' SG is a variant  of Gated Linear Units (GLU) [10] as evident from the fol-  lowing equations 1 and 2  GLU(X, f, g, σ) = f(X) ⊙ σ(g(X))  (1)  S impleGate(X, Y) = X ⊙ Y  (2)  and a replacement for GELU[18] activation function be-  cause of the similarity between GLU and GELU (Equa-  tion 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  GELU(x) = xφ(x)  (3)  In Simple Gate, the feature maps are divided into two parts  along the channel dimension and then multiplied as shown  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Another novelty introduced in this block is  Simplified Channel Attention (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Channel Attention  (CA) can be expressed as:  CA(X) = X ⊗ σ(W2max(0, W1pool(X)))  (4)  where X represents the feature map, pool indicates the  global average pooling operation,σ is Sigmoid, W1, W2 are  fully-connected layers and ⊗ is a channel-wise product op-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This can be taken as a special case of GLU from  W  W  W  H  H  H  C/2  C/2  C/2  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Simple Gate as represented by Equation 2 ⊗ denotes  channel-wise multiplicaWere  which we can derivate the equation for Simplified Channel  Attention:  SCA(X) = X ⊗ Wpool(X)  (5)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2  Level Attention Module  Once we have extracted features from all the NAF Groups,  we concatenate them and pass them through the Level At-  tention Module (LAM) [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This module learns attention  weights for features obtained at different levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In LAM, each feature map is first reshaped to a 2D matrix  of the size K × HWC, where K, H, W, and C are the no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' of  NAF Groups, height, width, and no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' of channels of the fea-  ture maps respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We find a correlation matrix of this  2D matrix by multiplying it with its transpose matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Fi-  nally, we multiply the 2D matrix with this correlation ma-  trix and reshape it to K × H × W × C tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Inspired by  residual learning, this tensor is substituted for residual and  is added to the original concatenated feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The re-  sultant features are then reshaped to H × W × KC, passing  through 1 × 1 convolution operation to get the H × W × C  feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This is passed through some post-processing  convolutions to get the final enhanced output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We include  its architecture diagram in the supplementary material for a  better understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Loss Functions  Four loss functions, namely, reconstruction loss, perceptual  loss, edge loss [19], and FFT loss[7], have been used to  supervise the task of image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The total loss L is defined in Equation 6, where λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='04,  λ2 = 1 and λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  L = Ls + λ1Lp + λ2Le + λ3Lf  (6)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1  Reconstruction Loss:  The restored clear output image is compared with its ground  truth value in the spatial domain using a standard l1 loss as  5  demonstrated in Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We use l1 loss instead of l2 loss  as it does not over-penalize the errors and leads to better  image restoration performance [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Ls = 1  N  n  �  i=1  ∥ xgt  i − NAFNet(xdark,hazy  i  ) ∥1  (7)  In the above equation, xgt  i refers to the ground truth clear im-  age, and NAFNet(xdark,hazy  i  ) denotes the output of our pro-  posed NAFNet when a dark and hazy image is fed to the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2  Perceptual Loss:  To reduce the perceptual loss and improve the image’s vi-  sual quality, we utilize the features of the pre-trained VGG-  19 network [38] obtained from the output of one of the  ReLU activation layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It is defined in Equation 8, where  wi j, hij, and cij refer to the dimensions of the respective  feature maps inside the VGG-19 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' φij denotes  the feature maps outputted from the jth convolutional layer  inside the i-th block in the VGG network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Lp =  1  wijhijci j  wij  �  x=1  hij  �  y=1  cij  �  z=1  ∥ φij(Igt)xyz − φij(Iout)xyz ∥  (8)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='3  Edge Loss:  To recover the high-frequency details lost because of the in-  herent noise in dark and hazy images, we have an additional  edge loss to constrain the high-frequency components be-  tween the ground truth and the recovered image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Le =  �  (∇2(Igt) − ∇2(Iout))2 + ϵ2  (9)  In Equation 9, ∇2 refers to the Laplacian operation [22],  which is then applied to the ground truth and the predicted  clean image to get the edge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='4  FFT Loss:  To supervise the haze-free results in the frequency domain,  we add another loss called Fast Fourier transform (FFT) loss  (denoted by Lf in Equation 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It calculates the loss of both  amplitude and phase using the l1 loss function without ad-  ditional inference cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Axgt  i , Pxgt  i = FFT(xgt  i ),  (10)  Axout  i , Pxout  i  = FFT(xout  i ),  (11)  L f = 1  N  n  �  i=1  (∥ Axgt  i − Axout  i  ∥1 + ∥ Pxgt  i − Pxout  i  ∥1)  (12)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Experimental Results  To demonstrate the outcomes of our model’s approach to-  wards image enhancement under low-visibility conditions,  this section contains a detailed description of the dataset  generated and used in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1, the experimental set-  tings in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2, the metrics used for evaluation in Sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='3 and a discussion on the results obtained in Sec-  tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Dataset Details  Due to the lack of available datasets that meet our require-  ments, we generate a new one using the AFO dataset [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The dataset generation process has been elaborated below,  and the final images have been shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Haze effect - To add fog, imgaug [21], a well-known  python library was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' A random integral value be-  tween 3, 4, 5 was selected, representing the fog’s sever-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' For each image, this random number was chosen  and pre-defined functions within the package were uti-  lized to add a layer of fog to the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Low-light Effect - Given a normal image, our goal is  to output a low-lit image while preserving the underly-  ing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We follow the pipeline introduced [8],  which parametrically models the low light-degrading  transformation by observing the image signal process-  ing (ISP) pipeline between the sensor measurement  system and the final image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The low-illumination-  degrading pipeline is a three-stage process:  – Unprocessing procedure - This part aims to syn-  thesize RAW format images from input sRGB  images by invert tone mapping, invert gamma  correction, and the transformation of the image  from sRGB space to cRGB space, and invert  white balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  – Low Light Corruption - This aims at adding shot  and read noises to the output of the unprocess-  ing procedure, as these are common in-camera  imaging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Shot noise is a type of noise  generated by the random arrival of photons in  a camera, which is a fundamental limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Read noise occurs during the charge conversion  of electrons into voltage in the output amplifier,  which can be approximated using a Gaussian ran-  dom variable with zero mean and fixed variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  – ISP Pipeline - RAW image processing is done  after the lowlight corruption process in the fol-  lowing order: add quantization noise, white bal-  ancing from cRGB to sRGB, and gamma correc-  tion, which finally outputs a degraded low-light  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  6  Ground Truth Images  Ground Truth Images  Generated Images  Generated Images  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Visual illustration of a few sample images from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Columns 1 and 3 show original images taken from AFO Dataset  [15], whereas Columns 2 and 4 show their corresponding images generated as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1 simulating low-visibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Combination of Haze and Low-light Effect - Re-  sults of implementing the low-light generation algo-  rithm described above on foggy images generated us-  ing img-aug are shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' It can be seen that com-  bining the two (fog and low light) has introduced ad-  versity in finding the location of the objects in the wa-  ter bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Moreover, finding a unique solution for  such a combination has not been explored to date  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Experimental Settings  The images were resized to get the resultant dimensions as  256 × 456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Adam optimizer with an initial learning rate of  1e−4, β1, and β2 with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='9 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='999 were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The batch size was fixed as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We have used 3 groups in all  our experiments, each with 16 blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Pytorch backend was  used to compile the model and train it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Evaluation Metrics  We reported the results we obtained using two standard im-  age restoration metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=', PSNR and SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' These met-  rics will help us quantitatively evaluate the performance of  our model in terms of feature colors and structure similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  High PSNR and SSIM values if indicative of good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Experimental Results  The architecture used is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This section  gives a detailed analysis of the results obtained by the pro-  posed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Discussion and Comparison  In this subsection, we discuss the evaluation results ob-  tained by the proposed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Previous methods were  Method  Year  PSNR  SSIM  Zero-DCE[13]  2020  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='323  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='529  SGZNet[49]  2022  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='578  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='519  BPPNet[39]  2022  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='507  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='755  DehazeNet[3]  2016  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='710  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='391  Star-DCE[47]  2021  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='651  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='539  FFANet[7]  2020  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='582  MSBDN-DFF[16]  2020  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='686  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='689  LVRNet (Ours)  2022  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='744  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='905  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Quantitative comparison of our proposed network  with previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The best results and the second-best results  have been highlighted with red color and blue colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  trained on the newly generated dataset and tested to com-  pare their metrics with our model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' These  methods were built to enhance the low-light image or obtain  a clear image from a hazy one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The results are mentioned  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  We observe a huge increase in the PSNR value as compared  to Zero-DCE[13], which enhances the low-light image as a  curve estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' However, it introduces an even  amplified noise leading to color degradation as seen in Fig-  ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Notwithstanding its fast processing speed, Zero-DCE  has limited noise suppression and haze removal capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Star-DCE[47], which uses a transformer backbone instead  of a CNN one in the Zero-DCE network, shows a 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='12%  increase in PSNR value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Owing to the added LAM struc-  7  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Reconstruction Loss  Perceptual Loss  Edge Loss  FFT Loss  PSNR  SSIM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  \x13  \x13  \x17  \x17  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='870  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  \x13  \x13  \x17  \x13  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='455  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='903  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  \x13  \x13  \x13  \x17  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='624  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='897  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  \x13  \x17  \x13  \x13  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='719  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='900  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  \x13  \x13  \x13  \x13  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='744  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='905  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ablation experiments: We train our model using different combinations of loss functions to understand the importance of  individual losses for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The best results are obtained when the model is trained using all the loss functions mentioned in  this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  ture, using which our model can focus on more important  feature maps, we can achieve a 54% higher PSNR value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  SGZNet[49] uses pretrained networks for enhancement fac-  tor estimation, thus their result is dependent on those pre-  trained weights, leading to a lower PSNR value of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='578  on LowVis-AFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' From Figure 1, we observe that the result  obtained from SGZNet is still degraded by excessive noise  and lacks saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' DehazeNet[3] is limited by the net-  work’s depth and cannot generalize to real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Hence, it results in a low PSNR of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Methods like  BPPNet[39] and FFANet[33] are end-to-end deep learning  methods for image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' BPPNet[39] distorts the color  distribution in the recovered image as it cannot remove the  dark regions, whereas FFA-Net[33] produces image with a  lower perceptual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  We propose an end-to-end deep learning pipeline (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='43M  parameters) that can perform image dehazing and low-light  image enhancement with a significant decrease in the num-  ber of parameters as compared to MSBDN-DFF [16] (31M  parameters) and FFA-Net[33] (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='45M parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The supplementary material has provided a discussion  on the number of parameters of other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  We also  trained the model for 10 epochs with fewer NAF blocks to  prove that we achieved better results than the lighter results,  not due to an increase in parameters but because of the self-  sufficiency of the added LAM module, non-linear activation  networks, and residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The results of these ex-  periments are reported in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ablation Studies  To prove the importance of the perceptual loss, edge loss,  and fft-loss, added to supervise the training procedure, we  conducted experiments excluding each of them and reported  the values of PSNR and SSIM in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We keep the l1  loss function constant in all experiments as it is critical in  image restoration tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We observe an increase in metric  values in the lower rows compared to row 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' As a result  of more supervision in the unchanged architecture, there is  an increase in the quality of clear images obtained, which  are demonstrated in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' There is  also an increase in PSNR value (which depends on per-pixel  distance) in row 3, once we train the model without percep-  tual loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' This is seen as perceptual loss doesn’t compare  individual pixel values but the high-level features obtained  from a pretrained network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In row 4, we get a lower PSNR  value on excluding edge loss compared to row 5, as we get  lesser edge supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Overall, we get the best perfor-  mance when we include all the loss functions, as seen in  row 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Conclusion  In this work, we have presented Low-Visibility Restora-  tion Network (LVRNet), a new lightweight deep learning  architecture for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  We also introduce a  new dataset, LowVis-AFO, that includes a diverse combi-  nation of synthetic darkness and haze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We also performed  benchmarking experiments on our generated dataset and  surpassed the results obtained using the previous image  restoration network by a significant margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Qualitative and  quantitative comparison with previous work has demon-  strated the effectiveness of LVRNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' We believe our work  will motivate more research, focused on dealing with a com-  bination of adverse effects such as haze, rain, snowfall, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  rather than considering a single factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In our future work,  we plan to extend LVRNet for image restoration tasks where  more factors, that negatively impact the image quality, are  taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  8  Supplementary Material  To make our submission self-contained and given the page  limitation, this supplementary material provides additional  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Section 1 gives an overview of the number of pa-  rameters and PSNR obtained by different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Sec-  tion 2 contains visual results that highlight the significance  of the loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Section 3 contains the ablation ex-  periment with lesser blocks, and Section 4 demonstrates the  architecture diagram of the level attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' PSNR vs Parameters  Figure 6 presents the PSNR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Parameters plot that the  previous methods and our method achieved on the testing  set of LowVis-AFO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Our model outperforms the state-of-  the-art image dehazing and low-light image enhancement  methods by a good margin while having a lesser number of  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The PSNR vs Number of Parameters of recent image  restoration methods on the newly proposed LowVis-AFO dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  #Blocks  PSNR  SSIM  #params  Runtime(s)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  14  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='3432  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='8626  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='38M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='035  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  12  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='4302  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='8488  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='33M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='029  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  10  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='2965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='8494  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='28M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='024  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Results of the experiments conducted on a lesser num-  ber of NAF blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The training was done for 10 epochs and the  metrics were obtained on the test set thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ablation Experiment on Different Loss  Functions  Figure 8 demonstrates the visual results obtained when  we conducted experiments excluding some loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  The motivation behind the experiment is to highlight the  importance of the extra loss functions (perceptual loss, edge  loss, fft-loss) added to supervise our pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' The quanti-  tative results are given in Table 2 in the main manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ablation Experiment with Lesser Number of  Blocks  To prove the self-sufficiency of the individual components  included in our architecture such as LAM, we conduct ex-  periments with a lesser number of NAF blocks [5] and re-  ported the PSNR and SSIM obtained in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Seeing  the results, we can conclude that our model achieves better  results, not because of an increase in the number of param-  eters as compared to the lighter model, but because of the  entire pipeline adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Level Attention Module  As mentioned in the main text, the diagram for LAM[45]  has been provided here in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' (re-  fer Figure 7)  References  [1] Codruta O Ancuti, Cosmin Ancuti, Chris Hermans, and  Philippe Bekaert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' A fast semi-inverse approach to detect and  remove the haze from a single image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Asian Conference  on Computer Vision, pages 501–514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Springer, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [2] Julian Besag, Jeremy York, and Annie Mollié.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Bayesian im-  age restoration, with two applications in spatial statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' An-  nals of the institute of statistical mathematics, 43(1):1–20,  1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [3] Bolun Cai, Xiangmin Xu, Kui Jia, Chunmei Qing, and  Dacheng Tao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Dehazenet: An end-to-end system for single  image haze removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE Transactions on Image Process-  ing, 25(11):5187–5198, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [4] Dongdong Chen, Mingming He, Qingnan Fan, Jing Liao, Li-  heng Zhang, Dongdong Hou, Lu Yuan, and Gang Hua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Gated  context aggregation network for image dehazing and derain-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In 2019 IEEE winter conference on applications of com-  puter vision (WACV), pages 1375–1383.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [5] Liangyu Chen, Xiaojie Chu, Xiangyu Zhang, and Jian Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Simple baselines for image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='04676, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [6] Rongsen Chen and Edmund M-K Lai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Convolutional autoen-  coder for single image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In ICIP, pages 4464–4468,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [7] Sung-Jin Cho, Seo-Won Ji, Jun-Pyo Hong, Seung-Won Jung,  and Sung-Jea Ko.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Rethinking coarse-to-fine approach in sin-  gle image deblurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF inter-  national conference on computer vision, pages 4641–4650,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [8] Ziteng Cui, Guo-Jun Qi, Lin Gu, Shaodi You, Zenghui  Zhang, and Tatsuya Harada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Multitask aet with orthogonal  tangent regularity for dark object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings  9  26  米  Zero-DCE  SGZNet  24  BPPNet  DehazeNet  22  Star-DCE  FFA-Net  MSBDN-DFF  PSNR  20  Ours  18  16  14  12  0  10M  20M  30M  NumberofParametersStacked  Feature Maps  K x H x W x C  K x HWC  Reshape  Transpose  Softmax  K x HWC  K x H x W x C  Feature Maps  Reshape  H x W x KC  Reshape  H x W x C  1 x 1  Conv  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Visual illustration of operations performed by Level Attention Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  of the IEEE/CVF International Conference on Computer Vi-  sion, pages 2553–2562, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [9] Sourya Dipta Das and Saikat Dutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Fast deep multi-patch  hierarchical network for nonhomogeneous image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In Proceedings of the IEEE/CVF Conference on Computer  Vision and Pattern Recognition Workshops, pages 482–483,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [10] Yann N Dauphin, Angela Fan, Michael Auli, and David  Grangier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Language modeling with gated convolutional net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In International conference on machine learning,  pages 933–941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [11] Raanan Fattal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Dehazing using color-lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' ACM transac-  tions on graphics (TOG), 34(1):1–14, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [12] Stuart Geman and Donald Geman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Stochastic relaxation,  gibbs distributions, and the bayesian restoration of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  IEEE Transactions on pattern analysis and machine intelli-  gence, (6):721–741, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [13] Chunle Guo, Chongyi Li, Jichang Guo, Chen Change Loy,  Junhui Hou, Sam Kwong, and Runmin Cong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Zero-reference  deep curve estimation for low-light image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF Conference on Computer Vi-  sion and Pattern Recognition, pages 1780–1789, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [14] Tiantong Guo, Venkateswararao Cherukuri, and Vishal  Monga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Dense123’color enhancement dehazing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF Conference on Computer Vi-  sion and Pattern Recognition Workshops, pages 0–0, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [15] Jan G˛asienica-Józkowy, Mateusz Knapik, and Boguslaw Cy-  ganek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' An ensemble deep learning method with optimized  weights for drone-based water rescue and surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Inte-  grated Computer-Aided Engineering, pages 1–15, 01 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [16] Dong Hang, Pan Jinshan, Hu Zhe, Lei Xiang, Zhang Xinyi,  Wang Fei, and Yang Ming-Hsuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Multi-scale boosted de-  hazing network with dense feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In CVPR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [17] Kaiming He, Jian Sun, and Xiaoou Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Single image haze  removal using dark channel prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE transactions on pat-  tern analysis and machine intelligence, 33(12):2341–2353,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [18] Dan Hendrycks and Kevin Gimpel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Gaussian error linear  units (gelus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' arXiv preprint arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='08415, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [19] Kui Jiang, Zhongyuan Wang, Peng Yi, Chen Chen, Baojin  Huang, Yimin Luo, Jiayi Ma, and Junjun Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Multi-scale  progressive fusion network for single image deraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF conference on computer vision  and pattern recognition, pages 8346–8355, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [20] Yifan Jiang, Xinyu Gong, Ding Liu, Yu Cheng, Chen Fang,  Xiaohui Shen, Jianchao Yang, Pan Zhou, and Zhangyang  Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Enlightengan:  Deep light enhancement without  paired supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE Transactions on Image Process-  ing, 30:2340–2349, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [21] Alexander B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Jung, Kentaro Wada, Jon Crall, Satoshi  Tanaka, Jake Graving, Christoph Reinders, Sarthak Ya-  dav, Joy Banerjee, Gábor Vecsei, Adam Kraft, Zheng Rui,  Jirka Borovec, Christian Vallentin, Semen Zhydenko, Kil-  ian Pfeiffer, Ben Cook, Ismael Fernández, François-Michel  De Rainville, Chi-Hung Weng, Abner Ayala-Acevedo,  Raphael Meudec, Matias Laporte, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' imgaug.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' https:  //github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='com/aleju/imgaug, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' accessed  01-Feb-2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [22] Behzad Kamgar-Parsi and Azriel Rosenfeld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Optimally  isotropic laplacian operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE Transactions on Image  Processing, 8(10):1467–1472, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [23] Yunan Li, Qiguang Miao, Wanli Ouyang, Zhenxin Ma, Hui-  juan Fang, Chao Dong, and Yining Quan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Lap-net: Level-  aware progressive network for image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceed-  ings of the IEEE/CVF International Conference on Computer  Vision, pages 3276–3285, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [24] Zhuwen Li, Ping Tan, Robby T Tan, Danping Zou, Steven  Zhiying Zhou, and Loong-Fah Cheong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Simultaneous video  defogging and stereo reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings of the  IEEE conference on computer vision and pattern recogni-  tion, pages 4988–4997, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [25] Xiao Liang, Runde Li, and Jinhui Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Selective attention  network for image dehazing and deraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings  of the ACM Multimedia Asia, pages 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [26] Xiaohong Liu, Yongrui Ma, Zhihao Shi, and Jun Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Grid-  dehazenet: Attention-based multi-scale network for image  dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In Proceedings of the IEEE/CVF international  conference on computer vision, pages 7314–7323, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [27] Kin Gwn Lore, Adedotun Akintayo, and Soumik Sarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ll-  net: A deep autoencoder approach to natural low-light image  enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Pattern Recognition, 61:650–662, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [28] Kun Lu and Lihong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Tbefn: A two-branch exposure-  fusion network for low-light image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  IEEE  Transactions on Multimedia, 23:4093–4105, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [29] Feifan Lv, Bo Liu, and Feng Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Fast enhancement for non-  uniform illumination images using light-weight cnns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Pro-  ceedings of the 28th ACM International Conference on Mul-  timedia, pages 1450–1458, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  10  [30] Feifan Lv, Feng Lu, Jianhua Wu, and Chongsoon Lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Mbllen: Low-light image/video enhancement using cnns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In  BMVC, volume 220, page 4, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [31] Earl J McCartney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Optics of the atmosphere: scattering by  molecules and particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' New York, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [32] Seungjun Nah, Tae Hyun Kim, and Kyoung Mu Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Deep  multi-scale convolutional neural network for dynamic scene  deblurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In Proceedings of the IEEE conference on  computer vision and pattern recognition, pages 3883–3891,  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [33] Xu Qin, Zhilin Wang, Yuanchao Bai, Xiaodong Xie, and  Huizhu Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Ffa-net: Feature fusion attention network for sin-  gle image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings of the AAAI Conference  on Artificial Intelligence, volume 34, pages 11908–11915,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [34] Wenqi Ren, Si Liu, Hua Zhang, Jinshan Pan, Xiaochun Cao,  and Ming-Hsuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Single image dehazing via multi-  scale convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In European conference  on computer vision, pages 154–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Springer, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [35] Wenqi Ren, Sifei Liu, Lin Ma, Qianqian Xu, Xiangyu Xu,  Xiaochun Cao, Junping Du, and Ming-Hsuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Low-  light image enhancement via a deep hybrid network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE  Transactions on Image Processing, 28(9):4364–4375, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [36] Wenqi Ren, Jinshan Pan, Hua Zhang, Xiaochun Cao, and  Ming-Hsuan Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Single image dehazing via multi-scale  convolutional neural networks with holistic edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Interna-  tional Journal of Computer Vision, 128(1):240–259, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [37] Olaf Ronneberger, Philipp Fischer, and Thomas Brox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' U-net:  Convolutional networks for biomedical image segmentation,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [38] Karen Simonyan and Andrew Zisserman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Very deep convo-  lutional networks for large-scale image recognition, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [39] Ayush Singh, Ajay Bhave, and Dilip K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Prasad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Single image  dehazing for a variety of haze scenarios using back projected  pyramid network, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [40] Chen Wei, Wenjing Wang, Wenhan Yang, and Jiaying  Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Deep retinex decomposition for low-light enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  arXiv preprint arXiv:1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='04560, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [41] Ke Xu, Xin Yang, Baocai Yin, and Rynson WH Lau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Learning to restore low-light images via decomposition-and-  enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings of the IEEE/CVF Conference  on Computer Vision and Pattern Recognition, pages 2281–  2290, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [42] Wenhan Yang, Wenjing Wang, Haofeng Huang, Shiqi Wang,  and Jiaying Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Sparse gradient regularized deep retinex  network for robust low-light image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  IEEE  Transactions on Image Processing, 30:2072–2086, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [43] Chia-Hung Yeh, Chih-Hsiang Huang, and Li-Wei Kang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Multi-scale deep residual learning-based single image haze  removal via image decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  IEEE Transactions on  Image Processing, 29:3153–3167, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [44] Kai Zhang, Wangmeng Zuo, Yunjin Chen, Deyu Meng, and  Lei Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Beyond a gaussian denoiser: Residual learning of  deep cnn for image denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE transactions on image  processing, 26(7):3142–3155, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [45] Kaihao Zhang, Wenhan Luo, Boheng Chen, Wenqi Ren,  Bjorn Stenger, Wei Liu, Hongdong Li, and Ming-Hsuan  Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Benchmarking deep deblurring algorithms: A large-  scale multi-cause dataset and a new baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' arXiv  preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='00234, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [46] Xiaoqin Zhang, Jinxin Wang, Tao Wang, and Runhua Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Hierarchical feature fusion with mixed convolution attention  for single image dehazing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE Transactions on Circuits  and Systems for Video Technology, 32(2):510–522, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [47] Zhaoyang Zhang, Yitong Jiang, Jun Jiang, Xiaogang Wang,  Ping Luo, and Jinwei Gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Star:  A structure-aware  lightweight transformer for real-time image enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In  Proceedings of the IEEE/CVF International Conference on  Computer Vision, pages 4106–4115, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [48] Hang Zhao, Orazio Gallo, Iuri Frosio, and Jan Kautz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Loss  functions for image restoration with neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE  Transactions on computational imaging, 3(1):47–57, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [49] Shen Zheng and Gaurav Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Semantic-guided zero-shot  learning for low-light image/video enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  In Pro-  ceedings of the IEEE/CVF Winter Conference on Applica-  tions of Computer Vision, pages 581–590, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [50] Zhuoran Zheng, Wenqi Ren, Xiaochun Cao, Xiaobin Hu,  Tao Wang, Fenglong Song, and Xiuyi Jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  Ultra-high-  definition image dehazing via multi-guided bilateral learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In 2021 IEEE/CVF Conference on Computer Vision and  Pattern Recognition (CVPR), pages 16180–16189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' IEEE,  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  [51] Minfeng Zhu, Pingbo Pan, Wei Chen, and Yi Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Eemefn:  Low-light image enhancement via edge-enhanced multi-  exposure fusion network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In Proceedings of the AAAI Con-  ference on Artificial Intelligence, volume 34, pages 13106–  13113, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  11  LP  LPF  LPE  LEF  LPEF  GT  Input  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' Qualitative results obtained from experiments conducted on different loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content=' In the figure, L = L1 loss, P = Perceptual  Loss, E = Edge loss and F = FFT loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
+page_content='  12' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ktE5T4oBgHgl3EQfGw6j/content/2301.05434v1.pdf'}
diff --git a/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/2301.13553v1.pdf.txt b/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/2301.13553v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..431291193886e908616290b1b37b56c5ec600a3e
--- /dev/null
+++ b/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/2301.13553v1.pdf.txt
@@ -0,0 +1,2047 @@
+1
+Millimetre-wave Radar for Low-Cost 3D Imaging:
+A Performance Study
+Han Cui, Jiacheng Wu and Naim Dahnoun
+Abstract—Millimetre-wave (mmWave) radars can generate 3D
+point clouds to represent objects in the scene. However, the accu-
+racy and density of the generated point cloud can be lower than
+a laser sensor. Although researchers have used mmWave radars
+for various applications, there are few quantitative evaluations
+on the quality of the point cloud generated by the radar and
+there is a lack of a standard on how this quality can be assessed.
+This work aims to fill the gap in the literature. A radar simulator
+is built to evaluate the most common data processing chains of
+3D point cloud construction and to examine the capability of the
+mmWave radar as a 3D imaging sensor under various factors. It
+will be shown that the radar detection can be noisy and have an
+imbalance distribution. To address the problem, a novel super-
+resolution point cloud construction (SRPC) algorithm is proposed
+to improve the spatial resolution of the point cloud and is shown
+to be able to produce a more natural point cloud and reduce
+outliers.
+Index Terms—mmWave radar, 3D imaging, point cloud
+I. INTRODUCTION
+Millimetre-wave (mmWave) radars have received increased
+popularity in many industries as an emerging type of sensor.
+The high bandwidth allows them to estimate the distance of an
+object at centimetre-level resolution, and the short wavelength
+and antenna size allow multiple antennas to be integrated into
+a single chip and measure the angle-of-arrival (AoA) of the ob-
+ject using multiple-input multiple-output (MIMO) techniques
+[1]. Combining the distance and AoA measurement, mmWave
+radars are able to construct a point cloud to represent the
+spatial shape of an object [2]. Therefore, mmWave radars can
+be used as a low-cost 3D imaging sensor, as an alternative to
+the traditional depth cameras and laser sensors. This allows
+many computer vision tasks, such as object detection [3],
+human tracking [4], [5], posture estimation [6], [7], and
+identification [4], to be addressed using mmWave radars as a
+non-intrusive solution. However, although many applications
+have been proposed that rely on the 3D imaging capability of
+mmWave radars, few researchers have attempted to evaluate
+the quality of mmWave radars’ detection quantitatively, and
+there is a lack of a standard on how this quality can be
+assessed.
+When detecting objects in a scene, the reflection signal can
+be seen as a time-delayed version of the transmitted signal.
+Combining the two signals gives an intermediate frequency
+(IF) signal, whose frequency and phase are determined by
+the time-of-flight (ToF) of the signal, and, equivalently, the
+distance between the object and the radar [2]. The distance
+can be estimated directly from the IF signal of one pair of
+transmitter and receiver at high resolution, whereas the AoA
+needs to be estimated from the signal phase over a linearly
+spaced antenna array. There is rich literature on antenna
+array-based AoA estimation for traditional radars [8], and the
+same concept also applies to mmWave radars. This paper
+discusses the AoA estimation in the context of mmWave
+radar 3D imaging. It reviews and discusses the 3D point
+cloud construction techniques that are commonly used with
+mmWave radars.
+This paper presents a purpose-built simulation system that
+simulates the data acquisition process of a mmWave radar
+when facing a scene. Radar data simulation allows researchers
+to focus on algorithm design and verification, instead of
+investing too much time in the hardware and real-world data
+collection. Existing radar simulators are often not designed
+for 3D imaging and have certain constraints. For example, the
+system in [9] generates range and Doppler information of the
+radar rather than the raw data, the system in [10] only supports
+single antenna data generation and cannot be used to estimate
+the AoA, and the system in [11] only supports up to four
+receivers in one direction and cannot be used for 3D imaging.
+In this research, a lightweight mmWave radar simulator is
+designed that supports raw data generation of a multi-antenna
+mmWave radar, configurable antenna parameters and layout,
+and customized scene construction using 3D human models
+with programmable motions.
+Using the simulation system, a quantitative evaluation of
+the radar’s capability of imaging a human subject is carried
+out, as well as an evaluation of the key factors that could
+affect the output quality, including the data processing chain
+(DPC), radar antenna configuration, chirp configuration, sub-
+ject velocity, and signal-to-noise ratio (SNR) in the scene.
+It will be shown that, although the radar can capture the
+spatial information of the subject’s body shape, the detected
+point cloud can be noisy, sparse and imbalanced, and can
+require further processing before being used for higher-level
+applications. Finally, a novel super-resolution point cloud
+construction (SRPC) algorithm is proposed to improve the
+spatial resolution of the point cloud and is shown to be able
+to produce a more natural point cloud and reduce outliers.
+The contribution of this paper can be summarized as fol-
+lows:
+• It presents a simulator of mmWave radar that can simulate
+the radar data as if it is placed in a real scene. It supports
+customized 3D models to be imported as the ground truth
+and provides a framework for evaluating a 3D imaging
+algorithm quantitatively.
+• It presents a systematical study of 3D imaging algorithms
+using mmWave radars and an evaluation of the key factors
+arXiv:2301.13553v1  [eess.SP]  31 Jan 2023
+
+2
+that could affect the radar detection. It highlights the
+challenges of the noisy and imbalanced point cloud.
+• It presents a novel SRPC algorithm that can be inserted
+into the traditional point cloud construction DPC and can
+improve the quality of the point cloud.
+The rest of the paper is organized as follows. Section II
+discusses the background and related work. Section III in-
+troduces the preliminaries of mmWave radars. Section IV
+presents the details of the simulator. Section V discusses the
+3D imaging DPC using mmWave radars. Section VI presents
+the experimental results based on the simulation system.
+Section VII presents the novel SRPC algorithm and shows how
+it can improve the radar detection. Section VIII concludes the
+work.
+II. BACKGROUND
+Traditionally, 3D imaging systems often use depth cameras
+(like stereo cameras or RGBD cameras) [12] or laser sensors
+[13], which are able to provide a dense and accurate 3D
+model of the object in front. However, camera-based systems
+can be intrusive and limited by the lighting conditions, and
+laser sensors are often constrained by their high cost (when
+compared with the cost of a mmWave radar which is only
+around £10). Radar-based 2D imaging has also been used
+widely in applications like security, but they provide limited
+depth information and often rely on a dense antenna array
+that has a fixed region-of-interest [14]. Radar-based 3D object
+detection uses radio frequency (RF) signals at certain fre-
+quencies to detect objects, and the resolution of the detection
+largely depends on the available bandwidth. For example,
+WiFi devices operating at 5 GHz with a 40 MHz bandwidth
+can locate people with sub-meter level resolution [15], and
+ultra-wide band (UWB) devices at higher frequency bands
+can achieve centimetre level resolution [16]. With mmWave
+radars operating at above 60 GHz, the range resolution can
+be below 5 cm [2] and even micrometre level when pointing
+to a corner reflector [17]. Therefore, the high resolution has
+gained mmWave radars great popularity in automotive driving
+applications, and researchers are actively investigating their
+usage in computer vision tasks.
+Although mmWave radars can be used as 3D imaging
+sensors, the point cloud is often less accurate and noisier than
+the traditional systems [5]. Many methods have been proposed
+to improve the detection quality of a mmWave radar, such as
+[18], [19]. However, as these methods often use different radar
+configurations and scene setup, it is hard to carry out a quan-
+titative comparison between them, and there lacks a standard
+on how to define the quality of the radar detection. This work
+aims to address the problem by providing a simulation system
+and a framework for a systematic evaluation of a 3D imaging
+algorithm.
+Radar-based 3D imaging requires measuring the distance
+and AoA of the object. The distance is often measured through
+the ToF of the signal, whereas the AoA measurement relies
+on the use of an antenna array. Since the antennas in the array
+will have different physical locations, the ToF at each antenna
+will be different, and the AoA of the object can be estimated
+by investigating the signal difference. This process has been
+studied in depth for traditional long-range radars [8], and the
+same principle can be applied to mmWave radars on a smaller
+scale. There are many algorithms designed for estimating the
+AoA based on a linearly spaced antenna array, such as the FFT-
+based method, beamforming method and subspace method
+(more details in Section III-C). These algorithms provide
+a trade-off between the computational complexity and the
+angular resolution [8], [20]. However, in contrast to traditional
+radar systems where the signal sources are often well-defined
+and uncorrelated, signal sources in 3D imaging can be one
+object with a continuous surface, which can require a different
+DPC. This paper discusses the traditional AoA estimation
+algorithms in the context of 3D imaging using a mmWave
+radar and investigates the key factors that would affect the
+detection result.
+III. MMWAVE RADAR PRELIMINARIES
+Commercial mmWave radars often implement the frequency
+modulated continuous wave (FMCW) model. The radar sends
+a modulated chirp signal, detects the signal reflection from
+any object, processes the signal and determines the range,
+velocity, and AoA of the object. The principle of the FMCW
+radar model has been documented in detail in the literature
+(e.g. [5]). This section will give a brief discussion of the
+fundamentals that are necessary for understanding this paper,
+with a particular focus on the AoA estimation.
+The radar sends an FMCW signal and receives its reflection
+from the object in the scene, where the reflection will be a
+time-delayed version of the transmitted signal. The two signals
+are mixed to produce an IF signal, as shown in Equation (1)
+(more details in Appendix A):
+IF(t) = Aej(ωbt+φb) where ωb = 2πSτ, φb = 2πf0τ
+(1)
+where S is the slope of the chirp, τ is the ToF of the signal,
+f0 is the starting frequency of the chirp, and A represents the
+amplitude of the signal. After obtaining the IF signal, a DPC
+will be applied to determine the presence of any object.
+A. Distance and Velocity Estimation
+For a single object, the frequency ωb will be a constant
+value and the distance of the object can be calculated as
+d = τc
+2 = ωbc
+4πS
+(2)
+When there are multiple refection sources, the frequencies can
+be found by applying an FFT over the IF signal, which is
+referred to as the range-FFT. The velocity can be measured by
+transmitting multiple chirps at a known interval and calculating
+the phase difference between the chirps. Assuming the radar
+transmits a chirp every Tc seconds and a phase difference ∆φ
+is observed between successive chirps, then the velocity v of
+the object can be estimated as:
+v =
+∆φc
+4πTcf0
+(3)
+When there are multiple reflection sources moving at different
+velocities, they can be found by applying another FFT over
+the chirp phases, which is referred to as the Doppler-FFT.
+
+3
+Fig. 1: Phase difference between two receivers from one signal
+source.
+B. AoA Estimation Principle
+The AoA of the object can be estimated by having multiple
+antennas operating concurrently and by comparing the phase
+difference between neighbouring receivers. Due to the spatial
+location difference between the receivers, the signal received at
+each receiver will have a slight phase difference depending on
+the relative position of the receivers and the AoA. The AoA
+can be computed in both azimuth and elevation directions,
+given that there exists more than one antenna in each direction.
+The azimuth and elevation angles will be denoted as θa and
+θe, respectively, or θ(a,e) when referring to both of them.
+Assuming there are Na×Ne linearly spaced receivers in the
+azimuth and elevation directions, and M objects in different
+directions θ(a,e)m, then each object can be viewed as a signal
+source and the receiving antenna array will receive a signal
+(denoted as x) as a weighted sum of the M data source:
+x(Na×Ne) =
+M
+�
+m=1
+αms(θ(a,e)m) + n
+(4)
+where s(θ(a,e)m) is the steering vector that represents the
+phase difference between receivers when a signal arrives with
+angle θ(a,e)m, α is an unknown parameter that models the
+signal transmission from the data source to the receivers,
+and n is the noise. The AoA estimation can be modelled as
+estimating the values of θ(a,e) for each object m, given a set
+of receiver data (x).
+For linearly spaced arrays, the receivers are often separated
+by a small distance l that is equal to half of the signal wave-
+length, i.e. l = λ
+2 , to maximize the angle-of-view (AoV) [8].
+When using an array of Na azimuth receivers and Ne elevation
+receivers, each subsequent receiver beyond the first one will
+receive an additional phase change that can be expressed using
+a 2D steering vector (more details in Appendix B):
+s(θ(a,e), Na, Ne) =
+�
+���
+1,
+...,
+ej(Na−1)∆φa
+ej∆φe,
+...,
+ej(∆φe+(Na−1)∆φa)
+...,
+...,
+...
+ej(Ne−1)∆φe,
+...,
+ej((Ne−1)∆φe+(Na−1)∆φa)
+�
+���
+(5)
+3D point cloud construction requires the x-y-z coordinates
+of the object instead of the azimuth and elevation angles.
+Therefore, the calculation of the exact value of θa and θe is
+often not required. Let d denote the distance of the object, then
+the 3D coordinates of the object can be calculated as (more
+details in Appendix B):
+x = d∆φa
+π
+, z = d∆φe
+π , y =
+�
+d2 − x2 − z2
+(6)
+Given that d can be obtained from the range-FFT as discussed
+in Section III-A, the x-y-z coordinates can be obtained if the
+phase differences ∆φa and ∆φe are known. Therefore, the
+AoA estimation of an object can be considered equivalently
+as searching for the best matching steering vector s(θ(a,e)m)
+of the object.
+C. AoA Estimation Algorithms
+In the following sections, some of the most widely-used
+AoA estimation algorithms will be discussed, including the
+FFT-based method, conventional beamforming (also known as
+the Bartlett beamforming or the delay-and-sum beamforming),
+the minimum variance distortionless response (MVDR) beam-
+forming (also known as the Capon beamforming) [21], and
+the multiple signal classification (MUSIC) subspace method
+[22]. The angle-FFT method is a single-snapshot method
+that can make an estimate based on a single chirp, whereas
+the other methods are multi-snapshot methods that require a
+few chirps to make one estimate. The performance of the
+algorithms depends on several factors, including the antenna
+layout, number of antennas, chirp configuration, number of
+snapshots, SNR, environment, etc.
+1) Angle-FFT Method: The simplest way of estimating
+s(θ(a,e)m) of an object m in Equation (4) is by using cor-
+relation between the receiver data x and the steering vector
+from the candidate angles. A set of candidate steering vectors
+s(¯θ(a,e)) is defined for θa ∈ [−π, π], θe ∈ [−π, π], and the
+correlation is calculated as s(¯θ(a,e)) · x, which will yield a
+peak output when ¯θ(a,e) equals to θ(a,e)m. This process is
+equivalent to applying an FFT over the receiver data x, since
+the steering vector can be considered the same as a set of FFT
+coefficients, which gives the frequency components in terms of
+∆φa and ∆φe. This FFT is also referred to as the angle-FFT.
+As an example, Figure 2 shows the antenna layout of the
+TI IWR6843 radar. It has three transmitters and four receivers,
+which can form a 12-receiver array when using MIMO tech-
+niques [1]. The phase of each virtual receiver is also shown,
+where ϕ is the random initial phase of the first receiver.
+The azimuth receivers will form a signal ej(∆φan+ϕ) and the
+elevation receivers will form a signal ej(∆φan+2∆φa+ϕ+∆φe),
+where n is the receiver index in each direction. The value
+of ∆φa can be obtained by applying an azimuth-FFT over
+the azimuth receivers (RX1-RX4 and RX9-RX12), which will
+give the frequency ∆φa and phase ϕ. The value of ∆φa can
+be obtained by applying an FFT over the elevation receivers,
+which will give the frequency ∆φa and phase 2∆φa+ϕ+∆φe.
+Hence, the value of ∆φe can also be calculated given ∆φa
+and ϕ.
+An alternative approach to calculate ∆φa is by applying an
+elevation-FFT over a set of receivers in the elevation direction.
+For example, Figure 3 shows the layout of the TI overhead
+detection sensor (ODS) model, where the receivers form a
+near-square shape and allows a 2D angle-FFT to be performed.
+
+Object
+d
+Ad
+0
+0
+~/=入2
+RXO
+RX1
+RXO
+RX1
+TX1
+Approximation4
+Fig. 2: IWR6843 radar antenna layout, the virtual receiver
+array and the received phases.
+The ODS models allow a higher elevation resolution at the cost
+of reduced azimuth resolution.
+Fig. 3: IWR6843ODS radar antenna layout, the virtual receiver
+array and the received phases.
+2) Beamforming Method: Beamforming methods calculate
+a set of weights w(Nrx×Θ) for the Nrx virtual receivers in the
+array (both azimuth and elevation), and for all possible angles
+θ(a,e) ∈ Θ where θa ∈ [−π, π], θe ∈ [−π, π]. When applying
+a column of weights to the receiver data x, the signal from the
+direction θ will receive a constructive inference. By searching
+all possible angles θ(a,e), a power spectrum p with size Θ can
+be obtained, where a high power in the spectrum indicates that
+there is a data source in that direction:
+p = wHx
+(7)
+where wH is the Hermitian transposition of w. The angles of
+the M objects can be obtained by taking the M highest peaks
+in p and finding the corresponding entries in w.
+In the data model shown in Equation (4), signals reflected
+from objects will be correlated when being received at each
+receiver, whereas the noise will be uncorrelated. Therefore,
+one way to extract signal information from x is by calculating
+a sensor covariance matrix Rx [8]:
+Rx = E{xHx} ≈ 1
+N
+N
+�
+t=1
+xH(t)x(t)
+(8)
+where E represents the statistical expectation and x(t) repre-
+sents one snapshot (or one frame) of the receiver data x. When
+evaluating the beamforming power spectrum using multiple
+snapshots, the overall power spectrum becomes the statistical
+expectation of p in Equation (7) over the snapshots, which
+gives:
+P = E{|wHx|2} = 1
+N
+N
+�
+t=1
+wHx(t)xH(t)w = wHRxw (9)
+Once the beamforming power spectrum is computed, the peaks
+in the spectrum will correspond to the signal from the objects.
+There are many algorithms designed for calculating the
+weights w. The conventional beamforming uses the steering
+vector directly as the weights, which is conceptually equivalent
+to the angle-FFT method (or correlation-based method) in
+Section III-C1:
+Pconventional = sHRxs
+(10)
+where s is the candidate steering vector in the format of
+Equation (5).
+There are also adaptive beamforming algorithms that calcu-
+late the weights using the signal information embedded in the
+covariance matrix. For example, the MVDR algorithm aims at
+minimizing the variance from non-interested directions while
+keeping the signal from the candidate direction distortionless
+[21]:
+Pmvdr =
+1
+sHR−1
+x s
+(11)
+3) Subspace Method: The core of the subspace method is
+that, since the signal x should contain M correlated signals
+and uncorrelated noise, the covariance matrix Rx should have
+M non-zero eigenvalues and N − M zero eigenvalues, where
+N is the rank of Rx that is equal to the number of receivers.
+The eigenvectors corresponding to the M eigenvalues form the
+signal subspace, and the eigenvectors corresponding to the zero
+eigenvalues form the noise subspace. The signal subspace and
+the noise subspace are orthogonal. One of the most widely-
+used subspace-based algorithms is the MUSIC algorithm [22].
+It searches for steering vectors that are orthogonal to the noise
+subspace. The power spectrum of the MUSIC algorithm can
+be written as:
+Pmusic =
+1
+sHUU Hs
+(12)
+where U is the set of eigenvectors corresponding to the zero
+eigenvalues.
+IV. MMWAVE RADAR SIMULATOR
+A simulator is designed to verify the discussed algorithms
+and evaluate the theoretical capability of using a mmWave
+radar as a 3D sensor. The simulator simulates mmWave radars
+with one transmitter and one receiving antenna array, which
+is practically equivalent to a multi-transmitter multi-receiver
+radar using an appropriate modulation scheme [1]. Any two
+neighbouring receivers in the array are separated by λ0/2,
+where λ0 (approximately 3.9 mm) is the wavelength of the
+mmWave signal at its chirp starting frequency (77 GHz).
+The simulator simulates the IF signal at each receiver of a
+mmWave radar when pointing toward a scene. The scene is
+modelled to have M points, where each point has a unique x-
+y-z coordinate and represents the spatial location of the object
+in the scene. Each point is modelled as a corner reflector and
+reflects the mmWave signal sent out by the radar with the
+same reflectivity. The reflection area of the object is modelled
+by the number of points, i.e. a large object would have a
+higher number of points. The IF signal at a receiver during
+one chirp is modelled using Equation (1). Given a certain chirp
+
+Virtual Antenna Array
+Receivers
+Transmitters
+Elevation
+RX5
+RX6
+RX7
+RX8
+TX2
+Antennas
+TX3
+TX1
+2
+RX2 RX3 RX4
+RX1
+X/2
+Azimuth
+RX4
+RX1
+RX2
+RX3
+RX9
+RX11
+RX12
+RX10
+Antennas
+V2
+Received Phase
+(5Apa+
+340a+
+24pa+
+44g.+
+Elevation
+(p+Ape
+p+Ape
+p+Ape
+p+△pe
+Antennas
+C
+07
+54ba+
+340a+
+24ga+
+6Apa+
+7Apa+
+(4△pa+
+Azimuth
+Apa+p
+Antennas
+CIEVirtual Antenna Array
+Received Phase
+Transmitters
+Receivers
+-b)
+p-
+RX4
+-△pa
+RX2
+RX6
+RX8
+24ba
+3△ba
+RX2 RX4
+TX2
+TX1
+RX1 RX3
+入/2
+入/2
+-d
+-d)
+p-pA
+2△A
+X3
+3△ΦA
+RX1
+RX3
+RX5
+RX7
+-Ade
+-Ape
+-Ade
+-Dde
+N/2
+N/2
+V2
+入/2
+p-LpA
+RX10
+RX12
+-2pe
+-24e
+N/2
+β-△A
+R212O
+RX9
+RX11
+-3△be
+-34be
+25
+configuration, the frequency and phase of the IF signal from
+one point are determined by the distance d between the point
+and the receiver. The amplitude of the IF signal is set to be
+inversely proportional to d4, to simulate the power loss due
+to distances according to the radar range equation [23]. The
+final IF signal at a receiver is the accumulated IF signals from
+all M points in the scene, with an additional white Gaussian
+noise n, as shown in Equation (13).
+IF(t) =
+M
+�
+i=1
+1
+d4
+i
+ej(2πSτi·t+2πf0τi) + n
+(13)
+where τi is the ToF of the signal from the transmitter to
+the point i and then to the receiver, and S is the slope of
+the chirp. The amplitude of the noise n is controlled by
+the desired SNR during the experiment. The signal IF(t)
+is sampled into a digital signal of length Ns, where Ns =
+(duration of the chirp) × (ADC sampling rate). During one
+chirp, the radar receives a signal that can be represented as
+a 2D matrix of size Nrx × Ns, where Nrx is the number of
+receivers in the array. One frame includes Nc chirps that form
+a 3D matrix of size Nrx ×Nc ×Ns, which becomes the input
+matrix of the point cloud construction algorithm, as shown as
+the input block in Figure 4.
+The design of the simulation system makes two assump-
+tions. First, the multipath effect is not considered in this
+system. While the multipath effect is a long-standing issue that
+can cause power fading and ghost targets, it highly depends
+on the scene and the reflectivity of the objects and is hard to
+incorporate in the model, so it is left as future work. Second,
+a practical radar often uses multiple transmitters and receivers
+and an appropriate signal modulation scheme to separate the
+signal from different transmitters, such as time demultiplexing
+modulation and binary phase modulation [1], to achieve an
+equivalent single-transmitter-multi-receiver system. The simu-
+lation system assumes a perfect signal modulation scheme for
+this purpose and ignores any error or SNR loss that may be
+introduced during the modulation process.
+V. POINT CLOUD CONSTRUCTION ALGORITHM
+The construction of a point cloud takes an input matrix of
+size Nrx × Nc × Ns and outputs a 2D matrix PCK of size
+K × 3 (referred to as the output point cloud), where K is
+the number of detected points and 3 is the x-y-z coordinates.
+This section studies one of the most common DPCs used on
+mmWave radars and its variant, which have shown success in
+many HAR systems, like in [4], [6], [24].
+A. Data Processing Chains
+Two DPCs are implemented that differ in using a Doppler-
+FFT or not, as shown in Figure 4. Both DPCs require a
+range-FFT over the raw data. The range-FFT identifies the
+frequency components in the IF signal that correspond to the
+distance of an object. It transforms the input matrix X of size
+Nrx ×Nc ×Ns into a range matrix R of size Nrx ×Nc ×N ∗
+s ,
+where N ∗
+s is the length of the range-FFT. The first DPC applies
+a Doppler-FFT on the data from all the chirps and generates
+Fig. 4: Two possible DPCs for mmWave radar point cloud
+construction.
+a Range-Doppler heatmap of size Nrx × N ∗
+c × N ∗
+s , where
+N ∗
+c is the length of the Doppler-FFT. Then, it searches for
+peaks in the Range-Doppler heatmap (using the average of
+all receivers), extracts the receivers’ data for each peak and
+generates a 2D matrix of size K × Nrx, where K is the
+number of detected peaks and, equivalently, the number of
+detected points. A constant false alarm rate (CFAR) algorithm
+is used for detecting peaks from the Range-Doppler heatmap.
+The parameters of the CFAR control the sensitivity of the
+peak detection and are considered the hyperparameters of the
+system. Finally, a single-snapshot AoA estimation is applied
+to each point in the matrix for a total of K times, to obtain the
+x-y-z coordinates of all detected points. The AoA estimation
+algorithm can be any of the angle-FFT, beamforming or
+subspace methods. Although the beamforming and subspace
+methods are multi-snapshot algorithms, the Doppler-FFT im-
+plicitly uses the information from all chirps and allows a good
+estimate of the covariance matrix at the AoA estimation stage.
+The second DPC does not include a Doppler-FFT. Instead, it
+considers the chirps as different snapshots and performs one
+multi-snapshot estimation for each range bin for a total of
+N ∗
+s times. More specifically, the input range matrix of size
+Nrx ×Nc ×N ∗
+s is re-arranged into N ∗
+s instances of Nrx ×Nc
+matrix, and the AoA estimation is applied to each Nrx × Nc
+matrix using Nc snapshots. The AoA estimation algorithm
+can be any of the beamforming or subspace methods. Finally,
+the points detected at each range bin are concatenated into
+one point cloud. In this research, the angle-FFT, conventional
+beamforming, MVDR beamforming and MUSIC subspace
+methods described in Section III-C are being studied.
+B. Model Order Estimation
+As described in Section III-C2 and Section III-C3, the
+beamforming and subspace methods include an angle power
+spectrum computation step, where each peak in the spectrum
+corresponds to an incoming signal from a point. However,
+in both DPCs, the expected number of incoming signals will
+be unknown in practice. Therefore, this number needs to
+be estimated from the signal data. This step is referred to
+as model order estimation. For this purpose, the covariance
+matrix of the signal data and its eigenvalues are computed. As
+
+Data Processing Chain 1
+口
+Nc
+Average all rx +
+Extract data for
+口
+Input
+peak detection
+each peak
+7
+Ns
+Chirp 1
+Doppler-FFT at
+Nc
+each distance
+. :
+Re-arrange
+K single-
+Range-Doppler
+K
+ snapshot AoA
+Nc
+Nc
+Chirp Nc
+Heatmap
+Nx
+estimations
+Ns
+NX
+INX
+N
+Nx}
+Ns
++
+Data Processing Chain 2
+Range-FFT
+indno
+Ns
+on each chirp
+用
+#
+#
+K
+Nc
+Range FFT 1
+Re-arrange
+Re-arrange
+Nrx
+-Nrx
+Nc
+X-y-Z
+Nc
+Ns multi-snapshot AoA estimations
+Range FFT Nc
++ peak detection
+Nx
+Ns
+Ns
+用
+用
+曲
+K1
+Ks
+4
+Concatenate6
+(a)
+(b)
+(c)
+Fig. 5: Three approaches when searching for the steering
+vectors. (a) An azimuth search (red) followed by an elevation
+search (black). (b) A full 2D azimuth-elevation search. (c) A
+2D azimuth-elevation search using sub-grids.
+described in Section III-C3, the covariance matrix should have
+a size of Nrx × Nrx and has a full rank equal to Nrx. There
+should be M large eigenvalues that correspond to the number
+of incoming signals and Nrx − M zeros corresponding to
+noise. In practice, due to the presence of noise, the difference
+between these eigenvalues may not be significant. Therefore,
+the minimum descriptive length (MDL) algorithm [25] is used
+for estimating the value of M. It fits a statistical model using
+the eigenvalues and searches for the optimal value of M that
+minimizes a cost function. The MDL algorithm is used in both
+DPCs to estimate the number of incoming signals in the AoA
+estimation stage. Once the angle power spectrum is calculated,
+all the local maxima will be found and the largest Mmdl peaks
+will be taken as the output, where Mmdl is the value found
+from the MDL algorithm.
+C. Steering Vector Searching
+The beamforming and subspace methods search for the
+steering vectors that maximize a power function. This process
+can be carried out using three approaches: an azimuth search
+followed by an elevation search, a 2D azimuth-elevation search
+or a 2D search using sub-grids. An example of the three
+approaches is shown in Figure 5. In the example, the power
+spectrum shows the incoming direction of the signal. The
+space of the spectrum is sampled into a 17×17 grid and each
+vertex on the grid represents a candidate AoA to be tested.
+In the first approach, an azimuth AoA search is performed
+using the data from azimuth receivers and steering vectors that
+only consider the azimuth angle. Then, based on the azimuth
+AoA output, a secondary search is performed in the elevation
+direction using the data from all receivers. This approach has
+the least computational cost (34 searches in the example), but
+the performance can be suboptimal as the azimuth search may
+not cover the actual AoA. The second approach performs a 2D
+search that considers all possible combinations of the azimuth
+and elevation directions and uses data from all receivers. It
+is computationally expensive (289 searches in the example)
+but provides the most accurate estimate. The third approach
+defines several levels of grids and performs the AoA search
+at different granularities. It starts the searching with a sparse
+grid, finds the peaks, defines a denser grid around each peak
+and performs the next search. The process can be performed
+iteratively until the desired resolution is achieved. It reduces
+Fig. 6: Some examples of the mesh models and point clouds
+from the FAUST dataset.
+the computational cost of the second approach significantly as
+it skips certain regions in the spectrum (50 searches in the
+example), at the cost of the potential possibility of missing
+some peaks.
+VI. EVALUATION
+A. Dataset
+The FAUST dataset [26] was used to serve as the ground
+truth for the simulator, to evaluate the point cloud construction
+algorithms described. The datasets contain human models in
+the form of watertight triangulated meshes. The meshes were
+generated from a high-resolution camera system containing
+stereo cameras, RGB cameras and speckle projectors. The
+FAUST dataset contains 10 subjects and 30 static postures
+per subject, of which 10 postures are provided with aligned
+watertight models, giving 100 models in total.
+In the simulation, the models were placed at 2 m from the
+radar and facing towards the radar. The height of the radar
+was set to be in the middle of each model. A ground truth
+point cloud was constructed from each model by randomly
+sampling M points from the surface of the mesh model,
+where each point was assumed to be a corner reflector. Some
+examples of the mesh models and point clouds are shown in
+Figure 6. The simulator computed a signal matrix for each
+point cloud to simulate the IF signal that would be received
+by the radar when placed towards a subject, as described by
+Equation (13). The entire dataset containing the 100 models
+was split into 80 training data and 20 test data, where the
+training data was used for hyperparameters searching in the
+point cloud construction algorithms, and the test data was used
+for evaluating the algorithms.
+When generating the IF signal matrix, there are two sources
+of randomness: the noise term n introduced in Equation (13)
+and the random sampling of the ground truth point cloud
+from the mesh model. Therefore, all the evaluation processes
+were repeated 10 times for each mesh model and the average
+metrics were reported, to minimize any potential effect of the
+randomness.
+B. Evaluation Metrics
+To evaluate the quality of the point cloud constructed by
+an algorithm, it is necessary to define the evaluation metrics
+for comparing the output point cloud against the ground truth
+point cloud. Let PCM denote the ground truth point cloud and
+PCK denote the point cloud generated by the radar, which are
+
++
+C
++
++
+DO
+C
+O
+O
+O
+O
+O
+O
+O7
+a M ×3 matrix and a K×3 matrix, respectively. It is important
+to note that the point cloud construction algorithm can provide
+an uncertain number of points that might be different to the
+ground truth (M ̸= K), and PCK can have a non-uniform
+distribution while PCM is distributed uniformly on the mesh
+model. The evaluation metrics should take the two point clouds
+PCM and PCK as input and measure the similarity between
+them. First, two points are defined to be close to each other
+if their Euclidean distance is less than a certain distance D.
+In this research, D is set to 10 cm as an empirical estimation
+of the error tolerance of a HAR system. Then, the following
+terms and metrics are defined:
+• Precision: Number of points in PCK that has at least one
+close point from PCM, divided by K. It evaluates how
+many points in PCK are considered to be accurate.
+• Sensitivity/Recall: Number of points in PCM that has
+at least one close point from PCK, divided by M. It
+evaluates how well PCK can cover the space of PCM.
+• Fowlkes–Mallows index (FMI): the geometric mean of
+precision and sensitivity, i.e. √precision × sensitivity.
+• Intersection over Union (IoU): Establish two regular 3D
+voxel grids for PCK and PCM with the voxel size set to
+10 cm × 10 cm × 10 cm, consider a voxel to be occupied
+if there is at least one point present in the voxel, then the
+IoU is calculated as the number of overlapping voxels
+of the two voxel grid, divided by the union. The IoU
+evaluates the similarity of the two point clouds at the
+granularity of the voxel size.
+An ideal system should have both high precision and high
+sensitivity, whereas the relative importance of the two depends
+on the application. In this section, the FMI, i.e. the geometric
+mean of precision and sensitivity, is used to indicate the
+performance of the system. The IoU also provides a good
+indication of how the generated point cloud can represent
+the scene. However, as the calculation of the IoU is highly
+sensitive to the voxel size and outliers, it is used as a secondary
+metric.
+C. Data Processing Chain and Algorithms
+In the first experiment, the two DPCs combined with
+different AoA algorithms were evaluated and compared, in
+terms of the quality of the estimated point cloud and the
+computational cost. A baseline radar and scene configuration
+were designed to approximate a typical setup in a common
+indoor environment as follows:
+• One transmitter and a 4 × 4 uniform receiver array.
+• The chirp frequency is 77 GHz to 81 GHz, the slope
+is 40 MHz/us, the chirp duration is 100 us, the ADC
+sampling rate is 15 MHz, each frame is 50 ms with 50
+chirps, and each chirp has 1500 samples (as shown in
+Figure 7).
+• Each human mesh model is sampled into 512 points and
+placed at 2 m away from the radar.
+• SNR is 30 dB.
+• The subject has a velocity of 0.05 m/s moving away from
+the radar.
+Fig. 7: Chirp configuration of one frame in the baseline setup.
+TABLE I: FMI (standard deviation in parentheses) comparison
+between the algorithms when using a 4 × 4 receiver array and
+a subject velocity of 0.05 m/s.
+FMI in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
+68.3
+(7.5)
+68.2
+(7.9)
+60.6
+(8.7)
+67.2
+(7.6)
+67.7
+(7.6)
+74.5
+(6.7)
+69.7
+(6.9)
+77.0
+(6.2)
+DPC2
+NA
+43.7
+(7.8)
+46.5
+(7.1)
+50.2
+(7.6)
+53.1
+(7.4)
+52.7
+(7.4)
+53.2
+(7.0)
+• The AoA algorithm uses 512 bins to cover the ±90° AoV,
+i.e. the angular resolution is 0.35°.
+The velocity of the subject is introduced following the
+assumption that a real person cannot stay absolutely stationary
+during the measurement. At a velocity of 0.05 m/s and a frame
+time of 50 ms, the total displacement will be 2.5 mm and is
+considered negligible. The velocity provides a variation on
+the signal received at different chirps, as otherwise the multi-
+snapshot AoA estimation algorithms would receive an identi-
+cal signal at all chirps and would yield a poor performance.
+Combining the two DPCs with different AoA estimation
+algorithms, there are 14 methods in total to be evaluated.
+For each method, both the 1D search approach and the 2D
+sub-grid approach described in Section V-C are included. For
+the 2D angle-FFT method, the full-grid approach is used
+instead of the sub-grid approach, since the benefit of the
+lower computational cost is less significant for FFTs. The
+algorithms will be referred to using the format “DPC-Method-
+1D/2D” throughout the paper. For example, DPC1-Conv-2D
+refers to the conventional beamforming method in DPC1 that
+uses a 2D steering vector search. The angle-FFT method is
+not applicable in DPC2 as it is not a multi-snapshot algorithm.
+Algorithms in DPC1 include a CFAR peak detection step on
+the Range-Doppler heatmap, where the optimal parameters for
+the CFAR were searched on the training dataset. Then, the
+performance of the algorithms on the test dataset was evaluated
+and compared. The results are shown in Table I and Table II
+as FMI and IoU (in % and with the standard deviation in
+parentheses), respectively.
+TABLE II: IoU (standard deviation in parentheses) comparison
+between the algorithms when using a 4 × 4 receiver array and
+a subject velocity of 0.05 m/s.
+IoU in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
+21.2
+(4.3)
+22.5
+(4.6)
+14.6
+(3.9)
+20.6
+(4.1)
+18.0
+(4.4)
+23.4
+(4.1)
+19.0
+(4.1)
+22.7
+(3.5)
+DPC2
+NA
+11.2
+(3.2)
+12.2
+(3.0)
+13.2
+(3.3)
+14.7
+(3.4)
+14.6
+(3.2)
+14.6
+(3.3)
+
+Frequency
+(GHz)
+Chirp 0
+Chirp 1
+Chirp 2
+Chirp 49
+81
+77
+0
+0.1
+1
+1.1
+2
+2.1
+49
+49.1
+50
+<
+Time (ms)
+1500 ADC samples
+50 x 1500 samples per receiver per frame8
+Fig. 8: Examples of the radar detection using the different
+algorithms, when using a 4 × 4 receiver array and a subject
+velocity of 0.05 m/s.
+There are a few important observations from the experiment.
+Even though the subject had a low velocity, the DPC1 with a
+Doppler-FFT outperformed the other significantly. One main
+reason is that, as the number of receivers is much lower than
+the number of signals, the AoA estimation algorithm can fail to
+distinguish points with a close angle. Instead, these points will
+be identified as one strong signal source. On the contrary, the
+CFAR peak detection step in DPC1 picks a set of points around
+the peak that are above the CFAR threshold. As these points
+also contribute to the point cloud, the output becomes denser
+and the sensitivity is improved. This effect can be observed
+from the example detection shown in Figure 8.
+In terms of the different algorithms, the MVDR and MU-
+SIC methods outperformed the angle-FFT and conventional
+methods, at the expense of higher complexity. Meanwhile, all
+the 2D methods outperformed the 1D methods due to a more
+fine-grained resolution (as shown earlier in Figure 5). The
+best performance was achieved with the DPC1-MVDR-2D
+and DPC1-MUSIC-2D methods, with an FMI of 74.5% and
+77.0%, respectively. However, the IoU metrics show that the
+point clouds were still far from the objective of high-accuracy
+scene reconstruction, as the highest IoU was only 23.4%. It
+can be seen from Figure 8 that, while the distribution of the
+point cloud mostly fitted the subject, the distribution was not
+even and there were body parts (like the hands) that received
+fewer points. Therefore, there is still a big gap before the radar
+output can be directly used by applications that require high
+quality data.
+Table III compares the algorithms in terms of execution
+time. The algorithms were run using the same dataset and
+parameters multiple times. The algorithms were written in
+Python without any processor-specific optimization and were
+run on one Intel i7-9700K CPU core. The result is shown
+as the relative execution time of each algorithm when com-
+pared with the DPC1-FFT-1D method (the most lightweight
+method) and normalized with the number of detected points,
+to give an indication of their relative complexity. All the
+2D methods have a higher complexity than the 1D methods.
+For algorithms in DPC1, the 1D angle-FFT method has the
+TABLE III: Normalized execution time comparison between
+the algorithms using the baseline setup.
+Normalized
+Complexity
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
+1.00
+13.42
+4.38
+9.32
+3.51
+8.99
+4.02
+8.85
+DPC2
+NA
+5.69
+12.38
+5.31
+10.89
+5.67
+10.61
+TABLE IV: Relative FMI difference of the algorithms when
+using a 4 × 4 receiver array and a subject velocity of 0.5 m/s
+in comparison to 0.05 m/s.
+FMI in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
++8.3
++11.4
++12.1
++10.7
++10.4
++8.8
++10.1
++7.7
+DPC2
+NA
++4.6
++2.6
++5.3
++4.6
++8.4
++5.6
+lowest computational cost. With the sub-grid optimization, the
+complexity of the 2D beamforming and MUSIC methods can
+be kept at around twice the 1D methods. The complexity
+without the sub-grid optimization is expected to be much
+higher, as can be estimated from the difference between the
+2D and 1D angle-FFT methods. When considering both the
+complexity and the performance, the DPC1-FFT-1D method
+provides a good trade-off between them. The MVDR methods
+and MUSIC methods in DPC1 give the best performance at
+the cost of 9x execution time and require additional efforts
+on the hardware and implementation. It is worth noting that
+many mmWave radar systems, like [6], [7], [27], are built
+based on the DPC1-FFT-1D method. Therefore, these systems
+can potentially benefit from a more complex AoA estimation
+algorithm.
+D. Subject Velocity
+The motion of the subject being sensed has a significant
+impact on the detection output. In DPC1, a higher velocity
+makes a subject easier to be identified in the Range-Doppler
+heatmap. Due to the relative position difference between the
+body parts of the subject, they will have a different radial
+velocity with respect to the radar, making them distinguishable
+in the Range-Doppler heatmap. In DPC2, a higher velocity
+increases the variance of the signal between chirps and allows
+a better estimate of the data covariance matrix. To verify
+the theorem, an experiment was carried out using the same
+configuration as the baseline setup, except that the velocity of
+the subject was set to different values from 0.1 m/s to 1 m/s.
+The ground truth point cloud was taken as the average position
+of the subject during the motion.
+Table IV and Table V show two examples of the experiment
+where the subject velocity was set to 0.5 m/s and 1 m/s,
+respectively. When compared with Table I, all algorithms
+achieved a 2.6% to 14.5% improvement in terms of the FMI
+when the subject had an increased velocity. Figure 9 shows the
+FMI and IoU of the DPC1-MUSIC-2D method with different
+subject velocities from 0.1 m/s to 1 m/s. An overall positive
+correlation can be observed between the subject velocity and
+the detection performance, and the impact is the most obvious
+at lower velocities (around 0.5 m/s). Some examples of the
+detection at 1 m/s are shown in Figure 10.
+
+Conventional
+MVDR
+MUSIC
+FFT
+Beamforming
+Beamforming
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+Front View
+DPC1
+Left View
+Front View
+DPC2
+Left View9
+TABLE V: Relative FMI difference of the algorithms when
+using a 4 × 4 receiver array and a subject velocity of 1 m/s
+in comparison to 0.05 m/s.
+FMI in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
++9.7
++13.0
++14.5
++12.8
++12.0
++9.8
++11.6
++8.2
+DPC2
+NA
++5.3
++2.6
++4.3
++3.4
++9.3
++5.8
+Fig. 9: FMI and IoU (with errors) of the DPC1-MUSIC-2D
+algorithm with different subject velocities.
+E. SNR
+In a practical environment, a radar system can experience
+noise from different sources, such as the thermal noise of the
+radar chip. The SNR also depends on the distance between
+the radar and the subject, as the signal power drops quickly
+along with the distance. In the simulator, the SNR can be
+controlled by the power of the noise term n in Equation (13).
+In this section, the performance of the algorithms between a
+high SNR environment (40 dB) and a lower SNR environment
+(5 dB) is compared. Two experiments were carried with the
+subject velocity set to 0.05 m/s and 0.5 m/s, respectively. The
+results are shown in Table VI and Table VII.
+In the low SNR environment, all the algorithms in DPC1
+experienced a similar drop in performance, as expected. How-
+ever, the algorithms in the DPC2 showed a higher perfor-
+mance. The reason is that the higher noise affected the model
+order estimation step and the system tends to report a higher
+number of points. Taking the DPC2-Conv-2D method as an
+example, the average size of the detected point cloud was
+Fig. 10: Examples of the radar detection using the different
+algorithms, when using a 4 × 4 receiver array and a subject
+velocity of 1 m/s.
+TABLE VI: Performance difference when using a 4 × 4
+receiver array and a subject velocity of 0.05 m/s in a low
+SNR environment (5 dB in comparison to 30 dB).
+FMI in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
+-8.1
+-5.8
+-5.5
+-5.6
+-6.4
+-6.3
+-5.7
+-6.2
+DPC2
+NA
++2.2
++2.8
++1.7
++2.7
++1.6
++2.4
+TABLE VII: Performance difference when using a 4 × 4
+receiver array and a subject velocity of 0.5 m/s in a low SNR
+environment (5 dB in comparison to 30 dB).
+FMI in %
+Angle-FFT
+Conv. BF
+MVDR BF
+MUSIC
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+DPC1
+-7.8
+-6.2
+-7.2
+-6.1
+-8.3
+-7.5
+-8.9
+-7.4
+DPC2
+NA
++2.5
++2.8
++1.0
++0.8
+-0.7
++0.9
+found to be 20.3% higher in a low SNR environment than in a
+higher SNR environment. However, this was still insufficient
+to reach a similar performance as DPC1.
+F. Antenna Layout
+Theoretically, the antenna layout determines the angular
+resolution that an AoA estimation algorithm can achieve. The
+more receivers in one direction, the higher resolution the radar
+can measure [1]. However, this is questionable when the signal
+sources are spatially close and continuous. Meanwhile, having
+more antennas also increases the cost of the hardware, as more
+circuit components, processing units and memory would be
+required. Therefore, it is beneficial to study the relationship
+between the antenna layout and the output quality and find
+the optimal trade-off for an application.
+Common commercial mmWave radars use up to three
+transmitters and up to four receivers, giving up to twelve
+virtual receivers as a receiving array. Some radar models are
+designed for automotive applications and prioritize the azimuth
+direction, while others are designed for general purpose appli-
+cations and have a similar resolution in both the azimuth and
+elevation directions. In this section, common antenna layouts
+implemented on the TI radars are evaluated and compared,
+as well as a few square-shape antenna layouts that are more
+common in research projects, as listed in Figure 11. The same
+radar configuration and scene setup in Section VI-C were used.
+The experiment compares the antenna layouts using the DPC1-
+MUSIC-2D algorithm (the best performing algorithm). The
+result is shown in Table VIII.
+Fig. 11: The list of receiver layouts being evaluated. (a)-(d) are
+square antenna arrays. (e)-(f) are non-regular antenna arrays
+implemented on TI radars.
+
+1.0
+FMI
+0.8
+Performance
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Subject velocity (m/s)Conventional
+MVDR
+MUSIC
+FFT
+Beamforming
+Beamforming
+1D
+2D
+1D
+2D
+1D
+2D
+1D
+2D
+Front View
+DPC1
+Left View
+Front View
+DPC2
+Left View0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+C
+(a) 3x3
+0
+0
+0
+0
+(b) 4x4
+0
+0
+0
+0
+0
+0
+0
+(c) 6x6
+O
+O
+(d) 8x8
+O
+O
+(e) IWR6843A0P
+(f) AWR1843AOP
+(g) IWR184310
+TABLE VIII: Performance comparison between different an-
+tenna layouts using the baseline configuration and the DPC1-
+MUSIC-2D algorithm (standard deviation in parentheses).
+Antenna
+Layouts
+a
+b
+c
+d
+e
+f
+g
+FMI
+in %
+76.7
+(6.4)
+77.0
+(6.2)
+76.8
+(4.8)
+77.9
+(4.5)
+72.4
+(6.3)
+77.8
+(5.9)
+65.0
+(6.0)
+IoU
+in %
+23.4
+(4.1)
+22.7
+(3.5)
+20.5
+(2.8)
+18.8
+(2.7)
+20.8
+(4.0)
+23.9
+(4.2)
+17.0
+(3.5)
+Fig. 12: Examples of the radar detection using the different
+antenna layouts with the baseline setup.
+It can be seen that most antenna layouts had similar perfor-
+mance, except the layout (g) which had a worse performance
+as it is designed for automotive applications. The layout (e)
+has a non-uniform antenna distribution that slightly affected its
+performance. All other layouts showed a similar performance
+regardless of the antenna size. Therefore, considering the
+increased hardware cost and computational cost of increasing
+the number of antennas, a small antenna size can be preferable
+for 3D sensing applications. Figure 12 shows some examples
+of the detection using different antenna layouts.
+G. Chirp Configuration
+The chirp configuration can have various effects on the
+distance detection and velocity detection. These factors can
+indirectly affect the quality of the final point cloud. In this
+section, three different chirp configurations were tested and
+compared against the baseline configuration in Section VI-C.
+The details of the three configurations (named A, B and C) and
+the performance are shown in Table IX. Each configuration
+has certain parameter cut to 80% to evaluate the effect on
+the output. Configuration A had an 80% reduced chirp slope
+and, hence, a reduced effective bandwidth from 4 GHz to
+3.2 GHz. Configuration B had an 80% reduced ADC sampling
+rate that reduced the samples per chirp from 1500 to 1200.
+Configuration C had an 80% reduced number of chirps per
+frame, from 50 to 40. All other parameters were kept the same
+as the baseline with the DPC1-MUSIC-2D algorithm.
+The result shows that the performance can be strongly
+affected by the effective bandwidth and the number of chirps.
+The former affects the distance resolution of the detection,
+and the latter affects the Doppler resolution. Reducing either
+of these parameters reduces the accuracy of the range-Doppler
+heatmap and the estimation of the covariance matrix. On the
+other hand, the effect of reducing the ADC sampling rate and
+the number of samples per chirp is much less significant.
+TABLE IX: FMI (standard deviation in parentheses) com-
+parison between four chirp configurations using the DPC1-
+MUSIC-2D algorithm.
+Chirp Configuration
+Baseline
+A
+B
+C
+Slope of the chirp (MHz/us)
+40
+32
+40
+40
+ADC sampling rate (MHz)
+15
+15
+12
+15
+Chirps per frame
+50
+50
+50
+40
+FMI in %
+77.0
+(6.2)
+71.1
+(6.8)
+76.5
+(6.0)
+70.2
+(6.0)
+(a) Points detected without SRPC.
+(b) Points detected with SRPC.
+Fig. 13: Using SRPC algorithm to improve the resolution and
+distribution of the data.
+VII. SUPER-RESOLUTION POINT CLOUD CONSTRUCTION
+ALGORITHM
+It can be seen from Figure 8 and Figure 10 that the
+constructed point clouds can be noisy and the distribution of
+the points can be imbalanced. One major reason is that the
+point cloud construction relies on the peak detection result
+over the range-Doppler-FFT spectrum, so the distribution of
+the points will be limited by the resolution of the FFT, and the
+points will have a discrete distribution in the range domain (as
+the curve-like data from the left view). Although it is possible
+to improve this resolution, such as zero padding the data before
+applying the FFT, it would also increase the computational cost
+and memory consumption. Meanwhile, there are false detected
+points due to the outliers from the peak detection stage. To
+address the mentioned issue and improve the quality of the
+constructed point cloud, a novel super-resolution point cloud
+construction (SRPC) algorithm is proposed.
+The SRPC algorithm aims to improve the distribution of
+the point cloud and make it span more naturally in the spatial
+space. The rationale is shown in Figure 13. When detecting
+peaks in a range-Doppler spectrum or an angle spectrum,
+a common approach is taking all points above a static or
+dynamic threshold, where the distribution of the points is
+limited by the resolution of the original data. An example of
+this effect is shown in Figure 13a, where the grid represents
+the resolution of the data and all the detected points must
+fall on the grid. The SRPC algorithm aims to return a set
+of points that have a higher resolution than the original data
+and fall more naturally on the distribution curve, as shown in
+Figure 13b.
+The algorithm can be broken down into the following
+steps. First, the power spectrum is upsampled into the desired
+resolution using linear interpolation. Then, for each of the
+originally detected points i ∈ [1..K], the algorithm randomly
+samples ni points around it with a probability distribution
+
+Front View
+Left View
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+(g)
+3x3
+4x4
+6x6
+8x8
+IWR6843AOP
+AWR1843AOP
+IWR1843Power spectrum
+Threshold
+Detected pointsPower spectrum
+Threshold
+Detected points11
+being the amplitude of the upsampled power spectrum. The
+value of ni is calculated as:
+ni = pi · αSRP C
+th
+(14)
+where pi is the power of the point, th is the threshold of the
+peak detection algorithm, and αSRP C is a global hyperpa-
+rameter that controls the aggressiveness of the algorithm. The
+term pi ensures that a point with higher power will be sampled
+into more points, as the power indicates the confidence that a
+point can represent a real signal source. The parameter αSRP C
+amplifies the importance of pi, where a higher αSRP C pushes
+the distribution of the points towards the peak of the spectrum
+and gives a more dense distribution. The sampling process is
+repeated for each point i to form a new point list. Finally, K
+points (the population of the original detection) are randomly
+selected from the new point list, so that the total number of
+detected points is kept the same and the computational cost
+of the rest of the system is not affected. Since the algorithm
+tends to sample more points at higher power, the distribution
+of the final points will also tend to be around higher powers,
+and, hence, gives a more natural distribution regarding the
+power spectrum and overcomes the limitation of the original
+data resolution. The time complexity of the SRPC algorithm
+is approximately O(K · ni), where a typical value of ni can
+fall between 2 and 8.
+When constructing the point cloud, the SRPC is applied
+when detecting peaks from the range-FFT spectrum and de-
+tecting peaks from the angle spectrum in the AoA estimation
+step. The former improves the data distribution in the range
+domain and eliminates the curve-like effect when looking at
+the point cloud from the left view. The latter improves the
+data distribution in the angle domain so that the points tend to
+span into the space rather than appearing as a dense cluster.
+Meanwhile, since the points will be distributed around higher
+powers, the probability of outliers will be reduced.
+To evaluate the proposed SRPC algorithm, it was inserted
+into the DPC1-FFT-1D and DPC1-MUSIC-2D methods men-
+tioned in Section VI-C when using the baseline setup. The two
+methods were chosen as they represent the most lightweight
+algorithm and the most accurate algorithm, respectively. Since
+the SRPC is likely to produce point clouds with different
+sizes and to ensure a fair comparison, a fixed number of 512
+points were randomly taken from the point cloud generated
+by each algorithm for the evaluation. The result is shown in
+Figure 14. After applying the SRPC algorithm, the distribution
+of the point cloud appeared to be more natural and better
+distributed around the ground truth, and the outliers in the
+original detection were reduced. A quantitative evaluation is
+shown in Table X. The performance without SRPC dropped
+slightly when compared with Table I because the output
+size was forced to be 512, but both metrics have improved
+after applying SRPC. Therefore, it is shown that the SRPC
+algorithm can successfully improve the data point distribution,
+reduce the outliers and produce a more natural point cloud
+that can be potentially preferable for higher-level applications.
+Future work of this research includes an efficient hardware
+implementation of this algorithm using the radar on-chip
+Fig. 14: Examples of point clouds constructed with and
+without the SRPC algorithm.
+TABLE X: Performance comparison of two algorithms with
+and without SRPC.
+DPC1-FFT-1D
+DPC1-MUSIC-2D
+FMI
+IoU
+FMI
+IoU
+Without SRPC
+64.9
+20.2
+72.1
+22.9
+With SRPC
+69.5
+23.6
+72.9
+25.9
+processors so that it can be further verified in real-world
+scenarios, as well as an evaluation of its effectiveness in
+higher-level applications like posture estimation.
+VIII. CONCLUSION
+In this paper, a mmWave radar simulator is presented. The
+system is used to evaluate the ability of the mmWave radar as
+a 3D imaging sensor. A mmWave radar dataset is constructed
+using the FAUST dataset as the ground truth to provide 3D
+mesh models of human subjects, from which mmWave radar
+IF signals are simulated and used to evaluate different point
+cloud construction algorithms. The FMI and IoU metrics are
+defined to evaluate the quality of the generated point cloud.
+The evaluation is performed regarding a set of different factors,
+including the DPCs, AoA estimation algorithms, subject veloc-
+ity, SNR, antenna layout and chirp configuration. It was found
+that the DPC combining a range-Doppler-FFT and a single-
+snapshot AoA estimation algorithm gives better performance.
+Among all the AoA estimation algorithms, the angle-FFT
+method gives a good trade-off between high performance and
+low computational cost, whereas the more advanced AoA
+estimation algorithms, like MVDR and MUSIC, give the best
+performance at up to 9x higher execution time. The velocity
+of the subject helps significantly in the detection, as the
+algorithms are better at detecting a moving subject than a
+stationary object. When comparing common antenna layouts,
+large square antenna arrays give the best performance, but the
+advantage is not significant in a 3D imaging application when
+the data sources are spatially close and continuous. It is shown
+that the performance of the point cloud detection benefits from
+higher effective bandwidth and a higher number of chirps
+per frame. Finally, a novel SRPC algorithm is proposed for
+improving the resolution and distribution of the point cloud
+and reducing the probability of outliers. The algorithm applies
+
+DPC1-FFT-1D
+DPC1-MUSIC-2D
+Front View
+Left View
+Front View
+Left View
+Before SRPC
+After SRPC12
+to the range-Doppler-FFT peak detection stage and the AoA
+estimation stage and detects points at a higher resolution that
+fits the power spectrum better. When evaluating the algorithm
+using the simulation system, it has been shown that the
+algorithm can successfully improve the data distribution and
+produces a more natural point cloud.
+REFERENCES
+[1] S. Rao, “MIMO radar,” Texas Instruments, Tech. Rep., 2017.
+[2] C. Iovescu and S. Rao, “The fundamentals of millimeter wave sensors,”
+Texas Instruments, 2017.
+[3] T. Liu, Y. Zhao, Y. Wei, Y. Zhao, and S. Wei, “Concealed object
+detection for activate millimeter wave image,” IEEE Transactions on
+Industrial Electronics, vol. 66, no. 12, pp. 9909–9917, 2019.
+[4] P. Zhao, C. X. Lu, J. Wang, C. Chen, W. Wang, N. Trigoni, and
+A. Markham, “mid: Tracking and identifying people with millimeter
+wave radar,” in 2019 15th International Conference on Distributed
+Computing in Sensor Systems (DCOSS).
+IEEE, 2019, pp. 33–40.
+[5] H. Cui and N. Dahnoun, “High precision human detection and tracking
+using millimeter-wave radars,” IEEE Aerospace and Electronic Systems
+Magazine, vol. 36, no. 1, pp. 22–32, 2021.
+[6] A. Sengupta, F. Jin, R. Zhang, and S. Cao, “mm-pose: Real-time human
+skeletal posture estimation using mmwave radars and cnns,” IEEE
+Sensors Journal, 2020.
+[7] H. Cui and N. Dahnoun, “Real-time short-range human posture estima-
+tion using mmwave radars and neural networks,” IEEE Sensors Journal,
+vol. 22, no. 1, pp. 535–543, 2022.
+[8] Z. Chen, G. Gokeda, and Y. Yu, Introduction to Direction-of-arrival
+Estimation.
+Artech House, 2010.
+[9] M. Dudek, D. Kissinger, R. Weigel, and G. Fischer, “A millimeter-
+wave fmcw radar system simulator for automotive applications including
+nonlinear component models,” in 2011 8th European Radar Conference.
+IEEE, 2011, pp. 89–92.
+[10] C. Stetco, B. Ubezio, S. M¨uhlbacher-Karrer, and H. Zangl, “Radar
+sensors in collaborative robotics: Fast simulation and experimental
+validation,” in 2020 IEEE International Conference on Robotics and
+Automation (ICRA).
+IEEE, 2020, pp. 10 452–10 458.
+[11] C. Sch¨offmann, B. Ubezio, C. B¨ohm, S. M¨uhlbacher-Karrer, and
+H. Zangl, “Virtual radar: Real-time millimeter-wave radar sensor sim-
+ulation for perception-driven robotics,” IEEE Robotics and Automation
+Letters, vol. 6, no. 3, pp. 4704–4711, 2021.
+[12] J. Han, L. Shao, D. Xu, and J. Shotton, “Enhanced computer vision with
+microsoft kinect sensor: A review,” IEEE transactions on cybernetics,
+vol. 43, no. 5, pp. 1318–1334, 2013.
+[13] T. Raj, F. H. Hashim, A. B. Huddin, M. F. Ibrahim, and A. Hussain,
+“A survey on lidar scanning mechanisms,” Electronics, vol. 9, no. 5, p.
+741, 2020.
+[14] R. Appleby, D. A. Robertson, and D. Wikner, “Millimeter wave imaging:
+a historical review,” in Passive and Active Millimeter-Wave Imaging XX,
+vol. 10189.
+SPIE, 2017, p. 1018902.
+[15] M. Kotaru, K. Joshi, D. Bharadia, and S. Katti, “Spotfi: Decimeter level
+localization using wifi,” in Proceedings of the 2015 ACM Conference
+on Special Interest Group on Data Communication, 2015, pp. 269–282.
+[16] K. Witrisal, S. Hinteregger, J. Kulmer, E. Leitinger, and P. Meissner,
+“High-accuracy positioning for indoor applications: Rfid, uwb, 5g, and
+beyond,” in 2016 IEEE International Conference on RFID (RFID).
+IEEE, 2016, pp. 1–7.
+[17] M. Z. Ikram, A. Ahmad, and D. Wang, “High-accuracy distance mea-
+surement using millimeter-wave radar,” in 2018 IEEE Radar Conference
+(RadarConf18).
+IEEE, 2018, pp. 1296–1300.
+[18] Y. Sun, Z. Huang, H. Zhang, Z. Cao, and D. Xu, “3drimr: 3d recon-
+struction and imaging via mmwave radar based on deep learning,” in
+2021 IEEE International Performance, Computing, and Communications
+Conference (IPCCC).
+IEEE, 2021, pp. 1–8.
+[19] Y. Cheng, J. Su, M. Jiang, and Y. Liu, “A novel radar point cloud gen-
+eration method for robot environment perception,” IEEE Transactions
+on Robotics, 2022.
+[20] E. Gentilho, P. R. Scalassara, and T. Abr˜ao, “Direction-of-arrival estima-
+tion methods: A performance-complexity tradeoff perspective,” Journal
+of Signal Processing Systems, vol. 92, no. 2, pp. 239–256, 2020.
+[21] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,”
+Proceedings of the IEEE, vol. 57, no. 8, pp. 1408–1418, 1969.
+[22] R. Schmidt, “Multiple emitter location and signal parameter estimation,”
+IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp.
+276–280, 1986.
+[23] D. K. Barton, “Modern radar system analysis,” Norwood, 1988.
+[24] A. D. Singh, S. S. Sandha, L. Garcia, and M. Srivastava, “Radhar:
+Human activity recognition from point clouds generated through a
+millimeter-wave radar,” in Proceedings of the 3rd ACM Workshop on
+Millimeter-wave Networks and Sensing Systems.
+ACM, 2019, pp. 51–
+56.
+[25] J. Rissanen, “Modeling by shortest data description,” Automatica,
+vol. 14, no. 5, pp. 465–471, 1978.
+[26] F. Bogo, J. Romero, M. Loper, and M. J. Black, “FAUST: Dataset and
+evaluation for 3D mesh registration,” in Proceedings IEEE Conf. on
+Computer Vision and Pattern Recognition (CVPR).
+Piscataway, NJ,
+USA: IEEE, Jun. 2014.
+[27] A. D. Singh, S. S. Sandha, L. Garcia, and M. Srivastava, “Radhar:
+Human activity recognition from point clouds generated through a
+millimeter-wave radar,” in Proceedings of the 3rd ACM Workshop on
+Millimeter-wave Networks and Sensing Systems, 2019, pp. 51–56.
+
+13
+APPENDIX A
+IF SIGNAL
+In a typical FMCW radar model, the transmitter sends a
+chirp signal Stx (a signal with frequency increasing linearly
+with time) to detect any object in front of the radar. When
+Stx is reflected by the object, the signal is received as Srx.
+Assuming the signal has an initial frequency f0 and a slope
+of S, then the frequency of Stx is a function of t:
+ftx(t) = f0 + S · t
+(15)
+The instantaneous phase of the signal is a function of t and is
+the integral of ftx:
+φtx(t) =
+� t
+τ=0
+2π · ftx(τ) dτ
+=
+� t
+τ=0
+2π · (f0 + S · τ) dτ
+= 2π · f0 · t +
+� t
+τ=0
+2π · S · τ dτ
+= 2π · f0 · t + 2π · 1
+2S · t2
+= 2π · f0 · t + π · S · t2
+(16)
+The transmitted signal Stx can be written as a sinusoid signal:
+Stx(t) = A · cos(2πf0t + πSt2)
+(17)
+where A is the transmission power. The received signal is a
+delayed and downscaled version of Stx:
+Srx(t) = αA · cos
+�
+2πf0(t − τ) + πS(t − τ)2�
+(18)
+where τ is the ToF of the signal and indicates the distance
+of the object, and α is the downscale factor that models the
+transmission loss. The two signals, Stx and Srx, are combined
+through a mixer (a multiplier) to generate one signal with both
+the sum frequency and the difference frequency:
+Stx(t)·Srx(t)
+= αA2
+2
+�
+cos
+�
+(2πf0t + πSt2) + (2πf0(t − τ) + πS(t − τ)2)
+�
++
+cos
+�
+(2πf0t + πSt2) − (2πf0(t − τ) + πS(t − τ)2)
+��
+= αA2
+2
+�
+cos
+�
+2π(2f0 − Sτ)t + 2πSt2 + πSτ 2 − 2πf0τ
+�
++
+cos
+�
+2π(Sτ)t + 2πf0τ − πSτ 2��
+(19)
+There are two cos terms in the result. The first one has a
+frequency of 2f0 and will be removed by a low pass filter.
+The second one is called the IF signal or the beat frequency.
+The IF signal has the equation:
+IF(t) = B · cos
+�
+2π(Sτ)t + 2πf0τ − πSτ 2�
+(20)
+where B = αA2
+2 . The signal has a frequency Sτ, i.e. the slope
+of the chirp multiplied by the ToF. Therefore, the frequency
+of the IF signal is directly proportional to the ToF. Given that
+the slope of the chirp S is known, the distance of the object
+can be calculated from the frequency of the IF signal. The
+phase of the IF signal, (2πf0τ − πSτ 2), can be simplified to
+(2πf0τ), as the second term is negligible: S has an order of
+1012, τ has an order of 10−8, so (πSτ 2) will have a negligible
+order of 10−4. In summary, the IF signal can be written as:
+IF(t) = B · cos(ωbt + φb)
+(21)
+where the angular frequency ωb and the phase of the signal
+φb are:
+ωb = 2π · Sτ,
+φb = 2πf0τ
+(22)
+The above equations assume that the object is stationary. If
+the object is moving, the ToF τ will be varying with respect
+to t. However, considering that this variation is limited to a
+single chirp time, it is unlikely to produce a big change in the
+frequency. The change in phase can be more significant, but
+will only affect certain applications where the phase informa-
+tion is critical, such as vital sign monitoring. In such cases,
+the phase can be written as a function of t as φb(t) = 4πd(t)
+λ0
+,
+where d(t) describes the displacement of the object during the
+chirp time and λ0 is the signal wavelength.
+Note that the TI mmWave radar uses a complex band
+architecture. It uses a complex mixer (an IQ mixer) to multiply
+the two signals Stx and Srx, which has several advantages like
+a lower noise figure. When in complex form, the IF signal in
+Equation (21) can be written as:
+IF(t) = B · ej(ωbt+φb)
+(23)
+which has the same frequency and phase as in Equation (22).
+APPENDIX B
+STEERING VECTOR AND AOA ESTIMATION
+Figure 15 shows the AoA of an object (point A) to the
+radar (point O). The azimuth angle θa is defined to be the
+angle between the object’s projection on the horizontal plane
+and the front direction of the radar. The line of incidence of
+the object OA is projected onto the horizontal plane as OB,
+and the angle between OB and the y-axis is the azimuth angle
+θa. The elevation angle θe is defined to be the angle between
+the object and the horizontal plane (between line OA and the
+x-y plane).
+Fig. 15: The azimuth and elevation angle of an object.
+The steering vector is a function of the receiver layout
+and the incident angle. To introduce the concept of steering
+vectors, it is easier to start with the one-dimensional situation.
+Assuming there are two receivers separated by l, a signal will
+travel an additional distance ∆d to reach the second receiver,
+
+Elevation
+A
+Z
+B
+0
+X
+Azimuth14
+where the following approximation can be made (as shown in
+Figure 1):
+∆d = l · sin(θ)
+(24)
+Given that the phase of a sinusoid signal travelled over any
+distance ∆d will have a phase
+2π∆d
+λ
+, the phase difference
+between the two neighbouring receivers will be:
+∆φ = 2π · ∆d
+λ = π · sin(θ)
+(25)
+When using a receiver array with N azimuth receivers,
+each subsequent receiver beyond the first one will receive an
+additional phase change of ∆φ, which can be written as a
+steering vector:
+s(θ, N) = [1, ejπ·sin(θ), e2jπ·sin(θ), ..., e(N−1)jπ·sin(θ)] (26)
+(a) Elevation angle.
+(b) Azimuth angle.
+Fig. 16: The AoA can be estimated from the phase difference
+between adjacent receivers.
+When considering the AoA in both azimuth and elevation
+directions, the situation is shown in Figure 16. Distances ∆da
+and ∆de represent the extra distance travelled by the signal to
+reach receiver RX0 when compared with the azimuth receiver
+RX1 and the elevation receiver RX2 respectively. Similar to
+Equation (25), the estimation of the elevation angle θe is given
+by:
+sin(θe) = ∆de
+l
+= ∆φe
+π
+(27)
+where ∆φe is the phase difference between RX0 and RX2.
+The azimuth angle requires a projection from the object’s
+3D location to the horizontal plane. As shown in Figure 16b,
+the projection from ∆da to ∆da’ gives:
+∆da = ∆da’ · cos(θe)
+(28)
+Then, the angle θa can be calculated as:
+sin(θa) = ∆da’
+l
+=
+∆da
+l · cos(θe)
+=
+∆φa
+π · cos(θe)
+(29)
+where ∆φa is the phase difference between RX0 and RX1.
+Extending Equation (27) and Equation (29) with multiple
+receivers as a 2D array gives Equation (5).
+Once the values of θa and θe are found, then according to
+Figure 1, it can be shown that:
+x = OB · sin(θa) = OA · cos(θe)sin(θa) = OA · ∆φa
+π
+z = OA · sin(θe) = OA · ∆φe
+π
+y =
+�
+OA2 − x2 − z2
+(30)
+where OA is the distance of the object and can be found from
+the range-FFT.
+
+Elevation
+Z
+RX2
+RXO
+RX1
+-X-
+AzimuthElevation
+Z
+RX2
+Ada
+RXO
+RX1
+-X-
+Azimuth
\ No newline at end of file
diff --git a/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/load_file.txt b/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..eba2c03fc978350619fd855e8d6b7cd0b589b370
--- /dev/null
+++ b/l9FRT4oBgHgl3EQfZjdA/content/tmp_files/load_file.txt
@@ -0,0 +1,974 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf,len=973
+page_content='1  Millimetre-wave Radar for Low-Cost 3D Imaging:  A Performance Study  Han Cui, Jiacheng Wu and Naim Dahnoun  Abstract—Millimetre-wave (mmWave) radars can generate 3D  point clouds to represent objects in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, the accu-  racy and density of the generated point cloud can be lower than  a laser sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Although researchers have used mmWave radars  for various applications, there are few quantitative evaluations  on the quality of the point cloud generated by the radar and  there is a lack of a standard on how this quality can be assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  This work aims to fill the gap in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A radar simulator  is built to evaluate the most common data processing chains of  3D point cloud construction and to examine the capability of the  mmWave radar as a 3D imaging sensor under various factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It  will be shown that the radar detection can be noisy and have an  imbalance distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' To address the problem, a novel super-  resolution point cloud construction (SRPC) algorithm is proposed  to improve the spatial resolution of the point cloud and is shown  to be able to produce a more natural point cloud and reduce  outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Index Terms—mmWave radar, 3D imaging, point cloud  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' INTRODUCTION  Millimetre-wave (mmWave) radars have received increased  popularity in many industries as an emerging type of sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The high bandwidth allows them to estimate the distance of an  object at centimetre-level resolution, and the short wavelength  and antenna size allow multiple antennas to be integrated into  a single chip and measure the angle-of-arrival (AoA) of the ob-  ject using multiple-input multiple-output (MIMO) techniques  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Combining the distance and AoA measurement, mmWave  radars are able to construct a point cloud to represent the  spatial shape of an object [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, mmWave radars can  be used as a low-cost 3D imaging sensor, as an alternative to  the traditional depth cameras and laser sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This allows  many computer vision tasks, such as object detection [3],  human tracking [4], [5], posture estimation [6], [7], and  identification [4], to be addressed using mmWave radars as a  non-intrusive solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, although many applications  have been proposed that rely on the 3D imaging capability of  mmWave radars, few researchers have attempted to evaluate  the quality of mmWave radars’ detection quantitatively, and  there is a lack of a standard on how this quality can be  assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  When detecting objects in a scene, the reflection signal can  be seen as a time-delayed version of the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Combining the two signals gives an intermediate frequency  (IF) signal, whose frequency and phase are determined by  the time-of-flight (ToF) of the signal, and, equivalently, the  distance between the object and the radar [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The distance  can be estimated directly from the IF signal of one pair of  transmitter and receiver at high resolution, whereas the AoA  needs to be estimated from the signal phase over a linearly  spaced antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' There is rich literature on antenna  array-based AoA estimation for traditional radars [8], and the  same concept also applies to mmWave radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This paper  discusses the AoA estimation in the context of mmWave  radar 3D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It reviews and discusses the 3D point  cloud construction techniques that are commonly used with  mmWave radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  This paper presents a purpose-built simulation system that  simulates the data acquisition process of a mmWave radar  when facing a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Radar data simulation allows researchers  to focus on algorithm design and verification, instead of  investing too much time in the hardware and real-world data  collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Existing radar simulators are often not designed  for 3D imaging and have certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For example, the  system in [9] generates range and Doppler information of the  radar rather than the raw data, the system in [10] only supports  single antenna data generation and cannot be used to estimate  the AoA, and the system in [11] only supports up to four  receivers in one direction and cannot be used for 3D imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In this research, a lightweight mmWave radar simulator is  designed that supports raw data generation of a multi-antenna  mmWave radar, configurable antenna parameters and layout,  and customized scene construction using 3D human models  with programmable motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Using the simulation system, a quantitative evaluation of  the radar’s capability of imaging a human subject is carried  out, as well as an evaluation of the key factors that could  affect the output quality, including the data processing chain  (DPC), radar antenna configuration, chirp configuration, sub-  ject velocity, and signal-to-noise ratio (SNR) in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  It will be shown that, although the radar can capture the  spatial information of the subject’s body shape, the detected  point cloud can be noisy, sparse and imbalanced, and can  require further processing before being used for higher-level  applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Finally, a novel super-resolution point cloud  construction (SRPC) algorithm is proposed to improve the  spatial resolution of the point cloud and is shown to be able  to produce a more natural point cloud and reduce outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The contribution of this paper can be summarized as fol-  lows:  It presents a simulator of mmWave radar that can simulate  the radar data as if it is placed in a real scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It supports  customized 3D models to be imported as the ground truth  and provides a framework for evaluating a 3D imaging  algorithm quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  It presents a systematical study of 3D imaging algorithms  using mmWave radars and an evaluation of the key factors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='13553v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='SP]  31 Jan 2023  2  that could affect the radar detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It highlights the  challenges of the noisy and imbalanced point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  It presents a novel SRPC algorithm that can be inserted  into the traditional point cloud construction DPC and can  improve the quality of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section II  discusses the background and related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section III in-  troduces the preliminaries of mmWave radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section IV  presents the details of the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section V discusses the  3D imaging DPC using mmWave radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section VI presents  the experimental results based on the simulation system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Section VII presents the novel SRPC algorithm and shows how  it can improve the radar detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Section VIII concludes the  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BACKGROUND  Traditionally, 3D imaging systems often use depth cameras  (like stereo cameras or RGBD cameras) [12] or laser sensors  [13], which are able to provide a dense and accurate 3D  model of the object in front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, camera-based systems  can be intrusive and limited by the lighting conditions, and  laser sensors are often constrained by their high cost (when  compared with the cost of a mmWave radar which is only  around £10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Radar-based 2D imaging has also been used  widely in applications like security, but they provide limited  depth information and often rely on a dense antenna array  that has a fixed region-of-interest [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Radar-based 3D object  detection uses radio frequency (RF) signals at certain fre-  quencies to detect objects, and the resolution of the detection  largely depends on the available bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For example,  WiFi devices operating at 5 GHz with a 40 MHz bandwidth  can locate people with sub-meter level resolution [15], and  ultra-wide band (UWB) devices at higher frequency bands  can achieve centimetre level resolution [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' With mmWave  radars operating at above 60 GHz, the range resolution can  be below 5 cm [2] and even micrometre level when pointing  to a corner reflector [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, the high resolution has  gained mmWave radars great popularity in automotive driving  applications, and researchers are actively investigating their  usage in computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Although mmWave radars can be used as 3D imaging  sensors, the point cloud is often less accurate and noisier than  the traditional systems [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Many methods have been proposed  to improve the detection quality of a mmWave radar, such as  [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, as these methods often use different radar  configurations and scene setup, it is hard to carry out a quan-  titative comparison between them, and there lacks a standard  on how to define the quality of the radar detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This work  aims to address the problem by providing a simulation system  and a framework for a systematic evaluation of a 3D imaging  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Radar-based 3D imaging requires measuring the distance  and AoA of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The distance is often measured through  the ToF of the signal, whereas the AoA measurement relies  on the use of an antenna array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Since the antennas in the array  will have different physical locations, the ToF at each antenna  will be different, and the AoA of the object can be estimated  by investigating the signal difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This process has been  studied in depth for traditional long-range radars [8], and the  same principle can be applied to mmWave radars on a smaller  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' There are many algorithms designed for estimating the  AoA based on a linearly spaced antenna array, such as the FFT-  based method, beamforming method and subspace method  (more details in Section III-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' These algorithms provide  a trade-off between the computational complexity and the  angular resolution [8], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, in contrast to traditional  radar systems where the signal sources are often well-defined  and uncorrelated, signal sources in 3D imaging can be one  object with a continuous surface, which can require a different  DPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This paper discusses the traditional AoA estimation  algorithms in the context of 3D imaging using a mmWave  radar and investigates the key factors that would affect the  detection result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' MMWAVE RADAR PRELIMINARIES  Commercial mmWave radars often implement the frequency  modulated continuous wave (FMCW) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The radar sends  a modulated chirp signal, detects the signal reflection from  any object, processes the signal and determines the range,  velocity, and AoA of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The principle of the FMCW  radar model has been documented in detail in the literature  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This section will give a brief discussion of the  fundamentals that are necessary for understanding this paper,  with a particular focus on the AoA estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The radar sends an FMCW signal and receives its reflection  from the object in the scene, where the reflection will be a  time-delayed version of the transmitted signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The two signals  are mixed to produce an IF signal, as shown in Equation (1)  (more details in Appendix A):  IF(t) = Aej(ωbt+φb) where ωb = 2πSτ, φb = 2πf0τ  (1)  where S is the slope of the chirp, τ is the ToF of the signal,  f0 is the starting frequency of the chirp, and A represents the  amplitude of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' After obtaining the IF signal, a DPC  will be applied to determine the presence of any object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Distance and Velocity Estimation  For a single object, the frequency ωb will be a constant  value and the distance of the object can be calculated as  d = τc  2 = ωbc  4πS  (2)  When there are multiple refection sources, the frequencies can  be found by applying an FFT over the IF signal, which is  referred to as the range-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The velocity can be measured by  transmitting multiple chirps at a known interval and calculating  the phase difference between the chirps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Assuming the radar  transmits a chirp every Tc seconds and a phase difference ∆φ  is observed between successive chirps, then the velocity v of  the object can be estimated as:  v =  ∆φc  4πTcf0  (3)  When there are multiple reflection sources moving at different  velocities, they can be found by applying another FFT over  the chirp phases, which is referred to as the Doppler-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1: Phase difference between two receivers from one signal  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' AoA Estimation Principle  The AoA of the object can be estimated by having multiple  antennas operating concurrently and by comparing the phase  difference between neighbouring receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Due to the spatial  location difference between the receivers, the signal received at  each receiver will have a slight phase difference depending on  the relative position of the receivers and the AoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The AoA  can be computed in both azimuth and elevation directions,  given that there exists more than one antenna in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The azimuth and elevation angles will be denoted as θa and  θe, respectively, or θ(a,e) when referring to both of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Assuming there are Na×Ne linearly spaced receivers in the  azimuth and elevation directions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' and M objects in different  directions θ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e)m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' then each object can be viewed as a signal  source and the receiving antenna array will receive a signal  (denoted as x) as a weighted sum of the M data source:  x(Na×Ne) =  M  �  m=1  αms(θ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e)m) + n  (4)  where s(θ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e)m) is the steering vector that represents the  phase difference between receivers when a signal arrives with  angle θ(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e)m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' α is an unknown parameter that models the  signal transmission from the data source to the receivers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  and n is the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The AoA estimation can be modelled as  estimating the values of θ(a,e) for each object m, given a set  of receiver data (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  For linearly spaced arrays, the receivers are often separated  by a small distance l that is equal to half of the signal wave-  length, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' l = λ  2 , to maximize the angle-of-view (AoV) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  When using an array of Na azimuth receivers and Ne elevation  receivers, each subsequent receiver beyond the first one will  receive an additional phase change that can be expressed using  a 2D steering vector (more details in Appendix B):  s(θ(a,e), Na, Ne) =  �  ���  1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=',  ej(Na−1)∆φa  ej∆φe,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=',  ej(∆φe+(Na−1)∆φa)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=',  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=',  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  ej(Ne−1)∆φe,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=',  ej((Ne−1)∆φe+(Na−1)∆φa)  �  ���  (5)  3D point cloud construction requires the x-y-z coordinates  of the object instead of the azimuth and elevation angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Therefore, the calculation of the exact value of θa and θe is  often not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Let d denote the distance of the object, then  the 3D coordinates of the object can be calculated as (more  details in Appendix B):  x = d∆φa  π  , z = d∆φe  π , y =  �  d2 − x2 − z2  (6)  Given that d can be obtained from the range-FFT as discussed  in Section III-A, the x-y-z coordinates can be obtained if the  phase differences ∆φa and ∆φe are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, the  AoA estimation of an object can be considered equivalently  as searching for the best matching steering vector s(θ(a,e)m)  of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' AoA Estimation Algorithms  In the following sections, some of the most widely-used  AoA estimation algorithms will be discussed, including the  FFT-based method, conventional beamforming (also known as  the Bartlett beamforming or the delay-and-sum beamforming),  the minimum variance distortionless response (MVDR) beam-  forming (also known as the Capon beamforming) [21], and  the multiple signal classification (MUSIC) subspace method  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The angle-FFT method is a single-snapshot method  that can make an estimate based on a single chirp, whereas  the other methods are multi-snapshot methods that require a  few chirps to make one estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The performance of the  algorithms depends on several factors, including the antenna  layout, number of antennas, chirp configuration, number of  snapshots, SNR, environment, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  1) Angle-FFT Method: The simplest way of estimating  s(θ(a,e)m) of an object m in Equation (4) is by using cor-  relation between the receiver data x and the steering vector  from the candidate angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A set of candidate steering vectors  s(¯θ(a,e)) is defined for θa ∈ [−π, π], θe ∈ [−π, π], and the  correlation is calculated as s(¯θ(a,e)) · x, which will yield a  peak output when ¯θ(a,e) equals to θ(a,e)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This process is  equivalent to applying an FFT over the receiver data x, since  the steering vector can be considered the same as a set of FFT  coefficients, which gives the frequency components in terms of  ∆φa and ∆φe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This FFT is also referred to as the angle-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  As an example, Figure 2 shows the antenna layout of the  TI IWR6843 radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It has three transmitters and four receivers,  which can form a 12-receiver array when using MIMO tech-  niques [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The phase of each virtual receiver is also shown,  where ϕ is the random initial phase of the first receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The azimuth receivers will form a signal ej(∆φan+ϕ) and the  elevation receivers will form a signal ej(∆φan+2∆φa+ϕ+∆φe),  where n is the receiver index in each direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The value  of ∆φa can be obtained by applying an azimuth-FFT over  the azimuth receivers (RX1-RX4 and RX9-RX12), which will  give the frequency ∆φa and phase ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The value of ∆φa can  be obtained by applying an FFT over the elevation receivers,  which will give the frequency ∆φa and phase 2∆φa+ϕ+∆φe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Hence, the value of ∆φe can also be calculated given ∆φa  and ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  An alternative approach to calculate ∆φa is by applying an  elevation-FFT over a set of receivers in the elevation direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  For example, Figure 3 shows the layout of the TI overhead  detection sensor (ODS) model, where the receivers form a  near-square shape and allows a 2D angle-FFT to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Object  d  Ad  0  0  ~/=入2  RXO  RX1  RXO  RX1  TX1  Approximation4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 2: IWR6843 radar antenna layout, the virtual receiver  array and the received phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The ODS models allow a higher elevation resolution at the cost  of reduced azimuth resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 3: IWR6843ODS radar antenna layout, the virtual receiver  array and the received phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  2) Beamforming Method: Beamforming methods calculate  a set of weights w(Nrx×Θ) for the Nrx virtual receivers in the  array (both azimuth and elevation), and for all possible angles  θ(a,e) ∈ Θ where θa ∈ [−π, π], θe ∈ [−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When applying  a column of weights to the receiver data x, the signal from the  direction θ will receive a constructive inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' By searching  all possible angles θ(a,e), a power spectrum p with size Θ can  be obtained, where a high power in the spectrum indicates that  there is a data source in that direction:  p = wHx  (7)  where wH is the Hermitian transposition of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The angles of  the M objects can be obtained by taking the M highest peaks  in p and finding the corresponding entries in w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In the data model shown in Equation (4), signals reflected  from objects will be correlated when being received at each  receiver, whereas the noise will be uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore,  one way to extract signal information from x is by calculating  a sensor covariance matrix Rx [8]:  Rx = E{xHx} ≈ 1  N  N  �  t=1  xH(t)x(t)  (8)  where E represents the statistical expectation and x(t) repre-  sents one snapshot (or one frame) of the receiver data x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When  evaluating the beamforming power spectrum using multiple  snapshots, the overall power spectrum becomes the statistical  expectation of p in Equation (7) over the snapshots, which  gives:  P = E{|wHx|2} = 1  N  N  �  t=1  wHx(t)xH(t)w = wHRxw (9)  Once the beamforming power spectrum is computed, the peaks  in the spectrum will correspond to the signal from the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  There are many algorithms designed for calculating the  weights w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The conventional beamforming uses the steering  vector directly as the weights, which is conceptually equivalent  to the angle-FFT method (or correlation-based method) in  Section III-C1:  Pconventional = sHRxs  (10)  where s is the candidate steering vector in the format of  Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  There are also adaptive beamforming algorithms that calcu-  late the weights using the signal information embedded in the  covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the MVDR algorithm aims at  minimizing the variance from non-interested directions while  keeping the signal from the candidate direction distortionless  [21]:  Pmvdr =  1  sHR−1  x s  (11)  3) Subspace Method: The core of the subspace method is  that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' since the signal x should contain M correlated signals  and uncorrelated noise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the covariance matrix Rx should have  M non-zero eigenvalues and N − M zero eigenvalues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' where  N is the rank of Rx that is equal to the number of receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The eigenvectors corresponding to the M eigenvalues form the  signal subspace, and the eigenvectors corresponding to the zero  eigenvalues form the noise subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The signal subspace and  the noise subspace are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' One of the most widely-  used subspace-based algorithms is the MUSIC algorithm [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  It searches for steering vectors that are orthogonal to the noise  subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The power spectrum of the MUSIC algorithm can  be written as:  Pmusic =  1  sHUU Hs  (12)  where U is the set of eigenvectors corresponding to the zero  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' MMWAVE RADAR SIMULATOR  A simulator is designed to verify the discussed algorithms  and evaluate the theoretical capability of using a mmWave  radar as a 3D sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The simulator simulates mmWave radars  with one transmitter and one receiving antenna array, which  is practically equivalent to a multi-transmitter multi-receiver  radar using an appropriate modulation scheme [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Any two  neighbouring receivers in the array are separated by λ0/2,  where λ0 (approximately 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9 mm) is the wavelength of the  mmWave signal at its chirp starting frequency (77 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The simulator simulates the IF signal at each receiver of a  mmWave radar when pointing toward a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The scene is  modelled to have M points, where each point has a unique x-  y-z coordinate and represents the spatial location of the object  in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Each point is modelled as a corner reflector and  reflects the mmWave signal sent out by the radar with the  same reflectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The reflection area of the object is modelled  by the number of points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' a large object would have a  higher number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The IF signal at a receiver during  one chirp is modelled using Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Given a certain chirp  Virtual Antenna Array  Receivers  Transmitters  Elevation  RX5  RX6  RX7  RX8  TX2  Antennas  TX3  TX1  2  RX2 RX3 RX4  RX1  X/2  Azimuth  RX4  RX1  RX2  RX3  RX9  RX11  RX12  RX10  Antennas  V2  Received Phase  (5Apa+  340a+  24pa+  44g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='+  Elevation  (p+Ape  p+Ape  p+Ape  p+△pe  Antennas  C  07  54ba+  340a+  24ga+  6Apa+  7Apa+  (4△pa+  Azimuth  Apa+p  Antennas  CIEVirtual Antenna Array  Received Phase  Transmitters  Receivers  b)  p-  RX4  △pa  RX2  RX6  RX8  24ba  3△ba  RX2 RX4  TX2  TX1  RX1 RX3  入/2  入/2  d  d)  p-pA  2△A  X3  3△ΦA  RX1  RX3  RX5  RX7  Ade  Ape  Ade  Dde  N/2  N/2  V2  入/2  p-LpA  RX10  RX12  2pe  24e  N/2  β-△A  R212O  RX9  RX11  3△be  34be  25  configuration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the frequency and phase of the IF signal from  one point are determined by the distance d between the point  and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The amplitude of the IF signal is set to be  inversely proportional to d4, to simulate the power loss due  to distances according to the radar range equation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  final IF signal at a receiver is the accumulated IF signals from  all M points in the scene, with an additional white Gaussian  noise n, as shown in Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IF(t) =  M  �  i=1  1  d4  i  ej(2πSτi·t+2πf0τi) + n  (13)  where τi is the ToF of the signal from the transmitter to  the point i and then to the receiver, and S is the slope of  the chirp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The amplitude of the noise n is controlled by  the desired SNR during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The signal IF(t)  is sampled into a digital signal of length Ns, where Ns =  (duration of the chirp) × (ADC sampling rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' During one  chirp, the radar receives a signal that can be represented as  a 2D matrix of size Nrx × Ns, where Nrx is the number of  receivers in the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' One frame includes Nc chirps that form  a 3D matrix of size Nrx ×Nc ×Ns, which becomes the input  matrix of the point cloud construction algorithm, as shown as  the input block in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The design of the simulation system makes two assump-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' First, the multipath effect is not considered in this  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' While the multipath effect is a long-standing issue that  can cause power fading and ghost targets, it highly depends  on the scene and the reflectivity of the objects and is hard to  incorporate in the model, so it is left as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Second,  a practical radar often uses multiple transmitters and receivers  and an appropriate signal modulation scheme to separate the  signal from different transmitters, such as time demultiplexing  modulation and binary phase modulation [1], to achieve an  equivalent single-transmitter-multi-receiver system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The simu-  lation system assumes a perfect signal modulation scheme for  this purpose and ignores any error or SNR loss that may be  introduced during the modulation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' POINT CLOUD CONSTRUCTION ALGORITHM  The construction of a point cloud takes an input matrix of  size Nrx × Nc × Ns and outputs a 2D matrix PCK of size  K × 3 (referred to as the output point cloud), where K is  the number of detected points and 3 is the x-y-z coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  This section studies one of the most common DPCs used on  mmWave radars and its variant, which have shown success in  many HAR systems, like in [4], [6], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Data Processing Chains  Two DPCs are implemented that differ in using a Doppler-  FFT or not, as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Both DPCs require a  range-FFT over the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The range-FFT identifies the  frequency components in the IF signal that correspond to the  distance of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It transforms the input matrix X of size  Nrx ×Nc ×Ns into a range matrix R of size Nrx ×Nc ×N ∗  s ,  where N ∗  s is the length of the range-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The first DPC applies  a Doppler-FFT on the data from all the chirps and generates  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 4: Two possible DPCs for mmWave radar point cloud  construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  a Range-Doppler heatmap of size Nrx × N ∗  c × N ∗  s , where  N ∗  c is the length of the Doppler-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Then, it searches for  peaks in the Range-Doppler heatmap (using the average of  all receivers), extracts the receivers’ data for each peak and  generates a 2D matrix of size K × Nrx, where K is the  number of detected peaks and, equivalently, the number of  detected points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A constant false alarm rate (CFAR) algorithm  is used for detecting peaks from the Range-Doppler heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The parameters of the CFAR control the sensitivity of the  peak detection and are considered the hyperparameters of the  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Finally, a single-snapshot AoA estimation is applied  to each point in the matrix for a total of K times, to obtain the  x-y-z coordinates of all detected points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The AoA estimation  algorithm can be any of the angle-FFT, beamforming or  subspace methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Although the beamforming and subspace  methods are multi-snapshot algorithms, the Doppler-FFT im-  plicitly uses the information from all chirps and allows a good  estimate of the covariance matrix at the AoA estimation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The second DPC does not include a Doppler-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Instead, it  considers the chirps as different snapshots and performs one  multi-snapshot estimation for each range bin for a total of  N ∗  s times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' More specifically, the input range matrix of size  Nrx ×Nc ×N ∗  s is re-arranged into N ∗  s instances of Nrx ×Nc  matrix, and the AoA estimation is applied to each Nrx × Nc  matrix using Nc snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The AoA estimation algorithm  can be any of the beamforming or subspace methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Finally,  the points detected at each range bin are concatenated into  one point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In this research, the angle-FFT, conventional  beamforming, MVDR beamforming and MUSIC subspace  methods described in Section III-C are being studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Model Order Estimation  As described in Section III-C2 and Section III-C3, the  beamforming and subspace methods include an angle power  spectrum computation step, where each peak in the spectrum  corresponds to an incoming signal from a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However,  in both DPCs, the expected number of incoming signals will  be unknown in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, this number needs to  be estimated from the signal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This step is referred to  as model order estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For this purpose, the covariance  matrix of the signal data and its eigenvalues are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' As  Data Processing Chain 1  口  Nc  Average all rx +  Extract data for  口  Input  peak detection  each peak  7  Ns  Chirp 1  Doppler-FFT at  Nc  each distance  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' :  Re-arrange  K single-  Range-Doppler  K  snapshot AoA  Nc  Nc  Chirp Nc  Heatmap  Nx  estimations  Ns  NX  INX  N  Nx}  Ns  +  Data Processing Chain 2  Range-FFT  indno  Ns  on each chirp  用  #  #  K  Nc  Range FFT 1  Re-arrange  Re-arrange  Nrx  Nrx  Nc  X-y-Z  Nc  Ns multi-snapshot AoA estimations  Range FFT Nc  + peak detection  Nx  Ns  Ns  用  用  曲  K1  Ks  4  Concatenate6  (a)  (b)  (c)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 5: Three approaches when searching for the steering  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' (a) An azimuth search (red) followed by an elevation  search (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' (b) A full 2D azimuth-elevation search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' (c) A  2D azimuth-elevation search using sub-grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  described in Section III-C3, the covariance matrix should have  a size of Nrx × Nrx and has a full rank equal to Nrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' There  should be M large eigenvalues that correspond to the number  of incoming signals and Nrx − M zeros corresponding to  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In practice, due to the presence of noise, the difference  between these eigenvalues may not be significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore,  the minimum descriptive length (MDL) algorithm [25] is used  for estimating the value of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It fits a statistical model using  the eigenvalues and searches for the optimal value of M that  minimizes a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The MDL algorithm is used in both  DPCs to estimate the number of incoming signals in the AoA  estimation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Once the angle power spectrum is calculated,  all the local maxima will be found and the largest Mmdl peaks  will be taken as the output, where Mmdl is the value found  from the MDL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Steering Vector Searching  The beamforming and subspace methods search for the  steering vectors that maximize a power function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This process  can be carried out using three approaches: an azimuth search  followed by an elevation search, a 2D azimuth-elevation search  or a 2D search using sub-grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' An example of the three  approaches is shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In the example, the power  spectrum shows the incoming direction of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  space of the spectrum is sampled into a 17×17 grid and each  vertex on the grid represents a candidate AoA to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In the first approach, an azimuth AoA search is performed  using the data from azimuth receivers and steering vectors that  only consider the azimuth angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Then, based on the azimuth  AoA output, a secondary search is performed in the elevation  direction using the data from all receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This approach has  the least computational cost (34 searches in the example), but  the performance can be suboptimal as the azimuth search may  not cover the actual AoA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The second approach performs a 2D  search that considers all possible combinations of the azimuth  and elevation directions and uses data from all receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It  is computationally expensive (289 searches in the example)  but provides the most accurate estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The third approach  defines several levels of grids and performs the AoA search  at different granularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It starts the searching with a sparse  grid, finds the peaks, defines a denser grid around each peak  and performs the next search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The process can be performed  iteratively until the desired resolution is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It reduces  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 6: Some examples of the mesh models and point clouds  from the FAUST dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  the computational cost of the second approach significantly as  it skips certain regions in the spectrum (50 searches in the  example), at the cost of the potential possibility of missing  some peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' EVALUATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Dataset  The FAUST dataset [26] was used to serve as the ground  truth for the simulator, to evaluate the point cloud construction  algorithms described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The datasets contain human models in  the form of watertight triangulated meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The meshes were  generated from a high-resolution camera system containing  stereo cameras, RGB cameras and speckle projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  FAUST dataset contains 10 subjects and 30 static postures  per subject, of which 10 postures are provided with aligned  watertight models, giving 100 models in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In the simulation, the models were placed at 2 m from the  radar and facing towards the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The height of the radar  was set to be in the middle of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A ground truth  point cloud was constructed from each model by randomly  sampling M points from the surface of the mesh model,  where each point was assumed to be a corner reflector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Some  examples of the mesh models and point clouds are shown in  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The simulator computed a signal matrix for each  point cloud to simulate the IF signal that would be received  by the radar when placed towards a subject, as described by  Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The entire dataset containing the 100 models  was split into 80 training data and 20 test data, where the  training data was used for hyperparameters searching in the  point cloud construction algorithms, and the test data was used  for evaluating the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  When generating the IF signal matrix, there are two sources  of randomness: the noise term n introduced in Equation (13)  and the random sampling of the ground truth point cloud  from the mesh model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, all the evaluation processes  were repeated 10 times for each mesh model and the average  metrics were reported, to minimize any potential effect of the  randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Evaluation Metrics  To evaluate the quality of the point cloud constructed by  an algorithm, it is necessary to define the evaluation metrics  for comparing the output point cloud against the ground truth  point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Let PCM denote the ground truth point cloud and  PCK denote the point cloud generated by the radar, which are  +  C  +  +  DO  C  O  O  O  O  O  O  O7  a M ×3 matrix and a K×3 matrix, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It is important  to note that the point cloud construction algorithm can provide  an uncertain number of points that might be different to the  ground truth (M ̸= K), and PCK can have a non-uniform  distribution while PCM is distributed uniformly on the mesh  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The evaluation metrics should take the two point clouds  PCM and PCK as input and measure the similarity between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' First, two points are defined to be close to each other  if their Euclidean distance is less than a certain distance D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In this research, D is set to 10 cm as an empirical estimation  of the error tolerance of a HAR system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Then, the following  terms and metrics are defined:  Precision: Number of points in PCK that has at least one  close point from PCM, divided by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It evaluates how  many points in PCK are considered to be accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Sensitivity/Recall: Number of points in PCM that has  at least one close point from PCK, divided by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It  evaluates how well PCK can cover the space of PCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fowlkes–Mallows index (FMI): the geometric mean of  precision and sensitivity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' √precision × sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Intersection over Union (IoU): Establish two regular 3D  voxel grids for PCK and PCM with the voxel size set to  10 cm × 10 cm × 10 cm, consider a voxel to be occupied  if there is at least one point present in the voxel, then the  IoU is calculated as the number of overlapping voxels  of the two voxel grid, divided by the union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The IoU  evaluates the similarity of the two point clouds at the  granularity of the voxel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  An ideal system should have both high precision and high  sensitivity, whereas the relative importance of the two depends  on the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In this section, the FMI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the geometric  mean of precision and sensitivity, is used to indicate the  performance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The IoU also provides a good  indication of how the generated point cloud can represent  the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, as the calculation of the IoU is highly  sensitive to the voxel size and outliers, it is used as a secondary  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Data Processing Chain and Algorithms  In the first experiment, the two DPCs combined with  different AoA algorithms were evaluated and compared, in  terms of the quality of the estimated point cloud and the  computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A baseline radar and scene configuration  were designed to approximate a typical setup in a common  indoor environment as follows:  One transmitter and a 4 × 4 uniform receiver array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The chirp frequency is 77 GHz to 81 GHz, the slope  is 40 MHz/us, the chirp duration is 100 us, the ADC  sampling rate is 15 MHz, each frame is 50 ms with 50  chirps, and each chirp has 1500 samples (as shown in  Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Each human mesh model is sampled into 512 points and  placed at 2 m away from the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  SNR is 30 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The subject has a velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s moving away from  the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 7: Chirp configuration of one frame in the baseline setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  TABLE I: FMI (standard deviation in parentheses) comparison  between the algorithms when using a 4 × 4 receiver array and  a subject velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  FMI in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5)  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9)  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6)  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6)  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7)  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  DPC2  NA  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8)  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1)  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6)  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4)  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4)  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  The AoA algorithm uses 512 bins to cover the ±90° AoV,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the angular resolution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='35°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The velocity of the subject is introduced following the  assumption that a real person cannot stay absolutely stationary  during the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' At a velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s and a frame  time of 50 ms, the total displacement will be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 mm and is  considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The velocity provides a variation on  the signal received at different chirps, as otherwise the multi-  snapshot AoA estimation algorithms would receive an identi-  cal signal at all chirps and would yield a poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Combining the two DPCs with different AoA estimation  algorithms, there are 14 methods in total to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  For each method, both the 1D search approach and the 2D  sub-grid approach described in Section V-C are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For  the 2D angle-FFT method, the full-grid approach is used  instead of the sub-grid approach, since the benefit of the  lower computational cost is less significant for FFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  algorithms will be referred to using the format “DPC-Method-  1D/2D” throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' For example, DPC1-Conv-2D  refers to the conventional beamforming method in DPC1 that  uses a 2D steering vector search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The angle-FFT method is  not applicable in DPC2 as it is not a multi-snapshot algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Algorithms in DPC1 include a CFAR peak detection step on  the Range-Doppler heatmap, where the optimal parameters for  the CFAR were searched on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Then, the  performance of the algorithms on the test dataset was evaluated  and compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The results are shown in Table I and Table II  as FMI and IoU (in % and with the standard deviation in  parentheses), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  TABLE II: IoU (standard deviation in parentheses) comparison  between the algorithms when using a 4 × 4 receiver array and  a subject velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IoU in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4)  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5)  DPC2  NA  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3)  Frequency  (GHz)  Chirp 0  Chirp 1  Chirp 2  Chirp 49  81  77  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  49  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  50  <  Time (ms)  1500 ADC samples  50 x 1500 samples per receiver per frame8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 8: Examples of the radar detection using the different  algorithms, when using a 4 × 4 receiver array and a subject  velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  There are a few important observations from the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Even though the subject had a low velocity, the DPC1 with a  Doppler-FFT outperformed the other significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' One main  reason is that, as the number of receivers is much lower than  the number of signals, the AoA estimation algorithm can fail to  distinguish points with a close angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Instead, these points will  be identified as one strong signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' On the contrary, the  CFAR peak detection step in DPC1 picks a set of points around  the peak that are above the CFAR threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' As these points  also contribute to the point cloud, the output becomes denser  and the sensitivity is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' This effect can be observed  from the example detection shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In terms of the different algorithms, the MVDR and MU-  SIC methods outperformed the angle-FFT and conventional  methods, at the expense of higher complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Meanwhile, all  the 2D methods outperformed the 1D methods due to a more  fine-grained resolution (as shown earlier in Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  best performance was achieved with the DPC1-MVDR-2D  and DPC1-MUSIC-2D methods, with an FMI of 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5% and  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, the IoU metrics show that the  point clouds were still far from the objective of high-accuracy  scene reconstruction, as the highest IoU was only 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It  can be seen from Figure 8 that, while the distribution of the  point cloud mostly fitted the subject, the distribution was not  even and there were body parts (like the hands) that received  fewer points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, there is still a big gap before the radar  output can be directly used by applications that require high  quality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Table III compares the algorithms in terms of execution  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The algorithms were run using the same dataset and  parameters multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The algorithms were written in  Python without any processor-specific optimization and were  run on one Intel i7-9700K CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The result is shown  as the relative execution time of each algorithm when com-  pared with the DPC1-FFT-1D method (the most lightweight  method) and normalized with the number of detected points,  to give an indication of their relative complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' All the  2D methods have a higher complexity than the 1D methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  For algorithms in DPC1, the 1D angle-FFT method has the  TABLE III: Normalized execution time comparison between  the algorithms using the baseline setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Normalized  Complexity  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='00  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='42  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='38  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='51  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='99  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='02  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='85  DPC2  NA  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='69  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='38  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='31  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='89  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='67  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='61  TABLE IV: Relative FMI difference of the algorithms when  using a 4 × 4 receiver array and a subject velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 m/s  in comparison to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  FMI in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  +11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  +7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  DPC2  NA  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  lowest computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' With the sub-grid optimization, the  complexity of the 2D beamforming and MUSIC methods can  be kept at around twice the 1D methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The complexity  without the sub-grid optimization is expected to be much  higher, as can be estimated from the difference between the  2D and 1D angle-FFT methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When considering both the  complexity and the performance, the DPC1-FFT-1D method  provides a good trade-off between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The MVDR methods  and MUSIC methods in DPC1 give the best performance at  the cost of 9x execution time and require additional efforts  on the hardware and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It is worth noting that  many mmWave radar systems, like [6], [7], [27], are built  based on the DPC1-FFT-1D method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, these systems  can potentially benefit from a more complex AoA estimation  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Subject Velocity  The motion of the subject being sensed has a significant  impact on the detection output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In DPC1, a higher velocity  makes a subject easier to be identified in the Range-Doppler  heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Due to the relative position difference between the  body parts of the subject, they will have a different radial  velocity with respect to the radar, making them distinguishable  in the Range-Doppler heatmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In DPC2, a higher velocity  increases the variance of the signal between chirps and allows  a better estimate of the data covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' To verify  the theorem, an experiment was carried out using the same  configuration as the baseline setup, except that the velocity of  the subject was set to different values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1 m/s to 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The ground truth point cloud was taken as the average position  of the subject during the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Table IV and Table V show two examples of the experiment  where the subject velocity was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 m/s and 1 m/s,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When compared with Table I, all algorithms  achieved a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6% to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5% improvement in terms of the FMI  when the subject had an increased velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Figure 9 shows the  FMI and IoU of the DPC1-MUSIC-2D method with different  subject velocities from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1 m/s to 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' An overall positive  correlation can be observed between the subject velocity and  the detection performance, and the impact is the most obvious  at lower velocities (around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Some examples of the  detection at 1 m/s are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Conventional  MVDR  MUSIC  FFT  Beamforming  Beamforming  1D  2D  1D  2D  1D  2D  1D  2D  Front View  DPC1  Left View  Front View  DPC2  Left View9  TABLE V: Relative FMI difference of the algorithms when  using a 4 × 4 receiver array and a subject velocity of 1 m/s  in comparison to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  FMI in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  +14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  +11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  DPC2  NA  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  +9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 9: FMI and IoU (with errors) of the DPC1-MUSIC-2D  algorithm with different subject velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' SNR  In a practical environment, a radar system can experience  noise from different sources, such as the thermal noise of the  radar chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The SNR also depends on the distance between  the radar and the subject, as the signal power drops quickly  along with the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In the simulator, the SNR can be  controlled by the power of the noise term n in Equation (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In this section, the performance of the algorithms between a  high SNR environment (40 dB) and a lower SNR environment  (5 dB) is compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Two experiments were carried with the  subject velocity set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 m/s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  results are shown in Table VI and Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  In the low SNR environment, all the algorithms in DPC1  experienced a similar drop in performance, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' How-  ever, the algorithms in the DPC2 showed a higher perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The reason is that the higher noise affected the model  order estimation step and the system tends to report a higher  number of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Taking the DPC2-Conv-2D method as an  example, the average size of the detected point cloud was  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 10: Examples of the radar detection using the different  algorithms, when using a 4 × 4 receiver array and a subject  velocity of 1 m/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  TABLE VI: Performance difference when using a 4 × 4  receiver array and a subject velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='05 m/s in a low  SNR environment (5 dB in comparison to 30 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  FMI in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  DPC2  NA  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  TABLE VII: Performance difference when using a 4 × 4  receiver array and a subject velocity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5 m/s in a low SNR  environment (5 dB in comparison to 30 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  FMI in %  Angle-FFT  Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' BF  MVDR BF  MUSIC  1D  2D  1D  2D  1D  2D  1D  2D  DPC1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  DPC2  NA  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  found to be 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3% higher in a low SNR environment than in a  higher SNR environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, this was still insufficient  to reach a similar performance as DPC1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Antenna Layout  Theoretically, the antenna layout determines the angular  resolution that an AoA estimation algorithm can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  more receivers in one direction, the higher resolution the radar  can measure [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, this is questionable when the signal  sources are spatially close and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Meanwhile, having  more antennas also increases the cost of the hardware, as more  circuit components, processing units and memory would be  required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, it is beneficial to study the relationship  between the antenna layout and the output quality and find  the optimal trade-off for an application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Common commercial mmWave radars use up to three  transmitters and up to four receivers, giving up to twelve  virtual receivers as a receiving array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Some radar models are  designed for automotive applications and prioritize the azimuth  direction, while others are designed for general purpose appli-  cations and have a similar resolution in both the azimuth and  elevation directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In this section, common antenna layouts  implemented on the TI radars are evaluated and compared,  as well as a few square-shape antenna layouts that are more  common in research projects, as listed in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The same  radar configuration and scene setup in Section VI-C were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The experiment compares the antenna layouts using the DPC1-  MUSIC-2D algorithm (the best performing algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  result is shown in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 11: The list of receiver layouts being evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' (a)-(d) are  square antenna arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' (e)-(f) are non-regular antenna arrays  implemented on TI radars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  FMI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  Performance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Subject velocity (m/s)Conventional  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='MVDR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='MUSIC  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='FFT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Beamforming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Beamforming  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Front View  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='DPC1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Left View  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Front View  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='DPC2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='Left View0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(a) 3x3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(b) 4x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(c) 6x6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(d) 8x8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='O  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(e) IWR6843A0P  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(f) AWR1843AOP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='(g) IWR184310  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='TABLE VIII: Performance comparison between different an-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='tenna layouts using the baseline configuration and the DPC1-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='MUSIC-2D algorithm (standard deviation in parentheses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Antenna  Layouts  a  b  c  d  e  f  g  FMI  in %  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5)  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='3)  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9)  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  IoU  in %  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='4  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1)  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8)  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='7)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 12: Examples of the radar detection using the different  antenna layouts with the baseline setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  It can be seen that most antenna layouts had similar perfor-  mance, except the layout (g) which had a worse performance  as it is designed for automotive applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The layout (e)  has a non-uniform antenna distribution that slightly affected its  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' All other layouts showed a similar performance  regardless of the antenna size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, considering the  increased hardware cost and computational cost of increasing  the number of antennas, a small antenna size can be preferable  for 3D sensing applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Figure 12 shows some examples  of the detection using different antenna layouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Chirp Configuration  The chirp configuration can have various effects on the  distance detection and velocity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' These factors can  indirectly affect the quality of the final point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In this  section, three different chirp configurations were tested and  compared against the baseline configuration in Section VI-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The details of the three configurations (named A, B and C) and  the performance are shown in Table IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Each configuration  has certain parameter cut to 80% to evaluate the effect on  the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Configuration A had an 80% reduced chirp slope  and, hence, a reduced effective bandwidth from 4 GHz to  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Configuration B had an 80% reduced ADC sampling  rate that reduced the samples per chirp from 1500 to 1200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Configuration C had an 80% reduced number of chirps per  frame, from 50 to 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' All other parameters were kept the same  as the baseline with the DPC1-MUSIC-2D algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The result shows that the performance can be strongly  affected by the effective bandwidth and the number of chirps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The former affects the distance resolution of the detection,  and the latter affects the Doppler resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Reducing either  of these parameters reduces the accuracy of the range-Doppler  heatmap and the estimation of the covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' On the  other hand, the effect of reducing the ADC sampling rate and  the number of samples per chirp is much less significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  TABLE IX: FMI (standard deviation in parentheses) com-  parison between four chirp configurations using the DPC1-  MUSIC-2D algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Chirp Configuration  Baseline  A  B  C  Slope of the chirp (MHz/us)  40  32  40  40  ADC sampling rate (MHz)  15  15  12  15  Chirps per frame  50  50  50  40  FMI in %  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2)  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='8)  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='0)  (a) Points detected without SRPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  (b) Points detected with SRPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 13: Using SRPC algorithm to improve the resolution and  distribution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' SUPER-RESOLUTION POINT CLOUD CONSTRUCTION  ALGORITHM  It can be seen from Figure 8 and Figure 10 that the  constructed point clouds can be noisy and the distribution of  the points can be imbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' One major reason is that the  point cloud construction relies on the peak detection result  over the range-Doppler-FFT spectrum, so the distribution of  the points will be limited by the resolution of the FFT, and the  points will have a discrete distribution in the range domain (as  the curve-like data from the left view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Although it is possible  to improve this resolution, such as zero padding the data before  applying the FFT, it would also increase the computational cost  and memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Meanwhile, there are false detected  points due to the outliers from the peak detection stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' To  address the mentioned issue and improve the quality of the  constructed point cloud, a novel super-resolution point cloud  construction (SRPC) algorithm is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The SRPC algorithm aims to improve the distribution of  the point cloud and make it span more naturally in the spatial  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The rationale is shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When detecting  peaks in a range-Doppler spectrum or an angle spectrum,  a common approach is taking all points above a static or  dynamic threshold, where the distribution of the points is  limited by the resolution of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' An example of  this effect is shown in Figure 13a, where the grid represents  the resolution of the data and all the detected points must  fall on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The SRPC algorithm aims to return a set  of points that have a higher resolution than the original data  and fall more naturally on the distribution curve, as shown in  Figure 13b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The algorithm can be broken down into the following  steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' First, the power spectrum is upsampled into the desired  resolution using linear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Then, for each of the  originally detected points i ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='.K], the algorithm randomly  samples ni points around it with a probability distribution  Front View  Left View  (a)  (b)  (c)  (d)  (e)  (f)  (g)  3x3  4x4  6x6  8x8  IWR6843AOP  AWR1843AOP  IWR1843Power spectrum  Threshold  Detected pointsPower spectrum  Threshold  Detected points11  being the amplitude of the upsampled power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  value of ni is calculated as:  ni = pi · αSRP C  th  (14)  where pi is the power of the point, th is the threshold of the  peak detection algorithm, and αSRP C is a global hyperpa-  rameter that controls the aggressiveness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  term pi ensures that a point with higher power will be sampled  into more points, as the power indicates the confidence that a  point can represent a real signal source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The parameter αSRP C  amplifies the importance of pi, where a higher αSRP C pushes  the distribution of the points towards the peak of the spectrum  and gives a more dense distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The sampling process is  repeated for each point i to form a new point list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Finally, K  points (the population of the original detection) are randomly  selected from the new point list, so that the total number of  detected points is kept the same and the computational cost  of the rest of the system is not affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Since the algorithm  tends to sample more points at higher power, the distribution  of the final points will also tend to be around higher powers,  and, hence, gives a more natural distribution regarding the  power spectrum and overcomes the limitation of the original  data resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The time complexity of the SRPC algorithm  is approximately O(K · ni), where a typical value of ni can  fall between 2 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  When constructing the point cloud, the SRPC is applied  when detecting peaks from the range-FFT spectrum and de-  tecting peaks from the angle spectrum in the AoA estimation  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The former improves the data distribution in the range  domain and eliminates the curve-like effect when looking at  the point cloud from the left view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The latter improves the  data distribution in the angle domain so that the points tend to  span into the space rather than appearing as a dense cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Meanwhile, since the points will be distributed around higher  powers, the probability of outliers will be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  To evaluate the proposed SRPC algorithm, it was inserted  into the DPC1-FFT-1D and DPC1-MUSIC-2D methods men-  tioned in Section VI-C when using the baseline setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The two  methods were chosen as they represent the most lightweight  algorithm and the most accurate algorithm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Since  the SRPC is likely to produce point clouds with different  sizes and to ensure a fair comparison, a fixed number of 512  points were randomly taken from the point cloud generated  by each algorithm for the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The result is shown in  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' After applying the SRPC algorithm, the distribution  of the point cloud appeared to be more natural and better  distributed around the ground truth, and the outliers in the  original detection were reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A quantitative evaluation is  shown in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The performance without SRPC dropped  slightly when compared with Table I because the output  size was forced to be 512, but both metrics have improved  after applying SRPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, it is shown that the SRPC  algorithm can successfully improve the data point distribution,  reduce the outliers and produce a more natural point cloud  that can be potentially preferable for higher-level applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Future work of this research includes an efficient hardware  implementation of this algorithm using the radar on-chip  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 14: Examples of point clouds constructed with and  without the SRPC algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  TABLE X: Performance comparison of two algorithms with  and without SRPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  DPC1-FFT-1D  DPC1-MUSIC-2D  FMI  IoU  FMI  IoU  Without SRPC  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='2  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  With SRPC  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='5  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='6  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='9  processors so that it can be further verified in real-world  scenarios, as well as an evaluation of its effectiveness in  higher-level applications like posture estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' CONCLUSION  In this paper, a mmWave radar simulator is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  system is used to evaluate the ability of the mmWave radar as  a 3D imaging sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A mmWave radar dataset is constructed  using the FAUST dataset as the ground truth to provide 3D  mesh models of human subjects, from which mmWave radar  IF signals are simulated and used to evaluate different point  cloud construction algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The FMI and IoU metrics are  defined to evaluate the quality of the generated point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The evaluation is performed regarding a set of different factors,  including the DPCs, AoA estimation algorithms, subject veloc-  ity, SNR, antenna layout and chirp configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It was found  that the DPC combining a range-Doppler-FFT and a single-  snapshot AoA estimation algorithm gives better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Among all the AoA estimation algorithms, the angle-FFT  method gives a good trade-off between high performance and  low computational cost, whereas the more advanced AoA  estimation algorithms, like MVDR and MUSIC, give the best  performance at up to 9x higher execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The velocity  of the subject helps significantly in the detection, as the  algorithms are better at detecting a moving subject than a  stationary object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When comparing common antenna layouts,  large square antenna arrays give the best performance, but the  advantage is not significant in a 3D imaging application when  the data sources are spatially close and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It is shown  that the performance of the point cloud detection benefits from  higher effective bandwidth and a higher number of chirps  per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Finally, a novel SRPC algorithm is proposed for  improving the resolution and distribution of the point cloud  and reducing the probability of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The algorithm applies  DPC1-FFT-1D  DPC1-MUSIC-2D  Front View  Left View  Front View  Left View  Before SRPC  After SRPC12  to the range-Doppler-FFT peak detection stage and the AoA  estimation stage and detects points at a higher resolution that  fits the power spectrum better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When evaluating the algorithm  using the simulation system, it has been shown that the  algorithm can successfully improve the data distribution and  produces a more natural point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Rao, “MIMO radar,” Texas Instruments, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [2] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Iovescu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Rao, “The fundamentals of millimeter wave sensors,”  Texas Instruments, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zhao, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wei, “Concealed object  detection for activate millimeter wave image,” IEEE Transactions on  Industrial Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 66, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 9909–9917, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [4] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zhao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Trigoni, and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Markham, “mid: Tracking and identifying people with millimeter  wave radar,” in 2019 15th International Conference on Distributed  Computing in Sensor Systems (DCOSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 33–40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [5] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Cui and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Dahnoun, “High precision human detection and tracking  using millimeter-wave radars,” IEEE Aerospace and Electronic Systems  Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 22–32, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Sengupta, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zhang, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Cao, “mm-pose: Real-time human  skeletal posture estimation using mmwave radars and cnns,” IEEE  Sensors Journal, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Cui and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Dahnoun, “Real-time short-range human posture estima-  tion using mmwave radars and neural networks,” IEEE Sensors Journal,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 22, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 535–543, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [8] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Gokeda, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Yu, Introduction to Direction-of-arrival  Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Artech House, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Dudek, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Kissinger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Weigel, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Fischer, “A millimeter-  wave fmcw radar system simulator for automotive applications including  nonlinear component models,” in 2011 8th European Radar Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2011, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 89–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [10] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Stetco, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Ubezio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' M¨uhlbacher-Karrer, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zangl, “Radar  sensors in collaborative robotics: Fast simulation and experimental  validation,” in 2020 IEEE International Conference on Robotics and  Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 10 452–10 458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Sch¨offmann, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Ubezio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' B¨ohm, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' M¨uhlbacher-Karrer, and  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zangl, “Virtual radar: Real-time millimeter-wave radar sensor sim-  ulation for perception-driven robotics,” IEEE Robotics and Automation  Letters, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 4704–4711, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [12] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Han, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Shao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Shotton, “Enhanced computer vision with  microsoft kinect sensor: A review,” IEEE transactions on cybernetics,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1318–1334, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [13] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Raj, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Hashim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Huddin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Ibrahim, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Hussain,  “A survey on lidar scanning mechanisms,” Electronics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 9, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  741, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Appleby, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Robertson, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wikner, “Millimeter wave imaging:  a historical review,” in Passive and Active Millimeter-Wave Imaging XX,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 10189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  SPIE, 2017, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1018902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Kotaru, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Joshi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Bharadia, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Katti, “Spotfi: Decimeter level  localization using wifi,” in Proceedings of the 2015 ACM Conference  on Special Interest Group on Data Communication, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 269–282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [16] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Witrisal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Hinteregger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Kulmer, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Leitinger, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Meissner,  “High-accuracy positioning for indoor applications: Rfid, uwb, 5g, and  beyond,” in 2016 IEEE International Conference on RFID (RFID).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [17] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Ikram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Ahmad, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Wang, “High-accuracy distance mea-  surement using millimeter-wave radar,” in 2018 IEEE Radar Conference  (RadarConf18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1296–1300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [18] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Cao, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Xu, “3drimr: 3d recon-  struction and imaging via mmwave radar based on deep learning,” in  2021 IEEE International Performance, Computing, and Communications  Conference (IPCCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  IEEE, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [19] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Su, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Jiang, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Liu, “A novel radar point cloud gen-  eration method for robot environment perception,” IEEE Transactions  on Robotics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [20] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Gentilho, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Scalassara, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Abr˜ao, “Direction-of-arrival estima-  tion methods: A performance-complexity tradeoff perspective,” Journal  of Signal Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 92, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 239–256, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Capon, “High-resolution frequency-wavenumber spectrum analysis,”  Proceedings of the IEEE, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 57, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 1408–1418, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Schmidt, “Multiple emitter location and signal parameter estimation,”  IEEE Transactions on Antennas and Propagation, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  276–280, 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Barton, “Modern radar system analysis,” Norwood, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [24] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Sandha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Garcia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Srivastava, “Radhar:  Human activity recognition from point clouds generated through a  millimeter-wave radar,” in Proceedings of the 3rd ACM Workshop on  Millimeter-wave Networks and Sensing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  ACM, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 51–  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Rissanen, “Modeling by shortest data description,” Automatica,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 465–471, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [26] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Bogo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Romero, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Loper, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Black, “FAUST: Dataset and  evaluation for 3D mesh registration,” in Proceedings IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' on  Computer Vision and Pattern Recognition (CVPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Piscataway, NJ,  USA: IEEE, Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Sandha, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Garcia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Srivastava, “Radhar:  Human activity recognition from point clouds generated through a  millimeter-wave radar,” in Proceedings of the 3rd ACM Workshop on  Millimeter-wave Networks and Sensing Systems, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 51–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  13  APPENDIX A  IF SIGNAL  In a typical FMCW radar model, the transmitter sends a  chirp signal Stx (a signal with frequency increasing linearly  with time) to detect any object in front of the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When  Stx is reflected by the object, the signal is received as Srx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Assuming the signal has an initial frequency f0 and a slope  of S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' then the frequency of Stx is a function of t:  ftx(t) = f0 + S · t  (15)  The instantaneous phase of the signal is a function of t and is  the integral of ftx:  φtx(t) =  � t  τ=0  2π · ftx(τ) dτ  =  � t  τ=0  2π · (f0 + S · τ) dτ  = 2π · f0 · t +  � t  τ=0  2π · S · τ dτ  = 2π · f0 · t + 2π · 1  2S · t2  = 2π · f0 · t + π · S · t2  (16)  The transmitted signal Stx can be written as a sinusoid signal:  Stx(t) = A · cos(2πf0t + πSt2)  (17)  where A is the transmission power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The received signal is a  delayed and downscaled version of Stx:  Srx(t) = αA · cos  �  2πf0(t − τ) + πS(t − τ)2�  (18)  where τ is the ToF of the signal and indicates the distance  of the object, and α is the downscale factor that models the  transmission loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The two signals, Stx and Srx, are combined  through a mixer (a multiplier) to generate one signal with both  the sum frequency and the difference frequency:  Stx(t)·Srx(t)  = αA2  2  �  cos  �  (2πf0t + πSt2) + (2πf0(t − τ) + πS(t − τ)2)  �  +  cos  �  (2πf0t + πSt2) − (2πf0(t − τ) + πS(t − τ)2)  ��  = αA2  2  �  cos  �  2π(2f0 − Sτ)t + 2πSt2 + πSτ 2 − 2πf0τ  �  +  cos  �  2π(Sτ)t + 2πf0τ − πSτ 2��  (19)  There are two cos terms in the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The first one has a  frequency of 2f0 and will be removed by a low pass filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The second one is called the IF signal or the beat frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The IF signal has the equation:  IF(t) = B · cos  �  2π(Sτ)t + 2πf0τ − πSτ 2�  (20)  where B = αA2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The signal has a frequency Sτ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the slope  of the chirp multiplied by the ToF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Therefore, the frequency  of the IF signal is directly proportional to the ToF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Given that  the slope of the chirp S is known, the distance of the object  can be calculated from the frequency of the IF signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The  phase of the IF signal, (2πf0τ − πSτ 2), can be simplified to  (2πf0τ), as the second term is negligible: S has an order of  1012, τ has an order of 10−8, so (πSτ 2) will have a negligible  order of 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In summary, the IF signal can be written as:  IF(t) = B · cos(ωbt + φb)  (21)  where the angular frequency ωb and the phase of the signal  φb are:  ωb = 2π · Sτ,  φb = 2πf0τ  (22)  The above equations assume that the object is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' If  the object is moving, the ToF τ will be varying with respect  to t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' However, considering that this variation is limited to a  single chirp time, it is unlikely to produce a big change in the  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The change in phase can be more significant, but  will only affect certain applications where the phase informa-  tion is critical, such as vital sign monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' In such cases,  the phase can be written as a function of t as φb(t) = 4πd(t)  λ0  ,  where d(t) describes the displacement of the object during the  chirp time and λ0 is the signal wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Note that the TI mmWave radar uses a complex band  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' It uses a complex mixer (an IQ mixer) to multiply  the two signals Stx and Srx, which has several advantages like  a lower noise figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' When in complex form, the IF signal in  Equation (21) can be written as:  IF(t) = B · ej(ωbt+φb)  (23)  which has the same frequency and phase as in Equation (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  APPENDIX B  STEERING VECTOR AND AOA ESTIMATION  Figure 15 shows the AoA of an object (point A) to the  radar (point O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The azimuth angle θa is defined to be the  angle between the object’s projection on the horizontal plane  and the front direction of the radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The line of incidence of  the object OA is projected onto the horizontal plane as OB,  and the angle between OB and the y-axis is the azimuth angle  θa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' The elevation angle θe is defined to be the angle between  the object and the horizontal plane (between line OA and the  x-y plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 15: The azimuth and elevation angle of an object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The steering vector is a function of the receiver layout  and the incident angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' To introduce the concept of steering  vectors, it is easier to start with the one-dimensional situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Assuming there are two receivers separated by l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' a signal will  travel an additional distance ∆d to reach the second receiver,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Elevation  A  Z  B  0  X  Azimuth14  where the following approximation can be made (as shown in  Figure 1):  ∆d = l · sin(θ)  (24)  Given that the phase of a sinusoid signal travelled over any  distance ∆d will have a phase  2π∆d  λ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' the phase difference  between the two neighbouring receivers will be:  ∆φ = 2π · ∆d  λ = π · sin(θ)  (25)  When using a receiver array with N azimuth receivers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  each subsequent receiver beyond the first one will receive an  additional phase change of ∆φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' which can be written as a  steering vector:  s(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' N) = [1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' ejπ·sin(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' e2jπ·sin(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=', e(N−1)jπ·sin(θ)] (26)  (a) Elevation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  (b) Azimuth angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' 16: The AoA can be estimated from the phase difference  between adjacent receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  When considering the AoA in both azimuth and elevation  directions, the situation is shown in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Distances ∆da  and ∆de represent the extra distance travelled by the signal to  reach receiver RX0 when compared with the azimuth receiver  RX1 and the elevation receiver RX2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' Similar to  Equation (25), the estimation of the elevation angle θe is given  by:  sin(θe) = ∆de  l  = ∆φe  π  (27)  where ∆φe is the phase difference between RX0 and RX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  The azimuth angle requires a projection from the object’s  3D location to the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content=' As shown in Figure 16b,  the projection from ∆da to ∆da’ gives:  ∆da = ∆da’ · cos(θe)  (28)  Then, the angle θa can be calculated as:  sin(θa) = ∆da’  l  =  ∆da  l · cos(θe)  =  ∆φa  π · cos(θe)  (29)  where ∆φa is the phase difference between RX0 and RX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Extending Equation (27) and Equation (29) with multiple  receivers as a 2D array gives Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Once the values of θa and θe are found, then according to  Figure 1, it can be shown that:  x = OB · sin(θa) = OA · cos(θe)sin(θa) = OA · ∆φa  π  z = OA · sin(θe) = OA · ∆φe  π  y =  �  OA2 − x2 − z2  (30)  where OA is the distance of the object and can be found from  the range-FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
+page_content='  Elevation  Z  RX2  RXO  RX1  X-  AzimuthElevation  Z  RX2  Ada  RXO  RX1  X-  Azimuth' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/l9FRT4oBgHgl3EQfZjdA/content/2301.13553v1.pdf'}
diff --git a/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/2301.11711v1.pdf.txt b/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/2301.11711v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d969bf3999eab91ded719936e8f025121897a45c
--- /dev/null
+++ b/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/2301.11711v1.pdf.txt
@@ -0,0 +1,1521 @@
+AN ADAPTIVE-DISCARD-GRAPH FOR ONLINE ERROR CONTROL
+Lasse Fischer
+Competence Center for Clinical Trials Bremen
+University of Bremen
+fischer1@uni-bremen.de
+Marta Bofill Roig
+Center for Medical Data Science
+Medical University of Vienna
+marta.bofillroig@meduniwien.ac.at
+Werner Brannath
+Competence Center for Clinical Trials Bremen
+University of Bremen
+brannath@uni-bremen.de
+January 30, 2023
+ABSTRACT
+In recent years, graphical multiple testing procedures have gained popularity due to their generality
+and ease of interpretation. In contemporary research, online error control is often required, where
+an error criterion, such as familywise error rate (FWER) or false discovery rate (FDR), shall remain
+under control while testing an a priori unbounded sequence of hypotheses. Although the classical
+graphical procedure can be extended to the online setting, previous work has shown that it leads
+to low power, and other approaches, such as Adaptive-Discard (ADDIS) procedures, are preferred
+instead. In this paper, we introduce an ADDIS-Graph with FWER control and its extension for the
+FDR setting. These graphical ADDIS procedures combine the good interpretability of graphical
+procedures with the high online power of ADDIS procedures. Moreover, they can be adapted to a
+local dependence structure and an asynchronous testing setup, leading to power improvements over
+the current state-of-art methods. Consequently, the proposed methods are useful for a wide range of
+applications, including innovative complex trial designs, such as platform trials, and large-scale test
+designs, such as in the evaluation of A/B tests for marketing research.
+Keywords graphical testing procedures · false discovery rate · familywise error rate · online multiple testing.
+1
+Introduction
+In online multiple testing, an infinite stream of hypotheses (Hi)i∈N is tested sequentially (Foster and Stine, 2008).
+This means, at each step i ∈ N, a decision has to be made on the current hypothesis Hi while having access only
+to the previous hypotheses and decisions. Since the number of hypotheses to be tested in the future is unknown,
+an infinite number is usually assumed. Due to the testing of multiple hypotheses, the probability of making false
+discoveries increases and a multiplicity correction is required. Usually, either the familywise error rate (FWER)
+or the false discovery rate (FDR) are used as error criteria. While controlling the FWER refers to maintaining the
+probability of rejecting at least one true null hypothesis below some pre-specified threshold, the FDR controls the
+expected proportion of true hypotheses among the rejections, and thus allows making false discoveries (Benjamini
+and Hochberg, 1995). The choice of error criterion depends on the study and the associated stringency towards false
+rejections. In some online applications, the less conservative FDR is preferred, as the number of hypotheses is very
+large. Examples can be found in the marketing of large internet companies, which conduct a sequence of so-called A/B
+tests to improve websites (Kohavi et al., 2013). However, there are also applications where false discoveries need to
+be avoided and control of the FWER is necessary. For instance, specific platform trials can be embedded in the online
+multiple testing setting (Robertson et al., 2022a) and in such clinical trials FWER control may be required. In genetic
+studies, an a priori unbounded sequence of hypotheses is tested (Muñoz-Fuentes et al., 2018), often with an interest in
+arXiv:2301.11711v1  [stat.ME]  27 Jan 2023
+
+A PREPRINT - JANUARY 30, 2023
+the FWER. Another example where online FWER is essential is the updating of a machine learning algorithm (Feng
+et al., 2022).
+In classical “offline” multiple testing, graphical approaches have been suggested to facilitate the visualization and
+communication of multiple testing procedures. In the seminal paper, Bretz et al. (2009) proposed the representation
+of multiple testing procedures by directed graphs, where the null hypotheses are represented by nodes accompanied
+by their initial significance levels and connected by weighted vertices, illustrating error propagation if the hypotheses
+are rejected. The graphical procedures provide FWER control, facilitate the illustration of the study objectives and
+are very popular as many test procedures can be represented by this graphical structure (Bretz et al., 2009). The
+graphical approaches were later extended to adaptive designs (Klinglmueller et al., 2014) and to other error measures
+(Robertson et al., 2020), and more recently Tian and Ramdas (2021) extended the graphical approach to the online
+case. Fischer et al. (2022) presented a new online closure principle which ensures that the resulting closed procedure
+can be applied in the online setting and, in particular, showed how the online version of the graphical procedure can be
+written as an online closed procedure based on the Alpha-Spending (Foster and Stine, 2008). Feng et al. (2022) also
+used an online version of the graphical procedure for a specific online multiple testing problem, namely, the updating
+of a machine learning algorithm. In addition, they included the correlation structure between the p-values in order
+to improve the algorithm. However, one disadvantage the previously mentioned online graphical approaches have in
+common is that the individual significance levels are typically rather low due to a large number of hypotheses in online
+multiple testing. Since the significance level is only distributed to the future hypotheses when a p-value is below
+an individual significance level, an update of the levels is unlikely, which results in low power. A more promising
+approach to online FWER control is the Adaptive-Discard (ADDIS) principle by Tian and Ramdas (2021). It allows
+to preserve the individual significance level of a hypothesis Hi for the future testing process if the p-value Pi lies
+outside of an interval (λi, τi] with 0 ≤ λi < τi ≤ 1. Tian and Ramdas (2021) proposed the ADDIS-Spending as a
+concrete ADDIS procedure. In the case of Pi ≤ λi or Pi > τi, the ADDIS-Spending ignores the hypothesis Hi in the
+future testing process and adjusts the future significance levels accordingly. The price for this improvement is a testing
+factor (τi − λi), which needs to be multiplied by the individual significance level before comparing it with the p-
+value. In comparison to FWER, a variety of approaches have been proposed for FDR control (Foster and Stine, 2008;
+Ramdas et al., 2017, 2018; Javanmard and Montanari, 2018; Tian and Ramdas, 2019). Also, in this case, the ADDIS∗
+procedure proposed by Tian and Ramdas (2019), which is based on an ADDIS principle for FDR control, seems to be
+the most promising in terms of online power (Robertson et al., 2022b). However, ADDIS-Spending and ADDIS∗ lack
+generality and interpretability, which also leads to power loss in certain frameworks, such as local dependency and an
+asynchronous test structure. Our main contribution in this paper is the so-called ADDIS-Graph. The ADDIS-Graph
+allows to distribute significance level to future hypotheses whenever a p-value Pi is less or equal than λi or greater
+than τi. Consequently, the ADDIS-Graph combines the good interpretability of graphical procedures with the high
+online power of ADDIS procedures.
+In Section 2, we formally describe the setting and present the online version of the graphical procedure, the ADDIS
+principle for FWER control and the ADDIS-Spending (Tian and Ramdas, 2021). Based on these concepts, we derive
+the ADDIS-Graph for FWER control and show that it contains all other online procedures satisfying the ADDIS
+principle (Section 3). The visual representation of the ADDIS-Graph clarifies the dependencies between an individual
+significance level and the outcomes of previous tests. This allows the ADDIS-Graph to adapt to complex situations,
+resulting in high efficiency. We illustrate this by showing that the ADDIS-Graph leads to an improvement over the
+ADDIS-Spending under local dependence (Section 4). In Section 5, we transfer the ADDIS-Graph approach to the
+FDR setting, resulting in the FDR-ADDIS-Graph. Here, we show superiority over ADDIS∗ with the example of an
+asynchronous testing setup, where a test does not necessarily start and finish at the same step. In Sections 6 and 7, we
+compare our proposals with the procedures proposed by Tian and Ramdas (2019, 2021) through a simulation study
+and application to real data, respectively. All formal proofs of the theoretical assertions are in the Appendix.
+2
+Preliminaries
+2.1
+Setting and notation
+Let I0 be the index set of true hypotheses, R(i) be the index set of rejected hypotheses up to step i ∈ N and V (i) =
+I0 ∩ R(i) denote the index set of falsely rejected hypotheses up to step i. We aim to control the familywise error rate
+FWER(i) := P(|V (i)| > 0)
+(1)
+at each step i ∈ N, where P denotes the probability under the true configuration of true and false hypotheses. Since
+FWER(i) is nondecreasing, it is sufficient to control FWER := P(v > 0), where v := lim
+i→∞ |V (i)|. The FWER is
+controlled strongly at level α, if FWER ≤ α for any configuration of true and false null hypotheses. In contrast, weak
+2
+
+A PREPRINT - JANUARY 30, 2023
+control only provides that FWER ≤ α under the global null hypothesis, which assumes that all hypotheses are true
+(I0 = N). In this paper, we focus on strong control.
+Denoting by (Pi)i∈N the p-values corresponding to the hypotheses (Hi)i∈N. Each null p-value Pi, i ∈ I0, is assumed
+to be valid, meaning P(Pi ≤ x) ≤ x for all x ∈ [0, 1]. A hypothesis Hi is rejected, if Pi ≤ αi, where αi ∈
+[0, 1) is the individual significance level of Hi. ADDIS algorithms also require additional parameters (τi)i∈N and
+(λi)i∈N with values in (0, 1] and [0, τi), respectively. In order to apply a multiple testing procedure in the online
+setting, these parameters are only allowed to depend on information about previous p-values. Mathematically, (αi)i∈N,
+(τi)i∈N and (λi)i∈N are sequences of random variables such that αi, τi and λi are measurable with respect to Gi−1 :=
+σ({R1, S1, C1, . . . , Ri−1, Si−1, Ci−1}), where Rj = 1Pj≤αj, Sj = 1Pj≤τj and Cj = 1Pj≤λj.
+2.2
+Online-Graph and ADDIS procedures
+In the following, we present essential concepts for constructing an ADDIS-Graph. We start with the Online-Graph,
+which was introduced by Tian and Ramdas (2021) (named Online-Fallback procedure in their paper) as the online
+version of the graphical procedure by Bretz et al. (2009). Fischer et al. (2022) have shown how the Online-Graph can
+be obtained by the online closure principle. Afterwards, we present the ADDIS principle and ADDIS-Spending by
+Tian and Ramdas (2021).
+The procedures considered in this paper involve a non-negative sequence (γi)i∈N with �
+i∈N γi ≤ 1, which can be
+interpreted as the initial allocation of the significance level α. The graphical procedures additionally require non-
+negative weights (gj,i)∞
+i=j+1 for all j ∈ N with �∞
+i=j+1 gj,i ≤ 1, which determine the updating of the individual
+significance levels during the testing process. With this, the Online-Graph is defined as
+αi = αγi +
+i−1
+�
+j=1
+gj,iRjαj.
+(2)
+The Online-Graph is illustrated in Figure 1. The initial individual significance level is below each hypothesis. After
+the rejection of a hypothesis Hj, j ∈ N, the individual significance level of Hj is distributed to the future hypotheses
+according to the weights (gj,i)∞
+i=j+1. The rectangles below the nodes can be ignored for the Online-Graph and the
+dots at the end refer to the fact that there is an infinite number of future hypotheses.
+Due to the ease of interpretation, graphical multiple testing procedures are very popular. However, except for un-
+realistic extreme cases (e.g. that each hypothesis can be rejected), the individual significance levels (αi)i∈N of the
+Online-Graph tend to 0 for i to infinity. This means that the probability of distributing a significance level to the future
+hypotheses becomes enormously unlikely at a late stage of the testing process, which results in low power. For this
+reason, Tian and Ramdas (2021) have proposed an ADDIS principle that preserves the significance level of Hi for
+the future testing process, if Pi ≤ λi or Pi > τi. In this way, the decrease of the significance levels can be slowed
+down and thus the power increased. However, this improvement also has its cost. In order to control the FWER, it is
+required that the null p-values are independent of each other and the non-nulls. In addition, it is assumed that the null
+p-values are uniformly valid, which means P(Pi ≤ xy|Pi ≤ y) ≤ x for all x, y ∈ [0, 1] and i ∈ I0 (Zhao et al., 2019).
+Furthermore, in case of λi < Pi ≤ τi, the level αi/(τi − λi) is lost instead of just αi.
+Theorem 2.1 (ADDIS principle for FWER control (Tian and Ramdas, 2021)). Assume the null p-values are uniformly
+valid and independent from each other and the non-nulls. Every multiple testing procedure controls the FWER in the
+strong sense when the individual significance levels (αi)i∈N satisfy
+i
+�
+j=1
+αj
+τj − λj
+(Sj − Cj) ≤ α
+for all i ∈ N.
+(3)
+The ADDIS principle combines the two concepts of adaptivity and discarding. The idea of adaptivity is based on the
+fact that false hypotheses cannot lead to a type I error and thus testing false hypotheses does not increase the FWER.
+Hence, λi is used to estimate whether a hypothesis is true or false. The discarding approach uses the fact that null
+p-values are often conservative, meaning P(Pi ≤ x) < x or equivalently P(Pi > x) > 1 − x for some x ∈ [0, 1]
+and i ∈ I0. Discarding exploits this by accepting large p-values without testing, which leads to higher significance
+levels for the remaining hypotheses. Note that one could also consider the adaptivity and discarding part separately by
+setting τi = 1 or λi = 0 respectively for all i ∈ N. In case of τi = 1, the assumption about the null p-values being
+uniformly valid can be dropped (Tian and Ramdas, 2021).
+Online multiple testing procedures that follow Theorem 2.1 are called ADDIS procedures. As a concrete ADDIS
+procedure, Tian and Ramdas (2021) proposed the ADDIS-Spending. The idea of ADDIS-Spending is to ignore a
+3
+
+A PREPRINT - JANUARY 30, 2023
+hypothesis Hi in case of Pi ≤ λi or Pi > τi in the future testing process and adjust the significance levels accordingly.
+This results in the individual significance level
+αi = α(τi − λi)γt(i),
+where t(i) = 1 +
+i−1
+�
+j=1
+(Sj − Cj).
+(4)
+3
+ADDIS-Graph for FWER control
+Tian and Ramdas (2021) showed by means of simulations that the ADDIS-Spending (in equation (4)) leads to a
+substantially higher power than the one achieved when using the Online-Graph. However, the interpretation of the
+Online-Graph is easier, which makes it simpler to use. To this end, we bring together the approaches of the Online-
+Graph (in (2)) and the ADDIS principle (Theorem 2.1), resulting in what we call the ADDIS-Graph.
+Definition 3.1 (ADDIS-Graph). Let (γi)i∈N and (gj,i)∞
+i=j+1, j ∈ N, be non-negative sequences that sum to at most
+one. In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding Gi−1 for all i ∈ N. The ADDIS-Graph tests
+each hypothesis Hi at significance level
+αi = (τi − λi)
+�
+�αγi +
+i−1
+�
+j=1
+gj,i(Cj − Sj + 1)
+αj
+τj − λj
+�
+� .
+(5)
+Theorem 3.2. The ADDIS-Graph satisfies the ADDIS principle (Definition 2.1) and thus controls the FWER in the
+strong sense when the the null p-values are uniformly valid and independent from each other and the non-nulls.
+In order to represent this ADDIS-Graph as a graph, consider ˜αi = αi
+1
+τi−λi for all i ∈ N, where αi is the significance
+level obtained by the ADDIS-Graph. Equation (5) gives us ˜αi = αi
+1
+τi−λi = αγi + �i−1
+j=1 gj,i(Cj − Sj + 1)˜αj.
+Comparing this with the Online-Graph (2), the (˜αi)i∈N can be interpreted as the significance levels we would obtain
+by a graph with initial levels (αγi)i∈N that updates the future levels whenever a p-value Pj, j ∈ N, is less or equal
+than λj or greater than τj. Thus, one can first determine ˜αi using this graph and then compute the level of the ADDIS-
+Graph as αi = ˜αi(τi −λi). This fact is used to illustrate the ADDIS-Graph in Figure 1. It can be interpreted just as the
+Online-Graph, with two subtle differences. First, we can choose at each step j ∈ N limits τj ∈ [0, 1) and λj ∈ (0, τj]
+for the p-value Pj that determine when the significance level of the j-th hypothesis is distributed among the future
+hypotheses. Second, we need to include an additional testing factor based on these parameters, which is illustrated in
+the rectangle below each hypothesis. This testing factor is only multiplied with the individual significance level when
+the corresponding hypothesis is tested, but it is not involved in the updating process with the graph.
+Figure 1: Illustration of the ADDIS-Graph. Ignoring the rectangles the figure can also be interpreted as the Online-
+Graph.
+4
+
+A PREPRINT - JANUARY 30, 2023
+When defining the ADDIS-Graph, we considered the parameters (γi)i∈N and (gj,i)∞
+i=j+1, j ∈ N as fixed. However,
+we do not need this assumption to satisfy the conditions of Theorem 2.1 and thus to control the FWER. Consequently,
+γi and gj,i could also be random variables that are measurable regarding Gi−1. With this, the procedures become more
+flexible. It can even be shown that, in this case, the ADDIS-Graph is the general ADDIS procedure, thus containing
+all procedures satisfying the ADDIS principle (Theorem 2.1).
+Theorem 3.3. Let γi (i ∈ N) and gj,i (j ∈ N, i > j) be measurable with respect to Gi−1. Then, any procedure
+satisfying the ADDIS principle (Theorem 2.1) can be written as an ADDIS-Graph (Definition 3.1).
+Note that the ADDIS-Graph is more general than the ADDIS-Spending, as Theorem 3.3 does not hold for ADDIS-
+Spending. To see this, suppose we choose a fix λi = λ and τi = τ for all i ∈ N. Then Pi ≤ λ or Pi > τ directly
+implies αi = αi+1 (see (4)) which does not need to hold for every other ADDIS procedure.
+Remark. For a positive and nonincreasing (γi)i∈N, one could write the ADDIS-Spending as an ADDIS-Graph by
+choosing gj,i = (γt(j)+i−j−1 − γt(j)+i−j)/γt(j), where t(j) = 1 + �j−1
+k=1(Sk − Ck). If (γi)i∈N is increasing, it
+becomes more complex, as the (γi)i∈N used in the ADDIS-Graph would need an adjustment as well.
+For the sake of simplicity, we consider (γi)i∈N and (gj,i)j∈N,i>j as fixed parameters in the remainder of this paper.
+In the following section, we show that the ADDIS-Graph can handle local dependence structures and argue why it
+provides a major improvement over the ADDIS-Spending under local dependence.
+4
+ADDIS-Graph under local dependence
+Procedures based on the ADDIS principle (Theorem 2.1) only control the FWER when the p-values are independent.
+In practice, this assumption is often violated. For example, when the same control group is used to test experimental
+groups in different hypotheses or the formulation of new hypotheses is based on the previous test outcomes. On the
+other hand, it is unlikely that p-values from the distant past have any influence on the current testing, as the data
+and context of the data might have changed. For this reason, Zrnic et al. (2021) have proposed a local dependence
+structure. Assume that a fixed sequence of lags (Li)i∈N with Li ∈ {0, 1, . . . , i − 1} and Li+1 ≤ Li + 1 for all i ∈ N
+is given. Then, the p-values (Pi)i∈N are called locally dependent, if ∀i ∈ N holds:
+Pi ⊥ Pi−Li−1, Pi−Li−2, . . . , P1.
+For every Pi, this local dependency structure specifies up to which point of time the previous p-values are independent
+of Pi. Note that local dependence contains independence (Li = 0 ∀i ∈ N) and arbitrary dependence (Li = i − 1
+∀i ∈ N) as special cases. Although we consider the lags as fixed, they do not need to be known before the evaluation.
+However, Li has to be determined at the beginning of step i ∈ N and must not depend on the data itself. For
+example, the lags could be based on content-related information about the data. Tian and Ramdas (2021) showed
+that local dependence can be incorporated into the ADDIS principle (Theorem 2.1) by ignoring the dependent p-
+values and making pessimistic assumptions instead. Mathematically, αi, λi and τi need to be measurable regarding
+Gi−Li−1 = σ({P1, . . . , Pi−Li−1}). Tian and Ramdas (2021) used this to adjust their ADDIS-Spending (4) to the local
+dependence by requiring (γi)i∈N to be nonincreasing and setting
+αi = α(τi − λi)γt(i),
+where t(i) = 1 + Li +
+i−Li−1
+�
+j=1
+(Sj − Cj).
+(6)
+Note that this procedure, which we refer to as ADDIS-Spendinglocal, loses significance level due to local dependence.
+To see that, suppose α∗
+i = α(τi−λi)γt∗(i), where t∗(i) = 1+�i−1
+j=1(Sj −Cj). Then, α∗
+i can be interpreted as the level
+we would obtain under independence (Li = 0). It is easy to see that t∗(i) ≤ t(i), and often even strictly smaller. For
+example, suppose P1 and P2 depend on each other (L2 = 1). If P1 ≤ λ1 or P1 > τ1, we have t∗(2) = 1 < 2 = t(2)
+and if additionally γ1 > γ2, we also have α∗
+2 > α2. Since (γi)i∈N needs to be decreasing at some steps (unless it is
+constant 0) and often is at every step, the power loss is inevitable and can get high when the lags (Li)i∈N are large. In
+the following, we will see that this power loss can be avoided using the ADDIS-Graph.
+First, we adjust the ADDIS-Graph to the local dependence structure such that FWER control is preserved. After that,
+we show how the weights of the ADDIS-Graph can be used such that no significance level is lost. A simple way to
+account for local dependence in the ADDIS-Graph (Figure 1) is to remove the arrows connecting dependent p-values
+and adjust the individual significance levels of the ADDIS-Graph (Definition 3.1) to
+αi = (τi − λi)
+�
+�αγi +
+i−Li−1
+�
+j=1
+gj,i(Cj − Sj + 1)
+αj
+τj − λj
+�
+� .
+5
+
+A PREPRINT - JANUARY 30, 2023
+In Figure 2, the ADDIS-Graph is illustrated for a specific local dependence structure. In this example, L2 = 1,
+meaning P1 and P2 depend on each other. This is illustrated by the dotted line. Hence, the link g1,2 is removed
+as no significance level of the first hypothesis can be allocated to the second. Furthermore, L3 was chosen equal to
+zero, which is why no further adjustment of the graph was needed. Note that by removing the weight g1,2 potential
+significance level is lost as well as in ADDIS-Spending. However, the ADDIS-Graph allows to adjust the weights to
+the given local dependence structure. For example, by adding g1,2 to one of the other weights g1,i, i > 2. In that case,
+it may be that not the same hypotheses benefit from the first hypothesis, but the same amount of significance level is
+distributed as under independence.
+Figure 2: Adjustment of the ADDIS-Graph (Figure 1) to a local dependence structure in which P1 and P2 depend on
+each other.
+In order to formalise such strategies, note that Li+1 ≤ Li + 1 for all i ∈ N implies that i − Li is nondecreasing in
+i. Hence, if we define dj := min{i ∈ N : i − Li > j} (we set min(∅) = ∞) as the index of the first future p-value
+that does not depend on Pj, all Pk with k > dj are independent from Pj as well. Thus, the idea is to distribute the
+entire significance level of Hj in case of Pj ≤ λj or Pj > τj only to hypotheses Hk with k ≥ dj. For this, we
+propose to remove the weights between dependent hypotheses and standardise the remaining weights, which leads to
+the following ADDIS-Graph under local dependence.
+Definition 4.1 (ADDIS-Graphlocal). Assume local dependence with the lags (Li)i∈N. Let (γi)i∈N be a non-negative
+sequence that sums up to 1 and (gj,i)∞
+i=j+1 be a non-negative sequence for all j ∈ N such that �k
+i=j+1 gj,i < 1 for all
+k > j. In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding Gi−Li−1 = σ({P1, . . . , Pi−Li−1}). The
+ADDIS-Graphlocal tests each hypothesis Hi at significance level
+αi = (τi − λi)
+�
+�αγi +
+i−Li−1
+�
+j=1
+g∗
+j,i(Cj − Sj + 1)
+αj
+τj − λj
+�
+� ,
+where g∗
+j,i = gj,i
+� �
+1 − �dj−1
+k=j+1 gj,k
+�
+if i ≥ dj and g∗
+j,i = 0 otherwise.
+The FWER control of the ADDIS-Graphlocal comes directly by Theorem 3.2 and the ADDIS principle under local
+dependence (Tian and Ramdas, 2021). Importantly, note that, for all j ∈ N, it holds
+dj < ∞ and
+∞
+�
+i=j+1
+gj,i = 1
+=⇒
+∞
+�
+i=j+1
+g∗
+j,i = 1,
+which implies that there are no uniformly larger weights than (g∗
+j,i)∞
+i=j+1, that are suitable for an ADDIS-Graph. Since
+αi = α∗
+i := (τi − λi)
+�
+αγi + �i−1
+j=1 g∗
+j,i(Cj − Sj + 1)αj
+1
+τj−λj
+�
+, we do not lose significance level compared to the
+6
+
+A PREPRINT - JANUARY 30, 2023
+independent case, indicating superiority over the ADDIS-Spending, where significance level is lost when the p-values
+are locally dependent.
+Remark.
+• The ADDIS-Graphlocal does not lose significance due to local dependence, if dj < ∞ for all j ∈ N, which
+is equivalent to lim
+i→∞ i − Li = ∞. In particular, this is satisfied if (Li)i∈N has an upper bound, which indeed
+covers many cases that occur in practice. For example, when the hypotheses are tested in finite batches. That
+means, there are disjoint groups of p-values B1 = {P1, . . . , Pj} (j ∈ N), B2 = {Pj, . . . , Pk} (k > j),
+B3 = {Pk, . . . , Pl} (l > k) and so on, such that p-values from the same batch may depend on each other, but
+hypotheses from different batches are independent.
+• There are many other possible ADDIS-Graphs for local dependence.
+We decided to present
+ADDIS-Graphlocal because g∗
+j,i/g∗
+j,k = gj,i/gj,k for all j ∈ N and i, k ≥ dj. Hence, compared to (gj,i)∞
+i=j+1,
+the weights (g∗
+j,i)∞
+i=j+1 are increased by the same factor for each j ∈ N. Simulations showed that an ex-
+tremely uneven allocation of the significance levels leads to low power.
+In the same manner as for local dependence, the ADDIS-Graph can be adjusted to an asynchronous testing setup (Zrnic
+et al., 2021). This is a generalisation of the online multiple testing framework in which the test for hypothesis Hi is not
+finished at step i ∈ N but at a random time Ei ≥ i. It is assumed that Ei is independent of the p-values and thus can be
+interpreted as fixed but unknown before time Ei. One has to determine a significance level for a hypothesis Hj at step
+j ∈ N without using information about tests that are not finished before step j. Thus, we can adjust the ADDIS-Graph
+to the asynchronous setting by removing arrows connecting hypotheses where the testing process overlaps in time.
+By standardizing the remaining weights we do not lose any significance level which again leads to a superiority of
+the ADDIS-Graph over the ADDIS-Spending. A more formal construction of an ADDIS-Graph for the asynchronous
+setting can be found in the next section, where we derive an ADDIS-Graph with FDR control.
+5
+ADDIS-Graph for FDR control
+In this section, we focus on FDR control, where
+FDR(i) := E
+�
+|V (i)|
+|R(i)| ∨ 1
+�
+.
+(7)
+In order to control FDR(i) at any time i ∈ N using ADDIS procedures, we need the additional assumptions that
+λi ≥ αi for all i ∈ N and that αi, λi and 1 − τi are monotonic functions of the past. This means that they are coordi-
+natewise nondecreasing functions in R1:(i−1) := (R1, . . . , Ri−1) and C1:(i−1) := (C1, . . . , Ci−1) and nonincreasing
+in S1:(i−1) := (S1, . . . , Si−1). Under these assumptions, Tian and Ramdas (2019) showed that the FDR is controlled
+if the condition of the ADDIS principle (3) for FWER control (Definition 2.1) is replaced with
+�i
+j=1
+αj
+τj−λj (Sj − Cj)
+|R(i)| ∨ 1
+≤ α
+for all i ∈ N.
+(8)
+Remark. If τi = 1 for all i ∈ N, the ADDIS principle for FDR control reduces to the SAFFRON principle by Ramdas
+et al. (2018). In this case, the uniformly validity assumption of the null p-values can be dropped.
+The only difference between the conditions of the ADDIS principle for the FDR (in (8)) and for the FWER case (in
+(3)) is the denominator |R(i)| ∨ 1. Bringing it on the other side, it can be interpreted as if an additional level α is
+gained after each rejection except for the first one. This can be incorporated into the ADDIS-Graph by distributing
+an additional α to future hypotheses in case of rejection according to non-negative weights (hj,i)∞
+i=j+1 such that
+�∞
+i=j+1 hj,i ≤ 1 for all j ∈ N. For example, one could just choose hj,i = gj,i.
+Since no significance level is gained for the first rejection, FDR procedures often assume that a lower overall signif-
+icance level of W0 ≤ α is available at the beginning of the testing process such that (α − W0) can be gained after
+the first rejection. To differentiate between the first and other rejections, we additionally define the indicator Ti with
+Ti = 1, if the first rejection happened within the first i − 1 steps and Ti = 0, otherwise. We also set T c
+i = 1 − Ti.
+With this, the ADDIS-Graph for FDR control can be defined as follows.
+Definition 5.1 (FDR-ADDIS-Graph). Let (γi)i∈N, (gj,i)j∈N,i>j, (τi)i∈N and (λi)i∈N be as in ADDIS-Graph (Defini-
+tion 3.1) such that τi and λi are monotonic functions of the past. In addition, let W0 ≤ α and (hj,i)∞
+i=j+1, j ∈ N, be a
+7
+
+A PREPRINT - JANUARY 30, 2023
+non-negative sequence such that �∞
+i=j+1 hi,j ≤ 1. The FDR-ADDIS-Graph tests each hypothesis Hi at significance
+level αi = min(ˆαi, λi), where
+ˆαi = (τi − λi)
+�
+�W0γi +
+i−1
+�
+j=1
+gj,i(Cj − Sj + 1)
+αj
+τi − λj
++
+i−1
+�
+j=1
+hj,iRj[αTj + (α − W0)T c
+j ]
+�
+� .
+(9)
+Obviously, αi is a monotonic function of the past for all i ∈ N, which leads to the following conclusion.
+Theorem 5.2. The FDR-ADDIS-Graph satisfies equation (8) and thus controls the FDR strongly, when the null p-
+values are uniformly conservative and independent from each other and the non-nulls.
+The FDR-ADDIS-Graph is illustrated in Figure 3. Note that the figure only contains (ˆαi)i∈N and one needs to set
+αi = min(ˆαi, λi) after using the graph. The FDR-ADDIS-Graph can be interpreted just as the ADDIS-Graph for
+FWER control (Figure 1). The additional grey arrows are activated if the corresponding hypothesis is rejected. In case
+of the first rejection, the level α − W0 is distributed to the future hypotheses according to the weights (hj,i)∞
+i=j+1,
+j ∈ N, and in case of any other rejection, the level α is distributed.
+Figure 3: Illustration of the FDR-ADDIS-Graph.
+The benefit of the FDR-ADDIS-Graph compared to the proposal of Tian and Ramdas (2019), the ADDIS∗ algorithm,
+is similar as for the ADDIS-Graph for FWER control and the ADDIS-Spending. Due to its graphical structure, the
+FDR-ADDIS-Graph is easier to interpret. In particular, the dependencies between the previous test outcomes and
+individual significance levels become clearer.
+The FDR-ADDIS-Graph can be easily adapted to an asynchronous setting by removing the arrows connecting timely
+overlapping hypotheses. By standardizing the remaining weights, no significance level is lost, which leads to an
+improvement over the ADDIS∗
+async by Tian and Ramdas (2019). In the same way, the FDR-ADDIS-Graph can be
+adjusted to a local dependence structure. However, in case of local dependence, only control of the marginal false
+discovery rate (mFDR) is provided (Zrnic et al., 2021), which is defined as
+mFDR(i) :=
+E(|V (i)|)
+E(|R(i)| ∨ 1).
+(10)
+8
+
+A PREPRINT - JANUARY 30, 2023
+For this reason, we only present an extension of the FDR-ADDIS-Graph to the asynchronous setting. However, if
+one is interested in mFDR control under local dependence, the same adjustments to the FDR-ADDIS-Graph can be
+made as we presented in Section 4 for the FWER controlling ADDIS-Graph. To derive an FDR-ADDIS-Graph for the
+asynchronous setting, we proceed as described at the end of Section 4. To this end, let Ei ≥ i, i ∈ N, be the stopping
+times given from the asynchronous testing. The idea is to distribute the significance level of Hj in case of Pj ≤ λj or
+Pj > τj only to hypotheses Hi with i > Ej, which leads to the following FDR-ADDIS-Graph for the asynchronous
+setting.
+Definition 5.3 (FDR-ADDIS-Graphasync). Let W0 ≤ α, (γi)i∈N be a non-negative sequence that sums up to 1 and
+(gj,i)∞
+i=j+1 and (hj,i)∞
+i=j+1 be non-negative sequences for all j ∈ N such that �k
+i=j+1 gj,i < 1 and �k
+i=j+1 hj,i < 1
+for all k > j. In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding GE
+i = σ({Pj : Ej < i}). We define
+g∗
+j,i = gj,i
+� �
+1 − �Ej
+k=j+1 gj,k
+�
+, h∗
+j,i = hj,i
+� �
+1 − �Ej
+k=j+1 hj,k
+�
+if i > Ej and g∗
+j,i = 0, h∗
+j,i = 0, otherwise. The
+FDR-ADDIS-Graphasync tests each hypothesis Hi at significance level αi = min(ˆαi, λi), where
+ˆαi = (τi − λi)
+�
+�W0γi +
+i−1
+�
+j=1
+g∗
+j,i(Cj − Sj + 1)
+αj
+τi − λj
++
+i−1
+�
+j=1
+h∗
+j,iRj[αTj + (α − W0)T c
+j ]
+�
+� .
+(11)
+The FDR control of FDR-ADDIS-Graphasync is directly implied by Theorem 5.2.
+6
+Simulations
+We investigate the power and error control of the proposed ADDIS-Graphs by means of simulations. In Subsection
+6.1, we compare the FWER controlling procedures ADDIS-Graph and ADDIS-Spending (Tian and Ramdas, 2021)
+under local dependence and in Subsection 6.2, we compare the FDR-ADDIS-Graph with the ADDIS* algorithm (Tian
+and Ramdas, 2019) in an asynchronous testing setup.
+6.1
+Simulations for FWER control
+We consider n
+=
+1000 hypotheses to be tested, whose corresponding p-values (Pi)i∈{1,...,n} arrive in fi-
+nite batches B1, . . . , Bn/b with the same batch-size b ∈ {1, 10, 25, 50} for every batch.
+Let Xbj+1:b(j+1) =
+(Xbj+1, . . . , Xb(j+1))T ∼Nb(µ, Σ), j ∈ {0, . . . , n/b − 1}, be b-dimensional i.i.d random vectors, where µ =
+(0, . . . , 0)T ∈ Rb and Σ = (σij)i,j=1,...,b ∈ Rb×b with σii = 1 and σij = ρ ∈ (0, 1) for all i ∈ {1, . . . , b} and
+j ̸= i. For each Hi, i ∈ {1, . . . , n}, we test the null hypothesis Hi : µi ≤ 0 with µi = E[Zi], where Zi = Xi + µA,
+µA > 0, with probability πA ∈ (0, 1) and Zi = Xi + µN, µN < 0, otherwise. Since the test statistics follow a
+standard gaussian distribution under the null hypothesis, a z-test can be used. The parameter µA can be interpreted
+as the strength of the alternative, πA as probability of a hypothesis being false and µN as the conservativeness of null
+p-values.
+In this subsection, we use an overall level α = 0.2 and estimate the FWER and power of the ADDIS-Graphlocal and
+ADDIS-Spendinglocal (Tian and Ramdas, 2021) by averaging over 2000 independent trials. Thereby, the proportion
+of rejected hypotheses among the false hypotheses is used as empirical power. We set µA = 4, µN = −0.5 and
+ρ = 0.8 in all simulations within this subsection, thus obtaining slightly conservative null p-values. Since both
+procedures are based on the same ADDIS principle and therefore exploit the conservativeness of null p-values in the
+same manner, no more parameter configurations are necessary. As recommended by Tian and Ramdas (2021), we
+choose the τi = 0.8 and λi = ατi = 0.16 for all i ∈ N. In the first simulation (Figure 4), we also use the same
+γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+as Tian and Ramdas (2021) in their simulations. However, in Figures 5 and 6, we use
+γi ∝ 1/i1.6 and γi = 6/(π2i2), as the procedures are very sensitive to the choice of (γi)i∈N. For the weights of the
+ADDIS-Graph, we always set gj,i = γi−j, j ∈ N and i > j.
+The plots indicate that ADDIS-Spendinglocal and ADDIS-Graphlocal perform quite similar under independence of the
+p-values. However, when the p-values become locally dependent the power of the ADDIS-Spendinglocal decreases
+systematically in all cases, while the power of the ADDIS-Graphlocal either remains similar (Figure 4) or even increases
+(Figures 5 and 6). To understand the power behavior of the ADDIS-Graphlocal note that the larger the batch-size, the
+9
+
+A PREPRINT - JANUARY 30, 2023
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Spendinglocal
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Graphlocal
+Batch−size
+1
+10
+25
+50
+Figure 4: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different
+batch-sizes and proportions of false null hypotheses (πA). Lines above the overall level α = 0.2 correspond to power
+and lines below to FWER. The p-values were generated as described in the text with parameters µN = −0.5, µA = 4
+and ρ = 0.8. Both procedures were applied with parameters τi = 0.8, λi = 0.16 and γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+,
+In addition, gj,i = γi−j was used in ADDIS-Graphlocal.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Spendinglocal
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Graphlocal
+Batch−size
+1
+10
+25
+50
+Figure 5: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different
+batch-sizes and proportions of false null hypotheses (πA). Lines above the overall level α = 0.2 correspond to power
+and lines below to FWER. The p-values were generated as described in the text with parameters µN = −0.5, µA = 4
+and ρ = 0.8. Both procedures were applied with parameters τi = 0.8, λi = 0.16 and γi ∝ 1/i1.6. In addition,
+gj,i = γi−j was used in ADDIS-Graphlocal.
+10
+
+A PREPRINT - JANUARY 30, 2023
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Spendinglocal
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FWER / Power
+ADDIS−Graphlocal
+Batch−size
+1
+10
+25
+50
+Figure 6: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different
+batch-sizes and proportions of false null hypotheses (πA). Lines above the overall level α = 0.2 correspond to power
+and lines below to FWER. The p-values were generated as described in the text with parameters µN = −0.5, µA = 4
+and ρ = 0.8. Both procedures were applied with parameters τi = 0.8, λi = 0.16 and γi = 6/(π2i2). In addition,
+gj,i = γi−j was used in ADDIS-Graphlocal.
+further into the future the significance level is distributed by the weights (g∗
+j,i)∞
+i=j+1 (see Definition 4.1). In these
+simulations γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+(Figure 4) decreases the slowest and γi = 6/(π2i2) (Figure 6) decreases
+the fastest. If (γi)i∈N decreases slow and the batch-size is large, ADDIS-Graphlocal distributes a lot of significance
+level to hypotheses in the far future. However, since the testing process is finite in this case, these hypotheses may
+never be tested, which leads to a power loss. On the other hand, if (γi)i∈N decreases fast, ADDIS-Graphlocal allocates
+the individual significance levels more evenly under a larger batch-size, which results in a higher power. Thus, in order
+to obtain the optimal power for each batch-size, one could choose a faster decreasing (γi)i∈N the larger the batch-size.
+6.2
+Simulations for FDR control
+In this subsection we consider the same simulation setup as described in Subsection 6.1, but for independent p-values
+(b = 1). However, applying the procedures, it is assumed that the hypotheses are tested in an asynchronous manner.
+Thus, for each hypotheses Hi, i ∈ N, we have a stopping time Ei ≥ i. We assume that Ei = i + e for some constant
+test duration e ∈ N0. In the following simulations we compare the FDR-ADDIS-Graphasync and ADDIS∗
+async (Tian
+and Ramdas, 2019) in terms of power and FDR for e ∈ {0, 2, 5, 10}. Since FDR is less conservative than FWER,
+we change the overall level to α = 0.05 and strength of the alternative to µA = 3. As recommended by Tian and
+Ramdas (2019), we choose τi = 0.5 and λi = 0.25 for all i ∈ N, but use the same (γi)i∈N and (gj,i)∞
+j=i+1, j ∈ N, as
+before. Furthermore, we set W0 = α. For the additional weights (hj,i)∞
+j=i+1 of the FDR-ADDIS-Graphasync, we fix
+hj,i = gj,i for all j ∈ N and i > j. The results obtained by averaging over 200 independent trials can be found in the
+Figures 7-9.
+The results are similar as for the FWER controlling procedures (Subsection 6.1). The power of ADDIS∗
+async decreases
+enormously for an increasing test duration. This decrease can be decelerated by the FDR-ADDIS-Graphasync (Figures
+7 and 8) or even stopped (Figure 9), if a faster decreasing (γi)i∈N is chosen.
+7
+Application to International Mouse Phenotyping Consortium data
+We illustrate how the proposed ADDIS-Graphlocal could be used by using the International Mouse Phenotyping (IMPC)
+data and compare the results obtained with those obtained using the ADDIS-Spendinglocal. The IMPC coordinates a
+large study which aims to identify the function of every protein coding gene. To this regard, each of the 20000
+genes is systematically knocked out and the impact on the phenotype explored. In our evaluation, we use the first
+5000 observations of the database at the Zenodo repository https://zenodo.org/record/2396572, organized by
+Robertson et al. (2019). It contains p-values that resulted from the analysis by Karp et al. (2017). These p-values
+11
+
+A PREPRINT - JANUARY 30, 2023
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+ADDIS*async
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+FDR−ADDIS−Graphasync
+Test duration
+0
+2
+5
+10
+Figure 7: Comparison of ADDIS∗
+async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-
+tions and proportions of false null hypotheses (πA). Lines above the overall level α = 0.05 correspond to power and
+lines below to FDR. The p-values were generated as described in the text with parameters µN = −0.5 and µA = 3.
+Both procedures were applied with parameters τi = 0.5, λi = 0.25, γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+and W0 = α. In
+addition, gj,i = γi−j and hj,i = gj,i were used in FDR-ADDIS-Graphasync.
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+ADDIS*async
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+FDR−ADDIS−Graphasync
+Test duration
+0
+2
+5
+10
+Figure 8: Comparison of ADDIS∗
+async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-
+tions and proportions of false null hypotheses (πA). Lines above the overall level α = 0.05 correspond to power and
+lines below to FDR. The p-values were generated as described in the text with parameters µN = −0.5 and µA = 3.
+Both procedures were applied with parameters τi = 0.5, λi = 0.25, γi ∝ 1/i1.6 and W0 = α. In addition, gj,i = γi−j
+and hj,i = gj,i were used in FDR-ADDIS-Graphasync.
+12
+
+A PREPRINT - JANUARY 30, 2023
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+ADDIS*async
+4
+4
+4
+4
+4
+4
+4
+4
+4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+πA
+FDR / Power
+FDR−ADDIS−Graphasync
+Test duration
+0
+2
+5
+10
+Figure 9: Comparison of ADDIS∗
+async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-
+tions and proportions of false null hypotheses (πA). Lines above the overall level α = 0.05 correspond to power and
+lines below to FDR. The p-values were generated as described in the text with parameters µN = −0.5 and µA = 3.
+Both procedures were applied with parameters τi = 0.5, λi = 0.25, γi = 6/(π2i2) and W0 = α. In addition,
+gj,i = γi−j and hj,i = gj,i were used in FDR-ADDIS-Graphasync.
+follow a natural local dependence structure: the hypotheses are tested in batches and within each batch the same mice
+are used for each hypothesis. Thus, the p-values within a batch depend on each other, but p-values from different
+batches are independent.
+We applied ADDIS-Spendinglocal and ADDIS-Graphlocal with the same parameters as in Subsection 6.1. The (γi)i∈N
+was chosen such that γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+for all i ∈ N. The results can be found in figure 10. The left plot
+shows the number of rejections achieved by the two procedures with respect to the FWER level α considered. Similar
+to Subsection 6.1, we can see that the ADDIS-Graph allows a significantly larger number of hypotheses to be rejected
+than the ADDIS-Spendinglocal. The right plot shows the individual significance levels obtained by the two procedures
+for the FWER level α = 0.2. Note however that for ease of illustration, we omitted the first 100 levels that yielded
+much higher individual significance levels. It can be seen that the ADDIS-Graphlocal tests each hypothesis using
+higher significance levels than the ADDIS-Spendinglocal. In fact, for each FWER level and hypothesis, the individual
+significance level obtained by the ADDIS-Graph is greater than or equal to the ADDIS-Spending level. That means,
+the ADDIS-Graph rejects all hypotheses that are rejected by the ADDIS-Spending, but additionally some more.
+8
+Conclusion
+In this work, we presented a graphical approach to exploit the ADDIS principles for FWER (Tian and Ramdas, 2021)
+and FDR (Tian and Ramdas, 2019) control. We started with the construction of an FWER controlling ADDIS-Graph.
+This proposal enhances the interpretability of the ADDIS-Spending and also enlarges the family of procedures that it
+includes, as we show that by means of the ADDIS-Graph all procedures that satisfy the ADDIS principle for FWER
+control can be obtained. Furthermore, the ADDIS-Graph can easily be adapted to a local dependence structure and an
+asynchronous testing setup without losing significance level. For both situation, in the considered simulation scenarios
+we show that the ADDIS-Graph leads to a large power gain as compared to the ADDIS-Spending. Moreover, we
+extend the ADDIS-Graph to the FDR control setting resulting in an FDR-ADDIS-Graph. It has the same advantages
+as the ADDIS-Graph and is superior to the currently used ADDIS method with FDR control, the ADDIS∗ algorithm.
+Robertson et al. (2022b) claimed that the individual significance levels assigned by asynchronous online procedures
+are more conservative. We have illustrated with the ADDIS-Graph that this is not necessarily the case. Although the
+significance level of a hypothesis depends on pessimistic assumptions about the outcomes of tests that are still running,
+future hypotheses can take advantage of this conservatism and achieve higher significance levels such that no level is
+lost overall.
+We wonder whether the ADDIS-Graph can be extended to further multiple testing frameworks. For example, Zrnic
+et al. (2020) considered online batched-testing, where at each step not only one, but a batch of several hypotheses
+13
+
+A PREPRINT - JANUARY 30, 2023
+150
+180
+210
+240
+270
+300
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+FWER level
+Number of rejections
+0.00000
+0.00005
+0.00010
+0.00015
+0.00020
+0.00025
+0.00030
+0
+1000
+2000
+3000
+4000
+5000
+Index of hypothesis
+Individual significance level
+Procedure
+ADDIS−Spendinglocal
+ADDIS−Graphlocal
+Figure 10: The left plot shows the number of rejections for different FWER levels α and the right plot the individual
+significance levels (for α = 0.2) obtained by ADDIS-Spendinglocal and ADDIS-Graphlocal. Both procedures were
+applied with parameters τi = 0.8, λi = 0.16 and γi ∝ 1/
+�
+(i + 1) log(i + 1)2�
+. In addition, gj,i = γi−j was used in
+ADDIS-Graphlocal.
+is tested at the same time. One could think of an ADDIS-Graph that distributes significance level to all hypotheses
+within the considered batch, but outside the batch only to future hypotheses.
+Another task for future work is the optimal choice of the parameters (γi)i∈N, (gj,i)j∈N,i>j and (hj,i)j∈N,i>j. In our
+simulations (Section 6), we chose (γi)i∈N as in the literature for the comparison procedures and set (gj,i)∞
+i=j+1 and
+(hj,i)∞
+i=j+1 related to (γi)i∈N for each j ∈ N. However, the large number of parameters allows for many further
+possibilities that can strongly influence the performance of the ADDIS-Graphs. For instance, we saw that a faster
+decreasing (γi)i∈N would be useful when the batch-size or test duration is large. Many more such recommendations
+could be made through simulations and theoretical results. In addition, one could study time-varying choices of (τi)i∈N
+and (λi)i∈N that may depend on the previous test outcomes.
+14
+
+A PREPRINT - JANUARY 30, 2023
+Appendix
+Proof of Theorem 3.2. Let (αi)i∈N be given by the ADDIS-Graph. We need to show that for any i ∈ N, S1:i :=
+(S1, . . . , Si)T ∈ {0, 1}i−1 and C1:i := (C1, . . . , Ci)T ∈ {0, 1}i:
+i
+�
+j=1
+αi
+τi − λi
+(Si − Ci) ≤ α.
+(12)
+We define Uj := Cj − Sj + 1 for all j ∈ N. Then 1 − Uj = Sj − Cj and since Cj ≤ Sj, it holds Uj ∈ {0, 1}. Now
+let i ∈ N and U1:i = (U1, . . . , Ui)T ∈ {0, 1}i be arbitrary but fixed. With this, (12) is equivalent to
+Fi(U1:i) :=
+i
+�
+j=1
+�
+αγj +
+j−1
+�
+k=1
+gk,jUkαk(U1:(k−1))
+1
+τk − λk
+�
+(1 − Uj) ≤ α.
+(13)
+Note that we only wrote the dependence of αk on U1:(k−1) = (U1, . . . , Uk−1)T , although the parameters λk and
+τk could depend on it as well. That is, because these parameters could also be fixed, meaning if we change the
+U1:(k−1) they would still be valid parameters for an ADDIS-Graph. In contrast, the αk changes by definition. It is
+difficult to show the validity of (13) directly. However, we will see that there exists ˜U1:i ∈ {0, 1}i that obviously fulfil
+Fi( ˜U1:i) ≤ α. Therefore, the idea is to determine such a ˜U1:i that additionally satisfies Fi(U1:i) ≤ Fi( ˜U1:i).
+Let l = max{j ∈ {1, . . . , i} : Uj = 1} (we set max(∅) = 0) and U l
+1:i = (U l
+1, . . . , U l
+i)T , where U l
+j = Uj for all j ̸= l
+and U l
+l = 0. We assume that l > 0 (if l = 0, we later see Fi(U1:i) ≤ α anyway). In the next step we want to show that
+Fi(U1:i) ≤ Fi(U l
+1:i). For shorter notation we write αj = αj(U1:(j−1)) and αl
+j = αj(U l
+1:(j−1)). Since for all j ≤ i:
+U l
+j = Uj (j ̸= l), U l
+j = 0 (j ≥ l), Uj = 0 (j ≥ l + 1) and αl
+j = αj (j ≤ l), we have:
+Fi(U l
+1:i) − Fi(U1:i)
+=
+i
+�
+j=1
+αγj(1 − U l
+j) −
+i
+�
+j=1
+αγj(1 − Uj) +
+i
+�
+j=1
+�j−1
+�
+k=1
+gk,jU l
+kαl
+k
+1
+τk − λk
+�
+(1 − U l
+j)
+−
+i
+�
+j=1
+�j−1
+�
+k=1
+gk,jUkαk
+1
+τk − λk
+�
+(1 − Uj)
+= αγl +
+i
+�
+j=1
+�j−1
+�
+k=1
+gk,jU l
+kαl
+k
+1
+τk − λk
+�
+(1 − U l
+j) −
+i
+�
+j=1
+�j−1
+�
+k=1
+gk,jUkαk
+1
+τk − λk
+�
+(1 − Uj)
+= αγl +
+l−1
+�
+k=1
+gk,lU l
+kαl
+k
+1
+τk − λk
++
+i
+�
+j=l+1
+l−1
+�
+k=1
+gk,jU l
+kαl
+k
+1
+τk − λk
+−
+i
+�
+j=l+1
+l
+�
+k=1
+gk,jUkαk
+1
+τk − λk
+= αγl +
+l−1
+�
+k=1
+gk,lUkαk
+1
+τk − λk
+−
+i
+�
+j=l+1
+gl,jαl
+1
+τl − λl
+≥ αγl +
+l−1
+�
+k=1
+gk,lUkαk
+1
+τk − λk
+− αl
+1
+τl − λl
+Def.3.1
+=
+0,
+where we used in the inequality that the sequence (gl,j)∞
+j=l+1 is non-negative and sums to at most 1 for all l ∈ N.
+Since the U1:i ∈ {0, 1}i was arbitrary, this shows Fi(U1:i) ≤ Fi(U 0
+1:i) for all U1:i ∈ {0, 1}i, where U 0
+1:i =
+(0, . . . , 0)T ∈ {0, 1}i. Next, we deduce that Fi(U 0
+1:i) ≤ α and conclude the proof. For this, just recognize that
+U 0
+1:i means Uj = 0 for all j ≤ i. Hence, we obtain
+Fi(U 0
+1:i) =
+i
+�
+j=1
+αγj ≤ α.
+15
+
+A PREPRINT - JANUARY 30, 2023
+Proof of Theorem 3.3. Let G1:i = (R1, C1, S1, . . . , Ri, Ci, Si) ∈ {0, 1}3i, then every procedure satisfying the AD-
+DIS principle (Theorem 2.1) is a sequence of non-negative functions (αi(G1:(i−1)))i∈N such that
+�
+j≤i
+αj(G1:(j−1))
+τj − λj
+(Sj − Cj) ≤ α
+for all i ∈ N.
+(14)
+Note that the function αi(G1:(i−1)) is fully determined through the information until step i − 1, hence pessimistic
+assumptions about Si and Ci need to be made in order to satisfy equation (14). Consequently, the condition of the
+ADDIS principle is equivalent to
+0 ≤ αi(G1:(i−1)) ≤ (τi − λi)
+�
+�α −
+�
+j≤i−1
+αj(G1:(j−1))
+τj − λj
+(Sj − Cj)
+�
+�
+for all i ∈ N.
+(15)
+Let i ∈ N and G1:(i−1) ∈ {0, 1}3(i−1) be arbitrary but fixed. In addition, let (αj)j<i be levels obtained by an
+ADDIS-Graph with parameters (γj)j<i and (gk,j)k<j<i. We want to prove that
+αi(γi, (gj,i)j<i)) = (τi − λi)
+�
+�αγi +
+i−1
+�
+j=1
+gj,i(Cj − Sj + 1)
+αj
+τi − λj
+�
+� ,
+where γi ∈
+�
+0, 1 − �i−1
+j=1 γj
+�
+and gj,i ∈
+�
+0, 1 − �i−1
+k=j+1 gj,k
+�
+, j ∈ {1, . . . , i − 1}, can take any value in the interval
+�
+0, (τi − λi)
+�
+α − �
+j≤i−1
+αj
+τj−λj (Sj − Cj)
+��
+. Since αi is continuous in γi and (gj,i)j<i, it is sufficient to show that
+αi(0, (0)j<i) = 0 and αi
+�
+1 − �i−1
+j=1 γj,
+�
+1 − �i−1
+k=j+1 gj,k
+�
+j<i
+�
+= (τi − λi)
+�
+α − �
+j≤i−1
+αj
+τj−λj (Sj − Cj)
+�
+.
+The first equation follows immediately, hence we only need to show the second (we set Uj = Cj − Sj + 1 for all
+j ∈ N):
+αi
+�
+�1 −
+i−1
+�
+j=1
+γj,
+�
+�1 −
+i−1
+�
+k=j+1
+gj,k
+�
+�
+j<i
+�
+� − (τi − λi)
+�
+�α −
+i−1
+�
+j=1
+αj
+τj − λj
+(1 − Uj)
+�
+�
+= (τi − λi)
+�
+�α
+�
+�1 −
+i−1
+�
+j=1
+γj
+�
+� +
+i−1
+�
+j=1
+�
+�1 −
+i−1
+�
+k=j+1
+gj,k
+�
+� Uj
+αj
+τi − λj
+�
+�
+− (τi − λi)
+�
+�α −
+i−1
+�
+j=1
+�
+αγj +
+j−1
+�
+k=1
+gk,jUk
+αk
+τk − λk
+�
+(1 − Uj)
+�
+�
+= (τi − λi)
+�
+�−α
+i−1
+�
+j=1
+γjUj +
+i−1
+�
+j=1
+Uj
+αj
+τi − λj
+−
+i−1
+�
+j=1
+i−1
+�
+k=j+1
+gj,kUj
+αj
+τi − λj
++
+i−1
+�
+j=1
+j−1
+�
+k=1
+gk,jUk
+αk
+τk − λk
+−
+i−1
+�
+j=1
+�j−1
+�
+k=1
+gk,jUk
+αk
+τk − λk
+�
+Uj
+�
+�
+= (τi − λi)
+�
+�
+i−1
+�
+j=1
+Uj
+αj
+τi − λj
+−
+i−1
+�
+j=1
+Uj
+�
+αγj +
+j−1
+�
+k=1
+gk,jUk
+αk
+τk − λk
+��
+� = 0.
+16
+
+A PREPRINT - JANUARY 30, 2023
+Proof of Theorem 5.2. Let α0
+j = W0γj + �j−1
+k=1 hk,jRk[αTk + (α − W0)T c
+k] for all j ∈ N. Note that
+i
+�
+j=1
+α0
+j =
+i
+�
+j=1
+W0γj +
+i
+�
+j=1
+j−1
+�
+k=1
+hk,jRk[αTk + (α − W0)T c
+k]
+≤ W0 +
+i−1
+�
+k=1
+Rk[αTk + (α − W0)T c
+k]
+i
+�
+j=k+1
+hk,j
+≤ W0 +
+i−1
+�
+k=1
+Rk[αTk + (α − W0)T c
+k]
+= W0 + (α − W0)Ti + α(|R(i − 1)| − 1)Ti
+≤ α(|R(i)| ∨ 1)
+With this, the proof can be performed as for Theorem 3.2 by replacing αγj with α0
+j on the left side in equation (13)
+and α with α(|R(i)| ∨ 1) on the right side.
+Funding
+L. Fischer acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –
+Project number 281474342/GRK2224/2.
+M. Bofill Roig is a member of the EU Patient-centric clinical trial platform (EU-PEARL). EU-PEARL has received
+funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No 853966. This Joint
+Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme and EF-
+PIA and Children’s Tumor Foundation, Global Alliance for TB Drug Development non-profit organization, Spring-
+works Therapeutics Inc. This publication reflects the author’s views. Neither IMI nor the European Union, EFPIA, or
+any Associated Partners are responsible for any use that may be made of the information contained herein.
+Supplementary Material
+The R code for the simulations and case study can be found at the GitHub repository https://github.com/
+fischer23/Adaptive-Discard-Graph.
+References
+Benjamini, Y. and Hochberg, Y. (1995). Controlling the false discovery rate: a practical and powerful approach to
+multiple testing. Journal of the Royal statistical society: series B (Methodological), 57(1):289–300.
+Bretz, F., Maurer, W., Brannath, W., and Posch, M. (2009). A graphical approach to sequentially rejective multiple
+test procedures. Statistics in medicine, 28(4):586–604.
+Feng, J., Pennllo, G., Petrick, N., Sahiner, B., Pirracchio, R., and Gossmann, A. (2022). Sequential algorithmic
+modification with test data reuse. In Uncertainty in Artificial Intelligence, pages 674–684. PMLR.
+Fischer, L., Roig, M. B., and Brannath, W. (2022). The online closure principle. arXiv preprint arXiv:2211.11400.
+Foster, D. P. and Stine, R. A. (2008). α-investing: A procedure for sequential control of expected false discoveries.
+Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(2):429–444.
+Javanmard, A. and Montanari, A. (2018). Online rules for control of false discovery rate and false discovery ex-
+ceedance. The Annals of statistics, 46(2):526–554.
+Karp, N. A., Mason, J., Beaudet, A. L., Benjamini, Y., Bower, L., Braun, R. E., Brown, S. D., Chesler, E. J., Dickinson,
+M. E., Flenniken, A. M., et al. (2017). Prevalence of sexual dimorphism in mammalian phenotypic traits. Nature
+communications, 8(1):1–12.
+Klinglmueller, F., Posch, M., and Koenig, F. (2014). Adaptive graph-based multiple testing procedures. Pharmaceu-
+tical Statistics, 13(6):345–356.
+Kohavi, R., Deng, A., Frasca, B., Walker, T., Xu, Y., and Pohlmann, N. (2013). Online controlled experiments at
+large scale. In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data
+mining, pages 1168–1176.
+17
+
+A PREPRINT - JANUARY 30, 2023
+Muñoz-Fuentes, V., Cacheiro, P., Meehan, T. F., Aguilar-Pimentel, J. A., Brown, S. D., Flenniken, A. M., Flicek,
+P., Galli, A., Mashhadi, H. H., Hrabˇe de Angelis, M., et al. (2018). The international mouse phenotyping consor-
+tium (impc): a functional catalogue of the mammalian genome that informs conservation. Conservation genetics,
+19(4):995–1005.
+Ramdas, A., Yang, F., Wainwright, M. J., and Jordan, M. I. (2017). Online control of the false discovery rate with
+decaying memory. Advances in neural information processing systems, 30.
+Ramdas, A., Zrnic, T., Wainwright, M., and Jordan, M. (2018). Saffron: an adaptive algorithm for online control of
+the false discovery rate. In International conference on machine learning, pages 4286–4294. PMLR.
+Robertson, D. S., Wason, J., König, F., Posch, M., and Jaki, T. (2022a). Online error control for platform trials. arXiv
+preprint arXiv:2202.03838.
+Robertson, D. S., Wason, J., and Ramdas, A. (2022b). Online multiple hypothesis testing for reproducible research.
+arXiv preprint arXiv:2208.11418.
+Robertson, D. S., Wason, J. M., and Bretz, F. (2020). Graphical approaches for the control of generalized error rates.
+Statistics in medicine, 39(23):3135–3155.
+Robertson, D. S., Wildenhain, J., Javanmard, A., and Karp, N. A. (2019). onlinefdr: an r package to control the false
+discovery rate for growing data repositories. Bioinformatics, 35(20):4196–4199.
+Tian, J. and Ramdas, A. (2019). Addis: an adaptive discarding algorithm for online fdr control with conservative nulls.
+Advances in neural information processing systems, 32.
+Tian, J. and Ramdas, A. (2021). Online control of the familywise error rate. Statistical Methods in Medical Research,
+30(4):976–993.
+Zhao, Q., Small, D. S., and Su, W. (2019). Multiple testing when many p-values are uniformly conservative, with
+application to testing qualitative interaction in educational interventions. Journal of the American Statistical Asso-
+ciation, 114(527):1291–1304.
+Zrnic, T., Jiang, D., Ramdas, A., and Jordan, M. (2020). The power of batching in multiple hypothesis testing. In
+International Conference on Artificial Intelligence and Statistics, pages 3806–3815. PMLR.
+Zrnic, T., Ramdas, A., and Jordan, M. I. (2021). Asynchronous online testing of multiple hypotheses. J. Mach. Learn.
+Res., 22:33–1.
+18
+
diff --git a/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/load_file.txt b/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..96b911ea2015654502ab4d8e360df460012e3bb7
--- /dev/null
+++ b/ldFKT4oBgHgl3EQfDS1x/content/tmp_files/load_file.txt
@@ -0,0 +1,935 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf,len=934
+page_content='AN ADAPTIVE-DISCARD-GRAPH FOR ONLINE ERROR CONTROL  Lasse Fischer  Competence Center for Clinical Trials Bremen  University of Bremen  fischer1@uni-bremen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='de  Marta Bofill Roig  Center for Medical Data Science  Medical University of Vienna  marta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='bofillroig@meduniwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='at  Werner Brannath  Competence Center for Clinical Trials Bremen  University of Bremen  brannath@uni-bremen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='de  January 30, 2023  ABSTRACT  In recent years, graphical multiple testing procedures have gained popularity due to their generality  and ease of interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In contemporary research, online error control is often required, where  an error criterion, such as familywise error rate (FWER) or false discovery rate (FDR), shall remain  under control while testing an a priori unbounded sequence of hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Although the classical  graphical procedure can be extended to the online setting, previous work has shown that it leads  to low power, and other approaches, such as Adaptive-Discard (ADDIS) procedures, are preferred  instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In this paper, we introduce an ADDIS-Graph with FWER control and its extension for the  FDR setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' These graphical ADDIS procedures combine the good interpretability of graphical  procedures with the high online power of ADDIS procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Moreover, they can be adapted to a  local dependence structure and an asynchronous testing setup, leading to power improvements over  the current state-of-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Consequently, the proposed methods are useful for a wide range of  applications, including innovative complex trial designs, such as platform trials, and large-scale test  designs, such as in the evaluation of A/B tests for marketing research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Keywords graphical testing procedures · false discovery rate · familywise error rate · online multiple testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  1  Introduction  In online multiple testing, an infinite stream of hypotheses (Hi)i∈N is tested sequentially (Foster and Stine, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  This means, at each step i ∈ N, a decision has to be made on the current hypothesis Hi while having access only  to the previous hypotheses and decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since the number of hypotheses to be tested in the future is unknown,  an infinite number is usually assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Due to the testing of multiple hypotheses, the probability of making false  discoveries increases and a multiplicity correction is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Usually, either the familywise error rate (FWER)  or the false discovery rate (FDR) are used as error criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' While controlling the FWER refers to maintaining the  probability of rejecting at least one true null hypothesis below some pre-specified threshold, the FDR controls the  expected proportion of true hypotheses among the rejections, and thus allows making false discoveries (Benjamini  and Hochberg, 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The choice of error criterion depends on the study and the associated stringency towards false  rejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In some online applications, the less conservative FDR is preferred, as the number of hypotheses is very  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Examples can be found in the marketing of large internet companies, which conduct a sequence of so-called A/B  tests to improve websites (Kohavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, there are also applications where false discoveries need to  be avoided and control of the FWER is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For instance, specific platform trials can be embedded in the online  multiple testing setting (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2022a) and in such clinical trials FWER control may be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In genetic  studies, an a priori unbounded sequence of hypotheses is tested (Muñoz-Fuentes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2018), often with an interest in  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='11711v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='ME]  27 Jan 2023  A PREPRINT - JANUARY 30, 2023  the FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Another example where online FWER is essential is the updating of a machine learning algorithm (Feng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In classical “offline” multiple testing, graphical approaches have been suggested to facilitate the visualization and  communication of multiple testing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the seminal paper, Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2009) proposed the representation  of multiple testing procedures by directed graphs, where the null hypotheses are represented by nodes accompanied  by their initial significance levels and connected by weighted vertices, illustrating error propagation if the hypotheses  are rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The graphical procedures provide FWER control, facilitate the illustration of the study objectives and  are very popular as many test procedures can be represented by this graphical structure (Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The  graphical approaches were later extended to adaptive designs (Klinglmueller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2014) and to other error measures  (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2020), and more recently Tian and Ramdas (2021) extended the graphical approach to the online  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022) presented a new online closure principle which ensures that the resulting closed procedure  can be applied in the online setting and, in particular, showed how the online version of the graphical procedure can be  written as an online closed procedure based on the Alpha-Spending (Foster and Stine, 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022) also  used an online version of the graphical procedure for a specific online multiple testing problem, namely, the updating  of a machine learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, they included the correlation structure between the p-values in order  to improve the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, one disadvantage the previously mentioned online graphical approaches have in  common is that the individual significance levels are typically rather low due to a large number of hypotheses in online  multiple testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since the significance level is only distributed to the future hypotheses when a p-value is below  an individual significance level, an update of the levels is unlikely, which results in low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A more promising  approach to online FWER control is the Adaptive-Discard (ADDIS) principle by Tian and Ramdas (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It allows  to preserve the individual significance level of a hypothesis Hi for the future testing process if the p-value Pi lies  outside of an interval (λi, τi] with 0 ≤ λi < τi ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Tian and Ramdas (2021) proposed the ADDIS-Spending as a  concrete ADDIS procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the case of Pi ≤ λi or Pi > τi, the ADDIS-Spending ignores the hypothesis Hi in the  future testing process and adjusts the future significance levels accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The price for this improvement is a testing  factor (τi − λi), which needs to be multiplied by the individual significance level before comparing it with the p-  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In comparison to FWER, a variety of approaches have been proposed for FDR control (Foster and Stine, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Ramdas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2017, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Javanmard and Montanari, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Tian and Ramdas, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Also, in this case, the ADDIS∗  procedure proposed by Tian and Ramdas (2019), which is based on an ADDIS principle for FDR control, seems to be  the most promising in terms of online power (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, ADDIS-Spending and ADDIS∗ lack  generality and interpretability, which also leads to power loss in certain frameworks, such as local dependency and an  asynchronous test structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Our main contribution in this paper is the so-called ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The ADDIS-Graph  allows to distribute significance level to future hypotheses whenever a p-value Pi is less or equal than λi or greater  than τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Consequently, the ADDIS-Graph combines the good interpretability of graphical procedures with the high  online power of ADDIS procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In Section 2, we formally describe the setting and present the online version of the graphical procedure, the ADDIS  principle for FWER control and the ADDIS-Spending (Tian and Ramdas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Based on these concepts, we derive  the ADDIS-Graph for FWER control and show that it contains all other online procedures satisfying the ADDIS  principle (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The visual representation of the ADDIS-Graph clarifies the dependencies between an individual  significance level and the outcomes of previous tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This allows the ADDIS-Graph to adapt to complex situations,  resulting in high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We illustrate this by showing that the ADDIS-Graph leads to an improvement over the  ADDIS-Spending under local dependence (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In Section 5, we transfer the ADDIS-Graph approach to the  FDR setting, resulting in the FDR-ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Here, we show superiority over ADDIS∗ with the example of an  asynchronous testing setup, where a test does not necessarily start and finish at the same step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In Sections 6 and 7, we  compare our proposals with the procedures proposed by Tian and Ramdas (2019, 2021) through a simulation study  and application to real data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' All formal proofs of the theoretical assertions are in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  2  Preliminaries  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1  Setting and notation  Let I0 be the index set of true hypotheses, R(i) be the index set of rejected hypotheses up to step i ∈ N and V (i) =  I0 ∩ R(i) denote the index set of falsely rejected hypotheses up to step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We aim to control the familywise error rate  FWER(i) := P(|V (i)| > 0)  (1)  at each step i ∈ N, where P denotes the probability under the true configuration of true and false hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since  FWER(i) is nondecreasing, it is sufficient to control FWER := P(v > 0), where v := lim  i→∞ |V (i)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The FWER is  controlled strongly at level α, if FWER ≤ α for any configuration of true and false null hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In contrast, weak  2  A PREPRINT - JANUARY 30, 2023  control only provides that FWER ≤ α under the global null hypothesis, which assumes that all hypotheses are true  (I0 = N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In this paper, we focus on strong control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Denoting by (Pi)i∈N the p-values corresponding to the hypotheses (Hi)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Each null p-value Pi, i ∈ I0, is assumed  to be valid, meaning P(Pi ≤ x) ≤ x for all x ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A hypothesis Hi is rejected, if Pi ≤ αi, where αi ∈  [0, 1) is the individual significance level of Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' ADDIS algorithms also require additional parameters (τi)i∈N and  (λi)i∈N with values in (0, 1] and [0, τi), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In order to apply a multiple testing procedure in the online  setting, these parameters are only allowed to depend on information about previous p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Mathematically, (αi)i∈N,  (τi)i∈N and (λi)i∈N are sequences of random variables such that αi, τi and λi are measurable with respect to Gi−1 :=  σ({R1, S1, C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ri−1, Si−1, Ci−1}), where Rj = 1Pj≤αj, Sj = 1Pj≤τj and Cj = 1Pj≤λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  Online-Graph and ADDIS procedures  In the following, we present essential concepts for constructing an ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We start with the Online-Graph,  which was introduced by Tian and Ramdas (2021) (named Online-Fallback procedure in their paper) as the online  version of the graphical procedure by Bretz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022) have shown how the Online-Graph can  be obtained by the online closure principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Afterwards, we present the ADDIS principle and ADDIS-Spending by  Tian and Ramdas (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The procedures considered in this paper involve a non-negative sequence (γi)i∈N with �  i∈N γi ≤ 1, which can be  interpreted as the initial allocation of the significance level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The graphical procedures additionally require non-  negative weights (gj,i)∞  i=j+1 for all j ∈ N with �∞  i=j+1 gj,i ≤ 1, which determine the updating of the individual  significance levels during the testing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' With this, the Online-Graph is defined as  αi = αγi +  i−1  �  j=1  gj,iRjαj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (2)  The Online-Graph is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The initial individual significance level is below each hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' After  the rejection of a hypothesis Hj, j ∈ N, the individual significance level of Hj is distributed to the future hypotheses  according to the weights (gj,i)∞  i=j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The rectangles below the nodes can be ignored for the Online-Graph and the  dots at the end refer to the fact that there is an infinite number of future hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Due to the ease of interpretation, graphical multiple testing procedures are very popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, except for un-  realistic extreme cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' that each hypothesis can be rejected), the individual significance levels (αi)i∈N of the  Online-Graph tend to 0 for i to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This means that the probability of distributing a significance level to the future  hypotheses becomes enormously unlikely at a late stage of the testing process, which results in low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For this  reason, Tian and Ramdas (2021) have proposed an ADDIS principle that preserves the significance level of Hi for  the future testing process, if Pi ≤ λi or Pi > τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In this way, the decrease of the significance levels can be slowed  down and thus the power increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, this improvement also has its cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In order to control the FWER, it is  required that the null p-values are independent of each other and the non-nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, it is assumed that the null  p-values are uniformly valid, which means P(Pi ≤ xy|Pi ≤ y) ≤ x for all x, y ∈ [0, 1] and i ∈ I0 (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Furthermore, in case of λi < Pi ≤ τi, the level αi/(τi − λi) is lost instead of just αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 (ADDIS principle for FWER control (Tian and Ramdas, 2021)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Assume the null p-values are uniformly  valid and independent from each other and the non-nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Every multiple testing procedure controls the FWER in the  strong sense when the individual significance levels (αi)i∈N satisfy  i  �  j=1  αj  τj − λj  (Sj − Cj) ≤ α  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (3)  The ADDIS principle combines the two concepts of adaptivity and discarding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The idea of adaptivity is based on the  fact that false hypotheses cannot lead to a type I error and thus testing false hypotheses does not increase the FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Hence, λi is used to estimate whether a hypothesis is true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The discarding approach uses the fact that null  p-values are often conservative, meaning P(Pi ≤ x) < x or equivalently P(Pi > x) > 1 − x for some x ∈ [0, 1]  and i ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Discarding exploits this by accepting large p-values without testing, which leads to higher significance  levels for the remaining hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note that one could also consider the adaptivity and discarding part separately by  setting τi = 1 or λi = 0 respectively for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In case of τi = 1, the assumption about the null p-values being  uniformly valid can be dropped (Tian and Ramdas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Online multiple testing procedures that follow Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 are called ADDIS procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' As a concrete ADDIS  procedure, Tian and Ramdas (2021) proposed the ADDIS-Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The idea of ADDIS-Spending is to ignore a  3  A PREPRINT - JANUARY 30, 2023  hypothesis Hi in case of Pi ≤ λi or Pi > τi in the future testing process and adjust the significance levels accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  This results in the individual significance level  αi = α(τi − λi)γt(i),  where t(i) = 1 +  i−1  �  j=1  (Sj − Cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (4)  3  ADDIS-Graph for FWER control  Tian and Ramdas (2021) showed by means of simulations that the ADDIS-Spending (in equation (4)) leads to a  substantially higher power than the one achieved when using the Online-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, the interpretation of the  Online-Graph is easier, which makes it simpler to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To this end, we bring together the approaches of the Online-  Graph (in (2)) and the ADDIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1), resulting in what we call the ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 (ADDIS-Graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let (γi)i∈N and (gj,i)∞  i=j+1, j ∈ N, be non-negative sequences that sum to at most  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding Gi−1 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The ADDIS-Graph tests  each hypothesis Hi at significance level  αi = (τi − λi)  �  �αγi +  i−1  �  j=1  gj,i(Cj − Sj + 1)  αj  τj − λj  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (5)  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The ADDIS-Graph satisfies the ADDIS principle (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) and thus controls the FWER in the  strong sense when the the null p-values are uniformly valid and independent from each other and the non-nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In order to represent this ADDIS-Graph as a graph, consider ˜αi = αi  1  τi−λi for all i ∈ N, where αi is the significance  level obtained by the ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Equation (5) gives us ˜αi = αi  1  τi−λi = αγi + �i−1  j=1 gj,i(Cj − Sj + 1)˜αj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Comparing this with the Online-Graph (2), the (˜αi)i∈N can be interpreted as the significance levels we would obtain  by a graph with initial levels (αγi)i∈N that updates the future levels whenever a p-value Pj, j ∈ N, is less or equal  than λj or greater than τj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thus, one can first determine ˜αi using this graph and then compute the level of the ADDIS-  Graph as αi = ˜αi(τi −λi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This fact is used to illustrate the ADDIS-Graph in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It can be interpreted just as the  Online-Graph, with two subtle differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' First, we can choose at each step j ∈ N limits τj ∈ [0, 1) and λj ∈ (0, τj]  for the p-value Pj that determine when the significance level of the j-th hypothesis is distributed among the future  hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Second, we need to include an additional testing factor based on these parameters, which is illustrated in  the rectangle below each hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This testing factor is only multiplied with the individual significance level when  the corresponding hypothesis is tested, but it is not involved in the updating process with the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Figure 1: Illustration of the ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Ignoring the rectangles the figure can also be interpreted as the Online-  Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  4  A PREPRINT - JANUARY 30, 2023  When defining the ADDIS-Graph, we considered the parameters (γi)i∈N and (gj,i)∞  i=j+1, j ∈ N as fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However,  we do not need this assumption to satisfy the conditions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 and thus to control the FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Consequently,  γi and gj,i could also be random variables that are measurable regarding Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' With this, the procedures become more  flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It can even be shown that, in this case, the ADDIS-Graph is the general ADDIS procedure, thus containing  all procedures satisfying the ADDIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let γi (i ∈ N) and gj,i (j ∈ N, i > j) be measurable with respect to Gi−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Then, any procedure  satisfying the ADDIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) can be written as an ADDIS-Graph (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Note that the ADDIS-Graph is more general than the ADDIS-Spending, as Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='3 does not hold for ADDIS-  Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To see this, suppose we choose a fix λi = λ and τi = τ for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Then Pi ≤ λ or Pi > τ directly  implies αi = αi+1 (see (4)) which does not need to hold for every other ADDIS procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For a positive and nonincreasing (γi)i∈N, one could write the ADDIS-Spending as an ADDIS-Graph by  choosing gj,i = (γt(j)+i−j−1 − γt(j)+i−j)/γt(j), where t(j) = 1 + �j−1  k=1(Sk − Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' If (γi)i∈N is increasing, it  becomes more complex, as the (γi)i∈N used in the ADDIS-Graph would need an adjustment as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  For the sake of simplicity, we consider (γi)i∈N and (gj,i)j∈N,i>j as fixed parameters in the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In the following section, we show that the ADDIS-Graph can handle local dependence structures and argue why it  provides a major improvement over the ADDIS-Spending under local dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  4  ADDIS-Graph under local dependence  Procedures based on the ADDIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) only control the FWER when the p-values are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In practice, this assumption is often violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For example, when the same control group is used to test experimental  groups in different hypotheses or the formulation of new hypotheses is based on the previous test outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' On the  other hand, it is unlikely that p-values from the distant past have any influence on the current testing, as the data  and context of the data might have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For this reason, Zrnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2021) have proposed a local dependence  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Assume that a fixed sequence of lags (Li)i∈N with Li ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , i − 1} and Li+1 ≤ Li + 1 for all i ∈ N  is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Then, the p-values (Pi)i∈N are called locally dependent, if ∀i ∈ N holds:  Pi ⊥ Pi−Li−1, Pi−Li−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  For every Pi, this local dependency structure specifies up to which point of time the previous p-values are independent  of Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note that local dependence contains independence (Li = 0 ∀i ∈ N) and arbitrary dependence (Li = i − 1  ∀i ∈ N) as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Although we consider the lags as fixed, they do not need to be known before the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  However, Li has to be determined at the beginning of step i ∈ N and must not depend on the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For  example, the lags could be based on content-related information about the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Tian and Ramdas (2021) showed  that local dependence can be incorporated into the ADDIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) by ignoring the dependent p-  values and making pessimistic assumptions instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Mathematically, αi, λi and τi need to be measurable regarding  Gi−Li−1 = σ({P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Pi−Li−1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Tian and Ramdas (2021) used this to adjust their ADDIS-Spending (4) to the local  dependence by requiring (γi)i∈N to be nonincreasing and setting  αi = α(τi − λi)γt(i),  where t(i) = 1 + Li +  i−Li−1  �  j=1  (Sj − Cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (6)  Note that this procedure, which we refer to as ADDIS-Spendinglocal, loses significance level due to local dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  To see that, suppose α∗  i = α(τi−λi)γt∗(i), where t∗(i) = 1+�i−1  j=1(Sj −Cj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Then, α∗  i can be interpreted as the level  we would obtain under independence (Li = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It is easy to see that t∗(i) ≤ t(i), and often even strictly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For  example, suppose P1 and P2 depend on each other (L2 = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' If P1 ≤ λ1 or P1 > τ1, we have t∗(2) = 1 < 2 = t(2)  and if additionally γ1 > γ2, we also have α∗  2 > α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since (γi)i∈N needs to be decreasing at some steps (unless it is  constant 0) and often is at every step, the power loss is inevitable and can get high when the lags (Li)i∈N are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In  the following, we will see that this power loss can be avoided using the ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  First, we adjust the ADDIS-Graph to the local dependence structure such that FWER control is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' After that,  we show how the weights of the ADDIS-Graph can be used such that no significance level is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A simple way to  account for local dependence in the ADDIS-Graph (Figure 1) is to remove the arrows connecting dependent p-values  and adjust the individual significance levels of the ADDIS-Graph (Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) to  αi = (τi − λi)  �  �αγi +  i−Li−1  �  j=1  gj,i(Cj − Sj + 1)  αj  τj − λj  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  5  A PREPRINT - JANUARY 30, 2023  In Figure 2, the ADDIS-Graph is illustrated for a specific local dependence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In this example, L2 = 1,  meaning P1 and P2 depend on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This is illustrated by the dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Hence, the link g1,2 is removed  as no significance level of the first hypothesis can be allocated to the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Furthermore, L3 was chosen equal to  zero, which is why no further adjustment of the graph was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note that by removing the weight g1,2 potential  significance level is lost as well as in ADDIS-Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, the ADDIS-Graph allows to adjust the weights to  the given local dependence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For example, by adding g1,2 to one of the other weights g1,i, i > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In that case,  it may be that not the same hypotheses benefit from the first hypothesis, but the same amount of significance level is  distributed as under independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Figure 2: Adjustment of the ADDIS-Graph (Figure 1) to a local dependence structure in which P1 and P2 depend on  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In order to formalise such strategies, note that Li+1 ≤ Li + 1 for all i ∈ N implies that i − Li is nondecreasing in  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Hence, if we define dj := min{i ∈ N : i − Li > j} (we set min(∅) = ∞) as the index of the first future p-value  that does not depend on Pj, all Pk with k > dj are independent from Pj as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thus, the idea is to distribute the  entire significance level of Hj in case of Pj ≤ λj or Pj > τj only to hypotheses Hk with k ≥ dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For this, we  propose to remove the weights between dependent hypotheses and standardise the remaining weights, which leads to  the following ADDIS-Graph under local dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 (ADDIS-Graphlocal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Assume local dependence with the lags (Li)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let (γi)i∈N be a non-negative  sequence that sums up to 1 and (gj,i)∞  i=j+1 be a non-negative sequence for all j ∈ N such that �k  i=j+1 gj,i < 1 for all  k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding Gi−Li−1 = σ({P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Pi−Li−1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The  ADDIS-Graphlocal tests each hypothesis Hi at significance level  αi = (τi − λi)  �  �αγi +  i−Li−1  �  j=1  g∗  j,i(Cj − Sj + 1)  αj  τj − λj  �  � ,  where g∗  j,i = gj,i  � �  1 − �dj−1  k=j+1 gj,k  �  if i ≥ dj and g∗  j,i = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The FWER control of the ADDIS-Graphlocal comes directly by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 and the ADDIS principle under local  dependence (Tian and Ramdas, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Importantly, note that, for all j ∈ N, it holds  dj < ∞ and  ∞  �  i=j+1  gj,i = 1  =⇒  ∞  �  i=j+1  g∗  j,i = 1,  which implies that there are no uniformly larger weights than (g∗  j,i)∞  i=j+1, that are suitable for an ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since  αi = α∗  i := (τi − λi)  �  αγi + �i−1  j=1 g∗  j,i(Cj − Sj + 1)αj  1  τj−λj  �  , we do not lose significance level compared to the  6  A PREPRINT - JANUARY 30, 2023  independent case, indicating superiority over the ADDIS-Spending, where significance level is lost when the p-values  are locally dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The ADDIS-Graphlocal does not lose significance due to local dependence, if dj < ∞ for all j ∈ N, which  is equivalent to lim  i→∞ i − Li = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In particular, this is satisfied if (Li)i∈N has an upper bound, which indeed  covers many cases that occur in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For example, when the hypotheses are tested in finite batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' That  means, there are disjoint groups of p-values B1 = {P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Pj} (j ∈ N), B2 = {Pj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Pk} (k > j),  B3 = {Pk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Pl} (l > k) and so on, such that p-values from the same batch may depend on each other, but  hypotheses from different batches are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  There are many other possible ADDIS-Graphs for local dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  We decided to present  ADDIS-Graphlocal because g∗  j,i/g∗  j,k = gj,i/gj,k for all j ∈ N and i, k ≥ dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Hence, compared to (gj,i)∞  i=j+1,  the weights (g∗  j,i)∞  i=j+1 are increased by the same factor for each j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Simulations showed that an ex-  tremely uneven allocation of the significance levels leads to low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In the same manner as for local dependence, the ADDIS-Graph can be adjusted to an asynchronous testing setup (Zrnic  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This is a generalisation of the online multiple testing framework in which the test for hypothesis Hi is not  finished at step i ∈ N but at a random time Ei ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It is assumed that Ei is independent of the p-values and thus can be  interpreted as fixed but unknown before time Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' One has to determine a significance level for a hypothesis Hj at step  j ∈ N without using information about tests that are not finished before step j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thus, we can adjust the ADDIS-Graph  to the asynchronous setting by removing arrows connecting hypotheses where the testing process overlaps in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  By standardizing the remaining weights we do not lose any significance level which again leads to a superiority of  the ADDIS-Graph over the ADDIS-Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A more formal construction of an ADDIS-Graph for the asynchronous  setting can be found in the next section, where we derive an ADDIS-Graph with FDR control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  5  ADDIS-Graph for FDR control  In this section, we focus on FDR control, where  FDR(i) := E  �  |V (i)|  |R(i)| ∨ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (7)  In order to control FDR(i) at any time i ∈ N using ADDIS procedures, we need the additional assumptions that  λi ≥ αi for all i ∈ N and that αi, λi and 1 − τi are monotonic functions of the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This means that they are coordi-  natewise nondecreasing functions in R1:(i−1) := (R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ri−1) and C1:(i−1) := (C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ci−1) and nonincreasing  in S1:(i−1) := (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Si−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Under these assumptions, Tian and Ramdas (2019) showed that the FDR is controlled  if the condition of the ADDIS principle (3) for FWER control (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) is replaced with  �i  j=1  αj  τj−λj (Sj − Cj)  |R(i)| ∨ 1  ≤ α  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (8)  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' If τi = 1 for all i ∈ N, the ADDIS principle for FDR control reduces to the SAFFRON principle by Ramdas  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In this case, the uniformly validity assumption of the null p-values can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The only difference between the conditions of the ADDIS principle for the FDR (in (8)) and for the FWER case (in  (3)) is the denominator |R(i)| ∨ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Bringing it on the other side, it can be interpreted as if an additional level α is  gained after each rejection except for the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This can be incorporated into the ADDIS-Graph by distributing  an additional α to future hypotheses in case of rejection according to non-negative weights (hj,i)∞  i=j+1 such that  �∞  i=j+1 hj,i ≤ 1 for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For example, one could just choose hj,i = gj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Since no significance level is gained for the first rejection, FDR procedures often assume that a lower overall signif-  icance level of W0 ≤ α is available at the beginning of the testing process such that (α − W0) can be gained after  the first rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To differentiate between the first and other rejections, we additionally define the indicator Ti with  Ti = 1, if the first rejection happened within the first i − 1 steps and Ti = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We also set T c  i = 1 − Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  With this, the ADDIS-Graph for FDR control can be defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1 (FDR-ADDIS-Graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let (γi)i∈N, (gj,i)j∈N,i>j, (τi)i∈N and (λi)i∈N be as in ADDIS-Graph (Defini-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) such that τi and λi are monotonic functions of the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, let W0 ≤ α and (hj,i)∞  i=j+1, j ∈ N, be a  7  A PREPRINT - JANUARY 30, 2023  non-negative sequence such that �∞  i=j+1 hi,j ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The FDR-ADDIS-Graph tests each hypothesis Hi at significance  level αi = min(ˆαi, λi), where  ˆαi = (τi − λi)  �  �W0γi +  i−1  �  j=1  gj,i(Cj − Sj + 1)  αj  τi − λj  +  i−1  �  j=1  hj,iRj[αTj + (α − W0)T c  j ]  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (9)  Obviously, αi is a monotonic function of the past for all i ∈ N, which leads to the following conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The FDR-ADDIS-Graph satisfies equation (8) and thus controls the FDR strongly, when the null p-  values are uniformly conservative and independent from each other and the non-nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The FDR-ADDIS-Graph is illustrated in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note that the figure only contains (ˆαi)i∈N and one needs to set  αi = min(ˆαi, λi) after using the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The FDR-ADDIS-Graph can be interpreted just as the ADDIS-Graph for  FWER control (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The additional grey arrows are activated if the corresponding hypothesis is rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In case  of the first rejection, the level α − W0 is distributed to the future hypotheses according to the weights (hj,i)∞  i=j+1,  j ∈ N, and in case of any other rejection, the level α is distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Figure 3: Illustration of the FDR-ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The benefit of the FDR-ADDIS-Graph compared to the proposal of Tian and Ramdas (2019), the ADDIS∗ algorithm,  is similar as for the ADDIS-Graph for FWER control and the ADDIS-Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Due to its graphical structure, the  FDR-ADDIS-Graph is easier to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In particular, the dependencies between the previous test outcomes and  individual significance levels become clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The FDR-ADDIS-Graph can be easily adapted to an asynchronous setting by removing the arrows connecting timely  overlapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' By standardizing the remaining weights, no significance level is lost, which leads to an  improvement over the ADDIS∗  async by Tian and Ramdas (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the same way, the FDR-ADDIS-Graph can be  adjusted to a local dependence structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, in case of local dependence, only control of the marginal false  discovery rate (mFDR) is provided (Zrnic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 2021), which is defined as  mFDR(i) :=  E(|V (i)|)  E(|R(i)| ∨ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (10)  8  A PREPRINT - JANUARY 30, 2023  For this reason, we only present an extension of the FDR-ADDIS-Graph to the asynchronous setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, if  one is interested in mFDR control under local dependence, the same adjustments to the FDR-ADDIS-Graph can be  made as we presented in Section 4 for the FWER controlling ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To derive an FDR-ADDIS-Graph for the  asynchronous setting, we proceed as described at the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To this end, let Ei ≥ i, i ∈ N, be the stopping  times given from the asynchronous testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The idea is to distribute the significance level of Hj in case of Pj ≤ λj or  Pj > τj only to hypotheses Hi with i > Ej, which leads to the following FDR-ADDIS-Graph for the asynchronous  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='3 (FDR-ADDIS-Graphasync).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let W0 ≤ α, (γi)i∈N be a non-negative sequence that sums up to 1 and  (gj,i)∞  i=j+1 and (hj,i)∞  i=j+1 be non-negative sequences for all j ∈ N such that �k  i=j+1 gj,i < 1 and �k  i=j+1 hj,i < 1  for all k > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, let τi ∈ (0, 1] and λi ∈ [0, τi) be measurable regarding GE  i = σ({Pj : Ej < i}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We define  g∗  j,i = gj,i  � �  1 − �Ej  k=j+1 gj,k  �  , h∗  j,i = hj,i  � �  1 − �Ej  k=j+1 hj,k  �  if i > Ej and g∗  j,i = 0, h∗  j,i = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The  FDR-ADDIS-Graphasync tests each hypothesis Hi at significance level αi = min(ˆαi, λi), where  ˆαi = (τi − λi)  �  �W0γi +  i−1  �  j=1  g∗  j,i(Cj − Sj + 1)  αj  τi − λj  +  i−1  �  j=1  h∗  j,iRj[αTj + (α − W0)T c  j ]  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (11)  The FDR control of FDR-ADDIS-Graphasync is directly implied by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  6  Simulations  We investigate the power and error control of the proposed ADDIS-Graphs by means of simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In Subsection  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1, we compare the FWER controlling procedures ADDIS-Graph and ADDIS-Spending (Tian and Ramdas, 2021)  under local dependence and in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2, we compare the FDR-ADDIS-Graph with the ADDIS* algorithm (Tian  and Ramdas, 2019) in an asynchronous testing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1  Simulations for FWER control  We consider n  =  1000 hypotheses to be tested, whose corresponding p-values (Pi)i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=',n} arrive in fi-  nite batches B1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Bn/b with the same batch-size b ∈ {1, 10, 25, 50} for every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Let Xbj+1:b(j+1) =  (Xbj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Xb(j+1))T ∼Nb(µ, Σ), j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , n/b − 1}, be b-dimensional i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='d random vectors, where µ =  (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , 0)T ∈ Rb and Σ = (σij)i,j=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=',b ∈ Rb×b with σii = 1 and σij = ρ ∈ (0, 1) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , b} and  j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For each Hi, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , n}, we test the null hypothesis Hi : µi ≤ 0 with µi = E[Zi], where Zi = Xi + µA,  µA > 0, with probability πA ∈ (0, 1) and Zi = Xi + µN, µN < 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since the test statistics follow a  standard gaussian distribution under the null hypothesis, a z-test can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The parameter µA can be interpreted  as the strength of the alternative, πA as probability of a hypothesis being false and µN as the conservativeness of null  p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  In this subsection, we use an overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 and estimate the FWER and power of the ADDIS-Graphlocal and  ADDIS-Spendinglocal (Tian and Ramdas, 2021) by averaging over 2000 independent trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thereby, the proportion  of rejected hypotheses among the false hypotheses is used as empirical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We set µA = 4, µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5 and  ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8 in all simulations within this subsection, thus obtaining slightly conservative null p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since both  procedures are based on the same ADDIS principle and therefore exploit the conservativeness of null p-values in the  same manner, no more parameter configurations are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' As recommended by Tian and Ramdas (2021), we  choose the τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8 and λi = ατi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='16 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the first simulation (Figure 4), we also use the same  γi ∝ 1/  �  (i + 1) log(i + 1)2�  as Tian and Ramdas (2021) in their simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, in Figures 5 and 6, we use  γi ∝ 1/i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6 and γi = 6/(π2i2), as the procedures are very sensitive to the choice of (γi)i∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For the weights of the  ADDIS-Graph, we always set gj,i = γi−j, j ∈ N and i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The plots indicate that ADDIS-Spendinglocal and ADDIS-Graphlocal perform quite similar under independence of the  p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, when the p-values become locally dependent the power of the ADDIS-Spendinglocal decreases  systematically in all cases, while the power of the ADDIS-Graphlocal either remains similar (Figure 4) or even increases  (Figures 5 and 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To understand the power behavior of the ADDIS-Graphlocal note that the larger the batch-size, the  9  A PREPRINT - JANUARY 30, 2023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Spendinglocal  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Graphlocal  Batch−size  1  10  25  50  Figure 4: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different  batch-sizes and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 correspond to power  and lines below to FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, µA = 4  and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='16 and γi ∝ 1/  �  (i + 1) log(i + 1)2�  ,  In addition, gj,i = γi−j was used in ADDIS-Graphlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Spendinglocal  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Graphlocal  Batch−size  1  10  25  50  Figure 5: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different  batch-sizes and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 correspond to power  and lines below to FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, µA = 4  and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='16 and γi ∝ 1/i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition,  gj,i = γi−j was used in ADDIS-Graphlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  10  A PREPRINT - JANUARY 30, 2023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Spendinglocal  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FWER / Power  ADDIS−Graphlocal  Batch−size  1  10  25  50  Figure 6: Comparison of ADDIS-Spendinglocal and ADDIS-Graphlocal in terms of power and FWER for different  batch-sizes and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 correspond to power  and lines below to FWER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, µA = 4  and ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='16 and γi = 6/(π2i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition,  gj,i = γi−j was used in ADDIS-Graphlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  further into the future the significance level is distributed by the weights (g∗  j,i)∞  i=j+1 (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In these  simulations γi ∝ 1/  �  (i + 1) log(i + 1)2�  (Figure 4) decreases the slowest and γi = 6/(π2i2) (Figure 6) decreases  the fastest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' If (γi)i∈N decreases slow and the batch-size is large, ADDIS-Graphlocal distributes a lot of significance  level to hypotheses in the far future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, since the testing process is finite in this case, these hypotheses may  never be tested, which leads to a power loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' On the other hand, if (γi)i∈N decreases fast, ADDIS-Graphlocal allocates  the individual significance levels more evenly under a larger batch-size, which results in a higher power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thus, in order  to obtain the optimal power for each batch-size, one could choose a faster decreasing (γi)i∈N the larger the batch-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  Simulations for FDR control  In this subsection we consider the same simulation setup as described in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1, but for independent p-values  (b = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, applying the procedures, it is assumed that the hypotheses are tested in an asynchronous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Thus, for each hypotheses Hi, i ∈ N, we have a stopping time Ei ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We assume that Ei = i + e for some constant  test duration e ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the following simulations we compare the FDR-ADDIS-Graphasync and ADDIS∗  async (Tian  and Ramdas, 2019) in terms of power and FDR for e ∈ {0, 2, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since FDR is less conservative than FWER,  we change the overall level to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='05 and strength of the alternative to µA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' As recommended by Tian and  Ramdas (2019), we choose τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5 and λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='25 for all i ∈ N, but use the same (γi)i∈N and (gj,i)∞  j=i+1, j ∈ N, as  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Furthermore, we set W0 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For the additional weights (hj,i)∞  j=i+1 of the FDR-ADDIS-Graphasync, we fix  hj,i = gj,i for all j ∈ N and i > j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The results obtained by averaging over 200 independent trials can be found in the  Figures 7-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The results are similar as for the FWER controlling procedures (Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The power of ADDIS∗  async decreases  enormously for an increasing test duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This decrease can be decelerated by the FDR-ADDIS-Graphasync (Figures  7 and 8) or even stopped (Figure 9), if a faster decreasing (γi)i∈N is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  7  Application to International Mouse Phenotyping Consortium data  We illustrate how the proposed ADDIS-Graphlocal could be used by using the International Mouse Phenotyping (IMPC)  data and compare the results obtained with those obtained using the ADDIS-Spendinglocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The IMPC coordinates a  large study which aims to identify the function of every protein coding gene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' To this regard, each of the 20000  genes is systematically knocked out and the impact on the phenotype explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In our evaluation, we use the first  5000 observations of the database at the Zenodo repository https://zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='org/record/2396572, organized by  Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It contains p-values that resulted from the analysis by Karp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' These p-values  11  A PREPRINT - JANUARY 30, 2023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  ADDIS*async  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  FDR−ADDIS−Graphasync  Test duration  0  2  5  10  Figure 7: Comparison of ADDIS∗  async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-  tions and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='05 correspond to power and  lines below to FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5 and µA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='25, γi ∝ 1/  �  (i + 1) log(i + 1)2�  and W0 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In  addition, gj,i = γi−j and hj,i = gj,i were used in FDR-ADDIS-Graphasync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  ADDIS*async  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  FDR−ADDIS−Graphasync  Test duration  0  2  5  10  Figure 8: Comparison of ADDIS∗  async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-  tions and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='05 correspond to power and  lines below to FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5 and µA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='25, γi ∝ 1/i1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6 and W0 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, gj,i = γi−j  and hj,i = gj,i were used in FDR-ADDIS-Graphasync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  12  A PREPRINT - JANUARY 30, 2023  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  ADDIS*async  4  4  4  4  4  4  4  4  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8  πA  FDR / Power  FDR−ADDIS−Graphasync  Test duration  0  2  5  10  Figure 9: Comparison of ADDIS∗  async and FDR-ADDIS-Graphasync in terms of power and FDR for different test dura-  tions and proportions of false null hypotheses (πA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Lines above the overall level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='05 correspond to power and  lines below to FDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The p-values were generated as described in the text with parameters µN = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5 and µA = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Both procedures were applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='5, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='25, γi = 6/(π2i2) and W0 = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition,  gj,i = γi−j and hj,i = gj,i were used in FDR-ADDIS-Graphasync.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  follow a natural local dependence structure: the hypotheses are tested in batches and within each batch the same mice  are used for each hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Thus, the p-values within a batch depend on each other, but p-values from different  batches are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  We applied ADDIS-Spendinglocal and ADDIS-Graphlocal with the same parameters as in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The (γi)i∈N  was chosen such that γi ∝ 1/  �  (i + 1) log(i + 1)2�  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The results can be found in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The left plot  shows the number of rejections achieved by the two procedures with respect to the FWER level α considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Similar  to Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1, we can see that the ADDIS-Graph allows a significantly larger number of hypotheses to be rejected  than the ADDIS-Spendinglocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The right plot shows the individual significance levels obtained by the two procedures  for the FWER level α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note however that for ease of illustration, we omitted the first 100 levels that yielded  much higher individual significance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It can be seen that the ADDIS-Graphlocal tests each hypothesis using  higher significance levels than the ADDIS-Spendinglocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In fact, for each FWER level and hypothesis, the individual  significance level obtained by the ADDIS-Graph is greater than or equal to the ADDIS-Spending level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' That means,  the ADDIS-Graph rejects all hypotheses that are rejected by the ADDIS-Spending, but additionally some more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  8  Conclusion  In this work, we presented a graphical approach to exploit the ADDIS principles for FWER (Tian and Ramdas, 2021)  and FDR (Tian and Ramdas, 2019) control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We started with the construction of an FWER controlling ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  This proposal enhances the interpretability of the ADDIS-Spending and also enlarges the family of procedures that it  includes, as we show that by means of the ADDIS-Graph all procedures that satisfy the ADDIS principle for FWER  control can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Furthermore, the ADDIS-Graph can easily be adapted to a local dependence structure and an  asynchronous testing setup without losing significance level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For both situation, in the considered simulation scenarios  we show that the ADDIS-Graph leads to a large power gain as compared to the ADDIS-Spending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Moreover, we  extend the ADDIS-Graph to the FDR control setting resulting in an FDR-ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It has the same advantages  as the ADDIS-Graph and is superior to the currently used ADDIS method with FDR control, the ADDIS∗ algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022b) claimed that the individual significance levels assigned by asynchronous online procedures  are more conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We have illustrated with the ADDIS-Graph that this is not necessarily the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Although the  significance level of a hypothesis depends on pessimistic assumptions about the outcomes of tests that are still running,  future hypotheses can take advantage of this conservatism and achieve higher significance levels such that no level is  lost overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  We wonder whether the ADDIS-Graph can be extended to further multiple testing frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For example, Zrnic  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2020) considered online batched-testing, where at each step not only one, but a batch of several hypotheses  13  A PREPRINT - JANUARY 30, 2023  150  180  210  240  270  300  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='35  FWER level  Number of rejections  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='00030  0  1000  2000  3000  4000  5000  Index of hypothesis  Individual significance level  Procedure  ADDIS−Spendinglocal  ADDIS−Graphlocal  Figure 10: The left plot shows the number of rejections for different FWER levels α and the right plot the individual  significance levels (for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2) obtained by ADDIS-Spendinglocal and ADDIS-Graphlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Both procedures were  applied with parameters τi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='8, λi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='16 and γi ∝ 1/  �  (i + 1) log(i + 1)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, gj,i = γi−j was used in  ADDIS-Graphlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  is tested at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' One could think of an ADDIS-Graph that distributes significance level to all hypotheses  within the considered batch, but outside the batch only to future hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Another task for future work is the optimal choice of the parameters (γi)i∈N, (gj,i)j∈N,i>j and (hj,i)j∈N,i>j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In our  simulations (Section 6), we chose (γi)i∈N as in the literature for the comparison procedures and set (gj,i)∞  i=j+1 and  (hj,i)∞  i=j+1 related to (γi)i∈N for each j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, the large number of parameters allows for many further  possibilities that can strongly influence the performance of the ADDIS-Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For instance, we saw that a faster  decreasing (γi)i∈N would be useful when the batch-size or test duration is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Many more such recommendations  could be made through simulations and theoretical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, one could study time-varying choices of (τi)i∈N  and (λi)i∈N that may depend on the previous test outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  14  A PREPRINT - JANUARY 30, 2023  Appendix  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let (αi)i∈N be given by the ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We need to show that for any i ∈ N, S1:i :=  (S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Si)T ∈ {0, 1}i−1 and C1:i := (C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ci)T ∈ {0, 1}i:  i  �  j=1  αi  τi − λi  (Si − Ci) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (12)  We define Uj := Cj − Sj + 1 for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Then 1 − Uj = Sj − Cj and since Cj ≤ Sj, it holds Uj ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Now  let i ∈ N and U1:i = (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ui)T ∈ {0, 1}i be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' With this, (12) is equivalent to  Fi(U1:i) :=  i  �  j=1  �  αγj +  j−1  �  k=1  gk,jUkαk(U1:(k−1))  1  τk − λk  �  (1 − Uj) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (13)  Note that we only wrote the dependence of αk on U1:(k−1) = (U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Uk−1)T , although the parameters λk and  τk could depend on it as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' That is, because these parameters could also be fixed, meaning if we change the  U1:(k−1) they would still be valid parameters for an ADDIS-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In contrast, the αk changes by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' It is  difficult to show the validity of (13) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' However, we will see that there exists ˜U1:i ∈ {0, 1}i that obviously fulfil  Fi( ˜U1:i) ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Therefore, the idea is to determine such a ˜U1:i that additionally satisfies Fi(U1:i) ≤ Fi( ˜U1:i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Let l = max{j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , i} : Uj = 1} (we set max(∅) = 0) and U l  1:i = (U l  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , U l  i)T , where U l  j = Uj for all j ̸= l  and U l  l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We assume that l > 0 (if l = 0, we later see Fi(U1:i) ≤ α anyway).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In the next step we want to show that  Fi(U1:i) ≤ Fi(U l  1:i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For shorter notation we write αj = αj(U1:(j−1)) and αl  j = αj(U l  1:(j−1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since for all j ≤ i:  U l  j = Uj (j ̸= l),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' U l  j = 0 (j ≥ l),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Uj = 0 (j ≥ l + 1) and αl  j = αj (j ≤ l),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' we have:  Fi(U l  1:i) − Fi(U1:i)  =  i  �  j=1  αγj(1 − U l  j) −  i  �  j=1  αγj(1 − Uj) +  i  �  j=1  �j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jU l  kαl  k  1  τk − λk  �  (1 − U l  j)  −  i  �  j=1  �j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUkαk  1  τk − λk  �  (1 − Uj)  = αγl +  i  �  j=1  �j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jU l  kαl  k  1  τk − λk  �  (1 − U l  j) −  i  �  j=1  �j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUkαk  1  τk − λk  �  (1 − Uj)  = αγl +  l−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='lU l  kαl  k  1  τk − λk  +  i  �  j=l+1  l−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jU l  kαl  k  1  τk − λk  −  i  �  j=l+1  l  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUkαk  1  τk − λk  = αγl +  l−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='lUkαk  1  τk − λk  −  i  �  j=l+1  gl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jαl  1  τl − λl  ≥ αγl +  l−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='lUkαk  1  τk − λk  − αl  1  τl − λl  Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1  =  0,  where we used in the inequality that the sequence (gl,j)∞  j=l+1 is non-negative and sums to at most 1 for all l ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Since the U1:i ∈ {0, 1}i was arbitrary, this shows Fi(U1:i) ≤ Fi(U 0  1:i) for all U1:i ∈ {0, 1}i, where U 0  1:i =  (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , 0)T ∈ {0, 1}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Next, we deduce that Fi(U 0  1:i) ≤ α and conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' For this, just recognize that  U 0  1:i means Uj = 0 for all j ≤ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Hence, we obtain  Fi(U 0  1:i) =  i  �  j=1  αγj ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  15  A PREPRINT - JANUARY 30, 2023  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let G1:i = (R1, C1, S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , Ri, Ci, Si) ∈ {0, 1}3i, then every procedure satisfying the AD-  DIS principle (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='1) is a sequence of non-negative functions (αi(G1:(i−1)))i∈N such that  �  j≤i  αj(G1:(j−1))  τj − λj  (Sj − Cj) ≤ α  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (14)  Note that the function αi(G1:(i−1)) is fully determined through the information until step i − 1, hence pessimistic  assumptions about Si and Ci need to be made in order to satisfy equation (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Consequently, the condition of the  ADDIS principle is equivalent to  0 ≤ αi(G1:(i−1)) ≤ (τi − λi)  �  �α −  �  j≤i−1  αj(G1:(j−1))  τj − λj  (Sj − Cj)  �  �  for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  (15)  Let i ∈ N and G1:(i−1) ∈ {0, 1}3(i−1) be arbitrary but fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In addition, let (αj)j<i be levels obtained by an  ADDIS-Graph with parameters (γj)j<i and (gk,j)k<j<i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' We want to prove that  αi(γi, (gj,i)j<i)) = (τi − λi)  �  �αγi +  i−1  �  j=1  gj,i(Cj − Sj + 1)  αj  τi − λj  �  � ,  where γi ∈  �  0, 1 − �i−1  j=1 γj  �  and gj,i ∈  �  0, 1 − �i−1  k=j+1 gj,k  �  , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' , i − 1}, can take any value in the interval  �  0, (τi − λi)  �  α − �  j≤i−1  αj  τj−λj (Sj − Cj)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Since αi is continuous in γi and (gj,i)j<i, it is sufficient to show that  αi(0, (0)j<i) = 0 and αi  �  1 − �i−1  j=1 γj,  �  1 − �i−1  k=j+1 gj,k  �  j<i  �  = (τi − λi)  �  α − �  j≤i−1  αj  τj−λj (Sj − Cj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  The first equation follows immediately,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' hence we only need to show the second (we set Uj = Cj − Sj + 1 for all  j ∈ N):  αi  �  �1 −  i−1  �  j=1  γj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  �  �1 −  i−1  �  k=j+1  gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='k  �  �  j<i  �  � − (τi − λi)  �  �α −  i−1  �  j=1  αj  τj − λj  (1 − Uj)  �  �  = (τi − λi)  �  �α  �  �1 −  i−1  �  j=1  γj  �  � +  i−1  �  j=1  �  �1 −  i−1  �  k=j+1  gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='k  �  � Uj  αj  τi − λj  �  �  − (τi − λi)  �  �α −  i−1  �  j=1  �  αγj +  j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUk  αk  τk − λk  �  (1 − Uj)  �  �  = (τi − λi)  �  �−α  i−1  �  j=1  γjUj +  i−1  �  j=1  Uj  αj  τi − λj  −  i−1  �  j=1  i−1  �  k=j+1  gj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='kUj  αj  τi − λj  +  i−1  �  j=1  j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUk  αk  τk − λk  −  i−1  �  j=1  �j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUk  αk  τk − λk  �  Uj  �  �  = (τi − λi)  �  �  i−1  �  j=1  Uj  αj  τi − λj  −  i−1  �  j=1  Uj  �  αγj +  j−1  �  k=1  gk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='jUk  αk  τk − λk  ��  � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  16  A PREPRINT - JANUARY 30, 2023  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Let α0  j = W0γj + �j−1  k=1 hk,jRk[αTk + (α − W0)T c  k] for all j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Note that  i  �  j=1  α0  j =  i  �  j=1  W0γj +  i  �  j=1  j−1  �  k=1  hk,jRk[αTk + (α − W0)T c  k]  ≤ W0 +  i−1  �  k=1  Rk[αTk + (α − W0)T c  k]  i  �  j=k+1  hk,j  ≤ W0 +  i−1  �  k=1  Rk[αTk + (α − W0)T c  k]  = W0 + (α − W0)Ti + α(|R(i − 1)| − 1)Ti  ≤ α(|R(i)| ∨ 1)  With this, the proof can be performed as for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='2 by replacing αγj with α0  j on the left side in equation (13)  and α with α(|R(i)| ∨ 1) on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Funding  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Fischer acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) –  Project number 281474342/GRK2224/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Bofill Roig is a member of the EU Patient-centric clinical trial platform (EU-PEARL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' EU-PEARL has received  funding from the Innovative Medicines Initiative 2 Joint Undertaking under grant agreement No 853966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This Joint  Undertaking receives support from the European Union’s Horizon 2020 research and innovation programme and EF-  PIA and Children’s Tumor Foundation, Global Alliance for TB Drug Development non-profit organization, Spring-  works Therapeutics Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' This publication reflects the author’s views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Neither IMI nor the European Union, EFPIA, or  any Associated Partners are responsible for any use that may be made of the information contained herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Supplementary Material  The R code for the simulations and case study can be found at the GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='com/  fischer23/Adaptive-Discard-Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  References  Benjamini, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' and Hochberg, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Controlling the false discovery rate: a practical and powerful approach to  multiple testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Journal of the Royal statistical society: series B (Methodological), 57(1):289–300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Bretz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Maurer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Brannath, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Posch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A graphical approach to sequentially rejective multiple  test procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Statistics in medicine, 28(4):586–604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Pennllo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Petrick, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Sahiner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Pirracchio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Gossmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Sequential algorithmic  modification with test data reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In Uncertainty in Artificial Intelligence, pages 674–684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Fischer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Roig, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Brannath, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The online closure principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='11400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Foster, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' and Stine, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' α-investing: A procedure for sequential control of expected false discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 70(2):429–444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Javanmard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' and Montanari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online rules for control of false discovery rate and false discovery ex-  ceedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The Annals of statistics, 46(2):526–554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Karp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Mason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Beaudet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Benjamini, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Bower, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Braun, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Brown, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Chesler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Dickinson,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Flenniken, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Prevalence of sexual dimorphism in mammalian phenotypic traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Nature  communications, 8(1):1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Klinglmueller, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Posch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Koenig, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Adaptive graph-based multiple testing procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Pharmaceu-  tical Statistics, 13(6):345–356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Kohavi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Deng, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Frasca, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Walker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Xu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Pohlmann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online controlled experiments at  large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data  mining, pages 1168–1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  17  A PREPRINT - JANUARY 30, 2023  Muñoz-Fuentes, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Cacheiro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Meehan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Aguilar-Pimentel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Brown, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Flenniken, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Flicek,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Galli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Mashhadi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Hrabˇe de Angelis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The international mouse phenotyping consor-  tium (impc): a functional catalogue of the mammalian genome that informs conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Conservation genetics,  19(4):995–1005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Yang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wainwright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online control of the false discovery rate with  decaying memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Advances in neural information processing systems, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Zrnic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wainwright, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Saffron: an adaptive algorithm for online control of  the false discovery rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In International conference on machine learning, pages 4286–4294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Robertson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', König, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Posch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Jaki, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online error control for platform trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' arXiv  preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='03838.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Robertson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online multiple hypothesis testing for reproducible research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='11418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Robertson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wason, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Bretz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Graphical approaches for the control of generalized error rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Statistics in medicine, 39(23):3135–3155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Robertson, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Wildenhain, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Javanmard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Karp, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' onlinefdr: an r package to control the false  discovery rate for growing data repositories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Bioinformatics, 35(20):4196–4199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' and Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Addis: an adaptive discarding algorithm for online fdr control with conservative nulls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Advances in neural information processing systems, 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Tian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' and Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Online control of the familywise error rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Statistical Methods in Medical Research,  30(4):976–993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Zhao, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Small, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Su, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Multiple testing when many p-values are uniformly conservative, with  application to testing qualitative interaction in educational interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Journal of the American Statistical Asso-  ciation, 114(527):1291–1304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Zrnic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Jiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' The power of batching in multiple hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' In  International Conference on Artificial Intelligence and Statistics, pages 3806–3815.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Zrnic, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', Ramdas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Asynchronous online testing of multiple hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content=', 22:33–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldFKT4oBgHgl3EQfDS1x/content/2301.11711v1.pdf'}
diff --git a/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/2301.00204v1.pdf.txt b/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/2301.00204v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d8d0875368acc18043c8c922b4cead14c9106efc
--- /dev/null
+++ b/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/2301.00204v1.pdf.txt
@@ -0,0 +1,4052 @@
+arXiv:2301.00204v1  [math.SP]  31 Dec 2022
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK
+JACOBI MATRICES
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Abstract. It is well-known for self-adjoint scalar Jacobi operators that their absolute continuity and the
+absolute continuous spectrum in a subset of the real line can be characterized by non-existence of subordinate
+generalized eigenvectors. In our work we explore to what extent a similar relation is true for block Jacobi
+operators. In this setup, under some uniformity conditions, we show that nonsubordinacy in the sense of
+matrix generalized eigenvectors also implies the absolute continuity, similarly as in the case of the scalar
+Jacobi operator. Namely, we get the absolute continuity of the matrix spectral measure with the invertibility
+of its density almost everywhere. We also present an example showing that the reverse implication in general
+does not hold. We extend some sufficient conditions for nonsubordinacy from the scalar to the block case.
+Finally, we give applications of our results to some classes of block Jacobi matrices.
+Contents
+1.
+Introduction
+2
+Acknowledgment
+7
+2.
+Preliminaries
+7
+2.1.
+Introductory notation and notions
+7
+2.2.
+Some asymptotic symbols and the affine interpolation
+9
+2.3.
+Some analogs of “J–L continuous interpolation” of a discrete family of semi-norms
+10
+2.4.
+Some matrix norm inequalities
+11
+3.
+Block Jacobi matrix and operator(s). A spectral representation and the associated difference
+equations
+11
+3.1.
+The finite-cyclicity
+12
+3.2.
+The spectral matrix measure 퐸퐽, �휑 and the representation of 퐽 as the multiplication operator. 14
+3.3.
+The two associated difference equations and “the solution extensions to -1”
+15
+3.4.
+Transfer matrices and the Liouville–Ostrogradsky formulae
+18
+4.
+The Weyl function
+19
+4.1.
+ℓ2 matrix solutions and the matrix Weyl function 푊
+20
+4.2.
+The Cauchy transform of the spectral matrix measure and the matrix Weyl function
+22
+4.3.
+The boundary limits of the matrix Weyl function and the properties of the spectral trace
+measure
+23
+4.4.
+The boundary limits and spectral consequences for 퐽
+24
+5.
+An analog of the Jitomirskaya–Last’s approach
+24
+5.1.
+The vector and matrix nonsubordinacy
+24
+5.2.
+Barriers and barrier nonsubordinacy
+26
+5.3.
+The main result and its consequences
+29
+5.4.
+The proof of the main result
+31
+6.
+Sufficient conditions for absolute continuity
+35
+6.1.
+GLS condition
+35
+6.2.
+GBS condition
+36
+6.3.
+H class
+36
+7.
+Examples and applications
+37
+7.1.
+Vector nonsubordinacy does not characterise invertibility of 푀
+37
+7.2.
+GLS does not characterise the a.c. spectrum
+38
+2020 Mathematics Subject Classification. Primary 47B36.
+Key words and phrases. Block Jacobi matrix, absolutely continuous spectrum, matrix measures, Weyl function.
+1
+
+2
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+7.3.
+Application to some classes of block Jacobi operators
+39
+Appendix A.
+Vector and matrix measures — selected basic notions
+41
+References
+45
+1. Introduction
+A Jacobi matrix is a complex semi-infinite Hermitian matrix of the form
+(1.1)
+J =
+������
+�
+푏0
+푎0
+푎0
+푏1
+푎1
+푎1
+푏2
+...
+...
+...
+������
+�
+with 푎푛 ≠ 0 and 푏푛 ∈ R for all 푛 ∈ N0 = {0, 1, 2, . . .}. The action of J is well-defined on the linear
+space ℓ(N0, C) of all complex-valued sequences, and it is natural to define the operator 퐽 as the restriction
+of J to the standard Hilbert space ℓ2(N0, C) of square summable sequences. Namely, we define 퐽푥 = J푥
+for any 푥 ∈ ℓ2(N0, C) such that also J푥 ∈ ℓ2(N0, C). The operator 퐽 is called the (maximal) Jacobi
+operator. It does not have to be self-adjoint. If it is, then there exists a probability Borel measure 휇 on the
+real line, such that 퐽 is unitary equivalent to the operator acting by multiplication by the identity function
+on 퐿2(휇), see e.g. [53, Theorem 5.14 and Theorem 6.16]. The well-known sufficient condition for the
+self-adjointness of 퐽 is the Carleman condition
+(1.2)
++∞
+�
+푛=0
+1
+|푎푛| = ∞,
+see e.g. [53, Corollary 6.19] in the case of 푎푛 > 0 for all 푛.
+The interest in Jacobi operators comes from their close relation to the classical moment problem as
+well as the theory of orthogonal polynomials on the real line, see e.g. [54]. As every self-adjoint operator
+having a “non-degenerate” cyclic vector is unitary equivalent to a Jacobi operator, the Jacobi operators
+are basic building blocks of self-adjoint operators. Some types of Jacobi operators are related to random
+walks and birth–death processes, see e.g. [31,32]. Finally, Jacobi matrices are very useful in numerical
+analysis in the construction of Gaussian quadratures, see e.g. [18].
+A method of spectral analysis, the theory of subordinacy, due to Gilbert–Pearson [16] and later, due
+to Khan–Pearson, in its Jacobi variant (see [33]) started to be more and more prominent during the last
+three decades. Given 휆 ∈ C a sequence 푢 = (푢푛)푛∈N0 is called generalized eigenvector (associated with
+휆), 푢 ∈ GEV(휆), if it satisfies the recurrence relation
+(1.3)
+휆푢푛 = 푎푛−1푢푛−1 + 푏푛푢푛 + 푎푛푢푛+1,
+푛 ≥ 1
+with some initial conditions (푢0, 푢1). A non-zero sequence 푢 ∈ GEV(휆) is subordinate if for any linearly
+independent 푣 ∈ GEV(휆)
+(1.4)
+lim
+푛→∞
+∥푢∥ [0,푛]
+∥푣∥ [0,푛]
+= 0,
+where for a sequence 푥 ∈ ℓ(N0, C) and each 푛 ≥ 0 the seminorm ∥·∥ [0,푛] is given by
+(1.5)
+∥푥∥ [0,푛] :=
+�
+� 푛
+�
+푘=0
+|푥푘|2.
+Let us decompose the measure 휇 as
+휇 = 휇ac + 휇sing,
+where 휇ac and 휇sing denote the absolutely
+continuous and the singular part of 휇 with respect to the Lebesgue measure, respectively. The main
+result, [33, Theorem 3], defines two sets
+푆ac = {휆 ∈ R : no 푢 ∈ GEV(휆) is subordinate}
+푆sing = {휆 ∈ R : a non-zero 푢 ∈ GEV(휆) such that 푎0푢1 = (휆 − 푏0)푢0 is subordinate}
+and states that:
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+3
+• 푆ac is an essential support of 휇ac with respect to the Lebesgue measure,
+• 푆sing is a support of 휇sing and its Lebesgue measure is zero.
+The above measure theory assertions on 퐽 leads to purely spectral consequences of nonsubordinacy.
+Namely, it can be proved that for any 퐺 ∈ Bor(R)
+(i) If 퐺 ⊂ R \ 푆sing, then 퐽 is absolutely continuous in 퐺.
+(ii) If 퐺 ⊂ 푆ac, then 퐽 is absolutely continuous in 퐺 (see Definition 2.1) and clLe(퐺) ⊂ 휎ac(퐽).
+clLe(퐺) denotes here the Lebesgue closure of 퐺 — see (2.3). So, e.g., if moreover 퐺 is open, then its
+closure cl(퐺) ⊂ 휎ac(퐽).
+Since we often have some idea about asymptotic behaviour of generalized eigenvectors, this theory
+turned out to be very successful in spectral analysis of various classes of Jacobi matrices, see e.g. [37].
+Because of this similar theories exist for several classes of operators: continuous one-dimensional
+Schödinger operators on the real half-line (see [16]) and on the whole real line (see [15]), CMV matrices
+one-sided (see [17]) and two-sided (see [19]), one-dimensional Dirac operators (see [1]), Sturm–Liouville
+operators (see [4, 52]), canonical systems (see [20]), and Jacobi matrices on some types of graphs
+(see [38]). More detailed information on generalized eigenvectors allowed to obtain even more subtle
+spectral information on 퐽, e.g. uniform bounds on the density of 휇ac (see [4]) and absolute continuity
+with respect to Hausdorff measures (see [30]). An excellent survey containing more information on this
+subject is [14].
+In this article we extend some parts of Gilbert–Peason–Khan subordinacy theory to the setup of block
+Jacobi matrices. Thus, we consider block semi-infinite tridiagonal Hermitian matrices of the form (cf.
+(1.1))
+(1.6)
+J =
+������
+�
+퐵0
+퐴0
+퐴∗
+0
+퐵1
+퐴1
+퐴∗
+1
+퐵2
+...
+...
+...
+������
+�
+,
+where 퐴푛 and 퐵푛 are “blocks”, i.e., 푑 × 푑 complex matrices with all the 퐴푛 invertible and Hermitian 퐵푛.
+As before, the action of J is well-defined on the linear space ℓ(N0, C푑) of all C푑-valued sequences, and
+similarly to the scalar case block Jacobi operator 퐽 is simply the restriction of J to the Hilbert space
+ℓ2(N0, C푑), where
+ℓ2(N0, C푑) =
+�
+푥 ∈ ℓ(N0, C푑) :
++∞
+�
+푛=0
+∥푥푛∥2 < ∞
+�
+.
+If 퐽 is self-adjoint then there exists a non-negative matrix measure 푀 defined on the Borel subsets of
+the real line such that 퐽 is unitary equivalent to the operator acting by the multiplication by the identity
+function on the space 퐿2(푀) of C푑-valued functions, see Section 3.2 for details. Also the block variant
+of Carleman condition for self-adjointness of 퐽 looks similarly:
+(1.7)
++∞
+�
+푛=0
+1
+∥퐴푛∥ = ∞,
+see e.g. [2, Theorem VII-2.9].
+The interest in block Jacobi operators comes from their close relation to the matrix moment problem
+(see e.g. [3,10]) as well as from the theory of matrix orthogonal polynomials on the real line, see e.g. [6].
+They are useful for analysis of difference equations of finite order, see [11]. Some types of block Jacobi
+operators are related to random walks and level dependent quasi–birth–death processes, see e.g. [7]. For
+further applications we refer to [58].
+Spectral analysis of block Jacobi operators is not well-developed yet. In [51] the Mourre’s commutator
+method was applied to study compact perturbations of constant sequences 퐴푛, 퐵푛. Under some regularity
+hypotheses it was shown there that the singular continuous spectrum is empty. The commutator method
+was also applied in [27, 28] for obtaining a bound on the entries of the resolvent. In [61], by analysing
+Turán determinants, the continuous spectrum of bounded and unbounded Jacobi operators was studied.
+Finally, in [36] the problem of localisation of the essential spectrum of unbounded block Jacobi operators
+was studied by estimating the quadratic form associated to 퐽.
+
+4
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+It seems that the switch from the scalar to the 푑 > 1 case is especially problematic for the subordinacy
+theory. As far as we know, there is still a lack of any good understanding of how a “full analog” of this
+theory for the block Jacobi matrices might look like. In this article we concentrate ourselves only on “a
+second half” of this difficult question. – On the “negative” one.
+The general remark is, however, that – by several reasons of the algebraic character (see, e.g., Propo-
+sition 3.12) – it is a good idea to deal rather with some 푑 × 푑-matrix analogs of scalar objects from
+푑 = 1 theory, instead of their C푑-vector analogs, despite the fact that the Hilbert space for 퐽 consists of
+sequences of C푑-vectors (and not of 푑 × 푑-matrices).
+So, a sequence 푈 = (푈푛)푛∈N0 of 푀푑(C) matrices is called (matrix) generalized eigenvector for 퐽 and
+휆, i.e., 푈 ∈ MGEV(휆), if it satisfies the recurrence relation
+(1.8)
+휆푈푛 = 퐴∗
+푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1,
+푛 ≥ 1
+with some initial conditions (푈0,푈1).
+A natural condition, being a kind of “nonsubordinacy” (which in the 푑 = 1 case means exactly
+non-existence of subordinate solutions for 휆) is: for any non-zero 푈,푉 ∈ MGEV(휆)
+(1.9)
+lim inf
+푛→∞
+∥푈∥[0,푛]
+∥푉∥[0,푛]
+< ∞
+(cf. (1.4)), where for any sequence 푋 ∈ ℓ(N0, 푀푑(C)) and any 푛 ∈ N0 we define
+(1.10)
+∥푋∥[0,푛] =
+�
+� 푛
+�
+푘=0
+∥푋푘 ∥2,
+and we use the operator norm for matrices.
+In fact, motivated by the approach of Jitomirskaya–Last in [30], we use a slight reformulated version
+of (1.9). We define1 퐴−1 := − I and “extrapolating (1.8) to 푛 = 0” we “formally compute” 푈−1. Hence
+now we consider N−1 = {−1, 0, 1, . . .} as the index-set for extended generalized eigenvectors.
+If
+푈 ∈ MGEV(휆), then we say that such extended sequence (푈푛)푛∈N−1 belongs to MGEV−1 (휆). Next, for
+any 푋 ∈ ℓ(N−1, 푀푑(C))
+(1.11)
+∥푋∥ [0,푡] =
+�
+�
+� ⌊푡⌋
+�
+푘=0
+∥푋푘 ∥2 + {푡}
+��푋⌊푡⌋+1
+��2,
+where ⌊푡⌋ and {푡} are the integer and the fractional part of 푡, respectively. Then we say that 퐽 satisfies
+(matrix) nonsubordinacy condition (at 휆 ∈ R) if for any non-zero 푈,푉 ∈ MGEV−1 (휆)
+(1.12)
+lim inf
+푡→+∞
+∥푈∥[0,푡]
+∥푉∥ [0,푡]
+< +∞.2
+Let us mention that in Section 5.1 we also define the notion of vector nonsubordinacy, but it turns out to
+be equivalent to the matrix one. As we show in Theorem 5.3 the condition (1.12) implies that 휆 belongs
+to the continuous spectrum of 퐽.
+We shall be interested in a more quantitative version of (1.12). Namely, let 퐺 ⊂ R be non-empty. We
+say that a function 픟 : 퐺 × [1, +∞) → R is a barrier if for any 휆 ∈ 퐺 and any 푈,푉 ∈ MGEV−1 (휆)
+normalized by ∥푈−1∥2 + ∥푈0∥2 = ∥푉−1∥2 + ∥푉0∥2 = 1 we have
+� ∥푈∥ [0,푡]
+∥푉∥ [0,푡]
+�2
+≤ 픟(휆, 푡),
+푡 ≥ 1.
+In Proposition 5.7 we show that there is always an optimal barrier but it might not be the easiest one to
+deal with. Given a barrier 픟 we say that 퐽 is 픟-nonsubordinate on 퐺 if
+(1.13)
+lim inf
+푡→∞ 픟(휆, 푡) < +∞,
+휆 ∈ 퐺.
+1By I we denote the identity matrix of the appropriate dimension.
+2By symmetry we can define this equivalently, by requiring
+0 < lim sup푡→+∞
+∥푈 ∥ [0,푡]
+∥푉 ∥[0,푡]
+for any non-zero
+푈, 푉 ∈
+MGEV−1 (휆).
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+5
+If this condition holds uniformly on 퐺, namely
+(1.14)
+sup
+휆∈퐺
+lim inf
+푡→∞ 픟(휆, 푡) < +∞,
+then we say that 퐽 is uniformly 픟-nonsubordinate on 퐺.
+Our main abstract spectral result can be summarized as follows.
+Theorem 1.1. (see Theorem 5.13)
+Assume that 퐽 is self-adjoint, 퐺 ⊂ Bor(R) and 픟 is a barrier for 퐽
+on 퐺. If 퐽 is 픟-nonsubordinate on 퐺, then
+(a) 푀 is absolutely continuous on 퐺.
+(b) there exists a density 퐷 of 푀 on 퐺 which is an invertible matrix a.e. on 퐺.
+(c) 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).
+If, moreover, 퐽 is uniformly 픟-nonsubordinate on 퐺, then there exist 푐1, 푐2 > 0 such that the above density
+퐷 satisfies
+푐1 I ≤ 퐷(휆) ≤ 푐2 I
+for a.e. 휆 ∈ 퐺.
+The above density of 푀 and “a.e.” are both with respect to the Lebesgue measure.
+We have to emphasize that Theorem 1.1, unlike Khan–Pearson theory for the 푑 = 1 case, gives only a
+sufficient condition for the absolute continuity of 푀. As we constructively show in Example 7.1, there
+exist such block Jacobi matrices, that 푀 is absolutely continuous on R, 푀(퐵) is invertible for all non-
+empty open 퐵 ⊂ R but 퐽 does not satisfy (1.12) for any 휆 ∈ R. Nevertheless, as we show in Section 7.3,
+Theorem 1.1 is applicable to some explicit Jacobi matrices considered in the literature.
+In Section 6 we adapt to our setup some of the sufficient conditions implying non-existence of
+subordinate solutions, which were useful in the case 푑 = 1. In particular, the one introduced by Last–
+Simon in [37]. This conditions are formulated in terms of transfer matrices. Recall: if 푈 ∈ MGEV−1 (휆),
+then
+�
+푈푛
+푈푛+1
+�
+= 푇푛(휆)
+�
+푈푛−1
+푈푛
+�
+,
+푛 ≥ 0,
+where 푇푛(휆) is the (1-step) transfer matrix and
+푇푛(휆) =
+�
+0
+I
+−퐴−1
+푛 퐴∗
+푛−1
+퐴−1
+푛 (휆 I−퐵푛)
+�
+,
+푛 ≥ 0,
+where we have set 퐴−1 = − I. Then the 푛-step transfer matrix 푅푛(휆) := 푇푛−1(휆) . . .푇0(휆) satisfies
+�
+푈푛−1
+푈푛
+�
+= 푅푛(휆)
+�
+푈−1
+푈0
+�
+,
+푛 ≥ 1.
+Assume that the Carleman condition (1.7) holds and set
+휌푛 :=
+푛−1
+�
+푘=0
+1
+∥퐴푘 ∥,
+푛 ≥ 1.
+Thus 휌푛 → ∞. We shall say that 퐽 satisfies Generalized Last–Simon condition (GLS in short) on some
+Borel 퐺 ⊂ R if
+(1.15)
+lim inf
+푛→∞
+1
+휌푛
+푛
+�
+푘=1
+∥푅푘 (휆)∥2 < +∞,
+휆 ∈ 퐺.
+This condition was introduced in [37] for 푑 = 1 and 퐴푛 ≡ 1. It was proved there, that the maximal set
+퐺 ⊂ R, where (1.15) is satisfied is a minimal support of 휇ac. Recently, an analogous conclusion for 푑 ≥ 1
+was established in [48, Theorem 1.2] under the assumption that 퐴푛 = 퐴∗
+푛 and
+(1.16)
+sup
+푛≥0
+� ∥퐴푛∥ +
+��퐴−1
+푛
+�� � < ∞.
+We show in Example 7.2 that the Jacobi matrix corresponding to the Laguerre polynomials satisfies
+휎ac(퐽) = [0, ∞), yet, the condition (1.15) is violated for any non-empty 퐺 ⊂ (0, ∞). So, even for 푑 = 1,
+one cannot hope to obtain the part concerning the minimal support when (1.16) is not satisfied. However,
+we show in Theorem 1.1 that GLS condition implies the hypotheses of Theorem 1.1 for a suitably chosen
+barrier 픟 (see (5.20)). Let us recall that in [48, Theorem 1.2], under the assumption (1.16) and 퐴푛 = 퐴∗
+푛,
+
+6
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+a characterisation of multiplicities of the a.c. spectrum of 퐽 is provided in terms of asymptotic properties
+of singular values of the sequences 푃(휆), 푄(휆) ∈ MGEV−1 (휆) corresponding to the initial values
+(1.17)
+�
+푄−1(휆) = I
+푄0(휆) = 0,
+�
+푃−1(휆) = 0
+푃0(휆) = I,
+respectively. The approach used there is different than in the present work.
+Our approach to prove Theorem 1.1 is based on linking the asymptotic properties of MGEV−1 (푧) to
+the boundary values of the Cauchy transform of the matrix measure 푀. It is defined by
+푊(푧) =
+∫
+R
+d푀(휆)
+휆 − 푧 ,
+푧 ∈ C \ R
+and it turns out to be the matrix analogue of the well-known Weyl function. Since3 Im푊(푧) ≥ 0 for
+Im 푧 > 0, it is a matrix Herglotz function and its boundary values are closely related to properties of 푀,
+see Section 4.3 for details.
+Inspired by [30] we prove in Proposition 5.11 that if 퐽 is self-adjoint, then given 휆 ∈ R there exists a
+unique function ℓ휆 : R+ → R+ satisfying
+∥푃(휆)∥[0,ℓ휆 (휖 )] ∥푄(휆)∥[0,ℓ휆 (휖 )] = 1
+2휖 .
+Moreover, lim휖 →0+ ℓ휆(휖) = +∞.
+Adapting the approach of [30] and utilizing our novel object, namely the barrier function 픟, we prove
+our main result “on controlling the boundary limits of the matrix Weyl function”, being probably the most
+important result of this work.
+Theorem 1.2. (see Theorem 5.12)
+Assume that 퐽 is self-adjoint and 퐺 ⊂ R. If 픟 is a barrier for 퐽 on
+퐺, then for any 휆 ∈ 퐺 and any 휖 > 0 with ℓ휆(휖) ≥ 1
+(1.18)
+�8픟�휆, ℓ휆(휖)��−1 I ≤ Im푊(휆 + 푖휖)
+and
+(1.19)
+푠−(휆, 휖) ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+(휆, 휖),
+where
+(1.20)
+푠±(휆, 휖) := 4푑픟�휆, ℓ휆(휖)� ±
+��
+4푑픟�휆, ℓ휆(휖)��2
+− 1.
+Let us recall that Jitomirskaya–Last in [30] established a variant of the inequality (1.19) for 푑 = 1 with
+푠′
+± instead of 푠±, where
+푠′
+±(휆, 휖) := �5 ±
+√
+24� ∥푄(휆)∥ [0,ℓ휆(휖 )]
+∥푃(휆)∥[0,ℓ휆 (휖 )]
+.
+Then Khan–Pearson theory can be obtained by the rank-one perturbation theory applied to the corre-
+sponding Jacobi operator, see e.g. [52, Section 2] for details in the case of Sturm–Liouville operators.
+Let us point out that, under the hypotheses of Theorem 1.2, the last argument can be avoided once one
+uses our new inequality (1.18). Recently, an analogue of Jitomirskaya–Last inequality, expressed as a
+quotient of partial norms of 푄(휆) and 푃(휆) (and its singular values), has been obtained in [48] for 푑 ≥ 1.
+However, we were not able to use it to prove that GLS condition implies absolute continuity of 퐽. In
+contrast, our Theorem 1.2 allows us to prove Theorem 1.1, which as we have mentioned already, leads us
+to the proof of absolute continuity under GLS condition.
+The article is organized as follows. In Section 2 we fix our notation and prove basic results concerning
+the family of semi-norms (1.11). Next, in Section 3, we show that 퐽 is finitely cyclic, and as a consequence,
+it is unitary equivalent to the multiplication operator on the space 퐿2(푀) of C푑-valued functions for some
+matrix measure 푀. We also show an approach to Jacobi operators based on generalized eigenvectors and
+transfer matrices. In Section 4 we study the matrix Weyl function 푊 and its relation to properties of 푀.
+Section 5 is devoted to the proof of our main results: Theorem 1.2 — the main result of the paper and
+its spectral consequence: Theorem 1.1. In Section 6 we extend some well-known conditions implying
+3For any square matrix 푋 we define Im 푋 := 1
+2푖 (푋 − 푋∗).
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+7
+nonsubordinacy from 푑 = 1 to the general case. In particular, we cover: GLS (Generalized Last–Simon)
+condition, GBS (Generalized Behncke–Stolz) condition and 퐻 (homogenous) class condition. Finally,
+in Section 7, we show some examples and counterexamples illustrating the applicability of our results.
+For the sake of self-containment in Appendix A we collected basic notions concerning vector and matrix
+measures.
+Acknowledgment. The author Grzegorz Świderski was supported by long term structural funding –
+Methusalem grant of the Flemish Government. Part of this work was done while he was a postdoctoral
+fellow at KU Leuven. The author Marcin Moszyński wishes to thank:
+• Anna Moszyńska (IPEVP, Warsaw) – his wife – for extraordinary patience and for valuable
+linguistic help,
+• Nadia V. Zaleska (EIMI, St. Petersburg) – his friend – for some wise hints and for invaluable
+moral support,
+• Grzegorz Świderski – the co-author – for several years of confidence in success and for the
+barriers.
+2. Preliminaries
+We collect and fix here some general notation for in the paper, and we also introduce here several
+convenient tools, which will be important in the main sections.
+2.1. Introductory notation and notions. We use here the following symbols for some sets of scalars:
+C+ := {푧 ∈ C : Im(푧) > 0},
+R+ := {푡 ∈ R : 푡 > 0},
+N푘 := {푛 ∈ Z : 푛 ≥ 푘}
+for 푘 ∈ Z,
+so, e.g.,
+N = N1,
+N0 = N ∪ {0},
+N−1 = N ∪ {−1, 0}.
+Let us fix here some 푑 ∈ N. The vectors of the standard base in C푑 are denoted by 푒1, . . . , 푒푑.
+By 푀푑(C) we denote the space of all 푑 × 푑 complex matrices, with the usual matrix/operator norm.
+We identify any 퐴 ∈ 푀푑(C) with the appropriate linear transformation of C푑 induced by matrix 퐴.
+In particular, we shall use alternatively both the operator and the matrix notation for the action of 퐴
+on vectors from C푑, namely, for 푣 ∈ C푑 we typically use 퐴푣, but sometimes we also write 퐴푣T. For
+푖, 푗 ∈ {1, . . . , 푑} the term of 퐴 from its 푖-th row and 푗-th column is denoted as usual by 퐴푖, 푗 and 푣 푗
+is the 푗-th term of 푣. The symbol 퐴{ 푗} denotes the 푗-th column of 퐴 (usually we treat columns as
+C푑-vectors and not as one-column matrices). Similarly for 퐴{푖} — the 푖-th row of 퐴. Moreover, for
+vectors 푣(1), . . . , 푣(푑) ∈ C푑 the matrix 퐴 with 퐴{ 푗} = 푣( 푗) for any 푗 is denoted by [푣(1), . . . , 푣(푑)].
+If 푋 is a linear space, then by ℓ(N푘, 푋) we denote the linear space of all the sequences 푥 = (푥푛)푛∈N푘
+with terms in 푋 and ℓfin(N푘, 푋) is the subspace of ℓ(N푘, 푋) consisting of all the “finite” sequences, i.e.,
+of such 푥 that 푥푛 = 0 for 푛 sufficiently large.
+For sequences of vectors: if 푢(1), . . . , 푢(푑) ∈ ℓ(N푘, C푑) with 푢( 푗) = (푢( 푗)
+푛 )푛∈N푘 for 푗 = 1, . . . , 푑, then
+the symbol [푢(1), . . . , 푢(푑)], used already above for vectors and not for sequences of vectors, denotes the
+matrix sequence 푈 ∈ ℓ(N푘, 푀푑(C)) with 푈푛 := [푢(1)
+푛 , . . . , 푢(푑)
+푛 ] for any 푛 ≥ 푘.
+The symbols ∥·∥푋 and ⟨·, ·⟩푋 denote here the norm in a normed space 푋 and the scalar product in a
+Hilbert space 푋, respectively, but we often omit the subscript ‘푋’. This applies also to operator norms
+which we use by default for the bounded operators (mainly matrices from 푀푑(C), here), if no other choice
+is made. Our general rule here is: “to possibly omit the subscript ‘푋’ in almost all cases except the case
+of norm or the scalar product for C푑:
+∥·∥C푑, ⟨·, ·⟩C푑”.
+For a linear operator 퐴 : 푋 −→ 푋 in a normed space 푋 ≠ {0} we define its minimum modulus by
+(2.1)
+⇃|퐴|⇂:= inf
+∥푥 ∥=1 ∥퐴푥∥.
+Obviously, if dim 푋 = 1, then ⇃|퐴|⇂= ∥퐴∥. Recall that if 퐴 is invertible, then
+(2.2)
+⇃|퐴|⇂=
+1
+∥퐴−1∥,
+and for 0 < dim 푋 < +∞ 퐴 is invertible iff ⇃|퐴|⇂> 0.
+
+8
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+We use the symbols
+cl(퐺),
+clLe(퐺)
+for the “usual” (topological) closure and the “Lebesgue” closure of the set 퐺 ⊂ R, respectively (휆
+is
+used here for the complex conjugation of 휆 ∈ C). Recall, that
+(2.3)
+clLe(퐺) := {푡 ∈ R : ∀휀>0 |퐺 ∩ (푡 − 휀; 푡 + 휀)| > 0}
+for 퐺 ∈ Bor(R) and | · | is here the standard 1-dimensional Lebesgue measure on Bor(R) but, as usual,
+it will denote also the absolute value.
+If 휇 is a measure4 on 픐 – a 휎-algebra of subsets of some Ω, and 푝 ∈ [1, +∞), then 퐿 푝(휇) (without
+the “universum” Ω and the 휎-algebra for the measure, for short) denotes the standard 퐿 푝 Banach space
+of the classes of the appropriate complex functions on the “universum” for the measure 휇. We need
+sometimes to distinguish here 퐿 푝(휇) from the appropriate space of functions (and not classes) denoted
+here by L 푝(휇). For 푝 = 1 and 푋 := R푑 we also use L1
+푋 (휇) to denote the space of integrable functions
+from Ω into 푋 in the standard coordinatewise sense of the integral and the integrability.
+Moreover, for a matrix measure 푀 by 퐿2(푀) we denote the appropriate 퐿2-Hilbert space induced by
+this matrix measure (see Section 3.2 and, e.g., [44] for more details).
+If 푋 is a norm space, then
+ℓ2(N0, 푋) :=
+�
+푥 ∈ ℓ(N0, 푋) :
++∞
+�
+푛=0
+∥푥푛∥2
+푋 < ∞
+�
+,
+is a normed space with the norm defined for 푥 ∈ ℓ2(N0, 푋) by
+∥푥∥ :=
+�
+� +∞
+�
+푛=0
+∥푥푛∥2
+푋.
+If, moreover, 푋 is a Hilbert space, then ℓ2(N0, 푋) is a Hilbert space with the scalar product given for
+푥, 푦 ∈ ℓ2(N0, 푋) by
+(2.4)
+⟨푥, 푦⟩ :=
++∞
+�
+푛=0
+⟨푥푛, 푦푛⟩푋.
+Here, the most important case for us is the Hilbert space space ℓ2(N0, C푑). As the standard orthonormal
+basis of ℓ2(N0, C푑) we consider
+�
+훿푛(푒푖)
+�
+(푖,푛)∈{1,...,푑}×N0,
+where for any vector 푣 ∈ C푑 and 푛 ∈ N0 we define the sequence 훿푛(푣) ∈ ℓfin(N0, C푑) by
+(훿푛(푣)) 푗 :=
+�
+푣
+if 푗 = 푛,
+0
+otherwise.
+Moreover we have
+(2.5)
+ℓfin(N0, C푑) = lin{훿푛(푣) : 푣 ∈ C푑, 푛 ∈ N0},
+and
+cl(ℓfin(N0, C푑)) = ℓ2(N0, C푑).
+If H is a Hilbert space and 퐴 — a self-adjoint operator (possibly unbounded) in H, then the projection-
+valued spectral measure (“the resolution of identity”) for 퐴 is denoted by 퐸퐴.
+In particular 퐸퐴 :
+Bor(R) −→ B(H), where Bor(R) is the Borel 휎-algebra of R and B(H) denotes, as usual, the space of
+bounded operators on H. If 푥, 푦 ∈ H, then 퐸퐴,푥,푦 denotes the spectral measure for 퐴, 푥 and 푦, i.e. the
+complex measure given by
+(2.6)
+퐸퐴,푥,푦 (휔) := ⟨퐸퐴(휔)푥, 푦⟩ ,
+휔 ∈ Bor(R),
+4Here the name “measure” without some extra adjectives / names of the type vector, matrix, complex, real, spectral etc.,
+denotes always a classical Lebesgue-type measure with values in [0, +∞], without necessity of adding “non-negative”. The
+remaining ones, all “adjective (of the above type) + measures”, belongs to a wide class of vector measures (— see the definition
+on page 41) for an appropriate measure vector space; e.g, equal to R for real measure. So, measure can be, but also can be not
+(e.g. the Bor(R) Lebesgue measure), a vector measure. It simply depends on it finiteness. And “unfortunately”, from the point
+of view of the abuse of terminology, a vector measure is “usually” not a measure, here.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+9
+and 퐸퐴,푥 := 퐸퐴,푥,푥 (the spectral measure for 퐴 and 푥). Denote also
+Hac(퐴) := {푥 ∈ H : 퐸퐴,푥 is a.c. with respect to the Lebesgue measure on Bor(R)}.
+If and 퐺 ∈ Bor(R), then the symbol (퐴)퐺 denotes the part of 퐴 in the (invariant reducing) subspace
+H퐺(퐴) := Ran 퐸퐴(퐺).
+Recall the key notion for this paper, of the absolute continuity of 퐴:
+Definition 2.1. 퐴 is absolutely continuous (a.c.) in 퐺
+iff H퐺(퐴) ⊂ Hac(퐴).
+퐴 is absolutely continuous iff 퐴 is absolutely continuous in R.
+We apply here also some other “more or less, but not totally” common abbreviations and symbols:
+•
+iff
+for:
+if and only if
+•
+TFCAE
+for:
+the following conditions are (mutually) equivalent
+•
+w.r.t.
+for:
+with respect to
+•
+s.a.
+for:
+self-adjoint (for operators)
+•
+a.c.
+for:
+absolutely continuous (for operators, measures etc.)
+•
+sing.
+for:
+singular (as above)
+•
+a.e.
+for:
+almost everywhere
+•
+JM, BJM
+for:
+Jacobi matrix, block Jacobi matrix, respectively
+•
+JO, BJO
+for:
+Jacobi operator, block Jacobi operator, respectively
+•
+lin푌
+for:
+the linear subspace generated by a subset 푌 of a linear space
+•
+퐹↾푌
+for:
+the restriction of function 퐹 to the subset 푌 of the domain
+•
+퐹(푌)
+for:
+the image of subset 푌 with respect to function 퐹
+•
+Dom(퐴)
+for:
+the domain of linear operator 퐴
+•
+Dom(퐴∞)
+for:
+the intersection of all Dom(퐴푛) for 푛 ∈ N.
+Some further notation is introduced successively in next subsections.
+2.2. Some asymptotic symbols and the affine interpolation. Let 푆 be an arbitrary set and 푓 , 푔 : 푆 −→
+C. We define the symbol ≍ of “asymptotic similarity” of functions:
+(2.7)
+푓 ≍ 푔
+⇐⇒
+∃푐,퐶∈R+∀푠∈푆 푐|푔(푠)| ≤ | 푓 (푠)| ≤ 퐶|푔(푠)|
+(note the presence of the absolute value in this definition).
+We shall use also alternative notation:
+푓 (푠) ≍푠 푔(푠).
+If 푆 = N푘 for some 푘 ∈ Z, then 푓 is just a sequence from ℓ(N푘, C), i.e. 푓 = ( 푓 (푛))푛∈N푘 = ( 푓푛)푛∈N푘,
+and we shall consider its affine interpolation aff ( 푓 ) : [푘, +∞) −→ C, uniquely defined by the conditions:
+(a) aff ( 푓 )↾N푘= 푓 ,
+(b) for each 푛 ∈ N푘
+aff ( 푓 )↾[푛,푛+1] is an affine function, i.e. a function of the form
+[푛, 푛 + 1] ∋ 푡 ↦→ 푎푡 + 푐 ∈ C for some 푎, 푐 ∈ C (depending here also on 푛).
+In particular aff ( 푓 ) is a continuous interpolation of 푓 , and one can easily check that it can be also
+given by the explicit formula:
+(2.8)
+aff ( 푓 ) (푡) = 푓⌊푡⌋ + {푡} � 푓⌊푡⌋+1 − 푓⌊푡⌋
+� ,
+푡 ∈ [0, ∞)
+where {푡} = 푡 − ⌊푡⌋ is the fractional part of 푡.
+The following result joining the asymptotic similarity of sequences and of their affine interpolations,
+will be convenient in the proof of our main result.
+Let us prove first the following “slightly unexpected” result on quotients of two affine functions.
+Lemma 2.2. Suppose that 훼, 훽, 푎1, 푐1, 푎2, 푐2 ∈ R, 훼 < 훽 and 푎2푡 + 푐2 ≠ 0 for any 푡 ∈ [훼, 훽]. Let
+휑 : [훼, 훽] −→ R be given by
+휑(푡) := 푎1푡 + 푐1
+푎2푡 + 푐2
+,
+푡 ∈ [훼, 훽].
+Then 휑 is monotonic and, in particular, its maximal and minimal value is attained on the set {훼, 훽}.
+Proof. We have:
+휑′(푡) := 푎1푐2 − 푎2푐1
+(푎2푡 + 푐2)2
+and thus, we have only two cases:
+
+10
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+• 푎1푐2 − 푎2푐1 = 0, hence 휑 is constant;
+• 푎1푐2 − 푎2푐1 ≠ 0, so 휑′ has no zeros.
+In both cases our assertion holds.
+□
+2.3. Some analogs of “J–L continuous interpolation” of a discrete family of semi-norms. The main
+idea of the Jitomirskaya–Last’s new approachin [30] was “technically” based on a continuous interpolation
+of the discrete family of semi-norms {∥·∥푛}푛∈N0 in ℓ(N0, C) to the family {∥·∥푡}푡 ∈[0,+∞).
+For our purposes we shall extend this notion in two ways. Namely, let 푉 be a normed space, 푛0 ∈ Z
+and assume that 푋 ∈ ℓ(N푛0,푉).
+For any (푛1, 푡) ∈ Z × R such that 푛0 ≤ 푛1 ≤ 푡, we define
+(2.9)
+∥푋∥ [푛1,푡] :=
+�
+⌊푡⌋
+�
+푘=푛1
+∥푋푘 ∥2 + {푡}
+��푋⌊푡⌋+1
+��2
+�1/2
+,
+and for 푡 = ∞
+(2.10)
+∥푋∥[푛1,∞] :=
+� +∞
+�
+푘=푛1
+∥푋푘 ∥2
+�1/2
+(which can possibly be +∞).
+Moreover, if 푉 = 푀푑(C)) for some 푑 ≥ 1, then similarly to (2.9) we define a continuous family based
+on minimum modulus (see (2.1)) instead of matrix norm. So, for any (푛1, 푡) ∈ Z×R such that 푛0 ≤ 푛1 ≤ 푡
+we consider
+(2.11)
+⇃|푋|⇂[푛1,푡]=
+�
+⌊푡⌋
+�
+푘=푛1
+⇃|푋푘|⇂2 +{푡}⇃|푋⌊푡⌋+1|⇂2
+�1/2
+.
+By (2.8) we can relate the above constructions with the notion of affine extension introduced in the
+previous subsection. The squares of the “new objects” are just the affine extensions of their discrete
+counterparts (see (2.12) and (2.13) below), which are somewhat more natural and simpler for the context
+of operators “acting on sequences” considered in this paper.
+Observation 2.3. The notions defined by (2.9) and (2.11) (i.e, for 푡 < +∞) satisfy respectively
+∥푋∥2
+[푛1,푡] = aff �푆푋,푛1
+� (푡)
+and
+⇃|푋|⇂2
+[푛1,푡]= aff �ˇ푆푋,푛1
+� (푡)
+for 푡 ≥ 푛1, where 푆푋,푛1, ˇ푆푋,푛1 : N푛1 −→ R are given for 푛 ≥ 푛1 by
+(2.12)
+푆푋,푛1 (푛) := ∥푋∥2
+[푛1,푛] =
+푛
+�
+푘=푛1
+∥푋푘∥2 ,
+(2.13)
+ˇ푆푋,푛1 (푛) :=⇃|푋|⇂2
+[푛1,푛]=
+푛
+�
+푘=푛1
+⇃|푋푘|⇂2 .
+Note also that in the scalar case 푑 = 1 for 푋 ∈ ℓ(N푛0, 푀푑(C)) we simply have ∥푋∥[푛1,푡] =⇃|푋|⇂[푛1,푡]
+and 푆푋,푛1 = ˇ푆푋,푛1.
+Consider a second sequence 푌 ∈ ℓ(N푛0,푉). Using the above Observation and Lemma 2.2 we get:
+Corollary 2.4. Suppose that 푛1 ∈ Z, 푡 ∈ R, 푛0 ≤ 푛1 ≤ 푡. Let 푛 := ⌊푡⌋ be such that ∥푌 ∥[푛1,푛] ≠ 0. Then
+there exist such 푛, 푛 ∈ {푛, 푛 + 1} that
+∥푋∥[푛1,푛]
+∥푌 ∥[푛1,푛]
+≤
+∥푋∥[푛1,푡]
+∥푌 ∥[푛1,푡]
+≤
+∥푋∥ [푛1,푛]
+∥푌 ∥[푛1,푛]
+.
+If 푉 = 푀푑(C) then the analogous result with all the “∥·∥” replaced by “⇃|·|⇂” is also true.
+Proof. It suffices to use Observation 2.3 and then Lemma 2.2 to the function 휑 given on [푛, 푛 + 1] by the
+formula 휑(푡) :=
+� ∥푋 ∥ [푛1,푡]
+∥푌 ∥ [푛1,푡]
+�2
+.
+□
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+11
+2.4. Some matrix norm inequalities. Let us recall some notions related in particular to 푑 × 푑 matrices.
+For 푋 ∈ 푀푑(C):
+• its Hilbert–Schmidt norm is defined by
+(2.14)
+∥푋∥HS =
+�
+푑
+�
+푖=1
+∥푋푒푖∥2
+C푑
+�1/2
+.
+• its real and imaginary parts Re(푋) and Im(푋) (in the adjoint, and not the complex conjugation
+sense) are given by
+Re(푋) := 1
+2 (푋 + 푋∗),
+Im(푋) := 1
+2푖 (푋 − 푋∗).
+For 푣 ∈ C푑 we shall use the symbol 퐸 푣 to denote the matrix / operator [푣, 0, . . . , 0] ∈ 푀푑(C) , i.e,
+(2.15)
+퐸 푣(푒 푗) =
+� 푣
+for 푗 = 1
+0
+for 푗 > 1,
+푗 = 1, . . . , 푑.
+Proposition 2.5. Let 푋 ∈ 푀푑(C) and 푣 ∈ C푑. Then:
+(i) ∥푋∥ ≤ ∥푋∥HS.
+(ii) ∥Im(푋)∥ ≤ ∥푋∥ , ∥Re(푋)∥ ≤ ∥푋∥.
+(iii) ∥푋퐸 푣∥ = ∥푋푣∥.
+(iv) If 푋 ≥ 0, then
+∥푋∥ ≤ tr 푋 ≤ 푑 ∥푋∥.
+Proof. Part (i) is a classical result (easy to get by the Schwarz inequality), and (ii) follows directly from
+the definitions of Re, Im.
+To get (iii) observe first that 푋퐸 푣 = 퐸푋 푣 by (2.15), so it suffices to consider 푋 = 퐼. But using (i) we
+get ∥퐸 푣∥ ≤ ∥푣∥ and ∥퐸 푣∥ ≥ ∥퐸 푣푒1∥ = ∥푣∥, so the equality holds.
+Suppose that 푋 ≥ 0 and let 휆1 . . . , 휆푑 be all the eigenvalues of 푋 repeated according to their multi-
+plicities. We thus get (iv) by
+∥푋∥ = max
+1≤푖≤푑 휆푖 ≤
+푑
+�
+푖=1
+휆푖 = tr 푋 ≤ 푑 max
+1≤푖≤푑 휆푖 = 푑 ∥푋∥
+□.
+3. Block Jacobi matrix and operator(s). A spectral representation and the associated
+difference equations
+We denote “the size of the block” for BJM by 푑 ∈ N, and for the whole paper we assume that (퐴푛)푛∈N0
+and (퐵푛)푛∈N0 are sequences of matrices from 푀푑(C) such that
+(3.1)
+det 퐴푛 ≠ 0,
+퐵푛 = 퐵∗
+푛,
+푛 ∈ N0.
+Now we define a block Jacobi matrix
+J =
+������
+�
+퐵0
+퐴0
+퐴∗
+0
+퐵1
+퐴1
+퐴∗
+1
+퐵2
+...
+...
+...
+������
+�
+,
+and the pair of the sequences 퐴 = (퐴푛)푛∈N0 and 퐵 = (퐵푛)푛∈N0 will be called Jacobi parameters of J.
+In fact, we mean that J is the linear operator (“the formal block Jacobi operator”) acting on the space
+ℓ(N0, C푑) of all the C푑 sequences, whose action is well-defined via formal matrix multiplication:
+J : ℓ(N0, C푑) −→ ℓ(N0, C푑),
+i.e., for any 푢 ∈ ℓ(N0, C푑)
+(3.2)
+(J푢)푛 :=
+� 퐵0푢0 + 퐴0푢1
+for 푛 = 0
+퐴∗
+푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1
+for 푛 ∈ N.
+Recall now two important operators related to J and acting in the Hilbert space ℓ2(N0, C푑):
+퐽min
+and 퐽 (which will coincide in most of our further considerations). The operator 퐽min (the minimal block
+Jacobi operator) is simply the closure in ℓ2(N0, C푑) of the operator in ℓ2(N0, C푑) being the restriction:
+
+12
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+J↾ℓfin(N0,C푑). And to define 퐽 (the maximal block Jacobi operator) we first choose its domain in a usual
+way for “maximal-type” operators:
+(3.3)
+Dom(퐽) :=
+�
+푢 ∈ ℓ2(N0, C푑) : J푢 ∈ ℓ2(N0, C푑)
+�
+,
+and then we define 퐽 := J↾Dom(퐽).
+When both (퐴푛)푛∈N0 and (퐵푛)푛∈N0 are bounded sequences, then obviously we have only one operator
+퐽min = 퐽 ∈ B(ℓ2(N0, C푑)), which is symmetric, so self-adjoint. But here we consider also unbounded
+cases, so let us recall the following result
+Fact 3.1. (퐽min)∗ = 퐽. Moreover, TFCAE:
+(i) 퐽min is s.a.,
+(ii) 퐽 is s.a.;
+and if one of the above conditions holds, then 퐽min = 퐽.
+See [2, Chapter VII.§2.5] for5 퐽∗
+min = 퐽, and the remaining part follows from this by [56, formula
+(7.1.22)].
+For the main results of the theory presented here the maximality of the domain is crucial. Moreover,
+we study mainly “the self-adjoint case”, so (by Fact 3.1) the best choice here is to use just only the operator
+퐽, later on.
+By (3.3) and (3.2) we get
+(3.4)
+ℓfin(N0, C푑) ⊂ Dom(퐽),
+퐽
+�
+ℓfin(N0, C푑)
+�
+⊂ ℓfin(N0, C푑),
+so in particular 퐽 is densely defined by (2.5). Moreover, by (3.2) we compute
+(3.5)
+퐽 (훿푛(푣)) = 훿푛−1(퐴푛−1푣) + 훿푛(퐵푛푣) + 훿푛+1(퐴∗
+푛푣),
+푣 ∈ C푑, 푛 ∈ N0,
+where we additionally denote
+훿−1(푣) := 0,
+푣 ∈ C푑.
+3.1. The finite-cyclicity. In the scalar 푑 = 1 case Jacobi operator 퐽 is cyclic with a cyclic vector
+휑 := 훿0(푒1) (the canonical cyclic vector for 퐽), which means that the space
+(3.6)
+lin{퐽푛휑 : 푛 ∈ N0}
+is dense in ℓ2(N0, C) (here, for 푑 = 1, we simply have 푒1 = 1 ∈ C). — Indeed, this is known that the
+above space is just equal to ℓfin(N0, C) for such 휑. Cyclicity plus self-adjointness provides a very simple
+spectral representation of the operator.
+So, suppose now, that “the scalar” 퐽 is s.a. and consider:
+x — the identity function on R,
+x(푡) = 푡
+for 푡 ∈ R, and 휇 — “the scalar” spectral measure for 퐽 and 휑, i.e.
+휇 = 퐸퐽,휑 (see, Section 2.1 for the
+spectral notation).
+By the well-known spectral result, 퐽, as a cyclic s.a. operator, is unitary equivalent to the operator of
+the multiplication by
+x in the space 퐿2(휇).
+This one-dimensional result has also its analog for the block-Jacobi case with arbitrary 푑. First of
+all, instead of cyclicity, we consider here the so-called finite-cyclicity notion (see [44]). It means that
+for some 푘 ∈ N there exists a cyclic system �휑 = (휑1, . . . , 휑푘) for 퐽, i.e., a system of such vectors from
+Dom(퐽∞), that
+(3.7)
+lin{퐽푛휑 푗 : 푛 ∈ N0, 푗 = 1, . . . , 푘}
+is dense in ℓ2(N0, C푑). In our general 푑-dimensional case the choice of �휑 can be done in an analogical
+way, as it was for 푑 = 1, namely define
+(3.8)
+�휑 := (휑1, . . . , 휑푑),
+휑 푗 := 훿0(푒 푗),
+푗 = 1, . . . , 푑.
+The following result is true.
+Proposition 3.2. The system �휑 is a cyclic system for 퐽.
+5Actually, in [2, Chapter VII.§2.5] only the case 퐴푛 = 퐴∗푛 for all 푛 ≥ 0 was considered, but the proof in the general case is
+similar.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+13
+Proof. For each 푛 ∈ N0 denote6
+(3.9)
+푋푛 :=
+�
+푥 ∈ ℓ2(N0, C푑) : ∀푗≠푛 푥 푗 = 0
+�
+,
+푌푛 :=
+푛
+�
+푘=0
+푋푘.
+So 푋푛,푌푛 ⊂ ℓfin(N0, C푑) and
+(3.10)
+푋푛 =
+�
+훿푛(푤) : 푤 ∈ C푑�
+= lin {훿푛(푒푠) :
+푠 = 1, . . . , 푑} ,
+푛 ∈ N0.
+In particular
+(3.11)
+푋0 = lin {휑푠 :
+푠 = 1, . . . , 푑} .
+Using (3.10) and (3.5) we get
+퐽(푋푛) ⊂ 푌푛+1,
+푛 ∈ N0,
+so also
+(3.12)
+퐽(푌푛) ⊂ 푌푛+1,
+푛 ∈ N0.
+Using this by the obvious induction we obtain
+(3.13)
+퐽푛(푋0) ⊂ 푌푛,
+푛 ∈ N0.
+Denote 푌−1 := {0} and for 푘 ∈ N0 denote also
+(3.14)
+퐶푘 :=
+�
+(퐴0 · · · · · 퐴푘−1)∗
+for 푘 > 0
+퐼
+for 푘 = 0,
+so in particular 퐶푘 is invertible. To continue the proof, we shall first prove the following two results.
+Lemma 3.3. For any 푘 ∈ N0
+(3.15)
+∀푤 ∈C푑 ∃푢∈푌푘−1 퐽푘훿0(푤) = 푢 + 훿푘 (퐶푘푤).
+Proof of Lemma 3.3. For 푘 = 0 and 푤 ∈ C푑 we have 퐽푘훿0(푤) = 훿0(푤) = 0 + 훿0(퐶0푤), so (3.15) holds
+for 푘 = 0. Suppose now that it holds for some 푘 ∈ N0. Then by (3.15) for 푘 and by (3.5), for 푤 ∈ C푑
+퐽푘+1훿0(푤) = 퐽(퐽푘훿0(푤)) = 퐽(푢) + 퐽 (훿푘 (퐶푘푤)) = 퐽(푢) + 푢′ + 훿푘+1(퐴∗
+푘퐶푘푤),
+with some 푢 ∈ 푌푘−1, 푢′ ∈ 푌푘. Finally, by (3.12) and (3.14)
+퐽푘+1훿0(푤) = 푢′′ + 훿푘+1(퐴∗
+푘퐶푘푤) = 푢′′ + 훿푘+1(퐶푘+1푤),
+with some 푢′′ ∈ 푌푘. So (3.15) holds for 푘 + 1, and we obtain our assertion by the induction.
+□
+We shall use this lemma to prove the result below.
+Fact 3.4. For any 푛 ∈ N0
+(3.16)
+푌푛 =
+푛
+�
+푗=0
+퐽 푗 (푋0).
+Proof of Fact 3.4. We get “⊃” from (3.13). Let us prove “⊂” by induction. For 푛 = 0 the assertion is
+obvious. Now consider 푛 ∈ N0 and suppose that 푌푛 ⊂ �푛
+푗=0 퐽 푗 (푋0). Then
+lin
+�
+푌푛 ∪ 퐽푛+1(푋0)
+�
+⊂
+푛+1
+�
+푗=0
+퐽 푗 (푋0),
+by the linearity of the RHS. Thus, by (3.9), to get 푌푛+1 ⊂ �푛+1
+푗=0 퐽 푗 (푋0), it suffices to prove
+(3.17)
+푋푛+1 ⊂ lin
+�
+푌푛 ∪ 퐽푛+1(푋0)
+�
+.
+Indeed, if 푤 ∈ C푑, then by Lemma 3.3
+훿푛+1(푤) = 푣 + 퐽푛+1훿0(퐶−1
+푛+1푤),
+6Hereweusethestandardoperationof thesum ofsubsets ofa linear space, i.e., �푛
+푘=0 퐷푘 := {�푛
+푘=0 푥푘 : 푥푘 ∈ 퐷푘 for any 푘 =
+0, . . . 푛} for subsets 퐷0, . . . 퐷푛.
+
+14
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+with some 푣 ∈ 푌푛. Therefore, by (3.10) we get (3.17).
+□
+Let us continue the proof of Proposition 3.2. Using again the standard argumentation concerning the
+linear spaces generated by a subset, we see by (3.11) that
+(3.18)
+lin{퐽 푗휑푠 : 푗 = 0, . . . , 푛, 푠 = 1, . . . , 푑} =
+푛
+�
+푗=0
+퐽 푗 (푋0),
+푛 ∈ N0.
+So, by Fact 3.4 and (3.18)
+lin{퐽푛휑푠 : 푛 ∈ N0, 푠 = 1, . . . , 푑} =
++∞
+�
+푛=0
+lin{퐽 푗휑푠 : 푗 = 0, . . . , 푛, 푠 = 1, . . . , 푑} =
++∞
+�
+푛=0
+푌푛 = ℓfin(N0, C푑),
+and the density of ℓfin(N0, C푑) in ℓ2(N0, C푑) finishes the proof.
+□
+Surely, this choice of a cyclic system for 퐽 is not unique, but this particular �휑 defined by (3.8) is called
+canonical for 퐽.
+3.2. The spectral matrix measure 퐸퐽, �휑 and the representation of 퐽 as the multiplication operator.
+The next notion which should be generalized, when we are moving from the scalar to the block case,
+is the classical 퐿2- type Hilbert space with “the scalar non-negative” spectral measure 휇 for 퐽 (and for
+the canonical cyclic vector 휑). It should be replaced by the less-known 퐿2-type Hilbert space with the
+so-called spectral matrix measure for the operator 퐽 and for its canonical cyclic system �휑, described in
+Section 3.1. The spectral matrix measure for 퐽 is a particular example of the general notion of matrix
+measure (see Appendix A). Similarly to the “scalar” spectral measure for the Jacobi (the scalar one) case,
+is defined on Bor(R) and is tightly related to 퐽. E.g., 퐽 can be recovered from its spectral matrix measure
+up to a unitary equivalence, as follows from Theorem 3.6 below.
+Recall (see [44]):
+Definition 3.5. The spectral matrix measure 퐸퐽, �휑 for 퐽 is given by
+(3.19)
+퐸퐽, �휑 : Bor(R) → 푀푑(C),
+퐸퐽, �휑(휔) :=
+�
+퐸퐽,휑 푗,휑푖 (휔)
+�
+푖, 푗=1,...,푑 ∈ 푀푑(C),
+휔 ∈ Bor(R)
+(see (2.6) for the symbol 퐸퐽,휑 푗,휑푖), where �휑 = (휑1, . . . , 휑푘) is canonical cyclic system for 퐽.
+The analog of the scalar-Jacobi unitary representation result, mentioned above, can be formulated for
+our block-Jacobi case in a short way, as follows.
+Theorem 3.6. 퐽 is unitary equivalent to the operator of the multiplication by
+x in the space 퐿2(푀).
+For the full formulation and the proof in the general finitely cyclic case see [44,
+xMUE Theorem]7.
+In particular, the above result means that the spectral matrix measure of 퐽 “contains” all the important
+spectral information about 퐽, similarly to the spectral measure 휇 in the scalar Jacobi case. For instance,
+such typically studied information, as: the absolute continuity or the singularity in some subset of R, the
+location of the spectrum and of some particular kinds of spectra of 퐽, etc.
+On the other hand, “the nice properties” of the trace measure tr푀 (see A.4) for matrix measure 푀
+suggest that instead of dealing with spectral matrix measure, somewhat sophisticated at times, it could
+be more useful to deal with its trace measure. Especially, when we try to “control” the above mentioned
+spectral properties of 퐽 related, e.g., to the absolute continuity or singularity. In Fact A.5, Fact A.6,
+and Lemma A.7 from Appendix A we discussed this problem in more details for its abstract (vector)
+measure theory aspect. Their main “spectral operator theory” consequences for 퐽 are Proposition 4.8 and
+Proposition 4.7 in Section 4.
+7Also the detailed definition of the multiplication by a function operator in 퐿2-matrix measure spaces is presented there.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+15
+3.3. The two associated difference equations and “the solution extensions to -1”. For any 푧 ∈ C let
+us consider two difference equations tightly related to J. The first — “the vector” one — is the infinite
+system of equations for a sequence 푢 = (푢푛)푛∈N0 ∈ ℓ(N0, C푑):
+(3.20)
+(J푢)푛 = 푧푢푛,
+푛 ≥ 1.
+Each such a vector sequence 푢 is called generalized eigenvector (for 퐽 and 푧)8 — “gev” for short. By
+(3.2) equivalently its explicit form can be written
+(3.21)
+퐴∗
+푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1 = 푧푢푛,
+푛 ≥ 1.
+The second difference equation — “the matricial” one — is the analog equation (with the right-side
+multiplication choice9) for a matrix sequence 푈 = (푈푛)푛∈N0 ∈ ℓ(N0, 푀푑(C)):
+(3.22)
+퐴∗
+푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1 = 푧푈푛,
+푛 ≥ 1,
+and each such a matrix sequence 푈 is called matrix generalized eigenvector (for 퐽 and 푧) — “mgev” for
+short.
+Having our BJM J fixed, for 푧 ∈ C we denote
+GEV(푧) := {푢 ∈ ℓ(N0, C푑) : 푢 is a gev for 퐽 and 푧}
+and parallelly
+MGEV(푧) := {푈 ∈ ℓ(N0, 푀푑(C)) : 푈 is a mgev for 퐽 and 푧},
+being obviously liner subspaces of ℓ(N0, C푑) and ℓ(N0, 푀푑(C)), respectively. By (3.1), both recurrence
+relations are of degree 2, in the sense that for any initial condition �퐶0, 퐶1
+� ∈ (푀푑(C))2 there is a unique
+sequence 푈 satisfying (3.22) with 푈0 = 퐶0, 푈1 = 퐶1, and analogously for (3.21). More precisely, one
+easily check the following.
+Fact 3.7. For any 푧 ∈ C the map Ini푧;0,1 : GEV(푧) −→ �C푑�2, given by
+Ini푧;0,1(푢) = (푢0, 푢1),
+푢 ∈ GEV(푧) ,
+is a linear isomorphism. The analogous result is true for MGEV (푧) and (푀푑(C))2. In particular
+dim GEV(푧) = 2푑 and dim MGEV(푧) = 2푑2.
+However, it is often more convenient to use another kind of “initial conditions”, namely “at −1 and
+0” instead of 0 and 1. To formulate this properly, we shall define first the appropriate extension of each
+solution (in both, vector and matrix cases), which is tightly related to the “a priori choice” of 퐴−1:
+(3.23)
+퐴−1 := − I .
+The informal idea of the extension is simply to “extend to 푛 = 0” the system (3.22) (and (3.21) analogously)
+and to “compute the “value at −1”, using our choice made in (3.23), i.e., we get “푈−1 := (퐵0 − 푧 I)푈0 +
+퐴0푈1” for the matrix case. Unfortunately, as one can see, such a definition seems to depend explicitly on
+the parameter 푧, and not only on 푈. So, at the first sight it seems that the notation for the extension “to −1”
+of a solution 푈 has to contain always this parameter, which would be not very convenient. And YES — it is
+true, that we have really this problem, extending in such a way any sequence 푈 ∈ ℓ(N0, 푀푑(C)) (similarly
+for the vector version). So we define first the following family {푧•}푧∈C of “extending transformations”
+푧• : ℓ(N0, 푀푑(C)) −→ ℓ(N−1, 푀푑(C)) given for 푈 ∈ ℓ(N0, 푀푑(C)) and 푧 ∈ C simply by
+(3.24)
+(푧•푈)푛 :=
+�
+(퐵0 − 푧 I)푈0 + 퐴0푈1
+for 푛 = −1
+푈푛
+for 푛 ∈ N0
+We shall use here the same notation for the vector sequences without any risk of confusion, i.e., we shall
+also write 푧•푢 for 푢 ∈ ℓ(N0, C푑) and 푧 ∈ C with the analogous meaning
+(3.25)
+(푧•푢)푛 :=
+�
+(퐵0 − 푧 I)푢0 + 퐴0푢1
+for 푛 = −1
+푢푛
+for 푛 ∈ N0.
+8Note here, that it is “generalized” for two reasons; the first, because 푢 may not belong to Dom(퐽), not even ℓ2(N0, C푑),
+and the second, since we do not require the equality for 푛 = 0 above.
+9The left-side one is also possible and used for several reasons, but we shall not consider it here.
+
+16
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Therefore, for any 푧 ∈ C both kinds of transformations are linear, and moreover:
+(3.26)
+푧•(푈퐶) = (푧•푈)퐶,
+for any 푈 ∈ ℓ(N0, 푀푑(C)), 퐶 ∈ 푀푑(C).
+Fortunately, the situation is much simpler if we restrict ourselves to the subspace of all the (M)GEV-s.
+Namely, we have:
+Fact 3.8. If 푧, 푤 ∈ C, 푧 ≠ 푤, then10
+GEV(푧) ∩ GEV(푤) = {0},
+MGEV(푧) ∩ MGEV (푤) = {0}.
+Proof. For the matrix case, consider 푈 satisfying both (3.22), and its analog for 푤. Then, subtracting,
+we get
+0 = (푧 − 푤)푈푛,
+푛 ≥ 1,
+hence 푈푛 = 0 for any 푛 ∈ N, but now again by (3.22) used only for 푛 = 1 we get also 푈0 = 0, i.e,
+푈 = 0.
+□
+This means, that for any non-zero vector or matrix solution of our equations, the parameter ’푧’ is in
+fact “coded in the solution”. On the other hand, the value of the extension •푧 for the zero sequence is
+obviously also the zero sequence (indexed from −1 already) by the linearity, independently of 푧. Hence
+denote
+GEV :=
+�
+푧∈C
+GEV(푧) ,
+MGEV :=
+�
+푧∈C
+MGEV(푧) ,
+and, thank to Fact 3.8, for any 푈 ∈ MGEV \ {0} (푢 ∈ GEV \ {0}) we can define Par(푈) (Par(푢)) as
+the unique number 푧 ∈ C satisfying
+푈 ∈ MGEV(푧) (푢 ∈ GEV(푧)).
+Finally, we can simplify our notation and omit the parameter ’푧’, defining
+• : MGEV −→ ℓ(N−1, 푀푑(C))
+given for 푈 ∈ MGEV simply by the formula
+(3.27)
+•푈 :=
+�
+Par(푈)•푈
+for 푈 ≠ 0
+0
+for 푈 = 0
+and analogically for 푢 ∈ GEV.
+Now, taking into account (3.23), let us consider “extensions” of the systems (3.21) and (3.22):
+(3.28)
+퐴∗
+푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1 = 푧푢푛,
+푛 ≥ 000
+for sequences 푢 = (푢푛)푛∈N−1 ∈ ℓ(N−1, C푑) and
+(3.29)
+퐴∗
+푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1 = 푧푈푛,
+푛 ≥ 000
+for sequences 푈 = (푈푛)푛∈N−1 ∈ ℓ(N−1, 푀푑(C)). Their solutions will be called extended generalized
+eigenvectors and extended matrix generalized eigenvectors, respectively, (for 퐽 and 푧) — “egev” and
+“emgev” for short.
+We denote also
+GEV−1 (푧) :=
+�
+푢 ∈ ℓ(N−1, C푑) : 푢 is an egev for 퐽 and 푧
+�
+and
+MGEV−1 (푧) :=
+�
+푈 ∈ ℓ(N−1, 푀푑(C)) : 푈 is an emgev for 퐽 and 푧
+�
+.
+Let us formulate explicitly some simple relations between all the above “extended” and “non-extended”
+notions and the • transformation.
+Fact 3.9. For any 푧 ∈ C the following assertions hold:
+(i)
+•↾GEV(푧) = 푧•↾GEV(푧),
+(ii)
+•↾GEV(푧): GEV(푧) −→ GEV−1 (푧) is a linear isomorphism between GEV(푧) and GEV−1 (푧),
+(iii) �
+•↾GEV(푧)
+�−1 푢 = 푢↾N0
+for any 푢 ∈ GEV−1 (푧),
+(iv) for any (푐−1, 푐0) ∈ (C푑)2 there is a unique 푢 ∈ GEV−1 (푧) with 푢−1 = 푐−1, 푢0 = 푐0,
+10Below 0 denotes the zero sequence both for the C푑-vector and for the matrix case.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+17
+and their obvious reformulations for the matrix sequences variants are also true.
+Proof. Let’s check, e.g., the C푑 vector version. Observe that (i) is in fact the definition of •. Using
+this and taking 푢 ∈ GEV(푧), 푣 ∈ GEV−1 (푧), we get obviously (•푢)↾N0= 푢 by (3.25), but we get also
+•(푣↾N0) = 푣, because 푣 satisfies (3.28) — in particular for 푛 = 0. Linearity is clear by the definition, so
+(ii) and (iii) hold. Part (iv) follows directly from (3.28) and (3.1).
+□
+It can be easily checked that (e)mgev-s and (e)gev-s are mutually related in the following simple ways.
+Fact 3.10. If 푈 ∈ ℓ(N−1, 푀푑(C)), 푧 ∈ C, then
+(i) 푈 is an emgev for 퐽 and 푧 iff for any 푗 = 1, . . . , 푑 the vector sequence 푈 { 푗} := �푈 { 푗}
+푛
+�
+푛∈N−1 ∈
+ℓ(N−1, C푑) is an egev for 퐽 and 푧;
+(ii) If 푈 is an emgev for 퐽 and 푧 then for any 푣 ∈ C푑 the vector sequence 푈푣 := (푈푛푣)푛∈N−1 ∈
+ℓ(N−1, C푑) is an egev for 퐽 and 푧.
+The analog result holds for sequences 푈 ∈ ℓ(N0, 푀푑(C)) and mgev-s and gev-s.
+According to Fact 3.9(iv) for the matrix case, for any 푧 ∈ C choose 푄(푧), 푃(푧) ∈ MGEV−1 (푧)
+corresponding to
+(3.30)
+�
+푄−1(푧) = I
+푄0(푧) = 0,
+�
+푃−1(푧) = 0
+푃0(푧) = I,
+with the following general notation: for any sequence 푈(푝) = ((푈(푝))푛)푛∈N푘 depending on an extra
+“function variable type parameter” 푝:
+(3.31)
+푈푛(푝) := (푈(푝))푛,
+푛 ∈ N푘
+for any 푝. The two sequences of functions 푄, 푃 are the so-called the second and the first kind matrix
+orthogonal polynomials11.
+We can also use “the conditions in 0, 1” instead of those “in −1, 0”:
+(3.32)
+�
+푄0(푧) = 0
+푄1(푧) = 퐴−1
+0 ,
+�
+푃0(푧) = I
+푃1(푧) = 퐴−1
+0 (푧 I −퐵0).
+The linear space of matrix solutions MGEV−1 (푧) and the special solutions 푄(푧) and 푃(푧) have some
+important algebraic properties.
+Fact 3.11. Let 푧 ∈ C.
+(i) If 푈 ∈ MGEV−1 (푧), then for any 푉 ∈ 푀푑(C) also 푈푉 = (푈푛푉)푛∈N0 ∈ MGEV−1 (푧).
+(ii) Each 푈 ∈ MGEV−1 (푧) has the form
+(3.33)
+푈 = 푃(푧)푆 + 푄(푧)푇,
+for a unique pair (푆,푇) of matrices from 푀푑(C). This pair is given by
+(3.34)
+�
+푆 := 푈0
+푇 := 푈−1 = (퐵0 − 푧 I)푈0 + 퐴0푈1.
+(iii) For any 푆 ∈ 푀푑(C) the matrix sequence 퐻 := (푃(푧)푆)↾N0 satisfies “the formal matrix eigenequa-
+tion for J and 푧”, namely: 퐻 ∈ MGEV(푧) and
+(3.35)
+퐵0퐻0 + 퐴0퐻1 = 푧퐻0.
+So, for any 푣 ∈ C푑 \ {0} the C푑-sequence ℎ := (푃(푧)푣)↾N0 is an eigenvector of the formal
+operator J for 푧:
+(3.36)
+Jℎ = 푧ℎ,
+ℎ ≠ 0.
+11More precisely, this name and the orthogonality property belong to the appropriate two sequences (푄푛)푛∈N, (푃푛)푛∈N0 of
+matrix valued polynomial functions 푄푛, 푃푛 on R or on C with the values at each 푧 given by above defined 푄푛(푧), 푃푛(푧).
+
+18
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Proof. Part (i) is obvious. Hence, using it, by linearity, by (3.30) and by the unicity from Fact 3.9(iv),
+we get (3.33) with 푆 and 푇 given by (3.34). — I.e., we can simply assume that some 푆 and 푇 are given
+by (3.34) and then we see that the initial conditions of the solution on the RHS of (3.33) are just the pair
+(푈−1,푈0), which proves (3.33) by the unicity. So, to finish (ii) we should check that the choice of the pair
+(푆,푇) is also unique. By the linearity, it suffices to check only that if 푃(푧)푆 + 푄(푧)푇 is the zero solution,
+then 푆 = 푇 = 0. Indeed, in this case we have 0 = 푃0(푧)푆 + 푄0(푧)푇 = 푆 and 0 = 푃−1(푧)푆 + 푄−1(푧)푇 = 푇.
+Now, to get (iii), we can first use (i) with (ii) for 푈 of the form (3.33) with 푇 = 0, so, using also Fact 3.9
+(the matrix version), we get 퐻 ∈ MGEV(푧) with (3.35) obtained by (3.34) for 푇 = 0. Now we obtain
+(Jℎ)푛 = 푧ℎ푛 by (3.22) for 푛 ≥ 1 and separately for 푛 = 0 from (3.35) with 푆 = I. Finally ℎ ≠ 0, because
+by (3.30) ℎ0 = (푃(푧)푣)0 = 푃0푣 = 푣 ≠ 0.
+□
+3.4. Transfer matrices and the Liouville–Ostrogradsky formulae. In the scalar case 푑 = 1 the transfer
+matrix sequences turned out to be a very useful tool for describing spectral properties of the operator 퐽.
+As we shall see, this is the case of general dimension 푑.
+Let us fix here 푧 ∈ C. Our basic difference equations: the generalized eigenequation (3.20), its matrix
+analog (3.22), as well as their extended variants, can be written in equivalent forms with the use of the
+so-called (one step) transfer matrices (for 퐽 and 푧). The 푛-th transfer matrix 푇푛(푧) ∈ 푀2푑(C) has the
+block form, with blocks in 푀푑(C):
+(3.37)
+푇푛(푧) :=
+�
+0
+I
+−퐴−1
+푛 퐴∗
+푛−1
+퐴−1
+푛 (푧 I −퐵푛)
+�
+,
+푛 ≥ 0
+(for 푛 = 0 recall that 퐴−1 = − I by (3.23)). Hence, obviously, (3.21) ((3.28)) is equivalent to
+(3.38)
+� 푢푛
+푢푛+1
+�
+= 푇푛(푧)
+�푢푛−1
+푢푛
+�
+,
+푛 ≥ 1 (≥ 0).
+Similarly, (3.22) ((3.29)) is equivalent to
+(3.39)
+�
+푈푛
+푈푛+1
+�
+= 푇푛(푧)
+�
+푈푛−1
+푈푛
+�
+,
+푛 ≥ 1 (≥ 0).
+Let us observe that 푇푛(푧) is invertible and
+(3.40)
+�푇푛(푧)�−1 =
+��퐴∗
+푛−1
+�−1(푧 I −퐵푛)
+−�퐴∗
+푛−1
+�−1퐴푛
+I
+0
+�
+,
+푛 ≥ 0,
+which is clear by direct multiplying (or by expressing 푢푛−1 by 푢푛 and 푢푛−1 from (3.28)).
+Moreover we define the 푛-step transfer matrix by
+(3.41)
+푅푛(푧) = 푇푛−1(푧) . . . 푇0(푧),
+푛 ≥ 1.
+This name is justified, e.g., by the property
+(3.42)
+�
+푈푛−1
+푈푛
+�
+= 푅푛(푧)
+�
+푈−1
+푈0
+�
+,
+푛 ≥ 1,
+which we obtain from (3.39). Hence, by (3.42) and (3.30) we get
+(3.43)
+푅푛(푧) =
+�
+푄푛−1(푧)
+푃푛−1(푧)
+푄푛(푧)
+푃푛(푧)
+�
+,
+푛 ≥ 1,
+being simply a direct consequence of 푅푛(푧) = 푅푛(푧)
+�
+I
+0
+0
+I
+�
+.
+Presently we shall derive a formula for the inverse of 푅푛(푧) expressing it explicitly in terms of 푅푛(푧).
+Let us set
+(3.44)
+퐾푛 :=
+�
+퐴∗
+푛
+0
+0
+I
+�
+,
+푛 ≥ −1
+and
+(3.45)
+˜푇푛(푧) := 퐾푛푇푛(푧)퐾−1
+푛−1,
+푛 ≥ 0.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+19
+So by (3.37)
+˜푇푛(푧) =
+�
+0
+퐴∗
+푛
+−퐴−1
+푛
+퐴−1
+푛 (푧 I −퐵푛)
+�
+,
+푛 ≥ 0
+Therefore, defining
+Ω :=
+�
+0
+I
+− I
+0
+�
+,
+we verify by direct computations that
+(3.46)
+Ω = � ˜푇푛(푧)�∗Ω˜푇푛(푧),
+푛 ≥ 0.
+By using (3.45) we get
+(3.47)
+푅푛(푧) = (퐾−1
+푛−1 ˜푇푛−1(푧)퐾푛−2)(퐾−1
+푛−2 ˜푇푛−2(푧)퐾푛−3) . . . (퐾−1
+0 ˜푇0(푧)퐾−1) = 퐾−1
+푛−1 ˜푅푛(푧)퐾−1,
+where
+(3.48)
+˜푅푛(푧) := ˜푇푛−1(푧) ˜푇푛−2(푧) . . . ˜푇0(푧),
+푛 ≥ 1.
+We claim that
+(3.49)
+Ω = � ˜푅푛(푧)�∗Ω ˜푅푛(푧),
+푛 ≥ 1.
+We shall prove it inductively. By (3.48) we have ˜푅1(푧) = ˜푇0(푧) for any 푧 ∈ C. Thus, in view of (3.46)
+the formula (3.49) holds true for 푛 = 1. Next, if (3.49) holds for some 푛 ≥ 1, then by (3.46) we have
+Ω = � ˜푅푛(푧)�∗Ω ˜푅푛(푧)
+= � ˜푅푛(푧)�∗��˜푇푛(푧)�∗Ω˜푇푛(푧)
+�
+˜푅푛(푧)
+= � ˜푅푛+1(푧)�∗Ω ˜푅푛+1(푧),
+where in the last equality we have used (3.48). It ends the inductive step in the proof of (3.49). Thus, by
+multiplying both sides of (3.49) by Ω−1 on the left and then by � ˜푅푛(푧)�−1 on the right we arrive at
+� ˜푅푛(푧)�−1 = Ω−1� ˜푅푛(푧)�∗Ω.
+Consequently, using twice (3.47) we can derive
+�푅푛(푧)�−1 = 퐾−1
+−1Ω−1�퐾−1
+−1
+�∗�푅푛(푧)�∗퐾∗
+푛−1Ω퐾푛−1,
+so, by (3.44), finally
+(3.50)
+�푅푛(푧)�−1 =
+�
+0
+I
+− I
+0
+� �푅푛(푧)�∗
+�
+0
+퐴푛−1
+−퐴∗
+푛−1
+0
+�
+,
+푧 ∈ C.
+The following result is well-known, see e.g. [3, Theorem 5.2] and [46, Lemma 2.4]. Our proof seems
+to be new.
+Proposition 3.12 (Liouville–Ostrogradsky). For any 푤 ∈ C one has
+푄푘 (푤)�푃푘(푤)�∗ = 푃푘(푤)�푄푘 (푤)�∗,
+푘 ≥ 0
+(3.51)
+푄푘 (푤)�푃푘−1(푤)�∗ − 푃푘(푤)�푄푘−1(푤)�∗ = 퐴−1
+푘−1,
+푘 ≥ 1.
+(3.52)
+Proof. By (3.50) and (3.43) we have
+�I
+0
+0
+I
+�
+= 푅푘(푤)푅−1
+푘 (푤) =
+�−푃푘−1(푤)
+푄푘−1(푤)
+−푃푘(푤)
+푄푘 (푤)
+� �−�푄푘 (푤)�∗퐴∗
+푘−1
+�푄푘−1(푤)�∗퐴푘−1
+−�푃푘(푤)�∗퐴∗
+푘−1
+�푃푘−1(푤)�∗퐴푘−1
+�
+.
+Thus, the formulas (3.51) and (3.52) follows from computing the last row.
+□
+4. The Weyl function
+Similarly to the scalar Jacobi case, Weyl coefficient (being a matrix for the block case), is the main
+object in the method of subordinacy, which gives the link between generalized eigenvectors and the
+absolute continuous and the singular part of the spectral measure. And consequently – the absolutely
+continuous and the singular spectrum of 퐽.
+
+20
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+4.1. ℓ2 matrix solutions and the matrix Weyl function 푊. In the scalar Jacobi case “the scalar
+orthogonal polynomials” are used, with the common notation 푝(푧) := 푃(푧), 푞(푧) := 푄(푧) for 푧 ∈ C.
+That is, since 푑 = 1, we treat complex numbers as elements of 푀푑(C) and also as C푑-vectors, and
+푝(푧), 푞(푧) are solutions of both (3.28) and (3.29), being now just the same equation.
+It is also well-known for this case that if 퐽 is s.a. and 푧 ∈ C \ R, then neither 푝(푧)↾N0, nor 푞(푧)↾N0
+belong to the Hilbert space ℓ2(N0, C) in which Jacobi operator 퐽 acts, and there exists exactly one
+푤(푧) ∈ C such that (푤(푧)푝(푧) + 푞(푧))↾N0∈ ℓ2(N0, C). Surely, instead of making the restriction to N0,
+we can equivalently just claim here that 푝(푧), 푞(푧) ∉ ℓ2(N−1, C), and 푤(푧)푝(푧) + 푞(푧) ∈ ℓ2(N−1, C),
+respectively.
+The above unique 푤(푧) is called the Weyl coefficient (for 퐽 and 푧), and the appropriate function
+푤 : C \ R −→ C is called the Weyl function for 퐽.
+Let us recall here some less known generalisations of the above results and definitions for block Jacobi
+case.
+Observe first, that if 퐽 is s.a., and 푧 ∈ C \ R, then 푧 ∉ 휎(퐽), so denote:
+(4.1)
+푢( 푗) (푧) := (퐽 − 푧 I)−1훿0(푒 푗),
+푗 = 1, . . . , 푑.
+In particular, for any 푗 we have 푢( 푗) (푧) ∈ Dom(퐽) ⊂ ℓ2(N0, C푑) and
+J푢( 푗) (푧) = 푧푢( 푗) (푧) + 훿0(푒 푗),
+푗 = 1, . . . , 푑.
+Considering the terms 푛 ≥ 1 of the above equality of sequences we see that each 푢( 푗) (푧) is a gev for 퐽
+and 푧. Moreover, taking the term 푛 = 0 we get
+(4.2)
+푒 푗 =
+�
+(J − 푧 I)푢( 푗) (푧)
+�
+0 = (퐵0 − 푧 I)푢( 푗)
+0 (푧) + 퐴0푢( 푗)
+1 (푧).
+So, defining the matrix sequences
+(4.3)
+˜푈(푧) := [푢(1) (푧), . . . , 푢(푑) (푧)] ∈ ℓ(N0, 푀푑(C)) and 푈(푧) := •( ˜푈(푧)) ∈ ℓ(N−1, 푀푑(C))
+we have 푈(푧) ∈ ℓ2(N−1, 푀푑(C)), and by Fact 3.10 (the version for gev-s and mgev-s) and by Fact 3.9 we
+see that 푈(푧) is an emgev for 퐽 and 푧. We call it Weyl matrix solution for 퐽 and 푧. By (3.24) and (4.2) we
+obtain
+(4.4)
+I = (퐵0 − 푧 I)푈0(푧) + 퐴0푈1(푧) = 푈−1(푧).
+We can now formulate the expected result on “matrix ℓ2 solutions” (see [2, Theorem VII.2.8] for similar
+result with a sketch of the proof; we give here a full and simple proof for the sake of self-sufficiency).
+Proposition 4.1. Suppose that 퐽 is s.a. and 푧 ∈ C \ R. Then there exists exactly one 푊(푧) ∈ 푀푑(C) such
+that
+(4.5)
+푃(푧)푊(푧) + 푄(푧) ∈ ℓ2(N−1, 푀푑(C)).
+Moreover, with the above unique 푊(푧)
+(i) 푃(푧)푊(푧) + 푄(푧) is the Weyl matrix solution for 퐽 and 푧:
+(4.6)
+푃(푧)푊(푧) + 푄(푧) = 푈(푧);
+(ii) 푊(푧) = 푈0(푧),
+(iii) det푊(푧) ≠ 0.
+Proof. Fix 푧 ∉ R and consider the Weyl matrix solution 푈(푧) for 퐽 and 푧. By Fact 3.11(ii) 푈(푧) has
+the form (3.33), where 푆 = 푈0(푧) and 푇 = 푈−1(푧) = I, by (4.4). This proves the “exists”– part of the
+assertion and (ii).
+Let us prove the uniqueness, now. Suppose that for some 푧 ∉ R there exist two different matrices “푊(푧)”
+satisfying (4.5). Then, subtracting, we get a non-zero 퐶 ∈ 푀푑(C) such that 푃(푧)퐶 ∈ ℓ2(N−1, 푀푑(C)).
+Now, choosing 푤 ∈ C푑 such that 푣 := 퐶푤 ≠ 0 we get 푃(푧)푣 ∈ ℓ2(N−1, C푑). Thus, using Fact 3.11(iii),
+for ℎ := (푃(푧)푣)↾N0 we get J ℎ = 푧ℎ ∈ ℓ2(N0, C푑), which means that ℎ ∈ Dom(퐽). Moreover ℎ ≠ 0,
+because ℎ0 = 푃0(푧)푣 = I 푣 = 푣 ≠ 0. Thus ℎ is an eigenvector of 퐽 with the eigenvalue 푧 ∉ R — a
+contradiction with the assumption, that 퐽 is s.a.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+21
+To prove that 푊(푧) = 푈0(푧) is invertible, consider 푣 ∈ Ker푈0(푧) and the sequence 푤 := (푈(푧)푣)↾N0∈
+ℓ(N0, C푑). Then by Fact 3.10, by (3.2) and by (4.4) we see that 푤 is such a gev that
+((J − 푧 I)푤)0 = (퐵0 − 푧 I)푤0 + 퐴0푤1 = (퐵0 − 푧 I)푈0(푧)푣 + 퐴0푈1(푧)푣 = I 푣 = 푣 = (훿0(푣))0.
+Hence we have
+(a) (J − 푧 I)푤 = 훿0(푣),
+(b) 푤0 = 푈0(푧)푣 = 0,
+(c) 푤 ∈ ℓ2(N0, C푑).
+Now (a) gives J푤 = 푧푤 + 훿0(푣), so by (c) both 푤 and J푤 are in ℓ2(N0, C푑), i.e. 푤 ∈ Dom(퐽) and
+(4.7)
+(퐽 − 푧 I)푤 = (J − 푧 I)푤 = 훿0(푣).
+But (b) means in particular that 푤0 ⊥ 푣, which allows us to get the expected assertion 푣 = 0. — Indeed,
+by (4.7) we obtain ⟨훿0(푣), 푤⟩ = ⟨퐽푤, 푤⟩ − 푧 ∥푤∥2 and on the other hand ⟨훿0(푣), 푤⟩ = ⟨푣, 푤0⟩C푑 = 0.
+Hence, using the s.a. of 퐽, we get 푧 ∥푤∥2 = ⟨퐽푤, 푤⟩ ∈ R, but since 푧 ∉ R, 푤 has to be 0.
+□
+Corollary 4.2. Suppose that 퐽 is s.a. and 푧 ∈ C \ R. Then
+푃(푧), 푄(푧) ∉ ℓ2(N−1, 푀푑(C)).
+Proof. We have 푄(푧) = 푃(푧)0 + 푄(푧), so if 푄(푧) ∈ ℓ2(N−1, 푀푑(C)) then 푊(푧) = 0 by the uniqueness
+from Proposition 4.1. But it contradicts the condition det 푊(푧) ≠ 0 from (iii).
+If 푄(푧) ∈ ℓ2(N−1, 푀푑(C)) then using (4.5) and (iii) we get 푃(푧)+푄(푧)(푊(푧))−1 ∈ ℓ2(N−1, 푀푑(C)), so
+also 푄(푧)(푊(푧))−1 ∈ ℓ2(N−1, 푀푑(C)). But then also 푄(푧) = �푄(푧)(푊(푧))−1� 푊(푧) ∈ ℓ2(N−1, 푀푑(C)),
+which contradicts the part just proved.
+□
+Definition 4.3. Let 퐽 be s.a. For fixed 푧 ∈ C \ R such 푊(푧), that (4.5) holds is called the matrix Weyl
+coefficient (for 퐽 and 푧), and the appropriate function 푊 : C \ R −→ 푀푑(C) is called the matrix Weyl
+function (for 퐽).12
+We omit here the dependence on 퐽 in the notation, assuming that we consider a fixed 퐽.
+Here we present also a result being a stronger version of [61, Proposition 3]. Let us stress that we do
+not assume the self-adjointness of 퐽 at this moment.
+For 푧 ∈ C denote
+(4.8)
+GEVℓ2(푧) := GEV(푧) ∩ ℓ2(N0, C푑).
+Recall that by Fact 3.7 we have
+dim (GEV(푧)) = 2푑,
+and let us think about the dimension of its subspace GEVℓ2(푧). To find it in some cases, consider also
+EV(푧) — the eigenspace for 퐽 and 푧, which in the case of arbitrary 푧 ∈ C is defined by
+EV(푧) := {푢 ∈ Dom(퐽) : 퐽푢 = 푧푢}
+(and so, it is just the trivial zero space if 푧 is not an eigenvalue of 퐽). Since 퐽 is the maximal block Jacobi
+operator (see (3.3)), we have
+(4.9)
+EV(푧) = {푢 ∈ GEVℓ2(푧) : ((퐽 − 푧)푢)0 = 0}.
+Indeed, the above equality follows directly from
+GEVℓ2(푧) ⊂ Dom(퐽),
+which holds, because for 푢 ∈ GEVℓ2(푧) we have 푢 ∈ ℓ2(N0, C푑), so also 푧푢 ∈ ℓ2(N0, C푑), but J푢 and
+푧푢 differ at most at the zero term, hence J푢 ∈ ℓ2(N0, C푑).
+Proposition 4.4. If 푧 ∈ C \ 휎푝(퐽), then dim(GEVℓ2(푧)) ≤ 푑.
+If, moreover, 푧 ∈ C \ 휎(퐽), then
+dim(GEVℓ2(푧)) = 푑.
+12Using the argumentation from the proof of Proposition 4.1 and from the beginning of this subsection one can easily see
+that in fact it suffices here to assume that 푧 ∈ C \ 휎(퐽) to properly define the matrix Weyl coefficient for 퐽 and 푧. But note also
+that the invertibility of 푊(푧) from property (iii) is guaranteed only for 푧 ∈ C \ R.
+
+22
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Proof. Consider first an arbitrary 푧 ∈ C. and define Ψ : GEVℓ2(푧) −→ C푑 by
+Ψ(푢) := ((퐽 − 푧 I)푢)0 ,
+푢 ∈ GEVℓ2(푧) .
+It is a linear transformation and by (4.9) Ker Ψ = EV(푧). So, by the standard linear algebra result,
+dim (GEVℓ2(푧)) = dim (EV(푧)) + dim (Ran Ψ) .
+Hence, if 푧 ∈ C \ 휎푝(퐽), then dim(GEVℓ2(푧)) = dim(Ran Ψ) ≤ 푑. But, if moreover 푧 ∈ C \ 휎(퐽), then
+Ran(퐽 − 푧 I) = ℓ2(N0, C푑), so in particular, for any 푣 ∈ C푑 we have 훿0(푣) ∈ Ran(퐽 − 푧 I). Therefore, for
+some 푢 ∈ Dom(퐽)
+퐽푢 − 푧푢 = 훿0(푣),
+thus 푢 ∈ GEVℓ2(푧) and Ψ(푢) = 푣 for such 푢. So, Ran Ψ = C푑 and dim(GEVℓ2(푧)) = dim(Ran Ψ) =
+푑.
+□
+This result gives in particular dim(GEVℓ2(푧)) = 푑, when 퐽 is s.a. and 푧 ∈ C \ R. On the other hand,
+one can easily see that for such 푧
+GEVℓ2(푧) = lin{푢( 푗) (푧) : 푗 = 1, . . . , 푑},
+where 푢( 푗) (푧) are defined by (4.1), and they are just the successive 푗-th column sequences (푗 = 1, . . . , 푑)
+of the matrix sequence ˜푈(푧), being the restriction to N0 of the Weyl matrix solution 푈(푧) for 퐽 and 푧 (see
+(4.3)).
+4.2. The Cauchy transform of the spectral matrix measure and the matrix Weyl function. Let us
+assume also here that 퐽 is s.a. The Cauchy transform of the spectral matrix measure 푀 := 퐸퐽, �휑 from
+(3.19) of 퐽 is defined as C퐽 : C \ R −→ 푀푑(C), with
+(4.10)
+C퐽 (푧) :=
+∫
+R
+1
+휆 − 푧 d푀(휆),
+푧 ∈ C \ R,
+where the above integral is understood in the sense of (A.11). Consequently, (4.10) means just
+(4.11)
+C퐽 (푧) :=
+�∫
+R
+1
+휆 − 푧 d푀푖, 푗 (휆)
+�
+푖, 푗=1,...,푑 ,
+with
+푀푖, 푗 = 퐸퐽,휑 푗,휑푖,
+푖, 푗 = 1, . . . , 푑,
+and 휑 푗 given by (3.8).
+Hence, by spectral calculus for s.a. operators, for 푧 ∈ C \ R we get
+(4.12)
+C퐽 (푧) :=
+�∫
+R
+1
+휆 − 푧 d퐸퐽,휑 푗,휑푖 (휆)
+�
+푖, 푗=1,...,푑
+=
+��
+(퐽 − 푧 I)−1훿0(푒 푗), 훿0(푒푖)
+��
+푖, 푗=1,...,푑 .
+Now, by (4.1), (2.4) and (4.3) for any 푖, 푗 = 1, . . . , 푑
+(C퐽 (푧))푖, 푗 =
+�
+푢( 푗) (푧), 훿0(푒푖)
+�
+=
+�
+(푢( 푗) (푧))0, 푒푖
+�
+C푑 = (푈0(푧))푖, 푗
+for 푧 ∈ C \ R, where 푢( 푗) (푧) is given by (4.1) and 푈(푧) is Weyl matrix solution for 퐽 and 푧. Finally, by
+Proposition 4.1, we get
+Fact 4.5. If 퐽 is s.a., then the Cauchy transform of the spectral matrix measure of 퐽 is equal to the matrix
+Weyl function for 퐽 and moreover, for any 푧 ∈ C \ R
+C퐽 (푧) = 푊(푧) = 푈0(푧) =
+��
+(퐽 − 푧 I)−1훿0(푒 푗), 훿0(푒푖)
+��
+푖, 푗=1,...,푑.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+23
+4.3. The boundary limits of the matrix Weyl function and the properties of the spectral trace
+measure. Assume here the self-adjointness of 퐽, as before, and let us use the notation from the previous
+subsection, including 푀 := 퐸퐽, �휑.
+Fact 4.5 implies that 푊 is a holomorphic matrix-valued function. Also by Fact 4.5, by (4.10) and
+(A.3) for any 푧 ∈ C \ R
+(4.13)
+Im푊(푧) = Im �C퐽 (푧)� = 1
+2푖
+∫
+R
+�
+1
+푡 − 푧 −
+1
+푡 − 푧
+�
+d푀(푡) = Im(푧)
+∫
+R
+1
+|푡 − 푧|2 d푀(푡),
+which is a non-negative matrix on C+. Hence the restriction of 푊 to C+ is a matrix Herglotz function.
+Denote
+퐿(푊) :=
+�
+휆 ∈ R : lim
+휖 →0+ 푊(휆 + 푖휖) exists13
+�
+,
+and
+푊(휆 + 푖0) := lim
+휖 →0+ 푊(휆 + 푖휖),
+휆 ∈ 퐿(푊),
+and define the following sets:
+푆ac,r :=
+�
+휆 ∈ 퐿(푊) : rank� Im푊(휆 + 푖0)� = 푟
+�
+,
+1 ≤ 푟 ≤ 푑,
+(4.14)
+푆ac :=
+푑
+�
+푟=1
+푆ac,r ,
+(4.15)
+푆sing :=
+�
+휆 ∈ R : lim
+휖 →0+ Im � tr푊(휆 + 푖휖)� = +∞
+�
+.
+(4.16)
+In particular it follows that
+(4.17)
+푆sing ⊂ R \ 퐿(푊).
+Referring to (4.16), observe that for any 퐴 ∈ 푀푑(C)
+tr(Im 퐴) = Im(tr 퐴).
+It is important that (4.15) means simply
+(4.18)
+푆ac =
+�
+휆 ∈ 퐿(푊) : Im푊(휆 + 푖0) ≠ 0
+�
+.
+Let us recall now the crucial result, joining the above defined sets with properties of 푀ac and 푀sing
+— the a.c. and the sing. parts w.r.t. the Lebesgue measure | · | on Bor(R) (see Fact A.6) of the spectral
+matrix measure 푀 of 퐽. This theorem is obtained just as the direct use of the abstract result [13, Theorem
+6.1] to the spectral matrix measure 푀.
+Define 퐷 : 퐿(푊) −→ 푀푑(C) by
+(4.19)
+퐷(휆) := 1
+휋 Im푊(휆 + 푖0),
+휆 ∈ 퐿(푊).
+Theorem 4.6.
+(i) 푆sing is a support of 푀sing;
+(ii) |R \ 퐿(푊)| = 0;
+(iii) 퐷 is a density of 푀ac on 퐿(푊) w.r.t. | · |.
+So, this theorem shows that controlling of the boundary limits of the matrix Weyl function allows to
+get a lot of detailed information about the spectral matrix measure of 퐽 and of its sing. and a.c. parts.
+Combining the theorem with Lemma A.7 we get:
+Proposition 4.7. 푆ac is a minimal support of (tr푀)ac with respect to | · |. Moreover 푆sing is a support of
+(tr푀)sing and |푆sing| = 0. Moreover, 푆ac ∪ 훿 is also a minimal support of (tr푀)ac with respect to | · |
+for any Borel 훿 ⊂ R with |훿| = 0.
+13As the limit in 푀푑(C), i.e., in particular the limit must belong to 푀푑(C).
+
+24
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Proof. We use Lemma A.7 taking:
+휈 := | · | so, also Ω := R and 픐 := Bor(R),
+푆푎 := 푆ac ,
+푆푠 := 푆sing ,
+퐹 := 퐷↾푆ac
+and by (4.17) with (4.18) we see that 푆푎 ∩ 푆푠 = ∅, that is, the assumption (ii) of the lemma holds. Now,
+(ii) of Theorem 4.6 shows that 퐿(푊) is a support of 푀ac, since this matrix measure, by definition, is
+a.c. w.r.t. | · |. But (iii) of this theorem with (4.18) and (4.19) mean, that the restriction of 푀ac to
+퐿(푊) \ 푆ac is the zero matrix measure. Therefore 푆ac is also a support of 푀ac. This together with (i) of
+Theorem 4.6 prove that the assumption (i) of the lemma holds. The assumption (iii) also holds, by the
+fact that 푆ac ⊂ 퐿(푊) and by (iii) of Theorem 4.6, again. And (iv) of the lemma is obvious by (4.18).
+Therefore Lemma A.7 yields the first two assertions and |푆sing| = 0 by (4.17) with (iii) of the theorem.
+The last assertion is obvious just by the definition of minimal support.
+□
+4.4. The boundary limits and spectral consequences for 퐽. Assume the self-adjointness of 퐽, and let
+us hold the notation as above.
+Here we “translate” Theorem 4.6 into the spectral operator language via Proposition 4.7. Namely, we
+show:
+Proposition 4.8. Suppose that 퐽 is self-adjoint and 퐺 ∈ Bor(R).
+(i) If 퐺 ⊂ R \ 푆sing, then 퐽 is absolutely continuous in 퐺.
+(ii) If 퐺 ⊂ 푆ac ∪ (R \ (퐿(푊) ∪ 푆sing)), then 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).
+So, if moreover 퐺 is open, or if 퐺 is a sum of an arbitrary family of connected non-singletons in
+R, then cl(퐺) ⊂ 휎ac(퐽).
+Proof. First of all, by Proposition 3.2, 퐽 is s.a. finitely-cyclic operator, with �휑 being a cyclic system
+for 퐽 and with 푀 being the spectral matrix measure of 퐽 and �휑. Hence the initial assumptions of [44,
+Theorem C.2] hold for 퐽. So, using its assertion (2), we obtain our (i), because, by Proposition 4.7,
+(tr푀)sing(퐺) = 0 for 퐺 ⊂ R \ 푆sing.
+By Theorem 4.6(iii) we get |R \ (퐿(푊) ∪ 푆sing)| = 0. Hence, 푆ac ∪ �R \ (퐿(푊) ∪ 푆sing)� is a minimal
+support of (tr푀)ac with respect to | · | by Proposition 4.7. Moreover (tr푀)sing(퐺) = 0, again because
+푆ac ∪ �R \ (퐿(푊) ∪ 푆sing)� ⊂ R \ 푆sing and 푆sing is a support of (tr푀)sing by By Theorem 4.6(i).
+Now, using the assertion (3) of [44, Theorem C.2] The assertion for the special kinds of 퐺 follows
+form the property clLe(퐺) = cl(퐺), which holds for those 퐺 (see, e.g., [44, Fact C.3]).
+□
+5. An analog of the Jitomirskaya–Last’s approach
+This is the most important part of the article. — It contains the main new ideas allowing to get
+some analogies of the 1-dimensional subordinacy results also in the 푑-block case. However, one of key
+technical tools used in our proof of Theorem 5.12 is a generalisation of the Jitomirskaya–Last’s idea
+from [30].
+5.1. The vector and matrix nonsubordinacy. We start with a new notion, which seems natural from
+the context of the crucial notion of subordinated solutions in the 푑 = 1 case from Gilbert–Pearson–Khan
+subordination theory (see [16, 33]). Recall that the “interpolated” semi-norms ∥·∥[0,푡] were introduced
+here in Section 2.3.
+Definition 5.1. We say that 퐽 satisfies vector nonsubordinacy (condition) for 휆 ∈ R iff14 for each pair of
+non-zero 푢, 푣 ∈ GEV−1 (휆)
+(5.1)
+lim inf
+푡→+∞
+∥푢∥ [0,푡]
+∥푣∥[0,푡]
+< +∞.
+Nevertheless, the followingmatrixgev termsformulation seems to be more convenient for our purposes:
+퐽 satisfies matrix nonsubordinacy condition for 휆 ∈ R iff for each pair of non-zero 푈,푉 ∈ MGEV−1 (휆)
+(5.2)
+lim inf
+푡→+∞
+∥푈∥[0,푡]
+∥푉∥ [0,푡]
+< +∞.
+14Note that we consider here the function given by the fraction
+∥푢 ∥2
+[0,푡]
+∥푣 ∥2
+[0,푡]
+for 푡 > 0 only, and the denominator is positive since
+here sequences are not the zero sequence and they belong to GEV; Similarly for the definition below.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+25
+Note here that the choice of “liminf-s over 푡 > 0”, instead of more original Khan and Pearson’s like
+“liminf-s over 푛 ∈ N”, does not matter, e.a., is equivalent to this original. One direction of the implication
+follows directly from the definition of lim inf, and the other can be immediately obtained by Corollary 2.4.
+As we shall see soon, also the distinction “vector” / “matrix” is not very important here.
+We can use the symmetry w.r.t. 푢 and 푣 or 푈 and 푉, respectively, and we get:
+Fact 5.2. 퐽 satisfies vector nonsubordinacy for 휆 ∈ R iff for each pair of non-zero 푢, 푣 ∈ GEV−1 (휆)
+(5.3)
+lim sup
+푡→+∞
+∥푢∥ [0,푡]
+∥푣∥[0,푡]
+> 0.
+And analogically in the matrix nonsubordinacy case.
+Let us formulate now an important spectral consequence of vector nonsubordinacy.
+Recall that
+GEVℓ2(휆) was defined in (4.8).
+Theorem 5.3. Suppose that 퐽 is self-adjoint and it satisfies vector nonsubordinacy for some 휆 ∈ R. Then
+dim(GEVℓ2(휆)) = 0 and 휆 ∈ 휎(퐽) \ 휎p(퐽).
+Proof. Let 푢, 푣 ∈ GEV−1 (휆) be non-zero. By Fact 5.2 there exists a constant 푐 > 0 and a sequence
+(푡푘)푘 ∈N tending to +∞ such that
+∥푢∥[0,푡푘 ] ≥ 푐 ∥푣∥ [0,푡푘 ] .
+Consequently, taking the limit we get
+(5.4)
+1
+푐 ∥푢∥ [0,+∞] ≥ ∥푣∥ [0,+∞] .
+Thus, if it existed a non-trivial 푢 ∈ ℓ2(N0, C푑), then all 푣 would be also in ℓ2(N0, C푑). But then the
+operator 퐽 would not be self-adjoint, by [12, Theorem 1.3]. Thus each non-zero 푢 ∈ GEV−1 (휆) is not
+square-summable. Therefore Proposition 4.4 yields the last assertion.
+□
+As we already announced, vector and matrix nonsubordinacy are equivalent.
+Proposition 5.4. Let 휆 ∈ R. 퐽 satisfies matrix nonsubordinacy for 휆 iff it satisfies vector nonsubordinacy
+for 휆.
+Proof. (⇒) Take any non-zero 푢, 푣 ∈ GEV−1 (휆) and view them as a sequence of column vectors. Let
+us define
+푈푛 := 퐸푢푛,
+푉푛 := 퐸 푣푛,
+푛 ≥ −1,
+cf. (2.15), i.e. the first column of 푈푛 is equal to the column vector 푢푛 and the rest is zero, analogously for
+푉푛. By Fact 3.10 both 푈,푉 ∈ MGEV−1 (휆). By Proposition 2.5 for any 푛 ≥ −1 we have ∥푈푛∥ = ∥푢푛∥
+and ∥푉푛∥ = ∥푣푛∥. Consequently, ∥푈∥[0,푡] = ∥푢∥ [0,푡] and ∥푉∥ [0,푡] = ∥푣∥[0,푡] for any 푡 ≥ 0. Thus, the
+condition (5.2) implies (5.1).
+(⇐) Let us observe that for any 푋 ∈ MGEV−1 (휆) and any 푤 ∈ C푑 such that ∥푤∥ = 1 we have
+(5.5)
+∥푋푤∥2
+[0,푡] ≤ ∥푋∥2
+[0,푡] ≤
+푑
+�
+푖=1
+∥푋푒푖∥2
+[0,푡] ,
+푡 > 0.
+Let 푈,푉 ∈ MGEV−1 (휆) be non-zero. Then there exists 푤 ∈ C푑 such that ∥푤∥ = 1 and 푉푤 is non-zero.
+Then by (5.5) we have
+(5.6)
+∥푈∥2
+[0,푡]
+∥푉∥2
+[0,푡]
+≤
+푑
+�
+푖=1
+∥푈푒푖∥2
+[0,푡]
+∥푉푤∥2
+[0,푡]
+,
+푡 > 0.
+Now, by Fact 3.10, 푈푒푖 ∈ GEV−1 (휆) for 푖 = 1, . . . , 푑 and 푉푤 ∈ GEV−1 (휆). Thus, by (5.1) we get
+lim inf
+푡→+∞
+∥푈푒푖∥2
+[0,푡]
+∥푉푤∥2
+[0,푡]
+< +∞,
+푖 = 1, . . . , 푑,
+which together with (5.6) implies (5.2).
+□
+
+26
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+5.2. Barriers and barrier nonsubordinacy. In what follows, we need to make the concept of matrix
+nonsubordinacy more controllable. The goal is to control two things:
+• a bound on the ratio
+∥푈 ∥ [0,푡]
+∥푉 ∥ [0,푡]
+for any fixed “large” 푡 and 휆 ∈ 퐺, but joint for all such
+푈,푉 ∈ MGEV−1 (휆) which are “normalized” in a proper sense — the key one!
+• the size of the above bound as a function of 푡.
+So, we define:
+Definition 5.5. Let 퐺 ⊂ R be non-empty. A function 픟 : 퐺 × [1, ∞) −→ R is a barrier (on 퐺, for 퐽) iff
+for each 휆 ∈ 퐺 and each pair 푈,푉 ∈ MGEV−1 (휆) normalized by
+(5.7)
+∥푈−1∥2 + ∥푈0∥2 = ∥푉−1∥2 + ∥푉0∥2 = 1
+the estimate
+(5.8)
+� ∥푈∥[0,푡]
+∥푉∥ [0,푡]
+�2
+≤ 픟(휆, 푡)
+holds for all 푡 ≥ 1.
+To shorten the notation related to our normalisation let us denote
+S :=
+�
+(푋, 푋 ′) ∈ 푀푑(C) × 푀푑(C) : ∥푋∥2 + ∥푋 ′∥2 = 1
+�
+and, for 휆 ∈ C,
+MGEVnor (휆) := {푈 ∈ MGEV−1 (휆) : (푈−1,푈0) ∈ S},
+MGEV★ (휆) := MGEV−1 (휆) \ {0}.
+Moreover, for (푋, 푋 ′) ∈ 푀푑(C) × 푀푑(C) denote
+[(푋, 푋 ′)]퐼 := 푋,
+[(푋, 푋 ′)]퐼 퐼 := 푋 ′.
+By the homogeneity of all the norms and semi-norms, and by the use of the “symmetrization w.r.t. 푈
+and 푉” we easily obtain:
+Remark 5.6. Let ∅ ≠ 퐺 ⊂ R and 픟 : 퐺 × [1, ∞) −→ R.
+(i) If 픟 is a barrier, then 픟(휆, 푡) ≥ 1 for any 휆 ∈ 퐺, 푡 ≥ 1.
+(ii) If 픟 is strictly positive, then TFCAE:
+(a) 픟 is a barrier,
+(b) ∀휆∈퐺∀푈,푉 ∈MGEVnor(휆)∀푡 ≥1
+1
+픟(휆, 푡) ≤
+∥푈∥2
+[0,푡]
+∥푉∥2
+[0,푡]
+,
+(c) ∀휆∈퐺∀푈,푉 ∈MGEV★(휆)∀푡 ≥1
+1
+픟(휆, 푡)
+∥푈−1∥2 + ∥푈0∥2
+∥푉−1∥2 + ∥푉0∥2 ≤
+∥푈∥2
+[0,푡]
+∥푉∥2
+[0,푡]
+≤ 픟(휆, 푡) ∥푈−1∥2 + ∥푈0∥2
+∥푉−1∥2 + ∥푉0∥2 .
+Recall now (Section 3.4) that each 푈 ∈ MGEV−1 (휆) can be expressed in a convenient form by its
+initial values (푈−1,푈0) and 푛-step transfer 2푑 × 2푑 matrices 푅푛(휆) (see (3.41)), defined for 푛 ∈ N.
+Defining also
+푅0(휆) := I,
+we get in particular15
+(5.9)
+푈(푛) = [푅푛+1(휆)(푈−1,푈0)]퐼,
+푛 ∈ N−1;
+푈(푛) = [푅푛(휆)(푈−1,푈0)]퐼 퐼,
+푛 ∈ N0
+for any 푈 ∈ MGEV−1 (휆).
+The following investigations show that for any 퐽 there exists its smallest “universal” barrier (on each
+퐺). Consider the function 픟min : R × [1, +∞) → [1, +∞]
+given by the formula
+(5.10)
+픟min(휆, 푡) :=
+sup
+푈,푉 ∈MGEVnor(휆)
+� ∥푈∥[0,푡]
+∥푉∥ [0,푡]
+�2
+,
+휆 ∈ R, 푡 ≥ 1.
+15We simplify here the notation, and we use the row-block form instead of the “column-block” one.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+27
+Proposition 5.7. The function 픟min has only finite values and it is a barrier for 퐽 on R. Moreover it is
+the smallest barrier for 퐽 on any 퐺 ⊂ R, in the sense that for any barrier 픟 for 퐽 on 퐺
+픟min(휆, 푡) ≤ 픟(휆, 푡),
+휆 ∈ 퐺, 푡 ≥ 1.
+Proof. Fix 푡 ≥ 1 and 휆 ∈ R. Let 푈 ∈ MGEV−1 (휆). Observe that by (3.42) and (3.41) for any 푛 ∈ N−1 the
+value of the matrix 푈푛 is a continuous (even linear) function of its initial conditions (푈−1,푈0). Namely,
+by (5.9) we have 푈(푛) = [푅푛+1(휆)(푈−1,푈0)]퐼 . So, denoting by 푌 (훼) the sequence
+(5.11)
+푌 (훼) := ([푅푛+1(휆)훼]퐼)푛∈N−1
+for any 훼 ∈ 푀푑(C) × 푀푑(C), we get
+(5.12)
+푈 = 푌 (푈−1,푈0).
+By (2.9), also the function
+S ∋ 훼 ↦→ ∥푌 (훼)∥ [0,푡]
+is continuous and, moreover, it is strictly positive. Now, define a function 푓 : S × S → R by
+푓 (훼, 훼′) :=
+� ∥푌 (훼)∥ [0,푡]
+∥푌 (훼′)∥ [0,푡]
+�2
+.
+Obviously, 푓 is also continuous and S × S is compact, hence 푓 attains supremum of its values, and by
+(5.12) and (5.10)
+픟min(휆, 푡) = sup
+훼,훼′∈S
+푓 (훼, 훼′) = max
+훼,훼′∈S 푓 (훼, 훼′) ∈ R+.
+On the other hand, it is obvious that 픟min is a barrier, as well as that that it is the smallest one, directly
+from its definition (5.10), .
+□
+For any 퐺 ⊂ R we call 픟min↾퐺
+the minimal barrier for 퐽 on 퐺.
+Our next goal is now the construction of an another barrier for 퐺 = R in terms of the sequence of
+transfer matrices. It will be much more convenient in practice than the minimal one, because of its more
+explicit form. To do this, we need first:
+Proposition 5.8. Let 휆 ∈ R. If 푢 ∈ GEV−1 (휆), then for any 푡 ≥ 1
+(5.13)
+1
+2
+� ∥푢−1∥2 + ∥푢0∥2 � ⇃|푅(휆)|⇂2
+[1,푡]≤ ∥푢∥2
+[0,푡] ≤ � ∥푢−1∥2 + ∥푢0∥2 � ∥푅(휆)∥2
+[1,푡] .
+If 푈 ∈ MGEV−1 (휆), then for any 푡 ≥ 1
+(5.14)
+1
+4
+� ∥푈−1∥2 + ∥푈0∥2 � ⇃|푅(휆)|⇂2
+[1,푡]≤ ∥푈∥2
+[0,푡] ≤ 2� ∥푈−1∥2 + ∥푈0∥2 � ∥푅(휆)∥2
+[1,푡] .
+In particular, for any 푈,푉 ∈ MGEVnor (휆)
+(5.15)
+� ∥푈∥[0,푡]
+∥푉∥ [0,푡]
+�2
+≤ 8
+� ∥푅(휆)∥[1,푡]
+⇃|푅(휆)|⇂[1,푡]
+�2
+,
+푡 ≥ 1.
+Proof. If 푢−1 = 푢0 = 0, then 푢 = 0, and the first assertion holds, so assume that ∥푢−1∥2 + ∥푢0∥2 > 0. Set
+�푢푘 :=
+�
+푢푘−1
+푢푘
+�
+,
+푘 ≥ 0.
+Then
+�푢푘 = 푅푘 (휆) �푢0,
+푘 ≥ 1.
+Thus
+(5.16)
+∥�푢0∥2 ⇃|푅푘(휆)|⇂2≤ ∥�푢푘 ∥2 ≤ ∥�푢0∥2 ∥푅푘(휆)∥2 ,
+푘 ≥ 1.
+Then by Proposition 2.3 we get
+∥�푢0∥2 ⇃|푅(휆)|⇂2
+[1,푡]≤ ∥�푢∥2
+[1,푡] ≤ ∥�푢0∥2 ∥푅(휆)∥2
+[1,푡] ,
+푡 ≥ 1.
+Thus (5.13) follows, since we have
+∥푢∥2
+[0,푡] ≤ ∥�푢∥2
+[1,푡] ≤ 2 ∥푢∥2
+[0,푡]
+
+28
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+by direct computations. Now, consider 푈 ∈ MGEV−1 (휆). Then by Fact 3.10 for any 푣 ∈ C푑 the
+sequence 푈푣 is a vector generalized eigenvector. Thus (5.16) implies
+(5.17)
+� ∥푈−1푣∥2 + ∥푈0푣∥2 � ⇃|푅푘(휆)|⇂2≤
+����
+�푈푘−1푣
+푈푘푣
+�����
+2
+≤ � ∥푈−1푣∥2 + ∥푈0푣∥2 � ∥푅푘 (휆)∥2 .
+In particular,
+� ∥푈−1푣∥2 + ∥푈0푣∥2 � ⇃|푅푘(휆)|⇂2≤ � ∥푈푘−1∥2 + ∥푈푘∥2 � ∥푣∥2 .
+Thus
+(5.18)
+1
+2
+� ∥푈−1∥2 + ∥푈0∥2 � ⇃|푅푘(휆)|⇂2≤ ∥푈푘−1∥2 + ∥푈푘∥2 .
+On the other hand, by (5.17)
+∥푈푘−1푣∥2 + ∥푈푘푣∥2 ≤ � ∥푈−1∥2 + ∥푈0∥2 � ∥푣∥2 ∥푅푘(휆)∥2 ,
+and consequently,
+(5.19)
+∥푈푘−1∥2 + ∥푈푘 ∥2 ≤ 2� ∥푈−1∥2 + ∥푈0∥2 � ∥푅푘 (휆)∥2 .
+For any 푘 ≥ 0 let us define
+�푥푘 :=
+�
+∥푈푘−1∥
+∥푈푘 ∥
+�
+∈ C2.
+Therefore, by combining (5.18) and (5.19) we obtain
+1
+2 ∥�푥0∥2 ⇃|푅푘(휆)|⇂2≤ ∥�푥푘 ∥2 ≤ 2 ∥�푥0∥2 ∥푅푘 (휆)∥2 ,
+푘 ≥ 1,
+which is an analogue of (5.16). Now we get (5.14) by a similar manner.
+□
+In view of this result let us consider now 픟TR : R × [1, +∞) → R given by the formula16
+(5.20)
+픟TR(휆, 푡) := 8
+� ∥푅(휆)∥[1,푡]
+⇃|푅(휆)|⇂[1,푡]
+�2
+,
+휆 ∈ R, 푡 ≥ 1.
+The last assertion of the proposition yields exactly:
+Corollary 5.9. 픟TR is a barrier for 퐽 on R.
+For any 퐺 ⊂ R we call 픟TR↾퐺
+the transfer matrix barrier for 퐽 on 퐺.
+At the end of the subsection we introduce the most important notion of the paper.
+Definition 5.10. Suppose that 퐽 is self-adjoint, Let 퐺 ⊂ R be non-empty and let 픟 be a barrier on 퐺. We
+say that 퐽 is 픟-nonsubordinate on 퐺 if
+(5.21)
+lim inf
+푡→+∞ 픟(휆, 푡) < +∞,
+휆 ∈ 퐺.
+If in fact, this condition holds uniformly on 퐺, namely
+(5.22)
+sup
+휆∈퐺
+lim inf
+푡→+∞ 픟휆(푡) < +∞,
+then we say that 퐽 is uniformly 픟-nonsubordinate on 퐺.
+The convenience and the importance of constructing just such definitions (of the barrier and of the
+barrier-nonsubordinacy) as here, will be clearly visible in proofs in the next subsections.
+16Note that ⇃|푅(휆)|⇂[1,푡 ]> 0 for any 푡 ≥ 1, since 푅1(휆) = 푇0(휆) is invertible (see also (2.11) and (2.1).
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+29
+5.3. The main result and its consequences. Assume, as before, that 퐽 is s.a. and consider the matrix
+Weyl function 푊 for 퐽 (see Definition 4.3). Before we start to “control” in a sense its boundary limits we
+need to make use of the choice of Jitomirskaya–Last type semi-norms, and prove a result being a block
+case analog of the appropriate scalar case one result from [30].
+Proposition 5.11. Suppose that 퐽 is self-adjoint. For any 휆 ∈ R there exists a unique function
+ℓ휆 :
+R+ → R+ satisfying
+(5.23)
+∥푃(휆)∥[0,ℓ휆(휖 )] ∥푄(휆)∥[0,ℓ휆 (휖 )] = 1
+2휖 ,
+휖 > 0.
+Moreover, ℓ휆 is a strictly decreasing continuous function and satisfies
+(5.24)
+lim
+휖 →0+ ℓ휆(휖) = +∞,
+lim
+휖 →+∞ ℓ휆(휖) = 0.
+Consequently, its inverse ℓ−1
+휆 is also strictly decreasing continuous function and satisfies
+(5.25)
+lim
+푡→0+ ℓ−1
+휆 (푡) = +∞,
+lim
+푡→+∞ ℓ−1
+휆 (푡) = 0.
+Proof. Define a function 푓 : R+ → R+ by the formula
+푓 (푡) = ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡] .
+By (2.9) this function is continuous, non-negative and weakly increasing. Moreover, by (3.30) it is in
+fact positive. Let us observe that it is strictly increasing. Indeed, because if not, then there would exist
+푛 ∈ N0 such that both ∥푃(휆)∥ [0,푡] and ∥푄(휆)∥[0,푡] were constant for 푡 ∈ (푛, 푛 + 1). It would mean that
+∥푃푛+1(휆)∥ = ∥푄푛+1(휆)∥ = 0. Which by (3.43) would imply that 푅푛(휆) was singular. This contradicts
+(3.50). Next, observe that
+lim
+푡→0+ 푓 (푡) = 0,
+lim
+푡→+∞ 푓 (푡) = +∞.
+Indeed, the first limit follows from ∥푄0(휆)∥ = 0. The second follows from the fact that if ∥푃(휆)∥[0,+∞] <
++∞ and ∥푄(휆)∥[0,+∞] < +∞, then the operator 퐽 is not self-adjoint, see [12, Theorem 1.3]. Thus, we
+have shown that 푓 is continuous, strictly increasing and surjective. Thus its inverse 푓 −1 : R+ → R+ has
+the same properties. Consider the function 푔 : R+ → R+ defined by 푔(휖) = 1/(2휖). This function is
+strictly decreasing and surjective. Thus, if ℓ휆 function exists it is a solution of the equation
+푓 �ℓ휆(휖)� = 푔(휖).
+This equation has a unique solution given by
+ℓ휆(휖) = 푓 −1�푔(휖)�.
+It is immediate that this defines a function satisfying (5.23). From this representation it is immediate that
+ℓ휆 : R+ → R+ is a continuous strictly decreasing surjective function. It implies (5.24). Since again ℓ−1
+휆
+has analogous properties, we also obtain (5.25).
+□
+The unique function ℓ휆 described above will be called J-L function (for 퐽 and 휆).
+We are ready to formulate our main result “on controlling the boundary limits of the matrix Weyl
+function”, being probably the most important result of this work.
+Theorem 5.12. Assume that 퐽 is self-adjoint and 퐺 ⊂ R. If 픟 is a barrier for 퐽 on 퐺, then for any 휆 ∈ 퐺
+and any 휖 > 0 with ℓ휆(휖) ≥ 1
+(5.26)
+�8픟�휆, ℓ휆(휖)��−1 I ≤ Im푊(휆 + 푖휖)
+and
+(5.27)
+푠−(휆, 휖) ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+(휆, 휖),
+where
+(5.28)
+푠±(휆, 휖) := 4푑픟�휆, ℓ휆(휖)� ±
+��
+4푑픟�휆, ℓ휆(휖)��2
+− 1.
+
+30
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+We present the proof in the next subsection. But now let us remark only that the square root in (5.28)
+is at least 15, since 푑 ≥ 1 and 픟(휆, 푡) ≥ 1 for any 푡 ≥ 1. Consequently, both 푠−(휆, 휖) and 푠+(휆, 휖) are
+positive for the considered 휆 and 휖, hence both estimates in (5.27) could be of significant importance.
+Having Theorem 5.12 and all the delicate relations between various objects connected somehow to 퐽
+and described in the previous sections, we can finally formulate and prove the most important abstract
+spectral result of this article.
+Denote for short the spectral matrix measure for 퐽 by 푀, i.e.,
+푀 := 퐸퐽, �휑,
+where �휑 = (휑1, . . . , 휑푘) is canonical cyclic system (3.8) for 퐽. Recall that some important results and
+some notation related to the absolutely continuous and the singular part of 푀, such as Theorem 4.6,
+the set 퐿(푊) “with boundary limits for 푊” and the density 퐷 (see (4.19)) of 푀ac on 퐿(푊) w.r.t. the
+Lebesgue measure | · | are presented in Section 4.3. And some measure theory notions (including matrix
+measures) are collected in Appendix A.
+Theorem 5.13. Assume that 퐽 is self-adjoint, 퐺 ⊂ Bor(R) and 픟 is a barrier for 퐽 on 퐺. If 퐽 is
+픟-nonsubordinate on 퐺, then
+(a) 푀 is absolutely continuous on 퐺.
+(b) the density 퐷 of 푀ac is an invertible matrix at any 휆 ∈ 퐺 ∩ 퐿(푊).
+(c) 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).
+If, moreover, 퐽 is uniformly 픟-nonsubordinate on 퐺, then there exist 푐1, 푐2 > 0 such that
+(5.29)
+푐1 I ≤ 퐷(휆) ≤ 푐2 I,
+휆 ∈ 퐺 ∩ 퐿(푊).
+Proof. Suppose that 퐽 is 픟-nonsubordinate on 퐺. By Theorem 5.12
+(5.30)
+lim inf
+휖 →0+ ∥푊(휆 + 푖휖)∥ ≤ 8푑 lim inf
+휖 →0+ 픟�휆, ℓ휆(휖)�,
+휆 ∈ 퐺.
+By the definition of “lim inf” there exists a sequence (푡푘)푘 ∈N in [1, +∞) with lim푘→+∞ 푡푘 = +∞, such
+that
+lim inf
+푡→+∞ 픟(휆, 푡) = lim
+푘→+∞ 픟(휆, 푡푘).
+By Proposition 5.11 for each 푘 we define
+휖푘 := ℓ−1
+휆 (푡푘)
+and by (5.25) we get
+lim
+푘→+∞ 휖푘 = 0.
+Hence, by (5.21)
+(5.31)
+lim inf
+휖 →0+ 픟�휆, ℓ휆(휖)� ≤ lim
+푘→+∞ 픟�휆, ℓ휆(휖푘)� = lim
+푘→+∞ 픟(휆, 푡푘) = lim inf
+푡→+∞ 픟(휆, 푡) < +∞,
+which by (5.30) implies
+(5.32)
+lim inf
+휖 →0+ ∥푊(휆 + 푖휖)∥ < +∞,
+휆 ∈ 퐺,
+and by (4.16) this, in particular, yields17 퐺 ⊂ R\푆sing. Now, using Theorem 4.6(i), we obtain assertion (a):
+푀 is a.c. on 퐺.
+Observe now that (5.26) implies
+1
+8 lim inf 휖 →0+ 픟�휆, ℓ휆(휖)� ∥푣∥2 ≤ lim sup
+휖 →0+
+�� Im푊(휆 + 푖휖)�푣, 푣
+�
+,
+푣 ∈ C푑,
+which by (5.31) and 픟-nonsubordinacy yields
+(5.33)
+푐(휆) ∥푣∥2 ≤ lim sup
+휖 →0+
+�� Im푊(휆 + 푖휖)�푣, 푣
+�
+,
+푣 ∈ C푑,
+where
+(5.34)
+푐(휆) =
+�
+8 lim inf
+푡→+∞ 픟(휆, 푡)
+�−1
+> 0,
+휆 ∈ 퐺.
+17One can use here, e.g., Proposition 2.5(iv) to estimate:
+tr � Im푊(휆 + 푖휖)� ≤ 푑
+��Im푊(휆 + 푖휖)��� ≤ 푑 ∥푊(휆 + 푖휖)∥, but it
+follows also directly from the continuity of the norm, of the trace, and of the imaginary part.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+31
+Assume now that 휆 ∈ 퐺 ∩ 퐿(푊). To get assertion (b) in view of (4.19) it is enough to prove that the
+matrix Im �푊(휆 + 푖0)� is invertible. But now
+lim sup
+휖 →0+ Im푊(휆 + 푖휖) = lim
+휖 →0+ Im푊(휆 + 푖휖) = 푊(휆 + 푖0),
+so, by (5.33),
+(5.35)
+푐(휆) ≤ Im푊(휆 + 푖0)
+Therefore Im푊(휆 + 푖0) is invertible, by 5.34, and 퐺 ∩ 퐿(푊) ⊂ 푆ac,d ⊂ 푆ac.
+Now Proposition 4.8(ii) gives
+clLe(퐺 ∩ 퐿(푊)) ⊂ 휎ac(퐽).
+But clLe(퐺 ∩ 퐿(푊)) = clLe(퐺), and the conclusion (c) follows.
+In the uniform case, the LHS bound on 퐷(휆) follows directly from (5.35) and (5.34). And to get the
+RHS estimate, it suffices to use the fact that the norm of a matrix gives the upper bound for the quadratic
+form, and so it suffices to use (5.30).
+□
+5.4. The proof of the main result. Before we turn to the proof of Theorem 5.12 we need some
+preparations. For fixed 휆 ∈ R and 휖 > 0 let us define
+푈휖 := 푄(휆 + 푖휖) + 푃(휆 + 푖휖)푊(휆 + 푖휖)
+(5.36)
+푉휖 := 푄(휆) + 푃(휆)푊(휆 + 푖휖).
+(5.37)
+The following lemma is not new, see e.g. [35, Section 2]. For the sake of completeness we include its
+proof.
+Lemma 5.14. Let 퐹 ∈ ℓ(N0, 푀푑(C)). The unique solution of the recurrence relation
+(5.38)
+퐴푛푆푛+1 + 퐵푛푆푛 + 퐴∗
+푛−1푆푛−1 = 푧푆푛 + 퐹푛,
+푛 ≥ 0.
+with the initial conditions 푆−1 = 푆0 = 0 is equal to
+(5.39)
+푆푛 :=
+푛−1
+�
+푘=0
+�
+푄푛(푧)�푃푘(푧)�∗ − 푃푛(푧)�푄푘 (푧)�∗�
+퐹푘,
+푛 ≥ −1.
+Proof. Since all 퐴푘 are invertible it is clear that the solution of (5.38) with given initial conditions 푆−1
+and 푆0 is unique. It remains to prove that the sequence given by (5.39) satisfies it.
+It is immediate from (5.39) that 푆−1 = 푆0 = 0. Moreover, we get
+푆1 =
+�
+푄1(푧)�푃0(푧)�∗ − 푃1(푧)�푄0(푧)�∗�
+퐹0 = 퐴−1
+0 퐹0
+which is in agreement with (5.38) for 푛 = 0. So let us assume that 푛 ≥ 1. Since both 푃(푧) and 푄(푧)
+satisfy (3.22) we get
+퐴푛푆푛+1 = 퐴푛
+푛
+�
+푘=0
+�
+푄푛+1(푧)�푃푘(푧)�∗ − 푃푛+1(푧)�푄푘 (푧)�∗�
+퐹푘
+= (푧 I −퐵푛)
+푛
+�
+푘=0
+�
+푄푛(푧)�푃푘(푧)�∗ − 푃푛(푧)�푄푘 (푧)�∗�
+퐹푘
+− 퐴∗
+푛−1
+푛
+�
+푘=0
+�
+푄푛−1(푧)�푃푘(푧)�∗ − 푃푛−1(푧)�푄푘 (푧)�∗�
+퐹푘.
+Thus,
+퐴푛푆푛+1 = (푧 I −퐵푛)푆푛 − 퐴∗
+푛−1푆푛−1 + ˜퐹푛,
+where
+˜퐹푛 = (푧 I −퐵푛)
+�
+푄푛(푧)�푃푛(푧)�∗ − 푃푛(푧)�푄푛(푧)�∗�
+퐹푛
+(5.40)
+− 퐴∗
+푛−1
+�
+푄푛−1(푧)�푃푛(푧)�∗ − 푃푛−1(푧)�푄푛(푧)�∗�
+퐹푛
+− 퐴∗
+푛−1
+�
+푄푛−1(푧)�푃푛−1(푧)�∗ − 푃푛−1(푧)�푄푛−1(푧)�∗�
+퐹푛−1.
+
+32
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+It remains to prove that ˜퐹푛 = 퐹푛. To do so, let us apply (3.51) for 푤 = 푧 with 푘 = 푛 and 푘 = 푛 − 1. Then
+we get that on the right-hand side of (5.40) the first and the third lines are equal to 0. By considering
+(3.52) for 푤 = 푧 with 푘 = 푛 and taking the adjoint of both sides we get
+�퐴−1
+푛−1
+�∗ =
+�
+푄푛(푧)�푃푛−1(푧)�∗ − 푃푛(푧)�푄푛−1(푧)�∗�∗
+= 푃푛−1(푧)�푄푛(푧)�∗ − 푄푛−1(푧)�푃푛(푧)�∗,
+which gives that the second line on the right-hand side of (5.40) is equal to 퐹푛, and consequently, ˜퐹푛 = 퐹푛.
+It ends the proof.
+□
+Corollary 5.15. For 휆 ∈ R define an operator 퐿 : ℓ(N0, 푀푑(C)) → ℓ(N0, 푀푑(C)) by the formula
+(5.41)
+(퐿퐹)푚 =
+푚−1
+�
+푘=0
+�
+푄푚(휆)�푃푘(휆)�∗ − 푃푚(휆)�푄푘 (휆)�∗�
+퐹푘,
+푚 ≥ 0.
+Then the sequences 푈휖 and 푉휖 defined in (5.36) and (5.37) satisfy for any 푋 ∈ 푀푑(C)
+(5.42)
+(I −푖휖퐿)(푈휖 푋) = 푉휖 푋.
+Proof. Take 푋 ∈ 푀푑(C). Observe that 푈휖 푋 satisfies (5.38) with 푧 = 휆 and 퐹푛 = 푖휖(푈휖 )푛푋. Similarly,
+푉휖 푋 satisfies (5.38) with 푧 = 휆 and 퐹푛 ≡ 0. Since the initial conditions of 푉휖 푋 and 푈휖 푋 are the same
+we get that the sequence 푆 := 푈휖 푋 − 푉휖 푋 can be expressed in the form (5.39). Thus, according to the
+definition (5.41) we can write
+푈휖 푋 − 푉휖 푋 = 푖휖퐿(푈휖 푋),
+which is equivalent to (5.42).
+□
+Lemma 5.16. Let the operator 퐿 : ℓ(N0, 푀푑(C)) → ℓ(N0, 푀푑(C)) be defined in (5.41). Then for any
+푡 ∈ [0, +∞)
+(5.43)
+∥퐿퐹∥[0,푡] ≤ 2 ∥푃(휆)∥ [0,푡] ∥푄(휆)∥[0,푡] ∥퐹∥[0,푡] .
+Proof. Observe that by (5.41) for any 푚 ∈ N0 we have
+∥(퐿퐹)푚∥ ≤ ∥푄푚(휆)∥
+푚−1
+�
+푘=0
+∥푃푘(휆)∥ ∥퐹푘 ∥
++ ∥푃푚(휆)∥
+푚−1
+�
+푘=0
+∥푄푘 (휆)∥ ∥퐹푘∥ .
+Hence, by Cauchy–Schwarz inequality
+(5.44)
+∥(퐿퐹)푚∥ ≤
+�
+∥푄푚(휆)∥ ∥푃(휆)∥[0,푚−1] + ∥푃푚(휆)∥ ∥푄(휆)∥ [0,푚−1]
+�
+∥퐹∥[0,푚−1] .
+Let 푡 ∈ [0, +∞). Then 푡 ∈ [푛, 푛 + 1) for some 푛 ∈ N0. Therefore, by (5.44)
+∥(퐿퐹)푚∥ ≤ 푣푚 ∥퐹∥[0,푡] ,
+푚 = 0, 1, . . . , 푛 + 1,
+where
+(5.45)
+푣푚 := ∥푄푚(휆)∥ ∥푃(휆)∥ [0,푡] + ∥푃푚(휆)∥ ∥푄(휆)∥ [0,푡] ,
+푚 = 0, 1, . . . , 푛 + 1.
+Consequently,
+∥퐿퐹∥[0,푡] =
+�
+푛
+�
+푚=0
+∥(퐿퐹)푚∥2 + {푡} ∥(퐿퐹)푛+1∥2
+�1/2
+≤
+�
+푛
+�
+푚=0
+푣2
+푚 + {푡}푣2
+푛+1
+�1/2
+∥퐹∥[0,푡] .
+(5.46)
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+33
+Now, by triangle inequality in C푛+2 with Euclidean norm and (5.45) we get
+�
+푛
+�
+푚=0
+푣2
+푚 + {푡}푣2
+푛+1
+�1/2
+≤ ∥푃(휆)∥[0,푡]
+�
+푛
+�
+푚=0
+∥푄푚(휆)∥2 + {푡} ∥푄푛+1(휆)∥2
+�1/2
++ ∥푄(휆)∥ [0,푡]
+�
+푛
+�
+푚=0
+∥푃푚(휆)∥2 + {푡} ∥푃푛+1(휆)∥2
+�1/2
+= 2 ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡]
+which combined with (5.46) gives (5.43).
+□
+Proposition 5.17. For any 푋 ∈ 푀푑(C) and 푡 ∈ [0, +∞) one has
+(5.47)
+∥푈휖 푋∥[0,푡] ≥ ∥푉휖 푋∥ [0,푡] − 2휖 ∥푃(휆)∥ [0,푡] ∥푄(휆)∥[0,푡] ∥푈휖 푋∥ [0,푡] .
+Consequently, if ℓ휆 is J-L function, then
+(5.48)
+2 ∥푈휖 푋∥[0,ℓ휆 (휖 )] ≥ ∥푉휖 푋∥[0,ℓ휆 (휖 )] .
+Proof. By (5.42)
+(I −푖휖퐿)(푈휖 푋) = 푉휖 푋.
+Thus, by (5.43) we get
+∥푉휖 푋∥ [0,푡] ≤ �1 + 2휖 ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡]
+� ∥푈휖 푋∥[0,푡]
+which implies (5.47). Then the inequality (5.48) follows from (5.23).
+□
+Proposition 5.18. For any 푣 ∈ C푑, 휆 ∈ R and any 휖 > 0 one has
+(5.49)
+1
+휖
+�� Im푊(휆 + 푖휖)�푣, 푣
+�
+= ∥푈휖 푣∥2
+[0,+∞] .
+Moreover,
+(5.50)
+∥푈휖 ∥2
+[0,+∞] ≤ tr � Im푊(휆 + 푖휖)�
+휖
+.
+Proof. Let 푧 = 휆 + 푖휖 for some 휖 > 0 and 푣 ∈ C푑. By Proposition 4.1 and Fact 4.5 we have
+(5.51)
+푈(푧)푣 := 푄(푧)푣 + 푃(푧)푊(푧)푣 = (퐽 − 푧 I)−1훿0(푣).
+Thus,
+(퐽 − 푧 I)�푈(푧)푣� = 훿0(푣).
+By taking the scalar product of both sides with 푈(푧)푣 we get
+�
+푈(푧)푣, 훿0(푣)
+�
+=
+�
+푈(푧)푣, (퐽 − 푧 I)�푈(푧)푣��
+=
+�
+푈(푧)푣, 퐽�푈(푧)푣��
+− 푧 ∥푈(푧)푣∥2
+[0,+∞]
+Since 퐽 is self-adjoint by taking imaginary parts of both sides and using (5.51) we get
+Im⟨푊(푧)푣, 푣⟩ = Im 푧 ∥푈(푧)푣∥2 .
+Consequently,
+�� Im푊(푧)�푣, 푣
+�
+= Im 푧 ∥푈(푧)푣∥2 .
+In view of (5.36) and (5.51) we have 푈휖 = 푈(푧) and (5.49) follows.
+Now, apply (5.49) for 푣 ∈ {푒1, 푒2, . . . , 푒푑} and sum them up. Then, since Im푊(푧) ≥ 0 for 푧 ∈ C+,
+푑
+�
+푖=1
+∥푈휖 푒푖∥2 = 1
+휖
+푑
+�
+푖=1
+�� Im푊(휆 + 푖휖)�푒푖, 푒푖
+�
+= 1
+휖 tr � Im푊(휆 + 푖휖)�.
+By Proposition 2.5(i) and (2.14) we get
+푑
+�
+푖=1
+∥푈휖 푒푖∥2 =
++∞
+�
+푘=0
+푑
+�
+푖=1
+∥(푈휖 푒푖)푘 ∥2 =
++∞
+�
+푘=0
+∥(푈휖 )푘 ∥2
+HS ≥
+++∞
+�
+푘=0
+∥(푈휖 )푘 ∥2 = ∥푈휖 ∥2
+[0,+∞] .
+Hence
+∥푈휖 ∥2
+[0,+∞] ≤ 1
+휖 tr � Im푊(휆 + 푖휖)�.
+
+34
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+from which the result follows.
+□
+We are ready to prove our main result.
+Proof of Theorem 5.12. By Proposition 5.17 applied to 푋 = I and by (5.50) we have
+∥푉휖 ∥[0,ℓ휆 (휖 )] ≤ 2 ∥푈휖 ∥ [0,ℓ휆(휖 )]
+≤ 2 ∥푈휖 ∥ [0,+∞]
+≤ 2
+√휖
+�
+tr � Im푊(휆 + 푖휖)�.
+(5.52)
+Now, by Proposition 2.5 (iv)
+tr � Im푊(휆 + 푖휖)� ≤ 푑 ∥Im푊(휆 + 푖휖)∥ ≤ 푑 ∥푊(휆 + 푖휖)∥ .
+Thus,
+(5.53)
+∥푉휖 ∥2
+[0,ℓ휆 (휖 )] ≤ 4푑
+휖 ∥푊(휆 + 푖휖)∥ .
+On the other hand, by (5.6) and (5.23)
+∥푉휖 ∥2
+[0,ℓ휆 (휖 )] =
+∥푉휖 ∥[0,ℓ휆 (휖 )]
+∥푃(휆)∥[0,ℓ휆 (휖 )]
+∥푉휖 ∥ [0,ℓ휆(휖 )]
+∥푄(휆)∥ [0,ℓ휆(휖 )]
+∥푃(휆)∥ [0,ℓ휆(휖 )] ∥푄(휆)∥ [0,ℓ휆(휖 )]
+≥
+1
+픟(휆, ℓ휆(휖))
+�1 + ∥푊(휆 + 푖휖)∥2 � 1
+2휖 .
+Thus, by combining it with (5.53) we obtain
+(5.54)
+1 + ∥푊(휆 + 푖휖)∥2 ≤ 8푑픟(휆, ℓ휆(휖)) ∥푊(휆 + 푖휖)∥ .
+This inequality is equivalent to
+푠2 − 8푑픟�휆, ℓ휆(휖)�푠 + 1 ≤ 0,
+where 푠 = ∥푊(휆 + 푖휖)∥ ≥ 0. Its discriminant is equal to
+(5.55)
+Δ =
+�
+8푑픟�휆, ℓ휆(휖)��2
+− 4.
+Since 푑 ≥ 1 and 픟�휆, ℓ휆(휖)� ≥ 1, we have
+(5.56)
+Δ ≥ 82 − 4 = 60 > 0.
+Thus
+푠2 − 8푑픟�휆, ℓ휆(휖)�푠 + 1 = (푠 − 푠−)(푠 − 푠+) ≤ 0,
+where
+푠− = 8푑픟�휆, ℓ휆(휖)� −
+√
+Δ
+2
+and
+푠+ = 8푑픟�휆, ℓ휆(휖)� +
+√
+Δ
+2
+.
+By (5.56) we see that 푠−, 푠+ ∈ R. Consequently, by (5.55) we have 푠−, 푠+ > 0. Thus,
+푠− ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+
+and (5.27) follows.
+The proof of (5.26) is even simpler. Namely, let 푣 ∈ C푑. Then by Proposition 5.17 applied to 푋 = 퐸 푣
+(see (2.15)), Proposition 2.5 and (5.49)
+∥푉휖 푣∥2
+[0,ℓ휆 (휖 )] ≤ 4 ∥푈휖 푣∥2
+[0,ℓ휆(휖 )] = 4
+휖
+�� Im푊(휆 + 푖휖)�푣, 푣
+�
+.
+Now by Proposition 2.5, (5.6) and (5.23)
+∥푉휖 푣∥2
+[0,ℓ휆(휖 )] = ∥푉휖 (퐸 푣)∥2
+[0,ℓ휆(휖 )]
+=
+∥푉휖 (퐸 푣)∥ [0,ℓ휆(휖 )]
+∥푃(휆)∥[0,ℓ휆 (휖 )]
+∥푉휖 (퐸 푣)∥ [0,ℓ휆(휖 )]
+∥푄(휆)∥[0,ℓ휆 (휖 )]
+∥푃(휆)∥[0,ℓ휆 (휖 )] ∥푄(휆)∥ [0,ℓ휆(휖 )]
+≥
+1
+픟�휆, ℓ휆(휖)� ∥푣∥2 1
+2휖 .
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+35
+Therefore,
+1
+픟�휆, ℓ휆(휖)� ∥푣∥2 ≤ 8
+�� Im푊(휆 + 푖휖)�푣, 푣
+�
+from which (5.26) follows. The proof is complete.
+□
+6. Sufficient conditions for absolute continuity
+In this section we extend some well-known conditions implying nonsubordinacy from 푑 = 1 to the
+general case. In particular, in Section 6.1 we cover Generalized Last–Simon condition, in Section 6.2
+Generalized Behncke–Stolz condition and in Section 6.3 the homogenous class condition.
+6.1. GLS condition. Suppose that Carleman’s condition is satisfied. Then the sequence
+(6.1)
+휌푛 :=
+푛−1
+�
+푘=0
+1
+∥퐴푘 ∥
+is divergent and 퐽 is self-adjoint.
+Let 퐺 ⊂ R be non-empty. We say that Jacobi matrix 퐽 satisfies Generalized Last–Simon (GLS in
+short) condition on 퐺 if
+(6.2)
+lim inf
+푛→+∞
+1
+휌푛
+푛
+�
+푘=1
+∥푅푘 (휆)∥2 < +∞,
+휆 ∈ 퐺.
+Moreover, if (6.2) is finite even after taking the supremum over 휆 ∈ 퐺, we say that 퐽 satisfies uniform
+GLS condition on 퐺.
+GLS condition has been introduced in [37] for scalar Jacobi matrices with 퐴푛 ≡ 1. It was shown
+in [37, Theorem 1.1] that the set of 휆 ∈ R where (6.2) is fulfilled is a minimal support of the a.c. part of the
+spectral measure of 퐽. Similarly, a variant of it has been studied for bounded block Jacobi matrices in [48]
+with a similar conclusion. Thus this condition seems to be the right extension to possibly unbounded
+Jacobi matrices. However, we would like to stress that in the unbounded case the set where (6.2) is
+satisfied might no longer be a support of the a.c. part of the spectral measure of 퐽 even for 푑 = 1, see
+Example 7.2.
+Below, inTheorem 6.2, we will show that (uniform) GLS conditionimplies(uniform) 픟TR-nonsubordinacy
+on 퐺 for the barrier (5.20).
+We need the following observation, whose proof is an adaptation of the reasoning from [47, Lemma
+4.8].
+Proposition 6.1. For any 휆 ∈ R and any 푛 ∈ N we have
+∥푅(휆)∥2
+[1,푛]
+⇃|푅(휆)|⇂2
+[1,푛]
+≤
+�
+1
+휌푛
+푛
+�
+푘=1
+∥푅푘(휆)∥2
+�2
+.
+Proof. From the formula (3.50) we get
+��푅−1
+푘 (휆)
+�� ≤ ∥퐴푘−1∥ ∥푅푘 (휆)∥ .
+Thus, by (2.2),
+⇃|푅푘(휆)|⇂2=
+1
+��푅−1
+푘 (휆)
+��2 ≥
+1
+∥퐴푘−1∥2 ∥푅푘 (휆)∥2 .
+Thus,
+(6.3)
+휌푛
+�푛
+푘=1 ⇃|푅푘(푥)|⇂2 ≤
+휌푛
+�푛
+푘=1
+1
+∥ 퐴푘−1 ∥
+1
+∥퐴푘−1 ∥ ∥푅푘 (휆) ∥2
+.
+Since the weighted harmonic mean is always not greater than weighed arithmetic mean (see e.g. [39,
+Theorem 1, p.76]) we get
+휌푛
+�푛
+푘=1
+1
+∥퐴푘 ∥
+1
+∥퐴푘−1 ∥ ∥푅푘 (휆) ∥2
+≤ 1
+휌푛
+푛
+�
+푘=1
+1
+∥퐴푘−1∥ ∥퐴푘−1∥ ∥푅푘(휆)∥2 = 1
+휌푛
+푛
+�
+푘=1
+∥푅푘(휆)∥2,
+which combined with (6.3) ends the proof.
+□
+
+36
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Theorem 6.2. Let 퐺 ∈ Bor(R) be non-empty. Suppose that 퐽 satisfies (uniform) GLS condition on 퐺.
+Then 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺. Consequently, 퐽 is absolutely continuous
+in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).
+Proof. By Corollary 5.9 the function (5.20) is a barrier. Since by Proposition 6.1 we have
+lim inf
+푡→+∞ 픟(휆, 푡) < +∞
+the operator 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺. Thus, the result follows from
+Theorem 5.13.
+□
+6.2. GBS condition. Suppose that Carleman’s condition is satisfied and let 퐺 ⊂ R be non-empty. We
+say that 퐽 satisfies Generalized Behncke—Stolz (GBS in short) on 퐺 if
+(6.4)
+lim sup
+푛→+∞
+1
+휌푛
+푛
+�
+푘=1
+∥푅푘(휆)∥2 < +∞,
+휆 ∈ 퐺,
+where 휌푛 is defined in (6.1). Similarly, as in GLS condition, if (6.4) is finite even after taking the
+supremum over 휆 ∈ 퐺, we say that 퐽 satisfies uniform GBS condition on 퐺.
+GBS condition (6.4) has been introduced in [23, Remark 2.3] to study spectral properties of scalar
+unbounded Jacobi matrices. This condition turned out to be very convenient in the study of various
+classes of unbounded Jacobi matrices, see e.g. [21,23,24,29,40].
+We obviously have that (uniform) GBS condition implies (uniform) GLS condition. So the following
+result follows from Theorem 6.2
+Theorem 6.3. Let 퐺 ∈ Bor(R) be non-empty. Suppose that 퐽 satisfies (uniform) GBS condition on 퐺.
+Then 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺. Consequently, 퐽 is absolutely continuous
+in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).
+6.3. H class. For the arbitrary size 푚×푚 of matrices this class is given as follows (see, e.g., [43, Definition
+2.3]).
+Definition 6.4. Let (퐶푛)푛≥푛0 be a sequence of complex 푚 × 푚 matrices. We say that (퐶푛)푛≥푛0 ∈ 퐻 if
+there is a constant 푐 > 0 such that for any 푛 ≥ 푛0
+(6.5)
+∥퐶푛 · 퐶푛−1 · . . . · 퐶푛0∥ ≤ 푐
+푛
+�
+푘=푛0
+| det 퐶푘|1/푚.
+The following result, being “the sum” of [43, Corollary 2.4 and Theorem 1.7 (see also Definition 1.1)]
+explains the importance of this notion.
+Proposition 6.5. Let (퐶푛)푛≥푛0 be a sequence of complex invertible 푚 × 푚 matrices. For any 푛 ≥ 푛0
+define
+푅푛 := 퐶푛 · . . . · 퐶푛0.
+Then TFCAE:
+(퐶푛)푛≥푛0 ∈ 퐻,
+(6.6)
+∥푅푛∥ ≍푛 | det 푅푛|1/푚,
+(6.7)
+∥푅푛∥ ≍푛 ⇃|푅푛|⇂,
+(6.8)
+∀푣,푤 ∈C푚\{0}
+∥푅푛푣∥ ≍푛 ∥푅푛푤∥ .
+(6.9)
+The concept of 퐻 class has been introduced in [40] (see also [21, Lemma 1.8]) as a sufficient condition
+for GBS for scalar Jacobi matrices. It turned out that sometimes this stronger condition is somewhat
+easier to show than GBS, see [22, 40–42]. Later, in [43] this notion has been extended to block Jacobi
+matrices.
+Let us begin with an observation than under a condition, which is automatically satisfied for 푑 = 1,
+indeed 퐻 class implies GBS.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+37
+Proposition 6.6. Suppose that Carleman’s condition is satisfied. If �푇푛(휆)�
+푛∈N0 ∈ 퐻 for some 휆 ∈ R and
+(6.10)
+lim inf
+푛→+∞
+���� det
+� 퐴푛
+∥퐴푛∥
+����� > 0,
+then 퐽 satisfies GBS condition on {휆}.
+Proof. If �푅푛(휆)�
+푛∈N ∈ 퐻, then by (3.37) and (6.5) there exists a constant 푐 > 0 such that for any 푘 ≥ 1
+(6.11)
+∥푅푘(휆)∥2 ≤ 푐
+�� det 푅푘(휆)
+��1/푑.
+Now, by (3.41), (3.37) and (3.23) we have
+�� det 푅푘(휆)
+�� =
+1
+| det 퐴0|
+푘−1
+�
+푗=1
+| det 퐴∗
+푗−1|
+| det 퐴 푗|
+=
+1
+| det 퐴푘−1| .
+Using this we can rewrite (6.11) in the form
+∥푅푘(휆)∥2 ≤ 푐
+1
+∥퐴푘−1∥
+���� det
+� 퐴푘−1
+∥퐴푘−1∥
+�����
+−1/푑
+.
+Thus, if (6.10) is satisfied, then there exists a constant 푐′ > 0 such that for any 푘 ≥ 1
+∥푅푘(휆)∥2 ≤ 푐′
+1
+∥퐴푘−1∥ .
+Now by summing it for 푘 = 1, 2, . . . , 푛 it simply implies (6.4), what we needed to show.
+□
+In the following we show that in general 퐻 class implies 픟TR-nonsubordinacy condition, i.e., with the
+transfer matrix barrier (5.20).
+Theorem 6.7. Suppose that 퐽 is self-adjoint. Let 퐺 ∈ Bor(R) be non-empty. Suppose that �푇푛(휆)�
+푛∈N0 ∈
+퐻 for any 휆 ∈ 퐺. Then 퐽 satisfies 픟TR-nonsubordinacy condition on 퐺. Consequently, 퐽 is absolutely
+continuous in 퐺 and clLe(퐺) ⊂ 휎(퐽).
+Proof. Observe that by (6.8) there exists a constant 푐(휆) > 0 such that for any 푘 ≥ 1 we have
+푐(휆) ∥푅푘(휆)∥2 ≤⇃|푅푘(휆)|⇂2. Thus, for any 푡 ≥ 1 we get
+푐(휆) ∥푅(휆)∥2
+[1,푡] ≤⇃|푅(휆)|⇂2
+[1,푡] .
+In other words,
+∥푅(휆)∥2
+[1,푡]
+⇃|푅(휆)|⇂2
+[1,푡]
+≤
+1
+푐(휆)
+Thus
+lim sup
+푡→+∞ 픟TR(휆, 푡) ≤
+8
+푐(휆) < +∞.
+Consequently, the operator 퐽 satisfies 픟TR-nonsubordinacy condition on 퐺. Thus, the result follows from
+Theorem 5.13.
+□
+7. Examples and applications
+In this section we show some examples and counterexamples illustrating the applicability of our results.
+7.1. Vector nonsubordinacy does not characterise invertibility of 푀. The following example demon-
+strates that one cannot hope for the full characterisation of the a.c. spectrum of block Jacobi matrices in
+terms of vector nonsubordinacy condition. Let us recall that by Proposition 5.4 vector nonsubordinacy
+is equivalent to matrix nonsubordinacy. This shows that inevitably Theorem 1.1 provides only sufficient
+conditions for the absolute continuity of 푀.
+
+38
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Example 7.1. Let 푑 = 2. Define
+퐴푛 =
+�
+푎(1)
+푛
+0
+0
+푎(2)
+푛
+�
+,
+퐵푛 =
+�
+푏(1)
+푛
+0
+0
+푏(2)
+푛
+�
+,
+where 푎(푖) = �푎(푖)
+푛
+�
+푛∈N0, 푏(푖) = �푎(푖)
+푛
+�
+푛∈N0 are Jacobi parameters for 푖 = 1, 2. Let 퐽 (푖) be the Jacobi
+operator associated to 푎(푖), 푏(푖) for 푖 = 1, 2. Observe that
+(7.1)
+퐽 � 퐽 (1) ⊕ 퐽 (2).
+Let us take
+푎(푖)
+푛
+= (푛 + 1) 훼푖,
+푏(푖)
+푛
+= 0,
+푛 ≥ 0, 푖 = 1, 2,
+where 훼1, 훼2 ∈ (0, 1] and 훼1 > 훼2. Then by [59, Theorem 1] the corresponding measures 휇(1), 휇(2) are
+absolutely continuous on R with continuous positive densities. By (7.1)
+푀(·) �
+�
+휇(1) (·)
+0
+0
+휇(2) (·)
+�
+,
+and consequently, 푀(퐵) is invertible for any non-empty open 퐵 ⊂ R. We are going to show that 퐽
+does not satisfy vector nonsubordinacy condition for any 휆 ∈ R. Fix 휆 ∈ R. Let 푢(푖) ∈ GEV−1
+�퐽 (푖), 휆�
+for 푖 = 1, 2 be non-zero. Set 푢 := �푢(1)
+푛 푒1
+�
+푛∈N−1 and 푣 := �푢(2)
+푛 푒2
+�
+푛∈N−1. Then 푢, 푣 ∈ GEV−1 (퐽, 휆).
+According to [62, Theorem A]
+∥푢∥2
+[0,푛]
+�푛
+푘=0
+1
+푎(1)
+푛
+≍푛 1,
+∥푣∥2
+[0,푛]
+�푛
+푘=0
+1
+푎(2)
+푛
+≍푛 1.
+Thus,
+∥푢∥2
+[0,푛]
+∥푣∥2
+[0,푛]
+≍푛
+�푛
+푘=0
+1
+푎(2)
+푛
+�푛
+푘=0
+1
+푎(1)
+푛
+.
+By Stolz–Cesàro lemma
+lim
+푛→+∞
+�푛
+푘=0
+1
+푎(2)
+푛
+�푛
+푘=0
+1
+푎(1)
+푛
+= lim
+푛→+∞
+푎(1)
+푛
+푎(2)
+푛
+= +∞.
+Therefore,
+lim
+푛→+∞
+∥푢∥2
+[0,푛]
+∥푣∥2
+[0,푛]
+= +∞.
+In view of Corollary 2.4 it implies
+lim
+푡→+∞
+∥푢∥2
+[0,푡]
+∥푣∥2
+[0,푡]
+= +∞.
+Hence, 퐽 does not satisfy vector nonsubordinacy condition for 휆.
+7.2. GLS does not characterise the a.c. spectrum. In the following example we show that even for
+푑 = 1 the set where (6.2) is satisfied might not be a support of the a.c. part of 푀.
+Example 7.2. Let 푑 = 1 and 훼 > −1. Consider Jacobi parameters
+퐴푛 =
+�
+(푛 + 1)(푛 + 1 + 훼),
+퐵푛 = 2푛 + 1 + 훼.
+Then according to (3.43) we have
+푅푛(휆) =
+�
+1
+퐴0 ℓ(훼+1)
+푛−2
+(휆)
+ℓ(훼)
+푛−1(휆)
+1
+퐴0 ℓ(훼+1)
+푛−1
+(휆)
+ℓ(훼)
+푛
+(휆)
+�
+,
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+39
+where �ℓ(훼)
+푛
+�
+푛∈N0 is the sequence of orthonormal Laguerre polynomials of the order 훼, see e.g. [34,
+formula (9.12.4)].
+Consequently, we have 휎ac(퐽) = [0, +∞) and we are going to show that GLS
+condition is not satisfied for any 휆 ∈ (0, +∞). Obviously we have
+∥푅푛(휆)∥ ≥
+����푅푛(휆)
+�0
+1
+����� =
+�����
+�
+ℓ(훼)
+푛−1(휆)
+ℓ(훼)
+푛
+(휆)
+������ .
+Hence,
+푛
+�
+푘=1
+∥푅푘(휆)∥2 ≥
+푛
+�
+푘=1
+�����
+�
+ℓ(훼)
+푘−1(휆)
+ℓ(훼)
+푘
+(휆)
+������
+2
+.
+Let us set
+퐾푛(푥, 푦) =
+푛
+�
+푘=0
+ℓ(훼)
+푘
+(푥)ℓ(훼)
+푘
+(푦),
+˜휌푛 =
+푛
+�
+푘=0
+1
+√퐴푘
+.
+Then
+1
+˜휌푛
+푛
+�
+푘=1
+∥푅푘(휆)∥2 ≥ 1
+˜휌푛
+퐾푛(푥, 푥).
+Hence according to [64, Theorem B] the right-hand side of this inequality is bounded and positive for
+any 휆 ∈ (0, +∞). Hence, for some 푐 > 0
+1
+˜휌푛
+푛
+�
+푘=1
+∥푅푘 (휆)∥2 ≥ 푐,
+but by Stolz–Cesàro lemma
+lim
+푛→+∞
+˜휌푛
+휌푛
+= lim
+푛→+∞
+1
+√퐴푛
+1
+퐴푛
+= +∞.
+It implies that for any 휆 ∈ (0, +∞)
+lim
+푛→+∞
+1
+휌푛
+푛
+�
+푘=1
+∥푅푘 (휆)∥2 = +∞,
+and consequently, the condition (6.2) is not satisfied.
+7.3. Application to some classes of block Jacobi operators. In this section we would like to illustrate
+our results by showing that Jacobi matrices considered in [61, Theorem 2] are in fact absolutely continuous
+in some explicit region of the real line.
+Let us introduce a necessary notion. Let 푋 = (푋푛)푛∈N be a sequence from a normed space 푉. Let
+푁 ≥ 1 be a positive integer. We say that 푋 ∈ D푁
+1 if
++∞
+�
+푛=1
+∥푋푛+푁 − 푋푛∥ < +∞.
+Observe that if 푋 ∈ D푁
+1 , then for any 푖 ∈ {0, 1, . . . , 푁 − 1} the sequence (푋푘 푁 +푖)푘 ∈N satisfies Cauchy
+condition.
+Theorem 7.3. Let 퐽 be a block Jacobi matrix. Suppose that Jacobi parameters of 퐽 satisfy for some
+integer 푁 ≥ 1
+(7.2)
+�퐴−1
+푛
+�
+푛∈N, �퐴−1
+푛 퐵푛
+�
+푛∈N, �퐴−1
+푛 퐴∗
+푛−1
+�
+푛∈N ∈ D푁
+1 .
+Then for any 푖 ∈ {0, 1, . . . , 푁 − 1} and any 푧 ∈ C the limit
+(7.3)
+X푖(푧) :=
+lim
+푛→+∞
+푛≡푖 mod 푁
+푇푛+푁 −1(푧) . . . 푇푛+1(푧)푇푛(푧)
+exists, where 푇푛 is defined in (3.37). Suppose further that for some 푁-periodic sequence of invertible
+matrices (C푛)푛∈N0
+lim
+푛→+∞
+����
+푎푛
+∥푎푛∥ − C푛
+���� = 0.
+
+40
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+Define
+Λ =
+�
+휆 ∈ R : Re
+� �
+0
+−C푁 −1
+C∗
+푁 −1
+0
+�
+X0(휆)
+�
+is strictly positive or strictly negative on C2푑
+�
+.
+Then �푅푛(휆)�
+푛∈N ∈ 퐻 for any 휆 ∈ Λ. Consequently, if Carleman’s condition is satisfied, then 퐽 is
+absolutely continuous in Λ and cl(Λ) ⊂ 휎ac(퐽).
+Proof. First of all, let us observe that (7.2) implies that for any 푗 ∈ {0, 1, . . . , 푁 − 1} the sequence
+(푇푘 푁 +푗)푘 ∈N is convergent. Thus the limit (7.3) exists.
+We are going to show that the condition (6.8) is satisfied. Let us recall that according to [61, Theorem
+2] for any compact interval 퐾 ⊂ Λ there are constants 푐1 > 0, 푐2 > 0 such that for any normalized
+푢 ∈ GEV−1 (휆), where 휆 ∈ 퐾, and any 푛 ≥ 1 we have
+(7.4)
+푐1
+∥퐴푛∥ ≤
+����
+�
+푢푛−1
+푢푛
+�����
+2
+≤
+푐2
+∥퐴푛∥ .
+Since, for any 푢 ∈ GEV−1 (휆)
+�
+푢푛−1
+푢푛
+�
+= 푅푛(휆)
+�
+푢−1
+푢0
+�
+.
+Hence, having the notation above in mind we have
+⇃|푅푛(휆)|⇂2=
+inf
+∥푢−1 ∥2+∥푢0 ∥2=1
+푢∈GEV−1(휆)
+����
+�
+푢푛−1
+푢푛
+�����
+2
+,
+∥푅푛(휆)∥2 =
+sup
+∥푢−1 ∥2+∥푢0 ∥2=1
+푢∈GEV−1(휆)
+����
+�
+푢푛−1
+푢푛
+�����
+2
+.
+Thus, (7.4) implies
+(7.5)
+푐1
+∥퐴푛∥ ≤⇃|푅푛(휆)|⇂2≤ ∥푅푛(휆)∥2 ≤
+푐2
+∥퐴푛∥,
+which shows that (6.8) is satisfied. Therefore, by Proposition 6.5, �푅푛(휆)�
+푛∈N ∈ 퐻 for any 휆 ∈ Λ. Finally,
+if Carleman’s condition is satisfied, then the remaining conclusion follows from Theorem 6.7.
+□
+Remark 7.4. Let us observe that (7.5) immediately implies that 퐽 satisfies uniform GBS condition for
+any compact interval 퐾 ⊂ Λ.
+7.3.1. Asymptotically periodic Jacobi parameters. Let 푁 be a positive integer andlet (A푛)푛∈N0, (B푛)푛∈N0
+be 푁-periodic Jacobi parameters. For any 푖 ∈ {1, . . . , 푁} let us define
+(7.6)
+픛푖(푧) =
+푁 +푖−1
+�
+푗=푖
+픗푖(푥)
+where
+픗푖(푧) =
+�
+0
+I
+−A−1
+푗 A∗
+푗−1
+A−1
+푗 (푧 I −B푗)
+�
+.
+Let 퐽per be the Jacobi operator associated with the sequences (A푛)푛∈N0, (B푛)푛∈N0.
+Spectral properties of the operator 퐽per are well-known in the case 푑 = 1, see e.g. [55, Chapter
+5], [67, Chapter 2.1]. For 푑 > 1 we have a good understanding of 퐽per when its Jacobi parameters are all
+self-adjoint and 푁 = 1, see e.g. [69,70]. Then we have
+휎(퐽per) =
+�
+푡 ∈[−2,2]
+휎(A0푡 + B)
+and the spectrum is absolutely continuous, see [70, Corollary 1.1]. When the Jacobi parameters are not
+self-adjoint or 푁 > 1 a point spectrum is possible, see [26] for more details. Some results concerning
+absolute continuity might be extracted from [51, Section 2].
+It is natural to consider compact perturbations of 퐽per. Namely, we say that 퐽 has asymptotically
+푁-periodic Jacobi parameters if
+lim
+푛→+∞ ∥퐴푛 − A푛∥ = 0
+and
+lim
+푛→+∞ ∥퐵푛 − B푛∥ = 0.
+By Weyl’s perturbation theorem we have
+휎ess(퐽) = 휎ess(퐽per).
+Our Theorem 7.3 leads to
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+41
+Corollary 7.5. Let 푁 be a positive integer.
+Suppose that 퐽 has asymptotically 푁-periodic Jacobi
+parameters satisfying
+(퐴푛)푛∈N0, (퐵푛)푛∈N0 ∈ D푁
+1 .
+Define
+(7.7)
+Λ =
+�
+푥 ∈ R : Re
+� �
+0
+−A푁 −1
+A∗
+푁 −1
+0
+�
+픛푁 (푥)
+�
+is strictly positive or strictly negative on C푑
+�
+.
+Then 퐽 is absolutely continuous in Λ and cl(Λ) ⊂ 휎ac(퐽).
+7.3.2. Periodic modulations. Let 퐽 be a block Jacobi matrix and let 푁 ≥ 1 be an integer. We shall call 퐽
+a block Jacobi matrix with 푁-periodically modulated entries if there exists 푁-periodic Jacobi parameters
+(A푛)푛∈N0, (B푛)푛∈N0 such that the Jacobi parameters of 퐽 satisfy
+(7.8)
+lim
+푛→+∞
+��퐴−1
+푛
+�� = 0,
+lim
+푛→+∞
+��퐴−1
+푛 퐴∗
+푛−1 − A−1
+푛 A∗
+푛−1
+�� = 0
+and
+lim
+푛→+∞
+��퐴−1
+푛 퐵푛 − A−1
+푛 B푛
+�� = 0.
+In the scalar case, this class has been introduced in [25] and it is actively studied ever since, see
+e.g. [5, 9, 45, 49, 57, 60, 63, 65, 66]. Let us observe that given any positive scalar sequence (푐푛)푛∈N0
+satisfying
+(7.9)
+lim
+푛→+∞ 푐푛 = +∞
+and
+lim
+푛→+∞
+푐푛−1
+푐푛
+= 1
+leads to (7.8) for Jacobi parameters
+(7.10)
+퐴푛 = 푐푛A푛,
+퐵푛 = 푐푛B푛.
+Therefore, the class of block Jacobi matrices with periodically modulated entries might be thought of as
+a certain perturbation of the model example (7.9)–(7.10).
+Observe that due to (7.8), (3.37) and (7.6) we have for any 푖 ∈ {1, . . . , 푁} and 푧 ∈ C
+lim
+푘→+∞푇푘 푁 +푖(푧) = 픗푖(0),
+thus in view of (7.3) and (7.6),
+X0(푧) = lim
+푘→+∞푇푗푁 +푁 −1(푧) . . .푇푗푁 (푧) = 픛푁 (0).
+Therefore, our Theorem 7.3 leads to
+Corollary 7.6. Suppose that 퐽 is a block Jacobi matrix with 푁-periodically modulated entries for some
+푁 ≥ 1. Assume the Carleman’s condition (1.7) and
+�퐴−1
+푛 퐴∗
+푛−1
+�
+푛∈N, �퐴−1
+푛 퐵푛
+�
+푛∈N, �퐴−1
+푛
+�
+푛∈N ∈ D푁
+1 .
+If 0 ∈ Λ, where Λ is defined in (7.7), then 퐽 is absolutely continuous in R and 휎ac(퐽) = R.
+Appendix A. Vector and matrix measures — selected basic notions
+For self-consistency and some self-sufficiency of the paper we collect here selected definitions of some
+basic notions and some properties related to matrix measures and — more generally — vector measures.
+We omit here the more sophisticated construction of the appropriate 퐿2-type space for the matrix measure,
+referring the reader to the literature (see, e.g., [68, Section 8] or [44]18)
+We start from the general definition of vector measure.
+Consider a set Ω with 픐 — a 휎-algebra of subsets of Ω and a certain norm space 푋. Let 푉 : 픐 −→ 푋
+Definition A.1. 푉 is a vector measure (in 푋) iff 푉 is countably additive in the norm sense in 푋.
+Now, let us consider a special case 푋 := 푀푑(C) for some 푑 ∈ N (with a standard norm, say). And let
+푀 : 픐 −→ 푀푑(C).
+Definition A.2. 푀 is a (푑 × 푑) matrix measure iff
+(a) 푀 is a vector measure;
+(b) 푀(휔) ≥ 0 for any 휔 ∈ 픐.19
+18The second position contains the detailed definition of the Hilbert space 퐿2(푀) for matrix measures and the details of the
+abstract spectral theory for finitely cyclic s.a. operators, based on the matrix measure approach and multiplication by a function
+operators in 퐿2(푀) spaces.
+19So, in particular 푀(휔) is s.a..
+
+42
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+So, in particular, each matrix measure is a vector measure, but despite its name and due to the extra
+non-negativity property (b), matrix measure is “much more” than a vector measure in 푀푑(C).
+The above Ω, 픐 and 푑 are “fixed” below.
+We have to precise here some terminology (choosing it from various versions in literature) and to recall
+several basic facts related to vector measures, matrix measures and measures.
+Suppose that 휈 is a measure on 픐 and 푉 : 픐 −→ 푋 is a vector measure, where 푋 is a norm space.
+In several cases below we will also need to assume additionally that 푋 = C푘 for some 푘, including
+possible obvious identifications, as, e.g., 푀푑(C) ≡ C(푑2), to provide the clear and standard sense of the
+integral and of L1
+푋 (휈) functions (see the appropriate part of Section 2.1).
+For any 퐺 ∈ 픐 denote 픐퐺 := {휔 ∈ 픐 : 휔 ⊂ 퐺}. Surely 픐퐺 is a 휎-algebra of subsets of 퐺 and
+푉↾픐퐺 is a vector measure on 픐퐺 ("on 퐺"). We denote it by 푉퐺, i.e.
+푉퐺 := 푉↾픐퐺,
+but we also call it the restriction of 푉 to 퐺. Analogous situation (and notation, and terminology) is well
+known and valid here for measures.
+Suppose that 퐻 : Ω −→ 푋 = C푘 is a measurable function w.r.t. 픐. If, moreover, 퐻 ∈ L1
+푋 (휈), then
+we define a new function from 픐 into 푋 by the formula:
+(A.1)
+∫
+휔
+퐻 d휈,
+휔 ∈ 픐.
+It is obviously a vector measure, and we denote it by
+퐻 d휈,
+analogously as in the case of measures (when 퐻 should be a scalar non-negative measurable function,
+instead of 퐻 ∈ L1
+푋 (휈)). But in our main case 푋 = 푀푑(C)
+(identified with C(푑2)) the situation is
+somewhat similar and one easily checks the following result.
+Fact A.3. If 퐻 ∈ L1
+푋 (휈) and 퐻(푡) ≥ 0 for 휈-a.e. 푡 ∈ Ω, then the vector measure 퐻 d휈 is a matrix
+measure.
+Definition A.4. If a vector measure 푉 is such, that 푉 = 퐻 d휈 with some 퐻 ∈ L1
+푋 (휈), then we call 퐻 the
+density20 of 푉 with respect to 휈.
+Let 퐺 ∈ 픐.
+the density on 퐺 w.r.t.: The density of 푉 on 퐺 w.r.t. 휈 means: any density of 푉퐺 w.r.t. 휈퐺;
+a support: 퐺 is a support of 푉 iff 푉Ω\퐺 is the zero vector measure (on 픐Ω\퐺)21;
+a minimal support w.r.t.: 퐺 is a minimal support of 푉 w.r.t. 휈 iff 퐺 is a support of 푉 and for any
+support 퐺′ of 푉 included in 퐺
+휈(퐺 \ 퐺′) = 0;
+a.c. w.r.t.: 푉 is absolutely continuous (abbrev.: a.c.) w.r.t. 휈 iff for any 휔 ∈ 픐 if 휈(휔) = 0, then
+푉(휔) = 0 ;
+sing. w.r.t.: 푉 is singular (abbrev.: sing.) w.r.t. 휈 iff there exists such a support 푆 ∈ 픐 of 푉 that
+휈(푆) = 0;
+the a.c./sing. part w.r.t.: If 푉1,푉2 : 픐 −→ 푋 are two vector measures, such that
+(i) 푉1 is a.c. w.r.t. 휈 and 푉1 is sing. w.r.t. 휈,
+(ii) 푉 = 푉1 + 푉2,
+then we call 푉1 the a.c. part of 푉 w.r.t. 휈 and we denote it by
+푉ac,휈,
+and we call 푉2 the sing. part of 푉 w.r.t. 휈, and we denote it by
+푉sing,휈.
+20However, it can be not unique, as a function from L1
+푋 (휈).
+21 Generally, it is not sufficient here (contrary to measures) that 푉 (Ω \ 퐺) = 0 because the property of having zero measure
+is not inheritable into measurable subsets. However, for matrix measures it is sufficient by non-negativity.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+43
+Note here, that above two notions are well(uniquely)-defined, since the above decomposition, if
+exists, is unique22;
+a.c. on 퐺 w.r.t.: 푉 is a.c. on 퐺 w.r.t. 휈 iff 푉퐺 is a.c. w.r.t. 휈퐺;
+sing. on 퐺 w.r.t.: 푉 is sing. on 퐺 w.r.t. 휈 iff 푉퐺 is sing. w.r.t. 휈퐺.
+All adopt all the above definitions and names also for any measure 휇 instead of a vector measure 푉
+(recall that measure can be not a vector measure) just by interchanging the symbols 휇 and 푉, including
+the notation
+휇ac,휈,
+휇sing,휈,
+however they are mostly commonly known in the case of measures.
+We adopt here also the convention, that the part “w.r.t. 휈”, as well as “, 휈” in the appropriate symbols,
+as e.g., 푉ac,휈, 푉sing,휈, can be omitted to shorten the notation to, e.g.,
+푉ac, 푉sing, 휇ac, 휇sing
+etc, in the case when Ω = R, 픐 = Bor(R) and 휈 is the Lebesgue measure | · |.
+For a 푑 × 푑 matrix measure 푀 and 푖, 푗 = 1, . . . , 푑 let us define 푀푖 푗 : 픐 → C by
+(A.2)
+푀푖 푗(휔) := �푀(휔)�
+푖 푗 ,
+휔 ∈ 픐.
+By the point (a) of Definition A.2, each of the 푀푖 푗 is a complex measure on 픐. Moreover, non-negativity
+from (b) of a matrix means also its self-adjointness, so we have
+(A.3)
+푀 푗,푖 = 푀푖, 푗
+푖, 푗 = 1, . . . , 푑.
+By non-negativity, defining tr푀 : 픐 → C by the formula
+(A.4)
+tr푀 (휔) := tr(푀(휔)),
+휔 ∈ 픐,
+we get in fact a finite measure tr푀, called trace measure of 푀. This “classical” measure is much simpler
+mathematical object than the matrix measure 푀, but it contains a lot of important information on 푀.
+The results below are proved in [44, Section III.1].
+Fact A.5. For 푀 being a matrix measure as above:
+(i) 푀, as well as each 푀푖 푗 for 푖, 푗 = 1, . . . , 푑, are absolutely continuous with respect to tr푀;
+(ii)
+(A.5)
+푀(휔) = 0 ⇐⇒ tr푀 (휔) = 0,
+휔 ∈ 픐;
+(iii)
+(A.6)
+0 ≤ 푀(휔) ≤ tr푀 (휔) I
+휔 ∈ 픐;
+(iv) There exists a density 퐷 : Ω −→ 푀푑(C) of 푀 w.r.t. tr푀, such that for any 푡 ∈ Ω
+0 ≤ 퐷(푡) ≤ I , 23
+푡 ∈ Ω.
+Note that each density 퐷 of 푀 w.r.t. tr푀 is determined only up to tr푀-a.e. equality, and each one is
+called the trace density of 푀. The set of all the densities of 푀 w.r.t. tr푀 is denoted here by D푀, and by
+D•
+푀 we denote the set of those 퐷, which satisfy conditions from the point (iv) above.
+By the definition of density we have
+(A.7)
+푀(휔) =
+∫
+휔
+퐷 d tr푀,
+휔 ∈ 픐, 퐷 ∈ D푀.
+We assume here, to the end if this section, that 푀 : 픐 −→ 푀푑(C) is a matrix measure and
+휈 : 픐 −→ [0, +∞] is a 휎-finite measure.
+There exist several ways to get a decomposition of some vector measures 푉 into its a.c. and sing.
+parts w.r.t. a measure 휈. All they are based somehow on the Lebesgue–Radon–Nikodym Theorem (see,
+e.g. [50]) for a complex measure “w.r.t. a 휎-finite measure” version. We need it only for our matrix
+measure 푀 and it will be convenient to make it via the appropriate decomposition of tr푀 onto parts
+22Because any linear combination of vector measures being a.c. (sing.) w.r.t. 휈 is also a.c. (sing.) w.r.t. 휈; and a vector
+measure which is both a.c. and sing. w.r.t. 휈 is the zero measure.
+23In particular, 퐷(푡) is self-adjoint.
+
+44
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+(tr푀)ac,휈 and (tr푀)sing,휈, which exist by the Lebesgue–Radon–Nikodym Theorem. So, we just consider
+any 퐷 ∈ D푀 and two matrix measures:
+퐷 d(tr푀)ac,휈,
+퐷 d(tr푀)sing,휈
+(see notation (A.1))
+Fact A.6. The a.c. and the sing. parts of 푀 w.r.t. 휈 exist, and they satisfy
+(A.8)
+푀ac,휈 = 퐷 d(tr푀)ac,휈 ,
+푀sing,휈 = 퐷 d(tr푀)sing,휈 ,
+where 퐷 is an arbitrary density from D푀. Moreover, both 푀ac,휈, 푀sing,휈 are matrix measures24, and
+(A.9)
+tr푀ac,휈 = (tr푀)ac,휈,
+tr푀sing,휈 = (tr푀)sing,휈.
+In particular, if 푆 ∈ 픐, then TFCAE:
+• 푆 is a support (version 2.: minimal support w.r.t. 휈) of 푀ac,휈 ;
+• 푆 is a support (version 2.: minimal support w.r.t. 휈) of tr푀ac,휈 ;
+• 푆 is a support (version 2.: minimal support w.r.t. 휈) of (tr푀)ac,휈 ;
+and analogically for 푀sing,휈, tr푀sing,휈, (tr푀)sing,휈.
+Proof. Using “the short theory” presented above, one immediately checks that the pair of vector measures
+퐷푑(tr푀)ac,휈, 퐷푑(tr푀)sing,휈 satisfies the conditions from Definition A.4 of the parts 푀ac,휈 푀sing,휈. So,
+by the uniqueness of the decomposition, we get (A.8). Now, by the non-negativity of 퐷 and by Fact A.3
+we see that 푀ac,휈, 푀sing,휈 are matrix measures.
+To get the assertion (A.9), observe that the equality
+푀 = 푀ac,휈 + 푀sing,휈
+yields
+tr푀 =
+tr푀ac,휈 + tr푀sing,휈
+by the definition of the trace maesure.
+But 푀ac,휈, 푀sing,휈 are a.c.
+or, respec-
+tively, sing. w.r.t. 휈, hence also tr푀ac,휈, tr푀sing,휈 are a.c. or, respectively, sing. w.r.t. 휈, just by the use of
+the property (A.5) for both those matrix measures. So, we get the result just by the definitions of a.c. and
+sing. parts. And the last part follows directly from (A.9), (A.5) and by the observation that both notions:
+of support, as well as of minimal support w.r.t. a measure, are determined by zero vector measure sets,
+only.
+□
+Now we turn to a “technical” result concerning the notion of the minimal support.
+Lemma A.7. Consider a matrix measure 푀 : 픐 −→ 푀푑(C), a measure 휈 : 픐 −→ [0, +∞], sets
+푆푎, 푆푠 ∈ 픐 and a non-negative function 퐹 : 푆푎 −→ 푀푑(C). If
+(i)
+푆푎 is a support of 푀ac,휈 and 푆푠 is a support of 푀sing,휈,
+(ii)
+푆푎 ∩ 푆푠 = ∅,
+(iii) 퐹 is a density of 푀ac,휈 on 푆푎 w.r.t. 휈,
+(iv)
+휈 ({푡 ∈ 푆푎 : 퐹(푡) = 0}) = 0,
+then 푆푠 is a support of (푡푟푀)sing,휈 and 푆푎 is a minimal support of (푡푟푀)ac,휈 w.r.t. 휈.
+Proof. From Fact A.6 we immediately see that 푆푠 is a support of (tr푀)sing,휈 and 푆푎 is support of
+(tr푀)ac,휈. To prove the minimality, consider any 푆′ ⊂ 푆푎 which is also a support of (tr푀)ac,휈. Hence,
+again by Fact A.6
+(A.10)
+0 = (tr푀)ac,휈(푆푎 \ 푆′) = tr푀ac,휈 (푆푎 \ 푆′).
+On the other hand, by the assumption (iii), using (푆푎 \ 푆′) ⊂ 푆푎 and the non-negativity of tr 퐹(푡) for any
+푡 ∈ 푆푎, we have
+tr푀ac,휈 (푆푎 \ 푆′) = tr �푀ac,휈(푆푎 \ 푆′)� = tr
+�∫
+(푆푎\푆′)
+퐹 d휈
+�
+=
+∫
+(푆푎\푆′)
+tr 퐹(푡) d휈(푡).
+Thus, by (A.10) we have tr 퐹(푡) = 0 for 휈-a.e.
+푡 ∈ (푆푎 \ 푆′). Moreover, by Proposition 2.5(iv),
+tr 퐹(푡) = 0 ⇐⇒ 퐹(푡) = 0. So, by the assumption (iv), we get tr 퐹(푡) ≠ 0 also for 휈-a.e. 푡 ∈ (푆푎 \ 푆′).
+Thus 휈(푆푎 \ 푆′) = 0.
+□
+24By the definition, they are “only” vector measures in 푀푑(C).
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+45
+At the end of this section let us recall the definition of the integral of the scalar function w.r.t. a vector
+measure for some simplest case, but sufficient for our goals. Consider a vector measure 푉 : 픐 −→ 푋,
+where 푋 = C푘 (e.g. — a matrix measure, with 푘 = 푑2) and 푓 : Ω −→ C — a bounded 픐-measurable
+function. Then for any 푗 = 1, . . . , 푘 the function 푉푗 : 픐 −→ C given by
+푉푗(휔) := �푉(휔)�
+푗,
+휔 ∈ 픐,
+is a complex measure and therefore the integral
+∫
+Ω
+푓 d푉푗
+is well-defined (see, e.g., [8, Section I.1]). Thus, we define simply:
+(A.11)
+∫
+Ω
+푓 d푉 :=
+�∫
+Ω
+푓 d푉1, . . . ,
+∫
+Ω
+푓 d푉푘
+�
+∈ C푘.
+References
+[1] H. Behncke, Absolute continuity of Hamiltonians with von Neumann-Wigner potentials, Proc. Amer. Math. Soc. 111
+(1991), no. 2, 373–384.
+[2] Ju. M. Berezanski˘ı, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol.
+17, American Mathematical Society, Providence, R.I., 1968, Translated from the Russian by R. Bolstein, J. M. Danskin,
+J. Rovnyak and L. Shulman.
+[3] Ch. Berg, The matrix moment problem, Coimbra Lecture Notes on Orthogonal Polynomials, Nova Science Publishers Inc.,
+New York, 2008, pp. 1–57.
+[4] S. Clark and D. Hinton, Strong nonsubordinacy and absolutely continuous spectra for Sturm-Liouville equations, Differ-
+ential Integral Equations 6 (1993), no. 3, 573–586.
+[5] D. Damanik and S. Naboko, Unbounded Jacobi matrices at critical coupling, J. Approx. Theory 145 (2007), no. 2,
+221–236.
+[6] D. Damanik, A. Pushnitski, and B. Simon, The analytic theory of matrix orthogonal polynomials, Surv. Approx. Theory
+4 (2008), 1–85.
+[7] H. Dette, Reuther B., W. J. Studden, and M. Zygmunt, Matrix measures and random walks with a block tridiagonal
+transition matrix, SIAM J. Matrix Anal. A. 29 (2007), no. 1, 117–142.
+[8] J. Diestel and J. J. Uhl, Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence,
+R.I., 1977.
+[9] J. Dombrowski, J. Janas, M. Moszyński, and S. Pedersen, Spectral gaps resulting from periodic perturbations of a class
+of Jacobi operators, Constr. Approx. 20 (2004), no. 4, 585–601.
+[10] A. J. Durán and P. López-Rodrıguez, The matrix moment problem, Margarita Mathematica en memoria de José Javier
+Guadalupe, Universidad de la Rioja, 2001, pp. 333–348.
+[11] A.J. Durán and W. Van Assche, Orthogonal matrix polynomials and higher-order recurrence relations, Linear Algebra
+Appl. 219 (1995), 261–280.
+[12] Yu.M. Dyukarev, On conditions of complete indeterminacy for the matricial Hamburger moment problem, Complex
+function theory, operator theory, Schur analysis and systems theory—a volume in honor of V. E. Katsnelson, Oper. Theory
+Adv. Appl., vol. 280, Birkhäuser/Springer, Cham, 2020, pp. 327–353.
+[13] F. Gesztesy and E. Tsekanovskii, On matrix-valued Herglotz functions, Math. Nachr. 218 (2000), 61–138.
+[14] D. Gilbert, Asymptotic methods in the spectral analysis of Sturm-Liouville operators, Sturm-Liouville theory, Birkhäuser,
+Basel, 2005, pp. 121–136.
+[15] D.J. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc.
+Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 213–229.
+[16] D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J.
+Math. Anal. Appl. 128 (1987), no. 1, 30–56.
+[17] L. Golinskii and P. Nevai, Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle,
+Comm. Math. Phys. 223 (2001), no. 2, 223–259.
+[18] G.H. Golub and J.H. Welsch, Calculation of Gauss quadrature rules, Math. Comp. 23 (1969), 221–230.
+[19] Sh. Guo, D. Damanik, and D.C. Ong, Subordinacy theory for extended CMV matrices, Sci. China Math. 65 (2022), no. 3,
+539–558.
+[20] S. Hassi, Ch. Remling, and H. de Snoo, Subordinate solutions and spectral measures of canonical systems, Integral
+Equations Operator Theory 37 (2000), no. 1, 48–63.
+[21] J. Janas and M. Moszyński, Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights,
+Integral Equations Operator Theory 43 (2002), no. 4, 397–416.
+[22]
+, Spectral analysis of unbounded Jacobi operators with oscillating entries, Studia Math. 209 (2012), no. 2, 107–133.
+[23] J. Janas and S. Naboko, Jacobi matrices with power-like weights—grouping in blocks approach, J. Funct. Anal. 166 (1999),
+no. 2, 218–243.
+
+46
+MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI
+[24]
+, Multithreshold spectral phase transitions for a class of Jacobi matrices, Recent advances in operator theory
+(Groningen, 1998), Oper. Theory Adv. Appl., vol. 124, Birkhäuser Basel, 2001, pp. 267–285.
+[25]
+, Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries, J. Funct. Anal. 191 (2002),
+no. 2, 318–342.
+[26]
+, On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach, J. Difference Equ.
+Appl. 21 (2015), no. 11, 1103–1118.
+[27] J. Janas, S. Naboko, and L.O. Silva, Green matrix estimates of block Jacobi matrices I: Unbounded gap in the essential
+spectrum, Integral Equations Operator Theory 90 (2018), no. 4, Paper No. 49, 24.
+[28]
+, Green matrix estimates of block Jacobi matrices II: Bounded gap in the essential spectrum, Integral Equations
+Operator Theory 92 (2020), no. 3, Paper No. 21, 30.
+[29] J. Janas, S. Naboko, and G. Stolz, Spectral theory for a class of periodically perturbed unbounded Jacobi matrices:
+elementary methods, J. Comput. Appl. Math. 171 (2004), no. 1-2, 265–276.
+[30] S. Jitomirskaya and Y. Last, Power-law subordinacy and singular spectra I. Half-line operators, Acta Math. 183 (1999),
+no. 2, 171–189.
+[31] S. Karlin and J. McGregor, Random walks, Illinois J. Math. 3 (1959), 66–81.
+[32] S. Karlin and J.L. McGregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem,
+Trans. Amer. Math. Soc. 85 (1957), 489–546.
+[33] S. Khan and D.B. Pearson, Subordinacy and spectral theory for infinite matrices, Helv. Phys. Acta 65 (1992), no. 4,
+505–527.
+[34] R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric orthogonal polynomials and their 푞-analogues, Springer
+Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
+[35] A. G. Kostyuchenko and K. A. Mirzoev, Three-term recurrence relations with matrix coefficients. The completely indefinite
+case, Math. Notes 63 (1998), no. 5, 624–630.
+[36] S. Kupin and S. Naboko, On the instability of the essential spectrum for block Jacobi matrices, Constr. Approx. 48 (2018),
+no. 3, 473–500.
+[37] Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional
+Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329–367.
+[38] N. Levi, Subordinacy theory on star-like graphs, arXiv:2203.13548, 2022.
+[39] D.S. Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag,
+New York-Berlin, 1970, In cooperation with P. M. Vasić.
+[40] M. Moszyński, Spectral properties of some Jacobi matrices with double weights, J. Math. Anal. Appl. 280 (2003), no. 2,
+400–412.
+[41]
+, Slowly oscillating perturbations of periodic Jacobi operators in 푙2(N), Studia Math. 192 (2009), no. 3, 259–279.
+[42]
+, Weyl sequences and the essential spectrum of some Jacobi operators, J. Operator Theory 67 (2012), no. 1,
+237–256.
+[43]
+, Uni-asymptotic linear systems and Jacobi operators, Integral Equations Operator Theory 91 (2019), no. 3, Paper
+No. 23, 15.
+[44] M. Moszyński, Spectral Theory of Self-adjoint Finitely Cyclic Operators and Introduction to Matrix-measure 퐿2-spaces,
+arXiv:2212.13953, 2022.
+[45] S. Naboko, I. Pchelintseva, and L.O. Silva, Discrete spectrum in a critical coupling case of Jacobi matrices with spectral
+phase transitions by uniform asymptotic analysis, J. Approx. Theory 161 (2009), 314–336.
+[46] F.V. Oliveira and S.L. Carvalho, Kotani theory for ergodic matrix-like Jacobi operators, arXiv:2105.11524, 2021.
+[47]
+,
+Criteria
+for
+the
+absolutely
+continuous
+spectral
+components
+of
+matrix-valued
+Jacobi
+operators,
+arXiv:2108.12485v2, 2022.
+[48] F.V. Oliveira and S.L. de Carvalho, Criteria for the absolutely continuous spectral components of matrix-valued Jacobi
+operators, Rev. Math. Phys. 34 (2022), no. 10, Paper No. 2250037, 42.
+[49] I. Pchelintseva, A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices,
+Opuscula Math. 28 (2008), no. 2, 137–150.
+[50] W. Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.
+[51] J. Sahbani, Spectral theory of a class of block Jacobi matrices and applications, J. Math. Anal. Appl. 438 (2016), no. 1,
+93–118.
+[52] M. Schmied, R. Sims, and G. Teschl, On the absolutely continuous spectrum of Sturm-Liouville operators with applications
+to radial quantum trees, Oper. Matrices 2 (2008), no. 3, 417–434.
+[53] K. Schmüdgen, The moment problem, Graduate Texts in Mathematics, vol. 277, Springer, Cham, 2017.
+[54] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998), no. 1, 82–203.
+[55]
+, Szegő’s theorem and its descendants: Spectral theory for 퐿2 perturbations of orthogonal polynomials, Princeton
+University Press, 2010.
+[56]
+, Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI,
+2015.
+[57] S. Simonov, An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated
+weights, pp. 187–203, Birkhäuser Basel, 2007.
+[58] A. Sinap and W. Van Assche, Orthogonal matrix polynomials and applications, J. Comput. Appl. Math. 66 (1996), no. 1,
+27–52.
+
+NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES
+47
+[59] G. Świderski, Periodic perturbations of unbounded Jacobi matrices II: Formulas for density, J. Approx. Theory 216
+(2017), 67–85.
+[60]
+, Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime, J. Approx. Theory 233 (2018),
+1–36.
+[61]
+, Spectral properties of block Jacobi matrices, Constr. Approx. 48 (2018), no. 2, 301–335.
+[62] G. Świderski and B. Trojan, Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvec-
+tors, J. Approx. Theory 216 (2017), 38–66.
+[63]
+, Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients, J. Funct. Anal. 278 (2020),
+no. 3, 108326, 55.
+[64]
+, Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II, arXiv:
+2107.11154, 2021.
+[65]
+, About essential spectra of unbounded Jacobi matrices, J. Approx. Theory 278 (2022), Paper No. 105746, 47.
+[66]
+, Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case, accepted
+in Annales de l’Institut Fourier, 2022.
+[67] W. Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol. 1265, Springer-Verlag Berlin
+Heidelberg, 1987.
+[68] J. Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol. 1258, Springer-Verlag,
+Berlin, 1987.
+[69] M. J. Zygmunt, Generalized Chebyshev polynomials and discrete Schrödinger operators, J. Phys. A: Math. Gen. 34 (2001),
+no. 48, 10613–10618.
+[70] M.J. Zygmunt, Jacobi block matrices with constant matrix terms, Spectral methods for operators of mathematical physics,
+Oper. Theory Adv. Appl., vol. 154, Birkhäuser, Basel, 2004, pp. 233–238.
+Marcin Moszyński, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul. Stefana
+Banacha 2, 02-097 Warsaw, Poland
+Email address: mmoszyns@mimuw.edu.pl
+Grzegorz Świderski, Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-696 Warsaw,
+Poland
+Email address: gswiderski@impan.pl
+
diff --git a/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/load_file.txt b/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b7694cb21c1f81a70c6aaced7384cb1938a7540c
--- /dev/null
+++ b/ltAyT4oBgHgl3EQfYfdr/content/tmp_files/load_file.txt
@@ -0,0 +1,2903 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf,len=2902
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='00204v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='SP]  31 Dec 2022  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK  JACOBI MATRICES  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It is well-known for self-adjoint scalar Jacobi operators that their absolute continuity and the  absolute continuous spectrum in a subset of the real line can be characterized by non-existence of subordinate  generalized eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In our work we explore to what extent a similar relation is true for block Jacobi  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In this setup, under some uniformity conditions, we show that nonsubordinacy in the sense of  matrix generalized eigenvectors also implies the absolute continuity, similarly as in the case of the scalar  Jacobi operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, we get the absolute continuity of the matrix spectral measure with the invertibility  of its density almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We also present an example showing that the reverse implication in general  does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We extend some sufficient conditions for nonsubordinacy from the scalar to the block case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Finally, we give applications of our results to some classes of block Jacobi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Introduction  2  Acknowledgment  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Preliminaries  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Introductory notation and notions  7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Some asymptotic symbols and the affine interpolation  9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Some analogs of “J–L continuous interpolation” of a discrete family of semi-norms  10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Some matrix norm inequalities  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Block Jacobi matrix and operator(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A spectral representation and the associated difference  equations  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The finite-cyclicity  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The spectral matrix measure 퐸퐽, �휑 and the representation of 퐽 as the multiplication operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The two associated difference equations and “the solution extensions to -1”  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Transfer matrices and the Liouville–Ostrogradsky formulae  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The Weyl function  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  ℓ2 matrix solutions and the matrix Weyl function 푊  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The Cauchy transform of the spectral matrix measure and the matrix Weyl function  22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The boundary limits of the matrix Weyl function and the properties of the spectral trace  measure  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The boundary limits and spectral consequences for 퐽  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  An analog of the Jitomirskaya–Last’s approach  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The vector and matrix nonsubordinacy  24  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Barriers and barrier nonsubordinacy  26  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The main result and its consequences  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The proof of the main result  31  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Sufficient conditions for absolute continuity  35  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  GLS condition  35  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  GBS condition  36  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  H class  36  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Examples and applications  37  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Vector nonsubordinacy does not characterise invertibility of 푀  37  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  GLS does not characterise the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' spectrum  38  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Primary 47B36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Block Jacobi matrix, absolutely continuous spectrum, matrix measures, Weyl function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  1  2  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Application to some classes of block Jacobi operators  39  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Vector and matrix measures — selected basic notions  41  References  45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Introduction  A Jacobi matrix is a complex semi-infinite Hermitian matrix of the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  J =  ������  �  푏0  푎0  푎0  푏1  푎1  푎1  푏2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  ������  �  with 푎푛 ≠ 0 and 푏푛 ∈ R for all 푛 ∈ N0 = {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The action of J is well-defined on the linear  space ℓ(N0, C) of all complex-valued sequences, and it is natural to define the operator 퐽 as the restriction  of J to the standard Hilbert space ℓ2(N0, C) of square summable sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, we define 퐽푥 = J푥  for any 푥 ∈ ℓ2(N0, C) such that also J푥 ∈ ℓ2(N0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The operator 퐽 is called the (maximal) Jacobi  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It does not have to be self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If it is, then there exists a probability Borel measure 휇 on the  real line, such that 퐽 is unitary equivalent to the operator acting by multiplication by the identity function  on 퐿2(휇), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [53, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The well-known sufficient condition for the  self-adjointness of 퐽 is the Carleman condition  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  +∞  �  푛=0  1  |푎푛| = ∞,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [53, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19] in the case of 푎푛 > 0 for all 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The interest in Jacobi operators comes from their close relation to the classical moment problem as  well as the theory of orthogonal polynomials on the real line, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' As every self-adjoint operator  having a “non-degenerate” cyclic vector is unitary equivalent to a Jacobi operator, the Jacobi operators  are basic building blocks of self-adjoint operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some types of Jacobi operators are related to random  walks and birth–death processes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [31,32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally, Jacobi matrices are very useful in numerical  analysis in the construction of Gaussian quadratures, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  A method of spectral analysis, the theory of subordinacy, due to Gilbert–Pearson [16] and later, due  to Khan–Pearson, in its Jacobi variant (see [33]) started to be more and more prominent during the last  three decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Given 휆 ∈ C a sequence 푢 = (푢푛)푛∈N0 is called generalized eigenvector (associated with  휆), 푢 ∈ GEV(휆), if it satisfies the recurrence relation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  휆푢푛 = 푎푛−1푢푛−1 + 푏푛푢푛 + 푎푛푢푛+1,  푛 ≥ 1  with some initial conditions (푢0, 푢1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A non-zero sequence 푢 ∈ GEV(휆) is subordinate if for any linearly  independent 푣 ∈ GEV(휆)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  lim  푛→∞  ∥푢∥ [0,푛]  ∥푣∥ [0,푛]  = 0,  where for a sequence 푥 ∈ ℓ(N0, C) and each 푛 ≥ 0 the seminorm ∥·∥ [0,푛] is given by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  ∥푥∥ [0,푛] :=  �  � 푛  �  푘=0  |푥푘|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us decompose the measure 휇 as  휇 = 휇ac + 휇sing,  where 휇ac and 휇sing denote the absolutely  continuous and the singular part of 휇 with respect to the Lebesgue measure, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The main  result, [33, Theorem 3], defines two sets  푆ac = {휆 ∈ R : no 푢 ∈ GEV(휆) is subordinate}  푆sing = {휆 ∈ R : a non-zero 푢 ∈ GEV(휆) such that 푎0푢1 = (휆 − 푏0)푢0 is subordinate}  and states that:  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  3  푆ac is an essential support of 휇ac with respect to the Lebesgue measure,  푆sing is a support of 휇sing and its Lebesgue measure is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The above measure theory assertions on 퐽 leads to purely spectral consequences of nonsubordinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Namely, it can be proved that for any 퐺 ∈ Bor(R)  (i) If 퐺 ⊂ R \\ 푆sing, then 퐽 is absolutely continuous in 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) If 퐺 ⊂ 푆ac, then 퐽 is absolutely continuous in 퐺 (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1) and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  clLe(퐺) denotes here the Lebesgue closure of 퐺 — see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', if moreover 퐺 is open, then its  closure cl(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Since we often have some idea about asymptotic behaviour of generalized eigenvectors, this theory  turned out to be very successful in spectral analysis of various classes of Jacobi matrices, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Because of this similar theories exist for several classes of operators: continuous one-dimensional  Schödinger operators on the real half-line (see [16]) and on the whole real line (see [15]), CMV matrices  one-sided (see [17]) and two-sided (see [19]), one-dimensional Dirac operators (see [1]), Sturm–Liouville  operators (see [4, 52]), canonical systems (see [20]), and Jacobi matrices on some types of graphs  (see [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' More detailed information on generalized eigenvectors allowed to obtain even more subtle  spectral information on 퐽, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' uniform bounds on the density of 휇ac (see [4]) and absolute continuity  with respect to Hausdorff measures (see [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' An excellent survey containing more information on this  subject is [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In this article we extend some parts of Gilbert–Peason–Khan subordinacy theory to the setup of block  Jacobi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, we consider block semi-infinite tridiagonal Hermitian matrices of the form (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1))  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  J =  ������  �  퐵0  퐴0  퐴∗  0  퐵1  퐴1  퐴∗  1  퐵2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  ������  �  ,  where 퐴푛 and 퐵푛 are “blocks”, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 푑 × 푑 complex matrices with all the 퐴푛 invertible and Hermitian 퐵푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  As before, the action of J is well-defined on the linear space ℓ(N0, C푑) of all C푑-valued sequences, and  similarly to the scalar case block Jacobi operator 퐽 is simply the restriction of J to the Hilbert space  ℓ2(N0, C푑), where  ℓ2(N0, C푑) =  �  푥 ∈ ℓ(N0, C푑) :  +∞  �  푛=0  ∥푥푛∥2 < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 퐽 is self-adjoint then there exists a non-negative matrix measure 푀 defined on the Borel subsets of  the real line such that 퐽 is unitary equivalent to the operator acting by the multiplication by the identity  function on the space 퐿2(푀) of C푑-valued functions, see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Also the block variant  of Carleman condition for self-adjointness of 퐽 looks similarly:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  +∞  �  푛=0  1  ∥퐴푛∥ = ∞,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [2, Theorem VII-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The interest in block Jacobi operators comes from their close relation to the matrix moment problem  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [3,10]) as well as from the theory of matrix orthogonal polynomials on the real line, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  They are useful for analysis of difference equations of finite order, see [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some types of block Jacobi  operators are related to random walks and level dependent quasi–birth–death processes, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For  further applications we refer to [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Spectral analysis of block Jacobi operators is not well-developed yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In [51] the Mourre’s commutator  method was applied to study compact perturbations of constant sequences 퐴푛, 퐵푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Under some regularity  hypotheses it was shown there that the singular continuous spectrum is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The commutator method  was also applied in [27, 28] for obtaining a bound on the entries of the resolvent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In [61], by analysing  Turán determinants, the continuous spectrum of bounded and unbounded Jacobi operators was studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Finally, in [36] the problem of localisation of the essential spectrum of unbounded block Jacobi operators  was studied by estimating the quadratic form associated to 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  4  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  It seems that the switch from the scalar to the 푑 > 1 case is especially problematic for the subordinacy  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' As far as we know, there is still a lack of any good understanding of how a “full analog” of this  theory for the block Jacobi matrices might look like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In this article we concentrate ourselves only on “a  second half” of this difficult question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' – On the “negative” one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The general remark is, however, that – by several reasons of the algebraic character (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) – it is a good idea to deal rather with some 푑 × 푑-matrix analogs of scalar objects from  푑 = 1 theory, instead of their C푑-vector analogs, despite the fact that the Hilbert space for 퐽 consists of  sequences of C푑-vectors (and not of 푑 × 푑-matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, a sequence 푈 = (푈푛)푛∈N0 of 푀푑(C) matrices is called (matrix) generalized eigenvector for 퐽 and  휆, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 푈 ∈ MGEV(휆), if it satisfies the recurrence relation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  휆푈푛 = 퐴∗  푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1,  푛 ≥ 1  with some initial conditions (푈0,푈1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  A natural condition, being a kind of “nonsubordinacy” (which in the 푑 = 1 case means exactly  non-existence of subordinate solutions for 휆) is: for any non-zero 푈,푉 ∈ MGEV(휆)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  lim inf  푛→∞  ∥푈∥[0,푛]  ∥푉∥[0,푛]  < ∞  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)), where for any sequence 푋 ∈ ℓ(N0, 푀푑(C)) and any 푛 ∈ N0 we define  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  ∥푋∥[0,푛] =  �  � 푛  �  푘=0  ∥푋푘 ∥2,  and we use the operator norm for matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In fact, motivated by the approach of Jitomirskaya–Last in [30], we use a slight reformulated version  of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We define1 퐴−1 := − I and “extrapolating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) to 푛 = 0” we “formally compute” 푈−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence  now we consider N−1 = {−1, 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='} as the index-set for extended generalized eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If  푈 ∈ MGEV(휆), then we say that such extended sequence (푈푛)푛∈N−1 belongs to MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Next, for  any 푋 ∈ ℓ(N−1, 푀푑(C))  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  ∥푋∥ [0,푡] =  �  �  � ⌊푡⌋  �  푘=0  ∥푋푘 ∥2 + {푡}  ��푋⌊푡⌋+1  ��2,  where ⌊푡⌋ and {푡} are the integer and the fractional part of 푡, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then we say that 퐽 satisfies  (matrix) nonsubordinacy condition (at 휆 ∈ R) if for any non-zero 푈,푉 ∈ MGEV−1 (휆)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  lim inf  푡→+∞  ∥푈∥[0,푡]  ∥푉∥ [0,푡]  < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2  Let us mention that in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 we also define the notion of vector nonsubordinacy, but it turns out to  be equivalent to the matrix one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' As we show in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) implies that 휆 belongs  to the continuous spectrum of 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We shall be interested in a more quantitative version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, let 퐺 ⊂ R be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We  say that a function 픟 : 퐺 × [1, +∞) → R is a barrier if for any 휆 ∈ 퐺 and any 푈,푉 ∈ MGEV−1 (휆)  normalized by ∥푈−1∥2 + ∥푈0∥2 = ∥푉−1∥2 + ∥푉0∥2 = 1 we have  � ∥푈∥ [0,푡]  ∥푉∥ [0,푡]  �2  ≤ 픟(휆, 푡),  푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 we show that there is always an optimal barrier but it might not be the easiest one to  deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Given a barrier 픟 we say that 퐽 is 픟-nonsubordinate on 퐺 if  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  lim inf  푡→∞ 픟(휆, 푡) < +∞,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  1By I we denote the identity matrix of the appropriate dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  2By symmetry we can define this equivalently, by requiring  0 < lim sup푡→+∞  ∥푈 ∥ [0,푡]  ∥푉 ∥[0,푡]  for any non-zero  푈, 푉 ∈  MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  5  If this condition holds uniformly on 퐺, namely  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  sup  휆∈퐺  lim inf  푡→∞ 픟(휆, 푡) < +∞,  then we say that 퐽 is uniformly 픟-nonsubordinate on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Our main abstract spectral result can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  Assume that 퐽 is self-adjoint, 퐺 ⊂ Bor(R) and 픟 is a barrier for 퐽  on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 퐽 is 픟-nonsubordinate on 퐺, then  (a) 푀 is absolutely continuous on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (b) there exists a density 퐷 of 푀 on 퐺 which is an invertible matrix a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (c) 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If, moreover, 퐽 is uniformly 픟-nonsubordinate on 퐺, then there exist 푐1, 푐2 > 0 such that the above density  퐷 satisfies  푐1 I ≤ 퐷(휆) ≤ 푐2 I  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The above density of 푀 and “a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.” are both with respect to the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We have to emphasize that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1, unlike Khan–Pearson theory for the 푑 = 1 case, gives only a  sufficient condition for the absolute continuity of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' As we constructively show in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1, there  exist such block Jacobi matrices, that 푀 is absolutely continuous on R, 푀(퐵) is invertible for all non-  empty open 퐵 ⊂ R but 퐽 does not satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) for any 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Nevertheless, as we show in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 is applicable to some explicit Jacobi matrices considered in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In Section 6 we adapt to our setup some of the sufficient conditions implying non-existence of  subordinate solutions, which were useful in the case 푑 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In particular, the one introduced by Last–  Simon in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This conditions are formulated in terms of transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recall: if 푈 ∈ MGEV−1 (휆),  then  �  푈푛  푈푛+1  �  = 푇푛(휆)  �  푈푛−1  푈푛  �  ,  푛 ≥ 0,  where 푇푛(휆) is the (1-step) transfer matrix and  푇푛(휆) =  �  0  I  −퐴−1  푛 퐴∗  푛−1  퐴−1  푛 (휆 I−퐵푛)  �  ,  푛 ≥ 0,  where we have set 퐴−1 = − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then the 푛-step transfer matrix 푅푛(휆) := 푇푛−1(휆) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='푇0(휆) satisfies  �  푈푛−1  푈푛  �  = 푅푛(휆)  �  푈−1  푈0  �  ,  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Assume that the Carleman condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7) holds and set  휌푛 :=  푛−1  �  푘=0  1  ∥퐴푘 ∥,  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus 휌푛 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We shall say that 퐽 satisfies Generalized Last–Simon condition (GLS in short) on some  Borel 퐺 ⊂ R if  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)  lim inf  푛→∞  1  휌푛  푛  �  푘=1  ∥푅푘 (휆)∥2 < +∞,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  This condition was introduced in [37] for 푑 = 1 and 퐴푛 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It was proved there, that the maximal set  퐺 ⊂ R, where (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) is satisfied is a minimal support of 휇ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recently, an analogous conclusion for 푑 ≥ 1  was established in [48, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2] under the assumption that 퐴푛 = 퐴∗  푛 and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16)  sup  푛≥0  � ∥퐴푛∥ +  ��퐴−1  푛  �� � < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We show in Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 that the Jacobi matrix corresponding to the Laguerre polynomials satisfies  휎ac(퐽) = [0, ∞), yet, the condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) is violated for any non-empty 퐺 ⊂ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, even for 푑 = 1,  one cannot hope to obtain the part concerning the minimal support when (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' However,  we show in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 that GLS condition implies the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 for a suitably chosen  barrier 픟 (see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us recall that in [48, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2], under the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16) and 퐴푛 = 퐴∗  푛,  6  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  a characterisation of multiplicities of the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' spectrum of 퐽 is provided in terms of asymptotic properties  of singular values of the sequences 푃(휆), 푄(휆) ∈ MGEV−1 (휆) corresponding to the initial values  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17)  �  푄−1(휆) = I  푄0(휆) = 0,  �  푃−1(휆) = 0  푃0(휆) = I,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The approach used there is different than in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Our approach to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 is based on linking the asymptotic properties of MGEV−1 (푧) to  the boundary values of the Cauchy transform of the matrix measure 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It is defined by  푊(푧) =  ∫  R  d푀(휆)  휆 − 푧 ,  푧 ∈ C \\ R  and it turns out to be the matrix analogue of the well-known Weyl function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since3 Im푊(푧) ≥ 0 for  Im 푧 > 0, it is a matrix Herglotz function and its boundary values are closely related to properties of 푀,  see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Inspired by [30] we prove in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11 that if 퐽 is self-adjoint, then given 휆 ∈ R there exists a  unique function ℓ휆 : R+ → R+ satisfying  ∥푃(휆)∥[0,ℓ휆 (휖 )] ∥푄(휆)∥[0,ℓ휆 (휖 )] = 1  2휖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, lim휖 →0+ ℓ휆(휖) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Adapting the approach of [30] and utilizing our novel object, namely the barrier function 픟, we prove  our main result “on controlling the boundary limits of the matrix Weyl function”, being probably the most  important result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  Assume that 퐽 is self-adjoint and 퐺 ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 픟 is a barrier for 퐽 on  퐺, then for any 휆 ∈ 퐺 and any 휖 > 0 with ℓ휆(휖) ≥ 1  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18)  �8픟�휆, ℓ휆(휖)��−1 I ≤ Im푊(휆 + 푖휖)  and  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19)  푠−(휆, 휖) ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+(휆, 휖),  where  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20)  푠±(휆, 휖) := 4푑픟�휆, ℓ휆(휖)� ±  ��  4푑픟�휆, ℓ휆(휖)��2  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us recall that Jitomirskaya–Last in [30] established a variant of the inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19) for 푑 = 1 with  푠′  ± instead of 푠±, where  푠′  ±(휆, 휖) := �5 ±  √  24� ∥푄(휆)∥ [0,ℓ휆(휖 )]  ∥푃(휆)∥[0,ℓ휆 (휖 )]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then Khan–Pearson theory can be obtained by the rank-one perturbation theory applied to the corre-  sponding Jacobi operator, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [52, Section 2] for details in the case of Sturm–Liouville operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us point out that, under the hypotheses of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2, the last argument can be avoided once one  uses our new inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recently, an analogue of Jitomirskaya–Last inequality, expressed as a  quotient of partial norms of 푄(휆) and 푃(휆) (and its singular values), has been obtained in [48] for 푑 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  However, we were not able to use it to prove that GLS condition implies absolute continuity of 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In  contrast, our Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 allows us to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1, which as we have mentioned already, leads us  to the proof of absolute continuity under GLS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In Section 2 we fix our notation and prove basic results concerning  the family of semi-norms (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Next, in Section 3, we show that 퐽 is finitely cyclic, and as a consequence,  it is unitary equivalent to the multiplication operator on the space 퐿2(푀) of C푑-valued functions for some  matrix measure 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We also show an approach to Jacobi operators based on generalized eigenvectors and  transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In Section 4 we study the matrix Weyl function 푊 and its relation to properties of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Section 5 is devoted to the proof of our main results: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 — the main result of the paper and  its spectral consequence: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In Section 6 we extend some well-known conditions implying  3For any square matrix 푋 we define Im 푋 := 1  2푖 (푋 − 푋∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  7  nonsubordinacy from 푑 = 1 to the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In particular, we cover: GLS (Generalized Last–Simon)  condition, GBS (Generalized Behncke–Stolz) condition and 퐻 (homogenous) class condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally,  in Section 7, we show some examples and counterexamples illustrating the applicability of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For the sake of self-containment in Appendix A we collected basic notions concerning vector and matrix  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Acknowledgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The author Grzegorz Świderski was supported by long term structural funding –  Methusalem grant of the Flemish Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Part of this work was done while he was a postdoctoral  fellow at KU Leuven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The author Marcin Moszyński wishes to thank:  Anna Moszyńska (IPEVP, Warsaw) – his wife – for extraordinary patience and for valuable  linguistic help,  Nadia V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Zaleska (EIMI, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Petersburg) – his friend – for some wise hints and for invaluable  moral support,  Grzegorz Świderski – the co-author – for several years of confidence in success and for the  barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Preliminaries  We collect and fix here some general notation for in the paper, and we also introduce here several  convenient tools, which will be important in the main sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Introductory notation and notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We use here the following symbols for some sets of scalars:  C+ := {푧 ∈ C : Im(푧) > 0},  R+ := {푡 ∈ R : 푡 > 0},  N푘 := {푛 ∈ Z : 푛 ≥ 푘}  for 푘 ∈ Z,  so, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  N = N1,  N0 = N ∪ {0},  N−1 = N ∪ {−1, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us fix here some 푑 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The vectors of the standard base in C푑 are denoted by 푒1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푒푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By 푀푑(C) we denote the space of all 푑 × 푑 complex matrices, with the usual matrix/operator norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We identify any 퐴 ∈ 푀푑(C) with the appropriate linear transformation of C푑 induced by matrix 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular, we shall use alternatively both the operator and the matrix notation for the action of 퐴  on vectors from C푑, namely, for 푣 ∈ C푑 we typically use 퐴푣, but sometimes we also write 퐴푣T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For  푖, 푗 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} the term of 퐴 from its 푖-th row and 푗-th column is denoted as usual by 퐴푖, 푗 and 푣 푗  is the 푗-th term of 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The symbol 퐴{ 푗} denotes the 푗-th column of 퐴 (usually we treat columns as  C푑-vectors and not as one-column matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly for 퐴{푖} — the 푖-th row of 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, for  vectors 푣(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푣(푑) ∈ C푑 the matrix 퐴 with 퐴{ 푗} = 푣( 푗) for any 푗 is denoted by [푣(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푣(푑)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푋 is a linear space, then by ℓ(N푘, 푋) we denote the linear space of all the sequences 푥 = (푥푛)푛∈N푘  with terms in 푋 and ℓfin(N푘, 푋) is the subspace of ℓ(N푘, 푋) consisting of all the “finite” sequences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  of such 푥 that 푥푛 = 0 for 푛 sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For sequences of vectors: if 푢(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푢(푑) ∈ ℓ(N푘, C푑) with 푢( 푗) = (푢( 푗)  푛 )푛∈N푘 for 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑, then  the symbol [푢(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푢(푑)], used already above for vectors and not for sequences of vectors, denotes the  matrix sequence 푈 ∈ ℓ(N푘, 푀푑(C)) with 푈푛 := [푢(1)  푛 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푢(푑)  푛 ] for any 푛 ≥ 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The symbols ∥·∥푋 and ⟨·, ·⟩푋 denote here the norm in a normed space 푋 and the scalar product in a  Hilbert space 푋, respectively, but we often omit the subscript ‘푋’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This applies also to operator norms  which we use by default for the bounded operators (mainly matrices from 푀푑(C), here), if no other choice  is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Our general rule here is: “to possibly omit the subscript ‘푋’ in almost all cases except the case  of norm or the scalar product for C푑:  ∥·∥C푑, ⟨·, ·⟩C푑”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For a linear operator 퐴 : 푋 −→ 푋 in a normed space 푋 ≠ {0} we define its minimum modulus by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  ⇃|퐴|⇂:= inf  ∥푥 ∥=1 ∥퐴푥∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Obviously, if dim 푋 = 1, then ⇃|퐴|⇂= ∥퐴∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recall that if 퐴 is invertible, then  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  ⇃|퐴|⇂=  1  ∥퐴−1∥,  and for 0 < dim 푋 < +∞ 퐴 is invertible iff ⇃|퐴|⇂> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  8  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  We use the symbols  cl(퐺),  clLe(퐺)  for the “usual” (topological) closure and the “Lebesgue” closure of the set 퐺 ⊂ R, respectively (휆  is  used here for the complex conjugation of 휆 ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recall, that  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  clLe(퐺) := {푡 ∈ R : ∀휀>0 |퐺 ∩ (푡 − 휀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푡 + 휀)| > 0}  for 퐺 ∈ Bor(R) and | · | is here the standard 1-dimensional Lebesgue measure on Bor(R) but, as usual,  it will denote also the absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 휇 is a measure4 on 픐 – a 휎-algebra of subsets of some Ω, and 푝 ∈ [1, +∞), then 퐿 푝(휇) (without  the “universum” Ω and the 휎-algebra for the measure, for short) denotes the standard 퐿 푝 Banach space  of the classes of the appropriate complex functions on the “universum” for the measure 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We need  sometimes to distinguish here 퐿 푝(휇) from the appropriate space of functions (and not classes) denoted  here by L 푝(휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 푝 = 1 and 푋 := R푑 we also use L1  푋 (휇) to denote the space of integrable functions  from Ω into 푋 in the standard coordinatewise sense of the integral and the integrability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, for a matrix measure 푀 by 퐿2(푀) we denote the appropriate 퐿2-Hilbert space induced by  this matrix measure (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', [44] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푋 is a norm space, then  ℓ2(N0, 푋) :=  �  푥 ∈ ℓ(N0, 푋) :  +∞  �  푛=0  ∥푥푛∥2  푋 < ∞  �  ,  is a normed space with the norm defined for 푥 ∈ ℓ2(N0, 푋) by  ∥푥∥ :=  �  � +∞  �  푛=0  ∥푥푛∥2  푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If, moreover, 푋 is a Hilbert space, then ℓ2(N0, 푋) is a Hilbert space with the scalar product given for  푥, 푦 ∈ ℓ2(N0, 푋) by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  ⟨푥, 푦⟩ :=  +∞  �  푛=0  ⟨푥푛, 푦푛⟩푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Here, the most important case for us is the Hilbert space space ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' As the standard orthonormal  basis of ℓ2(N0, C푑) we consider  �  훿푛(푒푖)  �  (푖,푛)∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑}×N0,  where for any vector 푣 ∈ C푑 and 푛 ∈ N0 we define the sequence 훿푛(푣) ∈ ℓfin(N0, C푑) by  (훿푛(푣)) 푗 :=  �  푣  if 푗 = 푛,  0  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover we have  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  ℓfin(N0, C푑) = lin{훿푛(푣) : 푣 ∈ C푑, 푛 ∈ N0},  and  cl(ℓfin(N0, C푑)) = ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If H is a Hilbert space and 퐴 — a self-adjoint operator (possibly unbounded) in H, then the projection-  valued spectral measure (“the resolution of identity”) for 퐴 is denoted by 퐸퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular 퐸퐴 :  Bor(R) −→ B(H), where Bor(R) is the Borel 휎-algebra of R and B(H) denotes, as usual, the space of  bounded operators on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푥, 푦 ∈ H, then 퐸퐴,푥,푦 denotes the spectral measure for 퐴, 푥 and 푦, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the  complex measure given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  퐸퐴,푥,푦 (휔) := ⟨퐸퐴(휔)푥, 푦⟩ ,  휔 ∈ Bor(R),  4Here the name “measure” without some extra adjectives / names of the type vector, matrix, complex, real, spectral etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  denotes always a classical Lebesgue-type measure with values in [0, +∞], without necessity of adding “non-negative”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The  remaining ones, all “adjective (of the above type) + measures”, belongs to a wide class of vector measures (— see the definition  on page 41) for an appropriate measure vector space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g, equal to R for real measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, measure can be, but also can be not  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the Bor(R) Lebesgue measure), a vector measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It simply depends on it finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And “unfortunately”, from the point  of view of the abuse of terminology, a vector measure is “usually” not a measure, here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  9  and 퐸퐴,푥 := 퐸퐴,푥,푥 (the spectral measure for 퐴 and 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Denote also  Hac(퐴) := {푥 ∈ H : 퐸퐴,푥 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' with respect to the Lebesgue measure on Bor(R)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If and 퐺 ∈ Bor(R), then the symbol (퐴)퐺 denotes the part of 퐴 in the (invariant reducing) subspace  H퐺(퐴) := Ran 퐸퐴(퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall the key notion for this paper, of the absolute continuity of 퐴:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 퐴 is absolutely continuous (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') in 퐺  iff H퐺(퐴) ⊂ Hac(퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  퐴 is absolutely continuous iff 퐴 is absolutely continuous in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We apply here also some other “more or less, but not totally” common abbreviations and symbols: iff  for:  if and only if TFCAE  for:  the following conditions are (mutually) equivalent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  for:  with respect to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  for:  self-adjoint (for operators) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  for:  absolutely continuous (for operators, measures etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  for:  singular (as above) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  for:  almost everywhere JM, BJM  for:  Jacobi matrix, block Jacobi matrix, respectively JO, BJO  for:  Jacobi operator, block Jacobi operator, respectively lin푌  for:  the linear subspace generated by a subset 푌 of a linear space 퐹↾푌  for:  the restriction of function 퐹 to the subset 푌 of the domain 퐹(푌)  for:  the image of subset 푌 with respect to function 퐹 Dom(퐴)  for:  the domain of linear operator 퐴 Dom(퐴∞)  for:  the intersection of all Dom(퐴푛) for 푛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Some further notation is introduced successively in next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some asymptotic symbols and the affine interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푆 be an arbitrary set and 푓 , 푔 : 푆 −→  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We define the symbol ≍ of “asymptotic similarity” of functions:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  푓 ≍ 푔  ⇐⇒  ∃푐,퐶∈R+∀푠∈푆 푐|푔(푠)| ≤ | 푓 (푠)| ≤ 퐶|푔(푠)|  (note the presence of the absolute value in this definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We shall use also alternative notation:  푓 (푠) ≍푠 푔(푠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푆 = N푘 for some 푘 ∈ Z, then 푓 is just a sequence from ℓ(N푘, C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푓 = ( 푓 (푛))푛∈N푘 = ( 푓푛)푛∈N푘,  and we shall consider its affine interpolation aff ( 푓 ) : [푘, +∞) −→ C, uniquely defined by the conditions:  (a) aff ( 푓 )↾N푘= 푓 ,  (b) for each 푛 ∈ N푘  aff ( 푓 )↾[푛,푛+1] is an affine function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' a function of the form  [푛, 푛 + 1] ∋ 푡 ↦→ 푎푡 + 푐 ∈ C for some 푎, 푐 ∈ C (depending here also on 푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular aff ( 푓 ) is a continuous interpolation of 푓 , and one can easily check that it can be also  given by the explicit formula:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  aff ( 푓 ) (푡) = 푓⌊푡⌋ + {푡} � 푓⌊푡⌋+1 − 푓⌊푡⌋  � ,  푡 ∈ [0, ∞)  where {푡} = 푡 − ⌊푡⌋ is the fractional part of 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The following result joining the asymptotic similarity of sequences and of their affine interpolations,  will be convenient in the proof of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us prove first the following “slightly unexpected” result on quotients of two affine functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 훼, 훽, 푎1, 푐1, 푎2, 푐2 ∈ R, 훼 < 훽 and 푎2푡 + 푐2 ≠ 0 for any 푡 ∈ [훼, 훽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let  휑 : [훼, 훽] −→ R be given by  휑(푡) := 푎1푡 + 푐1  푎2푡 + 푐2  ,  푡 ∈ [훼, 훽].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then 휑 is monotonic and, in particular, its maximal and minimal value is attained on the set {훼, 훽}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We have:  휑′(푡) := 푎1푐2 − 푎2푐1  (푎2푡 + 푐2)2  and thus, we have only two cases:  10  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  푎1푐2 − 푎2푐1 = 0, hence 휑 is constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푎1푐2 − 푎2푐1 ≠ 0, so 휑′ has no zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In both cases our assertion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some analogs of “J–L continuous interpolation” of a discrete family of semi-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The main  idea of the Jitomirskaya–Last’s new approachin [30] was “technically” based on a continuous interpolation  of the discrete family of semi-norms {∥·∥푛}푛∈N0 in ℓ(N0, C) to the family {∥·∥푡}푡 ∈[0,+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For our purposes we shall extend this notion in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, let 푉 be a normed space, 푛0 ∈ Z  and assume that 푋 ∈ ℓ(N푛0,푉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For any (푛1, 푡) ∈ Z × R such that 푛0 ≤ 푛1 ≤ 푡, we define  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  ∥푋∥ [푛1,푡] :=  �  ⌊푡⌋  �  푘=푛1  ∥푋푘 ∥2 + {푡}  ��푋⌊푡⌋+1  ��2  �1/2  ,  and for 푡 = ∞  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  ∥푋∥[푛1,∞] :=  � +∞  �  푘=푛1  ∥푋푘 ∥2  �1/2  (which can possibly be +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, if 푉 = 푀푑(C)) for some 푑 ≥ 1, then similarly to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9) we define a continuous family based  on minimum modulus (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)) instead of matrix norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, for any (푛1, 푡) ∈ Z×R such that 푛0 ≤ 푛1 ≤ 푡  we consider  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  ⇃|푋|⇂[푛1,푡]=  �  ⌊푡⌋  �  푘=푛1  ⇃|푋푘|⇂2 +{푡}⇃|푋⌊푡⌋+1|⇂2  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) we can relate the above constructions with the notion of affine extension introduced in the  previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The squares of the “new objects” are just the affine extensions of their discrete  counterparts (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13) below), which are somewhat more natural and simpler for the context  of operators “acting on sequences” considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The notions defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e, for 푡 < +∞) satisfy respectively  ∥푋∥2  [푛1,푡] = aff �푆푋,푛1  � (푡)  and  ⇃|푋|⇂2  [푛1,푡]= aff �ˇ푆푋,푛1  � (푡)  for 푡 ≥ 푛1, where 푆푋,푛1, ˇ푆푋,푛1 : N푛1 −→ R are given for 푛 ≥ 푛1 by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  푆푋,푛1 (푛) := ∥푋∥2  [푛1,푛] =  푛  �  푘=푛1  ∥푋푘∥2 ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  ˇ푆푋,푛1 (푛) :=⇃|푋|⇂2  [푛1,푛]=  푛  �  푘=푛1  ⇃|푋푘|⇂2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Note also that in the scalar case 푑 = 1 for 푋 ∈ ℓ(N푛0, 푀푑(C)) we simply have ∥푋∥[푛1,푡] =⇃|푋|⇂[푛1,푡]  and 푆푋,푛1 = ˇ푆푋,푛1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consider a second sequence 푌 ∈ ℓ(N푛0,푉).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Using the above Observation and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 we get:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 푛1 ∈ Z, 푡 ∈ R, 푛0 ≤ 푛1 ≤ 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푛 := ⌊푡⌋ be such that ∥푌 ∥[푛1,푛] ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then  there exist such 푛, 푛 ∈ {푛, 푛 + 1} that  ∥푋∥[푛1,푛]  ∥푌 ∥[푛1,푛]  ≤  ∥푋∥[푛1,푡]  ∥푌 ∥[푛1,푡]  ≤  ∥푋∥ [푛1,푛]  ∥푌 ∥[푛1,푛]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푉 = 푀푑(C) then the analogous result with all the “∥·∥” replaced by “⇃|·|⇂” is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It suffices to use Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 and then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 to the function 휑 given on [푛, 푛 + 1] by the  formula 휑(푡) :=  � ∥푋 ∥ [푛1,푡]  ∥푌 ∥ [푛1,푡]  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some matrix norm inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us recall some notions related in particular to 푑 × 푑 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For 푋 ∈ 푀푑(C):  its Hilbert–Schmidt norm is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  ∥푋∥HS =  �  푑  �  푖=1  ∥푋푒푖∥2  C푑  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  its real and imaginary parts Re(푋) and Im(푋) (in the adjoint, and not the complex conjugation  sense) are given by  Re(푋) := 1  2 (푋 + 푋∗),  Im(푋) := 1  2푖 (푋 − 푋∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For 푣 ∈ C푑 we shall use the symbol 퐸 푣 to denote the matrix / operator [푣, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 0] ∈ 푀푑(C) , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)  퐸 푣(푒 푗) =  � 푣  for 푗 = 1  0  for 푗 > 1,  푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푋 ∈ 푀푑(C) and 푣 ∈ C푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then:  (i) ∥푋∥ ≤ ∥푋∥HS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) ∥Im(푋)∥ ≤ ∥푋∥ , ∥Re(푋)∥ ≤ ∥푋∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iii) ∥푋퐸 푣∥ = ∥푋푣∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iv) If 푋 ≥ 0, then  ∥푋∥ ≤ tr 푋 ≤ 푑 ∥푋∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Part (i) is a classical result (easy to get by the Schwarz inequality), and (ii) follows directly from  the definitions of Re, Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  To get (iii) observe first that 푋퐸 푣 = 퐸푋 푣 by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15), so it suffices to consider 푋 = 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But using (i) we  get ∥퐸 푣∥ ≤ ∥푣∥ and ∥퐸 푣∥ ≥ ∥퐸 푣푒1∥ = ∥푣∥, so the equality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Suppose that 푋 ≥ 0 and let 휆1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 휆푑 be all the eigenvalues of 푋 repeated according to their multi-  plicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We thus get (iv) by  ∥푋∥ = max  1≤푖≤푑 휆푖 ≤  푑  �  푖=1  휆푖 = tr 푋 ≤ 푑 max  1≤푖≤푑 휆푖 = 푑 ∥푋∥  □.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Block Jacobi matrix and operator(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A spectral representation and the associated  difference equations  We denote “the size of the block” for BJM by 푑 ∈ N, and for the whole paper we assume that (퐴푛)푛∈N0  and (퐵푛)푛∈N0 are sequences of matrices from 푀푑(C) such that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  det 퐴푛 ≠ 0,  퐵푛 = 퐵∗  푛,  푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now we define a block Jacobi matrix  J =  ������  �  퐵0  퐴0  퐴∗  0  퐵1  퐴1  퐴∗  1  퐵2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  ������  �  ,  and the pair of the sequences 퐴 = (퐴푛)푛∈N0 and 퐵 = (퐵푛)푛∈N0 will be called Jacobi parameters of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In fact, we mean that J is the linear operator (“the formal block Jacobi operator”) acting on the space  ℓ(N0, C푑) of all the C푑 sequences, whose action is well-defined via formal matrix multiplication:  J : ℓ(N0, C푑) −→ ℓ(N0, C푑),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', for any 푢 ∈ ℓ(N0, C푑)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  (J푢)푛 :=  � 퐵0푢0 + 퐴0푢1  for 푛 = 0  퐴∗  푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1  for 푛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall now two important operators related to J and acting in the Hilbert space ℓ2(N0, C푑):  퐽min  and 퐽 (which will coincide in most of our further considerations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The operator 퐽min (the minimal block  Jacobi operator) is simply the closure in ℓ2(N0, C푑) of the operator in ℓ2(N0, C푑) being the restriction:  12  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  J↾ℓfin(N0,C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And to define 퐽 (the maximal block Jacobi operator) we first choose its domain in a usual  way for “maximal-type” operators:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  Dom(퐽) :=  �  푢 ∈ ℓ2(N0, C푑) : J푢 ∈ ℓ2(N0, C푑)  �  ,  and then we define 퐽 := J↾Dom(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  When both (퐴푛)푛∈N0 and (퐵푛)푛∈N0 are bounded sequences, then obviously we have only one operator  퐽min = 퐽 ∈ B(ℓ2(N0, C푑)), which is symmetric, so self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But here we consider also unbounded  cases, so let us recall the following result  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (퐽min)∗ = 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, TFCAE:  (i) 퐽min is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  (ii) 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  and if one of the above conditions holds, then 퐽min = 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  See [2, Chapter VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5] for5 퐽∗  min = 퐽, and the remaining part follows from this by [56, formula  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For the main results of the theory presented here the maximality of the domain is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover,  we study mainly “the self-adjoint case”, so (by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1) the best choice here is to use just only the operator  퐽, later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  ℓfin(N0, C푑) ⊂ Dom(퐽),  퐽  �  ℓfin(N0, C푑)  �  ⊂ ℓfin(N0, C푑),  so in particular 퐽 is densely defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) we compute  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  퐽 (훿푛(푣)) = 훿푛−1(퐴푛−1푣) + 훿푛(퐵푛푣) + 훿푛+1(퐴∗  푛푣),  푣 ∈ C푑, 푛 ∈ N0,  where we additionally denote  훿−1(푣) := 0,  푣 ∈ C푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The finite-cyclicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In the scalar 푑 = 1 case Jacobi operator 퐽 is cyclic with a cyclic vector  휑 := 훿0(푒1) (the canonical cyclic vector for 퐽), which means that the space  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  lin{퐽푛휑 : 푛 ∈ N0}  is dense in ℓ2(N0, C) (here, for 푑 = 1, we simply have 푒1 = 1 ∈ C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' — Indeed, this is known that the  above space is just equal to ℓfin(N0, C) for such 휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Cyclicity plus self-adjointness provides a very simple  spectral representation of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, suppose now, that “the scalar” 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and consider:  x — the identity function on R,  x(푡) = 푡  for 푡 ∈ R, and 휇 — “the scalar” spectral measure for 퐽 and 휑, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  휇 = 퐸퐽,휑 (see, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 for the  spectral notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By the well-known spectral result, 퐽, as a cyclic s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' operator, is unitary equivalent to the operator of  the multiplication by  x in the space 퐿2(휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  This one-dimensional result has also its analog for the block-Jacobi case with arbitrary 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' First of  all, instead of cyclicity, we consider here the so-called finite-cyclicity notion (see [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It means that  for some 푘 ∈ N there exists a cyclic system �휑 = (휑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 휑푘) for 퐽, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', a system of such vectors from  Dom(퐽∞), that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  lin{퐽푛휑 푗 : 푛 ∈ N0, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푘}  is dense in ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In our general 푑-dimensional case the choice of �휑 can be done in an analogical  way, as it was for 푑 = 1, namely define  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  �휑 := (휑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 휑푑),  휑 푗 := 훿0(푒 푗),  푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The following result is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The system �휑 is a cyclic system for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  5Actually, in [2, Chapter VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='§2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5] only the case 퐴푛 = 퐴∗푛 for all 푛 ≥ 0 was considered, but the proof in the general case is  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  13  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For each 푛 ∈ N0 denote6  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  푋푛 :=  �  푥 ∈ ℓ2(N0, C푑) : ∀푗≠푛 푥 푗 = 0  �  ,  푌푛 :=  푛  �  푘=0  푋푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So 푋푛,푌푛 ⊂ ℓfin(N0, C푑) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  푋푛 =  �  훿푛(푤) : 푤 ∈ C푑�  = lin {훿푛(푒푠) :  푠 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} ,  푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  푋0 = lin {휑푠 :  푠 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) we get  퐽(푋푛) ⊂ 푌푛+1,  푛 ∈ N0,  so also  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  퐽(푌푛) ⊂ 푌푛+1,  푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Using this by the obvious induction we obtain  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  퐽푛(푋0) ⊂ 푌푛,  푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Denote 푌−1 := {0} and for 푘 ∈ N0 denote also  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  퐶푘 :=  �  (퐴0 · · · · · 퐴푘−1)∗  for 푘 > 0  퐼  for 푘 = 0,  so in particular 퐶푘 is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To continue the proof, we shall first prove the following two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푘 ∈ N0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)  ∀푤 ∈C푑 ∃푢∈푌푘−1 퐽푘훿0(푤) = 푢 + 훿푘 (퐶푘푤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 푘 = 0 and 푤 ∈ C푑 we have 퐽푘훿0(푤) = 훿0(푤) = 0 + 훿0(퐶0푤), so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) holds  for 푘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose now that it holds for some 푘 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) for 푘 and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5), for 푤 ∈ C푑  퐽푘+1훿0(푤) = 퐽(퐽푘훿0(푤)) = 퐽(푢) + 퐽 (훿푘 (퐶푘푤)) = 퐽(푢) + 푢′ + 훿푘+1(퐴∗  푘퐶푘푤),  with some 푢 ∈ 푌푘−1, 푢′ ∈ 푌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  퐽푘+1훿0(푤) = 푢′′ + 훿푘+1(퐴∗  푘퐶푘푤) = 푢′′ + 훿푘+1(퐶푘+1푤),  with some 푢′′ ∈ 푌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) holds for 푘 + 1, and we obtain our assertion by the induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  We shall use this lemma to prove the result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푛 ∈ N0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16)  푌푛 =  푛  �  푗=0  퐽 푗 (푋0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof of Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We get “⊃” from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us prove “⊂” by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 푛 = 0 the assertion is  obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now consider 푛 ∈ N0 and suppose that 푌푛 ⊂ �푛  푗=0 퐽 푗 (푋0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then  lin  �  푌푛 ∪ 퐽푛+1(푋0)  �  ⊂  푛+1  �  푗=0  퐽 푗 (푋0),  by the linearity of the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9), to get 푌푛+1 ⊂ �푛+1  푗=0 퐽 푗 (푋0), it suffices to prove  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17)  푋푛+1 ⊂ lin  �  푌푛 ∪ 퐽푛+1(푋0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Indeed, if 푤 ∈ C푑, then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3  훿푛+1(푤) = 푣 + 퐽푛+1훿0(퐶−1  푛+1푤),  6Hereweusethestandardoperationof thesum ofsubsets ofa linear space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', �푛  푘=0 퐷푘 := {�푛  푘=0 푥푘 : 푥푘 ∈ 퐷푘 for any 푘 =  0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푛} for subsets 퐷0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 퐷푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  14  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  with some 푣 ∈ 푌푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Let us continue the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Using again the standard argumentation concerning the  linear spaces generated by a subset, we see by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11) that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18)  lin{퐽 푗휑푠 : 푗 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푛, 푠 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} =  푛  �  푗=0  퐽 푗 (푋0),  푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18)  lin{퐽푛휑푠 : 푛 ∈ N0, 푠 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} =  +∞  �  푛=0  lin{퐽 푗휑푠 : 푗 = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푛, 푠 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑} =  +∞  �  푛=0  푌푛 = ℓfin(N0, C푑),  and the density of ℓfin(N0, C푑) in ℓ2(N0, C푑) finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Surely, this choice of a cyclic system for 퐽 is not unique, but this particular �휑 defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) is called  canonical for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The spectral matrix measure 퐸퐽, �휑 and the representation of 퐽 as the multiplication operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The next notion which should be generalized, when we are moving from the scalar to the block case,  is the classical 퐿2- type Hilbert space with “the scalar non-negative” spectral measure 휇 for 퐽 (and for  the canonical cyclic vector 휑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It should be replaced by the less-known 퐿2-type Hilbert space with the  so-called spectral matrix measure for the operator 퐽 and for its canonical cyclic system �휑, described in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The spectral matrix measure for 퐽 is a particular example of the general notion of matrix  measure (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly to the “scalar” spectral measure for the Jacobi (the scalar one) case,  is defined on Bor(R) and is tightly related to 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 퐽 can be recovered from its spectral matrix measure  up to a unitary equivalence, as follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall (see [44]):  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The spectral matrix measure 퐸퐽, �휑 for 퐽 is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19)  퐸퐽, �휑 : Bor(R) → 푀푑(C),  퐸퐽, �휑(휔) :=  �  퐸퐽,휑 푗,휑푖 (휔)  �  푖, 푗=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑 ∈ 푀푑(C),  휔 ∈ Bor(R)  (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) for the symbol 퐸퐽,휑 푗,휑푖), where �휑 = (휑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 휑푘) is canonical cyclic system for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The analog of the scalar-Jacobi unitary representation result, mentioned above, can be formulated for  our block-Jacobi case in a short way, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 퐽 is unitary equivalent to the operator of the multiplication by  x in the space 퐿2(푀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For the full formulation and the proof in the general finitely cyclic case see [44,  xMUE Theorem]7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular, the above result means that the spectral matrix measure of 퐽 “contains” all the important  spectral information about 퐽, similarly to the spectral measure 휇 in the scalar Jacobi case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For instance,  such typically studied information, as: the absolute continuity or the singularity in some subset of R, the  location of the spectrum and of some particular kinds of spectra of 퐽, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  On the other hand, “the nice properties” of the trace measure tr푀 (see A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) for matrix measure 푀  suggest that instead of dealing with spectral matrix measure, somewhat sophisticated at times, it could  be more useful to deal with its trace measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Especially, when we try to “control” the above mentioned  spectral properties of 퐽 related, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', to the absolute continuity or singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5, Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6,  and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 from Appendix A we discussed this problem in more details for its abstract (vector)  measure theory aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Their main “spectral operator theory” consequences for 퐽 are Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8 and  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7Also the detailed definition of the multiplication by a function operator in 퐿2-matrix measure spaces is presented there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The two associated difference equations and “the solution extensions to -1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푧 ∈ C let  us consider two difference equations tightly related to J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The first — “the vector” one — is the infinite  system of equations for a sequence 푢 = (푢푛)푛∈N0 ∈ ℓ(N0, C푑):  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20)  (J푢)푛 = 푧푢푛,  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Each such a vector sequence 푢 is called generalized eigenvector (for 퐽 and 푧)8 — “gev” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) equivalently its explicit form can be written  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21)  퐴∗  푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1 = 푧푢푛,  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The second difference equation — “the matricial” one — is the analog equation (with the right-side  multiplication choice9) for a matrix sequence 푈 = (푈푛)푛∈N0 ∈ ℓ(N0, 푀푑(C)):  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22)  퐴∗  푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1 = 푧푈푛,  푛 ≥ 1,  and each such a matrix sequence 푈 is called matrix generalized eigenvector (for 퐽 and 푧) — “mgev” for  short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Having our BJM J fixed, for 푧 ∈ C we denote  GEV(푧) := {푢 ∈ ℓ(N0, C푑) : 푢 is a gev for 퐽 and 푧}  and parallelly  MGEV(푧) := {푈 ∈ ℓ(N0, 푀푑(C)) : 푈 is a mgev for 퐽 and 푧},  being obviously liner subspaces of ℓ(N0, C푑) and ℓ(N0, 푀푑(C)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1), both recurrence  relations are of degree 2, in the sense that for any initial condition �퐶0, 퐶1  � ∈ (푀푑(C))2 there is a unique  sequence 푈 satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) with 푈0 = 퐶0, 푈1 = 퐶1, and analogously for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' More precisely, one  easily check the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푧 ∈ C the map Ini푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='0,1 : GEV(푧) −→ �C푑�2, given by  Ini푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='0,1(푢) = (푢0, 푢1),  푢 ∈ GEV(푧) ,  is a linear isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The analogous result is true for MGEV (푧) and (푀푑(C))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In particular  dim GEV(푧) = 2푑 and dim MGEV(푧) = 2푑2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  However, it is often more convenient to use another kind of “initial conditions”, namely “at −1 and  0” instead of 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To formulate this properly, we shall define first the appropriate extension of each  solution (in both, vector and matrix cases), which is tightly related to the “a priori choice” of 퐴−1:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23)  퐴−1 := − I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The informal idea of the extension is simply to “extend to 푛 = 0” the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) (and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21) analogously)  and to “compute the “value at −1”, using our choice made in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', we get “푈−1 := (퐵0 − 푧 I)푈0 +  퐴0푈1” for the matrix case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Unfortunately, as one can see, such a definition seems to depend explicitly on  the parameter 푧, and not only on 푈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, at the first sight it seems that the notation for the extension “to −1”  of a solution 푈 has to contain always this parameter, which would be not very convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And YES — it is  true, that we have really this problem, extending in such a way any sequence 푈 ∈ ℓ(N0, 푀푑(C)) (similarly  for the vector version).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So we define first the following family {푧•}푧∈C of “extending transformations”  푧• : ℓ(N0, 푀푑(C)) −→ ℓ(N−1, 푀푑(C)) given for 푈 ∈ ℓ(N0, 푀푑(C)) and 푧 ∈ C simply by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='24)  (푧•푈)푛 :=  �  (퐵0 − 푧 I)푈0 + 퐴0푈1  for 푛 = −1  푈푛  for 푛 ∈ N0  We shall use here the same notation for the vector sequences without any risk of confusion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', we shall  also write 푧•푢 for 푢 ∈ ℓ(N0, C푑) and 푧 ∈ C with the analogous meaning  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='25)  (푧•푢)푛 :=  �  (퐵0 − 푧 I)푢0 + 퐴0푢1  for 푛 = −1  푢푛  for 푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  8Note here, that it is “generalized” for two reasons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the first, because 푢 may not belong to Dom(퐽), not even ℓ2(N0, C푑),  and the second, since we do not require the equality for 푛 = 0 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  9The left-side one is also possible and used for several reasons, but we shall not consider it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  16  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Therefore, for any 푧 ∈ C both kinds of transformations are linear, and moreover:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='26)  푧•(푈퐶) = (푧•푈)퐶,  for any 푈 ∈ ℓ(N0, 푀푑(C)), 퐶 ∈ 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fortunately, the situation is much simpler if we restrict ourselves to the subspace of all the (M)GEV-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Namely, we have:  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푧, 푤 ∈ C, 푧 ≠ 푤, then10  GEV(푧) ∩ GEV(푤) = {0},  MGEV(푧) ∩ MGEV (푤) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For the matrix case, consider 푈 satisfying both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22), and its analog for 푤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then, subtracting,  we get  0 = (푧 − 푤)푈푛,  푛 ≥ 1,  hence 푈푛 = 0 for any 푛 ∈ N, but now again by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) used only for 푛 = 1 we get also 푈0 = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e,  푈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  This means, that for any non-zero vector or matrix solution of our equations, the parameter ’푧’ is in  fact “coded in the solution”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' On the other hand, the value of the extension •푧 for the zero sequence is  obviously also the zero sequence (indexed from −1 already) by the linearity, independently of 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence  denote  GEV :=  �  푧∈C  GEV(푧) ,  MGEV :=  �  푧∈C  MGEV(푧) ,  and, thank to Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8, for any 푈 ∈ MGEV \\ {0} (푢 ∈ GEV \\ {0}) we can define Par(푈) (Par(푢)) as  the unique number 푧 ∈ C satisfying  푈 ∈ MGEV(푧) (푢 ∈ GEV(푧)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Finally, we can simplify our notation and omit the parameter ’푧’, defining  : MGEV −→ ℓ(N−1, 푀푑(C))  given for 푈 ∈ MGEV simply by the formula  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='27)  푈 :=  �  Par(푈)•푈  for 푈 ≠ 0  0  for 푈 = 0  and analogically for 푢 ∈ GEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, taking into account (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23), let us consider “extensions” of the systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22):  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28)  퐴∗  푛−1푢푛−1 + 퐵푛푢푛 + 퐴푛푢푛+1 = 푧푢푛,  푛 ≥ 000  for sequences 푢 = (푢푛)푛∈N−1 ∈ ℓ(N−1, C푑) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='29)  퐴∗  푛−1푈푛−1 + 퐵푛푈푛 + 퐴푛푈푛+1 = 푧푈푛,  푛 ≥ 000  for sequences 푈 = (푈푛)푛∈N−1 ∈ ℓ(N−1, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Their solutions will be called extended generalized  eigenvectors and extended matrix generalized eigenvectors, respectively, (for 퐽 and 푧) — “egev” and  “emgev” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We denote also  GEV−1 (푧) :=  �  푢 ∈ ℓ(N−1, C푑) : 푢 is an egev for 퐽 and 푧  �  and  MGEV−1 (푧) :=  �  푈 ∈ ℓ(N−1, 푀푑(C)) : 푈 is an emgev for 퐽 and 푧  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us formulate explicitly some simple relations between all the above “extended” and “non-extended”  notions and the • transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푧 ∈ C the following assertions hold:  (i)  ↾GEV(푧) = 푧•↾GEV(푧),  (ii)  ↾GEV(푧): GEV(푧) −→ GEV−1 (푧) is a linear isomorphism between GEV(푧) and GEV−1 (푧),  (iii) �  ↾GEV(푧)  �−1 푢 = 푢↾N0  for any 푢 ∈ GEV−1 (푧),  (iv) for any (푐−1, 푐0) ∈ (C푑)2 there is a unique 푢 ∈ GEV−1 (푧) with 푢−1 = 푐−1, 푢0 = 푐0,  10Below 0 denotes the zero sequence both for the C푑-vector and for the matrix case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  17  and their obvious reformulations for the matrix sequences variants are also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let’s check, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', the C푑 vector version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that (i) is in fact the definition of •.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Using  this and taking 푢 ∈ GEV(푧), 푣 ∈ GEV−1 (푧), we get obviously (•푢)↾N0= 푢 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='25), but we get also  (푣↾N0) = 푣, because 푣 satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28) — in particular for 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Linearity is clear by the definition, so  (ii) and (iii) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Part (iv) follows directly from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  It can be easily checked that (e)mgev-s and (e)gev-s are mutually related in the following simple ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푈 ∈ ℓ(N−1, 푀푑(C)), 푧 ∈ C, then  (i) 푈 is an emgev for 퐽 and 푧 iff for any 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑 the vector sequence 푈 { 푗} := �푈 { 푗}  푛  �  푛∈N−1 ∈  ℓ(N−1, C푑) is an egev for 퐽 and 푧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) If 푈 is an emgev for 퐽 and 푧 then for any 푣 ∈ C푑 the vector sequence 푈푣 := (푈푛푣)푛∈N−1 ∈  ℓ(N−1, C푑) is an egev for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The analog result holds for sequences 푈 ∈ ℓ(N0, 푀푑(C)) and mgev-s and gev-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  According to Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9(iv) for the matrix case, for any 푧 ∈ C choose 푄(푧), 푃(푧) ∈ MGEV−1 (푧)  corresponding to  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30)  �  푄−1(푧) = I  푄0(푧) = 0,  �  푃−1(푧) = 0  푃0(푧) = I,  with the following general notation: for any sequence 푈(푝) = ((푈(푝))푛)푛∈N푘 depending on an extra  “function variable type parameter” 푝:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='31)  푈푛(푝) := (푈(푝))푛,  푛 ∈ N푘  for any 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The two sequences of functions 푄, 푃 are the so-called the second and the first kind matrix  orthogonal polynomials11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We can also use “the conditions in 0, 1” instead of those “in −1, 0”:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='32)  �  푄0(푧) = 0  푄1(푧) = 퐴−1  0 ,  �  푃0(푧) = I  푃1(푧) = 퐴−1  0 (푧 I −퐵0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The linear space of matrix solutions MGEV−1 (푧) and the special solutions 푄(푧) and 푃(푧) have some  important algebraic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (i) If 푈 ∈ MGEV−1 (푧), then for any 푉 ∈ 푀푑(C) also 푈푉 = (푈푛푉)푛∈N0 ∈ MGEV−1 (푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) Each 푈 ∈ MGEV−1 (푧) has the form  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33)  푈 = 푃(푧)푆 + 푄(푧)푇,  for a unique pair (푆,푇) of matrices from 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This pair is given by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34)  �  푆 := 푈0  푇 := 푈−1 = (퐵0 − 푧 I)푈0 + 퐴0푈1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iii) For any 푆 ∈ 푀푑(C) the matrix sequence 퐻 := (푃(푧)푆)↾N0 satisfies “the formal matrix eigenequa-  tion for J and 푧”, namely: 퐻 ∈ MGEV(푧) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='35)  퐵0퐻0 + 퐴0퐻1 = 푧퐻0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, for any 푣 ∈ C푑 \\ {0} the C푑-sequence ℎ := (푃(푧)푣)↾N0 is an eigenvector of the formal  operator J for 푧:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='36)  Jℎ = 푧ℎ,  ℎ ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  11More precisely, this name and the orthogonality property belong to the appropriate two sequences (푄푛)푛∈N, (푃푛)푛∈N0 of  matrix valued polynomial functions 푄푛, 푃푛 on R or on C with the values at each 푧 given by above defined 푄푛(푧), 푃푛(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  18  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Part (i) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence, using it, by linearity, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30) and by the unicity from Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9(iv),  we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33) with 푆 and 푇 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' — I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', we can simply assume that some 푆 and 푇 are given  by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34) and then we see that the initial conditions of the solution on the RHS of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33) are just the pair  (푈−1,푈0), which proves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33) by the unicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, to finish (ii) we should check that the choice of the pair  (푆,푇) is also unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By the linearity, it suffices to check only that if 푃(푧)푆 + 푄(푧)푇 is the zero solution,  then 푆 = 푇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Indeed, in this case we have 0 = 푃0(푧)푆 + 푄0(푧)푇 = 푆 and 0 = 푃−1(푧)푆 + 푄−1(푧)푇 = 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, to get (iii), we can first use (i) with (ii) for 푈 of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33) with 푇 = 0, so, using also Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9  (the matrix version), we get 퐻 ∈ MGEV(푧) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='35) obtained by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34) for 푇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now we obtain  (Jℎ)푛 = 푧ℎ푛 by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) for 푛 ≥ 1 and separately for 푛 = 0 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='35) with 푆 = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally ℎ ≠ 0, because  by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30) ℎ0 = (푃(푧)푣)0 = 푃0푣 = 푣 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Transfer matrices and the Liouville–Ostrogradsky formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In the scalar case 푑 = 1 the transfer  matrix sequences turned out to be a very useful tool for describing spectral properties of the operator 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  As we shall see, this is the case of general dimension 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us fix here 푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Our basic difference equations: the generalized eigenequation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20), its matrix  analog (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22), as well as their extended variants, can be written in equivalent forms with the use of the  so-called (one step) transfer matrices (for 퐽 and 푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The 푛-th transfer matrix 푇푛(푧) ∈ 푀2푑(C) has the  block form, with blocks in 푀푑(C):  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37)  푇푛(푧) :=  �  0  I  −퐴−1  푛 퐴∗  푛−1  퐴−1  푛 (푧 I −퐵푛)  �  ,  푛 ≥ 0  (for 푛 = 0 recall that 퐴−1 = − I by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence, obviously, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21) ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28)) is equivalent to  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38)  � 푢푛  푢푛+1  �  = 푇푛(푧)  �푢푛−1  푢푛  �  ,  푛 ≥ 1 (≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Similarly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) ((3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='29)) is equivalent to  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39)  �  푈푛  푈푛+1  �  = 푇푛(푧)  �  푈푛−1  푈푛  �  ,  푛 ≥ 1 (≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us observe that 푇푛(푧) is invertible and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='40)  �푇푛(푧)�−1 =  ��퐴∗  푛−1  �−1(푧 I −퐵푛)  −�퐴∗  푛−1  �−1퐴푛  I  0  �  ,  푛 ≥ 0,  which is clear by direct multiplying (or by expressing 푢푛−1 by 푢푛 and 푢푛−1 from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover we define the 푛-step transfer matrix by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41)  푅푛(푧) = 푇푛−1(푧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푇0(푧),  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  This name is justified, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', by the property  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42)  �  푈푛−1  푈푛  �  = 푅푛(푧)  �  푈−1  푈0  �  ,  푛 ≥ 1,  which we obtain from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30) we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43)  푅푛(푧) =  �  푄푛−1(푧)  푃푛−1(푧)  푄푛(푧)  푃푛(푧)  �  ,  푛 ≥ 1,  being simply a direct consequence of 푅푛(푧) = 푅푛(푧)  �  I  0  0  I  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Presently we shall derive a formula for the inverse of 푅푛(푧) expressing it explicitly in terms of 푅푛(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us set  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='44)  퐾푛 :=  �  퐴∗  푛  0  0  I  �  ,  푛 ≥ −1  and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='45)  ˜푇푛(푧) := 퐾푛푇푛(푧)퐾−1  푛−1,  푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  19  So by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37)  ˜푇푛(푧) =  �  0  퐴∗  푛  −퐴−1  푛  퐴−1  푛 (푧 I −퐵푛)  �  ,  푛 ≥ 0  Therefore, defining  Ω :=  �  0  I  − I  0  �  ,  we verify by direct computations that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='46)  Ω = � ˜푇푛(푧)�∗Ω˜푇푛(푧),  푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='45) we get  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='47)  푅푛(푧) = (퐾−1  푛−1 ˜푇푛−1(푧)퐾푛−2)(퐾−1  푛−2 ˜푇푛−2(푧)퐾푛−3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (퐾−1  0 ˜푇0(푧)퐾−1) = 퐾−1  푛−1 ˜푅푛(푧)퐾−1,  where  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='48)  ˜푅푛(푧) := ˜푇푛−1(푧) ˜푇푛−2(푧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' ˜푇0(푧),  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We claim that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49)  Ω = � ˜푅푛(푧)�∗Ω ˜푅푛(푧),  푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We shall prove it inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='48) we have ˜푅1(푧) = ˜푇0(푧) for any 푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, in view of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='46)  the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49) holds true for 푛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Next, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49) holds for some 푛 ≥ 1, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='46) we have  Ω = � ˜푅푛(푧)�∗Ω ˜푅푛(푧)  = � ˜푅푛(푧)�∗��˜푇푛(푧)�∗Ω˜푇푛(푧)  �  ˜푅푛(푧)  = � ˜푅푛+1(푧)�∗Ω ˜푅푛+1(푧),  where in the last equality we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It ends the inductive step in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, by  multiplying both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49) by Ω−1 on the left and then by � ˜푅푛(푧)�−1 on the right we arrive at  � ˜푅푛(푧)�−1 = Ω−1� ˜푅푛(푧)�∗Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, using twice (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='47) we can derive  �푅푛(푧)�−1 = 퐾−1  −1Ω−1�퐾−1  −1  �∗�푅푛(푧)�∗퐾∗  푛−1Ω퐾푛−1,  so, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='44), finally  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50)  �푅푛(푧)�−1 =  �  0  I  − I  0  � �푅푛(푧)�∗  �  0  퐴푛−1  −퐴∗  푛−1  0  �  ,  푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The following result is well-known, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [3, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2] and [46, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Our proof seems  to be new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12 (Liouville–Ostrogradsky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푤 ∈ C one has  푄푘 (푤)�푃푘(푤)�∗ = 푃푘(푤)�푄푘 (푤)�∗,  푘 ≥ 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51)  푄푘 (푤)�푃푘−1(푤)�∗ − 푃푘(푤)�푄푘−1(푤)�∗ = 퐴−1  푘−1,  푘 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='52)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43) we have  �I  0  0  I  �  = 푅푘(푤)푅−1  푘 (푤) =  �−푃푘−1(푤)  푄푘−1(푤)  −푃푘(푤)  푄푘 (푤)  � �−�푄푘 (푤)�∗퐴∗  푘−1  �푄푘−1(푤)�∗퐴푘−1  −�푃푘(푤)�∗퐴∗  푘−1  �푃푘−1(푤)�∗퐴푘−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='52) follows from computing the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The Weyl function  Similarly to the scalar Jacobi case, Weyl coefficient (being a matrix for the block case), is the main  object in the method of subordinacy, which gives the link between generalized eigenvectors and the  absolute continuous and the singular part of the spectral measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And consequently – the absolutely  continuous and the singular spectrum of 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  20  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' ℓ2 matrix solutions and the matrix Weyl function 푊.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In the scalar Jacobi case “the scalar  orthogonal polynomials” are used, with the common notation 푝(푧) := 푃(푧), 푞(푧) := 푄(푧) for 푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  That is, since 푑 = 1, we treat complex numbers as elements of 푀푑(C) and also as C푑-vectors, and  푝(푧), 푞(푧) are solutions of both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='29), being now just the same equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is also well-known for this case that if 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and 푧 ∈ C \\ R, then neither 푝(푧)↾N0, nor 푞(푧)↾N0  belong to the Hilbert space ℓ2(N0, C) in which Jacobi operator 퐽 acts, and there exists exactly one  푤(푧) ∈ C such that (푤(푧)푝(푧) + 푞(푧))↾N0∈ ℓ2(N0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Surely, instead of making the restriction to N0,  we can equivalently just claim here that 푝(푧), 푞(푧) ∉ ℓ2(N−1, C), and 푤(푧)푝(푧) + 푞(푧) ∈ ℓ2(N−1, C),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The above unique 푤(푧) is called the Weyl coefficient (for 퐽 and 푧), and the appropriate function  푤 : C \\ R −→ C is called the Weyl function for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us recall here some less known generalisations of the above results and definitions for block Jacobi  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Observe first, that if 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', and 푧 ∈ C \\ R, then 푧 ∉ 휎(퐽), so denote:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  푢( 푗) (푧) := (퐽 − 푧 I)−1훿0(푒 푗),  푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular, for any 푗 we have 푢( 푗) (푧) ∈ Dom(퐽) ⊂ ℓ2(N0, C푑) and  J푢( 푗) (푧) = 푧푢( 푗) (푧) + 훿0(푒 푗),  푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Considering the terms 푛 ≥ 1 of the above equality of sequences we see that each 푢( 푗) (푧) is a gev for 퐽  and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, taking the term 푛 = 0 we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  푒 푗 =  �  (J − 푧 I)푢( 푗) (푧)  �  0 = (퐵0 − 푧 I)푢( 푗)  0 (푧) + 퐴0푢( 푗)  1 (푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, defining the matrix sequences  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  ˜푈(푧) := [푢(1) (푧), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푢(푑) (푧)] ∈ ℓ(N0, 푀푑(C)) and 푈(푧) := •( ˜푈(푧)) ∈ ℓ(N−1, 푀푑(C))  we have 푈(푧) ∈ ℓ2(N−1, 푀푑(C)), and by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10 (the version for gev-s and mgev-s) and by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9 we  see that 푈(푧) is an emgev for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We call it Weyl matrix solution for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) we  obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  I = (퐵0 − 푧 I)푈0(푧) + 퐴0푈1(푧) = 푈−1(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We can now formulate the expected result on “matrix ℓ2 solutions” (see [2, Theorem VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8] for similar  result with a sketch of the proof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' we give here a full and simple proof for the sake of self-sufficiency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and 푧 ∈ C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then there exists exactly one 푊(푧) ∈ 푀푑(C) such  that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  푃(푧)푊(푧) + 푄(푧) ∈ ℓ2(N−1, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, with the above unique 푊(푧)  (i) 푃(푧)푊(푧) + 푄(푧) is the Weyl matrix solution for 퐽 and 푧:  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  푃(푧)푊(푧) + 푄(푧) = 푈(푧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) 푊(푧) = 푈0(푧),  (iii) det푊(푧) ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Fix 푧 ∉ R and consider the Weyl matrix solution 푈(푧) for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11(ii) 푈(푧) has  the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33), where 푆 = 푈0(푧) and 푇 = 푈−1(푧) = I, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This proves the “exists”– part of the  assertion and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us prove the uniqueness, now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that for some 푧 ∉ R there exist two different matrices “푊(푧)”  satisfying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then, subtracting, we get a non-zero 퐶 ∈ 푀푑(C) such that 푃(푧)퐶 ∈ ℓ2(N−1, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, choosing 푤 ∈ C푑 such that 푣 := 퐶푤 ≠ 0 we get 푃(푧)푣 ∈ ℓ2(N−1, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, using Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11(iii),  for ℎ := (푃(푧)푣)↾N0 we get J ℎ = 푧ℎ ∈ ℓ2(N0, C푑), which means that ℎ ∈ Dom(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover ℎ ≠ 0,  because ℎ0 = 푃0(푧)푣 = I 푣 = 푣 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus ℎ is an eigenvector of 퐽 with the eigenvalue 푧 ∉ R — a  contradiction with the assumption, that 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  21  To prove that 푊(푧) = 푈0(푧) is invertible, consider 푣 ∈ Ker푈0(푧) and the sequence 푤 := (푈(푧)푣)↾N0∈  ℓ(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) we see that 푤 is such a gev that  ((J − 푧 I)푤)0 = (퐵0 − 푧 I)푤0 + 퐴0푤1 = (퐵0 − 푧 I)푈0(푧)푣 + 퐴0푈1(푧)푣 = I 푣 = 푣 = (훿0(푣))0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence we have  (a) (J − 푧 I)푤 = 훿0(푣),  (b) 푤0 = 푈0(푧)푣 = 0,  (c) 푤 ∈ ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now (a) gives J푤 = 푧푤 + 훿0(푣), so by (c) both 푤 and J푤 are in ℓ2(N0, C푑), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푤 ∈ Dom(퐽) and  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  (퐽 − 푧 I)푤 = (J − 푧 I)푤 = 훿0(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  But (b) means in particular that 푤0 ⊥ 푣, which allows us to get the expected assertion 푣 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' — Indeed,  by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7) we obtain ⟨훿0(푣), 푤⟩ = ⟨퐽푤, 푤⟩ − 푧 ∥푤∥2 and on the other hand ⟨훿0(푣), 푤⟩ = ⟨푣, 푤0⟩C푑 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, using the s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' of 퐽, we get 푧 ∥푤∥2 = ⟨퐽푤, 푤⟩ ∈ R, but since 푧 ∉ R, 푤 has to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and 푧 ∈ C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then  푃(푧), 푄(푧) ∉ ℓ2(N−1, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We have 푄(푧) = 푃(푧)0 + 푄(푧), so if 푄(푧) ∈ ℓ2(N−1, 푀푑(C)) then 푊(푧) = 0 by the uniqueness  from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But it contradicts the condition det 푊(푧) ≠ 0 from (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푄(푧) ∈ ℓ2(N−1, 푀푑(C)) then using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) and (iii) we get 푃(푧)+푄(푧)(푊(푧))−1 ∈ ℓ2(N−1, 푀푑(C)), so  also 푄(푧)(푊(푧))−1 ∈ ℓ2(N−1, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But then also 푄(푧) = �푄(푧)(푊(푧))−1� 푊(푧) ∈ ℓ2(N−1, 푀푑(C)),  which contradicts the part just proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐽 be s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For fixed 푧 ∈ C \\ R such 푊(푧), that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) holds is called the matrix Weyl  coefficient (for 퐽 and 푧), and the appropriate function 푊 : C \\ R −→ 푀푑(C) is called the matrix Weyl  function (for 퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12  We omit here the dependence on 퐽 in the notation, assuming that we consider a fixed 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Here we present also a result being a stronger version of [61, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us stress that we do  not assume the self-adjointness of 퐽 at this moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For 푧 ∈ C denote  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  GEVℓ2(푧) := GEV(푧) ∩ ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall that by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 we have  dim (GEV(푧)) = 2푑,  and let us think about the dimension of its subspace GEVℓ2(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To find it in some cases, consider also  EV(푧) — the eigenspace for 퐽 and 푧, which in the case of arbitrary 푧 ∈ C is defined by  EV(푧) := {푢 ∈ Dom(퐽) : 퐽푢 = 푧푢}  (and so, it is just the trivial zero space if 푧 is not an eigenvalue of 퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since 퐽 is the maximal block Jacobi  operator (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)), we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  EV(푧) = {푢 ∈ GEVℓ2(푧) : ((퐽 − 푧)푢)0 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Indeed, the above equality follows directly from  GEVℓ2(푧) ⊂ Dom(퐽),  which holds, because for 푢 ∈ GEVℓ2(푧) we have 푢 ∈ ℓ2(N0, C푑), so also 푧푢 ∈ ℓ2(N0, C푑), but J푢 and  푧푢 differ at most at the zero term, hence J푢 ∈ ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푧 ∈ C \\ 휎푝(퐽), then dim(GEVℓ2(푧)) ≤ 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If, moreover, 푧 ∈ C \\ 휎(퐽), then  dim(GEVℓ2(푧)) = 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  12Using the argumentation from the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 and from the beginning of this subsection one can easily see  that in fact it suffices here to assume that 푧 ∈ C \\ 휎(퐽) to properly define the matrix Weyl coefficient for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But note also  that the invertibility of 푊(푧) from property (iii) is guaranteed only for 푧 ∈ C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  22  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider first an arbitrary 푧 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and define Ψ : GEVℓ2(푧) −→ C푑 by  Ψ(푢) := ((퐽 − 푧 I)푢)0 ,  푢 ∈ GEVℓ2(푧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is a linear transformation and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9) Ker Ψ = EV(푧).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, by the standard linear algebra result,  dim (GEVℓ2(푧)) = dim (EV(푧)) + dim (Ran Ψ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, if 푧 ∈ C \\ 휎푝(퐽), then dim(GEVℓ2(푧)) = dim(Ran Ψ) ≤ 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But, if moreover 푧 ∈ C \\ 휎(퐽), then  Ran(퐽 − 푧 I) = ℓ2(N0, C푑), so in particular, for any 푣 ∈ C푑 we have 훿0(푣) ∈ Ran(퐽 − 푧 I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore, for  some 푢 ∈ Dom(퐽)  퐽푢 − 푧푢 = 훿0(푣),  thus 푢 ∈ GEVℓ2(푧) and Ψ(푢) = 푣 for such 푢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, Ran Ψ = C푑 and dim(GEVℓ2(푧)) = dim(Ran Ψ) =  푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  This result gives in particular dim(GEVℓ2(푧)) = 푑, when 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and 푧 ∈ C \\ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' On the other hand,  one can easily see that for such 푧  GEVℓ2(푧) = lin{푢( 푗) (푧) : 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑},  where 푢( 푗) (푧) are defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1), and they are just the successive 푗-th column sequences (푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑)  of the matrix sequence ˜푈(푧), being the restriction to N0 of the Weyl matrix solution 푈(푧) for 퐽 and 푧 (see  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The Cauchy transform of the spectral matrix measure and the matrix Weyl function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us  assume also here that 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The Cauchy transform of the spectral matrix measure 푀 := 퐸퐽, �휑 from  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19) of 퐽 is defined as C퐽 : C \\ R −→ 푀푑(C), with  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  C퐽 (푧) :=  ∫  R  1  휆 − 푧 d푀(휆),  푧 ∈ C \\ R,  where the above integral is understood in the sense of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) means just  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  C퐽 (푧) :=  �∫  R  1  휆 − 푧 d푀푖, 푗 (휆)  �  푖, 푗=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑 ,  with  푀푖, 푗 = 퐸퐽,휑 푗,휑푖,  푖, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑,  and 휑 푗 given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, by spectral calculus for s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' operators, for 푧 ∈ C \\ R we get  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  C퐽 (푧) :=  �∫  R  1  휆 − 푧 d퐸퐽,휑 푗,휑푖 (휆)  �  푖, 푗=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑  =  ��  (퐽 − 푧 I)−1훿0(푒 푗), 훿0(푒푖)  ��  푖, 푗=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) for any 푖, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑  (C퐽 (푧))푖, 푗 =  �  푢( 푗) (푧), 훿0(푒푖)  �  =  �  (푢( 푗) (푧))0, 푒푖  �  C푑 = (푈0(푧))푖, 푗  for 푧 ∈ C \\ R, where 푢( 푗) (푧) is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1) and 푈(푧) is Weyl matrix solution for 퐽 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally, by  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1, we get  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', then the Cauchy transform of the spectral matrix measure of 퐽 is equal to the matrix  Weyl function for 퐽 and moreover, for any 푧 ∈ C \\ R  C퐽 (푧) = 푊(푧) = 푈0(푧) =  ��  (퐽 − 푧 I)−1훿0(푒 푗), 훿0(푒푖)  ��  푖, 푗=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The boundary limits of the matrix Weyl function and the properties of the spectral trace  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume here the self-adjointness of 퐽, as before, and let us use the notation from the previous  subsection, including 푀 := 퐸퐽, �휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5 implies that 푊 is a holomorphic matrix-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Also by Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) for any 푧 ∈ C \\ R  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  Im푊(푧) = Im �C퐽 (푧)� = 1  2푖  ∫  R  �  1  푡 − 푧 −  1  푡 − 푧  �  d푀(푡) = Im(푧)  ∫  R  1  |푡 − 푧|2 d푀(푡),  which is a non-negative matrix on C+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence the restriction of 푊 to C+ is a matrix Herglotz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Denote  퐿(푊) :=  �  휆 ∈ R : lim  휖 →0+ 푊(휆 + 푖휖) exists13  �  ,  and  푊(휆 + 푖0) := lim  휖 →0+ 푊(휆 + 푖휖),  휆 ∈ 퐿(푊),  and define the following sets:  푆ac,r :=  �  휆 ∈ 퐿(푊) : rank� Im푊(휆 + 푖0)� = 푟  �  ,  1 ≤ 푟 ≤ 푑,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  푆ac :=  푑  �  푟=1  푆ac,r ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)  푆sing :=  �  휆 ∈ R : lim  휖 →0+ Im � tr푊(휆 + 푖휖)� = +∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16)  In particular it follows that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17)  푆sing ⊂ R \\ 퐿(푊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Referring to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16), observe that for any 퐴 ∈ 푀푑(C)  tr(Im 퐴) = Im(tr 퐴).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is important that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15) means simply  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18)  푆ac =  �  휆 ∈ 퐿(푊) : Im푊(휆 + 푖0) ≠ 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us recall now the crucial result, joining the above defined sets with properties of 푀ac and 푀sing  — the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and the sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' parts w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the Lebesgue measure | · | on Bor(R) (see Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) of the spectral  matrix measure 푀 of 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This theorem is obtained just as the direct use of the abstract result [13, Theorem  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1] to the spectral matrix measure 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Define 퐷 : 퐿(푊) −→ 푀푑(C) by  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19)  퐷(휆) := 1  휋 Im푊(휆 + 푖0),  휆 ∈ 퐿(푊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (i) 푆sing is a support of 푀sing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) |R \\ 퐿(푊)| = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iii) 퐷 is a density of 푀ac on 퐿(푊) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, this theorem shows that controlling of the boundary limits of the matrix Weyl function allows to  get a lot of detailed information about the spectral matrix measure of 퐽 and of its sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Combining the theorem with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 we get:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푆ac is a minimal support of (tr푀)ac with respect to | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover 푆sing is a support of  (tr푀)sing and |푆sing| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, 푆ac ∪ 훿 is also a minimal support of (tr푀)ac with respect to | · |  for any Borel 훿 ⊂ R with |훿| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  13As the limit in 푀푑(C), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', in particular the limit must belong to 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  24  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We use Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 taking:  휈 := | · | so, also Ω := R and 픐 := Bor(R),  푆푎 := 푆ac ,  푆푠 := 푆sing ,  퐹 := 퐷↾푆ac  and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18) we see that 푆푎 ∩ 푆푠 = ∅, that is, the assumption (ii) of the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now,  (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6 shows that 퐿(푊) is a support of 푀ac, since this matrix measure, by definition, is  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But (iii) of this theorem with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19) mean, that the restriction of 푀ac to  퐿(푊) \\ 푆ac is the zero matrix measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore 푆ac is also a support of 푀ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This together with (i) of  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6 prove that the assumption (i) of the lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The assumption (iii) also holds, by the  fact that 푆ac ⊂ 퐿(푊) and by (iii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6, again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And (iv) of the lemma is obvious by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Therefore Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 yields the first two assertions and |푆sing| = 0 by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17) with (iii) of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The last assertion is obvious just by the definition of minimal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The boundary limits and spectral consequences for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume the self-adjointness of 퐽, and let  us hold the notation as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Here we “translate” Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6 into the spectral operator language via Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, we  show:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is self-adjoint and 퐺 ∈ Bor(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (i) If 퐺 ⊂ R \\ 푆sing, then 퐽 is absolutely continuous in 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) If 퐺 ⊂ 푆ac ∪ (R \\ (퐿(푊) ∪ 푆sing)), then 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, if moreover 퐺 is open, or if 퐺 is a sum of an arbitrary family of connected non-singletons in  R, then cl(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' First of all, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2, 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' finitely-cyclic operator, with �휑 being a cyclic system  for 퐽 and with 푀 being the spectral matrix measure of 퐽 and �휑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence the initial assumptions of [44,  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2] hold for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, using its assertion (2), we obtain our (i), because, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7,  (tr푀)sing(퐺) = 0 for 퐺 ⊂ R \\ 푆sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6(iii) we get |R \\ (퐿(푊) ∪ 푆sing)| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence, 푆ac ∪ �R \\ (퐿(푊) ∪ 푆sing)� is a minimal  support of (tr푀)ac with respect to | · | by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover (tr푀)sing(퐺) = 0, again because  푆ac ∪ �R \\ (퐿(푊) ∪ 푆sing)� ⊂ R \\ 푆sing and 푆sing is a support of (tr푀)sing by By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, using the assertion (3) of [44, Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2] The assertion for the special kinds of 퐺 follows  form the property clLe(퐺) = cl(퐺), which holds for those 퐺 (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', [44, Fact C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' An analog of the Jitomirskaya–Last’s approach  This is the most important part of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' — It contains the main new ideas allowing to get  some analogies of the 1-dimensional subordinacy results also in the 푑-block case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' However, one of key  technical tools used in our proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12 is a generalisation of the Jitomirskaya–Last’s idea  from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The vector and matrix nonsubordinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We start with a new notion, which seems natural from  the context of the crucial notion of subordinated solutions in the 푑 = 1 case from Gilbert–Pearson–Khan  subordination theory (see [16, 33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recall that the “interpolated” semi-norms ∥·∥[0,푡] were introduced  here in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We say that 퐽 satisfies vector nonsubordinacy (condition) for 휆 ∈ R iff14 for each pair of  non-zero 푢, 푣 ∈ GEV−1 (휆)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  lim inf  푡→+∞  ∥푢∥ [0,푡]  ∥푣∥[0,푡]  < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Nevertheless, the followingmatrixgev termsformulation seems to be more convenient for our purposes:  퐽 satisfies matrix nonsubordinacy condition for 휆 ∈ R iff for each pair of non-zero 푈,푉 ∈ MGEV−1 (휆)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  lim inf  푡→+∞  ∥푈∥[0,푡]  ∥푉∥ [0,푡]  < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  14Note that we consider here the function given by the fraction  ∥푢 ∥2  [0,푡]  ∥푣 ∥2  [0,푡]  for 푡 > 0 only, and the denominator is positive since  here sequences are not the zero sequence and they belong to GEV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly for the definition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  25  Note here that the choice of “liminf-s over 푡 > 0”, instead of more original Khan and Pearson’s like  “liminf-s over 푛 ∈ N”, does not matter, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', is equivalent to this original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' One direction of the implication  follows directly from the definition of lim inf, and the other can be immediately obtained by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  As we shall see soon, also the distinction “vector” / “matrix” is not very important here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We can use the symmetry w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푢 and 푣 or 푈 and 푉, respectively, and we get:  Fact 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 퐽 satisfies vector nonsubordinacy for 휆 ∈ R iff for each pair of non-zero 푢, 푣 ∈ GEV−1 (휆)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  lim sup  푡→+∞  ∥푢∥ [0,푡]  ∥푣∥[0,푡]  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  And analogically in the matrix nonsubordinacy case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us formulate now an important spectral consequence of vector nonsubordinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall that  GEVℓ2(휆) was defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is self-adjoint and it satisfies vector nonsubordinacy for some 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then  dim(GEVℓ2(휆)) = 0 and 휆 ∈ 휎(퐽) \\ 휎p(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푢, 푣 ∈ GEV−1 (휆) be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Fact 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2 there exists a constant 푐 > 0 and a sequence  (푡푘)푘 ∈N tending to +∞ such that  ∥푢∥[0,푡푘 ] ≥ 푐 ∥푣∥ [0,푡푘 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, taking the limit we get  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  1  푐 ∥푢∥ [0,+∞] ≥ ∥푣∥ [0,+∞] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, if it existed a non-trivial 푢 ∈ ℓ2(N0, C푑), then all 푣 would be also in ℓ2(N0, C푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But then the  operator 퐽 would not be self-adjoint, by [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus each non-zero 푢 ∈ GEV−1 (휆) is not  square-summable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 yields the last assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  As we already announced, vector and matrix nonsubordinacy are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 퐽 satisfies matrix nonsubordinacy for 휆 iff it satisfies vector nonsubordinacy  for 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (⇒) Take any non-zero 푢, 푣 ∈ GEV−1 (휆) and view them as a sequence of column vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let  us define  푈푛 := 퐸푢푛,  푉푛 := 퐸 푣푛,  푛 ≥ −1,  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the first column of 푈푛 is equal to the column vector 푢푛 and the rest is zero, analogously for  푉푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10 both 푈,푉 ∈ MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5 for any 푛 ≥ −1 we have ∥푈푛∥ = ∥푢푛∥  and ∥푉푛∥ = ∥푣푛∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, ∥푈∥[0,푡] = ∥푢∥ [0,푡] and ∥푉∥ [0,푡] = ∥푣∥[0,푡] for any 푡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, the  condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (⇐) Let us observe that for any 푋 ∈ MGEV−1 (휆) and any 푤 ∈ C푑 such that ∥푤∥ = 1 we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  ∥푋푤∥2  [0,푡] ≤ ∥푋∥2  [0,푡] ≤  푑  �  푖=1  ∥푋푒푖∥2  [0,푡] ,  푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let 푈,푉 ∈ MGEV−1 (휆) be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then there exists 푤 ∈ C푑 such that ∥푤∥ = 1 and 푉푤 is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  ∥푈∥2  [0,푡]  ∥푉∥2  [0,푡]  ≤  푑  �  푖=1  ∥푈푒푖∥2  [0,푡]  ∥푉푤∥2  [0,푡]  ,  푡 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10, 푈푒푖 ∈ GEV−1 (휆) for 푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑 and 푉푤 ∈ GEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1) we get  lim inf  푡→+∞  ∥푈푒푖∥2  [0,푡]  ∥푉푤∥2  [0,푡]  < +∞,  푖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑,  which together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  26  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Barriers and barrier nonsubordinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In what follows, we need to make the concept of matrix  nonsubordinacy more controllable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The goal is to control two things:  a bound on the ratio  ∥푈 ∥ [0,푡]  ∥푉 ∥ [0,푡]  for any fixed “large” 푡 and 휆 ∈ 퐺, but joint for all such  푈,푉 ∈ MGEV−1 (휆) which are “normalized” in a proper sense — the key one!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  the size of the above bound as a function of 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  So, we define:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐺 ⊂ R be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A function 픟 : 퐺 × [1, ∞) −→ R is a barrier (on 퐺, for 퐽) iff  for each 휆 ∈ 퐺 and each pair 푈,푉 ∈ MGEV−1 (휆) normalized by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  ∥푈−1∥2 + ∥푈0∥2 = ∥푉−1∥2 + ∥푉0∥2 = 1  the estimate  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  � ∥푈∥[0,푡]  ∥푉∥ [0,푡]  �2  ≤ 픟(휆, 푡)  holds for all 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  To shorten the notation related to our normalisation let us denote  S :=  �  (푋, 푋 ′) ∈ 푀푑(C) × 푀푑(C) : ∥푋∥2 + ∥푋 ′∥2 = 1  �  and, for 휆 ∈ C,  MGEVnor (휆) := {푈 ∈ MGEV−1 (휆) : (푈−1,푈0) ∈ S},  MGEV★ (휆) := MGEV−1 (휆) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, for (푋, 푋 ′) ∈ 푀푑(C) × 푀푑(C) denote  [(푋, 푋 ′)]퐼 := 푋,  [(푋, 푋 ′)]퐼 퐼 := 푋 ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By the homogeneity of all the norms and semi-norms, and by the use of the “symmetrization w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푈  and 푉” we easily obtain:  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let ∅ ≠ 퐺 ⊂ R and 픟 : 퐺 × [1, ∞) −→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (i) If 픟 is a barrier, then 픟(휆, 푡) ≥ 1 for any 휆 ∈ 퐺, 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii) If 픟 is strictly positive, then TFCAE:  (a) 픟 is a barrier,  (b) ∀휆∈퐺∀푈,푉 ∈MGEVnor(휆)∀푡 ≥1  1  픟(휆, 푡) ≤  ∥푈∥2  [0,푡]  ∥푉∥2  [0,푡]  ,  (c) ∀휆∈퐺∀푈,푉 ∈MGEV★(휆)∀푡 ≥1  1  픟(휆, 푡)  ∥푈−1∥2 + ∥푈0∥2  ∥푉−1∥2 + ∥푉0∥2 ≤  ∥푈∥2  [0,푡]  ∥푉∥2  [0,푡]  ≤ 픟(휆, 푡) ∥푈−1∥2 + ∥푈0∥2  ∥푉−1∥2 + ∥푉0∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Recall now (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) that each 푈 ∈ MGEV−1 (휆) can be expressed in a convenient form by its  initial values (푈−1,푈0) and 푛-step transfer 2푑 × 2푑 matrices 푅푛(휆) (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41)), defined for 푛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Defining also  푅0(휆) := I,  we get in particular15  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  푈(푛) = [푅푛+1(휆)(푈−1,푈0)]퐼,  푛 ∈ N−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푈(푛) = [푅푛(휆)(푈−1,푈0)]퐼 퐼,  푛 ∈ N0  for any 푈 ∈ MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The following investigations show that for any 퐽 there exists its smallest “universal” barrier (on each  퐺).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider the function 픟min : R × [1, +∞) → [1, +∞]  given by the formula  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  픟min(휆, 푡) :=  sup  푈,푉 ∈MGEVnor(휆)  � ∥푈∥[0,푡]  ∥푉∥ [0,푡]  �2  ,  휆 ∈ R, 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  15We simplify here the notation, and we use the row-block form instead of the “column-block” one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  27  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The function 픟min has only finite values and it is a barrier for 퐽 on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover it is  the smallest barrier for 퐽 on any 퐺 ⊂ R, in the sense that for any barrier 픟 for 퐽 on 퐺  픟min(휆, 푡) ≤ 픟(휆, 푡),  휆 ∈ 퐺, 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Fix 푡 ≥ 1 and 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푈 ∈ MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41) for any 푛 ∈ N−1 the  value of the matrix 푈푛 is a continuous (even linear) function of its initial conditions (푈−1,푈0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely,  by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9) we have 푈(푛) = [푅푛+1(휆)(푈−1,푈0)]퐼 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, denoting by 푌 (훼) the sequence  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  푌 (훼) := ([푅푛+1(휆)훼]퐼)푛∈N−1  for any 훼 ∈ 푀푑(C) × 푀푑(C), we get  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12)  푈 = 푌 (푈−1,푈0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9), also the function  S ∋ 훼 ↦→ ∥푌 (훼)∥ [0,푡]  is continuous and, moreover, it is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now, define a function 푓 : S × S → R by  푓 (훼, 훼′) :=  � ∥푌 (훼)∥ [0,푡]  ∥푌 (훼′)∥ [0,푡]  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Obviously, 푓 is also continuous and S × S is compact, hence 푓 attains supremum of its values, and by  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  픟min(휆, 푡) = sup  훼,훼′∈S  푓 (훼, 훼′) = max  훼,훼′∈S 푓 (훼, 훼′) ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  On the other hand, it is obvious that 픟min is a barrier, as well as that that it is the smallest one, directly  from its definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  For any 퐺 ⊂ R we call 픟min↾퐺  the minimal barrier for 퐽 on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Our next goal is now the construction of an another barrier for 퐺 = R in terms of the sequence of  transfer matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It will be much more convenient in practice than the minimal one, because of its more  explicit form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To do this, we need first:  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푢 ∈ GEV−1 (휆), then for any 푡 ≥ 1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13)  1  2  � ∥푢−1∥2 + ∥푢0∥2 � ⇃|푅(휆)|⇂2  [1,푡]≤ ∥푢∥2  [0,푡] ≤ � ∥푢−1∥2 + ∥푢0∥2 � ∥푅(휆)∥2  [1,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 푈 ∈ MGEV−1 (휆), then for any 푡 ≥ 1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14)  1  4  � ∥푈−1∥2 + ∥푈0∥2 � ⇃|푅(휆)|⇂2  [1,푡]≤ ∥푈∥2  [0,푡] ≤ 2� ∥푈−1∥2 + ∥푈0∥2 � ∥푅(휆)∥2  [1,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular, for any 푈,푉 ∈ MGEVnor (휆)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)  � ∥푈∥[0,푡]  ∥푉∥ [0,푡]  �2  ≤ 8  � ∥푅(휆)∥[1,푡]  ⇃|푅(휆)|⇂[1,푡]  �2  ,  푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 푢−1 = 푢0 = 0, then 푢 = 0, and the first assertion holds, so assume that ∥푢−1∥2 + ∥푢0∥2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Set  �푢푘 :=  �  푢푘−1  푢푘  �  ,  푘 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then  �푢푘 = 푅푘 (휆) �푢0,  푘 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16)  ∥�푢0∥2 ⇃|푅푘(휆)|⇂2≤ ∥�푢푘 ∥2 ≤ ∥�푢0∥2 ∥푅푘(휆)∥2 ,  푘 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 we get  ∥�푢0∥2 ⇃|푅(휆)|⇂2  [1,푡]≤ ∥�푢∥2  [1,푡] ≤ ∥�푢0∥2 ∥푅(휆)∥2  [1,푡] ,  푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13) follows, since we have  ∥푢∥2  [0,푡] ≤ ∥�푢∥2  [1,푡] ≤ 2 ∥푢∥2  [0,푡]  28  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  by direct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now, consider 푈 ∈ MGEV−1 (휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then by Fact 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10 for any 푣 ∈ C푑 the  sequence 푈푣 is a vector generalized eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16) implies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17)  � ∥푈−1푣∥2 + ∥푈0푣∥2 � ⇃|푅푘(휆)|⇂2≤  ����  �푈푘−1푣  푈푘푣  �����  2  ≤ � ∥푈−1푣∥2 + ∥푈0푣∥2 � ∥푅푘 (휆)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular,  � ∥푈−1푣∥2 + ∥푈0푣∥2 � ⇃|푅푘(휆)|⇂2≤ � ∥푈푘−1∥2 + ∥푈푘∥2 � ∥푣∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18)  1  2  � ∥푈−1∥2 + ∥푈0∥2 � ⇃|푅푘(휆)|⇂2≤ ∥푈푘−1∥2 + ∥푈푘∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  On the other hand, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17)  ∥푈푘−1푣∥2 + ∥푈푘푣∥2 ≤ � ∥푈−1∥2 + ∥푈0∥2 � ∥푣∥2 ∥푅푘(휆)∥2 ,  and consequently,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19)  ∥푈푘−1∥2 + ∥푈푘 ∥2 ≤ 2� ∥푈−1∥2 + ∥푈0∥2 � ∥푅푘 (휆)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For any 푘 ≥ 0 let us define  �푥푘 :=  �  ∥푈푘−1∥  ∥푈푘 ∥  �  ∈ C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Therefore, by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19) we obtain  1  2 ∥�푥0∥2 ⇃|푅푘(휆)|⇂2≤ ∥�푥푘 ∥2 ≤ 2 ∥�푥0∥2 ∥푅푘 (휆)∥2 ,  푘 ≥ 1,  which is an analogue of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now we get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14) by a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  In view of this result let us consider now 픟TR : R × [1, +∞) → R given by the formula16  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20)  픟TR(휆, 푡) := 8  � ∥푅(휆)∥[1,푡]  ⇃|푅(휆)|⇂[1,푡]  �2  ,  휆 ∈ R, 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The last assertion of the proposition yields exactly:  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 픟TR is a barrier for 퐽 on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For any 퐺 ⊂ R we call 픟TR↾퐺  the transfer matrix barrier for 퐽 on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  At the end of the subsection we introduce the most important notion of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is self-adjoint, Let 퐺 ⊂ R be non-empty and let 픟 be a barrier on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We  say that 퐽 is 픟-nonsubordinate on 퐺 if  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21)  lim inf  푡→+∞ 픟(휆, 푡) < +∞,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If in fact, this condition holds uniformly on 퐺, namely  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22)  sup  휆∈퐺  lim inf  푡→+∞ 픟휆(푡) < +∞,  then we say that 퐽 is uniformly 픟-nonsubordinate on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The convenience and the importance of constructing just such definitions (of the barrier and of the  barrier-nonsubordinacy) as here, will be clearly visible in proofs in the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  16Note that ⇃|푅(휆)|⇂[1,푡 ]> 0 for any 푡 ≥ 1, since 푅1(휆) = 푇0(휆) is invertible (see also (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The main result and its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume, as before, that 퐽 is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and consider the matrix  Weyl function 푊 for 퐽 (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Before we start to “control” in a sense its boundary limits we  need to make use of the choice of Jitomirskaya–Last type semi-norms, and prove a result being a block  case analog of the appropriate scalar case one result from [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 휆 ∈ R there exists a unique function  ℓ휆 :  R+ → R+ satisfying  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23)  ∥푃(휆)∥[0,ℓ휆(휖 )] ∥푄(휆)∥[0,ℓ휆 (휖 )] = 1  2휖 ,  휖 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, ℓ휆 is a strictly decreasing continuous function and satisfies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='24)  lim  휖 →0+ ℓ휆(휖) = +∞,  lim  휖 →+∞ ℓ휆(휖) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, its inverse ℓ−1  휆 is also strictly decreasing continuous function and satisfies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='25)  lim  푡→0+ ℓ−1  휆 (푡) = +∞,  lim  푡→+∞ ℓ−1  휆 (푡) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Define a function 푓 : R+ → R+ by the formula  푓 (푡) = ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9) this function is continuous, non-negative and weakly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30) it is in  fact positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us observe that it is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Indeed, because if not, then there would exist  푛 ∈ N0 such that both ∥푃(휆)∥ [0,푡] and ∥푄(휆)∥[0,푡] were constant for 푡 ∈ (푛, 푛 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It would mean that  ∥푃푛+1(휆)∥ = ∥푄푛+1(휆)∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Which by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43) would imply that 푅푛(휆) was singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This contradicts  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Next, observe that  lim  푡→0+ 푓 (푡) = 0,  lim  푡→+∞ 푓 (푡) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Indeed, the first limit follows from ∥푄0(휆)∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The second follows from the fact that if ∥푃(휆)∥[0,+∞] <  +∞ and ∥푄(휆)∥[0,+∞] < +∞, then the operator 퐽 is not self-adjoint, see [12, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, we  have shown that 푓 is continuous, strictly increasing and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus its inverse 푓 −1 : R+ → R+ has  the same properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider the function 푔 : R+ → R+ defined by 푔(휖) = 1/(2휖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This function is  strictly decreasing and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, if ℓ휆 function exists it is a solution of the equation  푓 �ℓ휆(휖)� = 푔(휖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  This equation has a unique solution given by  ℓ휆(휖) = 푓 −1�푔(휖)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is immediate that this defines a function satisfying (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' From this representation it is immediate that  ℓ휆 : R+ → R+ is a continuous strictly decreasing surjective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since again ℓ−1  휆  has analogous properties, we also obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  The unique function ℓ휆 described above will be called J-L function (for 퐽 and 휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We are ready to formulate our main result “on controlling the boundary limits of the matrix Weyl  function”, being probably the most important result of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume that 퐽 is self-adjoint and 퐺 ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 픟 is a barrier for 퐽 on 퐺, then for any 휆 ∈ 퐺  and any 휖 > 0 with ℓ휆(휖) ≥ 1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='26)  �8픟�휆, ℓ휆(휖)��−1 I ≤ Im푊(휆 + 푖휖)  and  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='27)  푠−(휆, 휖) ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+(휆, 휖),  where  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28)  푠±(휆, 휖) := 4푑픟�휆, ℓ휆(휖)� ±  ��  4푑픟�휆, ℓ휆(휖)��2  − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  30  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  We present the proof in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But now let us remark only that the square root in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='28)  is at least 15, since 푑 ≥ 1 and 픟(휆, 푡) ≥ 1 for any 푡 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, both 푠−(휆, 휖) and 푠+(휆, 휖) are  positive for the considered 휆 and 휖, hence both estimates in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='27) could be of significant importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Having Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12 and all the delicate relations between various objects connected somehow to 퐽  and described in the previous sections, we can finally formulate and prove the most important abstract  spectral result of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Denote for short the spectral matrix measure for 퐽 by 푀, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  푀 := 퐸퐽, �휑,  where �휑 = (휑1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 휑푘) is canonical cyclic system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) for 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Recall that some important results and  some notation related to the absolutely continuous and the singular part of 푀, such as Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6,  the set 퐿(푊) “with boundary limits for 푊” and the density 퐷 (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19)) of 푀ac on 퐿(푊) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' the  Lebesgue measure | · | are presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And some measure theory notions (including matrix  measures) are collected in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume that 퐽 is self-adjoint, 퐺 ⊂ Bor(R) and 픟 is a barrier for 퐽 on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 퐽 is  픟-nonsubordinate on 퐺, then  (a) 푀 is absolutely continuous on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (b) the density 퐷 of 푀ac is an invertible matrix at any 휆 ∈ 퐺 ∩ 퐿(푊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (c) 퐽 is absolutely continuous in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If, moreover, 퐽 is uniformly 픟-nonsubordinate on 퐺, then there exist 푐1, 푐2 > 0 such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='29)  푐1 I ≤ 퐷(휆) ≤ 푐2 I,  휆 ∈ 퐺 ∩ 퐿(푊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is 픟-nonsubordinate on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30)  lim inf  휖 →0+ ∥푊(휆 + 푖휖)∥ ≤ 8푑 lim inf  휖 →0+ 픟�휆, ℓ휆(휖)�,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By the definition of “lim inf” there exists a sequence (푡푘)푘 ∈N in [1, +∞) with lim푘→+∞ 푡푘 = +∞, such  that  lim inf  푡→+∞ 픟(휆, 푡) = lim  푘→+∞ 픟(휆, 푡푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11 for each 푘 we define  휖푘 := ℓ−1  휆 (푡푘)  and by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='25) we get  lim  푘→+∞ 휖푘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='21)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='31)  lim inf  휖 →0+ 픟�휆, ℓ휆(휖)� ≤ lim  푘→+∞ 픟�휆, ℓ휆(휖푘)� = lim  푘→+∞ 픟(휆, 푡푘) = lim inf  푡→+∞ 픟(휆, 푡) < +∞,  which by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30) implies  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='32)  lim inf  휖 →0+ ∥푊(휆 + 푖휖)∥ < +∞,  휆 ∈ 퐺,  and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16) this, in particular, yields17 퐺 ⊂ R\\푆sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now, using Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6(i), we obtain assertion (a):  푀 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Observe now that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='26) implies  1  8 lim inf 휖 →0+ 픟�휆, ℓ휆(휖)� ∥푣∥2 ≤ lim sup  휖 →0+  �� Im푊(휆 + 푖휖)�푣, 푣  �  ,  푣 ∈ C푑,  which by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='31) and 픟-nonsubordinacy yields  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33)  푐(휆) ∥푣∥2 ≤ lim sup  휖 →0+  �� Im푊(휆 + 푖휖)�푣, 푣  �  ,  푣 ∈ C푑,  where  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34)  푐(휆) =  �  8 lim inf  푡→+∞ 픟(휆, 푡)  �−1  > 0,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  17One can use here, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5(iv) to estimate:  tr � Im푊(휆 + 푖휖)� ≤ 푑  ��Im푊(휆 + 푖휖)��� ≤ 푑 ∥푊(휆 + 푖휖)∥, but it  follows also directly from the continuity of the norm, of the trace, and of the imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  31  Assume now that 휆 ∈ 퐺 ∩ 퐿(푊).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To get assertion (b) in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19) it is enough to prove that the  matrix Im �푊(휆 + 푖0)� is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But now  lim sup  휖 →0+ Im푊(휆 + 푖휖) = lim  휖 →0+ Im푊(휆 + 푖휖) = 푊(휆 + 푖0),  so, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='33),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='35)  푐(휆) ≤ Im푊(휆 + 푖0)  Therefore Im푊(휆 + 푖0) is invertible, by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34, and 퐺 ∩ 퐿(푊) ⊂ 푆ac,d ⊂ 푆ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8(ii) gives  clLe(퐺 ∩ 퐿(푊)) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  But clLe(퐺 ∩ 퐿(푊)) = clLe(퐺), and the conclusion (c) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In the uniform case, the LHS bound on 퐷(휆) follows directly from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='35) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And to get the  RHS estimate, it suffices to use the fact that the norm of a matrix gives the upper bound for the quadratic  form, and so it suffices to use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The proof of the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Before we turn to the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12 we need some  preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For fixed 휆 ∈ R and 휖 > 0 let us define  푈휖 := 푄(휆 + 푖휖) + 푃(휆 + 푖휖)푊(휆 + 푖휖)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='36)  푉휖 := 푄(휆) + 푃(휆)푊(휆 + 푖휖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37)  The following lemma is not new, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [35, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For the sake of completeness we include its  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐹 ∈ ℓ(N0, 푀푑(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The unique solution of the recurrence relation  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38)  퐴푛푆푛+1 + 퐵푛푆푛 + 퐴∗  푛−1푆푛−1 = 푧푆푛 + 퐹푛,  푛 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  with the initial conditions 푆−1 = 푆0 = 0 is equal to  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39)  푆푛 :=  푛−1  �  푘=0  �  푄푛(푧)�푃푘(푧)�∗ − 푃푛(푧)�푄푘 (푧)�∗�  퐹푘,  푛 ≥ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since all 퐴푘 are invertible it is clear that the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38) with given initial conditions 푆−1  and 푆0 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It remains to prove that the sequence given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39) satisfies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is immediate from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39) that 푆−1 = 푆0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, we get  푆1 =  �  푄1(푧)�푃0(푧)�∗ − 푃1(푧)�푄0(푧)�∗�  퐹0 = 퐴−1  0 퐹0  which is in agreement with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38) for 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So let us assume that 푛 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since both 푃(푧) and 푄(푧)  satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='22) we get  퐴푛푆푛+1 = 퐴푛  푛  �  푘=0  �  푄푛+1(푧)�푃푘(푧)�∗ − 푃푛+1(푧)�푄푘 (푧)�∗�  퐹푘  = (푧 I −퐵푛)  푛  �  푘=0  �  푄푛(푧)�푃푘(푧)�∗ − 푃푛(푧)�푄푘 (푧)�∗�  퐹푘  − 퐴∗  푛−1  푛  �  푘=0  �  푄푛−1(푧)�푃푘(푧)�∗ − 푃푛−1(푧)�푄푘 (푧)�∗�  퐹푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus,  퐴푛푆푛+1 = (푧 I −퐵푛)푆푛 − 퐴∗  푛−1푆푛−1 + ˜퐹푛,  where  ˜퐹푛 = (푧 I −퐵푛)  �  푄푛(푧)�푃푛(푧)�∗ − 푃푛(푧)�푄푛(푧)�∗�  퐹푛  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='40)  − 퐴∗  푛−1  �  푄푛−1(푧)�푃푛(푧)�∗ − 푃푛−1(푧)�푄푛(푧)�∗�  퐹푛  − 퐴∗  푛−1  �  푄푛−1(푧)�푃푛−1(푧)�∗ − 푃푛−1(푧)�푄푛−1(푧)�∗�  퐹푛−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  32  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  It remains to prove that ˜퐹푛 = 퐹푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To do so, let us apply (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51) for 푤 = 푧 with 푘 = 푛 and 푘 = 푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then  we get that on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='40) the first and the third lines are equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By considering  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='52) for 푤 = 푧 with 푘 = 푛 and taking the adjoint of both sides we get  �퐴−1  푛−1  �∗ =  �  푄푛(푧)�푃푛−1(푧)�∗ − 푃푛(푧)�푄푛−1(푧)�∗�∗  = 푃푛−1(푧)�푄푛(푧)�∗ − 푄푛−1(푧)�푃푛(푧)�∗,  which gives that the second line on the right-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='40) is equal to 퐹푛, and consequently, ˜퐹푛 = 퐹푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 휆 ∈ R define an operator 퐿 : ℓ(N0, 푀푑(C)) → ℓ(N0, 푀푑(C)) by the formula  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41)  (퐿퐹)푚 =  푚−1  �  푘=0  �  푄푚(휆)�푃푘(휆)�∗ − 푃푚(휆)�푄푘 (휆)�∗�  퐹푘,  푚 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then the sequences 푈휖 and 푉휖 defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='36) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37) satisfy for any 푋 ∈ 푀푑(C)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42)  (I −푖휖퐿)(푈휖 푋) = 푉휖 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Take 푋 ∈ 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that 푈휖 푋 satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38) with 푧 = 휆 and 퐹푛 = 푖휖(푈휖 )푛푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly,  푉휖 푋 satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='38) with 푧 = 휆 and 퐹푛 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since the initial conditions of 푉휖 푋 and 푈휖 푋 are the same  we get that the sequence 푆 := 푈휖 푋 − 푉휖 푋 can be expressed in the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, according to the  definition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41) we can write  푈휖 푋 − 푉휖 푋 = 푖휖퐿(푈휖 푋),  which is equivalent to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let the operator 퐿 : ℓ(N0, 푀푑(C)) → ℓ(N0, 푀푑(C)) be defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then for any  푡 ∈ [0, +∞)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43)  ∥퐿퐹∥[0,푡] ≤ 2 ∥푃(휆)∥ [0,푡] ∥푄(휆)∥[0,푡] ∥퐹∥[0,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41) for any 푚 ∈ N0 we have  ∥(퐿퐹)푚∥ ≤ ∥푄푚(휆)∥  푚−1  �  푘=0  ∥푃푘(휆)∥ ∥퐹푘 ∥  + ∥푃푚(휆)∥  푚−1  �  푘=0  ∥푄푘 (휆)∥ ∥퐹푘∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, by Cauchy–Schwarz inequality  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='44)  ∥(퐿퐹)푚∥ ≤  �  ∥푄푚(휆)∥ ∥푃(휆)∥[0,푚−1] + ∥푃푚(휆)∥ ∥푄(휆)∥ [0,푚−1]  �  ∥퐹∥[0,푚−1] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let 푡 ∈ [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then 푡 ∈ [푛, 푛 + 1) for some 푛 ∈ N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='44)  ∥(퐿퐹)푚∥ ≤ 푣푚 ∥퐹∥[0,푡] ,  푚 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푛 + 1,  where  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='45)  푣푚 := ∥푄푚(휆)∥ ∥푃(휆)∥ [0,푡] + ∥푃푚(휆)∥ ∥푄(휆)∥ [0,푡] ,  푚 = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푛 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently,  ∥퐿퐹∥[0,푡] =  �  푛  �  푚=0  ∥(퐿퐹)푚∥2 + {푡} ∥(퐿퐹)푛+1∥2  �1/2  ≤  �  푛  �  푚=0  푣2  푚 + {푡}푣2  푛+1  �1/2  ∥퐹∥[0,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='46)  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  33  Now, by triangle inequality in C푛+2 with Euclidean norm and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='45) we get  �  푛  �  푚=0  푣2  푚 + {푡}푣2  푛+1  �1/2  ≤ ∥푃(휆)∥[0,푡]  �  푛  �  푚=0  ∥푄푚(휆)∥2 + {푡} ∥푄푛+1(휆)∥2  �1/2  + ∥푄(휆)∥ [0,푡]  �  푛  �  푚=0  ∥푃푚(휆)∥2 + {푡} ∥푃푛+1(휆)∥2  �1/2  = 2 ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡]  which combined with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='46) gives (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푋 ∈ 푀푑(C) and 푡 ∈ [0, +∞) one has  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='47)  ∥푈휖 푋∥[0,푡] ≥ ∥푉휖 푋∥ [0,푡] − 2휖 ∥푃(휆)∥ [0,푡] ∥푄(휆)∥[0,푡] ∥푈휖 푋∥ [0,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, if ℓ휆 is J-L function, then  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='48)  2 ∥푈휖 푋∥[0,ℓ휆 (휖 )] ≥ ∥푉휖 푋∥[0,ℓ휆 (휖 )] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='42)  (I −푖휖퐿)(푈휖 푋) = 푉휖 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43) we get  ∥푉휖 푋∥ [0,푡] ≤ �1 + 2휖 ∥푃(휆)∥[0,푡] ∥푄(휆)∥[0,푡]  � ∥푈휖 푋∥[0,푡]  which implies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='48) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푣 ∈ C푑, 휆 ∈ R and any 휖 > 0 one has  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49)  1  휖  �� Im푊(휆 + 푖휖)�푣, 푣  �  = ∥푈휖 푣∥2  [0,+∞] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50)  ∥푈휖 ∥2  [0,+∞] ≤ tr � Im푊(휆 + 푖휖)�  휖  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푧 = 휆 + 푖휖 for some 휖 > 0 and 푣 ∈ C푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 and Fact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5 we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51)  푈(푧)푣 := 푄(푧)푣 + 푃(푧)푊(푧)푣 = (퐽 − 푧 I)−1훿0(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus,  (퐽 − 푧 I)�푈(푧)푣� = 훿0(푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By taking the scalar product of both sides with 푈(푧)푣 we get  �  푈(푧)푣, 훿0(푣)  �  =  �  푈(푧)푣, (퐽 − 푧 I)�푈(푧)푣��  =  �  푈(푧)푣, 퐽�푈(푧)푣��  − 푧 ∥푈(푧)푣∥2  [0,+∞]  Since 퐽 is self-adjoint by taking imaginary parts of both sides and using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51) we get  Im⟨푊(푧)푣, 푣⟩ = Im 푧 ∥푈(푧)푣∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently,  �� Im푊(푧)�푣, 푣  �  = Im 푧 ∥푈(푧)푣∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In view of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='36) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='51) we have 푈휖 = 푈(푧) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, apply (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49) for 푣 ∈ {푒1, 푒2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푒푑} and sum them up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then, since Im푊(푧) ≥ 0 for 푧 ∈ C+,  푑  �  푖=1  ∥푈휖 푒푖∥2 = 1  휖  푑  �  푖=1  �� Im푊(휆 + 푖휖)�푒푖, 푒푖  �  = 1  휖 tr � Im푊(휆 + 푖휖)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5(i) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='14) we get  푑  �  푖=1  ∥푈휖 푒푖∥2 =  +∞  �  푘=0  푑  �  푖=1  ∥(푈휖 푒푖)푘 ∥2 =  +∞  �  푘=0  ∥(푈휖 )푘 ∥2  HS ≥  ++∞  �  푘=0  ∥(푈휖 )푘 ∥2 = ∥푈휖 ∥2  [0,+∞] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence  ∥푈휖 ∥2  [0,+∞] ≤ 1  휖 tr � Im푊(휆 + 푖휖)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  34  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  from which the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  We are ready to prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17 applied to 푋 = I and by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50) we have  ∥푉휖 ∥[0,ℓ휆 (휖 )] ≤ 2 ∥푈휖 ∥ [0,ℓ휆(휖 )]  ≤ 2 ∥푈휖 ∥ [0,+∞]  ≤ 2  √휖  �  tr � Im푊(휆 + 푖휖)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='52)  Now, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5 (iv)  tr � Im푊(휆 + 푖휖)� ≤ 푑 ∥Im푊(휆 + 푖휖)∥ ≤ 푑 ∥푊(휆 + 푖휖)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='53)  ∥푉휖 ∥2  [0,ℓ휆 (휖 )] ≤ 4푑  휖 ∥푊(휆 + 푖휖)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  On the other hand, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23)  ∥푉휖 ∥2  [0,ℓ휆 (휖 )] =  ∥푉휖 ∥[0,ℓ휆 (휖 )]  ∥푃(휆)∥[0,ℓ휆 (휖 )]  ∥푉휖 ∥ [0,ℓ휆(휖 )]  ∥푄(휆)∥ [0,ℓ휆(휖 )]  ∥푃(휆)∥ [0,ℓ휆(휖 )] ∥푄(휆)∥ [0,ℓ휆(휖 )]  ≥  1  픟(휆, ℓ휆(휖))  �1 + ∥푊(휆 + 푖휖)∥2 � 1  2휖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, by combining it with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='53) we obtain  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='54)  1 + ∥푊(휆 + 푖휖)∥2 ≤ 8푑픟(휆, ℓ휆(휖)) ∥푊(휆 + 푖휖)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  This inequality is equivalent to  푠2 − 8푑픟�휆, ℓ휆(휖)�푠 + 1 ≤ 0,  where 푠 = ∥푊(휆 + 푖휖)∥ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Its discriminant is equal to  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='55)  Δ =  �  8푑픟�휆, ℓ휆(휖)��2  − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Since 푑 ≥ 1 and 픟�휆, ℓ휆(휖)� ≥ 1, we have  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='56)  Δ ≥ 82 − 4 = 60 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus  푠2 − 8푑픟�휆, ℓ휆(휖)�푠 + 1 = (푠 − 푠−)(푠 − 푠+) ≤ 0,  where  푠− = 8푑픟�휆, ℓ휆(휖)� −  √  Δ  2  and  푠+ = 8푑픟�휆, ℓ휆(휖)� +  √  Δ  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='56) we see that 푠−, 푠+ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='55) we have 푠−, 푠+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus,  푠− ≤ ∥푊(휆 + 푖휖)∥ ≤ 푠+  and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='27) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='26) is even simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, let 푣 ∈ C푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='17 applied to 푋 = 퐸 푣  (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='15)), Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5 and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='49)  ∥푉휖 푣∥2  [0,ℓ휆 (휖 )] ≤ 4 ∥푈휖 푣∥2  [0,ℓ휆(휖 )] = 4  휖  �� Im푊(휆 + 푖휖)�푣, 푣  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23)  ∥푉휖 푣∥2  [0,ℓ휆(휖 )] = ∥푉휖 (퐸 푣)∥2  [0,ℓ휆(휖 )]  =  ∥푉휖 (퐸 푣)∥ [0,ℓ휆(휖 )]  ∥푃(휆)∥[0,ℓ휆 (휖 )]  ∥푉휖 (퐸 푣)∥ [0,ℓ휆(휖 )]  ∥푄(휆)∥[0,ℓ휆 (휖 )]  ∥푃(휆)∥[0,ℓ휆 (휖 )] ∥푄(휆)∥ [0,ℓ휆(휖 )]  ≥  1  픟�휆, ℓ휆(휖)� ∥푣∥2 1  2휖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  35  Therefore,  1  픟�휆, ℓ휆(휖)� ∥푣∥2 ≤ 8  �� Im푊(휆 + 푖휖)�푣, 푣  �  from which (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='26) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Sufficient conditions for absolute continuity  In this section we extend some well-known conditions implying nonsubordinacy from 푑 = 1 to the  general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In particular, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 we cover Generalized Last–Simon condition, in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2  Generalized Behncke–Stolz condition and in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 the homogenous class condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' GLS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that Carleman’s condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then the sequence  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  휌푛 :=  푛−1  �  푘=0  1  ∥퐴푘 ∥  is divergent and 퐽 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let 퐺 ⊂ R be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We say that Jacobi matrix 퐽 satisfies Generalized Last–Simon (GLS in  short) condition on 퐺 if  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  lim inf  푛→+∞  1  휌푛  푛  �  푘=1  ∥푅푘 (휆)∥2 < +∞,  휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Moreover, if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) is finite even after taking the supremum over 휆 ∈ 퐺, we say that 퐽 satisfies uniform  GLS condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  GLS condition has been introduced in [37] for scalar Jacobi matrices with 퐴푛 ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It was shown  in [37, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1] that the set of 휆 ∈ R where (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) is fulfilled is a minimal support of the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part of the  spectral measure of 퐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly, a variant of it has been studied for bounded block Jacobi matrices in [48]  with a similar conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus this condition seems to be the right extension to possibly unbounded  Jacobi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' However, we would like to stress that in the unbounded case the set where (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) is  satisfied might no longer be a support of the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part of the spectral measure of 퐽 even for 푑 = 1, see  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Below, inTheorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2, we will show that (uniform) GLS conditionimplies(uniform) 픟TR-nonsubordinacy  on 퐺 for the barrier (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We need the following observation, whose proof is an adaptation of the reasoning from [47, Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 휆 ∈ R and any 푛 ∈ N we have  ∥푅(휆)∥2  [1,푛]  ⇃|푅(휆)|⇂2  [1,푛]  ≤  �  1  휌푛  푛  �  푘=1  ∥푅푘(휆)∥2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' From the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='50) we get  ��푅−1  푘 (휆)  �� ≤ ∥퐴푘−1∥ ∥푅푘 (휆)∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2),  ⇃|푅푘(휆)|⇂2=  1  ��푅−1  푘 (휆)  ��2 ≥  1  ∥퐴푘−1∥2 ∥푅푘 (휆)∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  휌푛  �푛  푘=1 ⇃|푅푘(푥)|⇂2 ≤  휌푛  �푛  푘=1  1  ∥ 퐴푘−1 ∥  1  ∥퐴푘−1 ∥ ∥푅푘 (휆) ∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Since the weighted harmonic mean is always not greater than weighed arithmetic mean (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [39,  Theorem 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='76]) we get  휌푛  �푛  푘=1  1  ∥퐴푘 ∥  1  ∥퐴푘−1 ∥ ∥푅푘 (휆) ∥2  ≤ 1  휌푛  푛  �  푘=1  1  ∥퐴푘−1∥ ∥퐴푘−1∥ ∥푅푘(휆)∥2 = 1  휌푛  푛  �  푘=1  ∥푅푘(휆)∥2,  which combined with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) ends the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  36  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐺 ∈ Bor(R) be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 satisfies (uniform) GLS condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, 퐽 is absolutely continuous  in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9 the function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20) is a barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Since by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 we have  lim inf  푡→+∞ 픟(휆, 푡) < +∞  the operator 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, the result follows from  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' GBS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that Carleman’s condition is satisfied and let 퐺 ⊂ R be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We  say that 퐽 satisfies Generalized Behncke—Stolz (GBS in short) on 퐺 if  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  lim sup  푛→+∞  1  휌푛  푛  �  푘=1  ∥푅푘(휆)∥2 < +∞,  휆 ∈ 퐺,  where 휌푛 is defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Similarly, as in GLS condition, if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) is finite even after taking the  supremum over 휆 ∈ 퐺, we say that 퐽 satisfies uniform GBS condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  GBS condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) has been introduced in [23, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3] to study spectral properties of scalar  unbounded Jacobi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This condition turned out to be very convenient in the study of various  classes of unbounded Jacobi matrices, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [21,23,24,29,40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We obviously have that (uniform) GBS condition implies (uniform) GLS condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So the following  result follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐺 ∈ Bor(R) be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 satisfies (uniform) GBS condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then 퐽 satisfies (uniform) 픟TR-nonsubordinacy condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, 퐽 is absolutely continuous  in 퐺 and clLe(퐺) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' H class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For the arbitrary size 푚×푚 of matrices this class is given as follows (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', [43, Definition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let (퐶푛)푛≥푛0 be a sequence of complex 푚 × 푚 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We say that (퐶푛)푛≥푛0 ∈ 퐻 if  there is a constant 푐 > 0 such that for any 푛 ≥ 푛0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  ∥퐶푛 · 퐶푛−1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' · 퐶푛0∥ ≤ 푐  푛  �  푘=푛0  | det 퐶푘|1/푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The following result, being “the sum” of [43, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7 (see also Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)]  explains the importance of this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let (퐶푛)푛≥푛0 be a sequence of complex invertible 푚 × 푚 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푛 ≥ 푛0  define  푅푛 := 퐶푛 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' · 퐶푛0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then TFCAE:  (퐶푛)푛≥푛0 ∈ 퐻,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  ∥푅푛∥ ≍푛 | det 푅푛|1/푚,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  ∥푅푛∥ ≍푛 ⇃|푅푛|⇂,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  ∀푣,푤 ∈C푚\\{0}  ∥푅푛푣∥ ≍푛 ∥푅푛푤∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  The concept of 퐻 class has been introduced in [40] (see also [21, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8]) as a sufficient condition  for GBS for scalar Jacobi matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' It turned out that sometimes this stronger condition is somewhat  easier to show than GBS, see [22, 40–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Later, in [43] this notion has been extended to block Jacobi  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us begin with an observation than under a condition, which is automatically satisfied for 푑 = 1,  indeed 퐻 class implies GBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  37  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that Carleman’s condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If �푇푛(휆)�  푛∈N0 ∈ 퐻 for some 휆 ∈ R and  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  lim inf  푛→+∞  ���� det  � 퐴푛  ∥퐴푛∥  ����� > 0,  then 퐽 satisfies GBS condition on {휆}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If �푅푛(휆)�  푛∈N ∈ 퐻, then by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) there exists a constant 푐 > 0 such that for any 푘 ≥ 1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  ∥푅푘(휆)∥2 ≤ 푐  �� det 푅푘(휆)  ��1/푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='41), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='23) we have  �� det 푅푘(휆)  �� =  1  | det 퐴0|  푘−1  �  푗=1  | det 퐴∗  푗−1|  | det 퐴 푗|  =  1  | det 퐴푘−1| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Using this we can rewrite (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11) in the form  ∥푅푘(휆)∥2 ≤ 푐  1  ∥퐴푘−1∥  ���� det  � 퐴푘−1  ∥퐴푘−1∥  �����  −1/푑  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, if (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) is satisfied, then there exists a constant 푐′ > 0 such that for any 푘 ≥ 1  ∥푅푘(휆)∥2 ≤ 푐′  1  ∥퐴푘−1∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now by summing it for 푘 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푛 it simply implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4), what we needed to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  In the following we show that in general 퐻 class implies 픟TR-nonsubordinacy condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', with the  transfer matrix barrier (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐺 ∈ Bor(R) be non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that �푇푛(휆)�  푛∈N0 ∈  퐻 for any 휆 ∈ 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then 퐽 satisfies 픟TR-nonsubordinacy condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, 퐽 is absolutely  continuous in 퐺 and clLe(퐺) ⊂ 휎(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) there exists a constant 푐(휆) > 0 such that for any 푘 ≥ 1 we have  푐(휆) ∥푅푘(휆)∥2 ≤⇃|푅푘(휆)|⇂2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, for any 푡 ≥ 1 we get  푐(휆) ∥푅(휆)∥2  [1,푡] ≤⇃|푅(휆)|⇂2  [1,푡] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In other words,  ∥푅(휆)∥2  [1,푡]  ⇃|푅(휆)|⇂2  [1,푡]  ≤  1  푐(휆)  Thus  lim sup  푡→+∞ 픟TR(휆, 푡) ≤  8  푐(휆) < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, the operator 퐽 satisfies 픟TR-nonsubordinacy condition on 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, the result follows from  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Examples and applications  In this section we show some examples and counterexamples illustrating the applicability of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Vector nonsubordinacy does not characterise invertibility of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The following example demon-  strates that one cannot hope for the full characterisation of the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' spectrum of block Jacobi matrices in  terms of vector nonsubordinacy condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us recall that by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 vector nonsubordinacy  is equivalent to matrix nonsubordinacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This shows that inevitably Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1 provides only sufficient  conditions for the absolute continuity of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  38  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푑 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Define  퐴푛 =  �  푎(1)  푛  0  0  푎(2)  푛  �  ,  퐵푛 =  �  푏(1)  푛  0  0  푏(2)  푛  �  ,  where 푎(푖) = �푎(푖)  푛  �  푛∈N0, 푏(푖) = �푎(푖)  푛  �  푛∈N0 are Jacobi parameters for 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐽 (푖) be the Jacobi  operator associated to 푎(푖), 푏(푖) for 푖 = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Observe that  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  퐽 � 퐽 (1) ⊕ 퐽 (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us take  푎(푖)  푛  = (푛 + 1) 훼푖,  푏(푖)  푛  = 0,  푛 ≥ 0, 푖 = 1, 2,  where 훼1, 훼2 ∈ (0, 1] and 훼1 > 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then by [59, Theorem 1] the corresponding measures 휇(1), 휇(2) are  absolutely continuous on R with continuous positive densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' By (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  푀(·) �  �  휇(1) (·)  0  0  휇(2) (·)  �  ,  and consequently, 푀(퐵) is invertible for any non-empty open 퐵 ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We are going to show that 퐽  does not satisfy vector nonsubordinacy condition for any 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Fix 휆 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푢(푖) ∈ GEV−1  �퐽 (푖), 휆�  for 푖 = 1, 2 be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Set 푢 := �푢(1)  푛 푒1  �  푛∈N−1 and 푣 := �푢(2)  푛 푒2  �  푛∈N−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then 푢, 푣 ∈ GEV−1 (퐽, 휆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  According to [62, Theorem A]  ∥푢∥2  [0,푛]  �푛  푘=0  1  푎(1)  푛  ≍푛 1,  ∥푣∥2  [0,푛]  �푛  푘=0  1  푎(2)  푛  ≍푛 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus,  ∥푢∥2  [0,푛]  ∥푣∥2  [0,푛]  ≍푛  �푛  푘=0  1  푎(2)  푛  �푛  푘=0  1  푎(1)  푛  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By Stolz–Cesàro lemma  lim  푛→+∞  �푛  푘=0  1  푎(2)  푛  �푛  푘=0  1  푎(1)  푛  = lim  푛→+∞  푎(1)  푛  푎(2)  푛  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Therefore,  lim  푛→+∞  ∥푢∥2  [0,푛]  ∥푣∥2  [0,푛]  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In view of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 it implies  lim  푡→+∞  ∥푢∥2  [0,푡]  ∥푣∥2  [0,푡]  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, 퐽 does not satisfy vector nonsubordinacy condition for 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' GLS does not characterise the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In the following example we show that even for  푑 = 1 the set where (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) is satisfied might not be a support of the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푑 = 1 and 훼 > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider Jacobi parameters  퐴푛 =  �  (푛 + 1)(푛 + 1 + 훼),  퐵푛 = 2푛 + 1 + 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then according to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='43) we have  푅푛(휆) =  �  1  퐴0 ℓ(훼+1)  푛−2  (휆)  ℓ(훼)  푛−1(휆)  1  퐴0 ℓ(훼+1)  푛−1  (휆)  ℓ(훼)  푛  (휆)  �  ,  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  39  where �ℓ(훼)  푛  �  푛∈N0 is the sequence of orthonormal Laguerre polynomials of the order 훼, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [34,  formula (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consequently, we have 휎ac(퐽) = [0, +∞) and we are going to show that GLS  condition is not satisfied for any 휆 ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Obviously we have  ∥푅푛(휆)∥ ≥  ����푅푛(휆)  �0  1  ����� =  �����  �  ℓ(훼)  푛−1(휆)  ℓ(훼)  푛  (휆)  ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence,  푛  �  푘=1  ∥푅푘(휆)∥2 ≥  푛  �  푘=1  �����  �  ℓ(훼)  푘−1(휆)  ℓ(훼)  푘  (휆)  ������  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us set  퐾푛(푥, 푦) =  푛  �  푘=0  ℓ(훼)  푘  (푥)ℓ(훼)  푘  (푦),  ˜휌푛 =  푛  �  푘=0  1  √퐴푘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then  1  ˜휌푛  푛  �  푘=1  ∥푅푘(휆)∥2 ≥ 1  ˜휌푛  퐾푛(푥, 푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence according to [64, Theorem B] the right-hand side of this inequality is bounded and positive for  any 휆 ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence, for some 푐 > 0  1  ˜휌푛  푛  �  푘=1  ∥푅푘 (휆)∥2 ≥ 푐,  but by Stolz–Cesàro lemma  lim  푛→+∞  ˜휌푛  휌푛  = lim  푛→+∞  1  √퐴푛  1  퐴푛  = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It implies that for any 휆 ∈ (0, +∞)  lim  푛→+∞  1  휌푛  푛  �  푘=1  ∥푅푘 (휆)∥2 = +∞,  and consequently, the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Application to some classes of block Jacobi operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' In this section we would like to illustrate  our results by showing that Jacobi matrices considered in [61, Theorem 2] are in fact absolutely continuous  in some explicit region of the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let us introduce a necessary notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푋 = (푋푛)푛∈N be a sequence from a normed space 푉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let  푁 ≥ 1 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We say that 푋 ∈ D푁  1 if  +∞  �  푛=1  ∥푋푛+푁 − 푋푛∥ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Observe that if 푋 ∈ D푁  1 , then for any 푖 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푁 − 1} the sequence (푋푘 푁 +푖)푘 ∈N satisfies Cauchy  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐽 be a block Jacobi matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that Jacobi parameters of 퐽 satisfy for some  integer 푁 ≥ 1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  �퐴−1  푛  �  푛∈N, �퐴−1  푛 퐵푛  �  푛∈N, �퐴−1  푛 퐴∗  푛−1  �  푛∈N ∈ D푁  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then for any 푖 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푁 − 1} and any 푧 ∈ C the limit  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  X푖(푧) :=  lim  푛→+∞  푛≡푖 mod 푁  푇푛+푁 −1(푧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푇푛+1(푧)푇푛(푧)  exists, where 푇푛 is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose further that for some 푁-periodic sequence of invertible  matrices (C푛)푛∈N0  lim  푛→+∞  ����  푎푛  ∥푎푛∥ − C푛  ���� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  40  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  Define  Λ =  �  휆 ∈ R : Re  � �  0  −C푁 −1  C∗  푁 −1  0  �  X0(휆)  �  is strictly positive or strictly negative on C2푑  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then �푅푛(휆)�  푛∈N ∈ 퐻 for any 휆 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consequently, if Carleman’s condition is satisfied, then 퐽 is  absolutely continuous in Λ and cl(Λ) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' First of all, let us observe that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2) implies that for any 푗 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푁 − 1} the sequence  (푇푘 푁 +푗)푘 ∈N is convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus the limit (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We are going to show that the condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us recall that according to [61, Theorem  2] for any compact interval 퐾 ⊂ Λ there are constants 푐1 > 0, 푐2 > 0 such that for any normalized  푢 ∈ GEV−1 (휆), where 휆 ∈ 퐾, and any 푛 ≥ 1 we have  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  푐1  ∥퐴푛∥ ≤  ����  �  푢푛−1  푢푛  �����  2  ≤  푐2  ∥퐴푛∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Since, for any 푢 ∈ GEV−1 (휆)  �  푢푛−1  푢푛  �  = 푅푛(휆)  �  푢−1  푢0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Hence, having the notation above in mind we have  ⇃|푅푛(휆)|⇂2=  inf  ∥푢−1 ∥2+∥푢0 ∥2=1  푢∈GEV−1(휆)  ����  �  푢푛−1  푢푛  �����  2  ,  ∥푅푛(휆)∥2 =  sup  ∥푢−1 ∥2+∥푢0 ∥2=1  푢∈GEV−1(휆)  ����  �  푢푛−1  푢푛  �����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4) implies  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  푐1  ∥퐴푛∥ ≤⇃|푅푛(휆)|⇂2≤ ∥푅푛(휆)∥2 ≤  푐2  ∥퐴푛∥,  which shows that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Therefore, by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5, �푅푛(휆)�  푛∈N ∈ 퐻 for any 휆 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Finally,  if Carleman’s condition is satisfied, then the remaining conclusion follows from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us observe that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) immediately implies that 퐽 satisfies uniform GBS condition for  any compact interval 퐾 ⊂ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Asymptotically periodic Jacobi parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푁 be a positive integer andlet (A푛)푛∈N0, (B푛)푛∈N0  be 푁-periodic Jacobi parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For any 푖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푁} let us define  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  픛푖(푧) =  푁 +푖−1  �  푗=푖  픗푖(푥)  where  픗푖(푧) =  �  0  I  −A−1  푗 A∗  푗−1  A−1  푗 (푧 I −B푗)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let 퐽per be the Jacobi operator associated with the sequences (A푛)푛∈N0, (B푛)푛∈N0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Spectral properties of the operator 퐽per are well-known in the case 푑 = 1, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [55, Chapter  5], [67, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 푑 > 1 we have a good understanding of 퐽per when its Jacobi parameters are all  self-adjoint and 푁 = 1, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [69,70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then we have  휎(퐽per) =  �  푡 ∈[−2,2]  휎(A0푡 + B)  and the spectrum is absolutely continuous, see [70, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' When the Jacobi parameters are not  self-adjoint or 푁 > 1 a point spectrum is possible, see [26] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Some results concerning  absolute continuity might be extracted from [51, Section 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is natural to consider compact perturbations of 퐽per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Namely, we say that 퐽 has asymptotically  푁-periodic Jacobi parameters if  lim  푛→+∞ ∥퐴푛 − A푛∥ = 0  and  lim  푛→+∞ ∥퐵푛 − B푛∥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By Weyl’s perturbation theorem we have  휎ess(퐽) = 휎ess(퐽per).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Our Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 leads to  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  41  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푁 be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Suppose that 퐽 has asymptotically 푁-periodic Jacobi  parameters satisfying  (퐴푛)푛∈N0, (퐵푛)푛∈N0 ∈ D푁  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Define  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  Λ =  �  푥 ∈ R : Re  � �  0  −A푁 −1  A∗  푁 −1  0  �  픛푁 (푥)  �  is strictly positive or strictly negative on C푑  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Then 퐽 is absolutely continuous in Λ and cl(Λ) ⊂ 휎ac(퐽).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Periodic modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 퐽 be a block Jacobi matrix and let 푁 ≥ 1 be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We shall call 퐽  a block Jacobi matrix with 푁-periodically modulated entries if there exists 푁-periodic Jacobi parameters  (A푛)푛∈N0, (B푛)푛∈N0 such that the Jacobi parameters of 퐽 satisfy  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  lim  푛→+∞  ��퐴−1  푛  �� = 0,  lim  푛→+∞  ��퐴−1  푛 퐴∗  푛−1 − A−1  푛 A∗  푛−1  �� = 0  and  lim  푛→+∞  ��퐴−1  푛 퐵푛 − A−1  푛 B푛  �� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In the scalar case, this class has been introduced in [25] and it is actively studied ever since, see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [5, 9, 45, 49, 57, 60, 63, 65, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let us observe that given any positive scalar sequence (푐푛)푛∈N0  satisfying  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  lim  푛→+∞ 푐푛 = +∞  and  lim  푛→+∞  푐푛−1  푐푛  = 1  leads to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8) for Jacobi parameters  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  퐴푛 = 푐푛A푛,  퐵푛 = 푐푛B푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Therefore, the class of block Jacobi matrices with periodically modulated entries might be thought of as  a certain perturbation of the model example (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)–(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Observe that due to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='37) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6) we have for any 푖 ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푁} and 푧 ∈ C  lim  푘→+∞푇푘 푁 +푖(푧) = 픗푖(0),  thus in view of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6),  X0(푧) = lim  푘→+∞푇푗푁 +푁 −1(푧) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='푇푗푁 (푧) = 픛푁 (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Therefore, our Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3 leads to  Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Suppose that 퐽 is a block Jacobi matrix with 푁-periodically modulated entries for some  푁 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Assume the Carleman’s condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7) and  �퐴−1  푛 퐴∗  푛−1  �  푛∈N, �퐴−1  푛 퐵푛  �  푛∈N, �퐴−1  푛  �  푛∈N ∈ D푁  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  If 0 ∈ Λ, where Λ is defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7), then 퐽 is absolutely continuous in R and 휎ac(퐽) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Vector and matrix measures — selected basic notions  For self-consistency and some self-sufficiency of the paper we collect here selected definitions of some  basic notions and some properties related to matrix measures and — more generally — vector measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We omit here the more sophisticated construction of the appropriate 퐿2-type space for the matrix measure,  referring the reader to the literature (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', [68, Section 8] or [44]18)  We start from the general definition of vector measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Consider a set Ω with 픐 — a 휎-algebra of subsets of Ω and a certain norm space 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Let 푉 : 픐 −→ 푋  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푉 is a vector measure (in 푋) iff 푉 is countably additive in the norm sense in 푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Now, let us consider a special case 푋 := 푀푑(C) for some 푑 ∈ N (with a standard norm, say).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And let  푀 : 픐 −→ 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푀 is a (푑 × 푑) matrix measure iff  (a) 푀 is a vector measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (b) 푀(휔) ≥ 0 for any 휔 ∈ 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='19  18The second position contains the detailed definition of the Hilbert space 퐿2(푀) for matrix measures and the details of the  abstract spectral theory for finitely cyclic s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' operators, based on the matrix measure approach and multiplication by a function  operators in 퐿2(푀) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  19So, in particular 푀(휔) is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='.  42  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  So, in particular, each matrix measure is a vector measure, but despite its name and due to the extra  non-negativity property (b), matrix measure is “much more” than a vector measure in 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The above Ω, 픐 and 푑 are “fixed” below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We have to precise here some terminology (choosing it from various versions in literature) and to recall  several basic facts related to vector measures, matrix measures and measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Suppose that 휈 is a measure on 픐 and 푉 : 픐 −→ 푋 is a vector measure, where 푋 is a norm space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In several cases below we will also need to assume additionally that 푋 = C푘 for some 푘, including  possible obvious identifications, as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 푀푑(C) ≡ C(푑2), to provide the clear and standard sense of the  integral and of L1  푋 (휈) functions (see the appropriate part of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For any 퐺 ∈ 픐 denote 픐퐺 := {휔 ∈ 픐 : 휔 ⊂ 퐺}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Surely 픐퐺 is a 휎-algebra of subsets of 퐺 and  푉↾픐퐺 is a vector measure on 픐퐺 ("on 퐺").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We denote it by 푉퐺, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푉퐺 := 푉↾픐퐺,  but we also call it the restriction of 푉 to 퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Analogous situation (and notation, and terminology) is well  known and valid here for measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Suppose that 퐻 : Ω −→ 푋 = C푘 is a measurable function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If, moreover, 퐻 ∈ L1  푋 (휈), then  we define a new function from 픐 into 푋 by the formula:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1)  ∫  휔  퐻 d휈,  휔 ∈ 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  It is obviously a vector measure, and we denote it by  퐻 d휈,  analogously as in the case of measures (when 퐻 should be a scalar non-negative measurable function,  instead of 퐻 ∈ L1  푋 (휈)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' But in our main case 푋 = 푀푑(C)  (identified with C(푑2)) the situation is  somewhat similar and one easily checks the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If 퐻 ∈ L1  푋 (휈) and 퐻(푡) ≥ 0 for 휈-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푡 ∈ Ω, then the vector measure 퐻 d휈 is a matrix  measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If a vector measure 푉 is such, that 푉 = 퐻 d휈 with some 퐻 ∈ L1  푋 (휈), then we call 퐻 the  density20 of 푉 with respect to 휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Let 퐺 ∈ 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  the density on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : The density of 푉 on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 means: any density of 푉퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈퐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  a support: 퐺 is a support of 푉 iff 푉Ω\\퐺 is the zero vector measure (on 픐Ω\\퐺)21;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  a minimal support w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : 퐺 is a minimal support of 푉 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 iff 퐺 is a support of 푉 and for any  support 퐺′ of 푉 included in 퐺  휈(퐺 \\ 퐺′) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : 푉 is absolutely continuous (abbrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 iff for any 휔 ∈ 픐 if 휈(휔) = 0, then  푉(휔) = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : 푉 is singular (abbrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 iff there exists such a support 푆 ∈ 픐 of 푉 that  휈(푆) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='/sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : If 푉1,푉2 : 픐 −→ 푋 are two vector measures, such that  (i) 푉1 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 and 푉1 is sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈,  (ii) 푉 = 푉1 + 푉2,  then we call 푉1 the a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part of 푉 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 and we denote it by  푉ac,휈,  and we call 푉2 the sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' part of 푉 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈, and we denote it by  푉sing,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  20However, it can be not unique, as a function from L1  푋 (휈).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  21 Generally, it is not sufficient here (contrary to measures) that 푉 (Ω \\ 퐺) = 0 because the property of having zero measure  is not inheritable into measurable subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' However, for matrix measures it is sufficient by non-negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  43  Note here, that above two notions are well(uniquely)-defined, since the above decomposition, if  exists, is unique22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : 푉 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 iff 푉퐺 is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈퐺;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' : 푉 is sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' on 퐺 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 iff 푉퐺 is sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈퐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  All adopt all the above definitions and names also for any measure 휇 instead of a vector measure 푉  (recall that measure can be not a vector measure) just by interchanging the symbols 휇 and 푉, including  the notation  휇ac,휈,  휇sing,휈,  however they are mostly commonly known in the case of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We adopt here also the convention, that the part “w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈”, as well as “, 휈” in the appropriate symbols,  as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 푉ac,휈, 푉sing,휈, can be omitted to shorten the notation to, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  푉ac, 푉sing, 휇ac, 휇sing  etc, in the case when Ω = R, 픐 = Bor(R) and 휈 is the Lebesgue measure | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  For a 푑 × 푑 matrix measure 푀 and 푖, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑 let us define 푀푖 푗 : 픐 → C by  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2)  푀푖 푗(휔) := �푀(휔)�  푖 푗 ,  휔 ∈ 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By the point (a) of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='2, each of the 푀푖 푗 is a complex measure on 픐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, non-negativity  from (b) of a matrix means also its self-adjointness, so we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3)  푀 푗,푖 = 푀푖, 푗  푖, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By non-negativity, defining tr푀 : 픐 → C by the formula  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4)  tr푀 (휔) := tr(푀(휔)),  휔 ∈ 픐,  we get in fact a finite measure tr푀, called trace measure of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' This “classical” measure is much simpler  mathematical object than the matrix measure 푀, but it contains a lot of important information on 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  The results below are proved in [44, Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' For 푀 being a matrix measure as above:  (i) 푀, as well as each 푀푖 푗 for 푖, 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푑, are absolutely continuous with respect to tr푀;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (ii)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5)  푀(휔) = 0 ⇐⇒ tr푀 (휔) = 0,  휔 ∈ 픐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iii)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6)  0 ≤ 푀(휔) ≤ tr푀 (휔) I  휔 ∈ 픐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  (iv) There exists a density 퐷 : Ω −→ 푀푑(C) of 푀 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' tr푀, such that for any 푡 ∈ Ω  0 ≤ 퐷(푡) ≤ I , 23  푡 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Note that each density 퐷 of 푀 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' tr푀 is determined only up to tr푀-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' equality, and each one is  called the trace density of 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The set of all the densities of 푀 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' tr푀 is denoted here by D푀, and by  D•  푀 we denote the set of those 퐷, which satisfy conditions from the point (iv) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  By the definition of density we have  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7)  푀(휔) =  ∫  휔  퐷 d tr푀,  휔 ∈ 픐, 퐷 ∈ D푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  We assume here, to the end if this section, that 푀 : 픐 −→ 푀푑(C) is a matrix measure and  휈 : 픐 −→ [0, +∞] is a 휎-finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  There exist several ways to get a decomposition of some vector measures 푉 into its a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  parts w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' a measure 휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' All they are based somehow on the Lebesgue–Radon–Nikodym Theorem (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' [50]) for a complex measure “w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' a 휎-finite measure” version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' We need it only for our matrix  measure 푀 and it will be convenient to make it via the appropriate decomposition of tr푀 onto parts  22Because any linear combination of vector measures being a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 is also a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' (sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=') w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and a vector  measure which is both a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 is the zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  23In particular, 퐷(푡) is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  44  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  (tr푀)ac,휈 and (tr푀)sing,휈, which exist by the Lebesgue–Radon–Nikodym Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, we just consider  any 퐷 ∈ D푀 and two matrix measures:  퐷 d(tr푀)ac,휈,  퐷 d(tr푀)sing,휈  (see notation (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1))  Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and the sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' parts of 푀 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈 exist, and they satisfy  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8)  푀ac,휈 = 퐷 d(tr푀)ac,휈 ,  푀sing,휈 = 퐷 d(tr푀)sing,휈 ,  where 퐷 is an arbitrary density from D푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, both 푀ac,휈, 푀sing,휈 are matrix measures24, and  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9)  tr푀ac,휈 = (tr푀)ac,휈,  tr푀sing,휈 = (tr푀)sing,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  In particular, if 푆 ∈ 픐, then TFCAE:  푆 is a support (version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=': minimal support w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈) of 푀ac,휈 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푆 is a support (version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=': minimal support w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈) of tr푀ac,휈 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푆 is a support (version 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=': minimal support w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈) of (tr푀)ac,휈 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  and analogically for 푀sing,휈, tr푀sing,휈, (tr푀)sing,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Using “the short theory” presented above, one immediately checks that the pair of vector measures  퐷푑(tr푀)ac,휈, 퐷푑(tr푀)sing,휈 satisfies the conditions from Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='4 of the parts 푀ac,휈 푀sing,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So,  by the uniqueness of the decomposition, we get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Now, by the non-negativity of 퐷 and by Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='3  we see that 푀ac,휈, 푀sing,휈 are matrix measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  To get the assertion (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9), observe that the equality  푀 = 푀ac,휈 + 푀sing,휈  yields  tr푀 =  tr푀ac,휈 + tr푀sing,휈  by the definition of the trace maesure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  But 푀ac,휈, 푀sing,휈 are a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  or, respec-  tively, sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈, hence also tr푀ac,휈, tr푀sing,휈 are a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' or, respectively, sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈, just by the use of  the property (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) for both those matrix measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, we get the result just by the definitions of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' and  sing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' And the last part follows directly from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='9), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5) and by the observation that both notions:  of support, as well as of minimal support w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' a measure, are determined by zero vector measure sets,  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  Now we turn to a “technical” result concerning the notion of the minimal support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider a matrix measure 푀 : 픐 −→ 푀푑(C), a measure 휈 : 픐 −→ [0, +∞], sets  푆푎, 푆푠 ∈ 픐 and a non-negative function 퐹 : 푆푎 −→ 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' If  (i)  푆푎 is a support of 푀ac,휈 and 푆푠 is a support of 푀sing,휈,  (ii)  푆푎 ∩ 푆푠 = ∅,  (iii) 퐹 is a density of 푀ac,휈 on 푆푎 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈,  (iv)  휈 ({푡 ∈ 푆푎 : 퐹(푡) = 0}) = 0,  then 푆푠 is a support of (푡푟푀)sing,휈 and 푆푎 is a minimal support of (푡푟푀)ac,휈 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' From Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6 we immediately see that 푆푠 is a support of (tr푀)sing,휈 and 푆푎 is support of  (tr푀)ac,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' To prove the minimality, consider any 푆′ ⊂ 푆푎 which is also a support of (tr푀)ac,휈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hence,  again by Fact A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='6  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10)  0 = (tr푀)ac,휈(푆푎 \\ 푆′) = tr푀ac,휈 (푆푎 \\ 푆′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  On the other hand, by the assumption (iii), using (푆푎 \\ 푆′) ⊂ 푆푎 and the non-negativity of tr 퐹(푡) for any  푡 ∈ 푆푎, we have  tr푀ac,휈 (푆푎 \\ 푆′) = tr �푀ac,휈(푆푎 \\ 푆′)� = tr  �∫  (푆푎\\푆′)  퐹 d휈  �  =  ∫  (푆푎\\푆′)  tr 퐹(푡) d휈(푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='10) we have tr 퐹(푡) = 0 for 휈-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  푡 ∈ (푆푎 \\ 푆′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moreover, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='5(iv),  tr 퐹(푡) = 0 ⇐⇒ 퐹(푡) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' So, by the assumption (iv), we get tr 퐹(푡) ≠ 0 also for 휈-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 푡 ∈ (푆푎 \\ 푆′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Thus 휈(푆푎 \\ 푆′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  □  24By the definition, they are “only” vector measures in 푀푑(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  45  At the end of this section let us recall the definition of the integral of the scalar function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' a vector  measure for some simplest case, but sufficient for our goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Consider a vector measure 푉 : 픐 −→ 푋,  where 푋 = C푘 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' — a matrix measure, with 푘 = 푑2) and 푓 : Ω −→ C — a bounded 픐-measurable  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Then for any 푗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' , 푘 the function 푉푗 : 픐 −→ C given by  푉푗(휔) := �푉(휔)�  푗,  휔 ∈ 픐,  is a complex measure and therefore the integral  ∫  Ω  푓 d푉푗  is well-defined (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', [8, Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Thus, we define simply:  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11)  ∫  Ω  푓 d푉 :=  �∫  Ω  푓 d푉1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' ,  ∫  Ω  푓 d푉푘  �  ∈ C푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  References  [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Behncke, Absolute continuity of Hamiltonians with von Neumann-Wigner potentials, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 111  (1991), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 373–384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [2] Ju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Berezanski˘ı, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  17, American Mathematical Society, Providence, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 1968, Translated from the Russian by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Bolstein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Danskin,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Rovnyak and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Shulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [3] Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Berg, The matrix moment problem, Coimbra Lecture Notes on Orthogonal Polynomials, Nova Science Publishers Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=',  New York, 2008, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Clark and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hinton, Strong nonsubordinacy and absolutely continuous spectra for Sturm-Liouville equations, Differ-  ential Integral Equations 6 (1993), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, 573–586.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Damanik and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, Unbounded Jacobi matrices at critical coupling, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 145 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2,  221–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Damanik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pushnitski, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Simon, The analytic theory of matrix orthogonal polynomials, Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory  4 (2008), 1–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Dette, Reuther B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Studden, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Zygmunt, Matrix measures and random walks with a block tridiagonal  transition matrix, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Matrix Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 29 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1, 117–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Diestel and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Uhl, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', Vector measures, Mathematical Surveys, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 15, American Mathematical Society, Providence,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Dombrowski, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Janas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moszyński, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pedersen, Spectral gaps resulting from periodic perturbations of a class  of Jacobi operators, Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 20 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 4, 585–601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Durán and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' López-Rodrıguez, The matrix moment problem, Margarita Mathematica en memoria de José Javier  Guadalupe, Universidad de la Rioja, 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 333–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Durán and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Van Assche, Orthogonal matrix polynomials and higher-order recurrence relations, Linear Algebra  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 219 (1995), 261–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [12] Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Dyukarev, On conditions of complete indeterminacy for the matricial Hamburger moment problem, Complex  function theory, operator theory, Schur analysis and systems theory—a volume in honor of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Katsnelson, Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 280, Birkhäuser/Springer, Cham, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 327–353.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [13] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Gesztesy and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Tsekanovskii, On matrix-valued Herglotz functions, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 218 (2000), 61–138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Gilbert, Asymptotic methods in the spectral analysis of Sturm-Liouville operators, Sturm-Liouville theory, Birkhäuser,  Basel, 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 121–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [15] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Edinburgh Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A 112 (1989), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3-4, 213–229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [16] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Gilbert and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 128 (1987), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1, 30–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Golinskii and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Nevai, Szegő difference equations, transfer matrices and orthogonal polynomials on the unit circle,  Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 223 (2001), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 223–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Golub and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Welsch, Calculation of Gauss quadrature rules, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 23 (1969), 221–230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [19] Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Guo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Damanik, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Ong, Subordinacy theory for extended CMV matrices, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 65 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3,  539–558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Hassi, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Remling, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' de Snoo, Subordinate solutions and spectral measures of canonical systems, Integral  Equations Operator Theory 37 (2000), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1, 48–63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Janas and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moszyński, Alternative approaches to the absolute continuity of Jacobi matrices with monotonic weights,  Integral Equations Operator Theory 43 (2002), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 4, 397–416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [22]  , Spectral analysis of unbounded Jacobi operators with oscillating entries, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 209 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 107–133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [23] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Janas and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, Jacobi matrices with power-like weights—grouping in blocks approach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 166 (1999),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 218–243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  46  MARCIN MOSZYŃSKI AND GRZEGORZ ŚWIDERSKI  [24]  , Multithreshold spectral phase transitions for a class of Jacobi matrices, Recent advances in operator theory  (Groningen, 1998), Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 124, Birkhäuser Basel, 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 267–285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [25]  , Spectral analysis of selfadjoint Jacobi matrices with periodically modulated entries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 191 (2002),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 318–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [26]  , On the point spectrum of periodic Jacobi matrices with matrix entries: elementary approach, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Difference Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 21 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 11, 1103–1118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [27] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Janas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Silva, Green matrix estimates of block Jacobi matrices I: Unbounded gap in the essential  spectrum, Integral Equations Operator Theory 90 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 4, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 49, 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [28]  , Green matrix estimates of block Jacobi matrices II: Bounded gap in the essential spectrum, Integral Equations  Operator Theory 92 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 21, 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Janas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Stolz, Spectral theory for a class of periodically perturbed unbounded Jacobi matrices:  elementary methods, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 171 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1-2, 265–276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Jitomirskaya and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Last, Power-law subordinacy and singular spectra I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Half-line operators, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 183 (1999),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 171–189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [31] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Karlin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' McGregor, Random walks, Illinois J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3 (1959), 66–81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Karlin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' McGregor, The differential equations of birth-and-death processes, and the Stieltjes moment problem,  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 85 (1957), 489–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [33] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Khan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pearson, Subordinacy and spectral theory for infinite matrices, Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Acta 65 (1992), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 4,  505–527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [34] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Koekoek, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Lesky, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Swarttouw, Hypergeometric orthogonal polynomials and their 푞-analogues, Springer  Monographs in Mathematics, Springer-Verlag, Berlin, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Kostyuchenko and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Mirzoev, Three-term recurrence relations with matrix coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' The completely indefinite  case, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Notes 63 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 5, 624–630.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [36] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Kupin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, On the instability of the essential spectrum for block Jacobi matrices, Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 48 (2018),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, 473–500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [37] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Last and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional  Schrödinger operators, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 135 (1999), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 329–367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [38] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Levi, Subordinacy theory on star-like graphs, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13548, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [39] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Mitrinović, Analytic inequalities, Die Grundlehren der mathematischen Wissenschaften, Band 165, Springer-Verlag,  New York-Berlin, 1970, In cooperation with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Vasić.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moszyński, Spectral properties of some Jacobi matrices with double weights, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 280 (2003), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2,  400–412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [41]  , Slowly oscillating perturbations of periodic Jacobi operators in 푙2(N), Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 192 (2009), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, 259–279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [42]  , Weyl sequences and the essential spectrum of some Jacobi operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Operator Theory 67 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1,  237–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [43]  , Uni-asymptotic linear systems and Jacobi operators, Integral Equations Operator Theory 91 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, Paper  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 23, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [44] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Moszyński, Spectral Theory of Self-adjoint Finitely Cyclic Operators and Introduction to Matrix-measure 퐿2-spaces,  arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='13953, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Naboko, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pchelintseva, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Silva, Discrete spectrum in a critical coupling case of Jacobi matrices with spectral  phase transitions by uniform asymptotic analysis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 161 (2009), 314–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [46] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Oliveira and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Carvalho, Kotani theory for ergodic matrix-like Jacobi operators, arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11524, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [47]  ,  Criteria  for  the  absolutely  continuous  spectral  components  of  matrix-valued  Jacobi  operators,  arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='12485v2, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [48] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Oliveira and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' de Carvalho, Criteria for the absolutely continuous spectral components of matrix-valued Jacobi  operators, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 34 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 10, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2250037, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [49] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Pchelintseva, A first-order spectral phase transition in a class of periodically modulated Hermitian Jacobi matrices,  Opuscula Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 28 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 137–150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [50] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Rudin, Real and complex analysis, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', McGraw-Hill Book Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', New York, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Sahbani, Spectral theory of a class of block Jacobi matrices and applications, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 438 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1,  93–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [52] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Schmied, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Sims, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Teschl, On the absolutely continuous spectrum of Sturm-Liouville operators with applications  to radial quantum trees, Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Matrices 2 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, 417–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [53] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Schmüdgen, The moment problem, Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 277, Springer, Cham, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [54] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Simon, The classical moment problem as a self-adjoint finite difference operator, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 137 (1998), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1, 82–203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [55]  , Szegő’s theorem and its descendants: Spectral theory for 퐿2 perturbations of orthogonal polynomials, Princeton  University Press, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [56]  , Operator theory, A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [57] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Simonov, An example of spectral phase transition phenomenon in a class of Jacobi matrices with periodically modulated  weights, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 187–203, Birkhäuser Basel, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [58] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Sinap and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Van Assche, Orthogonal matrix polynomials and applications, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 66 (1996), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1,  27–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  NONSUBORDINACY AND ABSOLUTELY CONTINUOUS SPECTRUM OF BLOCK JACOBI MATRICES  47  [59] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Świderski, Periodic perturbations of unbounded Jacobi matrices II: Formulas for density, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 216  (2017), 67–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [60]  , Periodic perturbations of unbounded Jacobi matrices III: The soft edge regime, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 233 (2018),  1–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [61]  , Spectral properties of block Jacobi matrices, Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 48 (2018), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 2, 301–335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [62] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Świderski and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Trojan, Periodic perturbations of unbounded Jacobi matrices I: Asymptotics of generalized eigenvec-  tors, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 216 (2017), 38–66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [63]  , Asymptotics of orthogonal polynomials with slowly oscillating recurrence coefficients, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 278 (2020),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 3, 108326, 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [64]  , Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case II, arXiv:  2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='11154, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [65]  , About essential spectra of unbounded Jacobi matrices, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory 278 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 105746, 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [66]  , Orthogonal polynomials with periodically modulated recurrence coefficients in the Jordan block case, accepted  in Annales de l’Institut Fourier, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [67] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Van Assche, Asymptotics for orthogonal polynomials, Lecture Notes in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1265, Springer-Verlag Berlin  Heidelberg, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [68] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Weidmann, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 1258, Springer-Verlag,  Berlin, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [69] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Zygmunt, Generalized Chebyshev polynomials and discrete Schrödinger operators, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 34 (2001),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 48, 10613–10618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  [70] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Zygmunt, Jacobi block matrices with constant matrix terms, Spectral methods for operators of mathematical physics,  Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Theory Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 154, Birkhäuser, Basel, 2004, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' 233–238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='  Marcin Moszyński, Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Stefana  Banacha 2, 02-097 Warsaw, Poland  Email address: mmoszyns@mimuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='pl  Grzegorz Świderski, Institute of Mathematics, Polish Academy of Sciences, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content=' Śniadeckich 8, 00-696 Warsaw,  Poland  Email address: gswiderski@impan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
+page_content='pl' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltAyT4oBgHgl3EQfYfdr/content/2301.00204v1.pdf'}
diff --git a/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/2301.03686v1.pdf.txt b/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/2301.03686v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3dfc296e35aea825b54aa9a21bf80f546f8a89e2
--- /dev/null
+++ b/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/2301.03686v1.pdf.txt
@@ -0,0 +1,1788 @@
+arXiv:2301.03686v1  [cond-mat.stat-mech]  9 Jan 2023
+The distribution of the number of cycles
+in directed and undirected random 2-regular graphs
+Ido Tishby, Ofer Biham, and Eytan Katzav
+Racah Institute of Physics, The Hebrew University, Jerusalem, 9190401, Israel
+Reimer K¨uhn
+Mathematics Department, King’s College London, Strand, London WC2R 2LS, UK
+Abstract
+We present analytical results for the distribution of the number of cycles in directed and undi-
+rected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node
+has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undi-
+rected links. Since all the nodes are of degree k = 2, the resulting networks consist of cycles.
+These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in
+a random network instance scales with ln N, while the length of the longest cycle scales with N.
+The number of cycles varies between different network instances in the ensemble, where the mean
+number of cycles ⟨S⟩ scales with ln N. Here we present exact analytical results for the distribution
+PN(S = s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed
+in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a
+Poisson distribution in the large N limit. The moments and cumulants of PN(S = s) are also
+calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of
+cycles in random permutations of N objects. In this context our results recover and extend known
+results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied
+before.
+PACS numbers: 02.10.Ox,64.60.aq,89.75.Da
+1
+
+I.
+INTRODUCTION
+Random networks (or graphs) consist of a set of N nodes that are connected to each
+other by edges in a way that is determined by some random process. They provide a useful
+conceptual framework for the study of a large variety of systems and processes in science,
+technology and society [1–9]. The structure of a random network can be characterized by
+the degree distribution P(k). Here we focus on a special class of random networks, called
+random regular graphs (RRGs), which exhibit degenerate degree distributions of the form
+P(k) = δk,c,
+(1)
+where δk,k′ is the Kronecker delta and c ≥ 1 is an integer. While the degrees of all the nodes
+in these networks are the same, their connectivity is random and uncorrelated.
+In that
+sense, the RRG is a special case of the class of configuration model networks, which exhibit
+a specified degree distribution P(k), but no degree-degree correlations [10–13]. While RRGs
+with c = 1 consist of isolated dimers, RRGs with c ≥ 3 form a giant component. Thus,
+RRGs with c = 2 are a marginal case, separating between the subcritical regime of c < 2 and
+supercritical regime of c > 2. Note that unlike some other configuration model networks that
+exhibit a coexistence of a giant component and finite tree components above the percolation
+transition, in RRGs with c ≥ 3 the giant component encompasses the whole network.
+RRGs with c = 2, referred to as random 2-regular graphs (2-RRGs), consist of isolated
+cycles of various lengths. The lengths of these cycles are not determined by the topology,
+but by entropic considerations. An important distinction is between directed 2-RRGs, in
+which each node has one inbound link and one outbound link, and undirected 2-RRGs, in
+which each node has two undirected links. In both cases the cycles can be considered as
+isolated components of the network, where the length of each cycle is equal to the size of
+the network component that consists of this cycle.
+In this paper we present exact analytical results for the distribution of the number of
+cycles in ensembles of directed and undirected 2-RRGs that consist of N nodes. The results
+are expressed in terms of the Stirling numbers of the first kind.
+We first calculate the
+joint probability distribution of cycle lengths. This is done by mapping the directed and
+undirected 2-RRGs into combinatorically equivalent permutation problems. From the joint
+distribution of cycle lengths we extract the distribution PN(S = s) of the number of cycles
+2
+
+in random instances of directed and undirected 2-RRGs. In both cases PN(S = s) converges
+to a Poisson distribution in the large N limit. The moments and cumulants of PN(S = s)
+are also calculated. The similarities and differences between the results obtained for the
+directed and undirected 2-RRGs are discussed. The statistical properties of directed 2-RRGs
+are equivalent to the combinatorics of cycles in random permutations of N objects. In this
+context our results recover and extend known results. In contrast, the statistical properties
+of cycles in undirected 2-RRGs have not been studied before. The results presented in this
+paper are derived specifically for c = 2 and do not apply to the more general case of RRGs
+with c ≥ 3.
+The paper is organized as follows. In Sec. II we present the directed and undirected
+2-RRGs. The joint distributions of cycle lengths are presented in Sec. III. In Sec. IV we
+calculate the distribution PN(S = s) of the number of cycles. In Sec. V we calculate the
+moments and cumulants of PN(S = s). The results are discussed in Sec. VI and summarized
+in Sec. VII.
+II.
+RANDOM 2-REGULAR GRAPHS
+In both directed and undirected 2-RRGs, each node has two links and the resulting
+network consists of a set of cycles. In the directed case each node has one inbound link and
+one outbound link, while in the undirected case each node has two undirected links. Below
+we briefly present the properties of directed and undirected 2-RRGs and their construction.
+A.
+Directed 2-RRGs
+To construct a directed 2-RRG one first assembles N nodes such that each node has one
+inbound stub and one outbound stub. During the network construction process, at each time
+step one picks a random outbound stub and a random inbound stub among the remaining
+open stubs and connects them to each other. This process is repeated N times, until no
+open stubs remain. This procedure follows the standard construction process of directed
+configuration model networks, in which pairs of outbound and inbound stubs rather than
+pairs of nodes are selected for connection. The resulting ensemble of networks obtained from
+this procedure is referred to as stub-labeled graphs [14]. During the construction process,
+3
+
+FIG. 1.
+Illustration of a single instance of a 2-RRG of N = 14 nodes, which consists of cycles of
+lengths s = 1, 2, 3, 3 and 5, in the directed case (a) and in the undirected case (b).
+directed chains of nodes of different lengths are formed. In case that the open outbound
+stub of one chain is selected to connect to the open inbound stub of another chain, they are
+connected and form a longer chain. In case that the outbound and the inbound stubs at
+both ends of the same chain are selected, their connection closes the chain and turns it into
+a cycle. Once a chain of nodes becomes a cycle it does not connect to other nodes and its
+structure remains unchanged. At the end of the construction, the whole network consists of
+cycles of different lengths.
+In Fig. 1(a) we present a single instance of a directed 2-RRG of N = 14 nodes, which
+consists of cycles of lengths s = 1, 2, 3, 3 and 5. In the 2-RRGs considered here we allow the
+outbound and inbound stubs of the same node to be connected to each other. In such case,
+one obtains a self-loop of length ℓ = 1. We also allow the connection of a pair of outbound
+and inbound stubs, which belong to nodes that are already connected. In such case, the
+resulting cycle is of length ℓ = 2. This choice simplifies the analysis.
+Consider a directed 2-RRG consisting of N distinguishable nodes, which are marked by
+the labels i = 1, 2, . . . , N. In the construction of such a network the first selected inbound
+stub has N possible outbound stubs to which it may connect. The second selected inbound
+stub has only N − 1 outbound stubs to which it may connect. Continuing this process we
+conclude that there are N! possible ways to connect the N nodes. In fact, the combinatorics
+of directed 2-RRGs consisting of N nodes is equivalent to the permutation problem of N
+objects [15–17]. This permutation problem can be described by a line of N cells labeled
+4
+
+by i = 1, 2, . . . , N, and a corresponding set of N labeled balls. The number of ways to
+distribute the balls between the cells, one ball in each cell, is
+RD = N!,
+(2)
+where each arrangement of the balls in the cells corresponds to a specific network instance.
+In this representation, a ball i′ located in cell i represents a directed link from i to i′.
+Similarly, a ball i′′ located in cell i′ represents a directed link from i′ to i′′. One can follow
+these directed links until the cell in which ball i resides is reached and the cycle is closed.
+Repeating this process for all the balls and cells provides the structure of cycles associated
+with the specific permutation. The length of each cycle is given by the number of balls
+included in the cycle.
+The statistical properties of cycles in random permutations of N objects have been studied
+extensively [18–20]. In particular, the properties of the longest cycle in each permutation
+received much attention.
+It was found that the expectation value of the length of the
+longest cycle in a random permutation of N objects is equal to λN, where λ = 0.6243... is
+the Golomb-Dickman constant [21–23]. Interestingly, this constant plays a role in the prime
+factorization of a random integer. More specifically, it was found that the asymptotic average
+number of digits in the largest prime factor of a random N-digit number is λN [21, 22, 24].
+In the other extreme, the average length of the shortest cycle in a random permutation of
+N objects was also calculated and was found to be e−γ ln N, where γ = 0.5772... is the
+Euler-Mascheroni constant [16, 22].
+B.
+Undirected 2-RRGs
+To construct an undirected 2-RRG one first assembles N nodes such that each node is
+connected to two undirected stubs. At each time step one selects a random pair of stubs
+among the remaining open stubs and connects them to each other. This process is repeated
+N times, until no open stubs remain. This procedure follows the standard construction
+process of undirected configuration model networks, in which pairs of stubs rather than pairs
+of nodes are selected for connection [10–13]. The resulting ensemble of networks obtained
+from this procedure is referred to as stub-labeled graphs [14]. Initially, one obtains linear
+chains of nodes of increasing lengths that eventually close and form cycles. Here we consider
+5
+
+undirected 2-RRGs in which we allow the two stubs of the same node to be connected to
+each other and form a self-loop of length ℓ = 1. We also allow the connection of a pair of
+stubs which belong to nodes that are already connected, resulting in a cycle of length ℓ = 2.
+In Fig. 1(b) we present a single instance of an undirected 2-RRG of N = 14 nodes, which
+consist of cycles of lengths s = 1, 2, 3, 3 and 5.
+Consider an ensemble of undirected 2-RRGs consisting of N nodes. In the construction
+of such network the first selected stub has 2N − 1 other stubs to which it may connect.
+The second selected stub has 2N − 3 other stubs to which it may connect. Continuing this
+process we conclude that there are
+RU = (2N − 1)!!.
+(3)
+possible ways to construct such network, where m!! is the double factorial of m. The sta-
+tistical properties of the resulting ensemble of networks can be mapped to the combina-
+torial problem described below. Consider a set of 2N balls, such that for each value of
+i = 1, 2, . . . , N there are two identical balls on which the label i is marked. The two balls
+labeled by a given value of i represent the two stubs of node i. In addition, there are N
+unlabeled boxes such that in each box there is room for two balls. The 2N balls are then
+distributed uniformly at random in the N boxes, where each box contains two balls. Each
+pair of balls that resides in the same box represents one edge of the network. For example,
+if a ball labeled by i and a ball labeled by i′ are in the same box, it means that there is an
+edge between the nodes i and i′. Similarly, if the other ball labeled by i′ is in the same box
+with a ball labeled i′′, it means that there is an edge between the nodes i′ and i′′. One can
+follow this chain until reaching the box in which the second ball labeled by i resides, thus
+closing the cycle. The number of possible ways to distribute the 2N balls in the N cells is
+given by
+RU = (2N)!
+2NN! ,
+(4)
+where the numerator accounts for the number of permutations of 2N balls, the N! term in
+the denominator accounts for the number of permutations of the N identical cells, and the
+term 2N accounts for the fact that the order in which the two balls are placed in each cell is
+unimportant. Note that (2N)! = (2N − 1)!!(2N)!! and 2NN! = (2N)!!. These two identities
+6
+
+establish the equivalence between Eqs. (3) and (4).
+III.
+THE JOINT DISTRIBUTION OF CYCLE LENGTHS
+Both versions of the 2-RRG, with directed and undirected links, consist of closed cycles
+of different lengths. The configuration of cycles in a given network instance can be described
+by the sequence of cycle lengths, ℓ1, ℓ2, . . . , ℓs, where 1 ≤ ℓi ≤ N, s is the number of cycles
+in the given network instance and
+s
+�
+i=1
+ℓi = N.
+(5)
+For convenience and uniqueness, we order the lengths in increasing order, such that ℓ1 ≤
+ℓ2 ≤ · · · ≤ ℓs.
+Another way to describe the configuration of cycles in a given network instance is in the
+form {gℓ}N
+ℓ=1 = {g1, g2, . . . , gN}, where gℓ is the number of cycles of length ℓ. The gℓ’s satisfy
+the condition
+N
+�
+ℓ=1
+ℓgℓ = N,
+(6)
+which is equivalent to Eq. (5). The number of cycles in such a network instance can be
+expressed by
+s =
+N
+�
+ℓ=1
+gℓ.
+(7)
+The joint distribution of cycle lengths in an ensemble of 2-RRGs consisting of N nodes is
+denoted by PN({Gℓ} = {gℓ}), under the condition of Eq. (6). For convenience we use the
+notation PN({gℓ}). Below we consider the joint distributions of the cycle lengths in directed
+and undirected 2-RRGs.
+A.
+Joint distribution of cycle lengths in directed 2-RRGs
+Consider an ensemble of directed 2-RRGs that consist of N nodes.
+The number of
+configurations of the form {gℓ} is given by
+7
+
+N({gℓ}) = N!
+N
+�
+ℓ=1
+1
+ℓgℓgℓ!δ� ℓgℓ,N.
+(8)
+To understand the first term in the denominator, consider a cycle of length ℓ. Such cycle
+exhibits ℓ cyclic permutations. This yields the first term in the denominator, which is raised
+to the power gℓ to account for the number of cycles of length ℓ. The term gℓ! accounts
+for the permutations between the gℓ degenerate cycles of length ℓ, which correspond to the
+same configuration, while the Kronecker delta imposes the condition of Eq. (6). Dividing
+the number of configurations N({gℓ}) by the total number of configurations RD, given by
+Eq. (2), we obtain the joint probability distribution of cycle lengths. It is given by
+PN({gℓ}) =
+N
+�
+ℓ=1
+1
+ℓgℓgℓ!δ� ℓgℓ,N.
+(9)
+It can be shown that this probability distribution is normalized, namely
+�
+g1
+�
+g2
+· · ·
+�
+gN
+PN({gℓ}) = 1,
+(10)
+where the summation is over all the configurations of {gℓ} that satisfy Eq. (6).
+B.
+Joint distribution of cycle lengths in undirected 2-RRGs
+Consider an ensemble of undirected 2-RRGs of N nodes. The number of configurations
+of the form {gℓ} is
+N({gℓ}) = N!
+N
+�
+ℓ=1
+2(ℓ−1)gℓ
+ℓgℓgℓ! δ� ℓgℓ,N.
+(11)
+This result can be understood in terms of the analogous combinatorial problem described
+above, which consists of N pairs of identical balls and N unlabled boxes. Inserting two
+random balls in each box, the factor of N! accounts for the number of permutations of the
+N pairs of indices marked on the balls. To account for the other factors, consider a cycle
+of length ℓ. There are 2ℓ possible ways to exchange the ℓ pairs of identical balls. However,
+due to the cyclic structure there is also a factor of 1/2, because in each cycle the lables
+marked on the balls can be listed either in the clockwise direction or in the counterclockwise
+direction. Taking this into account yields the factor of 2ℓ−1 in the numerator. This factor
+8
+
+is raised to the power gℓ to account for the fact that there are gℓ cycles of length ℓ. In the
+denominator, the ℓgℓ term accounts for the number of cyclic permutations of the indices in
+all the cycles of length ℓ, while the gℓ! term accounts for the number of permutations of gℓ
+degenerate cycles of the same length. The probability that a random network instance will
+have a cycle structure given by {gℓ} is given by
+PN({gℓ}) = N({gℓ})
+RU
+.
+(12)
+Inserting N({gℓ}) from Eq. (11), and RU from Eq. (3) into Eq. (12), we obtain
+PN({gℓ}) =
+N!
+(2N − 1)!!
+N
+�
+ℓ=1
+2(ℓ−1)gℓ
+ℓgℓgℓ! δ� ℓgℓ,N.
+(13)
+Using the relation 2N = 2
+�N
+ℓ=1 ℓgℓ, we obtain
+PN({gℓ}) =
+(2N)!!
+(2N − 1)!!
+N
+�
+ℓ=1
+1
+(2ℓ)gℓgℓ!δ� ℓgℓ,N.
+(14)
+This expression differs from the corresponding result for directed 2-RRGs in two ways: it has
+a factor of (2ℓ)gℓ in the denominator instead of ℓgℓ and there is a pre-factor that is required
+in order to maintain the normalization. The factor of 2gℓ in the denominator means that
+as the number of cycles is increased the configuration becomes exponentially less probable
+than the corresponding configuration of the directed 2-RRG.
+IV.
+THE DISTRIBUTION OF THE NUMBER OF CYCLES
+The probability PN(S = s) that a random network instance includes s cycles can be
+calculated by summing up over all the combinations of {gℓ} that consist of s cycles, namely
+PN(S = s) =
+�
+g1,...,gN
+PN({gℓ})δ�
+ℓ gℓ,s,
+(15)
+where the configurations {gℓ} satisfy the condition of Eq.
+(6).
+Below we calculate the
+distribution PN(S = s) for the directed and undirected 2-RRGs.
+A.
+Distribution of the number of cycles in directed 2-RRGs
+Inserting the expression for PN({gℓ}) from Eq. (9) into Eq. (15), we obtain
+9
+
+PN(S = s) =
+�
+g1,...,gN≥0
+N
+�
+ℓ=1
+1
+ℓgℓgℓ!δ� ℓgℓ,Nδ�
+ℓ gℓ,s.
+(16)
+For the analysis below it is convenient to perform a change of variables from gℓ, ℓ =
+1, 2, . . . , N to ℓi, i = 1, 2, . . . , s. This transformation is based on the identity
+�
+g1,...,gN≥0
+1
+g1!g2! . . . gN!δ�
+ℓ gℓ,s = Ns
+s! = 1
+s!
+N
+�
+ℓ1,ℓ2,...,ℓs=1
+1,
+(17)
+which is a result of the multinomial theorem (equation 26.4.9 in Ref. [25]). Inserting the
+constraints that �N
+ℓ=1 ℓgℓ = N = �s
+i=1 ℓi, obtained from Eqs. (5) and (6) we obtain the
+identity
+�
+g1,...,gN≥0
+1
+g1!g2! . . . gN!δ�
+ℓ gℓ,sδ� ℓgℓ,N = 1
+s!
+N
+�
+ℓ1,ℓ2,...,ℓs=1
+δ�
+i ℓi,N
+(18)
+Using this transformation, we express Eq. (16) in the form
+PN(S = s) = 1
+s!
+N
+�
+ℓ1,...,ℓs=1
+1
+ℓ1ℓ2 . . . ℓs
+δ�
+i ℓi,N.
+(19)
+Below we use the discrete Laplace transform, which is related to the one-sided Z-transform
+and to the starred transform [26], to evaluate the right hand side of Eq. (19). We denote
+the sum on the right hand side of Eq. (19) by
+fs(N) =
+�
+ℓ1,...,ℓs≥1
+1
+ℓ1ℓ2 . . . ℓs
+δ�s
+i=1 ℓi,N.
+(20)
+The discrete Laplace transform of fs(N) is given by
+�fs(z) =
+∞
+�
+N=0
+zNfs(N).
+(21)
+Inserting fs(N) from Eq. (20) into Eq. (21), we obtain
+�fs(z) =
+�
+ℓ1,...,ℓs≥1
+1
+ℓ1ℓ2 . . . ℓs
+∞
+�
+N=0
+zNδ�s
+i=1 ℓi,N,
+(22)
+where
+∞
+�
+N=0
+zNδ�s
+i=1 ℓi,N = z
+�s
+i=1 ℓi.
+(23)
+10
+
+Decomposing the multiple summation in Eq. (22) into a product of sums over the ℓi’s, we
+obtain
+�fs(z) =
+� ∞
+�
+ℓ=1
+zℓ
+ℓ
+�s
+.
+(24)
+Carrying out the summation, we obtain
+�fs(z) = [− ln (1 − z)]s .
+(25)
+The next step is to apply the inverse discrete Laplace transform on �fs(z) to obtain fs(N).
+To this end we use identity 26.8.8 from Ref. [25], which is given by
+[ln(1 + y)]k
+k!
+=
+∞
+�
+n=0
+s(n, k)yn
+n! ,
+(26)
+where s(n, k) is the Stirling number of the first kind. These Stirling numbers can be expressed
+in the form
+s(n, k) = (−1)n−k
+� n
+k
+�
+,
+(27)
+where
+� n
+k
+�
+is the unsigned Stirling number of the first kind [25]. Inserting s(n, k) from Eq.
+(27) into Eq. (26), we obtain
+[ln(1 + y)]k = k!
+∞
+�
+n=0
+(−1)n−k
+� n
+k
+�yn
+n! .
+(28)
+Inserting y = −z into Eq. (28) we rewrite �fs(z) in the form
+�fs(z) = s!
+∞
+�
+n=0
+1
+n!
+� n
+s
+�
+zn.
+(29)
+To obtain the inverse discrete Laplace transform of �fs(z) we use the fact that
+L−1 {zn} (N) = δN,n.
+(30)
+Applying the inverse discrete Laplace transform on Eq. (29), we obtain
+11
+
+fs(N) = s!
+N!
+� N
+s
+�
+.
+(31)
+Inserting fs(N) from Eq. (31) into Eq. (19), we obtain
+PN(S = s) = 1
+N!
+� N
+s
+�
+.
+(32)
+The normalization of the distribution PN(S = s) can be confirmed using identity 26.8.29 in
+Ref. [25]. In the context of permutations, the result expressed by Eq. (32) implies that
+� N
+s
+�
+counts the number of permutations with precisely s cycles among the N! permutations of
+N objects. This is consistent with the combinatorial interpretation of the unsigned Stirling
+number of the first kind [25]. The cumulative distribution of the number of cycles is given
+by
+PN(S ≤ s) = 1
+N!
+s
+�
+s′=1
+� N
+s′
+�
+.
+(33)
+In Fig. 2(a) we present exact analytical results (solid line) for the cumulative distribution
+PN(S ≤ s) of the number of cycles in a directed 2-RRG that consists of N = 10 nodes,
+obtained from Eq. (33). The analytical results are found to be in very good agreement with
+the results obtained from computer simulations (circles).
+While Eq.
+(32) provides an exact result for PN(S = s), it is useful to express this
+distribution in terms of more elementary functions. This is possible in the asymptotic limit
+of large N. The limit of N ≫ 1 corresponds to the limit of z → 1− of the discrete Laplace
+transform �fs(z) [27, 28]. If one replaces the variable z by e−x, then this limit corresponds to
+x ≪ 1 in �fs(e−x). In this limit the expression for �fs (e−x) in Eq. (25) can be approximated
+by
+�fs
+�
+e−x�
+≃ (− ln x)s .
+(34)
+In order to calculate the inverse Laplace transform of �fs (e−x), we use the relation
+L−1
+� 1
+xν (− ln x)s
+�
+=
+� d
+dν
+�s �Nν−1
+Γ(ν)
+�
+,
+(35)
+12
+
+0
+1
+2
+3
+4
+5
+6
+7
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+1
+2
+3
+4
+5
+6
+7
+0
+0.2
+0.4
+0.6
+0.8
+1
+FIG. 2.
+Exact analytical results (solid lines) for the cumulative distribution of the number of
+cycles in a directed 2-RRG (a) and an undirected 2-RRG (b) of N = 10 nodes, obtained from Eqs.
+(33) and (57), respectively. The analytical results are in very good agreement with the results
+obtained from computer simulations (circles).
+where Γ(ν) is the Gamma function [25].
+The inverse Laplace transform of Eq.
+(34) is
+obtained by taking the limit ν → 0 in Eq. (35). To this end we use the general Leibnitz
+rule (equation 1.4.12 in Ref. [25])
+� d
+dν
+�s �Nν−1
+Γ(ν)
+�
+=
+s
+�
+i=0
+�s
+i
+� �� d
+dν
+�s−i
+Nν−1
+� �� d
+dν
+�i �
+1
+Γ(ν)
+��
+.
+(36)
+Below we evaluate the derivatives that appear in the two terms on the right hand side of
+Eq. (36). The derivative in the first term is given by
+� d
+dν
+�i
+Nν−1 = Nν−1(ln N)i.
+(37)
+Thus, for ν → 0+ we obtain
+lim
+ν→0+
+�� d
+dν
+�i
+Nν−1
+�
+= (ln N)i
+N
+.
+(38)
+The derivative in the second term on the right hand side of Eq. (36) is denoted by
+hi = lim
+ν→0+
+�� d
+dν
+�i �
+1
+Γ(ν)
+��
+.
+(39)
+13
+
+TABLE I. The coefficients ai, i = 1, 2, . . . , 10, which appear in the power series of the reciprocal
+gamma function given by Eq. (42)
+i
+ai
+1
+1
+2
+0.5772156649
+3
+- 0.6558780715
+4
+- 0.0420026350
+5
+0.1665386114
+6
+- 0.0421977345
+7
+- 0.0096219715
+8
+0.0072189432
+9
+- 0.0011651675
+10
+- 0.0002152416
+By its definition, hi is the coefficient of the i’th power of ν in the Taylor expansion of 1/Γ(ν)
+around ν = 0, namely
+1
+Γ(ν) =
+∞
+�
+i=1
+hi
+i! νi.
+(40)
+This expansion is often written as a power-series of the form [29, 30]
+1
+Γ(ν) =
+∞
+�
+i=1
+aiνi,
+(41)
+where ai = hi/i!. The first two coefficients are given by a1 = 1 and a2 = γ, where γ is the
+Euler-Mascheroni constant [25]. Higher order coefficients can be obtained from the recursion
+equation
+ai =
+1
+i − 1
+�
+a2ai−1 −
+i−1
+�
+j=2
+(−1)jζ(j)ai−j
+�
+,
+(42)
+where ζ(j) is the Riemann zeta function. The coefficients ai for i = 3, 4, . . . , 10, obtained
+from Eq. (42), are presented in Table I. These coefficients can also be obtained from the
+integral representation [31]
+14
+
+an = (−1)n
+πn!
+� π
+0
+e−tIm [(ln t − iπ)n] dt.
+(43)
+Using this notation we obtain
+fs(N) ≃ s!
+N
+s
+�
+i=1
+ai
+(ln N)s−i
+(s − i)! ,
+(44)
+which leads to
+PN(S = s) ≃ 1
+N
+s
+�
+i=1
+ai
+(ln N)s−i
+(s − i)! .
+(45)
+Note that for sufficiently large N the sum is dominated by the first few terms for two reasons.
+First, apart from the first few terms, the coefficients ai become negligibly small. Second,
+the power of ln N decreases as i is increased.
+The cumulative distribution of the number of cycles is given by
+PN(S ≤ s) = 1
+N
+s
+�
+s′=1
+s′
+�
+i=1
+ai
+(ln N)s′−i
+(s′ − i)! .
+(46)
+In Fig. 3(a) we present analytical results (solid line) for the large N approximation of the
+cumulative distribution PN(S ≤ s) of the number of cycles in a directed 2-RRG of N = 104
+nodes, obtained from Eq. (46). The analytical results are found to be in very good agreement
+with the results obtained from computer simulations (circles).
+B.
+Distribution of the number of cycles in undirected 2-RRGs
+We now turn to calculate the distribution PN(S = s) of the number of cycles in undirected
+2-RRGs. Inserting the expression for PN({gℓ}) from Eq. (14) in Eq. (15), we obtain
+PN(S = s) =
+(2N)!!
+(2N − 1)!!
+�
+g1,...,gN≥0
+N
+�
+ℓ=1
+1
+(2ℓ)gℓgℓ!δ�
+ℓ ℓgℓ,Nδ�
+ℓ gℓ,s.
+(47)
+Using the change of variables presented in Eq. (18), we obtain
+PN(S = s) =
+(2N)!!
+(2N − 1)!!
+1
+2ss!
+N
+�
+ℓ1,...,ℓs=1
+1
+ℓ1ℓ2 . . . ℓs
+δ�
+i ℓi,N.
+(48)
+15
+
+0
+5
+10
+15
+20
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+15
+20
+0
+0.2
+0.4
+0.6
+0.8
+1
+FIG. 3.
+Analytical results (solid lines) for the cumulative distribution of the number of cycles in
+a directed 2-RRG (a) and an undirected 2-RRG (b) of N = 104 nodes, obtained from Eqs. (46)
+and (60), respectively. The analytical results are in very good agreement with the results obtained
+from computer simulations (circles).
+Note that the sum on the right hand side of Eq. (48) is equal to fs(N), given by Eq. (20).
+We thus obtain
+PN(S = s) =
+(2N)!!
+(2N − 1)!!
+1
+2sN!
+� N
+s
+�
+.
+(49)
+Below we show that the distribution PN(S = s) is properly normalized. To this end we use
+equation 26.8.7 from Ref. [25], which can be written in the form
+N
+�
+s=1
+(−1)N−s
+� N
+s
+�
+xs = (x − N + 1)N,
+(50)
+where (a)N is the Pochhammer symbol [25]. Inserting x = −1/2 in Eq. (50), we obtain
+N
+�
+s=1
+� N
+s
+� 1
+2s = (−1)N
+�1
+2 − N
+�
+N
+.
+(51)
+Expressing the Pochhammer symbol on the right hand side of Eq. (51) as a ratio between
+two Gamma functions, we obtain
+16
+
+N
+�
+s=1
+� N
+s
+� 1
+2s = (−1)NΓ(1/2)
+Γ(1/2 − N) .
+(52)
+Using Euler’s reflection formula [25]
+Γ
+�1
+2 − N
+�
+Γ
+�1
+2 + N
+�
+= (−1)Nπ,
+(53)
+and the Legendre duplication formula [25]
+Γ(1/2 + N) = 21−2N√πΓ(2N)
+Γ(N) ,
+(54)
+we obtain
+N
+�
+s=1
+� N
+s
+� 1
+2s = 21−2N Γ(2N)
+Γ(N) .
+(55)
+Expressing the Gamma functions of integer variables in terms of factorials and double fac-
+torials, we obtain
+N
+�
+s=1
+� N
+s
+� 1
+2s = (2N − 1)!!
+(2N)!!
+N!.
+(56)
+This confirms the normalization of PN(S = s), given by Eq. (49). From Eq. (49) we obtain
+the cumulative distribution of the number of cycles, which is given by
+PN(S ≤ s) =
+(2N)!!
+(2N − 1)!!
+1
+N!
+s
+�
+s′=1
+1
+2s′
+� N
+s′
+�
+.
+(57)
+In Fig. 2(b) we present exact analytical results (solid line) for the cumulative distribution
+P(S ≤ s) of the number of cycles in undirected 2-RRGs that consist of N = 10 nodes,
+obtained from Eq. (57). The analytical results are found to be in very good agreement with
+the results obtained from computer simulations (circles).
+While Eq.
+(49) provides an exact result for PN(S = s), it is useful to express this
+distribution in terms of more elementary functions. This is possible in the asymptotic limit
+of large N, where the ratio of double factorials can be approximated by
+(2N)!!
+(2N − 1)!! ≃
+√
+πN + O
+�
+N−1/2�
+.
+(58)
+17
+
+Using this result and inserting fs(N) from Eq. (44) into Eq. (48), we find that
+PN(S = s) ≃ 1
+2s
+� π
+N
+s
+�
+i=1
+ai
+(ln N)s−i
+(s − i)! .
+(59)
+From Eq. (59) we obtain the cumulative distribution of the number of cycles, which is given
+by
+PN(S ≤ s) ≃
+� π
+N
+s
+�
+s′=1
+1
+2s′
+s′
+�
+i=1
+ai
+(ln N)s′−i
+(s′ − i)! .
+(60)
+In Fig. 3(b) we present analytical results (solid line) for the cumulative distribution of
+the number of cycles in undirected 2-RRGs of N = 104 nodes, obtained from Eq. (60). The
+analytical results are found to be in very good agreement with the results obtained from
+computer simulations (circles).
+V.
+MOMENTS AND CUMULANTS
+In this section we calculate the moments and cumulants of the distribution of the number
+of cycles in 2-RRGs that consist of N nodes. To this end we introduce the moment generating
+function, which is given by
+M(t) = E
+�
+etS�
+.
+(61)
+The cumulant generating function is given by
+K(t) = ln M(t).
+(62)
+Using this function one can calculate the cumulants via differentiation according to
+κn = dnK(t)
+dtn
+����
+t=0
+.
+(63)
+A.
+Moments and cumulants in directed 2-RRGs
+The moment generating function of directed 2-RRGs is given by
+18
+
+M(t) = 1
+N!
+N
+�
+s=0
+ets
+� N
+s
+�
+.
+(64)
+Using Eq. (50) with x = −et, we obtain
+M(t) = (−1)N
+N!
+�
+−et − N + 1
+�
+N .
+(65)
+The moment generating function M(t) may also be written in the form
+M(t) = Γ(N + et)
+Γ(et)N! ,
+(66)
+in agreement with the results presented in Ref. [16]. The corresponding cumulant generating
+function is given by
+K(t) = ln
+�Γ(N + et)
+Γ(et)N!
+�
+.
+(67)
+Using Eq. (67) we obtain the first two cumulants, which are given by
+⟨S⟩ = κ1 = HN,
+(68)
+where HN is the harmonic number [32], and
+Var(S) = κ2 = HN − H(2)
+N ,
+(69)
+where H(m)
+N
+is the generalized harmonic number of order m [32]. Similarly, one can calculate
+higher order cumulants such as
+κ3 = HN − 3H(2)
+N + 2H(3)
+N
+(70)
+and
+κ4 = HN − 7H(2)
+N + 12H(3)
+N − 6H(4)
+N .
+(71)
+In the limit of large N we can use the asymptotic expression for the distribution PN(S = s),
+given by Eq. (45), and obtain
+M(t) ≃ 1
+N
+∞
+�
+s=0
+s
+�
+i=1
+estai
+(ln N)s−i
+(s − i)! .
+(72)
+19
+
+Exchanging the order of summations, we obtain
+M(t) ≃ 1
+N
+∞
+�
+i=1
+ai
+∞
+�
+s=i
+est(ln N)s−i
+(s − i)! .
+(73)
+Shifting the summation index in the second sum, we obtain
+M(t) ≃ 1
+N
+∞
+�
+i=1
+aieit
+∞
+�
+s=0
+est(ln N)s
+s!
+.
+(74)
+Using Eq. (41) we carry out the two summations and obtain
+M(t) ≃ 1
+N
+1
+Γ(et) exp
+�
+et ln N
+�
+.
+(75)
+Using Eq. (62) we obtain the cumulant generating function, which is given by
+K(t) ≃
+�
+et − 1
+�
+ln N − ln
+�
+Γ
+�
+et��
+.
+(76)
+Using Eq. (63) we obtain the cumulants, which take the form
+κn ≃ ln N − dn
+dtn ln
+�
+Γ
+�
+et�� ����
+t=0
+.
+(77)
+In order to calculate high order derivatives of ln [Γ (et)] we use the identity (equation A.4 in
+Ref. [32])
+dn
+dtnf(et) =
+n
+�
+m=1
+� n
+m
+�
+emtf (m)(et),
+(78)
+where
+� n
+m
+�
+is the Stirling number of the second kind and f (m)(x) is the mth derivative of
+f(x). We also use the fact that
+dn
+dzn ln Γ(z) = ψ(n−1)(z),
+(79)
+where ψ(n)(z) is the nth derivative of the digamma function [25]. Using these identities, we
+obtain
+dn
+dtn ln
+�
+Γ
+�
+et�� ����
+t=0
+=
+n
+�
+m=1
+� n
+m
+�
+ψ(m−1)(1).
+(80)
+20
+
+It is also known that (equation 5.4.12 in Ref. [25])
+ψ(0)(1) = −γ,
+(81)
+and that for m ≥ 1 (equation 5.15.2 in Ref. [25])
+ψ(m)(1) = (−1)m+1m!ζ(m + 1),
+(82)
+where ζ(m) is the Riemann zeta function [25]. Combining the results derived above, we
+obtain
+κn ≃ ln N + γ +
+n
+�
+m=2
+� n
+m
+�
+(−1)m−1(m − 1)!ζ(m).
+(83)
+Using Eq. (83) we write down explicitly the first few cumulants of PN(S = s) in the large
+N limit. They are given by
+κ1 ≃ ln N + γ
+κ2 ≃ ln N + γ − π2
+6
+κ3 ≃ ln N + γ − π2
+2 + 2ζ(3)
+κ4 ≃ ln N + γ − 7π2
+6
++ 12ζ(3) − π4
+15.
+(84)
+The results for κ1 and κ2 are in agreement with the classical results reported in Refs. [33–36].
+In the large N limit all the cumulants are of the form ln N + O(1). This essentially implies
+that in the large N limit the distribution PN(S = s) approaches a Poisson distribution with
+a parameter ln N.
+Comparing between Eqs. (68)-(71) and Eq. (83), and using the fact that for m ≥ 2
+lim
+N→∞ H(m)
+N
+= ζ(m),
+(85)
+we obtain a general expression for the cumulants at finite values of N, which is given by
+κn = HN +
+n
+�
+m=2
+� n
+m
+�
+(−1)m−1(m − 1)!H(m)
+N .
+(86)
+21
+
+101
+102
+103
+104
+105
+0
+2
+4
+6
+8
+10
+12
+FIG. 4.
+Analytical results for the mean number of cycles ⟨S⟩ as a function of the network size N,
+in directed 2-RRGs (solid line) and in undirected 2-RRGs (dashed line), obtained from Eqs. (68)
+and (91), respectively. The analytical results are in very good agreement with the results obtained
+from computer simulations (circles). To leading order, in directed 2-RRGs ⟨S⟩ ≃ ln N, while in
+undirected 2-RRGs ⟨S⟩ ≃ 1
+2 ln N.
+In Fig. 4 we present analytical results (solid line) for the mean number of cycles ⟨S⟩ in
+directed 2-RRGs as a function of the network size N, obtained from Eq. (68). The analytical
+results are in very good agreement with the results obtained from computer simulations
+(circles).
+In Fig. 5 we present analytical results (solid line) for the variance Var(S) in directed
+2-RRGs as a function of the network size N, obtained from Eq. (69). The analytical results
+are in very good agreement with the results obtained from computer simulations (circles).
+B.
+Moments and cumulants in undirected 2-RRGs
+The moment generating function of undirected 2-RRGs is given by
+M(t) =
+(2N)!!
+(2N − 1)!!
+1
+N!
+N
+�
+s=0
+ets
+2s
+� N
+s
+�
+.
+(87)
+Using Eq. (50) with x = −et/2, we obtain
+22
+
+101
+102
+103
+104
+105
+0
+2
+4
+6
+8
+10
+12
+FIG. 5.
+Analytical results for the variance of the distribution of the number of cycles as a function
+of the network size N, in directed 2-RRGs (solid line) and in undirected 2-RRGs (dashed line),
+obtained from Eqs. (69) and (92), respectively. The analytical results are in very good agreement
+with the results obtained from computer simulations (circles). To leading order, in directed 2-RRGs
+Var(S) ≃ ln N, while in undirected 2-RRGs Var(S) ≃ 1
+2 ln N.
+M(t) = (−1)N
+N!
+(2N)!!
+(2N − 1)!!
+�
+−et
+2 − N + 1
+�
+N
+.
+(88)
+The moment generating function M(t) may also be written in the form
+M(t) =
+(2N)!!
+(2N − 1)!!
+Γ
+�
+N + et
+2
+�
+Γ
+�et
+2
+�
+N! .
+(89)
+The corresponding cumulant generating function is given by
+K(t) = ln
+
+
+(2N)!!
+(2N − 1)!!
+Γ
+�
+N + et
+2
+�
+Γ
+�et
+2
+�
+N!
+
+ .
+(90)
+Using Eq. (63) we obtain the first four cumulants. The first cumulant is given by
+⟨S⟩ = κ1 = 1
+2HN− 1
+2 + ln 2,
+(91)
+where HN− 1
+2 is an Harmonic number at a half-integer value [37]. The second cumulant is
+given by
+23
+
+Var(S) = κ2 = 1
+2HN− 1
+2 + ln 2 − 1
+4
+�
+H(2)
+N− 1
+2 + 2ζ(2)
+�
+,
+(92)
+while the third and fourth cumulants are given by
+κ3 = 1
+2HN− 1
+2 + ln 2 − 3
+4
+�
+H(2)
+N− 1
+2 + 2ζ(2)
+�
++ 1
+4
+�
+H(3)
+N− 1
+2 + 6ζ(3)
+�
+(93)
+and
+κ4 = 1
+2HN− 1
+2 + ln 2 − 7
+4
+�
+H(2)
+N− 1
+2 + 2ζ(2)
+�
++ 3
+2
+�
+H(3)
+N− 1
+2 + 6ζ(3)
+�
+− 3
+8
+�
+H(4)
+N− 1
+2 + 14ζ(4)
+�
+. (94)
+In the limit of large N we can use the asymptotic expression for the distribution PN(S =
+s), given by Eq. (59). Inserting it into Eq. (61) we obtain an asymptotic expression for the
+moment generating function, which is given by
+M(t) ≃
+� π
+N
+∞
+�
+s=0
+s
+�
+i=1
+estai
+2i
+(ln N)s−i
+2s−i(s − i)!.
+(95)
+Exchanging the order of summations and shifting the summation index in the second sum,
+we obtain
+M(t) ≃
+� π
+N
+∞
+�
+i=1
+ai
+eit
+2i
+∞
+�
+s=0
+est(ln N)s
+2ss!
+.
+(96)
+Using Eq. (41) we carry out the two summations and obtain
+M(t) ≃
+� π
+N
+1
+Γ(et/2) exp
+�et
+2 ln N
+�
+.
+(97)
+Using Eq. (62) we obtain the cumulant generating function, which is given by
+K(t) ≃ et − 1
+2
+ln N − ln
+� 1
+√πΓ
+�et
+2
+��
+.
+(98)
+Using Eq. (63) we obtain the cumulants, which take the form
+κn ≃ 1
+2 ln N − dn
+dtn ln
+� 1
+√πΓ
+�et
+2
+�� ����
+t=0
+.
+(99)
+Using Eqs. (78) and (79), we obtain
+24
+
+dn
+dtn ln
+� 1
+√πΓ
+�et
+2
+�� ����
+t=0
+=
+n
+�
+m=1
+� n
+m
+�
+2−mψ(m−1)
+�1
+2
+�
+.
+(100)
+It is also known that [38]
+ψ(0)
+�1
+2
+�
+= −γ − ln 4,
+(101)
+and that for m ≥ 1 [38]
+ψ(m)
+�1
+2
+�
+= (−1)m+1m!
+�
+2m+1 − 1
+�
+ζ(m + 1).
+(102)
+Combining the results derived above, we obtain
+κn ≃ ln N
+2
++ γ + ln 4
+2
++
+n
+�
+m=2
+� n
+m
+�
+(−1)m−1 �
+1 − 2−m�
+(m − 1)!ζ(m),
+(103)
+which becomes exact in the large N limit. Using Eq. (103) we write down explicitly the
+first few cumulants of PN(S = s). They are given by
+κ1 ≃ ln N
+2
++ γ + ln 4
+2
+κ2 ≃ ln N
+2
++ γ + ln 4
+2
+− π2
+8
+κ3 ≃ ln N
+2
++ γ + ln 4
+2
+− 3π2
+8
++ 7
+4ζ(3)
+κ4 ≃ ln N
+2
++ γ + ln 4
+2
+− 7π2
+8
++ 21
+2 ζ(3) − π4
+16.
+(104)
+In the large N limit all the cumulants are of the form 1
+2 ln N +O(1). This essentially implies
+that in the large N limit the distribution PN(S = s) approaches a Poisson distribution with
+a parameter 1
+2 ln N.
+Using the fact that for m ≥ 2
+lim
+N→∞ H(m)
+N− 1
+2 = ζ(m),
+(105)
+we obtain a general expression for the cumulants at finite values of N, which is given by
+κn = 1
+2HN− 1
+2 + ln 2 +
+n
+�
+m=2
+� n
+m
+�
+(−1)m−12−m(m − 1)!
+�
+H(m)
+N− 1
+2 + (2m − 2)ζ(m)
+�
+.
+(106)
+25
+
+In Fig.
+4 we present analytical results (dashed line) for the mean number of cycles
+⟨S⟩ in undirected 2-RRGs as a function of the network size N, obtained from Eq. (91).
+The analytical results are in very good agreement with the results obtained from computer
+simulations (circles).
+In Fig. 5 we present analytical results (dashed line) for the variance Var(S) in undirected
+2-RRGs as a function of the network size N, obtained from Eq. (92). The analytical results
+are in very good agreement with the results obtained from computer simulations (circles).
+VI.
+DISCUSSION
+Below we discuss the similarities and differences between the directed and undirected
+2-RRGs. In directed 2-RRGs the mean number of cycles ⟨S⟩ scales with ln N, while in
+undirected 2-RRGs it scales with 1
+2 ln N. Thus, the expected number of cycles in directed
+2-RRGs is twice as large as in undirected 2-RRGs.
+This is due to the fact that in the
+construction of undirected 2-RRGs each end of a given chain may connect to both sides of
+any other linear chain, while in directed 2-RRGs it may only connect to the complementary
+side. As a result, in undirected 2-RRGs the connection of chains forming a longer chain is
+more probable than in directed 2-RRGs. Thus, in undirected 2-RRGs the competing process
+of closing a chain to form a cycle is less probable than in directed 2-RRGs. This implies that
+in undirected 2-RRGs the cycles are expected to be longer and their number is expected to
+be smaller than in directed 2-RRGs.
+2-RRGs are marginal networks that reside at the boundary between the subcritical regime
+and the supercritical regime. In the subcritical regime, configuration model networks consist
+of many finite tree components. The distribution of sizes of these tree components can be
+calculated using the framework of generating functions [39]. In this framework it is assumed
+that all the network components exhibit a tree structure. In the 2-RRG the topological
+constraint that all the nodes are of degree k = 2 imposes the formation of cycles. There-
+fore, the generating function formalism cannot be used to analyze the distribution of cycle
+lengths in 2-RRGs. A naive attempt to use the generating function formalism to obtain the
+distribution of cluster sizes (which are also the cycle lengths) in 2-RRGs fails to determine
+the distribution.
+RRGs with c ≥ 3 are supercritical. They consist of a giant component that encompasses
+26
+
+the whole network. While the local structure of the the network is typically tree-like, at
+larger scales it exhibits cycles with a broad distribution of cycle lengths. The length of a
+cycle is given by the number of nodes (or edges) that reside along the cycle. The longest
+possible cycle is a Hamiltonian cycle of length ℓ = N. The expected number of cycles of
+length ℓ in an undirected RRG that consists of N nodes of degree c ≥ 3, where ℓ ≪ ln N, is
+given by [40–42]
+⟨Gℓ⟩ = (c − 1)ℓ
+2ℓ
+.
+(107)
+This implies that for c ≥ 3 the number of cycles of length ℓ proliferates exponentially as ℓ
+is increased, as long as ℓ ≪ ln N. Although these results were not claimed to hold in the
+case of c = 2, it is interesting to examine their relevance to 2-RRGs. In the special case of
+an undirected 2-RRG, where c = 2, Eq. (107) is reduced to
+⟨Gℓ⟩ = 1
+2ℓ.
+(108)
+In Fig. 6 we present analytical results (solid lines) for the expected number ⟨Gℓ⟩ of cycles of
+length ℓ in undirected 2-RRGs, obtained from Eq. (108), as a function of ℓ for N = 10 (a)
+and N = 104 (b). We also present the results obtained from computer simulations (circles).
+It is found that for N = 10 there is a big difference between the analytical results obtained
+from Eq. (108) and the simulation results. In contrast, for N = 104 the analytical results are
+in very good agreement with the results of computer simulations for ℓ ≪ N. This implies
+that Eq. (108) is valid for 2-RRGs in the large network limit and for sufficiently short cycles.
+For larger values of ℓ Eq. (108) is no longer valid, as ⟨Gℓ⟩ becomes an increasing function of
+ℓ. Note that the simulation results for ⟨Gℓ⟩ exceed the values predicted by Eq. (108). The
+total number of nodes can be expressed in the form
+N =
+N
+�
+ℓ=1
+ℓ⟨Gℓ⟩,
+(109)
+which is obtained by averaging Eq. (6) over the ensemble. Inserting ⟨Gℓ⟩ from Eq. (108)
+into the right hand side of Eq. (109), it yields only N/2 nodes instead of N nodes. This
+implies that Eq. (108) is valid only as long as ℓ ≪ N. Indeed, Fig. 6 reveals that Eq. (108)
+misses the very long cycles whose length is of order N.
+27
+
+1
+2
+3
+4
+5
+6 7 8 910
+0.1
+0.2
+0.3
+0.4
+0.5
+100
+101
+102
+103
+104
+10-4
+10-3
+10-2
+10-1
+FIG. 6.
+Analytical results (solid lines) for the expected number ⟨Gℓ⟩ of cycles of length ℓ, in
+an undirected 2-RRG that consists of N nodes, for N = 10 (a) and for N = 104 (b), obtained
+from Eq. (108), on a log-log scale. We also present results obtained from computer simulations
+(circles). For N = 10 the simulation results deviate significantly from the prediction of Eq. (108).
+For N = 104 there is a very good agreement between the analytical results and the simulation
+results in the range of ℓ ≪ N. The agreement between the analytical results and the simulation
+results improves as N is increased.
+VII.
+SUMMARY
+2-RRGs are networks in which each node has two links. Therefore, these networks consist
+of a set of closed cycles whose lengths are determined by the random process of bond
+formation between the nodes. In this paper we have calculated the distributions PN(S =
+s) of the number of cycles in directed and undirected 2-RRGs.
+Starting from the joint
+distributions of cycle lengths PN({gℓ}) we obtained exact results for PN(S = s), which are
+expressed in terms of the Stirling numbers of the first kind. In sufficiently large networks
+these distributions can be expressed in terms of more elementary functions. We also derived
+closed-form expressions for the moments and cumulants of PN(S = s). It was found that to
+leading order, in directed 2-RRGs, the cumulants of all orders n = 1, 2, . . . satisfy κn ≃ ln N,
+while in undirected 2-RRGs they satisfy κn ≃ 1
+2 ln N. This implies that in the large N limit
+the distributions PN(S = s) converge towards the Poisson distribution.
+28
+
+This work was supported by the Israel Science Foundation grant no. 1682/18.
+[1] B. Bollob´as, Random Graphs, Second Edition (Cambridge University Press, Cambridge, 2001).
+[2] S.N. Dorogovtsev and J.F.F. Mendes, Evolution of Networks: From Biological Nets to the
+Internet and WWW (Oxford University Press, Oxford, 2003).
+[3] S. Havlin and R. Cohen, Complex Networks: Structure, Robustness and Function (Cambridge
+University Press, New York, 2010).
+[4] E. Estrada, The structure of complex networks: Theory and applications (Oxford University
+Press, Oxford, 2011).
+[5] A. Barrat, M. Barth´elemy and A. Vespignani, Dynamical Processes on Complex Networks
+(Cambridge University Press, Cambridge, 2012).
+[6] V. Latora, V. Nicosia and G. Russo, Complex Networks: Principles, Methods and Applications
+(Cambridge University Press, Cambridge, 2017).
+[7] M.E.J. Newman, Networks: an Introduction, Second Edition (Oxford University Press, Ox-
+ford, 2018).
+[8] R. van der Hofstad, Random graphs and complex networks (Cambridge University Press, Cam-
+bridge, 2016).
+[9] S.N. Dorogovtsev and J.F.F. Mendes, The Nature of Complex Networks (Oxford University
+Press, Oxford, 2022).
+[10] B. Bollob´as, A probabilistic proof of an asymptotic formula for the number of labelled regular
+graphs, Euro. J. Combin. 1, 311 (1980).
+[11] M. Molloy and A. Reed, A critical point for random graphs with a given degree sequence,
+Random Structures and Algorithms 6, 161 (1995).
+[12] M. Molloy and A. Reed, The size of the giant component of a random graph with a given
+degree sequence, Combin., Prob. and Comp. 7, 295 (1998).
+[13] M.E.J. Newman, S.H. Strogatz and D.J. Watts, Random graphs with arbitrary degree distri-
+butions and their applications, Phys. Rev. E 64, 026118 (2001).
+[14] B.K. Fosdick, D.B. Larremore, J. Nishimura and J. Ugander, Configuring random graph
+models with fixed degree sequences, SIAM Review 60, 315 (2018).
+[15] R. Arratia and S. Tavar´e, The cycle structure of random permutations, The Annals of Prob-
+29
+
+ability 20, 1567 (1992).
+[16] L.A. Shepp and S.P. Lloyd, Ordered cycle lengths in a random permutation, Transactions of
+the American Mathematical Society 121, 340 (1966).
+[17] P. Flajolet and A.M. Odlyzko, Random mapping statistics, Advances in Cryptography, J.J.
+Quisquater and J. Vandewalle (Eds.) Eurocrypt ’89, LNCS 434, 329 (1990)
+[18] S. W. Golomb, Random permutations, Bull. Amer. Math. Soc. 70, 747 (1964).
+[19] S. W. Golomb, Shift Register Sequences, Third Edition (World Scientific, Singapore, 2017).
+[20] M. B´ona, Combinatorics of Permutations, Second Edition (CRC Press, Boca Raton, 2012).
+[21] K. Dickman, On the frequency of numbers containing prime factors of a certain relative
+magnitude, Ark. Mat. Astron. Fysik 22A, 1 (1930).
+[22] S. R. Finch, Mathematical Constants, (Cambridge University Press, Cambridge, 2003).
+[23] S. W. Golomb and P. Gaal, On the number of permutations on n objects with greatest cycle
+length k, Adv. Appl. Math. 20, 98 (1998).
+[24] D. E. Knuth and L. Trabb Pardo, Analysis of a simple factorization algorithm, Theoret.
+Comput. Sci. 3, 321 (1976).
+[25] F.W.J. Olver, D.M. Lozier, R.F. Boisvert and C.W. Clark, NIST handbook of mathematical
+functions (Cambridge University Press, Cambridge, 2010).
+[26] C.L. Phillips, H.T. Nagle and A. Chakrabortty, Digital Control System: Analysis and Design,
+Fourth Edition (Pearson Education, Harlow, 2015).
+[27] O. Schl¨omilch, Recherches sur les coefficients des facult´es analytiques, Crelle 44, 344 (1852).
+[28] E.C. Titchmarsh, The Theory of Functions, Second Edition (Oxford University Press, Oxford,
+1939).
+[29] J.W. Wrench, Concerning two series for the gamma function, Mathematics of Computation
+22, 617 (1968).
+[30] J.W. Wrench, Erratum: Concerning two series for the gamma function, Mathematics of Com-
+putation 27, 681 (1973).
+[31] L. Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, HAL
+archives, https://hal.archives-ouvertes.fr/hal-01029331v1, arXiv:1407.5983.
+[32] K.N. Boyadzhiev, Notes on the Binomial Transform (World Scientific, Singapore, 2018).
+[33] W. Goncharov, Sur la distribution des cycles dans les permutations, C. R. (Dokl.) Acad. Sci.
+URSS 35, 267 (1942).
+30
+
+[34] W. Goncharov, On the field of combinatory analysis, Soviet Math. Izv., Ser. Math 8, 3 (1944).
+[35] R. E. Greenwood, The number of cycles associated with the elements of a permutation group,
+Amer. Math. Monthly 60, 407 (1953).
+[36] H. S. Wilf, generatingfunctionology, Third Edition (A K Peters, Wellesley, 2005).
+[37] A. Sofo, Hamonic numbers at half-integer values, Integral Transforms and Special Functions
+27, 430 (2016).
+[38] J. Choi and D. Cvijovi´c, Values of the polygamma functions at rational arguments, J. Phys.
+A 40, 15019 (2007).
+[39] M.E.J. Newman, Component sizes in networks with arbitrary degree distributions, Phys. Rev.
+E 76, 045101 (2007).
+[40] E. Marinari and R. Monasson, Circuits in random graphs: from local trees to global loops, J.
+Stat. Mech., P09004 (2004).
+[41] G. Bianconi and M. Marsili, Loops of any size and Hamilton cycles in random scale-free
+networks J. Stat. Mech., P06005 (2005).
+[42] E. Marinari and G. Semerjian, On the number of circuits in random graphs, J. Stat. Mech.,
+P06019 (2006).
+31
+
diff --git a/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/load_file.txt b/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7a3d8d30864c487dcca3168c1c21f4b004c8e2fc
--- /dev/null
+++ b/ltE2T4oBgHgl3EQfJAY7/content/tmp_files/load_file.txt
@@ -0,0 +1,927 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf,len=926
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='03686v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='stat-mech]  9 Jan 2023  The distribution of the number of cycles  in directed and undirected random 2-regular graphs  Ido Tishby, Ofer Biham, and Eytan Katzav  Racah Institute of Physics, The Hebrew University, Jerusalem, 9190401, Israel  Reimer K¨uhn  Mathematics Department, King’s College London, Strand, London WC2R 2LS, UK  Abstract  We present analytical results for the distribution of the number of cycles in directed and undi-  rected random 2-regular graphs (2-RRGs) consisting of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In directed 2-RRGs each node  has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undi-  rected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Since all the nodes are of degree k = 2, the resulting networks consist of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in  a random network instance scales with ln N, while the length of the longest cycle scales with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The number of cycles varies between different network instances in the ensemble, where the mean  number of cycles ⟨S⟩ scales with ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Here we present exact analytical results for the distribution  PN(S = s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed  in terms of the Stirling numbers of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In both cases the distributions converge to a  Poisson distribution in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The moments and cumulants of PN(S = s) are also  calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The statistical properties of directed 2-RRGs are equivalent to the combinatorics of  cycles in random permutations of N objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In this context our results recover and extend known  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  PACS numbers: 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='Ox,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='aq,89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='Da  1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  INTRODUCTION  Random networks (or graphs) consist of a set of N nodes that are connected to each  other by edges in a way that is determined by some random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' They provide a useful  conceptual framework for the study of a large variety of systems and processes in science,  technology and society [1–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The structure of a random network can be characterized by  the degree distribution P(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Here we focus on a special class of random networks, called  random regular graphs (RRGs), which exhibit degenerate degree distributions of the form  P(k) = δk,c,  (1)  where δk,k′ is the Kronecker delta and c ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' While the degrees of all the nodes  in these networks are the same, their connectivity is random and uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In that  sense, the RRG is a special case of the class of configuration model networks, which exhibit  a specified degree distribution P(k), but no degree-degree correlations [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' While RRGs  with c = 1 consist of isolated dimers, RRGs with c ≥ 3 form a giant component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Thus,  RRGs with c = 2 are a marginal case, separating between the subcritical regime of c < 2 and  supercritical regime of c > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Note that unlike some other configuration model networks that  exhibit a coexistence of a giant component and finite tree components above the percolation  transition, in RRGs with c ≥ 3 the giant component encompasses the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  RRGs with c = 2, referred to as random 2-regular graphs (2-RRGs), consist of isolated  cycles of various lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The lengths of these cycles are not determined by the topology,  but by entropic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' An important distinction is between directed 2-RRGs, in  which each node has one inbound link and one outbound link, and undirected 2-RRGs, in  which each node has two undirected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In both cases the cycles can be considered as  isolated components of the network, where the length of each cycle is equal to the size of  the network component that consists of this cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In this paper we present exact analytical results for the distribution of the number of  cycles in ensembles of directed and undirected 2-RRGs that consist of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The results  are expressed in terms of the Stirling numbers of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  We first calculate the  joint probability distribution of cycle lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This is done by mapping the directed and  undirected 2-RRGs into combinatorically equivalent permutation problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' From the joint  distribution of cycle lengths we extract the distribution PN(S = s) of the number of cycles  2  in random instances of directed and undirected 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In both cases PN(S = s) converges  to a Poisson distribution in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The moments and cumulants of PN(S = s)  are also calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The similarities and differences between the results obtained for the  directed and undirected 2-RRGs are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The statistical properties of directed 2-RRGs  are equivalent to the combinatorics of cycles in random permutations of N objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In this  context our results recover and extend known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In contrast, the statistical properties  of cycles in undirected 2-RRGs have not been studied before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The results presented in this  paper are derived specifically for c = 2 and do not apply to the more general case of RRGs  with c ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' II we present the directed and undirected  2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The joint distributions of cycle lengths are presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' IV we  calculate the distribution PN(S = s) of the number of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' V we calculate the  moments and cumulants of PN(S = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The results are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' VI and summarized  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  RANDOM 2-REGULAR GRAPHS  In both directed and undirected 2-RRGs, each node has two links and the resulting  network consists of a set of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the directed case each node has one inbound link and  one outbound link, while in the undirected case each node has two undirected links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Below  we briefly present the properties of directed and undirected 2-RRGs and their construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Directed 2-RRGs  To construct a directed 2-RRG one first assembles N nodes such that each node has one  inbound stub and one outbound stub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' During the network construction process, at each time  step one picks a random outbound stub and a random inbound stub among the remaining  open stubs and connects them to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This process is repeated N times, until no  open stubs remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This procedure follows the standard construction process of directed  configuration model networks, in which pairs of outbound and inbound stubs rather than  pairs of nodes are selected for connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The resulting ensemble of networks obtained from  this procedure is referred to as stub-labeled graphs [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' During the construction process,  3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Illustration of a single instance of a 2-RRG of N = 14 nodes, which consists of cycles of  lengths s = 1, 2, 3, 3 and 5, in the directed case (a) and in the undirected case (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  directed chains of nodes of different lengths are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In case that the open outbound  stub of one chain is selected to connect to the open inbound stub of another chain, they are  connected and form a longer chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In case that the outbound and the inbound stubs at  both ends of the same chain are selected, their connection closes the chain and turns it into  a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Once a chain of nodes becomes a cycle it does not connect to other nodes and its  structure remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' At the end of the construction, the whole network consists of  cycles of different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 1(a) we present a single instance of a directed 2-RRG of N = 14 nodes, which  consists of cycles of lengths s = 1, 2, 3, 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the 2-RRGs considered here we allow the  outbound and inbound stubs of the same node to be connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In such case,  one obtains a self-loop of length ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also allow the connection of a pair of outbound  and inbound stubs, which belong to nodes that are already connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In such case, the  resulting cycle is of length ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This choice simplifies the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Consider a directed 2-RRG consisting of N distinguishable nodes, which are marked by  the labels i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the construction of such a network the first selected inbound  stub has N possible outbound stubs to which it may connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The second selected inbound  stub has only N − 1 outbound stubs to which it may connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Continuing this process we  conclude that there are N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' possible ways to connect the N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In fact, the combinatorics  of directed 2-RRGs consisting of N nodes is equivalent to the permutation problem of N  objects [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This permutation problem can be described by a line of N cells labeled  4  by i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , N, and a corresponding set of N labeled balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The number of ways to  distribute the balls between the cells, one ball in each cell, is  RD = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',  (2)  where each arrangement of the balls in the cells corresponds to a specific network instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In this representation, a ball i′ located in cell i represents a directed link from i to i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Similarly, a ball i′′ located in cell i′ represents a directed link from i′ to i′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' One can follow  these directed links until the cell in which ball i resides is reached and the cycle is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Repeating this process for all the balls and cells provides the structure of cycles associated  with the specific permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The length of each cycle is given by the number of balls  included in the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The statistical properties of cycles in random permutations of N objects have been studied  extensively [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In particular, the properties of the longest cycle in each permutation  received much attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  It was found that the expectation value of the length of the  longest cycle in a random permutation of N objects is equal to λN, where λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' is  the Golomb-Dickman constant [21–23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Interestingly, this constant plays a role in the prime  factorization of a random integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' More specifically, it was found that the asymptotic average  number of digits in the largest prime factor of a random N-digit number is λN [21, 22, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In the other extreme, the average length of the shortest cycle in a random permutation of  N objects was also calculated and was found to be e−γ ln N, where γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='5772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' is the  Euler-Mascheroni constant [16, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Undirected 2-RRGs  To construct an undirected 2-RRG one first assembles N nodes such that each node is  connected to two undirected stubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' At each time step one selects a random pair of stubs  among the remaining open stubs and connects them to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This process is repeated  N times, until no open stubs remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This procedure follows the standard construction  process of undirected configuration model networks, in which pairs of stubs rather than pairs  of nodes are selected for connection [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The resulting ensemble of networks obtained  from this procedure is referred to as stub-labeled graphs [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Initially, one obtains linear  chains of nodes of increasing lengths that eventually close and form cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Here we consider  5  undirected 2-RRGs in which we allow the two stubs of the same node to be connected to  each other and form a self-loop of length ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also allow the connection of a pair of  stubs which belong to nodes that are already connected, resulting in a cycle of length ℓ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 1(b) we present a single instance of an undirected 2-RRG of N = 14 nodes, which  consist of cycles of lengths s = 1, 2, 3, 3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Consider an ensemble of undirected 2-RRGs consisting of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the construction  of such network the first selected stub has 2N − 1 other stubs to which it may connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The second selected stub has 2N − 3 other stubs to which it may connect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Continuing this  process we conclude that there are  RU = (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='.  (3)  possible ways to construct such network, where m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' is the double factorial of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The sta-  tistical properties of the resulting ensemble of networks can be mapped to the combina-  torial problem described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Consider a set of 2N balls, such that for each value of  i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , N there are two identical balls on which the label i is marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The two balls  labeled by a given value of i represent the two stubs of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In addition, there are N  unlabeled boxes such that in each box there is room for two balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The 2N balls are then  distributed uniformly at random in the N boxes, where each box contains two balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Each  pair of balls that resides in the same box represents one edge of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' For example,  if a ball labeled by i and a ball labeled by i′ are in the same box, it means that there is an  edge between the nodes i and i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Similarly, if the other ball labeled by i′ is in the same box  with a ball labeled i′′, it means that there is an edge between the nodes i′ and i′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' One can  follow this chain until reaching the box in which the second ball labeled by i resides, thus  closing the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The number of possible ways to distribute the 2N balls in the N cells is  given by  RU = (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  2NN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ,  (4)  where the numerator accounts for the number of permutations of 2N balls, the N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' term in  the denominator accounts for the number of permutations of the N identical cells, and the  term 2N accounts for the fact that the order in which the two balls are placed in each cell is  unimportant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Note that (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' = (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='(2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' and 2NN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' = (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' These two identities  6  establish the equivalence between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (3) and (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  THE JOINT DISTRIBUTION OF CYCLE LENGTHS  Both versions of the 2-RRG, with directed and undirected links, consist of closed cycles  of different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The configuration of cycles in a given network instance can be described  by the sequence of cycle lengths, ℓ1, ℓ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , ℓs, where 1 ≤ ℓi ≤ N, s is the number of cycles  in the given network instance and  s  �  i=1  ℓi = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (5)  For convenience and uniqueness, we order the lengths in increasing order, such that ℓ1 ≤  ℓ2 ≤ · · · ≤ ℓs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Another way to describe the configuration of cycles in a given network instance is in the  form {gℓ}N  ℓ=1 = {g1, g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , gN}, where gℓ is the number of cycles of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The gℓ’s satisfy  the condition  N  �  ℓ=1  ℓgℓ = N,  (6)  which is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The number of cycles in such a network instance can be  expressed by  s =  N  �  ℓ=1  gℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (7)  The joint distribution of cycle lengths in an ensemble of 2-RRGs consisting of N nodes is  denoted by PN({Gℓ} = {gℓ}), under the condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' For convenience we use the  notation PN({gℓ}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Below we consider the joint distributions of the cycle lengths in directed  and undirected 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Joint distribution of cycle lengths in directed 2-RRGs  Consider an ensemble of directed 2-RRGs that consist of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The number of  configurations of the form {gℓ} is given by  7  N({gℓ}) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ=1  1  ℓgℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ� ℓgℓ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (8)  To understand the first term in the denominator, consider a cycle of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Such cycle  exhibits ℓ cyclic permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This yields the first term in the denominator, which is raised  to the power gℓ to account for the number of cycles of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The term gℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' accounts  for the permutations between the gℓ degenerate cycles of length ℓ, which correspond to the  same configuration, while the Kronecker delta imposes the condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Dividing  the number of configurations N({gℓ}) by the total number of configurations RD, given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (2), we obtain the joint probability distribution of cycle lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' It is given by  PN({gℓ}) =  N  �  ℓ=1  1  ℓgℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ� ℓgℓ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (9)  It can be shown that this probability distribution is normalized, namely  �  g1  �  g2  · ·  �  gN  PN({gℓ}) = 1,  (10)  where the summation is over all the configurations of {gℓ} that satisfy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Joint distribution of cycle lengths in undirected 2-RRGs  Consider an ensemble of undirected 2-RRGs of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The number of configurations  of the form {gℓ} is  N({gℓ}) = N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ=1  2(ℓ−1)gℓ  ℓgℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' δ� ℓgℓ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (11)  This result can be understood in terms of the analogous combinatorial problem described  above, which consists of N pairs of identical balls and N unlabled boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting two  random balls in each box, the factor of N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' accounts for the number of permutations of the  N pairs of indices marked on the balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To account for the other factors, consider a cycle  of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' There are 2ℓ possible ways to exchange the ℓ pairs of identical balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' However,  due to the cyclic structure there is also a factor of 1/2, because in each cycle the lables  marked on the balls can be listed either in the clockwise direction or in the counterclockwise  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Taking this into account yields the factor of 2ℓ−1 in the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This factor  8  is raised to the power gℓ to account for the fact that there are gℓ cycles of length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the  denominator, the ℓgℓ term accounts for the number of cyclic permutations of the indices in  all the cycles of length ℓ, while the gℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' term accounts for the number of permutations of gℓ  degenerate cycles of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The probability that a random network instance will  have a cycle structure given by {gℓ} is given by  PN({gℓ}) = N({gℓ})  RU  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (12)  Inserting N({gℓ}) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (11), and RU from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (3) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (12), we obtain  PN({gℓ}) =  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ=1  2(ℓ−1)gℓ  ℓgℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' δ� ℓgℓ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (13)  Using the relation 2N = 2  �N  ℓ=1 ℓgℓ, we obtain  PN({gℓ}) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ=1  1  (2ℓ)gℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ� ℓgℓ,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (14)  This expression differs from the corresponding result for directed 2-RRGs in two ways: it has  a factor of (2ℓ)gℓ in the denominator instead of ℓgℓ and there is a pre-factor that is required  in order to maintain the normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The factor of 2gℓ in the denominator means that  as the number of cycles is increased the configuration becomes exponentially less probable  than the corresponding configuration of the directed 2-RRG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  THE DISTRIBUTION OF THE NUMBER OF CYCLES  The probability PN(S = s) that a random network instance includes s cycles can be  calculated by summing up over all the combinations of {gℓ} that consist of s cycles, namely  PN(S = s) =  �  g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',gN  PN({gℓ})δ�  ℓ gℓ,s,  (15)  where the configurations {gℓ} satisfy the condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Below we calculate the  distribution PN(S = s) for the directed and undirected 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Distribution of the number of cycles in directed 2-RRGs  Inserting the expression for PN({gℓ}) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (9) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (15), we obtain  9  PN(S = s) =  �  g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',gN≥0  N  �  ℓ=1  1  ℓgℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ� ℓgℓ,Nδ�  ℓ gℓ,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (16)  For the analysis below it is convenient to perform a change of variables from gℓ, ℓ =  1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , N to ℓi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This transformation is based on the identity  �  g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',gN≥0  1  g1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='g2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' gN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ�  ℓ gℓ,s = Ns  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' = 1  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs=1  1,  (17)  which is a result of the multinomial theorem (equation 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='9 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting the  constraints that �N  ℓ=1 ℓgℓ = N = �s  i=1 ℓi, obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (5) and (6) we obtain the  identity  �  g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',gN≥0  1  g1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='g2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' gN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ�  ℓ gℓ,sδ� ℓgℓ,N = 1  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ1,ℓ2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs=1  δ�  i ℓi,N  (18)  Using this transformation, we express Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (16) in the form  PN(S = s) = 1  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs=1  1  ℓ1ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ℓs  δ�  i ℓi,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (19)  Below we use the discrete Laplace transform, which is related to the one-sided Z-transform  and to the starred transform [26], to evaluate the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We denote  the sum on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (19) by  fs(N) =  �  ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs≥1  1  ℓ1ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ℓs  δ�s  i=1 ℓi,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (20)  The discrete Laplace transform of fs(N) is given by  �fs(z) =  ∞  �  N=0  zNfs(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (21)  Inserting fs(N) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (20) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (21), we obtain  �fs(z) =  �  ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs≥1  1  ℓ1ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ℓs  ∞  �  N=0  zNδ�s  i=1 ℓi,N,  (22)  where  ∞  �  N=0  zNδ�s  i=1 ℓi,N = z  �s  i=1 ℓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (23)  10  Decomposing the multiple summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (22) into a product of sums over the ℓi’s, we  obtain  �fs(z) =  � ∞  �  ℓ=1  zℓ  ℓ  �s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (24)  Carrying out the summation, we obtain  �fs(z) = [− ln (1 − z)]s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (25)  The next step is to apply the inverse discrete Laplace transform on �fs(z) to obtain fs(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  To this end we use identity 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25], which is given by  [ln(1 + y)]k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  =  ∞  �  n=0  s(n, k)yn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ,  (26)  where s(n, k) is the Stirling number of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' These Stirling numbers can be expressed  in the form  s(n, k) = (−1)n−k  � n  k  �  ,  (27)  where  � n  k  �  is the unsigned Stirling number of the first kind [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting s(n, k) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (27) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (26), we obtain  [ln(1 + y)]k = k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  ∞  �  n=0  (−1)n−k  � n  k  �yn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (28)  Inserting y = −z into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (28) we rewrite �fs(z) in the form  �fs(z) = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  ∞  �  n=0  1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  � n  s  �  zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (29)  To obtain the inverse discrete Laplace transform of �fs(z) we use the fact that  L−1 {zn} (N) = δN,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (30)  Applying the inverse discrete Laplace transform on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (29), we obtain  11  fs(N) = s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  � N  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (31)  Inserting fs(N) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (31) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (19), we obtain  PN(S = s) = 1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  � N  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (32)  The normalization of the distribution PN(S = s) can be confirmed using identity 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='29 in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the context of permutations, the result expressed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (32) implies that  � N  s  �  counts the number of permutations with precisely s cycles among the N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' permutations of  N objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This is consistent with the combinatorial interpretation of the unsigned Stirling  number of the first kind [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The cumulative distribution of the number of cycles is given  by  PN(S ≤ s) = 1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  s  �  s′=1  � N  s′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (33)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 2(a) we present exact analytical results (solid line) for the cumulative distribution  PN(S ≤ s) of the number of cycles in a directed 2-RRG that consists of N = 10 nodes,  obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are found to be in very good agreement with  the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (32) provides an exact result for PN(S = s), it is useful to express this  distribution in terms of more elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This is possible in the asymptotic limit  of large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The limit of N ≫ 1 corresponds to the limit of z → 1− of the discrete Laplace  transform �fs(z) [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' If one replaces the variable z by e−x, then this limit corresponds to  x ≪ 1 in �fs(e−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In this limit the expression for �fs (e−x) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (25) can be approximated  by  �fs  �  e−x�  ≃ (− ln x)s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (34)  In order to calculate the inverse Laplace transform of �fs (e−x), we use the relation  L−1  � 1  xν (− ln x)s  �  =  � d  dν  �s �Nν−1  Γ(ν)  �  ,  (35)  12  0  1  2  3  4  5  6  7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8  1  0  1  2  3  4  5  6  7  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Exact analytical results (solid lines) for the cumulative distribution of the number of  cycles in a directed 2-RRG (a) and an undirected 2-RRG (b) of N = 10 nodes, obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (33) and (57), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are in very good agreement with the results  obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  where Γ(ν) is the Gamma function [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The inverse Laplace transform of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (34) is  obtained by taking the limit ν → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To this end we use the general Leibnitz  rule (equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='12 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25])  � d  dν  �s �Nν−1  Γ(ν)  �  =  s  �  i=0  �s  i  � �� d  dν  �s−i  Nν−1  � �� d  dν  �i �  1  Γ(ν)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (36)  Below we evaluate the derivatives that appear in the two terms on the right hand side of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The derivative in the first term is given by  � d  dν  �i  Nν−1 = Nν−1(ln N)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (37)  Thus, for ν → 0+ we obtain  lim  ν→0+  �� d  dν  �i  Nν−1  �  = (ln N)i  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (38)  The derivative in the second term on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (36) is denoted by  hi = lim  ν→0+  �� d  dν  �i �  1  Γ(ν)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (39)  13  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The coefficients ai, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , 10, which appear in the power series of the reciprocal  gamma function given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (42)  i  ai  1  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='5772156649  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6558780715  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0420026350  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='1665386114  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0421977345  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0096219715  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0072189432  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0011651675  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='0002152416  By its definition, hi is the coefficient of the i’th power of ν in the Taylor expansion of 1/Γ(ν)  around ν = 0, namely  1  Γ(ν) =  ∞  �  i=1  hi  i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (40)  This expansion is often written as a power-series of the form [29, 30]  1  Γ(ν) =  ∞  �  i=1  aiνi,  (41)  where ai = hi/i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='. The first two coefficients are given by a1 = 1 and a2 = γ, where γ is the  Euler-Mascheroni constant [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Higher order coefficients can be obtained from the recursion  equation  ai =  1  i − 1  �  a2ai−1 −  i−1  �  j=2  (−1)jζ(j)ai−j  �  ,  (42)  where ζ(j) is the Riemann zeta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The coefficients ai for i = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' , 10, obtained  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (42), are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' These coefficients can also be obtained from the  integral representation [31]  14  an = (−1)n  πn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  � π  0  e−tIm [(ln t − iπ)n] dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (43)  Using this notation we obtain  fs(N) ≃ s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  s  �  i=1  ai  (ln N)s−i  (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ,  (44)  which leads to  PN(S = s) ≃ 1  N  s  �  i=1  ai  (ln N)s−i  (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (45)  Note that for sufficiently large N the sum is dominated by the first few terms for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  First, apart from the first few terms, the coefficients ai become negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Second,  the power of ln N decreases as i is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The cumulative distribution of the number of cycles is given by  PN(S ≤ s) = 1  N  s  �  s′=1  s′  �  i=1  ai  (ln N)s′−i  (s′ − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (46)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 3(a) we present analytical results (solid line) for the large N approximation of the  cumulative distribution PN(S ≤ s) of the number of cycles in a directed 2-RRG of N = 104  nodes, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are found to be in very good agreement  with the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Distribution of the number of cycles in undirected 2-RRGs  We now turn to calculate the distribution PN(S = s) of the number of cycles in undirected  2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting the expression for PN({gℓ}) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (14) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (15), we obtain  PN(S = s) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  g1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',gN≥0  N  �  ℓ=1  1  (2ℓ)gℓgℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='δ�  ℓ ℓgℓ,Nδ�  ℓ gℓ,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (47)  Using the change of variables presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (18), we obtain  PN(S = s) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  1  2ss!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',ℓs=1  1  ℓ1ℓ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ℓs  δ�  i ℓi,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (48)  15  0  5  10  15  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8  1  0  5  10  15  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Analytical results (solid lines) for the cumulative distribution of the number of cycles in  a directed 2-RRG (a) and an undirected 2-RRG (b) of N = 104 nodes, obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (46)  and (60), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are in very good agreement with the results obtained  from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Note that the sum on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (48) is equal to fs(N), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  We thus obtain  PN(S = s) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  1  2sN!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  � N  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (49)  Below we show that the distribution PN(S = s) is properly normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To this end we use  equation 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='7 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25], which can be written in the form  N  �  s=1  (−1)N−s  � N  s  �  xs = (x − N + 1)N,  (50)  where (a)N is the Pochhammer symbol [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting x = −1/2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (50), we obtain  N  �  s=1  � N  s  � 1  2s = (−1)N  �1  2 − N  �  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (51)  Expressing the Pochhammer symbol on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (51) as a ratio between  two Gamma functions, we obtain  16  N  �  s=1  � N  s  � 1  2s = (−1)NΓ(1/2)  Γ(1/2 − N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (52)  Using Euler’s reflection formula [25]  Γ  �1  2 − N  �  Γ  �1  2 + N  �  = (−1)Nπ,  (53)  and the Legendre duplication formula [25]  Γ(1/2 + N) = 21−2N√πΓ(2N)  Γ(N) ,  (54)  we obtain  N  �  s=1  � N  s  � 1  2s = 21−2N Γ(2N)  Γ(N) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (55)  Expressing the Gamma functions of integer variables in terms of factorials and double fac-  torials, we obtain  N  �  s=1  � N  s  � 1  2s = (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='.  (56)  This confirms the normalization of PN(S = s), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (49) we obtain  the cumulative distribution of the number of cycles, which is given by  PN(S ≤ s) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  s  �  s′=1  1  2s′  � N  s′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (57)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 2(b) we present exact analytical results (solid line) for the cumulative distribution  P(S ≤ s) of the number of cycles in undirected 2-RRGs that consist of N = 10 nodes,  obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are found to be in very good agreement with  the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (49) provides an exact result for PN(S = s), it is useful to express this  distribution in terms of more elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This is possible in the asymptotic limit  of large N, where the ratio of double factorials can be approximated by  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ≃  √  πN + O  �  N−1/2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (58)  17  Using this result and inserting fs(N) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (44) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (48), we find that  PN(S = s) ≃ 1  2s  � π  N  s  �  i=1  ai  (ln N)s−i  (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (59)  From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (59) we obtain the cumulative distribution of the number of cycles, which is given  by  PN(S ≤ s) ≃  � π  N  s  �  s′=1  1  2s′  s′  �  i=1  ai  (ln N)s′−i  (s′ − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (60)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 3(b) we present analytical results (solid line) for the cumulative distribution of  the number of cycles in undirected 2-RRGs of N = 104 nodes, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The  analytical results are found to be in very good agreement with the results obtained from  computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  MOMENTS AND CUMULANTS  In this section we calculate the moments and cumulants of the distribution of the number  of cycles in 2-RRGs that consist of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To this end we introduce the moment generating  function, which is given by  M(t) = E  �  etS�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (61)  The cumulant generating function is given by  K(t) = ln M(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (62)  Using this function one can calculate the cumulants via differentiation according to  κn = dnK(t)  dtn  ����  t=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (63)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Moments and cumulants in directed 2-RRGs  The moment generating function of directed 2-RRGs is given by  18  M(t) = 1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  s=0  ets  � N  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (64)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (50) with x = −et, we obtain  M(t) = (−1)N  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  −et − N + 1  �  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (65)  The moment generating function M(t) may also be written in the form  M(t) = Γ(N + et)  Γ(et)N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' ,  (66)  in agreement with the results presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The corresponding cumulant generating  function is given by  K(t) = ln  �Γ(N + et)  Γ(et)N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (67)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (67) we obtain the first two cumulants, which are given by  ⟨S⟩ = κ1 = HN,  (68)  where HN is the harmonic number [32], and  Var(S) = κ2 = HN − H(2)  N ,  (69)  where H(m)  N  is the generalized harmonic number of order m [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Similarly, one can calculate  higher order cumulants such as  κ3 = HN − 3H(2)  N + 2H(3)  N  (70)  and  κ4 = HN − 7H(2)  N + 12H(3)  N − 6H(4)  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (71)  In the limit of large N we can use the asymptotic expression for the distribution PN(S = s),  given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (45), and obtain  M(t) ≃ 1  N  ∞  �  s=0  s  �  i=1  estai  (ln N)s−i  (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (72)  19  Exchanging the order of summations, we obtain  M(t) ≃ 1  N  ∞  �  i=1  ai  ∞  �  s=i  est(ln N)s−i  (s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (73)  Shifting the summation index in the second sum, we obtain  M(t) ≃ 1  N  ∞  �  i=1  aieit  ∞  �  s=0  est(ln N)s  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (74)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (41) we carry out the two summations and obtain  M(t) ≃ 1  N  1  Γ(et) exp  �  et ln N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (75)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (62) we obtain the cumulant generating function, which is given by  K(t) ≃  �  et − 1  �  ln N − ln  �  Γ  �  et��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (76)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (63) we obtain the cumulants, which take the form  κn ≃ ln N − dn  dtn ln  �  Γ  �  et�� ����  t=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (77)  In order to calculate high order derivatives of ln [Γ (et)] we use the identity (equation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4 in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [32])  dn  dtnf(et) =  n  �  m=1  � n  m  �  emtf (m)(et),  (78)  where  � n  m  �  is the Stirling number of the second kind and f (m)(x) is the mth derivative of  f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also use the fact that  dn  dzn ln Γ(z) = ψ(n−1)(z),  (79)  where ψ(n)(z) is the nth derivative of the digamma function [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Using these identities, we  obtain  dn  dtn ln  �  Γ  �  et�� ����  t=0  =  n  �  m=1  � n  m  �  ψ(m−1)(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (80)  20  It is also known that (equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='12 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25])  ψ(0)(1) = −γ,  (81)  and that for m ≥ 1 (equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [25])  ψ(m)(1) = (−1)m+1m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='ζ(m + 1),  (82)  where ζ(m) is the Riemann zeta function [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Combining the results derived above, we  obtain  κn ≃ ln N + γ +  n  �  m=2  � n  m  �  (−1)m−1(m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='ζ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (83)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (83) we write down explicitly the first few cumulants of PN(S = s) in the large  N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' They are given by  κ1 ≃ ln N + γ  κ2 ≃ ln N + γ − π2  6  κ3 ≃ ln N + γ − π2  2 + 2ζ(3)  κ4 ≃ ln N + γ − 7π2  6  + 12ζ(3) − π4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (84)  The results for κ1 and κ2 are in agreement with the classical results reported in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' [33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In the large N limit all the cumulants are of the form ln N + O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This essentially implies  that in the large N limit the distribution PN(S = s) approaches a Poisson distribution with  a parameter ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Comparing between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (68)-(71) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (83), and using the fact that for m ≥ 2  lim  N→∞ H(m)  N  = ζ(m),  (85)  we obtain a general expression for the cumulants at finite values of N, which is given by  κn = HN +  n  �  m=2  � n  m  �  (−1)m−1(m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='H(m)  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (86)  21  101  102  103  104  105  0  2  4  6  8  10  12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Analytical results for the mean number of cycles ⟨S⟩ as a function of the network size N,  in directed 2-RRGs (solid line) and in undirected 2-RRGs (dashed line), obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (68)  and (91), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are in very good agreement with the results obtained  from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To leading order, in directed 2-RRGs ⟨S⟩ ≃ ln N, while in  undirected 2-RRGs ⟨S⟩ ≃ 1  2 ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 4 we present analytical results (solid line) for the mean number of cycles ⟨S⟩ in  directed 2-RRGs as a function of the network size N, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical  results are in very good agreement with the results obtained from computer simulations  (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 5 we present analytical results (solid line) for the variance Var(S) in directed  2-RRGs as a function of the network size N, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results  are in very good agreement with the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Moments and cumulants in undirected 2-RRGs  The moment generating function of undirected 2-RRGs is given by  M(t) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  N  �  s=0  ets  2s  � N  s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (87)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (50) with x = −et/2, we obtain  22  101  102  103  104  105  0  2  4  6  8  10  12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Analytical results for the variance of the distribution of the number of cycles as a function  of the network size N, in directed 2-RRGs (solid line) and in undirected 2-RRGs (dashed line),  obtained from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (69) and (92), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results are in very good agreement  with the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' To leading order, in directed 2-RRGs  Var(S) ≃ ln N, while in undirected 2-RRGs Var(S) ≃ 1  2 ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  M(t) = (−1)N  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  −et  2 − N + 1  �  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (88)  The moment generating function M(t) may also be written in the form  M(t) =  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Γ  �  N + et  2  �  Γ  �et  2  �  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (89)  The corresponding cumulant generating function is given by  K(t) = ln  \uf8ee  \uf8f0  (2N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (2N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Γ  �  N + et  2  �  Γ  �et  2  �  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (90)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (63) we obtain the first four cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The first cumulant is given by  ⟨S⟩ = κ1 = 1  2HN− 1  2 + ln 2,  (91)  where HN− 1  2 is an Harmonic number at a half-integer value [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The second cumulant is  given by  23  Var(S) = κ2 = 1  2HN− 1  2 + ln 2 − 1  4  �  H(2)  N− 1  2 + 2ζ(2)  �  ,  (92)  while the third and fourth cumulants are given by  κ3 = 1  2HN− 1  2 + ln 2 − 3  4  �  H(2)  N− 1  2 + 2ζ(2)  �  + 1  4  �  H(3)  N− 1  2 + 6ζ(3)  �  (93)  and  κ4 = 1  2HN− 1  2 + ln 2 − 7  4  �  H(2)  N− 1  2 + 2ζ(2)  �  + 3  2  �  H(3)  N− 1  2 + 6ζ(3)  �  − 3  8  �  H(4)  N− 1  2 + 14ζ(4)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (94)  In the limit of large N we can use the asymptotic expression for the distribution PN(S =  s), given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (61) we obtain an asymptotic expression for the  moment generating function, which is given by  M(t) ≃  � π  N  ∞  �  s=0  s  �  i=1  estai  2i  (ln N)s−i  2s−i(s − i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='.  (95)  Exchanging the order of summations and shifting the summation index in the second sum,  we obtain  M(t) ≃  � π  N  ∞  �  i=1  ai  eit  2i  ∞  �  s=0  est(ln N)s  2ss!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (96)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (41) we carry out the two summations and obtain  M(t) ≃  � π  N  1  Γ(et/2) exp  �et  2 ln N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (97)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (62) we obtain the cumulant generating function, which is given by  K(t) ≃ et − 1  2  ln N − ln  � 1  √πΓ  �et  2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (98)  Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (63) we obtain the cumulants, which take the form  κn ≃ 1  2 ln N − dn  dtn ln  � 1  √πΓ  �et  2  �� ����  t=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (99)  Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (78) and (79), we obtain  24  dn  dtn ln  � 1  √πΓ  �et  2  �� ����  t=0  =  n  �  m=1  � n  m  �  2−mψ(m−1)  �1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (100)  It is also known that [38]  ψ(0)  �1  2  �  = −γ − ln 4,  (101)  and that for m ≥ 1 [38]  ψ(m)  �1  2  �  = (−1)m+1m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  2m+1 − 1  �  ζ(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (102)  Combining the results derived above, we obtain  κn ≃ ln N  2  + γ + ln 4  2  +  n  �  m=2  � n  m  �  (−1)m−1 �  1 − 2−m�  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='ζ(m),  (103)  which becomes exact in the large N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (103) we write down explicitly the  first few cumulants of PN(S = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' They are given by  κ1 ≃ ln N  2  + γ + ln 4  2  κ2 ≃ ln N  2  + γ + ln 4  2  − π2  8  κ3 ≃ ln N  2  + γ + ln 4  2  − 3π2  8  + 7  4ζ(3)  κ4 ≃ ln N  2  + γ + ln 4  2  − 7π2  8  + 21  2 ζ(3) − π4  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (104)  In the large N limit all the cumulants are of the form 1  2 ln N +O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This essentially implies  that in the large N limit the distribution PN(S = s) approaches a Poisson distribution with  a parameter 1  2 ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Using the fact that for m ≥ 2  lim  N→∞ H(m)  N− 1  2 = ζ(m),  (105)  we obtain a general expression for the cumulants at finite values of N, which is given by  κn = 1  2HN− 1  2 + ln 2 +  n  �  m=2  � n  m  �  (−1)m−12−m(m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  �  H(m)  N− 1  2 + (2m − 2)ζ(m)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (106)  25  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  4 we present analytical results (dashed line) for the mean number of cycles  ⟨S⟩ in undirected 2-RRGs as a function of the network size N, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  The analytical results are in very good agreement with the results obtained from computer  simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 5 we present analytical results (dashed line) for the variance Var(S) in undirected  2-RRGs as a function of the network size N, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The analytical results  are in very good agreement with the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  DISCUSSION  Below we discuss the similarities and differences between the directed and undirected  2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In directed 2-RRGs the mean number of cycles ⟨S⟩ scales with ln N, while in  undirected 2-RRGs it scales with 1  2 ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Thus, the expected number of cycles in directed  2-RRGs is twice as large as in undirected 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  This is due to the fact that in the  construction of undirected 2-RRGs each end of a given chain may connect to both sides of  any other linear chain, while in directed 2-RRGs it may only connect to the complementary  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' As a result, in undirected 2-RRGs the connection of chains forming a longer chain is  more probable than in directed 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Thus, in undirected 2-RRGs the competing process  of closing a chain to form a cycle is less probable than in directed 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This implies that  in undirected 2-RRGs the cycles are expected to be longer and their number is expected to  be smaller than in directed 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  2-RRGs are marginal networks that reside at the boundary between the subcritical regime  and the supercritical regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the subcritical regime, configuration model networks consist  of many finite tree components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The distribution of sizes of these tree components can be  calculated using the framework of generating functions [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In this framework it is assumed  that all the network components exhibit a tree structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the 2-RRG the topological  constraint that all the nodes are of degree k = 2 imposes the formation of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' There-  fore, the generating function formalism cannot be used to analyze the distribution of cycle  lengths in 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' A naive attempt to use the generating function formalism to obtain the  distribution of cluster sizes (which are also the cycle lengths) in 2-RRGs fails to determine  the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  RRGs with c ≥ 3 are supercritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' They consist of a giant component that encompasses  26  the whole network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' While the local structure of the the network is typically tree-like, at  larger scales it exhibits cycles with a broad distribution of cycle lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The length of a  cycle is given by the number of nodes (or edges) that reside along the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The longest  possible cycle is a Hamiltonian cycle of length ℓ = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The expected number of cycles of  length ℓ in an undirected RRG that consists of N nodes of degree c ≥ 3, where ℓ ≪ ln N, is  given by [40–42]  ⟨Gℓ⟩ = (c − 1)ℓ  2ℓ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (107)  This implies that for c ≥ 3 the number of cycles of length ℓ proliferates exponentially as ℓ  is increased, as long as ℓ ≪ ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Although these results were not claimed to hold in the  case of c = 2, it is interesting to examine their relevance to 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In the special case of  an undirected 2-RRG, where c = 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (107) is reduced to  ⟨Gℓ⟩ = 1  2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  (108)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 6 we present analytical results (solid lines) for the expected number ⟨Gℓ⟩ of cycles of  length ℓ in undirected 2-RRGs, obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108), as a function of ℓ for N = 10 (a)  and N = 104 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also present the results obtained from computer simulations (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  It is found that for N = 10 there is a big difference between the analytical results obtained  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108) and the simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In contrast, for N = 104 the analytical results are  in very good agreement with the results of computer simulations for ℓ ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This implies  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108) is valid for 2-RRGs in the large network limit and for sufficiently short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  For larger values of ℓ Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108) is no longer valid, as ⟨Gℓ⟩ becomes an increasing function of  ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Note that the simulation results for ⟨Gℓ⟩ exceed the values predicted by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The  total number of nodes can be expressed in the form  N =  N  �  ℓ=1  ℓ⟨Gℓ⟩,  (109)  which is obtained by averaging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (6) over the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Inserting ⟨Gℓ⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108)  into the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (109), it yields only N/2 nodes instead of N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This  implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108) is valid only as long as ℓ ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Indeed, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 6 reveals that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108)  misses the very long cycles whose length is of order N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  27  1  2  3  4  5  6 7 8 910  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='5  100  101  102  103  104  10-4  10-3  10-2  10-1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Analytical results (solid lines) for the expected number ⟨Gℓ⟩ of cycles of length ℓ, in  an undirected 2-RRG that consists of N nodes, for N = 10 (a) and for N = 104 (b), obtained  from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108), on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also present results obtained from computer simulations  (circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' For N = 10 the simulation results deviate significantly from the prediction of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (108).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  For N = 104 there is a very good agreement between the analytical results and the simulation  results in the range of ℓ ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' The agreement between the analytical results and the simulation  results improves as N is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  SUMMARY  2-RRGs are networks in which each node has two links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Therefore, these networks consist  of a set of closed cycles whose lengths are determined by the random process of bond  formation between the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In this paper we have calculated the distributions PN(S =  s) of the number of cycles in directed and undirected 2-RRGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Starting from the joint  distributions of cycle lengths PN({gℓ}) we obtained exact results for PN(S = s), which are  expressed in terms of the Stirling numbers of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' In sufficiently large networks  these distributions can be expressed in terms of more elementary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' We also derived  closed-form expressions for the moments and cumulants of PN(S = s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' It was found that to  leading order, in directed 2-RRGs, the cumulants of all orders n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' satisfy κn ≃ ln N,  while in undirected 2-RRGs they satisfy κn ≃ 1  2 ln N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' This implies that in the large N limit  the distributions PN(S = s) converge towards the Poisson distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  28  This work was supported by the Israel Science Foundation grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 1682/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Bollob´as, Random Graphs, Second Edition (Cambridge University Press, Cambridge, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Dorogovtsev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mendes, Evolution of Networks: From Biological Nets to the  Internet and WWW (Oxford University Press, Oxford, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Havlin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Cohen, Complex Networks: Structure, Robustness and Function (Cambridge  University Press, New York, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Estrada, The structure of complex networks: Theory and applications (Oxford University  Press, Oxford, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Barrat, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Barth´elemy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Vespignani, Dynamical Processes on Complex Networks  (Cambridge University Press, Cambridge, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Latora, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Nicosia and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Russo, Complex Networks: Principles, Methods and Applications  (Cambridge University Press, Cambridge, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Newman, Networks: an Introduction, Second Edition (Oxford University Press, Ox-  ford, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' van der Hofstad, Random graphs and complex networks (Cambridge University Press, Cam-  bridge, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Dorogovtsev and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mendes, The Nature of Complex Networks (Oxford University  Press, Oxford, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Bollob´as, A probabilistic proof of an asymptotic formula for the number of labelled regular  graphs, Euro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 1, 311 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [11] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Molloy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Reed, A critical point for random graphs with a given degree sequence,  Random Structures and Algorithms 6, 161 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Molloy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Reed, The size of the giant component of a random graph with a given  degree sequence, Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=', Prob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' and Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 7, 295 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Newman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Strogatz and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Watts, Random graphs with arbitrary degree distri-  butions and their applications, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' E 64, 026118 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [14] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Fosdick, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Larremore, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Nishimura and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Ugander, Configuring random graph  models with fixed degree sequences, SIAM Review 60, 315 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Arratia and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Tavar´e, The cycle structure of random permutations, The Annals of Prob-  29  ability 20, 1567 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Shepp and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Lloyd, Ordered cycle lengths in a random permutation, Transactions of  the American Mathematical Society 121, 340 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [17] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Flajolet and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Odlyzko, Random mapping statistics, Advances in Cryptography, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Quisquater and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Vandewalle (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=') Eurocrypt ’89, LNCS 434, 329 (1990)  [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Golomb, Random permutations, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 70, 747 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Golomb, Shift Register Sequences, Third Edition (World Scientific, Singapore, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' B´ona, Combinatorics of Permutations, Second Edition (CRC Press, Boca Raton, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [21] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Dickman, On the frequency of numbers containing prime factors of a certain relative  magnitude, Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Fysik 22A, 1 (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Finch, Mathematical Constants, (Cambridge University Press, Cambridge, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Golomb and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Gaal, On the number of permutations on n objects with greatest cycle  length k, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 20, 98 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Knuth and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Trabb Pardo, Analysis of a simple factorization algorithm, Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' 3, 321 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [25] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Olver, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Lozier, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Boisvert and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Clark, NIST handbook of mathematical  functions (Cambridge University Press, Cambridge, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Phillips, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Nagle and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Chakrabortty, Digital Control System: Analysis and Design,  Fourth Edition (Pearson Education, Harlow, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [27] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Schl¨omilch, Recherches sur les coefficients des facult´es analytiques, Crelle 44, 344 (1852).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [28] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Titchmarsh, The Theory of Functions, Second Edition (Oxford University Press, Oxford,  1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Wrench, Concerning two series for the gamma function, Mathematics of Computation  22, 617 (1968).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Wrench, Erratum: Concerning two series for the gamma function, Mathematics of Com-  putation 27, 681 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Fekih-Ahmed, On the Power Series Expansion of the Reciprocal Gamma Function, HAL  archives, https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='fr/hal-01029331v1, arXiv:1407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='5983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [32] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Boyadzhiev, Notes on the Binomial Transform (World Scientific, Singapore, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [33] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Goncharov, Sur la distribution des cycles dans les permutations, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' (Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=') Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  URSS 35, 267 (1942).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  30  [34] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Goncharov, On the field of combinatory analysis, Soviet Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=', Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Math 8, 3 (1944).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Greenwood, The number of cycles associated with the elements of a permutation group,  Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Monthly 60, 407 (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [36] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Wilf, generatingfunctionology, Third Edition (A K Peters, Wellesley, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Sofo, Hamonic numbers at half-integer values, Integral Transforms and Special Functions  27, 430 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [38] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Choi and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Cvijovi´c, Values of the polygamma functions at rational arguments, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  A 40, 15019 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Newman, Component sizes in networks with arbitrary degree distributions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  E 76, 045101 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [40] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Marinari and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Monasson, Circuits in random graphs: from local trees to global loops, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=', P09004 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [41] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Bianconi and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Marsili, Loops of any size and Hamilton cycles in random scale-free  networks J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=', P06005 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  [42] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Marinari and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Semerjian, On the number of circuits in random graphs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content=',  P06019 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
+page_content='  31' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ltE2T4oBgHgl3EQfJAY7/content/2301.03686v1.pdf'}
diff --git a/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf b/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf
new file mode 100644
index 0000000000000000000000000000000000000000..15c7309d09531f72b50990b2074ed7449201e179
--- /dev/null
+++ b/mtE1T4oBgHgl3EQfhQRC/content/2301.03238v1.pdf
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6bc24e101368403ccb9520e6ac2b378f0870f02a0b6c21a11f1068ded40d5387
+size 173031
diff --git a/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/2301.04132v1.pdf.txt b/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/2301.04132v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1ec2b7e8fde782543ead91ca648b59bf1336ed37
--- /dev/null
+++ b/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/2301.04132v1.pdf.txt
@@ -0,0 +1,4554 @@
+arXiv:2301.04132v1  [physics.atom-ph]  10 Jan 2023
+Model-QED-operator approach to relativistic calculations of the
+nuclear recoil effect in many-electron atoms and ions
+I. S. Anisimova,1 A. V. Malyshev,1 D. A. Glazov,1 M. Y. Kaygorodov,1
+Y. S. Kozhedub,1 G. Plunien,2 and V. M. Shabaev1
+1Department of Physics, St. Petersburg State University,
+Universitetskaya 7/9, 199034 St. Petersburg, Russia
+2Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden,
+Mommsenstraße 13, D-01062 Dresden, Germany
+Abstract
+A model-operator approach to fully relativistic calculations of the nuclear recoil effect on energy
+levels in many-electron atomic systems is worked out. The one-electron part of the model operator
+for treating the normal mass shift beyond the Breit approximation is represented by a sum of
+semilocal and nonlocal potentials. The latter ones are constructed by employing the diagonal and
+off-diagonal matrix elements rigorously evaluated for hydrogenlike ions to first order in the electron-
+to-nucleus mass ratio. The specific mass shift beyond the lowest-order relativistic approximation
+has a form which can be directly employed in calculations. The capabilities of the method are
+probed by comparison of its predictions with the results of ab initio QED calculations.
+The
+proposed operator can be easily incorporated into any relativistic calculation based on the Dirac-
+Coulomb-Breit Hamiltonian.
+1
+
+I.
+INTRODUCTION
+An accurate description of the nuclear recoil effect is substantial for the proper analysis
+of a large number of spectroscopic experiments aimed to measure various atomic properties
+such as, e.g., binding and transition energies or bound-electron g factors. Obviously, this
+effect is most pronounced in isotope differences of the corresponding properties, see, e.g.
+Refs. [1–11].
+The nuclear recoil leads to the so-called mass shift.
+Along with the field
+shift caused by the finite-nuclear size, these effects constitute the dominant contribution to
+the isotope shifts. Joint high-precision theoretical and experimental studies of the isotope
+differences not only allow one to determine nuclear parameters, e.g., changes in the mean-
+square charge radii, but also pave the way in the search of new physics [12–22].
+Within the (m/M)(αZ)4mc2 approximation, where m and M are the masses of the elec-
+tron and nucleus, respectively, α is the fine-structure constant, and Z is the nuclear charge
+number, the nuclear recoil effect on binding energies can be described by the relativistic
+mass-shift operator [23–26]. The fully relativistic theory of the nuclear recoil effect to first
+order in m/M, to all orders in αZ, and to zeroth order in α can be formulated only within
+the framework of quantum electrodynamics (QED) [23, 24, 26], see also Refs. [27–29]. De-
+spite the smallness of the nuclear-strength parameter αZ for light atoms, the contribution
+of the higher orders may nevertheless be significant even in the case of hydrogen [30, 31].
+Therefore, an accurate treatment of the nuclear recoil effect demands a nonperturbative
+(in αZ) consideration. To date, the corresponding ab initio QED calculations have been
+performed only for few-electron systems, see, e.g. Refs. [4, 29, 32–34] and references therein.
+The computational difficulty of the rigorous methods rapidly increases with the number of
+electrons, which makes them practically infeasible at larger scales. A similar problem exists
+for evaluation of the radiative QED corrections associated with the electron self-energy and
+vacuum polarization. For this reason, approximate and efficient approaches for including
+both the QED and recoil effects within the methods based on the Dirac-Coulomb-Breit
+Hamiltonian [35–45] are urgent.
+For the case of the radiative QED corrections, our group suggested the model-QED-
+operator approach [46] which recently has been extended to the region of superheavy ele-
+ments [47]. The QEDMOD Fortran package to generate the operator was presented in Ref. [48].
+This operator was successfully applied to the approximate description of the QED effects
+2
+
+on binding and transition energies in various many-electron systems [49–60].
+The main
+goal of the present work is to design a similar approach for the QED calculations of the
+nuclear recoil effect on energy levels beyond the approximation corresponding to the mass-
+shift operator. The application of the proposed approach in combination with the standard
+electron-correlation methods should make possible the approximate QED treatment of this
+effect in systems where rigorous calculations are rather problematic at the moment.
+In view of the significant progress achieved over the past decades in the accuracy of g-
+factor measurements in Penning traps [6, 9, 61–67], high-precision evaluation of the nuclear
+recoil effect in the presence of an external magnetic field becomes essential as well. The
+QED theory of the nuclear recoil effect on the atomic g factor valid to all orders in αZ was
+elaborated in Ref. [68]. The corresponding ab initio calculations have been performed for
+few-electron ions in Refs. [6, 69–75]. In the case of more complicated systems, the nuclear
+recoil effect on the bound-electron g factor can be treated nowadays only within the lowest-
+order relativistic approximation by means of the effective four-component approach derived
+from the QED formalism in Ref. [70]. In this context, the model-operator approach devel-
+oped in the present work for the nuclear recoil effect on binding energies can be considered
+as a first step towards the construction of a more general operator suitable for studying the
+bound-electron g factors as well.
+The paper is organized as follows. In Sec. II, we give a brief description of the relativistic
+theory of the nuclear recoil effect on energy levels in atoms and ions. Sec. III is devoted
+to the construction of the model operator for QED calculations of the nuclear recoil effect.
+In Sec. IV, the numerical results are presented in a wide range of Z = 5 − 100, and the
+performance of the suggested approach is demonstrated by comparing its predictions with
+the results of ab initio calculations. The nonperturbative (in αZ) expressions for the one- and
+two-electron matrix elements, which describe the nuclear recoil effect on the binding energies,
+are derived in Appendix A. In Appendix B, these expressions are additionally transformed to
+make them convenient for practical calculations. Finally, Appendix C summarizes formulas
+necessary for the construction of the local effective potentials employed in the tests of the
+model-QED operator in Sec. IV.
+Relativistic units (ℏ = 1 and c = 1) and the Heaviside charge unit (e2 = 4πα, where
+e < 0 is the electron charge) are used throughout the paper.
+3
+
+II.
+QED THEORY OF THE NUCLEAR RECOIL EFFECT
+As is well known, within the nonrelativistic approximation the nuclear recoil contribution
+to the binding energy of a hydrogenlike atom can be found by replacing the electron mass m
+with the reduced mass, mr = mM/(m + M). For atoms with more than one electron, this
+recipe is insufficient, and the two-electron part of the nuclear recoil effect has to be taken
+into account [76]. The lowest-order relativistic (Breit) correction of first order in m/M can
+be obtained by employing the mass-shift Hamiltonian [23–25]. For N-electron system, this
+operator reads as
+HMS =
+1
+2M
+N
+�
+i,j=1
+�
+pi · pj − αZ
+ri
+�
+αi + (αi · ri)
+r2
+i
+ri
+�
+· pj
+�
+,
+(1)
+where the indices i and j enumerate the electrons, p = −i∇ is the momentum operator, r
+is the position vector, r = |r|, and α are the Dirac matrices. The first term in the square
+brackets in Eq. (1) corresponds to the nonrelativistic nuclear recoil operator, whereas the
+second term determines the leading relativistic correction.
+Following Hughes and Eckart [76], the nuclear recoil contribution to atomic spectra is
+usually divided into the normal (NMS) and specific (SMS) mass shifts. Accordingly, the
+Hamiltonian (1) can be represented as a sum
+HMS = HNMS + HSMS ,
+(2)
+where the first operator corresponds to the terms i = j in Eq. (1) and the second one
+corresponds to i ̸= j. For further discussion, it is useful to rewrite Eq. (1) in the form
+HMS =
+1
+2M
+N
+�
+i,j=1
+�
+pi · pj − 2Di(0) · pj
+�
+(3)
+by introducing the vector operator
+D(0) = αZ
+2r
+�
+α + (α · r)
+r2
+r
+�
+.
+(4)
+The mass-shift operator HMS yields the nuclear recoil corrections up to the order (m/M)(αZ)4mc2.
+This operator is widely used in relativistic calculations of atomic spectra and isotope shifts,
+where the nuclear recoil effect is particularly significant [4, 15, 41, 54, 77–94].
+The fully relativistic theory of the nuclear recoil effect on binding energies can be formu-
+lated only within the rigorous QED approach (beyond the Breit approximation). To first
+4
+
+(b)
+(c)
+(d)
+(a)
+FIG. 1. One-electron nuclear recoil diagrams: the Coulomb (a), one-transverse (b) and (c), and
+two-transverse (d) contributions. See the text and Ref. [26] for the description of the Feynman
+rules.
+order in m/M, to all orders in αZ, and to zeroth order in α the corresponding theory was
+developed in Refs. [23, 24, 26]. The formalism worked out in Ref. [26] is the most suitable
+for the goals of the present study. Within this formalism, the pure nuclear recoil effect is
+taken into account by modifying the standard QED Hamiltonian of the electron-positron
+field interacting with the quantized electromagnetic field and the classical Coulomb poten-
+tial of the nucleus V . Namely, an extra term is added to the interaction part of the QED
+Hamiltonian, see Ref. [26] for details. As a result, the pure nuclear recoil effect on energy
+levels can be obtained on equal footing with the non-recoil QED effects, e.g., the electron
+self-energy and vacuum polarization, by means of the perturbation theory in the interac-
+tion representation of the Furry picture [95]. A convenient approach to construct the QED
+perturbation series both for single and quasi-degenerate levels is provided by the two-time
+Green’s function (TTGF) method [96]. This method is employed in Appendix A to derive
+the formal expressions for the matrix elements describing the nuclear recoil effect on bind-
+ing energies. Within this approach, the zeroth-order one-electron wave functions |ψn⟩ and
+energies εn are assumed to be the solutions of the Dirac equation
+hD|ψn⟩ ≡
+�
+α · p + βm + V
+�
+|ψn⟩ = εn|ψn⟩ .
+(5)
+The all-order (in αZ) expressions describing the nuclear recoil effect can be divided
+into the NMS and SMS parts as well. The one-electron (NMS) and two-electron (SMS)
+contributions are given by the Feynman diagrams shown in Figs. 1 and 2, respectively. The
+exhaustive description of the additional diagram-technique rules, which arise in connection
+with the treatment of the nuclear recoil effect, can be found in Ref. [26], see also Ref. [96].
+5
+
+(a)
+(b)
+(c)
+(d)
+FIG. 2. Two-electron nuclear recoil diagrams: the Coulomb (a), one-transverse (b) and (c), and
+two-transverse (d) contributions.
+To explain the terminology employed throughout the paper, we briefly comment on these
+rules using Fig. 1 as an example. First of all, the double line and the vertex with a small dot
+in the figure correspond to the conventional diagram technique of the bound-state QED [96].
+Namely, the line denotes the electron propagator in the potential V , and the vertex arises
+from the standard interaction of the electron-positron and electromagnetic fields. All the
+other diagram elements originate due to the presence of the aforementioned extra term in
+the QED Hamiltonian. The Coulomb gauge established itself as the most appropriate one
+for the nuclear-recoil-effect studies [23, 24, 26, 28], and it leads to the natural terminology.
+In this gauge, the photon propagator Dµν(ω, r) is divided into the Coulomb,
+D00(ω, r) =
+1
+4πr ,
+(6)
+and transverse,
+Dlk(ω, r) = − 1
+4π
+�
+exp
+�
+i
+√
+ω2 + i0 r
+�
+r
+δlk + ∇l∇k
+exp
+�
+i
+√
+ω2 + i0 r
+�
+− 1
+ω2r
+�
+.
+(7)
+parts, while the remaining components of the photon propagator are equal to zero, Dl0 =
+D0l = 0 (l, k = 1, 2, 3). All the recoil contributions in Fig. 1 can be classified with respect to
+the number of the propagators (7) involved. There are three possibilities [26]: (i) the dotted
+line connecting two bold dots in Fig. 1(a) depicts the so-called “Coulomb recoil” interaction,
+which does not contain the transverse part of the photon propagator at all; (ii) the dashed
+line with the bold dot at one of the ends in Figs. 1(b) and 1(c) stands for the “one-transverse-
+photon recoil” interaction, which includes Dlk once; (iii) finally, the dashed line with the
+bold dot in the middle in Fig. 1(d) designates the “two-transverse-photon recoil” interaction,
+6
+
+which involves the product of two photon propagators. We note that the approach initially
+developed in Ref. [23] leads to the same result as the formalism of Ref. [26] but it implies the
+summation of the infinite sequences of Feynman diagrams describing the electron-nucleus
+interaction via photon exchange. In this case, the three discussed possibilities correspond to
+the summation of the diagrams with zero, one, and two transverse photons and an arbitrary
+number of the Coulomb photons.
+To all orders in αZ, the NMS contribution for the state |ψa⟩ can be expressed as fol-
+lows [23, 26]
+ENMS = ⟨ψa|P(εa)|ψa⟩ ,
+(8)
+where we have introduced the operator P(E) by
+⟨ψi|P(E)|ψk⟩ = i
+2π
+∞
+�
+−∞
+dω
+�
+n
+⟨ψiψn|R(ω)|ψnψk⟩
+E − ω − εn(1 − i0)
+(9)
+with
+R(ω) = 1
+M
+�
+p1 − D1(ω)
+�
+·
+�
+p2 − D2(ω)
+�
+.
+(10)
+In Eq. (9), |ψiψn⟩ = |ψi⟩|ψn⟩ is the direct product of the one-electron wave functions. The
+operator D(ω) in Eq. (10) is related to the transverse part of the photon propagator (7),
+and its kth Cartesian component, Dk(ω), is equal to
+Dk(ω) = −4παZαlDlk(ω) .
+(11)
+The ω → 0 limit of the operator D(ω) coincides with formula (4). The indices 1 and 2 in
+Eq. (10) designate the electron, on which the corresponding operators act. Let us rewrite
+R(ω) in the form:
+R(ω) = Rc + Rtr1(ω) + Rtr2(ω) ,
+(12)
+Rc = 1
+M p1 · p2 ,
+(13)
+Rtr1(ω) = − 1
+M
+�
+p1 · D2(ω) + D1(ω) · p2
+�
+,
+(14)
+Rtr2(ω) = 1
+M D1(ω) · D2(ω) .
+(15)
+Substituting Eq. (12) into Eq. (8), one arrives at the Coulomb, one-transverse-photon, and
+two-transverse-photon contributions to the NMS.
+7
+
+Let us turn to the discussion of the SMS contribution which corresponds to the Feynman
+diagrams in Fig. 2. For simplicity, we consider the case of a one-determinant unperturbed
+wave function,
+Ψab(r1, r2) = 1
+√
+2
+�
+P
+(−1)PψP a(r1)ψP b(r2) ,
+(16)
+where P is the permutation operator. The generalization to the case of a many-determinant
+wave function is straightforward.
+The nonperturbative (in αZ) expression for the SMS
+contribution reads as [24, 26]
+ESMS = −⟨ψbψa|R(∆)|ψaψb⟩ ,
+(17)
+where ∆ = εa − εb.
+The formula (17) gives the “exchange” term for the two-electron
+operator R. The “direct” one is equal to zero, since the matrix elements of the operators
+p and D are zeroes for states of the same parity.
+Substituting (12) into Eq. (17), one
+obtains the expansion of the SMS contribution into the Coulomb, one-transverse-photon,
+and two-transverse-photon parts.
+Over the past three decades, numerous QED calculations of the nuclear recoil effect
+on binding energies were carried out [4, 29, 31–34, 84, 88, 97, 98]. We should note that
+the expressions (8) and (17) with the operator D defined by Eq. (11) are derived for the
+point-nucleus case. In Ref. [33], it was argued that the dominant part of the finite-nuclear
+size (FNS) correction to the nuclear recoil effect can be accounted for by employing the po-
+tential of the extended nucleus in Eq. (5), and since that paper this prescription is usually
+used in the QED calculations of the mass shift. It was also found there that the treatment
+of the FNS correction to the nuclear recoil effect within the Breit approximation defined by
+the operator (1) leads to an artificial contribution of order (m/M)(αZ)5(Rnucl/λ)mc2 which
+even exceeds the main contribution of order (m/M)(αZ)4(Rnucl/λ)2mc2 (here λ = ℏ/(mc) is
+the Compton wavelength). This artificial contribution arises from the first (Coulomb) term
+in Eq. (1). However, it is completely cancelled by the corresponding FNS correction to the
+Coulomb part of the QED nuclear recoil effect, which means that the rigorous theory for
+the FNS contribution beyond the main (m/M)(αZ)4(Rnucl/λ)2mc2 term can be formulated
+only within the framework of QED. In Ref. [99], an additional FNS correction, which re-
+sults from modifying the Breit-approximation mass-shift operator (1) by inserting the form
+factor into the nuclear vertex [100, 101], was evaluated. This operator differs from the one
+obtained by considering the zero-frequency limit of the photon propagator in the modified
+8
+
+Coulomb gauge [102, 103]. The main difference between the results obtained using these
+two operators is due to a spurious contribution of the order (m/M)(αZ)5(Rnucl/λ)mc2 in
+the one-transverse-photon part, which occurs only in the calculation with the operator from
+Ref. [102, 103]. Like to the case of the Coulomb contribution, this spurious term is cancelled
+by the related FNS contribution to the one-transverse-photon QED correction, provided it
+is also calculated with the photon propagator in the modified Coulomb gauge [102, 103].
+In Refs. [34, 71, 119], the additional FNS correction from Ref. [99], which is free from the
+spurious term, was used to estimate the uncertainty of the calculations based on the pre-
+scription of Ref. [33]. To date, the most elaborated evaluation of the FNS correction to the
+nuclear recoil effect was performed within the QED approach in Ref. [102]. This calculation
+accomplished for the 1s state has confirmed that the dominant part of the FNS correction
+to the nuclear recoil effect is indeed covered by the recipe of Ref. [33], which we also follow
+here. The rigorous QED treatment of the total FNS correction to the nuclear recoil effect
+lies beyond the scope of the present work.
+Restricting Eqs. (8) and (17) by the lowest-order relativistic approximation leads to the
+NMS and SMS parts of the MS operator (1), respectively. Until recently, all nonperturba-
+tive (in αZ) calculations were limited by the zeroth order in 1/Z, i.e., the electron-electron
+interaction corrections to the nuclear recoil effect were considered at best only within the
+Breit approximation. In our recent works, we have advanced the QED theory of the nuclear
+recoil effect. Specifically, we have considered to all orders in αZ the electron-electron inter-
+action correction of first order in 1/Z to the one-electron [104] and two-electron [105] parts
+of the nuclear recoil effect on binding energies in atoms and ions. These higher-order QED
+contributions being also beyond the scope of the present work can be calculated additionally
+if needed.
+Finally, the radiative (∼ α) as well as the second-order (in m/M) recoil corrections
+are accessible nowadays only within the αZ-expansion approaches, see Refs. [106–108] and
+references therein. These contributions are also not considered in this paper.
+III.
+MODEL-QED OPERATOR FOR THE NUCLEAR RECOIL EFFECT
+The QED calculations for many-electron systems, becoming increasingly relevant in view
+of the considerable progress of the experiment, are complicated and in many cases currently
+9
+
+FIG. 3. Self-energy diagram.
+FIG. 4. Vacuum-polarization diagram.
+inaccessible. This is true for the radiative corrections as well as for the QED treatment of the
+nuclear recoil effect. For this reason, there is a vital need for a simple approximate approach
+for taking into account the QED corrections in various relativistic calculations. The conven-
+tional first-order QED corrections correspond to the self-energy (SE), vacuum-polarization
+(VP), and one-photon-exchange diagrams shown in Figs. 3, 4, and 5, respectively.
+The
+approximate model-QED-operator approach to evaluate these effects has been suggested re-
+cently by our group in Refs. [46–48]. In this section, the analogy will be traced that allows us
+FIG. 5. One-photon exchange diagram.
+10
+
+to construct a similar approach for the nuclear recoil contributions beyond the lowest-order
+relativistic approximation.
+The model-QED-operator approach [46] is worked out within the TTGF method. It is
+based on the fact that the QED effects to first order in α can be described by an effective
+Hamiltonian acting in the subspace which is spanned by all Slater determinants made up
+of the positive-energy solutions of the Dirac equation (5), see Ref. [109] for details. This
+effective Hamiltonian has the form
+H = Λ(+)
+� N
+�
+i
+�
+hD
+i + hSE
+i
++ hVP
+i
+�
++
+N
+�
+i<j
+h1ph
+ij
+�
+Λ(+) ,
+(18)
+where Λ(+) is the product of the one-electron projectors on the positive-energy eigenfunctions
+of the Dirac Hamiltonian hD, and the operators hSE, hVP, and h1ph arise from the diagrams
+in Figs. 3, 4, and 5, respectively. According to the QED theory of the nuclear recoil effect,
+described briefly in Sec. II, to first order in m/M and to all orders in αZ, the Hamiltonian
+H has to be supplemented by the term
+δH = Λ(+)
+� N
+�
+i
+hNMS
+i
++
+N
+�
+i<j
+hSMS
+ij
+�
+Λ(+) ,
+(19)
+where the operators hNMS and hSMS are associated with the diagrams in Figs. 1 and 2,
+respectively. Below we discuss how the part of the operator δH which is beyond the Breit
+approximation can be adapted for the practical relativistic electronic-structure calculations.
+We note that the entire formalism can be generalized to the case where the potential V in
+Eq. (5) along with the nuclear potential includes also some local screening potential Vscr
+modeling the interelectronic-interaction effects within the initial approximation, i.e., to the
+extended version of the Furry picture.
+Let us start with the analogy between the one-photon-exchange diagram in Fig. 5 and
+the two-electron nuclear recoil diagrams in Fig. 2. For the one-photon-exchange diagram,
+the TTGF method leads to the following symmetric expression [96, 109]
+h1ph =
+εi1,εi2,εk1,εk2>0
+�
+i1̸=i2,k1̸=k2
+|ψi1ψi2⟩⟨ψi1ψi2|1
+2
+�
+I(εi1 − εk1) + I(εi2 − εk2)
+�
+|ψk1ψk2⟩⟨ψk1ψk2| ,
+(20)
+where the indices i1, i2, k1, and k2 enumerate the positive-energy one-electron Dirac eigen-
+functions,
+I(ω) = e2αµ
+1αν
+2Dµν(ω, r12) ,
+(21)
+11
+
+αµ ≡ γ0γµ = (1, α) are the Dirac matrices, and the photon propagator in the Coulomb
+gauge is given by Eqs. (6) and (7). The matrix elements for the two-electron nuclear recoil
+diagrams are derived within the TTGF method in Appendix A, see Eq. (A21) for the final
+formula. As a result, the operator hSMS can be represented as
+hSMS =
+εi1,εi2,εk1,εk2>0
+�
+i1̸=i2,k1̸=k2
+|ψi1ψi2⟩⟨ψi1ψi2|1
+2
+�
+R(εi1 − εk1) + R(εi2 − εk2)
+�
+|ψk1ψk2⟩⟨ψk1ψk2| , (22)
+where the operator R(ω) is defined by Eq. (10). Therefore, the expressions (20) and (22)
+differ only by the replacement of I with R.
+Taking the photon propagator Dµν(ω, r12)
+in the Coulomb gauge at the zero-energy transfer (ω = 0), one obtains from Eq. (20)
+the interaction part of the Dirac-Coulomb-Breit Hamiltonian. In the Coulomb gauge, the
+formula (20) considered beyond the lowest-order relativistic approximation leads to the so-
+called frequency-dependent Breit interaction. The corresponding correction is readily taken
+into account within the electronic-structure calculations, see, e.g., Refs. [57, 60, 110–113], and
+does not require the construction of the model operator. When omitting the two-transverse-
+photon contribution and taking the ω → 0 limit in R(ω), the expression (22) boils down to
+the SMS part of the mass-shift operator (1). The remaining part of the contribution given
+by Eq. (22) can be treated on equal footing with the frequency-dependent Breit-interaction
+correction.
+Now we turn to the discussion of the one-electron contributions. The VP diagram in
+Fig. 4 does not have a counterpart in the QED theory of the nuclear recoil effect. Therefore,
+we will focus on the SE diagram in Fig. 3 and the one-electron nuclear recoil diagrams in
+Fig. 1. First, let us introduce the SE operator Σ(E),
+⟨ψi|Σ(E)|ψk⟩ = i
+2π
+∞
+�
+−∞
+dω
+�
+n
+⟨ψiψn|I(ω)|ψnψk⟩
+E − ω − εn(1 − i0) .
+(23)
+This formal expression suffers from ultraviolet divergences and has to be renormalized to-
+gether with the mass counterterm, see, e.g., Refs. [114–117]. Within the TTGF method,
+one can obtain the following symmetric expression for the operator hSE [46, 109]:
+hSE =
+εi,εk>0
+�
+i,k
+|ψi⟩⟨ψi|1
+2
+�
+ΣR(εi) + ΣR(εk)
+�
+|ψk⟩⟨ψk| ,
+(24)
+where ΣR(ε) is the renormalized SE operator. The corresponding diagonal and off-diagonal
+matrix elements are tabulated, e.g., in Refs. [46, 47]. The matrix elements for the one-
+electron nuclear recoil diagrams are considered in the framework of the TTGF method in
+12
+
+Appendix A, see Eq. (A14) for the final expression. In contrast to the SE diagram, this
+contribution is ultraviolet finite and does not require any regularization. Nevertheless, its
+evaluation is a complex task compared to the calculations with the mass-shift Hamilto-
+nian (1). The operator hNMS valid to all orders in αZ can be written in the form
+hNMS =
+εi,εk>0
+�
+i,k
+|ψi⟩⟨ψi|1
+2
+�
+P(εi) + P(εk)
+�
+|ψk⟩⟨ψk| ,
+(25)
+The analogy between Eqs. (24) and (25) is evident, and it is employed in the present work to
+develop the model-QED-operator approach for the one-electron nuclear recoil contribution.
+Since within the Breit approximation this contribution can be treated by means of the
+operator
+hNMS
+Breit =
+1
+2M
+�
+p2 − 2D(0) · p
+�
+,
+(26)
+we construct the model operator for the remaining (QED) part of hNMS, namely for
+hNMS
+h.o. ≡
+εi,εk>0
+�
+i,k
+|ψi⟩⟨ψi|
+�1
+2
+�
+P(εi) + P(εk)
+�
+− hNMS
+Breit
+�
+|ψk⟩⟨ψk| .
+(27)
+The model-operator approach should simultaneously solve two issues. First, since there is
+no simple-enough procedure for the ab initio evaluation of the P-operator matrix elements
+for arbitrary levels (including the continuum-spectrum levels), one has to terminate the
+summation in Eq. (27). Second, the short interaction range should be kept. To address
+these problems, similarly to Ref. [46], we approximate the QED recoil operator hNMS
+h.o.
+by a
+sum of semilocal (with respect to r) and nonlocal potentials
+˜hNMS
+h.o. = Vs.l. + Vn.l. .
+(28)
+The operator P, like the operator ΣR, conserves the relativistic angular quantum number κ =
+(−1)j+l+1/2(j + 1/2), where j and l are the total and orbital angular momenta, respectively.
+For this reason, the semilocal potential can be written as
+Vs.l. =
+�
+κ
+V κ
+s.l.Pκ ,
+(29)
+where the projector Pκ acts only on the angular variables, and its kernel is
+Pκ(n, n′) =
+
+
+
+�
+m
+Ωκm(n)Ω†
+κm(n′)
+0
+0
+�
+m
+Ω−κm(n)Ω†
+−κm(n′)
+
+
+
+(30)
+13
+
+with Ωκm(n) being the spherical spinor and n = r/r. In Eq. (29), we take
+V κ
+s.l.(r) = Aκ exp(−r/λ) ,
+(31)
+where the parameters Aκ are chosen to reproduce the ab initio values of the diagonal matrix
+elements of the operator hNMS
+h.o. for the state with the lowest principal quantum number n for
+a given κ. The nonlocal potential can be written in the form
+Vn.l. =
+n
+�
+j,l=1
+|φj⟩Bjl⟨φl| ,
+(32)
+where the functions {φi}n
+i=1, as in Ref. [46], are chosen to be
+φi(r) = 1
+2 [ I − (−1)siβ ] ρli(r)ψi(r) .
+(33)
+Here the index si = ni − li enumerates the positive-energy states for the given angular
+symmetry, I and β are the identity and the standard Dirac matrices, respectively, and the
+factors ρli(r) = exp [−2αZ(r/λ)/(1 + li)] provide the stronger localization of the functions
+{φi}n
+i=1 as compared to the eigenfunctions {ψi}n
+i=1 of the Dirac equation (5). Finally, the
+coefficients Bjl in Eq. (32) are determined from the condition that the matrix elements of the
+model-QED operator ˜hNMS
+h.o. evaluated in the space spanned by the functions {ψi}n
+i=1 coincide
+with the exact ones. This leads to the following equations
+n
+�
+j,l
+⟨ψi|φj⟩Bjl⟨φl|ψk⟩ = ⟨ψi|
+�1
+2
+�
+P(εi) + P(εk)
+�
+− hNMS
+Breit − Vs.l.
+�
+|ψk⟩
+(34)
+for i, k = 1 . . . n, see Ref. [46] for details. We note that the model-QED operator for the
+nuclear recoil effect is worked out in such a way that the QEDMOD Fortran package [48] can
+be readily adapted to incorporate it. We propose to refer to the resulting package as the
+RECMOD one.
+Therefore, to determine the model-QED operator we need to calculate the diagonal and
+off-diagonal matrix elements of the QED recoil operator hNMS
+h.o. with the Dirac-Coulomb wave
+functions. So far, only calculations of the diagonal matrix elements have been presented in
+the literature. In this work, the expressions derived within the TTGF method are employed
+for the ab initio QED evaluation of the one-electron nuclear recoil contributions. Concluding
+the description of the model-QED operator, we note that in practical calculations we con-
+struct the model operator employing the functions (33) which correspond to the ns states
+with the principal quantum number n ⩽ 3 and the np1/2, np3/2, nd3/2, and nd5/2 states
+with n ⩽ 4.
+14
+
+IV.
+TEST OF THE MODEL-QED OPERATOR FOR THE NUCLEAR RECOIL
+EFFECT
+In the present work, to obtain the diagonal and off-diagonal matrix elements of the
+higher-order operator hNMS
+h.o. , we first evaluate the corresponding values for the operator hNMS
+and then numerically subtract the Breit contribution given by the operator hNMS
+Breit. For the
+Coulomb part of the nuclear recoil effect, this subtraction is readily accomplished analytically
+by omitting the summation over the positive-energy states and doubling the contribution of
+the negative-energy continuum in Eq. (B2). This can be useful to avoid large cancellations
+for low values of Z. The calculations are performed for the ns, np1/2, np3/2, nd3/2, and
+nd5/2 states with the principal quantum number n ⩽ 5 in the wide range of Z = 5 − 100
+using the Dirac-Coulomb wave functions. The evaluation is carried out for the potential
+of the extended nucleus in Eq. (5).
+The nuclear-charge distribution is described by the
+homogeneously-charged-sphere model for Z ⩽ 14, and by the Fermi model otherwise. The
+nuclear-charge radii are taken from Refs. [118, 119]. The one-electron basis set to represent
+the electron Green’s function is constructed from B splines [120] within the dual-kinetic-
+balance approach [121]. The ω integration is performed in accordance with the equations
+presented in Appendix B.
+The results of the ab initio calculations are conveniently expressed in terms of the di-
+mensionless function Fnink(αZ) defined by
+⟨ψi|hNMS
+h.o. |ψk⟩ = m
+M
+(αZ)5
+(nink)3/2Fnink(αZ) mc2 .
+(35)
+Our data for the ns, np1/2, np3/2, nd3/2, and nd5/2 states are presented in Tables I, II, III, IV,
+and V, respectively. The individual Coulomb, one- and two-transverse-photon corrections
+as well as the total QED recoil contributions are shown.
+The uncertainties due to the
+approximate treatment of the nuclear-size correction are omitted, and with this in mind the
+values are accurate to all digits quoted. The function Fnink(αZ) for values of Z not listed
+in Tables I-V can be obtained using a polynomial fitting,
+Fnink(αZ) =
+N
+�
+n=1
+Fnink(αZn)
+�
+m̸=n
+Z − Zm
+Zn − Zm
+.
+(36)
+The model-QED operator for the nuclear recoil effect is constructed using the matrix
+elements for the ns states with n ⩽ 3 and the np and nd states with n ⩽ 4. By definition,
+15
+
+TABLE I. Matrix elements of the operator hNMS
+h.o.
+for the ns states calculated with the Dirac-
+Coulomb wave functions for the extended nuclei. Labels (ni, nk) stand for the function Fnink defined
+by Eq. (35). Rows labeled “c”, “tr1”, “tr2”, and “tot” contain the Coulomb, one-transverse-photon,
+two-transverse-photon and total nuclear recoil contributions, respectively.
+Z Term
+(1, 1)
+(1, 2)
+(1, 3)
+(1, 4)
+(1, 5)
+(2, 2)
+(2, 3)
+(2, 4)
+(2, 5)
+(3, 3)
+(3, 4)
+(3, 5)
+(4, 4)
+(4, 5)
+(5, 5)
+5
+c
+−0.385 22 −0.385 32 −0.385 31 −0.385 30 −0.385 28 −0.385 42 −0.385 40 −0.385 39 −0.385 38 −0.385 39 −0.385 38 −0.385 37 −0.385 37 −0.385 36 −0.385 35
+tr1
+2.732 91
+2.802 47
+2.807 94
+2.809 23
+2.809 68
+2.925 71
+2.941 92
+2.944 48
+2.945 25
+2.971 87
+2.978 25
+2.979 68
+2.989 97
+2.993 14
+2.998 90
+tr2
+−0.970 02 −0.939 91 −0.935 26 −0.933 65 −0.932 91 −0.930 21 −0.924 15 −0.922 37 −0.921 56 −0.922 21 −0.920 03 −0.919 17 −0.919 32 −0.918 30 −0.917 96
+tot
+1.377 67
+1.477 25
+1.487 37
+1.490 28
+1.491 49
+1.610 08
+1.632 36
+1.636 72
+1.638 31
+1.664 26
+1.672 84
+1.675 14
+1.685 28
+1.689 49
+1.695 60
+10
+c
+−0.359 34 −0.359 73 −0.359 70 −0.359 65 −0.359 61 −0.360 14 −0.360 10 −0.360 06 −0.360 02 −0.360 07 −0.360 02 −0.359 99 −0.359 98 −0.359 94 −0.359 90
+tr1
+2.227 48
+2.298 20
+2.303 46
+2.304 51
+2.304 78
+2.421 64
+2.437 53
+2.439 83
+2.440 40
+2.467 08
+2.473 17
+2.474 39
+2.484 57
+2.487 52
+2.493 05
+tr2
+−0.650 44 −0.619 34 −0.614 59 −0.612 92 −0.612 13 −0.607 24 −0.600 83 −0.598 94 −0.598 08 −0.598 33 −0.595 99 −0.595 06 −0.595 05 −0.593 93 −0.593 47
+tot
+1.217 70
+1.319 12
+1.329 16
+1.331 94
+1.333 04
+1.454 27
+1.476 60
+1.480 83
+1.482 30
+1.508 67
+1.517 16
+1.519 35
+1.529 54
+1.533 65
+1.539 68
+15
+c
+−0.340 02 −0.340 93 −0.340 87 −0.340 78 −0.340 70 −0.341 86 −0.341 80 −0.341 71 −0.341 63 −0.341 75 −0.341 65 −0.341 57 −0.341 55 −0.341 47 −0.341 39
+tr1
+1.952 62
+2.025 57
+2.030 60
+2.031 34
+2.031 37
+2.150 64
+2.166 24
+2.168 19
+2.168 49
+2.195 43
+2.201 16
+2.202 11
+2.212 17
+2.214 84
+2.220 07
+tr2
+−0.489 17 −0.457 02 −0.452 12 −0.450 37 −0.449 53 −0.442 51 −0.435 73 −0.433 74 −0.432 81 −0.432 67 −0.430 16 −0.429 16 −0.428 98 −0.427 77 −0.427 18
+tot
+1.123 43
+1.227 61
+1.237 60
+1.240 19
+1.241 15
+1.366 27
+1.388 70
+1.392 75
+1.394 05
+1.421 01
+1.429 36
+1.431 39
+1.441 63
+1.445 60
+1.451 49
+20
+c
+−0.324 33 −0.325 95 −0.325 87 −0.325 70 −0.325 57 −0.327 63 −0.327 54 −0.327 38 −0.327 25 −0.327 46 −0.327 30 −0.327 17 −0.327 14 −0.327 01 −0.326 87
+tr1
+1.772 26
+1.848 42
+1.853 20
+1.853 57
+1.853 29
+1.976 45
+1.991 78
+1.993 33
+1.993 29
+2.020 67
+2.025 99
+2.026 61
+2.036 55
+2.038 87
+2.043 73
+tr2
+−0.388 09 −0.354 77 −0.349 70 −0.347 85 −0.346 95 −0.337 81 −0.330 63 −0.328 52 −0.327 52 −0.327 01 −0.324 32 −0.323 24 −0.322 91 −0.321 60 −0.320 89
+tot
+1.059 85
+1.167 69
+1.177 64
+1.180 01
+1.180 77
+1.311 01
+1.333 61
+1.337 43
+1.338 52
+1.366 21
+1.374 37
+1.376 20
+1.386 49
+1.390 27
+1.395 97
+25
+c
+−0.312 16 −0.314 72 −0.314 60 −0.314 36 −0.314 16 −0.317 36 −0.317 25 −0.317 02 −0.316 81 −0.317 15 −0.316 91 −0.316 71 −0.316 68 −0.316 47 −0.316 27
+tr1
+1.644 47
+1.724 80
+1.729 33
+1.729 25
+1.728 60
+1.857 11
+1.872 22
+1.873 30
+1.872 86
+1.900 87
+1.905 70
+1.905 93
+1.915 76
+1.917 68
+1.922 11
+tr2
+−0.317 79 −0.283 13 −0.277 86 −0.275 91 −0.274 95 −0.263 62 −0.256 02 −0.253 79 −0.252 73 −0.251 83 −0.248 96 −0.247 82 −0.247 33 −0.245 93 −0.245 11
+tot
+1.014 53
+1.126 95
+1.136 87
+1.138 98
+1.139 50
+1.276 14
+1.298 94
+1.302 49
+1.303 32
+1.331 89
+1.339 83
+1.341 40
+1.351 75
+1.355 28
+1.360 73
+30
+c
+−0.302 86 −0.306 57 −0.306 42 −0.306 09 −0.305 81 −0.310 43 −0.310 30 −0.309 97 −0.309 69 −0.310 18 −0.309 85 −0.309 57 −0.309 52 −0.309 24 −0.308 96
+tr1
+1.550 87
+1.636 38
+1.640 65
+1.640 06
+1.638 98
+1.774 39
+1.789 29
+1.789 84
+1.788 94
+1.817 76
+1.822 05
+1.821 83
+1.831 53
+1.832 99
+1.836 91
+tr2
+−0.265 64 −0.229 46 −0.223 96 −0.221 91 −0.220 89 −0.207 25 −0.199 21 −0.196 85 −0.195 73 −0.194 43 −0.191 39 −0.190 19 −0.189 55 −0.188 06 −0.187 13
+tot
+0.982 37
+1.100 35
+1.110 27
+1.112 05
+1.112 29
+1.256 71
+1.279 78
+1.283 01
+1.283 52
+1.313 15
+1.320 81
+1.322 07
+1.332 46
+1.335 69
+1.340 83
+35
+c
+−0.296 03 −0.301 13 −0.300 96 −0.300 52 −0.300 14 −0.306 48 −0.306 33 −0.305 89 −0.305 51 −0.306 19 −0.305 75 −0.305 37 −0.305 31 −0.304 94 −0.304 56
+tr1
+1.481 88
+1.573 68
+1.577 68
+1.576 49
+1.574 93
+1.718 92
+1.733 66
+1.733 59
+1.732 16
+1.761 99
+1.765 67
+1.764 92
+1.774 49
+1.775 41
+1.778 76
+tr2
+−0.225 14 −0.187 23 −0.181 46 −0.179 31 −0.178 23 −0.162 10 −0.153 59 −0.151 12 −0.149 94 −0.148 22 −0.145 01 −0.143 75 −0.142 96 −0.141 39 −0.140 36
+tot
+0.960 71
+1.085 32
+1.095 26
+1.096 67
+1.096 56
+1.250 33
+1.273 73
+1.276 58
+1.276 70
+1.307 58
+1.314 90
+1.315 79
+1.326 22
+1.329 08
+1.333 84
+40
+c
+−0.291 95 −0.298 73 −0.298 52 −0.297 96 −0.297 47 −0.305 90 −0.305 73 −0.305 16 −0.304 66 −0.305 56 −0.305 00 −0.304 51 −0.304 43 −0.303 94 −0.303 45
+tr1
+1.432 10
+1.531 46
+1.535 16
+1.533 30
+1.531 16
+1.685 67
+1.700 27
+1.699 50
+1.697 44
+1.728 53
+1.731 49
+1.730 14
+1.739 58
+1.739 87
+1.742 55
+tr2
+−0.192 55 −0.152 64 −0.146 57 −0.144 32 −0.143 18 −0.124 34 −0.115 33 −0.112 73 −0.111 51 −0.109 34 −0.105 97 −0.104 66 −0.103 71 −0.102 08 −0.100 96
+tot
+0.947 60
+1.080 09
+1.090 06
+1.091 02
+1.090 51
+1.255 43
+1.279 22
+1.281 60
+1.281 27
+1.313 63
+1.320 53
+1.320 97
+1.331 44
+1.333 85
+1.338 13
+45
+c
+−0.289 62 −0.298 37 −0.298 13 −0.297 41 −0.296 80 −0.307 71 −0.307 52 −0.306 80 −0.306 17 −0.307 34 −0.306 62 −0.305 99 −0.305 90 −0.305 28 −0.304 65
+tr1
+1.398 15
+1.506 51
+1.509 91
+1.507 26
+1.504 46
+1.671 73
+1.686 23
+1.684 63
+1.681 85
+1.714 46
+1.716 60
+1.714 53
+1.723 83
+1.723 38
+1.725 28
+tr2
+−0.165 44 −0.123 23 −0.116 83 −0.114 47 −0.113 29 −0.091 42 −0.081 85 −0.079 15 −0.077 89 −0.075 22 −0.071 69 −0.070 34 −0.069 24 −0.067 55 −0.066 36
+tot
+0.943 09
+1.084 91
+1.094 94
+1.095 38
+1.094 38
+1.272 60
+1.296 85
+1.298 68
+1.297 79
+1.331 89
+1.338 29
+1.338 20
+1.348 68
+1.350 55
+1.354 27
+50
+c
+−0.289 66 −0.300 76 −0.300 49 −0.299 60 −0.298 82 −0.312 74 −0.312 53 −0.311 62 −0.310 83 −0.312 33 −0.311 43 −0.310 63 −0.310 53 −0.309 74 −0.308 95
+tr1
+1.378 25
+1.497 34
+1.500 41
+1.496 85
+1.493 28
+1.676 01
+1.690 42
+1.687 86
+1.684 22
+1.718 67
+1.719 85
+1.716 94
+1.726 07
+1.724 76
+1.725 74
+tr2
+−0.142 25 −0.097 35 −0.090 58 −0.088 11 −0.086 89 −0.061 59 −0.051 43 −0.048 62 −0.047 34 −0.044 12 −0.040 43 −0.039 06 −0.037 80 −0.036 07 −0.034 83
+tot
+0.946 35
+1.099 23
+1.109 34
+1.109 15
+1.107 57
+1.301 67
+1.326 46
+1.327 62
+1.326 06
+1.362 23
+1.367 99
+1.367 25
+1.377 74
+1.378 95
+1.381 97
+55
+c
+−0.291 87 −0.305 78 −0.305 46 −0.304 35 −0.303 39 −0.320 95 −0.320 71 −0.319 57 −0.318 58 −0.320 48 −0.319 36 −0.318 36 −0.318 23 −0.317 25 −0.316 26
+tr1
+1.371 39
+1.503 32
+1.506 02
+1.501 38
+1.496 90
+1.698 37
+1.712 73
+1.709 02
+1.704 35
+1.741 04
+1.741 07
+1.737 16
+1.746 09
+1.743 76
+1.743 64
+tr2
+−0.121 77 −0.073 73 −0.066 55 −0.063 97 −0.062 72 −0.033 46 −0.022 65 −0.019 74 −0.018 45 −0.014 61 −0.010 78 −0.009 40 −0.007 98 −0.006 23 −0.004 95
+tot
+0.957 75
+1.123 81
+1.134 01
+1.133 06
+1.130 80
+1.343 96
+1.369 37
+1.369 70
+1.367 32
+1.405 95
+1.410 94
+1.409 39
+1.419 88
+1.420 28
+1.422 43
+60
+c
+−0.296 55 −0.313 83 −0.313 46 −0.312 09 −0.310 91 −0.332 90 −0.332 62 −0.331 21 −0.329 97 −0.332 37 −0.330 97 −0.329 73 −0.329 57 −0.328 34 −0.327 11
+tr1
+1.377 42
+1.524 78
+1.527 07
+1.521 16
+1.515 60
+1.739 91
+1.754 21
+1.749 12
+1.743 20
+1.782 62
+1.781 27
+1.776 13
+1.784 85
+1.781 26
+1.779 83
+tr2
+−0.103 11 −0.051 36 −0.043 71 −0.041 03 −0.039 76 −0.005 86
+0.005 68
+0.008 66
+0.009 94
+0.014 51
+0.018 46
+0.019 82
+0.021 40
+0.023 16
+0.024 45
+tot
+0.977 76
+1.159 59
+1.169 91
+1.168 04
+1.164 93
+1.401 15
+1.427 26
+1.426 57
+1.423 17
+1.464 76
+1.468 77
+1.466 23
+1.476 68
+1.476 08
+1.477 17
+65
+c
+−0.303 46 −0.324 77 −0.324 33 −0.322 64 −0.321 18 −0.348 59 −0.348 27 −0.346 50 −0.344 96 −0.347 97 −0.346 22 −0.344 68 −0.344 48 −0.342 95 −0.341 42
+tr1
+1.396 53
+1.562 56
+1.564 38
+1.556 92
+1.550 05
+1.802 35
+1.816 60
+1.809 81
+1.802 38
+1.845 14
+1.842 08
+1.835 46
+1.843 89
+1.838 79
+1.835 75
+tr2
+−0.085 48 −0.029 30 −0.021 10 −0.018 34 −0.017 07
+0.022 39
+0.034 73
+0.037 78
+0.039 00
+0.044 43
+0.048 49
+0.049 79
+0.051 55
+0.053 27
+0.054 53
+tot
+1.007 59
+1.208 49
+1.218 94
+1.215 94
+1.211 81
+1.476 15
+1.503 07
+1.501 08
+1.496 41
+1.541 61
+1.544 36
+1.540 57
+1.550 96
+1.549 11
+1.548 86
+70
+c
+−0.312 32 −0.338 41 −0.337 87 −0.335 79 −0.333 99 −0.367 99 −0.367 59 −0.365 40 −0.363 47 −0.367 23 −0.365 05 −0.363 13 −0.362 88 −0.360 97 −0.359 08
+tr1
+1.429 20
+1.617 95
+1.619 19
+1.609 86
+1.601 40
+1.888 22
+1.902 39
+1.893 47
+1.884 16
+1.931 07
+1.925 88
+1.917 40
+1.925 49
+1.918 50
+1.913 49
+tr2
+−0.068 14 −0.006 59
+0.002 21
+0.005 04
+0.006 29
+0.052 51
+0.065 78
+0.068 83
+0.069 96
+0.076 44
+0.080 57
+0.081 76
+0.083 70
+0.085 34
+0.086 52
+tot
+1.048 75
+1.272 94
+1.283 53
+1.279 11
+1.273 70
+1.572 74
+1.600 58
+1.596 91
+1.590 65
+1.640 28
+1.641 40
+1.636 04
+1.646 32
+1.642 87
+1.640 94
+75
+c
+−0.325 49 −0.357 59 −0.356 91 −0.354 33 −0.352 11 −0.394 53 −0.394 02 −0.391 26 −0.388 84 −0.393 56 −0.390 82 −0.388 41 −0.388 09 −0.385 71 −0.383 33
+tr1
+1.478 73
+1.695 60
+1.696 15
+1.684 47
+1.674 03
+2.004 16
+2.018 20
+2.006 58
+1.994 87
+2.046 99
+2.039 09
+2.028 26
+2.035 92
+2.026 54
+2.019 05
+tr2
+−0.050 43
+0.017 69
+0.027 18
+0.030 06
+0.031 24
+0.085 88
+0.100 18
+0.103 20
+0.104 16
+0.111 93
+0.116 07
+0.117 10
+0.119 23
+0.120 71
+0.121 76
+tot
+1.102 81
+1.355 69
+1.366 42
+1.360 20
+1.353 16
+1.695 51
+1.724 36
+1.718 52
+1.710 20
+1.765 36
+1.764 35
+1.756 95
+1.767 06
+1.761 55
+1.757 47
+80
+c
+−0.342 01 −0.381 48 −0.380 61 −0.377 40 −0.374 64 −0.427 63 −0.426 97 −0.423 46 −0.420 41 −0.426 36 −0.422 88 −0.419 84 −0.419 44 −0.416 43 −0.413 44
+tr1
+1.546 61
+1.798 52
+1.798 17
+1.783 53
+1.770 61
+2.155 49
+2.169 28
+2.154 16
+2.139 39
+2.198 12
+2.186 70
+2.172 84
+2.179 94
+2.167 51
+2.156 85
+tr2
+−0.031 49
+0.044 81
+0.055 12
+0.057 99
+0.059 04
+0.124 35
+0.139 85
+0.142 75
+0.143 45
+0.152 85
+0.156 92
+0.157 67
+0.160 02
+0.161 25
+0.162 05
+tot
+1.173 10
+1.461 85
+1.472 68
+1.464 12
+1.455 01
+1.852 21
+1.882 17
+1.873 44
+1.862 42
+1.924 61
+1.920 74
+1.910 67
+1.920 53
+1.912 33
+1.905 46
+85
+c
+−0.363 06 −0.411 74 −0.410 59 −0.406 55 −0.403 11 −0.469 65 −0.468 73 −0.464 25 −0.460 37 −0.467 90 −0.463 45 −0.459 59 −0.459 05 −0.455 23 −0.451 44
+tr1
+1.637 02
+1.933 23
+1.931 69
+1.913 23
+1.897 11
+2.352 28
+2.365 60
+2.345 85
+2.327 07
+2.394 33
+2.378 27
+2.360 44
+2.366 80
+2.350 36
+2.335 60
+tr2
+−0.010 33
+0.076 35
+0.087 59
+0.090 38
+0.091 22
+0.170 37
+0.187 26
+0.189 89
+0.190 19
+0.201 69
+0.205 57
+0.205 90
+0.208 47
+0.209 30
+0.209 72
+tot
+1.263 63
+1.597 84
+1.608 68
+1.597 05
+1.585 23
+2.053 00
+2.084 13
+2.071 49
+2.056 88
+2.128 13
+2.120 39
+2.106 75
+2.116 22
+2.104 44
+2.093 89
+90
+c
+−0.387 02 −0.446 77 −0.445 20 −0.440 10 −0.435 78 −0.519 19 −0.517 86 −0.512 09 −0.507 14 −0.516 63 −0.510 91 −0.505 98 −0.505 25 −0.500 38 −0.495 56
+tr1
+1.751 33
+2.103 35
+2.100 12
+2.076 68
+2.056 45
+2.602 04
+2.614 45
+2.588 50
+2.564 41
+2.642 73
+2.620 44
+2.597 35
+2.602 71
+2.581 00
+2.560 88
+tr2
+0.014 21
+0.114 20
+0.126 52
+0.129 10
+0.129 60
+0.226 99
+0.245 50
+0.247 65
+0.247 34
+0.261 62
+0.265 08
+0.264 77
+0.267 58
+0.267 81
+0.267 63
+tot
+1.378 52
+1.770 78
+1.781 44
+1.765 68
+1.750 27
+2.309 84
+2.342 09
+2.324 07
+2.304 61
+2.387 71
+2.374 61
+2.356 14
+2.365 04
+2.348 43
+2.332 95
+95
+c
+−0.418 54 −0.492 61 −0.490 42 −0.483 89 −0.478 41 −0.584 20 −0.582 24 −0.574 69 −0.568 27 −0.580 41 −0.572 92 −0.566 54 −0.565 54 −0.559 25 −0.553 03
+tr1
+1.901 10
+2.325 75
+2.320 09
+2.289 96
+2.264 23
+2.929 69
+2.940 48
+2.905 96
+2.874 62
+2.967 70
+2.936 82
+2.906 55
+2.910 52
+2.881 65
+2.854 30
+tr2
+0.043 88
+0.161 36
+0.174 94
+0.177 10
+0.177 06
+0.299 16
+0.319 58
+0.320 91
+0.319 64
+0.337 68
+0.340 40
+0.339 10
+0.342 16
+0.341 44
+0.340 34
+tot
+1.526 44
+1.994 50
+2.004 61
+1.983 17
+1.962 88
+2.644 65
+2.677 82
+2.652 18
+2.625 99
+2.724 97
+2.704 30
+2.679 11
+2.687 14
+2.663 85
+2.641 62
+100
+c
+−0.461 28 −0.554 59 −0.551 47 −0.542 94 −0.535 85 −0.672 46 −0.669 50 −0.659 39 −0.650 90 −0.666 71 −0.656 70 −0.648 26 −0.646 85 −0.638 55 −0.630 36
+tr1
+2.100 97
+2.622 83
+2.613 55
+2.574 12
+2.540 84
+3.369 87
+3.377 77
+3.331 00
+3.289 46
+3.402 83
+3.359 72
+3.319 31
+3.321 30
+3.282 39
+3.244 99
+tr2
+0.081 50
+0.222 61
+0.237 65
+0.239 05
+0.238 14
+0.394 73
+0.417 40
+0.417 33
+0.414 57
+0.437 86
+0.439 26
+0.436 43
+0.439 73
+0.437 54
+0.435 00
+tot
+1.721 18
+2.290 85
+2.299 73
+2.270 24
+2.243 12
+3.092 14
+3.125 68
+3.088 93
+3.053 13
+3.173 98
+3.142 28
+3.107 48
+3.114 18
+3.081 38
+3.049 62
+16
+
+TABLE II. Matrix elements of the operator hNMS
+h.o.
+for the np1/2 states calculated with the Dirac-
+Coulomb wave functions for the extended nuclei. The notations are the same as in Table I.
+Z Term
+(2, 2)
+(2, 3)
+(2, 4)
+(2, 5)
+(3, 3)
+(3, 4)
+(3, 5)
+(4, 4)
+(4, 5)
+(5, 5)
+5
+c
+−0.000 21 −0.000 22 −0.000 23 −0.000 23 −0.000 24 −0.000 25 −0.000 25 −0.000 26 −0.000 26 −0.000 26
+tr1
+−0.047 48 −0.047 34 −0.049 21 −0.050 21 −0.041 34 −0.041 70 −0.042 96 −0.038 43 −0.038 63 −0.036 79
+tr2
+−0.038 02 −0.033 95 −0.033 02 −0.032 64 −0.035 70 −0.034 42 −0.034 00 −0.034 89 −0.034 32 −0.034 51
+tot
+−0.085 70 −0.081 51 −0.082 46 −0.083 08 −0.077 29 −0.076 37 −0.077 20 −0.073 57 −0.073 20 −0.071 56
+10
+c
+−0.000 76 −0.000 83 −0.000 85 −0.000 86 −0.000 90 −0.000 92 −0.000 93 −0.000 95 −0.000 96 −0.000 97
+tr1
+−0.048 58 −0.048 67 −0.050 59 −0.051 62 −0.042 94 −0.043 37 −0.044 65 −0.040 19 −0.040 42 −0.038 63
+tr2
+−0.023 13 −0.018 17 −0.017 02 −0.016 54 −0.018 55 −0.016 93 −0.016 39 −0.016 95 −0.016 22 −0.016 20
+tot
+−0.072 47 −0.067 67 −0.068 47 −0.069 02 −0.062 39 −0.061 22 −0.061 97 −0.058 08 −0.057 59 −0.055 80
+15
+c
+−0.001 61 −0.001 75 −0.001 80 −0.001 82 −0.001 91 −0.001 96 −0.001 98 −0.002 01 −0.002 03 −0.002 05
+tr1
+−0.048 87 −0.049 08 −0.051 02 −0.052 04 −0.043 48 −0.043 94 −0.045 22 −0.040 81 −0.041 05 −0.039 28
+tr2
+−0.008 21 −0.002 37 −0.001 00 −0.000 43 −0.001 38
+0.000 58
+0.001 24
+0.001 01
+0.001 90
+0.002 11
+tot
+−0.058 69 −0.053 20 −0.053 82 −0.054 29 −0.046 76 −0.045 32 −0.045 96 −0.041 81 −0.041 18 −0.039 22
+20
+c
+−0.002 74 −0.002 98 −0.003 06 −0.003 09 −0.003 24 −0.003 32 −0.003 36 −0.003 41 −0.003 45 −0.003 49
+tr1
+−0.048 49 −0.048 73 −0.050 64 −0.051 65 −0.043 15 −0.043 60 −0.044 86 −0.040 48 −0.040 71 −0.038 95
+tr2
+0.006 94
+0.013 66
+0.015 24
+0.015 89
+0.016 04
+0.018 32
+0.019 10
+0.019 20
+0.020 24
+0.020 65
+tot
+−0.044 30 −0.038 04 −0.038 46 −0.038 85 −0.030 36 −0.028 60 −0.029 12 −0.024 69 −0.023 92 −0.021 78
+25
+c
+−0.004 16 −0.004 51 −0.004 63 −0.004 68 −0.004 90 −0.005 02 −0.005 08 −0.005 15 −0.005 21 −0.005 26
+tr1
+−0.047 52 −0.047 68 −0.049 54 −0.050 52 −0.042 03 −0.042 43 −0.043 65 −0.039 28 −0.039 48 −0.037 71
+tr2
+0.022 47
+0.030 10
+0.031 87
+0.032 60
+0.033 88
+0.036 49
+0.037 37
+0.037 81
+0.038 99
+0.039 59
+tot
+−0.029 21 −0.022 09 −0.022 30 −0.022 60 −0.013 05 −0.010 96 −0.011 35 −0.006 62 −0.005 69 −0.003 38
+30
+c
+−0.005 87 −0.006 36 −0.006 52 −0.006 59 −0.006 90 −0.007 07 −0.007 15 −0.007 25 −0.007 33 −0.007 40
+tr1
+−0.045 96 −0.045 95 −0.047 72 −0.048 66 −0.040 12 −0.040 44 −0.041 60 −0.037 23 −0.037 39 −0.035 60
+tr2
+0.038 57
+0.047 13
+0.049 09
+0.049 89
+0.052 36
+0.055 30
+0.056 27
+0.057 05
+0.058 38
+0.059 16
+tot
+−0.013 26 −0.005 18 −0.005 15 −0.005 36
+0.005 34
+0.007 78
+0.007 52
+0.012 57
+0.013 67
+0.016 16
+35
+c
+−0.007 91 −0.008 57 −0.008 78 −0.008 87 −0.009 29 −0.009 52 −0.009 61 −0.009 75 −0.009 85 −0.009 94
+tr1
+−0.043 78 −0.043 50 −0.045 15 −0.046 03 −0.037 39 −0.037 59 −0.038 67 −0.034 30 −0.034 39 −0.032 57
+tr2
+0.055 45
+0.064 97
+0.067 11
+0.067 96
+0.071 72
+0.074 97
+0.076 03
+0.077 17
+0.078 62
+0.079 57
+tot
+0.003 77
+0.012 90
+0.013 18
+0.013 06
+0.025 04
+0.027 86
+0.027 74
+0.033 12
+0.034 39
+0.037 06
+40
+c
+−0.010 35 −0.011 21 −0.011 47 −0.011 58 −0.012 14 −0.012 42 −0.012 54 −0.012 71 −0.012 83 −0.012 96
+tr1
+−0.040 88 −0.040 23 −0.041 72 −0.042 52 −0.033 73 −0.033 78 −0.034 76 −0.030 36 −0.030 37 −0.028 51
+tr2
+0.073 35
+0.083 87
+0.086 17
+0.087 05
+0.092 21
+0.095 78
+0.096 90
+0.098 41
+0.099 98
+0.101 09
+tot
+0.022 11
+0.032 43
+0.032 98
+0.032 95
+0.046 34
+0.049 57
+0.049 60
+0.055 33
+0.056 77
+0.059 63
+45
+c
+−0.013 24 −0.014 33 −0.014 65 −0.014 78 −0.015 51 −0.015 86 −0.016 00 −0.016 22 −0.016 36 −0.016 51
+tr1
+−0.037 14 −0.036 00 −0.037 28 −0.037 99 −0.028 97 −0.028 83 −0.029 70 −0.025 26 −0.025 17 −0.023 26
+tr2
+0.092 53
+0.104 12
+0.106 55
+0.107 45
+0.114 16
+0.118 03
+0.119 20
+0.121 10
+0.122 76
+0.124 02
+tot
+0.042 14
+0.053 79
+0.054 62
+0.054 68
+0.069 67
+0.073 33
+0.073 50
+0.079 62
+0.081 23
+0.084 26
+50
+c
+−0.016 71 −0.018 07 −0.018 45 −0.018 60 −0.019 54 −0.019 96 −0.020 12 −0.020 39 −0.020 56 −0.020 73
+tr1
+−0.032 34 −0.030 56 −0.031 59 −0.032 19 −0.022 86 −0.022 49 −0.023 22 −0.018 72 −0.018 53 −0.016 55
+tr2
+0.113 34
+0.126 07
+0.128 61
+0.129 49
+0.137 94
+0.142 10
+0.143 29
+0.145 61
+0.147 35
+0.148 73
+tot
+0.064 29
+0.077 45
+0.078 57
+0.078 70
+0.095 54
+0.099 65
+0.099 95
+0.106 49
+0.108 26
+0.111 45
+55
+c
+−0.020 89 −0.022 56 −0.023 02 −0.023 19 −0.024 37 −0.024 87 −0.025 05 −0.025 38 −0.025 57 −0.025 76
+tr1
+−0.026 18 −0.023 57 −0.024 30 −0.024 76 −0.015 03 −0.014 38 −0.014 94 −0.010 38 −0.010 05 −0.008 01
+tr2
+0.136 21
+0.150 16
+0.152 78
+0.153 61
+0.164 02
+0.168 46
+0.169 64
+0.172 40
+0.174 18
+0.175 66
+tot
+0.089 14
+0.104 03
+0.105 47
+0.105 66
+0.124 62
+0.129 21
+0.129 64
+0.136 64
+0.138 56
+0.141 89
+60
+c
+−0.025 97 −0.028 01 −0.028 55 −0.028 74 −0.030 23 −0.030 82 −0.031 02 −0.031 42 −0.031 63 −0.031 84
+tr1
+−0.018 21 −0.014 56 −0.014 91 −0.015 23 −0.004 95 −0.003 96 −0.004 34
+0.000 32
+0.000 78
+0.002 90
+tr2
+0.161 66
+0.176 94
+0.179 60
+0.180 33
+0.193 01
+0.197 71
+0.198 82
+0.202 07
+0.203 86
+0.205 40
+tot
+0.117 48
+0.134 37
+0.136 14
+0.136 36
+0.157 82
+0.162 93
+0.163 46
+0.170 97
+0.173 01
+0.176 46
+65
+c
+−0.032 20 −0.034 68 −0.035 31 −0.035 50 −0.037 39 −0.038 07 −0.038 28 −0.038 76 −0.038 98 −0.039 20
+tr1
+−0.007 82 −0.002 83 −0.002 73 −0.002 87
+0.008 12
+0.009 52
+0.009 35
+0.014 12
+0.014 74
+0.016 93
+tr2
+0.190 40
+0.207 15
+0.209 79
+0.210 35
+0.225 67
+0.230 60
+0.231 59
+0.235 37
+0.237 11
+0.238 67
+tot
+0.150 39
+0.169 64
+0.171 75
+0.171 97
+0.196 40
+0.202 05
+0.202 66
+0.210 73
+0.212 87
+0.216 39
+70
+c
+−0.039 86 −0.042 88 −0.043 59 −0.043 79 −0.046 17 −0.046 94 −0.047 16 −0.047 73 −0.047 95 −0.048 17
+tr1
+0.005 89
+0.012 57
+0.013 22
+0.013 27
+0.025 24
+0.027 10
+0.027 17
+0.032 09
+0.032 86
+0.035 12
+tr2
+0.223 32
+0.241 72
+0.244 25
+0.244 57
+0.263 02
+0.268 13
+0.268 92
+0.273 29
+0.274 92
+0.276 42
+tot
+0.189 35
+0.211 41
+0.213 88
+0.214 05
+0.242 09
+0.248 29
+0.248 93
+0.257 64
+0.259 83
+0.263 37
+75
+c
+−0.049 60 −0.053 28 −0.054 08 −0.054 26 −0.057 29 −0.058 16 −0.058 36 −0.059 05 −0.059 25 −0.059 46
+tr1
+0.024 29
+0.033 18
+0.034 49
+0.034 74
+0.048 06
+0.050 47
+0.050 78
+0.055 90
+0.056 83
+0.059 12
+tr2
+0.261 68
+0.281 94
+0.284 26
+0.284 22
+0.306 42
+0.311 65
+0.312 14
+0.317 16
+0.318 58
+0.319 95
+tot
+0.236 37
+0.261 84
+0.264 67
+0.264 70
+0.297 20
+0.303 96
+0.304 56
+0.314 01
+0.316 16
+0.319 62
+80
+c
+−0.062 00 −0.066 50 −0.067 38 −0.067 51 −0.071 39 −0.072 35 −0.072 50 −0.073 33 −0.073 49 −0.073 65
+tr1
+0.049 34
+0.061 11
+0.063 19
+0.063 66
+0.078 85
+0.081 90
+0.082 45
+0.087 82
+0.088 88
+0.091 18
+tr2
+0.307 16
+0.329 55
+0.331 51
+0.330 95
+0.357 75
+0.362 99
+0.363 02
+0.368 79
+0.369 87
+0.370 98
+tot
+0.294 49
+0.324 16
+0.327 32
+0.327 10
+0.365 21
+0.372 54
+0.372 97
+0.383 28
+0.385 26
+0.388 51
+85
+c
+−0.078 11 −0.083 62 −0.084 56 −0.084 61 −0.089 62 −0.090 65 −0.090 71 −0.091 70 −0.091 76 −0.091 83
+tr1
+0.084 05
+0.099 64
+0.102 65
+0.103 31
+0.121 15
+0.124 92
+0.125 68
+0.131 39
+0.132 52
+0.134 74
+tr2
+0.362 14
+0.387 02
+0.388 39
+0.387 12
+0.419 60
+0.424 71
+0.424 08
+0.430 70
+0.431 24
+0.431 93
+tot
+0.368 07
+0.403 04
+0.406 48
+0.405 82
+0.451 13
+0.458 97
+0.459 05
+0.470 40
+0.472 00
+0.474 83
+90
+c
+−0.098 98 −0.105 73 −0.106 68 −0.106 56 −0.113 08 −0.114 12 −0.114 01 −0.115 19 −0.115 08 −0.114 98
+tr1
+0.132 64
+0.153 30
+0.157 37
+0.158 15
+0.179 78
+0.184 33
+0.185 22
+0.191 38
+0.192 44
+0.194 41
+tr2
+0.429 94
+0.457 76
+0.458 22
+0.455 91
+0.495 59
+0.500 32
+0.498 72
+0.506 33
+0.506 06
+0.506 05
+tot
+0.463 60
+0.505 33
+0.508 91
+0.507 50
+0.562 30
+0.570 53
+0.569 92
+0.582 51
+0.583 41
+0.585 49
+95
+c
+−0.127 26 −0.135 59 −0.136 45 −0.136 05 −0.144 66 −0.145 63 −0.145 22 −0.146 62 −0.146 22 −0.145 81
+tr1
+0.202 82
+0.230 40
+0.235 64
+0.236 35
+0.263 61
+0.268 94
+0.269 73
+0.276 49
+0.277 22
+0.278 67
+tr2
+0.515 74
+0.547 05
+0.546 10
+0.542 32
+0.591 28
+0.595 25
+0.592 23
+0.600 99
+0.599 48
+0.598 41
+tot
+0.591 30
+0.641 86
+0.645 29
+0.642 63
+0.710 23
+0.718 56
+0.716 74
+0.730 85
+0.730 49
+0.731 26
+100
+c
+−0.166 93 −0.177 34 −0.177 94 −0.177 03 −0.188 67 −0.189 38 −0.188 44 −0.190 12 −0.189 19 −0.188 27
+tr1
+0.307 52
+0.344 73
+0.351 15
+0.351 41
+0.387 25
+0.393 20
+0.393 45
+0.401 03
+0.400 91
+0.401 29
+tr2
+0.627 54
+0.663 10
+0.659 98
+0.654 01
+0.715 27
+0.717 85
+0.712 70
+0.722 81
+0.719 43
+0.716 67
+tot
+0.768 12
+0.830 49
+0.833 19
+0.828 39
+0.913 85
+0.921 67
+0.917 70
+0.933 72
+0.931 15
+0.929 69
+17
+
+TABLE III. Matrix elements of the operator hNMS
+h.o.
+for the np3/2 states calculated with the Dirac-
+Coulomb wave functions for the extended nuclei. The notations are the same as in Table I.
+Z
+Term
+(2, 2)
+(2, 3)
+(2, 4)
+(2, 5)
+(3, 3)
+(3, 4)
+(3, 5)
+(4, 4)
+(4, 5)
+(5, 5)
+5
+c
+−0.000 04 −0.000 04 −0.000 05 −0.000 05 −0.000 05 −0.000 05 −0.000 05 −0.000 05 −0.000 05 −0.000 05
+tr1
+−0.039 16 −0.038 65 −0.040 38 −0.041 32 −0.032 21 −0.032 43 −0.033 62 −0.029 01 −0.029 15 −0.027 24
+tr2
+−0.047 51 −0.043 89 −0.043 11 −0.042 79 −0.046 28 −0.045 17 −0.044 83 −0.045 85 −0.045 36 −0.045 65
+tot
+−0.086 71 −0.082 59 −0.083 54 −0.084 16 −0.078 54 −0.077 65 −0.078 50 −0.074 91 −0.074 56 −0.072 94
+10
+c
+−0.000 13 −0.000 15 −0.000 15 −0.000 15 −0.000 16 −0.000 16 −0.000 16 −0.000 17 −0.000 17 −0.000 17
+tr1
+−0.033 00 −0.032 50 −0.034 21 −0.035 13 −0.026 01 −0.026 22 −0.027 40 −0.022 79 −0.022 92 −0.021 01
+tr2
+−0.042 60 −0.038 59 −0.037 74 −0.037 40 −0.040 29 −0.039 04 −0.038 65 −0.039 47 −0.038 92 −0.039 09
+tot
+−0.075 73 −0.071 23 −0.072 10 −0.072 69 −0.066 46 −0.065 42 −0.066 22 −0.062 43 −0.062 01 −0.060 27
+15
+c
+−0.000 25 −0.000 28 −0.000 28 −0.000 29 −0.000 30 −0.000 31 −0.000 31 −0.000 31 −0.000 32 −0.000 32
+tr1
+−0.026 78 −0.026 30 −0.027 99 −0.028 90 −0.019 73 −0.019 93 −0.021 11 −0.016 48 −0.016 60 −0.014 68
+tr2
+−0.038 14 −0.033 77 −0.032 88 −0.032 51 −0.034 83 −0.033 44 −0.033 03 −0.033 66 −0.033 04 −0.033 11
+tot
+−0.065 17 −0.060 34 −0.061 15 −0.061 70 −0.054 85 −0.053 68 −0.054 44 −0.050 45 −0.049 96 −0.048 11
+20
+c
+−0.000 39 −0.000 42 −0.000 43 −0.000 44 −0.000 46 −0.000 47 −0.000 47 −0.000 48 −0.000 48 −0.000 49
+tr1
+−0.020 51 −0.020 04 −0.021 73 −0.022 64 −0.013 37 −0.013 57 −0.014 74 −0.010 08 −0.010 20 −0.008 26
+tr2
+−0.034 03 −0.029 34 −0.028 41 −0.028 03 −0.029 79 −0.028 28 −0.027 84 −0.028 28 −0.027 60 −0.027 58
+tot
+−0.054 93 −0.049 80 −0.050 57 −0.051 10 −0.043 61 −0.042 31 −0.043 05 −0.038 84 −0.038 29 −0.036 33
+25
+c
+−0.000 53 −0.000 58 −0.000 59 −0.000 59 −0.000 62 −0.000 64 −0.000 64 −0.000 65 −0.000 66 −0.000 67
+tr1
+−0.014 18 −0.013 74 −0.015 42 −0.016 33 −0.006 93 −0.007 12 −0.008 29 −0.003 59 −0.003 70 −0.001 74
+tr2
+−0.030 25 −0.025 26 −0.024 29 −0.023 90 −0.025 11 −0.023 48 −0.023 02 −0.023 28 −0.022 55 −0.022 43
+tot
+−0.044 97 −0.039 57 −0.040 30 −0.040 83 −0.032 66 −0.031 24 −0.031 95 −0.027 52 −0.026 91 −0.024 84
+30
+c
+−0.000 68 −0.000 73 −0.000 75 −0.000 76 −0.000 79 −0.000 81 −0.000 82 −0.000 83 −0.000 84 −0.000 84
+tr1
+−0.007 80 −0.007 37 −0.009 06 −0.009 98 −0.000 39 −0.000 57 −0.001 75
+0.003 02
+0.002 91
+0.004 90
+tr2
+−0.026 77 −0.021 49 −0.020 49 −0.020 10 −0.020 77 −0.019 03 −0.018 54 −0.018 63 −0.017 84 −0.017 63
+tot
+−0.035 25 −0.029 59 −0.030 31 −0.030 83 −0.021 95 −0.020 41 −0.021 11 −0.016 44 −0.015 77 −0.013 58
+35
+c
+−0.000 83 −0.000 89 −0.000 91 −0.000 92 −0.000 96 −0.000 98 −0.000 99 −0.001 01 −0.001 02 −0.001 03
+tr1
+−0.001 34 −0.000 92 −0.002 63 −0.003 56
+0.006 27
+0.006 10
+0.004 92
+0.009 77
+0.009 66
+0.011 68
+tr2
+−0.023 58 −0.018 02 −0.017 00 −0.016 60 −0.016 74 −0.014 89 −0.014 39 −0.014 30 −0.013 45 −0.013 16
+tot
+−0.025 75 −0.019 84 −0.020 55 −0.021 08 −0.011 43 −0.009 77 −0.010 46 −0.005 54 −0.004 81 −0.002 50
+40
+c
+−0.000 97 −0.001 05 −0.001 07 −0.001 08 −0.001 13 −0.001 16 −0.001 17 −0.001 18 −0.001 19 −0.001 20
+tr1
+0.005 20
+0.005 62
+0.003 89
+0.002 95
+0.013 08
+0.012 93
+0.011 73
+0.016 68
+0.016 58
+0.018 65
+tr2
+−0.020 67 −0.014 85 −0.013 81 −0.013 41 −0.013 02 −0.011 06 −0.010 54 −0.010 27 −0.009 38 −0.008 99
+tot
+−0.016 45 −0.010 28 −0.011 00 −0.011 54 −0.001 07
+0.000 71
+0.000 02
+0.005 22
+0.006 01
+0.008 45
+45
+c
+−0.001 12 −0.001 21 −0.001 23 −0.001 24 −0.001 30 −0.001 33 −0.001 34 −0.001 36 −0.001 37 −0.001 38
+tr1
+0.011 84
+0.012 28
+0.010 52
+0.009 56
+0.020 07
+0.019 94
+0.018 72
+0.023 80
+0.023 71
+0.025 82
+tr2
+−0.018 06 −0.011 98 −0.010 92 −0.010 51 −0.009 60 −0.007 54 −0.007 00 −0.006 56 −0.005 61 −0.005 14
+tot
+−0.007 34 −0.000 90 −0.001 64 −0.002 19
+0.009 17
+0.011 07
+0.010 38
+0.015 88
+0.016 73
+0.019 30
+50
+c
+−0.001 27 −0.001 36 −0.001 39 −0.001 40 −0.001 47 −0.001 50 −0.001 51 −0.001 53 −0.001 55 −0.001 56
+tr1
+0.018 61
+0.019 10
+0.017 29
+0.016 31
+0.027 27
+0.027 17
+0.025 93
+0.031 16
+0.031 08
+0.033 26
+tr2
+−0.015 74 −0.009 41 −0.008 34 −0.007 93 −0.006 48 −0.004 32 −0.003 77 −0.003 15 −0.002 16 −0.001 60
+tot
+0.001 60
+0.008 33
+0.007 57
+0.006 98
+0.019 32
+0.021 35
+0.020 64
+0.026 48
+0.027 38
+0.030 10
+55
+c
+−0.001 41 −0.001 52 −0.001 55 −0.001 56 −0.001 63 −0.001 67 −0.001 68 −0.001 71 −0.001 72 −0.001 74
+tr1
+0.025 52
+0.026 08
+0.024 24
+0.023 22
+0.034 73
+0.034 66
+0.033 40
+0.038 81
+0.038 75
+0.041 00
+tr2
+−0.013 74 −0.007 15 −0.006 06 −0.005 65 −0.003 69 −0.001 42 −0.000 86 −0.000 06
+0.000 98
+0.001 63
+tot
+0.010 37
+0.017 42
+0.016 63
+0.016 01
+0.029 40
+0.031 57
+0.030 86
+0.037 05
+0.038 01
+0.040 89
+60
+c
+−0.001 56 −0.001 67 −0.001 71 −0.001 72 −0.001 80 −0.001 84 −0.001 86 −0.001 88 −0.001 90 −0.001 92
+tr1
+0.032 61
+0.033 28
+0.031 39
+0.030 34
+0.042 48
+0.042 47
+0.041 18
+0.046 81
+0.046 77
+0.049 09
+tr2
+−0.012 07 −0.005 21 −0.004 12 −0.003 70 −0.001 22
+0.001 15
+0.001 72
+0.002 70
+0.003 79
+0.004 52
+tot
+0.018 99
+0.026 40
+0.025 57
+0.024 91
+0.039 46
+0.041 78
+0.041 04
+0.047 63
+0.048 66
+0.051 70
+65
+c
+−0.001 70 −0.001 83 −0.001 87 −0.001 88 −0.001 97 −0.002 02 −0.002 03 −0.002 06 −0.002 08 −0.002 10
+tr1
+0.039 90
+0.040 73
+0.038 79
+0.037 69
+0.050 59
+0.050 65
+0.049 33
+0.055 21
+0.055 20
+0.057 62
+tr2
+−0.010 74 −0.003 62 −0.002 51 −0.002 10
+0.000 89
+0.003 36
+0.003 95
+0.005 11
+0.006 24
+0.007 06
+tot
+0.027 46
+0.035 28
+0.034 41
+0.033 71
+0.049 51
+0.052 00
+0.051 25
+0.058 26
+0.059 36
+0.062 58
+70
+c
+−0.001 85 −0.001 99 −0.002 03 −0.002 04 −0.002 15 −0.002 19 −0.002 21 −0.002 24 −0.002 26 −0.002 28
+tr1
+0.047 42
+0.048 47
+0.046 47
+0.045 33
+0.059 10
+0.059 26
+0.057 91
+0.064 08
+0.064 10
+0.066 63
+tr2
+−0.009 78 −0.002 40 −0.001 27 −0.000 86
+0.002 62
+0.005 20
+0.005 80
+0.007 14
+0.008 32
+0.009 22
+tot
+0.035 80
+0.044 09
+0.043 17
+0.042 43
+0.059 58
+0.062 27
+0.061 50
+0.068 97
+0.070 16
+0.073 57
+75
+c
+−0.002 00 −0.002 15 −0.002 19 −0.002 21 −0.002 32 −0.002 37 −0.002 39 −0.002 43 −0.002 45 −0.002 47
+tr1
+0.055 20
+0.056 54
+0.054 49
+0.053 29
+0.068 09
+0.068 37
+0.066 99
+0.073 49
+0.073 56
+0.076 21
+tr2
+−0.009 21 −0.001 56 −0.000 42
+0.000 00
+0.003 95
+0.006 64
+0.007 25
+0.008 76
+0.010 00
+0.010 98
+tot
+0.044 00
+0.052 83
+0.051 88
+0.051 08
+0.069 72
+0.072 64
+0.071 84
+0.079 82
+0.081 11
+0.084 72
+80
+c
+−0.002 15 −0.002 31 −0.002 36 −0.002 37 −0.002 51 −0.002 56 −0.002 58 −0.002 62 −0.002 64 −0.002 66
+tr1
+0.063 29
+0.065 00
+0.062 90
+0.061 64
+0.077 62
+0.078 05
+0.076 64
+0.083 52
+0.083 65
+0.086 45
+tr2
+−0.009 05 −0.001 15
+0.000 01
+0.000 44
+0.004 84
+0.007 63
+0.008 25
+0.009 94
+0.011 22
+0.012 29
+tot
+0.052 09
+0.061 54
+0.060 56
+0.059 70
+0.079 95
+0.083 12
+0.082 31
+0.090 84
+0.092 23
+0.096 07
+85
+c
+−0.002 30 −0.002 48 −0.002 53 −0.002 55 −0.002 70 −0.002 76 −0.002 78 −0.002 82 −0.002 84 −0.002 87
+tr1
+0.071 71
+0.073 90
+0.071 75
+0.070 43
+0.087 76
+0.088 39
+0.086 95
+0.094 27
+0.094 48
+0.097 44
+tr2
+−0.009 36 −0.001 19
+0.000 00
+0.000 43
+0.005 24
+0.008 14
+0.008 77
+0.010 63
+0.011 96
+0.013 10
+tot
+0.060 05
+0.070 23
+0.069 22
+0.068 31
+0.090 30
+0.093 77
+0.092 94
+0.102 08
+0.103 59
+0.107 68
+90
+c
+−0.002 46 −0.002 66 −0.002 71 −0.002 73 −0.002 90 −0.002 96 −0.002 99 −0.003 03 −0.003 06 −0.003 08
+tr1
+0.080 50
+0.083 29
+0.081 11
+0.079 72
+0.098 59
+0.099 47
+0.098 01
+0.105 82
+0.106 12
+0.109 28
+tr2
+−0.010 14 −0.001 73 −0.000 51 −0.000 06
+0.005 10
+0.008 10
+0.008 75
+0.010 76
+0.012 14
+0.013 36
+tot
+0.067 89
+0.078 90
+0.077 89
+0.076 92
+0.100 79
+0.104 61
+0.103 77
+0.113 55
+0.115 21
+0.119 56
+95
+c
+−0.002 63 −0.002 85 −0.002 91 −0.002 93 −0.003 11 −0.003 18 −0.003 21 −0.003 26 −0.003 28 −0.003 31
+tr1
+0.089 69
+0.093 24
+0.091 04
+0.089 58
+0.110 20
+0.111 39
+0.109 92
+0.118 28
+0.118 71
+0.122 08
+tr2
+−0.011 46 −0.002 80 −0.001 55 −0.001 09
+0.004 36
+0.007 46
+0.008 12
+0.010 29
+0.011 71
+0.013 01
+tot
+0.075 61
+0.087 59
+0.086 59
+0.085 56
+0.111 45
+0.115 67
+0.114 83
+0.125 31
+0.127 14
+0.131 77
+100
+c
+−0.002 81 −0.003 05 −0.003 11 −0.003 13 −0.003 34 −0.003 42 −0.003 44 −0.003 50 −0.003 53 −0.003 56
+tr1
+0.099 35
+0.103 82
+0.101 63
+0.100 10
+0.122 68
+0.124 27
+0.122 80
+0.131 78
+0.132 36
+0.135 97
+tr2
+−0.013 34 −0.004 48 −0.003 18 −0.002 70
+0.002 95
+0.006 14
+0.006 82
+0.009 11
+0.010 58
+0.011 94
+tot
+0.083 20
+0.096 29
+0.095 34
+0.094 26
+0.122 30
+0.127 00
+0.126 17
+0.137 39
+0.139 41
+0.144 35
+18
+
+TABLE IV. Matrix elements of the operator hNMS
+h.o.
+for the nd3/2 states calculated with the Dirac-
+Coulomb wave functions for the extended nuclei. The notations are the same as in Table I.
+Z Term
+(3, 3)
+(3, 4)
+(3, 5)
+(4, 4)
+(4, 5)
+(5, 5)
+5
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.009 32 −0.009 18 −0.009 69 −0.008 14 −0.008 40 −0.007 46
+tr2
+−0.009 38 −0.007 96 −0.007 56 −0.009 08 −0.008 53 −0.008 94
+tot
+−0.018 70 −0.017 13 −0.017 26 −0.017 22 −0.016 93 −0.016 39
+10
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.008 96 −0.008 89 −0.009 44 −0.007 89 −0.008 17 −0.007 25
+tr2
+−0.008 14 −0.006 64 −0.006 24 −0.007 53 −0.006 94 −0.007 24
+tot
+−0.017 10 −0.015 53 −0.015 68 −0.015 41 −0.015 11 −0.014 50
+15
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.008 62 −0.008 64 −0.009 21 −0.007 66 −0.007 98 −0.007 07
+tr2
+−0.006 88 −0.005 31 −0.004 91 −0.005 96 −0.005 33 −0.005 54
+tot
+−0.015 50 −0.013 95 −0.014 12 −0.013 62 −0.013 31 −0.012 61
+20
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00 −0.000 01
+tr1
+−0.008 30 −0.008 40 −0.009 00 −0.007 45 −0.007 80 −0.006 91
+tr2
+−0.005 61 −0.003 97 −0.003 57 −0.004 38 −0.003 71 −0.003 82
+tot
+−0.013 92 −0.012 38 −0.012 58 −0.011 84 −0.011 51 −0.010 73
+25
+c
+−0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01
+tr1
+−0.007 99 −0.008 18 −0.008 81 −0.007 24 −0.007 63 −0.006 76
+tr2
+−0.004 33 −0.002 62 −0.002 22 −0.002 79 −0.002 08 −0.002 08
+tot
+−0.012 33 −0.010 81 −0.011 04 −0.010 05 −0.009 72 −0.008 85
+30
+c
+−0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 02 −0.000 02
+tr1
+−0.007 69 −0.007 97 −0.008 64 −0.007 05 −0.007 47 −0.006 61
+tr2
+−0.003 04 −0.001 25 −0.000 85 −0.001 18 −0.000 42 −0.000 32
+tot
+−0.010 74 −0.009 23 −0.009 51 −0.008 25 −0.007 91 −0.006 95
+35
+c
+−0.000 02 −0.000 02 −0.000 02 −0.000 02 −0.000 02 −0.000 03
+tr1
+−0.007 39 −0.007 76 −0.008 48 −0.006 85 −0.007 32 −0.006 46
+tr2
+−0.001 73
+0.000 13
+0.000 53
+0.000 45
+0.001 25
+0.001 45
+tot
+−0.009 13 −0.007 65 −0.007 97 −0.006 43 −0.006 09 −0.005 03
+40
+c
+−0.000 02 −0.000 03 −0.000 03 −0.000 03 −0.000 04 −0.000 04
+tr1
+−0.007 08 −0.007 56 −0.008 31 −0.006 65 −0.007 15 −0.006 30
+tr2
+−0.000 41
+0.001 53
+0.001 93
+0.002 10
+0.002 95
+0.003 26
+tot
+−0.007 51 −0.006 05 −0.006 41 −0.004 59 −0.004 24 −0.003 08
+45
+c
+−0.000 03 −0.000 04 −0.000 04 −0.000 05 −0.000 05 −0.000 05
+tr1
+−0.006 77 −0.007 34 −0.008 15 −0.006 43 −0.006 98 −0.006 12
+tr2
+0.000 94
+0.002 96
+0.003 35
+0.003 78
+0.004 68
+0.005 09
+tot
+−0.005 87 −0.004 42 −0.004 83 −0.002 70 −0.002 35 −0.001 08
+50
+c
+−0.000 05 −0.000 05 −0.000 06 −0.000 06 −0.000 07 −0.000 07
+tr1
+−0.006 44 −0.007 11 −0.007 97 −0.006 19 −0.006 78 −0.005 91
+tr2
+0.002 30
+0.004 40
+0.004 80
+0.005 49
+0.006 43
+0.006 96
+tot
+−0.004 18 −0.002 76 −0.003 23 −0.000 77 −0.000 41
+0.000 98
+55
+c
+−0.000 06 −0.000 07 −0.000 08 −0.000 08 −0.000 09 −0.000 09
+tr1
+−0.006 09 −0.006 86 −0.007 77 −0.005 92 −0.006 54 −0.005 66
+tr2
+0.003 69
+0.005 88
+0.006 27
+0.007 23
+0.008 23
+0.008 86
+tot
+−0.002 46 −0.001 05 −0.001 58
+0.001 24
+0.001 60
+0.003 11
+60
+c
+−0.000 08 −0.000 09 −0.000 10 −0.000 11 −0.000 11 −0.000 12
+tr1
+−0.005 70 −0.006 57 −0.007 55 −0.005 59 −0.006 26 −0.005 36
+tr2
+0.005 11
+0.007 38
+0.007 77
+0.009 02
+0.010 06
+0.010 81
+tot
+−0.000 68
+0.000 72
+0.000 13
+0.003 32
+0.003 69
+0.005 33
+65
+c
+−0.000 10 −0.000 12 −0.000 12 −0.000 13 −0.000 14 −0.000 15
+tr1
+−0.005 28 −0.006 24 −0.007 28 −0.005 21 −0.005 92 −0.004 99
+tr2
+0.006 55
+0.008 92
+0.009 31
+0.010 84
+0.011 94
+0.012 81
+tot
+0.001 17
+0.002 56
+0.001 90
+0.005 50
+0.005 88
+0.007 67
+70
+c
+−0.000 12 −0.000 14 −0.000 15 −0.000 17 −0.000 18 −0.000 19
+tr1
+−0.004 80 −0.005 85 −0.006 96 −0.004 75 −0.005 49 −0.004 52
+tr2
+0.008 02
+0.010 49
+0.010 87
+0.012 72
+0.013 86
+0.014 86
+tot
+0.003 10
+0.004 50
+0.003 76
+0.007 79
+0.008 20
+0.010 15
+75
+c
+−0.000 15 −0.000 18 −0.000 19 −0.000 20 −0.000 22 −0.000 23
+tr1
+−0.004 26 −0.005 38 −0.006 57 −0.004 20 −0.004 96 −0.003 94
+tr2
+0.009 53
+0.012 10
+0.012 48
+0.014 63
+0.015 84
+0.016 96
+tot
+0.005 12
+0.006 54
+0.005 73
+0.010 23
+0.010 66
+0.012 79
+80
+c
+−0.000 18 −0.000 21 −0.000 23 −0.000 25 −0.000 26 −0.000 28
+tr1
+−0.003 63 −0.004 82 −0.006 08 −0.003 52 −0.004 30 −0.003 22
+tr2
+0.011 07
+0.013 74
+0.014 12
+0.016 60
+0.017 87
+0.019 12
+tot
+0.007 26
+0.008 71
+0.007 81
+0.012 84
+0.013 30
+0.015 62
+85
+c
+−0.000 22 −0.000 26 −0.000 27 −0.000 30 −0.000 32 −0.000 33
+tr1
+−0.002 89 −0.004 13 −0.005 48 −0.002 68 −0.003 47 −0.002 31
+tr2
+0.012 64
+0.015 42
+0.015 80
+0.018 62
+0.019 94
+0.021 33
+tot
+0.009 52
+0.011 03
+0.010 05
+0.015 64
+0.016 16
+0.018 69
+90
+c
+−0.000 26 −0.000 31 −0.000 32 −0.000 36 −0.000 38 −0.000 40
+tr1
+−0.002 02 −0.003 29 −0.004 72 −0.001 65 −0.002 44 −0.001 17
+tr2
+0.014 23
+0.017 14
+0.017 51
+0.020 68
+0.022 07
+0.023 59
+tot
+0.011 95
+0.013 54
+0.012 47
+0.018 68
+0.019 26
+0.022 03
+95
+c
+−0.000 31 −0.000 36 −0.000 38 −0.000 42 −0.000 45 −0.000 47
+tr1
+−0.001 00 −0.002 26 −0.003 77 −0.000 38 −0.001 14
+0.000 25
+tr2
+0.015 85
+0.018 89
+0.019 25
+0.022 79
+0.024 24
+0.025 90
+tot
+0.014 55
+0.016 26
+0.015 10
+0.021 99
+0.022 65
+0.025 68
+100
+c
+−0.000 37 −0.000 43 −0.000 45 −0.000 50 −0.000 52 −0.000 55
+tr1
+0.000 23 −0.000 98 −0.002 57
+0.001 19
+0.000 48
+0.002 02
+tr2
+0.017 49
+0.020 65
+0.021 01
+0.024 92
+0.026 44
+0.028 25
+tot
+0.017 35
+0.019 24
+0.017 98
+0.025 62
+0.026 39
+0.029 72
+19
+
+TABLE V. Matrix elements of the operator hNMS
+h.o.
+for the nd5/2 states calculated with the Dirac-
+Coulomb wave functions for the extended nuclei. The notations are the same as in Table I.
+Z Term
+(3, 3)
+(3, 4)
+(3, 5)
+(4, 4)
+(4, 5)
+(5, 5)
+5
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.008 63 −0.008 46 −0.008 96 −0.007 37 −0.007 61 −0.006 65
+tr2
+−0.010 06 −0.008 65 −0.008 26 −0.009 84 −0.009 31 −0.009 74
+tot
+−0.018 69 −0.017 11 −0.017 22 −0.017 21 −0.016 92 −0.016 39
+10
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.007 56 −0.007 44 −0.007 94 −0.006 32 −0.006 56 −0.005 60
+tr2
+−0.009 51 −0.008 04 −0.007 66 −0.009 07 −0.008 51 −0.008 87
+tot
+−0.017 07 −0.015 48 −0.015 60 −0.015 39 −0.015 08 −0.014 48
+15
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.006 50 −0.006 42 −0.006 93 −0.005 27 −0.005 52 −0.004 56
+tr2
+−0.008 96 −0.007 43 −0.007 05 −0.008 31 −0.007 72 −0.008 01
+tot
+−0.015 45 −0.013 85 −0.013 98 −0.013 58 −0.013 25 −0.012 57
+20
+c
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+0.000 00
+tr1
+−0.005 43 −0.005 40 −0.005 92 −0.004 21 −0.004 48 −0.003 51
+tr2
+−0.008 41 −0.006 83 −0.006 46 −0.007 56 −0.006 94 −0.007 16
+tot
+−0.013 84 −0.012 23 −0.012 38 −0.011 77 −0.011 42 −0.010 67
+25
+c
+0.000 00
+0.000 00
+0.000 00 −0.000 01 −0.000 01 −0.000 01
+tr1
+−0.004 36 −0.004 38 −0.004 91 −0.003 16 −0.003 44 −0.002 46
+tr2
+−0.007 87 −0.006 24 −0.005 88 −0.006 81 −0.006 17 −0.006 32
+tot
+−0.012 24 −0.010 63 −0.010 79 −0.009 98 −0.009 61 −0.008 79
+30
+c
+−0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01
+tr1
+−0.003 29 −0.003 36 −0.003 90 −0.002 10 −0.002 39 −0.001 40
+tr2
+−0.007 34 −0.005 66 −0.005 30 −0.006 08 −0.005 41 −0.005 49
+tot
+−0.010 64 −0.009 03 −0.009 21 −0.008 18 −0.007 81 −0.006 90
+35
+c
+−0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 01
+tr1
+−0.002 22 −0.002 34 −0.002 89 −0.001 03 −0.001 33 −0.000 34
+tr2
+−0.006 82 −0.005 10 −0.004 74 −0.005 36 −0.004 67 −0.004 68
+tot
+−0.009 05 −0.007 44 −0.007 64 −0.006 40 −0.006 01 −0.005 03
+40
+c
+−0.000 01 −0.000 01 −0.000 01 −0.000 01 −0.000 02 −0.000 02
+tr1
+−0.001 14 −0.001 31 −0.001 87
+0.000 05 −0.000 26
+0.000 74
+tr2
+−0.006 32 −0.004 54 −0.004 20 −0.004 66 −0.003 94 −0.003 88
+tot
+−0.007 47 −0.005 87 −0.006 09 −0.004 62 −0.004 22 −0.003 16
+45
+c
+−0.000 01 −0.000 02 −0.000 02 −0.000 02 −0.000 02 −0.000 02
+tr1
+−0.000 06 −0.000 28 −0.000 85
+0.001 13
+0.000 81
+0.001 83
+tr2
+−0.005 83 −0.004 01 −0.003 67 −0.003 97 −0.003 23 −0.003 10
+tot
+−0.005 90 −0.004 30 −0.004 54 −0.002 85 −0.002 44 −0.001 29
+50
+c
+−0.000 02 −0.000 02 −0.000 02 −0.000 02 −0.000 03 −0.000 03
+tr1
+0.001 04
+0.000 76
+0.000 17
+0.002 23
+0.001 90
+0.002 93
+tr2
+−0.005 35 −0.003 49 −0.003 16 −0.003 30 −0.002 54 −0.002 34
+tot
+−0.004 33 −0.002 74 −0.003 01 −0.001 09 −0.000 66
+0.000 57
+55
+c
+−0.000 02 −0.000 03 −0.000 03 −0.000 03 −0.000 03 −0.000 03
+tr1
+0.002 14
+0.001 81
+0.001 20
+0.003 35
+0.003 00
+0.004 05
+tr2
+−0.004 89 −0.002 98 −0.002 67 −0.002 65 −0.001 86 −0.001 59
+tot
+−0.002 78 −0.001 20 −0.001 50
+0.000 67
+0.001 11
+0.002 43
+60
+c
+−0.000 03 −0.000 03 −0.000 03 −0.000 03 −0.000 04 −0.000 04
+tr1
+0.003 25
+0.002 87
+0.002 24
+0.004 48
+0.004 12
+0.005 19
+tr2
+−0.004 45 −0.002 50 −0.002 20 −0.002 01 −0.001 20 −0.000 87
+tot
+−0.001 23
+0.000 34
+0.000 01
+0.002 43
+0.002 88
+0.004 29
+65
+c
+−0.000 03 −0.000 04 −0.000 04 −0.000 04 −0.000 04 −0.000 04
+tr1
+0.004 37
+0.003 94
+0.003 29
+0.005 62
+0.005 25
+0.006 35
+tr2
+−0.004 03 −0.002 03 −0.001 74 −0.001 40 −0.000 57 −0.000 16
+tot
+0.000 31
+0.001 87
+0.001 51
+0.004 18
+0.004 64
+0.006 14
+70
+c
+−0.000 04 −0.000 04 −0.000 04 −0.000 05 −0.000 05 −0.000 05
+tr1
+0.005 50
+0.005 02
+0.004 34
+0.006 79
+0.006 41
+0.007 53
+tr2
+−0.003 63 −0.001 59 −0.001 31 −0.000 80
+0.000 05
+0.000 53
+tot
+0.001 83
+0.003 39
+0.002 99
+0.005 94
+0.006 41
+0.008 01
+75
+c
+−0.000 04 −0.000 05 −0.000 05 −0.000 05 −0.000 06 −0.000 06
+tr1
+0.006 65
+0.006 11
+0.005 41
+0.007 98
+0.007 58
+0.008 74
+tr2
+−0.003 25 −0.001 17 −0.000 90 −0.000 23
+0.000 65
+0.001 19
+tot
+0.003 35
+0.004 90
+0.004 46
+0.007 69
+0.008 17
+0.009 87
+80
+c
+−0.000 05 −0.000 05 −0.000 05 −0.000 06 −0.000 06 −0.000 06
+tr1
+0.007 81
+0.007 22
+0.006 50
+0.009 19
+0.008 78
+0.009 97
+tr2
+−0.002 90 −0.000 77 −0.000 51
+0.000 32
+0.001 22
+0.001 83
+tot
+0.004 86
+0.006 40
+0.005 93
+0.009 44
+0.009 94
+0.011 74
+85
+c
+−0.000 05 −0.000 06 −0.000 06 −0.000 07 −0.000 07 −0.000 07
+tr1
+0.008 98
+0.008 35
+0.007 59
+0.010 42
+0.010 01
+0.011 24
+tr2
+−0.002 57 −0.000 39 −0.000 15
+0.000 84
+0.001 76
+0.002 45
+tot
+0.006 37
+0.007 90
+0.007 38
+0.011 20
+0.011 71
+0.013 62
+90
+c
+−0.000 06 −0.000 06 −0.000 07 −0.000 07 −0.000 08 −0.000 08
+tr1
+0.010 18
+0.009 49
+0.008 71
+0.011 69
+0.011 27
+0.012 54
+tr2
+−0.002 26 −0.000 04
+0.000 19
+0.001 34
+0.002 29
+0.003 05
+tot
+0.007 86
+0.009 39
+0.008 83
+0.012 96
+0.013 48
+0.015 51
+95
+c
+−0.000 06 −0.000 07 −0.000 07 −0.000 08 −0.000 08 −0.000 09
+tr1
+0.011 39
+0.010 66
+0.009 84
+0.012 99
+0.012 56
+0.013 88
+tr2
+−0.001 98
+0.000 29
+0.000 50
+0.001 81
+0.002 78
+0.003 62
+tot
+0.009 35
+0.010 87
+0.010 26
+0.014 72
+0.015 26
+0.017 40
+100
+c
+−0.000 07 −0.000 08 −0.000 08 −0.000 09 −0.000 09 −0.000 10
+tr1
+0.012 63
+0.011 85
+0.010 99
+0.014 32
+0.013 88
+0.015 25
+tr2
+−0.001 73
+0.000 58
+0.000 78
+0.002 26
+0.003 25
+0.004 16
+tot
+0.010 82
+0.012 35
+0.011 69
+0.016 49
+0.017 04
+0.019 31
+20
+
+TABLE VI. The higher-order nuclear recoil correction for the 4s, 5s, 5p, and 5d states in terms of
+the function F defined by Eq. (35). The column labeled ⟨ψv|˜hNMS
+h.o. |ψv⟩ denotes the results obtained
+by means of the model QED operator, ⟨ψv|Vs.l.|ψv⟩ is the contribution of the semilocal part of the
+model operator, and “Exact” stands for the ab initio values. The ns states with n ⩽ 3 and the np
+and nd states with n ⩽ 4 are omitted since the operator ˜hNMS
+h.o. exactly reproduces the corresponding
+correction for them by construction.
+Z
+State
+⟨ψv|Vs.l.|ψv⟩ ⟨ψv|˜hNMS
+h.o. |ψv⟩
+Exact
+Z
+State
+⟨ψv|Vs.l.|ψv⟩ ⟨ψv|˜hNMS
+h.o. |ψv⟩
+Exact
+10
+4s
+1.2024
+1.5325
+1.5295
+60
+4s
+0.8494
+1.4804
+1.4767
+5s
+1.2017
+1.5445
+1.5397
+5s
+0.8394
+1.4833
+1.4772
+5p1/2
+−0.0922
+−0.0545
+−0.0558
+5p1/2
+0.1282
+0.1782
+0.1765
+5p3/2
+−0.0965
+−0.0590
+−0.0603
+5p3/2
+0.0221
+0.0540
+0.0517
+5d3/2
+−0.0279
+−0.0157
+−0.0145
+5d3/2
+−0.0010
+0.0058
+0.0053
+5d5/2
+−0.0278
+−0.0156
+−0.0145
+5d5/2
+−0.0019
+0.0044
+0.0043
+20
+4s
+1.0162
+1.3896
+1.3865
+70
+4s
+0.9227
+1.6503
+1.6463
+5s
+1.0141
+1.4010
+1.3960
+5s
+0.9088
+1.6475
+1.6409
+5p1/2
+−0.0555
+−0.0204
+−0.0218
+5p1/2
+0.1963
+0.2652
+0.2634
+5p3/2
+−0.0692
+−0.0349
+−0.0363
+5p3/2
+0.0406
+0.0762
+0.0736
+5d3/2
+−0.0225
+−0.0116
+−0.0107
+5d3/2
+0.0047
+0.0110
+0.0101
+5d5/2
+−0.0224
+−0.0116
+−0.0107
+5d5/2
+0.0028
+0.0084
+0.0080
+30
+4s
+0.9090
+1.3357
+1.3325
+80
+4s
+1.0683
+1.9249
+1.9205
+5s
+0.9053
+1.3460
+1.3408
+5s
+1.0478
+1.9125
+1.9055
+5p1/2
+−0.0162
+0.0177
+0.0162
+5p1/2
+0.2880
+0.3904
+0.3885
+5p3/2
+−0.0437
+−0.0119
+−0.0136
+5p3/2
+0.0578
+0.0990
+0.0961
+5d3/2
+−0.0172
+−0.0075
+−0.0069
+5d3/2
+0.0108
+0.0169
+0.0156
+5d5/2
+−0.0171
+−0.0076
+−0.0069
+5d5/2
+0.0073
+0.0124
+0.0117
+40
+4s
+0.8484
+1.3348
+1.3314
+90
+4s
+1.3278
+2.3698
+2.3650
+5s
+0.8430
+1.3436
+1.3381
+5s
+1.2952
+2.3406
+2.3329
+5p1/2
+0.0262
+0.0612
+0.0596
+5p1/2
+0.4244
+0.5875
+0.5855
+5p3/2
+−0.0200
+0.0103
+0.0084
+5p3/2
+0.0739
+0.1228
+0.1195
+5d3/2
+−0.0119
+−0.0033
+−0.0031
+5d3/2
+0.0173
+0.0236
+0.0220
+5d5/2
+−0.0119
+−0.0036
+−0.0032
+5d5/2
+0.0117
+0.0165
+0.0155
+50
+4s
+0.8280
+1.3813
+1.3777
+100
+4s
+1.7885
+3.1190
+3.1142
+5s
+0.8206
+1.3877
+1.3820
+5s
+1.7322
+3.0568
+3.0496
+5p1/2
+0.0733
+0.1131
+0.1115
+5p1/2
+0.6515
+0.9319
+0.9297
+5p3/2
+0.0019
+0.0322
+0.0301
+5p3/2
+0.0889
+0.1478
+0.1444
+5d3/2
+−0.0065
+0.0011
+0.0010
+5d3/2
+0.0246
+0.0317
+0.0297
+5d5/2
+−0.0068
+0.0004
+0.0006
+5d5/2
+0.0158
+0.0206
+0.0193
+21
+
+TABLE VII. The one-electron nuclear recoil correction beyond the Breit approximation for the
+valence ns electron in neutral alkali metals in terms of the function F defined by Eq. (35). The
+labels CH and xα = 0, 1/3, 2/3, and 1 correspond to the different effective potentials. See the text
+for details.
+Atom
+Approach
+CH
+xα = 0
+xα = 1/3
+xα = 2/3
+xα = 1
+Na 3s
+⟨ψv|Vs.l.|ψv⟩
+0.0502
+0.0446
+0.0437
+0.0473
+0.0575
+⟨ψv|˜hNMS
+h.o. |ψv⟩
+0.0581
+0.0516
+0.0508
+0.0553
+0.0676
+Exact
+0.0561
+0.0499
+0.0496
+0.0544
+0.0671
+K 4s
+⟨ψv|Vs.l.|ψv⟩
+0.0255
+0.0214
+0.0213
+0.0243
+0.0319
+⟨ψv|˜hNMS
+h.o. |ψv⟩
+0.0313
+0.0263
+0.0264
+0.0302
+0.0400
+Exact
+0.0311
+0.0261
+0.0263
+0.0302
+0.0401
+Rb 5s
+⟨ψv|Vs.l.|ψv⟩
+0.0100
+0.0080
+0.0082
+0.0098
+0.0136
+⟨ψv|˜hNMS
+h.o. |ψv⟩
+0.0141
+0.0112
+0.0116
+0.0140
+0.0195
+Exact
+0.0142
+0.0113
+0.0117
+0.0141
+0.0197
+Cs 6s
+⟨ψv|Vs.l.|ψv⟩
+0.00645
+0.00504
+0.00525
+0.00642
+0.00927
+⟨ψv|˜hNMS
+h.o. |ψv⟩
+0.01019
+0.00796
+0.00835
+0.01026
+0.01490
+Exact
+0.01028
+0.00803
+0.00841
+0.01034
+0.01500
+Fr 7s
+⟨ψv|Vs.l.|ψv⟩
+0.00586
+0.00416
+0.00457
+0.00595
+0.00906
+⟨ψv|˜hNMS
+h.o. |ψv⟩
+0.00999
+0.00708
+0.00782
+0.01022
+0.01563
+Exact
+0.01004
+0.00712
+0.00786
+0.01026
+0.01568
+the operator exactly reproduces the QED recoil contributions for these states.
+For this
+reason, the “predictive power” of the operator can be tested by applying it to evaluation
+of the corresponding corrections for the states with higher values of the principal quantum
+number. In Table VI, the nuclear recoil contributions beyond the Breit approximation are
+given for the 4s, 5s, 5p1/2, 5p3/2, 5d3/2, and 5d5/2 states in hydrogenlike ions. The columns
+labeled ⟨ψv|Vs.l.|ψv⟩ and ⟨ψv|˜hNMS
+h.o. |ψv⟩ contain the predictions obtained using the semilocal
+part (29) of the model-QED operator and the total model-QED operator (28), respectively.
+These values are compared with the ab initio results taken from Tables I-V and shown in the
+last column. As one can see, there is generally good agreement between the data, especially
+for the s states. We stress the importance of the nonlocal part of the model-QED operator.
+For the np and nd states the functions Fnink change the sign for the middle values of Z
+and, accordingly, have small absolute values there. As a result, the relative accuracy of the
+model-QED-operator predictions slightly decreases in these regions.
+22
+
+To date, the QED contribution to the NMS in many-electron systems was usually eval-
+uated within the independent-electron approximation by performing the calculations in the
+extended Furry picture, see, e.g., Refs. [4, 10, 84]. To demonstrate the performance of the
+developed model-QED-operator approach, we apply it to evaluation of the nuclear recoil
+effect on valence-electron energies in neutral alkali metals. First, we perform ab initio calcu-
+lations of the one-electron contribution for the ns states by using a local effective potential
+as V in Eq. (5). We use the core-Hartree (CH) and xα potentials, which are discussed in
+Appendix C. The results of ab initio calculations of the QED contribution to the NMS are
+presented in Table VII in rows labeled “Exact”. The model-QED-operator values are ob-
+tained by averaging the operator (28) with the valence-electron wave functions determined
+from the Dirac equation (5), in which the potential V is given by Eq. (C3). Along with
+the total model-operator predictions, ⟨ψv|˜hNMS
+h.o. |ψv⟩, the results ⟨ψv|Vs.l.|ψv⟩ for the semilocal
+part (29) only are shown as well. As it is seen from Table VII, for all the alkali metals there
+is good agreement between the approximate and exact values. Therefore, the model-QED
+operator based on the calculations for hydrogenlike ions works also reasonably well in the
+nonhydrogenic cases.
+To conclude the discussion of alkali metals, let us note that, as in Ref. [122], the strong
+dependence of the final results on the choice of the potential for the initial approximation
+takes place. The scatter of the results is related obviously with the approximate treatment
+of the interelectronic-interaction effects. We believe that the developed model-QED oper-
+ator for the nuclear recoil effect merged with the standard methods to treat the electron
+correlations will make it possible to perform much more accurate evaluation. However, such
+calculations are out of the scope of the present work. We reserve systematic calculations
+with the model-QED operator for the nuclear recoil effect for future research.
+V.
+CONCLUSION
+In the present paper, we have worked out the model-QED-operator approach to treat the
+nuclear recoil effect on binding energies in many-electron atomic systems beyond the Breit
+approximation. The approach is similar to the one proposed earlier for approximate calcula-
+tions of the radiative corrections to energy levels [46]. The developed operator can be readily
+included into any relativistic calculations based on the Dirac-Coulomb-Breit Hamiltonian.
+23
+
+The performance of the approach was demonstrated by comparing the model-QED-operator
+predictions with the results of the rigorous QED calculations.
+VI.
+ACKNOWLEDGEMENTS
+The work was supported by the Foundation for the Advancement of Theoretical Physics
+and Mathematics “BASIS” and by RFBR and ROSATOM according to the research project
+No. 20-21-00098. The work of I.S.A. was supported by the German-Russian Interdisciplinary
+Science Center (G-RISC) funded by the German Federal Foreign Office via the German
+Academic Exchange Service (DAAD).
+Appendix A: One- and two-electron contributions to the nuclear recoil effect on
+binding energies of quasi-degenerate levels
+In the present Appendix, we derive the fully relativistic expressions for the contributions
+of the nuclear recoil effect on binding energies within the two-time Green’s function (TTGF)
+method [96]. Let us denote the unperturbed wave functions of the two states under consid-
+eration as |ui⟩ and |uk⟩. During the derivation, we assume that the unperturbed energies
+E(0)
+i
+and E(0)
+k , which corresponds to these states, differ and, therefore, |ui⟩ ̸= |uk⟩. However,
+all the obtained expressions are valid for the coinciding energies and, of course, boil down to
+the expressions (8) and (17) in the case of diagonal matrix elements. In order to derive the
+formulas, we introduce a model subspace Ω, which is spanned by the states |ui⟩ and |uk⟩,
+and construct for these states the QED perturbation theory as for quasi-degenerate levels.
+The projector on Ω reads as
+P (0) = |ui⟩⟨ui| + |uk⟩⟨uk| .
+(A1)
+The derivation procedure can be readily generalized for an arbitrary number of quasi-
+degenerate levels.
+First, let us recall the basic ideas of the TTGF methods in the application to quasi-
+degenerate levels. The detailed description of the method can be found, e.g., in Refs. [96,
+109, 126]. The N-electron TTGF is defined as
+G(t′, t; r′
+1, . . . , r′
+N; r1, . . . , rN) = ⟨0|Tψ(x′
+1) · · ·ψ(x′
+N) ¯ψ(xN) · · · ¯ψ(x1)|0⟩
+�����t′0
+1 =...=t′0
+N≡t′
+t0
+1=...=t0
+N≡t
+, (A2)
+24
+
+where ψ is the electron-positron field operator in the Heisenberg representation, ¯ψ = ψ†γ0,
+x = (t0, r), and T is the time-ordering operator. Turning to the mixed representation, one
+obtains
+G(E; r′
+1, . . . , r′
+N; r1, . . . , rN)δ(E − E′)
+=
+1
+2πi
+1
+N!
+� ∞
+−∞
+dtdt′ eiE′t′−iEt G(t′, t; r′
+1, . . . , r′
+N; r1, . . . , rN) .
+(A3)
+Employing P (0), we can introduce the projection of the Green’s function (A3) on the sub-
+space Ω,
+g(E) = P (0)G(E)γ0
+1 . . . γ0
+NP (0) ,
+(A4)
+and then determine the ˆK and ˆP operators:
+ˆK ≡
+1
+2πi
+�
+Γ
+dE Eg(E) ,
+(A5)
+ˆP ≡
+1
+2πi
+�
+Γ
+dE g(E) .
+(A6)
+The anticlockwise oriented contour Γ in the complex E plane surrounds the poles corre-
+sponding to the quasi-degenerate levels and keeps outside all other singularities of g(E).
+The investigated system is fully described by the effective operator ˆH defined as
+ˆH = ˆP −1/2 ˆK ˆP −1/2 .
+(A7)
+The perturbation theory for the Green’s function (A2) leads to the perturbation series for
+the operator ˆH.
+To first order in m/M, the nuclear recoil effect is described by the one- and two-electron
+Feynman diagrams depicted in Figs. 1 and 2. The Feynman rules for these diagrams are
+formulated, e.g., in Ref. [26], see also Ref. [96]. The first-order contribution to the operator ˆH
+can be expressed as
+ˆH(1) = ˆK(1) − 1
+2
+ˆP (1) ˆK(0) − 1
+2
+ˆK(0) ˆP (1) ,
+(A8)
+where the superscripts denote the orders in the expansion parameter. The contributions of
+the nuclear recoil effect are determined by the matrix elements of the operator (A8) in the
+basis of the unperturbed functions |ui⟩ and |uk⟩:
+H(1)
+ik ≡ ⟨ui| ˆH(1)|uk⟩ .
+(A9)
+25
+
+To zeroth order, the matrix of the operator ˆK is diagonal, K(0)
+ik = E(0)
+i δik. Therefore, one
+can obtain
+H(1)
+ik = K(1)
+ik − E(0)
+i
++ E(0)
+k
+2
+P (1)
+ik .
+(A10)
+Let us now directly turn to the derivation of the desired nonperturbative (in αZ) formulas
+for the nuclear recoil effect on binding energies. We start from the one-electron (NMS) con-
+tribution corresponding to the diagrams in Fig. 1. In this case, N = 1 and the unperturbed
+wave functions |ui⟩, |uk⟩ and energies E(0)
+i , E(0)
+k
+are given by the Dirac eigenfunctions and
+eigenvalues, respectively, see Eq. (5). The present derivation is similar to the one for the
+self-energy diagram shown in Fig. 3. Employing the Feynman rules [26], one can obtain the
+following expression for the matrix element of the Green’s function ∆g(1)
+NMS(E) projected on
+the subspace Ω [96]:
+∆g(1)
+NMS,ik(E) =
+⟨ψi|P(E)|ψk⟩
+(E − εi)(E − εk) ,
+(A11)
+where the operator P is defined in Eq. (9). The corresponding contributions to the ˆK and
+ˆP operators read as:
+K(1)
+NMS,ik =
+1
+2πi
+�
+Γ
+dE E ∆g(1)
+NMS,ik(E) =
+εi
+εi − εk
+⟨ψi|P(εi)|ψk⟩ +
+εk
+εk − εi
+⟨ψi|P(εk)|ψk⟩ , (A12)
+P (1)
+NMS,ik =
+1
+2πi
+�
+Γ
+dE ∆g(1)
+NMS,ik(E) =
+1
+εi − εk
+⟨ψi|P(εi)|ψk⟩ +
+1
+εk − εi
+⟨ψi|P(εk)|ψk⟩ .
+(A13)
+Substituting Eqs. (A12) and (A13) into the formula (A10), one finally obtains the NMS
+contribution
+HNMS,ik = 1
+2
+�
+⟨ψi|P(εi)|ψk⟩ + ⟨ψi|P(εk)|ψk⟩
+�
+.
+(A14)
+The derivation of the nonperturbative (in αZ) expression for the two-electron (SMS)
+contribution in Fig. 2 is also straightforward but more tedious. In this case, N = 2 and
+we, for simplicity, assume that the unperturbed wave functions |ui⟩ and |uk⟩ are given
+by the one-determinant wave functions Ψi1i2 and Ψk1k2, respectively, see Eq. (16).
+The
+corresponding unperturbed energies are equal to E(0)
+i
+= εi1 + εi2 and E(0)
+k
+= εk1 + εk2. The
+derivation repeats the one for the one-photon-exchange diagram in Fig. 5. The Green’s
+26
+
+function ∆g(1)
+SMS(E) projected on the subspace Ω can be written as [26, 96]
+∆g(1)
+SMS,ik(E) =
+� i
+2π
+�2 �
+dp1dp′
+1
+�
+P
+(−1)P ⟨ψP i1ψP i2|R(p1 − p′
+1)|ψk1ψk2⟩
+×
+1
+(p′
+1 − εP i1 + i0)(E − p′
+1 − εP i2 + i0)
+1
+(p1 − εk1 + i0)(E − p1 − εk2 + i0) ,
+(A15)
+where the operator R is defined by Eq. (10). Using the identity
+1
+(p′
+1 − εP i1 + i0)(E − p′
+1 − εP i2 + i0) =
+1
+E − E(0)
+i
+�
+1
+p′
+1 − εP i1 + i0 +
+1
+E − p′
+1 − εP i2 + i0
+�
+(A16)
+and a similar one for the second pair of denominators in Eq. (A15), one can explicitly separate
+the poles of the Green’s function located inside the contour Γ. Then, the contribution to
+the ˆK operator is
+K(1)
+SMS,ik =
+�
+Γ
+dE E ∆g(1)
+SMS,ik(E)
+=
+� i
+2π
+�2 �
+dp1dp′
+1
+�
+P
+(−1)P ⟨ψP i1ψP i2|R(p1 − p′
+1)|ψk1ψk2⟩
+×
+�
+E(0)
+i
+E(0)
+i
+− E(0)
+k
+�2π
+i δ(p′
+1 − εP i1)
+� �
+1
+p1 − εk1 + i0 +
+1
+E(0)
+i
+− p1 − εk2 + i0
+�
++
+E(0)
+k
+E(0)
+k
+− E(0)
+i
+�
+1
+p′
+1 − εP i1 + i0 +
+1
+E(0)
+k
+− p′
+1 − εP i2 + i0
+� �2π
+i δ(p1 − εk1)
+� �
+,
+(A17)
+where the indentity
+1
+p − ε + i0 +
+1
+−p + ε + i0 = 2π
+i δ(p − ε)
+(A18)
+was employed. Defining the coefficients A and B so that K(1)
+SMS,ik ≡ AE(0)
+i
++BE(0)
+k , the contri-
+bution to the operator ˆP can be expressed as P (1)
+SMS,ik = A+B. Therefore, the formula (A10)
+27
+
+results in
+HSMS,ik = 1
+2
+i
+2π
+�
+dω
+�
+P
+(−1)P
+×
+�
+⟨ψP i1ψP i2|R(ω − εP i1)|ψk1ψk2⟩
+�
+1
+ω − εk1 + i0 +
+1
+E(0)
+i
+− ω − εk2 + i0
+�
++ ⟨ψP i1ψP i2|R(εk1 − ω)|ψk1ψk2⟩
+�
+1
+ω − εP i1 + i0 +
+1
+E(0)
+k
+− ω − εP i2 + i0
+� �
+.
+(A19)
+The integration over ω can be performed using the standard identity (ω1 < 0 < ω2):
+ω2
+�
+ω1
+dω f(ω)
+ω ± i0 = ∓iπf(0) + P.V.
+ω2
+�
+ω1
+dω f(ω)
+ω
+,
+(A20)
+where P.V means the principal-value integral. Indeed, applying the formula (A20) to all
+four terms in Eq. (A19) and taking into account that all the principal-value integrals vanish
+due to the fact that R(ω) is the even function of ω, one finally obtains the SMS contribution
+HSMS,ik = 1
+2
+�
+P
+(−1)P�
+⟨ψP i1ψP i2|R(∆1)|ψk1ψk2⟩ + ⟨ψP i1ψP i2|R(∆2)|ψk1ψk2⟩
+�
+,
+(A21)
+where ∆1 = εP i1 − εk1 and ∆2 = εP i2 − εk2.
+Appendix B: Computational formulas for the one-electron contributions
+In view of the definition (9), the expression (A14) obtained in Appendix A is given in a
+form that allows one to use the finite-basis-set methods [120, 121, 127] for its calculations.
+Therefore, the main difficulty is related with the evaluation of the integral over ω. For large
+real values of ω, the photon propagator (7) is a strongly oscillating function. It is convenient
+to perform the Wick’s rotation to overcome this obstacle. In the present Appendix, we
+discuss the related transformations for the contribution HNMS,ik.
+Employing Eq. (12), one can represent Eq. (A14) as the sum of the Coulomb, one-
+transverse-photon, and two-transverse-photon contributions
+HNMS,ik = Hc
+NMS,ik + Htr1
+NMS,ik + Htr2
+NMS,ik .
+(B1)
+28
+
+���������
+��������
+�
+���������
+��������
+�
+FIG. 6. The poles and the branch cuts of the integrand in the operator P(εx) for the one- and
+two-transverse-photon contributions. The integration contour: the original one oriented along the
+real axis (left panel); the rotated to the imaginary one (right panel).
+���������
+��������
+�
+FIG. 7. The poles and the branch cuts of the integrand in the operator P(εx) for the one- and
+two-transverse-photon contributions. The integration contour in the complex plane is chosen to
+avoid all the singularities of the integrand.
+In the Coulomb contribution, the integration over ω can be performed analytically by means
+of the identity (A20)
+Hc
+NMS,ik = 1
+2
+� εn>0
+�
+n
+⟨ψiψn|Rc|ψnψk⟩ −
+εn<0
+�
+n
+⟨ψiψn|Rc|ψnψk⟩
+�
+.
+(B2)
+The one- and two-transverse-photon contributions are handled identically. In what follows
+in this Appendix, we will refer to them together as the “transverse-photon” (tr) ones. In the
+left panel of Fig. 6, the original integration contour oriented along the real axis is shown. The
+poles and the branch cuts of the integrand are presented as well. Upon the Wick’s rotation
+to the imaginary axis, the bound-state poles of the electron Green’s function are picked up
+as the residues, see the right panel in Fig. 6. The final formulas for the transverse-photon
+29
+
+contribution can be written as a sum of three terms
+Htr
+NMS,ik = Htr(a)
+NMS,ik + Htr(b)
+NMS,ik + Htr(c)
+NMS,ik .
+(B3)
+The first term arises from the residues shown by the circles in Fig. 6 and reads as
+Htr(a)
+NMS,ik = 1
+2
+�
+x=i,k
+−1<εn<εx
+�
+n
+⟨ψiψn|Rtr(εn − εx)|ψnψk⟩ ,
+(B4)
+where for brevity we have introduced the summation over x in order to take into account
+the symmetric form of the expression with respect to the argument of P(E). The second
+term corresponds to the case of degeneracy between the states of the opposite parity. This
+term originates from the poles located at ω = 0 and has the form
+Htr(b)
+NMS,ik = 1
+4
+�
+x=i,k
+εn=εx
+�
+n
+⟨ψiψn|Rtr(0)|ψnψk⟩ .
+(B5)
+Finally, the third term corresponds to the integration over the imaginary axis
+Htr(c)
+NMS,ik = 1
+2
+�
+x=i,k
+εn̸=εx
+�
+n
+1
+π
+� ∞
+0
+dy
+εn − εx
+y2 + (εn − εx)2⟨ψiψn|Rtr(iy)|ψnψk⟩ .
+(B6)
+The expressions similar to (B2), (B4), (B5), and (B6) were derived, e.g., in Ref. [32].
+However, only the case of the diagonal matrix elements was considered. Moreover, in Ref. [32]
+the formulas for the higher-order QED corrections beyond the Breit approximation were
+given, while the present ones include the lowest-relativistic contributions as well.
+As an additional crosscheck of the ω-integration routine, we perform it also by employ-
+ing the contour schematically shown in Fig. 7. This contour is chosen to bypass all the
+singularities in the complex plane. The matrix elements HNMS,ik have been evaluated for
+both variants of the integration-contour rotation. The results are found to be in excellent
+agreement with each other.
+Appendix C: Local effective potentials
+In Sec. IV, in order to test the performance of the model-QED operator for the nuclear
+recoil effect, we need substitute the core-Hartree (CH) and xα potentials (see, e.g., Ref. [122])
+instead of V in Eq. (5). These potentials can be expressed via the charge densities of the
+valence electron,
+ρv(r) = g2
+v(r) + f 2
+v (r) ,
+(C1)
+30
+
+and the core,
+ρcore(r) =
+�
+c
+(2jc + 1)
+�
+g2
+c(r) + f 2
+c (r)
+�
+.
+(C2)
+Here the large, g, and small, f, components of the Dirac wave function in Eqs. (C1) and
+(C2) are determined self-consistently in the local potential
+V (r) = −αZeff(r)
+r
+,
+(C3)
+where Zeff(r) is the effective charge. The CH potential is given by
+ZCH
+eff (r) = Znucl(r) − r
+∞
+�
+0
+dr′
+ρcore(r′)
+max{r, r′} ,
+(C4)
+where Znucl(r) accounts for the finite size of the nucleus. Defining the total charge density,
+ρtot = ρcore + ρv, the effective charge for the xα potentials can be written in the form
+Zxα
+eff (r) = Znucl(r) − r
+∞
+�
+0
+dr′
+ρtot(r′)
+max{r, r′} + xα
+� 81
+32π2rρtot(r)
+�1/3
+.
+(C5)
+We use the values of xα equal to 0, 1/3, 2/3, and 1. The choice xα = 0 leads to the Dirac-
+Hartree potential, xα = 2/3 corresponds to the Kohn-Sham potential [123], while xα = 1 is
+the Dirac-Slater potential [124]. In order to restore the proper asymptotic behavior of the
+xα potentials, we introduce the Latter correction [125].
+[1] S. R. Elliott, P. Beiersdorfer, and M. H. Chen, Phys. Rev. Lett. 76, 1031 (1996); 77, 4278(E)
+(1996).
+[2] S. R. Elliott, P. Beiersdorfer, M. H. Chen, V. Decaux, and D. A. Knapp, Phys. Rev. C
+57, 583 (1998).
+[3] R. Schuch, E. Lindroth, S. Madzunkov, M. Fogle, T. Mohamed, and P. Indelicato, Phys.
+Rev. Lett. 95, 183003 (2005).
+[4] R. Soria Orts, Z. Harman, J. R. Crespo L´opez-Urrutia, A. N. Artemyev, H. Bruhns,
+A. J. G. Mart´ınez, U. D. Jentschura, C. H. Keitel, A. Lapierre, V. Mironov, V. M. Shabaev,
+H. Tawara, I. I. Tupitsyn, J. Ullrich, and A. V. Volotka, Phys. Rev. Lett. 97, 103002 (2006).
+31
+
+[5] C. Brandau, C. Kozhuharov, Z. Harman, A. M¨uller, S. Schippers, Y. S. Kozhedub, D. Bern-
+hardt, S. B¨ohm, J. Jacobi, E. W. Schmidt, P. H. Mokler, F. Bosch, H.-J. Kluge, Th. St¨ohlker,
+K. Beckert, P. Beller, F. Nolden, M. Steck, A. Gumberidze, R. Reuschl, U. Spillmann,
+F. J. Currell, I. I. Tupitsyn, V. M. Shabaev, U. D. Jentschura, C. H. Keitel, A. Wolf, and
+Z. Stachura, Phys. Rev. Lett. 100, 073201 (2008).
+[6] F. K¨ohler, K. Blaum, M. Block, S. Chenmarev, S. Eliseev, D. A. Glazov, M. Gon-
+charov, J. Hou, A. Kracke, D. A. Nesterenko, Y. N. Novikov, W. Quint, E. M. Ramirez,
+V. M. Shabaev, S. Sturm, A. V. Volotka, and G. Werth, Nat. Commun. 7, 10246 (2016).
+[7] B. Maaß, T. H¨uther, K. K¨onig, J. Kr¨amer, J. Krause, A. Lovato, P. M¨uller, K. Pachucki,
+M. Puchalski, R. Roth, R. S´anchez, F. Sommer, R. B. Wiringa, and W. N¨ortersh¨auser, Phys.
+Rev. Lett. 122, 182501 (2019).
+[8] ´A. Koszor´us, X. F. Yang, W. G. Jiang, S. J. Novario, S. W. Bai, J. Billowes, C. L. Binnersley,
+M. L. Bissell, T. E. Cocolios, B. S. Cooper, R. P. de Groote, A. Ekstr¨om, K. T. Flanagan,
+C. Forss´en, S. Franchoo, R. F. G. Ruiz, F. P. Gustafsson, G. Hagen, G. R. Jansen, A. Kanel-
+lakopoulos, M. Kortelainen, W. Nazarewicz, G. Neyens, T. Papenbrock, P.-G. Reinhard,
+C. M. Ricketts, B. K. Sahoo, A. R. Vernon, and S. G. Wilkins, Nat. Phys. 17, 439 (2021).
+[9] T. Sailer, V. Debierre, Z. Harman, F. Heiße, C. K¨onig, J. Morgner, B. Tu, A. V. Volotka,
+C. H. Keitel, K. Blaum, and S. Sturm, Nature 606, 479 (2022).
+[10] S. A. King, L. J. Spieß, P. Micke, A. Wilzewski, T. Leopold, E. Benkler, R. Lange,
+N. Huntemann, A. Surzhykov, V. A. Yerokhin, J. R. C. L´opez-Urrutia, and P. O. Schmidt,
+arXiv:2205.13053 [physics.atom-ph] (2022).
+[11] J. Z. Han, C. Pan, K. Y. Zhang, X. F. Yang, S. Q. Zhang, J. C. Berengut, S. Goriely,
+H. Wang, Y. M. Yu, J. Meng, J. W. Zhang, and L. J. Wang, Phys. Rev. Research 4, 033049
+(2022).
+[12] C. Frugiuele, E. Fuchs, G. Perez, and M. Schlaffer, Phys. Rev. D 96, 015011 (2017).
+[13] J. C. Berengut, D. Budker, C. Delaunay, V. V. Flambaum, C. Frugiuele, E. Fuchs, C. Gro-
+jean, R. Harnik, R. Ozeri, G. Perez, and Y. Soreq, Phys. Rev. Lett. 120, 091801 (2018).
+[14] V. V. Flambaum, A. J. Geddes, and A. V. Viatkina, Phys. Rev. A 97, 032510 (2018).
+[15] V. A. Yerokhin, R. A. M¨uller, A. Surzhykov, P. Micke, and P. O. Schmidt, Phys. Rev. A
+101, 012502 (2020).
+32
+
+[16] N.-H. Rehbehn, M. K. Rosner, H. Bekker, J. C. Berengut, P. O. Schmidt, S. A. King,
+P. Micke, M. F. Gu, R. M¨uller, A. Surzhykov, and J. R. C. L´opez-Urrutia, Phys. Rev. A
+103, L040801 (2021).
+[17] N. L. Figueroa, J. C. Berengut, V. A. Dzuba, V. V. Flambaum, D. Budker, and D. Antypas,
+Phys. Rev. Lett. 128, 073001 (2022).
+[18] K. Ono, Y. Saito, T. Ishiyama, T. Higomoto, T. Takano, Y. Takasu, Y. Yamamoto,
+M. Tanaka, and Y. Takahashi, Phys. Rev. X 12, 021033 (2022).
+[19] V. Debierre, C. H. Keitel, and Z. Harman, Physics Letters B 807, 135527 (2020).
+[20] V. Debierre and N. S. Oreshkina, Phys. Rev. A 104, 032825 (2021).
+[21] V. Debierre, C. H. Keitel, and Z. Harman,
+arXiv:2202.01668 [physics.atom-ph] (2022),
+arXiv:2202.01668.
+[22] V. Debierre, N. S. Oreshkina, I. A. Valuev, Z. Harman, and C. H. Keitel, rXiv:2207.04868
+[physics.atom-ph] (2022).
+[23] V. M. Shabaev, Theor. Math. Phys. 63, 588 (1985), [Teor. Mat. Fiz. 63, 394 (1985)].
+[24] V. M. Shabaev, Sov. J. Nucl. Phys. 47, 69 (1988), [Yad. Fiz. 47, 107 (1988)].
+[25] C. W. P. Palmer, J. Phys. B: At. Mol. Phys. 20, 5987 (1987).
+[26] V. M. Shabaev, Phys. Rev. A 57, 59 (1998).
+[27] K. Pachucki and H. Grotch, Phys. Rev. A 51, 1854 (1995).
+[28] A. S. Yelkhovsky, e-print hep-th/9403095 (1994); Zh. Eksp. Fiz. 110, 431 (1996).
+[29] G. S. Adkins, S. Morrison, and J. Sapirstein, Phys. Rev. A 76, 042508 (2007).
+[30] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. Lett. 115, 233002 (2015).
+[31] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A 93, 062514 (2016).
+[32] A. N. Artemyev, V. M. Shabaev, and V. A. Yerokhin, Phys. Rev. A 52, 1884 (1995).
+[33] V. M. Shabaev, A. N. Artemyev, T. Beier, G. Plunien, V. A. Yerokhin, and G. Soff, Phys.
+Rev. A 57, 4235 (1998).
+[34] A. V. Malyshev, R. V. Popov, V. M. Shabaev, and N. A. Zubova, J. Phys. B: At. Mol. Opt.
+Phys. 51, 085001 (2018).
+[35] I. P. Grant, Adv. Phys. 19, 747 (1970).
+[36] J. P. Desclaux, Comput. Phys. Commun. 9, 31 (1975).
+[37] V. F. Bratzev, G. B. Deyneka, and I. I. Tupitsyn, Bull. Acad. Sci. USSR, Phys. Ser. 41, 173
+(1977), [Izv. Acad. Nauk SSSR, Ser. Fiz. 41, 2655 (1977)].
+33
+
+[38] P. Indelicato and E. Lindroth, Phys. Rev. A 46, 2426 (1992).
+[39] V. A. Dzuba, V. V. Flambaum, and M. G. Kozlov, Phys. Rev. A 54, 3948 (1996).
+[40] M. S. Safronova, W. R. Johnson, and A. Derevianko, Phys. Rev. A 60, 4476 (1999).
+[41] I. I. Tupitsyn, V. M. Shabaev, J. R. Crespo L´opez-Urrutia, I. Dragani´c, R. Soria Orts, and
+J. Ullrich, Phys. Rev. A 68, 022511 (2003).
+[42] M. G. Kozlov, S. G. Porsev, M. S. Safronova, and I. I. Tupitsyn, Comput. Phys. Commun.
+195, 199 (2015).
+[43] V. A. Dzuba, J. C. Berengut, C. Harabati, and V. V. Flambaum, Phys. Rev. A 95, 012503
+(2017).
+[44] D. A. Glazov, A. V. Malyshev, A. V. Volotka, V. M. Shabaev, I. I. Tupitsyn, and G. Plunien,
+Nucl. Instrum. Methods Phys. Res., Sect. B 408, 46 (2017).
+[45] T. Saue, R. Bast, A. S. P. Gomes, H. J. A. Jensen, L. Visscher, I. A. Aucar, R. Di Remigio,
+K. G. Dyall, E. Eliav, E. Fasshauer, T. Fleig, L. Halbert, E. D. Hedeg˚ard, B. Helmich-Paris,
+M. Iliaˇs, C. R. Jacob, S. Knecht, J. K. Laerdahl, M. L. Vidal, M. K. Nayak, M. Olejniczak,
+J. M. H. Olsen, M. Pernpointner, B. Senjean, A. Shee, A. Sunaga, and J. N. P. van Stralen,
+J. Chem. Phys. 152, 204104 (2020).
+[46] V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin, Phys. Rev. A 88, 012513 (2013).
+[47] A. V. Malyshev, D. A. Glazov, V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, and V. A. Za-
+ytsev, Phys. Rev. A 106, 012806 (2022).
+[48] V. M. Shabaev, I. I. Tupitsyn, and V. A. Yerokhin,
+Comput. Phys. Commun. 189, 175
+(2015); 223, 69 (2018).
+[49] I. I. Tupitsyn, M. G. Kozlov, M. S. Safronova, V. M. Shabaev, and V. A. Dzuba, Phys. Rev.
+Lett. 117, 253001 (2016).
+[50] L. F. Paˇsteka, E. Eliav, A. Borschevsky, U. Kaldor, and P. Schwerdtfeger, Phys. Rev. Lett.
+118, 023002 (2017).
+[51] J. Machado, C. I. Szabo, J. P. Santos, P. Amaro, M. Guerra, A. Gumberidze, Guojie Bian,
+J. M. Isac, and P. Indelicato, Phys. Rev. A 97, 032517 (2018).
+[52] R. Si, X. L. Guo, T. Brage, C. Y. Chen, R. Hutton, and C. Froese Fischer, Phys. Rev. A
+98, 012504 (2018).
+[53] A. M¨uller, E. Lindroth, S. Bari, A. Borovik, Jr., P.-M. Hillenbrand, K. Holste, P. Indelicato,
+A. L. D. Kilcoyne, S. Klumpp, M. Martins, J. Viefhaus, P. Wilhelm, and S. Schippers, Phys.
+34
+
+Rev. A 98, 033416 (2018).
+[54] V. A. Zaytsev, I. A. Maltsev, I. I. Tupitsyn, and V. M. Shabaev, Phys. Rev. A 100, 052504
+(2019).
+[55] V. A. Yerokhin, M. Puchalski, and K. Pachucki, Phys. Rev. A 102, 042816 (2020).
+[56] V. M. Shabaev, I. I. Tupitsyn, M. Y. Kaygorodov, Y. S. Kozhedub, A. V. Malyshev, and
+D. V. Mironova, Phys. Rev. A 101, 052502 (2020).
+[57] I. I. Tupitsyn, S. V. Bezborodov, A. V. Malyshev, D. V. Mironova, and V. M. Shabaev, Opt.
+Spectrosc. 128, 21 (2020).
+[58] M. Y. Kaygorodov, L. V. Skripnikov, I. I. Tupitsyn, E. Eliav, Y. S. Kozhedub, A. V. Maly-
+shev, A. V. Oleynichenko, V. M. Shabaev, A. V. Titov, and A. V. Zaitsevskii, Phys. Rev. A
+104, 012819 (2021).
+[59] L. V. Skripnikov, J. Chem. Phys. 154, 201101 (2021).
+[60] M. Y. Kaygorodov, D. P. Usov, E. Eliav, Y. S. Kozhedub, A. V. Malyshev, A. V. Oleyn-
+ichenko, V. M. Shabaev, L. V. Skripnikov, A. V. Titov, I. I. Tupitsyn, and A. V. Zaitsevskii,
+Phys. Rev. A 105, 062805 (2022).
+[61] H. H¨affner, T. Beier, N. Hermanspahn, H.-J. Kluge, W. Quint, S. Stahl, J. Verd´u, and
+G. Werth, Phys. Rev. Lett. 85, 5308 (2000).
+[62] J. Verd´u, S. Djeki´c, S. Stahl, T. Valenzuela, M. Vogel, G. Werth, T. Beier, H.-J. Kluge, and
+W. Quint, Phys. Rev. Lett. 92, 093002 (2004).
+[63] S. Sturm, A. Wagner, B. Schabinger, J. Zatorski, Z. Harman, W. Quint, G. Werth, C. H. Kei-
+tel, and K. Blaum, Phys. Rev. Lett. 107, 023002 (2011).
+[64] A. Wagner, S. Sturm, F. K¨ohler, D. A. Glazov, A. V. Volotka, G. Plunien, W. Quint,
+G. Werth, V. M. Shabaev, and K. Blaum, Phys. Rev. Lett. 110, 033003 (2013).
+[65] S. Sturm, A. Wagner, M. Kretzschmar, W. Quint, G. Werth, and K. Blaum, Phys. Rev. A
+87, 030501(R) (2013).
+[66] D. A. Glazov, F. K¨ohler-Langes, A. V. Volotka, K. Blaum, F. Heiße, G. Plunien, W. Quint,
+S. Rau, V. M. Shabaev, S. Sturm, and G. Werth, Phys. Rev. Lett. 123, 173001 (2019).
+[67] I. Arapoglou, A. Egl, M. H¨ocker, T. Sailer, B. Tu, A. Weigel, R. Wolf, H. Cakir,
+V. A. Yerokhin, N. S. Oreshkina, V. A. Agababaev, A. V. Volotka, D. V. Zinenko,
+D. A. Glazov, Z. Harman, C. H. Keitel, S. Sturm, and K. Blaum, Phys. Rev. Lett. 122, 253001
+(2019).
+35
+
+[68] V. M. Shabaev, Phys. Rev. A 64, 052104 (2001).
+[69] V. M. Shabaev and V. A. Yerokhin, Phys. Rev. Lett. 88, 091801 (2002).
+[70] V. M. Shabaev, D. A. Glazov, A. V. Malyshev, and I. I. Tupitsyn,
+Phys. Rev. Lett.
+119, 263001 (2017).
+[71] A. V. Malyshev, V. M. Shabaev, D. A. Glazov, and I. I. Tupitsyn, JETP Lett. 106, 765
+(2017).
+[72] V. M. Shabaev, D. A. Glazov, A. V. Malyshev, and I. I. Tupitsyn, Phys. Rev. A 98, 032512
+(2018).
+[73] I. A. Aleksandrov, D. A. Glazov, A. V. Malyshev, V. M. Shabaev, and I. I. Tupitsyn, Phys.
+Rev. A 98, 062521 (2018).
+[74] A. V. Malyshev, D. A. Glazov, and V. M. Shabaev, Phys. Rev. A 101, 012513 (2020).
+[75] A. V. Malyshev, D. A. Glazov, I. A. Aleksandrov, I. I. Tupitsyn, and V. M. Shabaev, Opt.
+Spectrosc. 128, 297 (2020).
+[76] D. S. Hughes and C. Eckart, Phys. Rev. 36, 694 (1930).
+[77] V. A. Korol and M. G. Kozlov, Phys. Rev. A 76, 022103 (2007).
+[78] Y. S. Kozhedub, A. V. Volotka, A. N. Artemyev, D. A. Glazov, G. Plunien, V. M. Shabaev,
+I. I. Tupitsyn, and Th. St¨ohlker, Phys. Rev. A 81, 042513 (2010).
+[79] E. Gaidamauskas, C. Naz´e, P. Rynkun, G. Gaigalas, P. J¨onsson, and M. Godefroid, J. Phys.
+B: At. Mol. Opt. Phys. 44, 175003 (2011).
+[80] C. Naz´e, S. Verdebout, P. Rynkun, G. Gaigalas, M. Godefroid, and P. J¨onsson, At. Data
+Nucl. Data Tables 100, 1197 (2014).
+[81] N. A. Zubova, Y. S. Kozhedub, V. M. Shabaev, I. I. Tupitsyn, A. V. Volotka, G. Plunien,
+C. Brandau, and Th. St¨ohlker, Phys. Rev. A 90, 062512 (2014).
+[82] C. Naz´e, J. G. Li, and M. Godefroid, Phys. Rev. A 91, 032511 (2015).
+[83] C. F. Fischer, M. Godefroid, T. Brage, P. J¨onsson, and G. Gaigalas, J. Phys. B: At. Mol.
+Opt. Phys. 49, 182004 (2016).
+[84] N. A. Zubova, A. V. Malyshev, I. I. Tupitsyn, V. M. Shabaev, Y. S. Kozhedub, G. Plunien,
+C. Brandau, and Th. St¨ohlker, Phys. Rev. A 93, 052502 (2016).
+[85] L. Filippin, J. Biero´n, G. Gaigalas, M. Godefroid, and P. J¨onsson, Phys. Rev. A 96, 042502
+(2017).
+36
+
+[86] I. I. Tupitsyn, N. A. Zubova, V. M. Shabaev, G. Plunien, and Th. St¨ohlker, Phys. Rev. A
+98, 022517 (2018).
+[87] S. Gamrath, P. Palmeri, P. Quinet, S. Bouazza, and M. Godefroid, J. Quant. Spectrosc.
+Radiat. Transf. 218, 38 (2018).
+[88] N. A. Zubova, I. S. Anisimova, M. Y. Kaygorodov, Y. S. Kozhedub, A. V. Malyshev,
+V. M. Shabaev, I. I. Tupitsyn, G. Plunien, C. Brandau, and T. St¨ohlker, J. Phys. B: At.
+Mol. Opt. Phys. 52, 185001 (2019).
+[89] J. Ekman, P. J¨onsson, M. Godefroid, C. Naz´e, G. Gaigalas, and J. Biero´n, Comput. Phys.
+Commun. 235, 433 (2019).
+[90] N. A. Zubova, M. Y. Kaygorodov, Y. S. Kozhedub, A. V. Malyshev, R. V. Popov, I. M. Save-
+lyev, I. I. Tupitsyn, and V. M. Shabaev, Opt. Spectrosc. 128, 1090 (2020).
+[91] R. A. M¨uller, V. A. Yerokhin, A. N. Artemyev, and A. Surzhykov, Phys. Rev. A 104, L020802
+(2021).
+[92] R. Si, S. Schiffmann, K. Wang, C. Y. Chen, and M. Godefroid, Phys. Rev. A 104, 012802
+(2021).
+[93] J. S. Schelfhout and J. J. McFerran, Phys. Rev. A 104, 022806 (2021).
+[94] J. S. Schelfhout and J. J. McFerran, Phys. Rev. A 105, 022805 (2022).
+[95] W. H. Furry, Phys. Rev. 81, 115 (1951).
+[96] V. M. Shabaev, Phys. Rep. 356, 119 (2002).
+[97] A. N. Artemyev, V. M. Shabaev, and V. A. Yerokhin,
+J. Phys. B: At. Mol. Opt. Phys.
+28, 5201 (1995).
+[98] V. M. Shabaev, A. N. Artemyev, T. Beier, G. Plunien, V. A. Yerokhin, and G. Soff, Phys.
+Scr. T80, 493 (1999).
+[99] I. A. Aleksandrov, A. A. Shchepetnov, D. A. Glazov, and V. M. Shabaev, J. Phys. B: At.
+Mol. Opt. Phys. 48, 144004 (2015).
+[100] H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41, 350 (1969).
+[101] E. Borie and G. A. Rinker, Rev. Mod. Phys. 54, 67 (1982).
+[102] K. Pachucki and V. A. Yerokhin, arXiv:2208.13381 [physics.atom-ph] (2022).
+[103] A. Veitia and K. Pachucki, Phys. Rev. A 69, 042501 (2004).
+[104] A. V. Malyshev, I. S. Anisimova, D. V. Mironova, V. M. Shabaev, and G. Plunien, Phys.
+Rev. A 100, 012510 (2019).
+37
+
+[105] A. V. Malyshev, I. S. Anisimova, D. A. Glazov, M. Y. Kaygorodov, D. V. Mironova, G. Plu-
+nien, and V. M. Shabaev, Phys. Rev. A 101, 052506 (2020).
+[106] P. J. Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys. 84, 1527 (2012).
+[107] J. Sapirstein and D. R. Yennie,
+Theory of hydrogenic bound states,
+in Quantum
+Electrodynamics, edited by T. Kinoshita, p. 560, World Scientific, Singapore, 1990.
+[108] K. Pachucki and S. G. Karshenboim, J. Phys. B: At. Mol. Opt. Phys. 28, L221 (1995).
+[109] V. M. Shabaev, J. Phys. B: At. Mol. Opt. Phys. 26, 4703 (1993).
+[110] M. H. Chen and K. T. Cheng, Phys. Rev. A 55, 166 (1997).
+[111] K. T. Cheng, M. H. Chen, and W. R. Johnson, Phys. Rev. A 77, 052504 (2008).
+[112] V. A. Yerokhin and A. Surzhykov, Phys. Rev. A 86, 042507 (2012).
+[113] M. Y. Kaygorodov, Y. S. Kozhedub, I. I. Tupitsyn, A. V. Malyshev, D. A. Glazov, G. Plunien,
+and V. M. Shabaev, Phys. Rev. A 99, 032505 (2019).
+[114] P. J. Mohr, Ann. Phys. (N.Y.) 88, 26 (1974); 88, 52 (1974).
+[115] N. J. Snyderman, Ann. Phys. (N.Y.) 211, 43 (1991).
+[116] P. J. Mohr, G. Plunien, and G. Soff, Phys. Rep. 293, 227 (1998).
+[117] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A 60, 800 (1999).
+[118] I. Angeli and K. P. Marinova, At. Data Nucl. Data Tables 99, 69 (2013).
+[119] V. A. Yerokhin and V. M. Shabaev, J. Phys. Chem. Ref. Data 44, 033103 (2015).
+[120] W. R. Johnson, S. A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 307 (1988).
+[121] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plunien, and G. Soff, Phys. Rev. Lett.
+93, 130405 (2004).
+[122] J. Sapirstein and K. T. Cheng, Phys. Rev. A 66, 042501 (2002).
+[123] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
+[124] J. C. Slater, Phys. Rev. 81, 385 (1951).
+[125] R. Latter, Phys. Rev. 99, 510 (1955).
+[126] V. M. Shabaev, Phys. Rev. A 50, 4521 (1994).
+[127] S. Salomonson and P. ¨Oster, Phys. Rev. A 40, 5548 (1989).
+38
+
diff --git a/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/load_file.txt b/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8e86279fa32d66ce3e6bff7c84bca9258500742b
--- /dev/null
+++ b/ntE2T4oBgHgl3EQfzgi3/content/tmp_files/load_file.txt
@@ -0,0 +1,6145 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf,len=6144
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='04132v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='atom-ph]  10 Jan 2023  Model-QED-operator approach to relativistic calculations of the  nuclear recoil effect in many-electron atoms and ions  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Anisimova,1 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev,1 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov,1  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub,1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien,2 and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev1  1Department of Physics, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Petersburg State University,  Universitetskaya 7/9, 199034 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Petersburg, Russia  2Institut f¨ur Theoretische Physik, Technische Universit¨at Dresden,  Mommsenstraße 13, D-01062 Dresden, Germany  Abstract  A model-operator approach to fully relativistic calculations of the nuclear recoil effect on energy  levels in many-electron atomic systems is worked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The one-electron part of the model operator  for treating the normal mass shift beyond the Breit approximation is represented by a sum of  semilocal and nonlocal potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The latter ones are constructed by employing the diagonal and  off-diagonal matrix elements rigorously evaluated for hydrogenlike ions to first order in the electron-  to-nucleus mass ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The specific mass shift beyond the lowest-order relativistic approximation  has a form which can be directly employed in calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The capabilities of the method are  probed by comparison of its predictions with the results of ab initio QED calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The  proposed operator can be easily incorporated into any relativistic calculation based on the Dirac-  Coulomb-Breit Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  1  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  INTRODUCTION  An accurate description of the nuclear recoil effect is substantial for the proper analysis  of a large number of spectroscopic experiments aimed to measure various atomic properties  such as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', binding and transition energies or bound-electron g factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Obviously, this  effect is most pronounced in isotope differences of the corresponding properties, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [1–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The nuclear recoil leads to the so-called mass shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Along with the field  shift caused by the finite-nuclear size, these effects constitute the dominant contribution to  the isotope shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Joint high-precision theoretical and experimental studies of the isotope  differences not only allow one to determine nuclear parameters, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', changes in the mean-  square charge radii, but also pave the way in the search of new physics [12–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Within the (m/M)(αZ)4mc2 approximation, where m and M are the masses of the elec-  tron and nucleus, respectively, α is the fine-structure constant, and Z is the nuclear charge  number, the nuclear recoil effect on binding energies can be described by the relativistic  mass-shift operator [23–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The fully relativistic theory of the nuclear recoil effect to first  order in m/M, to all orders in αZ, and to zeroth order in α can be formulated only within  the framework of quantum electrodynamics (QED) [23, 24, 26], see also Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' De-  spite the smallness of the nuclear-strength parameter αZ for light atoms, the contribution  of the higher orders may nevertheless be significant even in the case of hydrogen [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Therefore, an accurate treatment of the nuclear recoil effect demands a nonperturbative  (in αZ) consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' To date, the corresponding ab initio QED calculations have been  performed only for few-electron systems, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [4, 29, 32–34] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The computational difficulty of the rigorous methods rapidly increases with the number of  electrons, which makes them practically infeasible at larger scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A similar problem exists  for evaluation of the radiative QED corrections associated with the electron self-energy and  vacuum polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For this reason, approximate and efficient approaches for including  both the QED and recoil effects within the methods based on the Dirac-Coulomb-Breit  Hamiltonian [35–45] are urgent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  For the case of the radiative QED corrections, our group suggested the model-QED-  operator approach [46] which recently has been extended to the region of superheavy ele-  ments [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The QEDMOD Fortran package to generate the operator was presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  This operator was successfully applied to the approximate description of the QED effects  2  on binding and transition energies in various many-electron systems [49–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The main  goal of the present work is to design a similar approach for the QED calculations of the  nuclear recoil effect on energy levels beyond the approximation corresponding to the mass-  shift operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The application of the proposed approach in combination with the standard  electron-correlation methods should make possible the approximate QED treatment of this  effect in systems where rigorous calculations are rather problematic at the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  In view of the significant progress achieved over the past decades in the accuracy of g-  factor measurements in Penning traps [6, 9, 61–67], high-precision evaluation of the nuclear  recoil effect in the presence of an external magnetic field becomes essential as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  QED theory of the nuclear recoil effect on the atomic g factor valid to all orders in αZ was  elaborated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The corresponding ab initio calculations have been performed for  few-electron ions in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [6, 69–75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In the case of more complicated systems, the nuclear  recoil effect on the bound-electron g factor can be treated nowadays only within the lowest-  order relativistic approximation by means of the effective four-component approach derived  from the QED formalism in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this context, the model-operator approach devel-  oped in the present work for the nuclear recoil effect on binding energies can be considered  as a first step towards the construction of a more general operator suitable for studying the  bound-electron g factors as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' II, we give a brief description of the relativistic  theory of the nuclear recoil effect on energy levels in atoms and ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' III is devoted  to the construction of the model operator for QED calculations of the nuclear recoil effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' IV, the numerical results are presented in a wide range of Z = 5 − 100, and the  performance of the suggested approach is demonstrated by comparing its predictions with  the results of ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The nonperturbative (in αZ) expressions for the one- and  two-electron matrix elements, which describe the nuclear recoil effect on the binding energies,  are derived in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Appendix B, these expressions are additionally transformed to  make them convenient for practical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Finally, Appendix C summarizes formulas  necessary for the construction of the local effective potentials employed in the tests of the  model-QED operator in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Relativistic units (ℏ = 1 and c = 1) and the Heaviside charge unit (e2 = 4πα, where  e < 0 is the electron charge) are used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  QED THEORY OF THE NUCLEAR RECOIL EFFECT  As is well known, within the nonrelativistic approximation the nuclear recoil contribution  to the binding energy of a hydrogenlike atom can be found by replacing the electron mass m  with the reduced mass, mr = mM/(m + M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For atoms with more than one electron, this  recipe is insufficient, and the two-electron part of the nuclear recoil effect has to be taken  into account [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The lowest-order relativistic (Breit) correction of first order in m/M can  be obtained by employing the mass-shift Hamiltonian [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For N-electron system, this  operator reads as  HMS =  1  2M  N  �  i,j=1  �  pi · pj − αZ  ri  �  αi + (αi · ri)  r2  i  ri  �  pj  �  ,  (1)  where the indices i and j enumerate the electrons, p = −i∇ is the momentum operator, r  is the position vector, r = |r|, and α are the Dirac matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The first term in the square  brackets in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (1) corresponds to the nonrelativistic nuclear recoil operator, whereas the  second term determines the leading relativistic correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Following Hughes and Eckart [76], the nuclear recoil contribution to atomic spectra is  usually divided into the normal (NMS) and specific (SMS) mass shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Accordingly, the  Hamiltonian (1) can be represented as a sum  HMS = HNMS + HSMS ,  (2)  where the first operator corresponds to the terms i = j in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (1) and the second one  corresponds to i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For further discussion, it is useful to rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (1) in the form  HMS =  1  2M  N  �  i,j=1  �  pi · pj − 2Di(0) · pj  �  (3)  by introducing the vector operator  D(0) = αZ  2r  �  α + (α · r)  r2  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (4)  The mass-shift operator HMS yields the nuclear recoil corrections up to the order (m/M)(αZ)4mc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  This operator is widely used in relativistic calculations of atomic spectra and isotope shifts,  where the nuclear recoil effect is particularly significant [4, 15, 41, 54, 77–94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The fully relativistic theory of the nuclear recoil effect on binding energies can be formu-  lated only within the rigorous QED approach (beyond the Breit approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' To first  4  (b)  (c)  (d)  (a)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' One-electron nuclear recoil diagrams: the Coulomb (a), one-transverse (b) and (c), and  two-transverse (d) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' See the text and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26] for the description of the Feynman  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  order in m/M, to all orders in αZ, and to zeroth order in α the corresponding theory was  developed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [23, 24, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The formalism worked out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26] is the most suitable  for the goals of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Within this formalism, the pure nuclear recoil effect is  taken into account by modifying the standard QED Hamiltonian of the electron-positron  field interacting with the quantized electromagnetic field and the classical Coulomb poten-  tial of the nucleus V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Namely, an extra term is added to the interaction part of the QED  Hamiltonian, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' As a result, the pure nuclear recoil effect on energy  levels can be obtained on equal footing with the non-recoil QED effects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', the electron  self-energy and vacuum polarization, by means of the perturbation theory in the interac-  tion representation of the Furry picture [95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A convenient approach to construct the QED  perturbation series both for single and quasi-degenerate levels is provided by the two-time  Green’s function (TTGF) method [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This method is employed in Appendix A to derive  the formal expressions for the matrix elements describing the nuclear recoil effect on bind-  ing energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Within this approach, the zeroth-order one-electron wave functions |ψn⟩ and  energies εn are assumed to be the solutions of the Dirac equation  hD|ψn⟩ ≡  �  α · p + βm + V  �  |ψn⟩ = εn|ψn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (5)  The all-order (in αZ) expressions describing the nuclear recoil effect can be divided  into the NMS and SMS parts as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The one-electron (NMS) and two-electron (SMS)  contributions are given by the Feynman diagrams shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1 and 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  exhaustive description of the additional diagram-technique rules, which arise in connection  with the treatment of the nuclear recoil effect, can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26], see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  5  (a)  (b)  (c)  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Two-electron nuclear recoil diagrams: the Coulomb (a), one-transverse (b) and (c), and  two-transverse (d) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  To explain the terminology employed throughout the paper, we briefly comment on these  rules using Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1 as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' First of all, the double line and the vertex with a small dot  in the figure correspond to the conventional diagram technique of the bound-state QED [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Namely, the line denotes the electron propagator in the potential V , and the vertex arises  from the standard interaction of the electron-positron and electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' All the  other diagram elements originate due to the presence of the aforementioned extra term in  the QED Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The Coulomb gauge established itself as the most appropriate one  for the nuclear-recoil-effect studies [23, 24, 26, 28], and it leads to the natural terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  In this gauge, the photon propagator Dµν(ω, r) is divided into the Coulomb,  D00(ω, r) =  1  4πr ,  (6)  and transverse,  Dlk(ω, r) = − 1  4π  �  exp  �  i  √  ω2 + i0 r  �  r  δlk + ∇l∇k  exp  �  i  √  ω2 + i0 r  �  − 1  ω2r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (7)  parts, while the remaining components of the photon propagator are equal to zero, Dl0 =  D0l = 0 (l, k = 1, 2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' All the recoil contributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1 can be classified with respect to  the number of the propagators (7) involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' There are three possibilities [26]: (i) the dotted  line connecting two bold dots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1(a) depicts the so-called “Coulomb recoil” interaction,  which does not contain the transverse part of the photon propagator at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (ii) the dashed  line with the bold dot at one of the ends in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1(b) and 1(c) stands for the “one-transverse-  photon recoil” interaction, which includes Dlk once;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (iii) finally, the dashed line with the  bold dot in the middle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1(d) designates the “two-transverse-photon recoil” interaction,  6  which involves the product of two photon propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We note that the approach initially  developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [23] leads to the same result as the formalism of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26] but it implies the  summation of the infinite sequences of Feynman diagrams describing the electron-nucleus  interaction via photon exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this case, the three discussed possibilities correspond to  the summation of the diagrams with zero, one, and two transverse photons and an arbitrary  number of the Coulomb photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  To all orders in αZ, the NMS contribution for the state |ψa⟩ can be expressed as fol-  lows [23, 26]  ENMS = ⟨ψa|P(εa)|ψa⟩ ,  (8)  where we have introduced the operator P(E) by  ⟨ψi|P(E)|ψk⟩ = i  2π  ∞  �  −∞  dω  �  n  ⟨ψiψn|R(ω)|ψnψk⟩  E − ω − εn(1 − i0)  (9)  with  R(ω) = 1  M  �  p1 − D1(ω)  � �  p2 − D2(ω)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (10)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (9), |ψiψn⟩ = |ψi⟩|ψn⟩ is the direct product of the one-electron wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  operator D(ω) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (10) is related to the transverse part of the photon propagator (7),  and its kth Cartesian component, Dk(ω), is equal to  Dk(ω) = −4παZαlDlk(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (11)  The ω → 0 limit of the operator D(ω) coincides with formula (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The indices 1 and 2 in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (10) designate the electron, on which the corresponding operators act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Let us rewrite  R(ω) in the form:  R(ω) = Rc + Rtr1(ω) + Rtr2(ω) ,  (12)  Rc = 1  M p1 · p2 ,  (13)  Rtr1(ω) = − 1  M  �  p1 · D2(ω) + D1(ω) · p2  �  ,  (14)  Rtr2(ω) = 1  M D1(ω) · D2(ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (15)  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (12) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (8), one arrives at the Coulomb, one-transverse-photon, and  two-transverse-photon contributions to the NMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  7  Let us turn to the discussion of the SMS contribution which corresponds to the Feynman  diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For simplicity, we consider the case of a one-determinant unperturbed  wave function,  Ψab(r1, r2) = 1  √  2  �  P  (−1)PψP a(r1)ψP b(r2) ,  (16)  where P is the permutation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The generalization to the case of a many-determinant  wave function is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The nonperturbative (in αZ) expression for the SMS  contribution reads as [24, 26]  ESMS = −⟨ψbψa|R(∆)|ψaψb⟩ ,  (17)  where ∆ = εa − εb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The formula (17) gives the “exchange” term for the two-electron  operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The “direct” one is equal to zero, since the matrix elements of the operators  p and D are zeroes for states of the same parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Substituting (12) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (17), one  obtains the expansion of the SMS contribution into the Coulomb, one-transverse-photon,  and two-transverse-photon parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Over the past three decades, numerous QED calculations of the nuclear recoil effect  on binding energies were carried out [4, 29, 31–34, 84, 88, 97, 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We should note that  the expressions (8) and (17) with the operator D defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (11) are derived for the  point-nucleus case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [33], it was argued that the dominant part of the finite-nuclear  size (FNS) correction to the nuclear recoil effect can be accounted for by employing the po-  tential of the extended nucleus in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5), and since that paper this prescription is usually  used in the QED calculations of the mass shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' It was also found there that the treatment  of the FNS correction to the nuclear recoil effect within the Breit approximation defined by  the operator (1) leads to an artificial contribution of order (m/M)(αZ)5(Rnucl/λ)mc2 which  even exceeds the main contribution of order (m/M)(αZ)4(Rnucl/λ)2mc2 (here λ = ℏ/(mc) is  the Compton wavelength).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This artificial contribution arises from the first (Coulomb) term  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' However, it is completely cancelled by the corresponding FNS correction to the  Coulomb part of the QED nuclear recoil effect, which means that the rigorous theory for  the FNS contribution beyond the main (m/M)(αZ)4(Rnucl/λ)2mc2 term can be formulated  only within the framework of QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [99], an additional FNS correction, which re-  sults from modifying the Breit-approximation mass-shift operator (1) by inserting the form  factor into the nuclear vertex [100, 101], was evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This operator differs from the one  obtained by considering the zero-frequency limit of the photon propagator in the modified  8  Coulomb gauge [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The main difference between the results obtained using these  two operators is due to a spurious contribution of the order (m/M)(αZ)5(Rnucl/λ)mc2 in  the one-transverse-photon part, which occurs only in the calculation with the operator from  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Like to the case of the Coulomb contribution, this spurious term is cancelled  by the related FNS contribution to the one-transverse-photon QED correction, provided it  is also calculated with the photon propagator in the modified Coulomb gauge [102, 103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  In Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [34, 71, 119], the additional FNS correction from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [99], which is free from the  spurious term, was used to estimate the uncertainty of the calculations based on the pre-  scription of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' To date, the most elaborated evaluation of the FNS correction to the  nuclear recoil effect was performed within the QED approach in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This calculation  accomplished for the 1s state has confirmed that the dominant part of the FNS correction  to the nuclear recoil effect is indeed covered by the recipe of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [33], which we also follow  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The rigorous QED treatment of the total FNS correction to the nuclear recoil effect  lies beyond the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Restricting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (8) and (17) by the lowest-order relativistic approximation leads to the  NMS and SMS parts of the MS operator (1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Until recently, all nonperturba-  tive (in αZ) calculations were limited by the zeroth order in 1/Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', the electron-electron  interaction corrections to the nuclear recoil effect were considered at best only within the  Breit approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In our recent works, we have advanced the QED theory of the nuclear  recoil effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Specifically, we have considered to all orders in αZ the electron-electron inter-  action correction of first order in 1/Z to the one-electron [104] and two-electron [105] parts  of the nuclear recoil effect on binding energies in atoms and ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' These higher-order QED  contributions being also beyond the scope of the present work can be calculated additionally  if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Finally, the radiative (∼ α) as well as the second-order (in m/M) recoil corrections  are accessible nowadays only within the αZ-expansion approaches, see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [106–108] and  references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' These contributions are also not considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  MODEL-QED OPERATOR FOR THE NUCLEAR RECOIL EFFECT  The QED calculations for many-electron systems, becoming increasingly relevant in view  of the considerable progress of the experiment, are complicated and in many cases currently  9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Self-energy diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Vacuum-polarization diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  inaccessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This is true for the radiative corrections as well as for the QED treatment of the  nuclear recoil effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For this reason, there is a vital need for a simple approximate approach  for taking into account the QED corrections in various relativistic calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The conven-  tional first-order QED corrections correspond to the self-energy (SE), vacuum-polarization  (VP), and one-photon-exchange diagrams shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The  approximate model-QED-operator approach to evaluate these effects has been suggested re-  cently by our group in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [46–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this section, the analogy will be traced that allows us  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' One-photon exchange diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  10  to construct a similar approach for the nuclear recoil contributions beyond the lowest-order  relativistic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The model-QED-operator approach [46] is worked out within the TTGF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' It is  based on the fact that the QED effects to first order in α can be described by an effective  Hamiltonian acting in the subspace which is spanned by all Slater determinants made up  of the positive-energy solutions of the Dirac equation (5), see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [109] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This  effective Hamiltonian has the form  H = Λ(+)  � N  �  i  �  hD  i + hSE  i  + hVP  i  �  +  N  �  i<j  h1ph  ij  �  Λ(+) ,  (18)  where Λ(+) is the product of the one-electron projectors on the positive-energy eigenfunctions  of the Dirac Hamiltonian hD, and the operators hSE, hVP, and h1ph arise from the diagrams  in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 3, 4, and 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' According to the QED theory of the nuclear recoil effect,  described briefly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' II, to first order in m/M and to all orders in αZ, the Hamiltonian  H has to be supplemented by the term  δH = Λ(+)  � N  �  i  hNMS  i  +  N  �  i<j  hSMS  ij  �  Λ(+) ,  (19)  where the operators hNMS and hSMS are associated with the diagrams in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1 and 2,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Below we discuss how the part of the operator δH which is beyond the Breit  approximation can be adapted for the practical relativistic electronic-structure calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  We note that the entire formalism can be generalized to the case where the potential V in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5) along with the nuclear potential includes also some local screening potential Vscr  modeling the interelectronic-interaction effects within the initial approximation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', to the  extended version of the Furry picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Let us start with the analogy between the one-photon-exchange diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 5 and  the two-electron nuclear recoil diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For the one-photon-exchange diagram,  the TTGF method leads to the following symmetric expression [96, 109]  h1ph =  εi1,εi2,εk1,εk2>0  �  i1̸=i2,k1̸=k2  |ψi1ψi2⟩⟨ψi1ψi2|1  2  �  I(εi1 − εk1) + I(εi2 − εk2)  �  |ψk1ψk2⟩⟨ψk1ψk2| ,  (20)  where the indices i1, i2, k1, and k2 enumerate the positive-energy one-electron Dirac eigen-  functions,  I(ω) = e2αµ  1αν  2Dµν(ω, r12) ,  (21)  11  αµ ≡ γ0γµ = (1, α) are the Dirac matrices, and the photon propagator in the Coulomb  gauge is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (6) and (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The matrix elements for the two-electron nuclear recoil  diagrams are derived within the TTGF method in Appendix A, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A21) for the final  formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' As a result, the operator hSMS can be represented as  hSMS =  εi1,εi2,εk1,εk2>0  �  i1̸=i2,k1̸=k2  |ψi1ψi2⟩⟨ψi1ψi2|1  2  �  R(εi1 − εk1) + R(εi2 − εk2)  �  |ψk1ψk2⟩⟨ψk1ψk2| , (22)  where the operator R(ω) is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Therefore, the expressions (20) and (22)  differ only by the replacement of I with R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Taking the photon propagator Dµν(ω, r12)  in the Coulomb gauge at the zero-energy transfer (ω = 0), one obtains from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (20)  the interaction part of the Dirac-Coulomb-Breit Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In the Coulomb gauge, the  formula (20) considered beyond the lowest-order relativistic approximation leads to the so-  called frequency-dependent Breit interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The corresponding correction is readily taken  into account within the electronic-structure calculations, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [57, 60, 110–113], and  does not require the construction of the model operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' When omitting the two-transverse-  photon contribution and taking the ω → 0 limit in R(ω), the expression (22) boils down to  the SMS part of the mass-shift operator (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The remaining part of the contribution given  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (22) can be treated on equal footing with the frequency-dependent Breit-interaction  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Now we turn to the discussion of the one-electron contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The VP diagram in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 4 does not have a counterpart in the QED theory of the nuclear recoil effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Therefore,  we will focus on the SE diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 3 and the one-electron nuclear recoil diagrams in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' First, let us introduce the SE operator Σ(E),  ⟨ψi|Σ(E)|ψk⟩ = i  2π  ∞  �  −∞  dω  �  n  ⟨ψiψn|I(ω)|ψnψk⟩  E − ω − εn(1 − i0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (23)  This formal expression suffers from ultraviolet divergences and has to be renormalized to-  gether with the mass counterterm, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [114–117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Within the TTGF method,  one can obtain the following symmetric expression for the operator hSE [46, 109]:  hSE =  εi,εk>0  �  i,k  |ψi⟩⟨ψi|1  2  �  ΣR(εi) + ΣR(εk)  �  |ψk⟩⟨ψk| ,  (24)  where ΣR(ε) is the renormalized SE operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The corresponding diagonal and off-diagonal  matrix elements are tabulated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The matrix elements for the one-  electron nuclear recoil diagrams are considered in the framework of the TTGF method in  12  Appendix A, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A14) for the final expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In contrast to the SE diagram, this  contribution is ultraviolet finite and does not require any regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nevertheless, its  evaluation is a complex task compared to the calculations with the mass-shift Hamilto-  nian (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The operator hNMS valid to all orders in αZ can be written in the form  hNMS =  εi,εk>0  �  i,k  |ψi⟩⟨ψi|1  2  �  P(εi) + P(εk)  �  |ψk⟩⟨ψk| ,  (25)  The analogy between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (24) and (25) is evident, and it is employed in the present work to  develop the model-QED-operator approach for the one-electron nuclear recoil contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Since within the Breit approximation this contribution can be treated by means of the  operator  hNMS  Breit =  1  2M  �  p2 − 2D(0) · p  �  ,  (26)  we construct the model operator for the remaining (QED) part of hNMS, namely for  hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' ≡  εi,εk>0  �  i,k  |ψi⟩⟨ψi|  �1  2  �  P(εi) + P(εk)  �  − hNMS  Breit  �  |ψk⟩⟨ψk| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (27)  The model-operator approach should simultaneously solve two issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' First, since there is  no simple-enough procedure for the ab initio evaluation of the P-operator matrix elements  for arbitrary levels (including the continuum-spectrum levels), one has to terminate the  summation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Second, the short interaction range should be kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' To address  these problems, similarly to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [46], we approximate the QED recoil operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  by a  sum of semilocal (with respect to r) and nonlocal potentials  ˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' = Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' + Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (28)  The operator P, like the operator ΣR, conserves the relativistic angular quantum number κ =  (−1)j+l+1/2(j + 1/2), where j and l are the total and orbital angular momenta, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  For this reason, the semilocal potential can be written as  Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' =  �  κ  V κ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='Pκ ,  (29)  where the projector Pκ acts only on the angular variables, and its kernel is  Pκ(n, n′) =  \uf8eb  \uf8ec  \uf8ed  �  m  Ωκm(n)Ω†  κm(n′)  0  0  �  m  Ω−κm(n)Ω†  −κm(n′)  \uf8f6  \uf8f7  \uf8f8  (30)  13  with Ωκm(n) being the spherical spinor and n = r/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (29), we take  V κ  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (r) = Aκ exp(−r/λ) ,  (31)  where the parameters Aκ are chosen to reproduce the ab initio values of the diagonal matrix  elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' for the state with the lowest principal quantum number n for  a given κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The nonlocal potential can be written in the form  Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' =  n  �  j,l=1  |φj⟩Bjl⟨φl| ,  (32)  where the functions {φi}n  i=1, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [46], are chosen to be  φi(r) = 1  2 [ I − (−1)siβ ] ρli(r)ψi(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (33)  Here the index si = ni − li enumerates the positive-energy states for the given angular  symmetry, I and β are the identity and the standard Dirac matrices, respectively, and the  factors ρli(r) = exp [−2αZ(r/λ)/(1 + li)] provide the stronger localization of the functions  {φi}n  i=1 as compared to the eigenfunctions {ψi}n  i=1 of the Dirac equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Finally, the  coefficients Bjl in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (32) are determined from the condition that the matrix elements of the  model-QED operator ˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' evaluated in the space spanned by the functions {ψi}n  i=1 coincide  with the exact ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This leads to the following equations  n  �  j,l  ⟨ψi|φj⟩Bjl⟨φl|ψk⟩ = ⟨ψi|  �1  2  �  P(εi) + P(εk)  �  − hNMS  Breit − Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  �  |ψk⟩  (34)  for i, k = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' n, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [46] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We note that the model-QED operator for the  nuclear recoil effect is worked out in such a way that the QEDMOD Fortran package [48] can  be readily adapted to incorporate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We propose to refer to the resulting package as the  RECMOD one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Therefore, to determine the model-QED operator we need to calculate the diagonal and  off-diagonal matrix elements of the QED recoil operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' with the Dirac-Coulomb wave  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' So far, only calculations of the diagonal matrix elements have been presented in  the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this work, the expressions derived within the TTGF method are employed  for the ab initio QED evaluation of the one-electron nuclear recoil contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Concluding  the description of the model-QED operator, we note that in practical calculations we con-  struct the model operator employing the functions (33) which correspond to the ns states  with the principal quantum number n ⩽ 3 and the np1/2, np3/2, nd3/2, and nd5/2 states  with n ⩽ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  14  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  TEST OF THE MODEL-QED OPERATOR FOR THE NUCLEAR RECOIL  EFFECT  In the present work, to obtain the diagonal and off-diagonal matrix elements of the  higher-order operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , we first evaluate the corresponding values for the operator hNMS  and then numerically subtract the Breit contribution given by the operator hNMS  Breit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For the  Coulomb part of the nuclear recoil effect, this subtraction is readily accomplished analytically  by omitting the summation over the positive-energy states and doubling the contribution of  the negative-energy continuum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This can be useful to avoid large cancellations  for low values of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The calculations are performed for the ns, np1/2, np3/2, nd3/2, and  nd5/2 states with the principal quantum number n ⩽ 5 in the wide range of Z = 5 − 100  using the Dirac-Coulomb wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The evaluation is carried out for the potential  of the extended nucleus in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The nuclear-charge distribution is described by the  homogeneously-charged-sphere model for Z ⩽ 14, and by the Fermi model otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  nuclear-charge radii are taken from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [118, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The one-electron basis set to represent  the electron Green’s function is constructed from B splines [120] within the dual-kinetic-  balance approach [121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The ω integration is performed in accordance with the equations  presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The results of the ab initio calculations are conveniently expressed in terms of the di-  mensionless function Fnink(αZ) defined by  ⟨ψi|hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψk⟩ = m  M  (αZ)5  (nink)3/2Fnink(αZ) mc2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (35)  Our data for the ns, np1/2, np3/2, nd3/2, and nd5/2 states are presented in Tables I, II, III, IV,  and V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The individual Coulomb, one- and two-transverse-photon corrections  as well as the total QED recoil contributions are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The uncertainties due to the  approximate treatment of the nuclear-size correction are omitted, and with this in mind the  values are accurate to all digits quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The function Fnink(αZ) for values of Z not listed  in Tables I-V can be obtained using a polynomial fitting,  Fnink(αZ) =  N  �  n=1  Fnink(αZn)  �  m̸=n  Z − Zm  Zn − Zm  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (36)  The model-QED operator for the nuclear recoil effect is constructed using the matrix  elements for the ns states with n ⩽ 3 and the np and nd states with n ⩽ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' By definition,  15  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Matrix elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  for the ns states calculated with the Dirac-  Coulomb wave functions for the extended nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Labels (ni, nk) stand for the function Fnink defined  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rows labeled “c”, “tr1”, “tr2”, and “tot” contain the Coulomb, one-transverse-photon,  two-transverse-photon and total nuclear recoil contributions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z Term  (1, 1)  (1, 2)  (1, 3)  (1, 4)  (1, 5)  (2, 2)  (2, 3)  (2, 4)  (2, 5)  (3, 3)  (3, 4)  (3, 5)  (4, 4)  (4, 5)  (5, 5)  5  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 35  tr1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='732 91  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='802 47  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='807 94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='809 23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='809 68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='925 71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='941 92  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='944 48  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='945 25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='971 87  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='978 25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='979 68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='989 97  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='993 14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='998 90  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='970 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='939 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='935 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='933 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='932 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='930 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='924 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='922 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='921 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='922 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='920 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='919 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='919 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='918 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='917 96  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='377 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='477 25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='487 37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='490 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='491 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='610 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='632 36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='636 72  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='638 31  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='664 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='672 84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='675 14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='685 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='689 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='695 60  10  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 90  tr1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='227 48  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 46  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 78  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='421 64  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='437 53  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='439 83  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='440 40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='467 08  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='473 17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='474 39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='484 57  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='487 52  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='493 05  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='650 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='619 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='614 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='612 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='612 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='607 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='600 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='598 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='598 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='598 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='595 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='595 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='595 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='593 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='593 47  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='217 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='333 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='454 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='476 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='480 83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='482 30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='508 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='517 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='519 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='529 54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='533 65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='539 68  15  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 39  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='952 62  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 57  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 60  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 37  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='150 64  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='166 24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='168 19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='168 49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='195 43  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='201 16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='202 11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='212 17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='214 84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='220 07  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='489 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='457 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='452 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='450 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='449 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='442 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='435 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='433 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='432 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='432 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='430 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='429 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='428 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='427 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='427 18  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='123 43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='227 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='237 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='240 19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='241 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='366 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='392 75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='394 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='421 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='429 36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='431 39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='441 63  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='445 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='451 49  20  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='326 87  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='772 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='848 42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='853 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='853 57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='853 29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='976 45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='991 78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='993 33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='993 29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 67  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 87  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 73  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='354 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='349 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='347 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='346 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='337 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='330 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='328 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='323 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='322 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='321 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 89  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='167 69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='180 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='180 77  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='311 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='333 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='337 43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='338 52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='366 21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='374 37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='376 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='386 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='390 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='395 97  25  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 27  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='644 47  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='724 80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='729 33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='729 25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='728 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='857 11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='872 22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='873 30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='872 86  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='900 87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='905 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='905 93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='915 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='917 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='922 11  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='283 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='277 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='275 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='274 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='263 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='256 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='253 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='252 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='251 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='248 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='247 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='247 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='245 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='245 11  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='126 95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='138 98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='139 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='276 14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='302 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 89  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='339 83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='351 75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='355 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 73  30  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='302 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='308 96  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='550 87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='636 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='640 65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='640 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='638 98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='774 39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='789 29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='789 84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='788 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='817 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='822 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='821 83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='831 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='832 99  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='836 91  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='265 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='229 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='223 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='221 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='220 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='207 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='199 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='196 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='195 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='194 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='191 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='190 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='189 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='188 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='187 13  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='982 37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='110 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='112 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='112 29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='256 71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='279 78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='283 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='283 52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='313 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='322 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='332 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='335 69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 83  35  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='296 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='301 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='300 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='300 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='300 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 56  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='481 88  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='573 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='577 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='576 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='574 93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='718 92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='733 66  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='733 59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='732 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='761 99  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='765 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='764 92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='774 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='775 41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='778 76  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='225 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='187 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='181 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='179 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='178 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='162 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='153 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='151 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='149 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='148 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='145 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='143 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='142 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='141 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='140 36  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='960 71  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='095 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='096 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='096 56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='250 33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='273 73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='276 58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='276 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='315 79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='326 22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='333 84  40  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='291 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='297 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='297 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 45  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='432 10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='531 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='535 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='533 30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='531 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='685 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='700 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='699 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='697 44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='728 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='731 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='730 14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='739 58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='739 87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='742 55  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='192 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='152 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='146 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='144 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='143 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='115 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='112 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='111 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='109 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='105 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='102 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 96  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='947 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='080 09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='255 43  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='279 22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='281 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='281 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='313 63  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 97  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='333 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='338 13  45  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='289 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='297 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='296 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 65  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='398 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='506 51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='509 91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='507 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='504 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='671 73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='686 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='684 63  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='681 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='714 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='716 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='714 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='723 83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='723 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='725 28  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='165 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='123 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='116 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='113 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='075 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='070 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 36  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='943 09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='094 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='095 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='094 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='272 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='296 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='297 79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 89  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='338 29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='338 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='348 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='350 55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='354 27  50  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='289 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='300 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='300 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='299 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='298 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='311 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='311 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='308 95  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='378 25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='497 34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='500 41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='496 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='493 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='676 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='690 42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='687 86  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='684 22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='718 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='719 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='716 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='726 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='724 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='725 74  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='142 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='097 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='088 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 83  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='946 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='109 34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='109 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='107 57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='301 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='326 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 62  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='326 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='362 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 99  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='377 74  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='378 95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='381 97  55  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='291 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='305 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='318 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='318 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='318 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 26  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='371 39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='503 32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='506 02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='501 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='496 90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='698 37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='712 73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='709 02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='704 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='741 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='741 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='737 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='746 09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='743 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='743 64  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='121 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='063 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 95  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='957 75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='123 81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='134 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='133 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='130 80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='343 96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='369 37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='369 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='405 95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='410 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='409 39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='419 88  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='420 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='422 43  60  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='296 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='313 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='313 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='310 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='332 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='332 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='332 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='330 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='328 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 11  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='377 42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='524 78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='527 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='521 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='515 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='739 91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='754 21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='749 12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='743 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='782 62  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='781 27  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='776 13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='784 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='781 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='779 83  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 45  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='977 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='159 59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='169 91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='168 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='164 93  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='401 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='427 26  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='426 57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='423 17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='464 76  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='468 77  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='466 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='476 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='476 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='477 17  65  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='322 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='321 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='348 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='348 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='346 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='344 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='347 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='346 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='344 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='344 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='342 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 42  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='396 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='562 56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='564 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='556 92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='550 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='802 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='816 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='809 81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='802 38  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='845 14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='842 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='835 46  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='843 89  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='838 79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='835 75  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 53  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 59  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='208 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='218 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='215 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='211 81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='476 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='503 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='501 08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='496 41  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='541 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='544 36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='540 57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='550 96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='549 11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='548 86  70  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='338 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='337 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='335 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='333 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='365 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='363 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='367 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='365 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='363 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='362 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 08  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='429 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='617 95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='619 19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='609 86  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='601 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='888 22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='902 39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='893 47  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='884 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='931 07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='925 88  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='917 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='925 49  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='918 50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='913 49  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='065 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='080 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 52  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='272 94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='283 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='279 11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='273 70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='572 74  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='600 58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='596 91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='590 65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='640 28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='641 40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='636 04  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='646 32  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='642 87  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='640 94  75  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='357 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='356 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='354 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='352 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='394 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='394 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='391 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='393 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='390 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='383 33  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='478 73  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='695 60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='696 15  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='684 47  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='674 03  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 58  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='994 87  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 09  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 26  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 92  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 54  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 05  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='111 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='116 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='117 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='119 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='120 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='121 76  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='102 81  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='355 69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='366 42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='353 16  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='695 51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='724 36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='718 52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='710 20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='765 36  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='764 35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='756 95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='767 06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='761 55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='757 47  80  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='342 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='381 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='380 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='377 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='374 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='427 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='426 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='423 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='420 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='426 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='422 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='419 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='419 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='416 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='413 44  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='546 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='798 52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='798 17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='783 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='770 61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='155 49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='169 28  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='154 16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='139 39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='198 12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='186 70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='172 84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='179 94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='167 51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='156 85  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='139 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='142 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='143 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='152 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='156 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='157 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='160 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='161 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='162 05  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='173 10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='461 85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='472 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='464 12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='455 01  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='852 21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='882 17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='873 44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='862 42  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='924 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='920 74  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='910 67  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='920 53  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='912 33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='905 46  85  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='363 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='411 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='410 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='406 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='403 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='469 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='468 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='464 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='460 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='467 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='463 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='459 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='459 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='455 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='451 44  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='637 02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='933 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='931 69  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='913 23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='897 11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='352 28  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='365 60  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='345 85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='394 33  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='378 27  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='360 44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='366 80  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='350 36  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='335 60  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='087 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='170 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='187 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='189 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='190 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='201 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='205 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='205 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='208 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='209 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='209 72  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='263 63  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='597 84  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='608 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='597 05  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='585 23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 49  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 88  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='128 13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='120 39  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='116 22  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='093 89  90  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='387 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='446 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='445 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='440 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='435 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='519 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='517 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='512 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='507 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='516 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='510 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='505 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='505 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='500 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='495 56  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='751 33  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 68  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='602 04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='614 45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='588 50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='564 41  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='642 73  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='620 44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='597 35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='602 71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='581 00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='560 88  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='126 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='129 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='129 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='226 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='245 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='247 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='247 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='261 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='265 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='264 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='267 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='267 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='267 63  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='378 52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='770 78  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='781 44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='765 68  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='750 27  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='309 84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='342 09  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 07  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='387 71  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='374 61  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='356 14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='365 04  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='348 43  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='332 95  95  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='418 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='492 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='490 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='483 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='478 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='584 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='582 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='574 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='568 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='580 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='572 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='566 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='565 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='559 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='553 03  tr1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='901 10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='325 75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 09  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='289 96  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='264 23  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='929 69  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='940 48  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='905 96  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='874 62  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='967 70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='936 82  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='906 55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='910 52  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='881 65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='854 30  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='161 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='174 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='299 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='320 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='337 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='339 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='342 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='341 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='340 34  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='526 44  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='994 50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 61  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='983 17  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='962 88  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='644 65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='677 82  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='652 18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='625 99  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='724 97  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='704 30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='679 11  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='687 14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='663 85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='641 62  100  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='461 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='554 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='551 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='542 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='535 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='672 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='669 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='659 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='650 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='666 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='656 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='648 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='646 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='638 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='630 36  tr1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 97  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='622 83  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='613 55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='574 12  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='540 84  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='369 87  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='377 77  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='289 46  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='402 83  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='359 72  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 31  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='321 30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='282 39  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='244 99  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='222 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='237 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='239 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='238 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='394 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='417 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='417 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='414 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='437 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='439 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='436 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='439 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='437 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='435 00  tot  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='721 18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='290 85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='299 73  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='270 24  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='243 12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='092 14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='125 68  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='088 93  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='173 98  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='142 28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='107 48  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 38  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 62  16  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Matrix elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  for the np1/2 states calculated with the Dirac-  Coulomb wave functions for the extended nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The notations are the same as in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z Term  (2, 2)  (2, 3)  (2, 4)  (2, 5)  (3, 3)  (3, 4)  (3, 5)  (4, 4)  (4, 5)  (5, 5)  5  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 79  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 51  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='082 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 56  10  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 97  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 63  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 20  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 80  15  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 05  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 28  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 11  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 22  20  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 49  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 95  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 65  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 78  25  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 26  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 71  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 59  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 38  30  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 40  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 60  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 16  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 16  35  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 94  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 57  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='064 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='074 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 57  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 06  40  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 96  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 51  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='087 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='092 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='095 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='096 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='098 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='101 09  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 63  45  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 51  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 26  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='092 53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='107 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='118 03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='119 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='121 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='122 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 02  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 26  50  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 73  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 55  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='113 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='126 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='128 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='129 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='137 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='142 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='143 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='145 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='147 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='148 73  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='064 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='095 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='108 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='111 45  55  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 76  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 01  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='150 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='152 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='153 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='164 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='168 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='169 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='172 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='174 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='175 66  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='089 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='105 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='105 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='129 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='129 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='138 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='141 89  60  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 84  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 90  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='161 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='176 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='179 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='180 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='193 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='197 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='198 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='202 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='203 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='205 40  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='117 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='134 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='157 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='162 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='163 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='170 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='173 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='176 46  65  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 20  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 93  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='190 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='207 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='209 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='210 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='225 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='230 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='231 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='235 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='237 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='238 67  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='150 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='169 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='171 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='171 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='196 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='202 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='202 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='210 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='212 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='216 39  70  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 17  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 12  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='223 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='241 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='244 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='244 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='263 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='268 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='268 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='273 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='274 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='276 42  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='189 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='211 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='213 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='214 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='242 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='248 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='248 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='257 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='259 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='263 37  75  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 46  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 12  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='261 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='281 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='284 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='284 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='306 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='311 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='312 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='317 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='318 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 95  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='236 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='261 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='264 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='264 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='297 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='303 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='304 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='314 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='316 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='319 62  80  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 65  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='063 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='063 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='082 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='087 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='088 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 18  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='329 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='331 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='330 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='357 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='362 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='363 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='368 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='369 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='370 98  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='294 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='324 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='327 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='365 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='372 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='372 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='383 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='385 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 51  85  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='089 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 83  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='102 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='121 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='125 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='131 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='132 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='134 74  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='362 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='387 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='388 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='387 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='419 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='424 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='424 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='430 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='431 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='431 93  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='368 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='403 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='406 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='405 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='451 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='458 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='459 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='470 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='472 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='474 83  90  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='098 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='105 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='113 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='115 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='115 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 98  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='132 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='153 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='157 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='158 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='179 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='184 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='185 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='191 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='192 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='194 41  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='429 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='457 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='458 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='455 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='495 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='500 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='498 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='506 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='506 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='506 05  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='463 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='505 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='508 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='507 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='562 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='570 53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='569 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='582 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='583 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='585 49  95  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='127 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='135 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='136 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='144 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='145 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='145 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='146 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='146 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='145 81  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='202 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='230 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='235 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='236 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='263 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='268 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='269 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='276 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='277 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='278 67  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='515 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='547 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='546 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='542 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='591 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='595 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='592 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='600 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='599 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='598 41  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='591 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='641 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='645 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='642 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='710 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='718 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='716 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='730 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='730 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='731 26  100  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='166 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='177 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='188 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='189 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='188 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='190 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='189 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='188 27  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='307 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='344 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='351 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='351 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='387 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='393 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='393 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='401 03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='400 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='401 29  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='627 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='663 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='659 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='654 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='715 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='717 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='712 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='722 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='719 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='716 67  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='768 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='830 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='833 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='828 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='913 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='921 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='917 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='933 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='931 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='929 69  17  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Matrix elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  for the np3/2 states calculated with the Dirac-  Coulomb wave functions for the extended nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The notations are the same as in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z  Term  (2, 2)  (2, 3)  (2, 4)  (2, 5)  (3, 3)  (3, 4)  (3, 5)  (4, 4)  (4, 5)  (5, 5)  5  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 24  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 65  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='082 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='074 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='074 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 94  10  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 17  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 01  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 09  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='075 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='065 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='060 27  15  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 68  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 11  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='065 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='060 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 11  20  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 49  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 26  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 58  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='036 33  25  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 67  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 74  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 43  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 84  30  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 84  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 90  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 63  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 58  35  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 03  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 68  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 16  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 50  40  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 20  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 65  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 99  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 45  45  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 38  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 82  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 14  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 30  50  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 56  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 26  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 60  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 10  55  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 74  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 00  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 63  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 89  60  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 92  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='032 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='031 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='030 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 09  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 52  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='041 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 70  65  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 10  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='039 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='040 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='038 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='037 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='050 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 62  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 06  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='027 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='034 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='033 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='049 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='058 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 58  70  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 28  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='047 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='048 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='046 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='045 33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='057 91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='064 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='064 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 63  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 22  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='035 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='043 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='042 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='070 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 57  75  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 47  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='055 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='056 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='054 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='053 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='066 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 21  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 98  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='044 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='051 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='072 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='084 72  80  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 66  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='063 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='065 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='062 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 45  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 29  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='052 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='061 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='060 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='059 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='082 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='092 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='096 07  85  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 87  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='073 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='071 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='070 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='087 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='088 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='094 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='094 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='097 44  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 10  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='060 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='070 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='069 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='068 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='090 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='093 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='092 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='102 08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='107 68  90  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 08  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='080 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='081 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='079 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='098 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='098 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='105 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='106 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='109 28  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 73 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 36  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='067 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='078 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='077 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='076 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='104 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='113 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='115 21  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='119 56  95  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 31  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='089 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='093 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='091 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='089 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='110 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='111 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='109 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='118 28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='118 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='122 08  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 09  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 01  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='075 61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='087 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='086 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='085 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='111 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='115 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='114 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='125 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='127 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='131 77  100  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 56  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='099 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='103 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='101 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='100 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='122 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='124 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='122 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='131 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='132 36  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='135 97  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 94  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='083 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='096 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='095 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='094 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='122 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='127 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='126 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='137 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='139 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='144 35  18  TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Matrix elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  for the nd3/2 states calculated with the Dirac-  Coulomb wave functions for the extended nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The notations are the same as in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z Term  (3, 3)  (3, 4)  (3, 5)  (4, 4)  (4, 5)  (5, 5)  5  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 46  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 94  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 39  10  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 25  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 24  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 53 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 50  15  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 07  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 54  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 95 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 61  20  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 91  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 71 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 82  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 73  25  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 99 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 76  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 08  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 85  30  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 61  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 95  35  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 46  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 45  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 03  40  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 30  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 26  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 08  45  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 12  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 09  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 08  50  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 91  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 96  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 76 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 98  55  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 66  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 86  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 11  60  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 12  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 70 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 55 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 36  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 77  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 81  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 33  65  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 99  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 81  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 67  70  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 12 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 19  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 80 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 75 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 52  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 86  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 15  75  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 19 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 94  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 53  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 96  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 79  80  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 28  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 22  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 12  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 62  85  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 33  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 68 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 31  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 33  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 69  90  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 40  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 17  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='023 59  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 03  95  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 47  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 25  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 90  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='022 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 68  100  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 55  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 57  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 02  tr2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='020 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='021 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='024 92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='028 25  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='025 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='026 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='029 72  19  TABLE V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Matrix elements of the operator hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  for the nd5/2 states calculated with the Dirac-  Coulomb wave functions for the extended nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The notations are the same as in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z Term  (3, 3)  (3, 4)  (3, 5)  (4, 4)  (4, 5)  (5, 5)  5  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 65  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 74  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='018 69 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 39  10  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 60  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 51 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 87  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 60 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 48  15  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 93 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 56  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 96 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 72 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 01  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 58 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 57  20  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 43 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 92 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 51  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 56 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 16  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 84 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 42 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 67  25  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 38 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 91 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 46  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 88 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 32  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 24 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 79 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 61 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 79  30  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 29 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 40  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 41 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 49  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 21 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 81 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 90  35  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 34 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 34  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 82 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 36 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 68  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 64 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 03  40  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 74  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 32 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 66 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 94 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 88  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 47 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 87 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 62 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 22 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 16  45  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02  tr1  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 83  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 83 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 97 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 10  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 85 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 29  50  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 93  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 35 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 49 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 30 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 54 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 34  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 33 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 57  55  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 05  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 89 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 98 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 67 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 65 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 86 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 59  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 78 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 67  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 43  60  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 19  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 45 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 01 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 20 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 87  tot  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 43  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 29  65  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 35  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 74 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 40 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 16  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 14  70  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 53  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 63 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 59 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 53  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 01  75  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 74  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 19  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 17  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 87  80  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 97  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 90 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 77 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 83  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='005 93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 74  85  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 24  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 57 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 39 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 45  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='006 37  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 62  90  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 71  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 54  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 26 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 05  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='007 86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='008 83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 51  95  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 39  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 66  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 84  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 56  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 88  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 81  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 62  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='009 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 87  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 40  100  c  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 10  tr1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='014 32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='013 88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='015 25  tr2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='001 73  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 58  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='000 78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='002 26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='003 25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='004 16  tot  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='010 82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='012 35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='011 69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='016 49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='017 04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='019 31  20  TABLE VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The higher-order nuclear recoil correction for the 4s, 5s, 5p, and 5d states in terms of  the function F defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The column labeled ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩ denotes the results obtained  by means of the model QED operator, ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩ is the contribution of the semilocal part of the  model operator, and “Exact” stands for the ab initio values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The ns states with n ⩽ 3 and the np  and nd states with n ⩽ 4 are omitted since the operator ˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' exactly reproduces the corresponding  correction for them by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Z  State  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩ ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  Exact  Z  State  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩ ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  Exact  10  4s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2024  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5325  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5295  60  4s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8494  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4804  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4767  5s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2017  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5445  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5397  5s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8394  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4833  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4772  5p1/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0922  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0545  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0558  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1282  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1765  5p3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0965  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0590  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0603  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0221  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0540  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0517  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0279  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0157  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0145  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0058  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0053  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0278  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0156  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0145  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0044  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0043  20  4s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0162  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3896  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3865  70  4s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9227  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='6503  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='6463  5s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0141  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4010  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3960  5s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9088  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='6475  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='6409  5p1/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0555  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0204  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0218  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1963  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2652  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2634  5p3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0692  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0349  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0363  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0406  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0762  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0736  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0225  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0116  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0107  5d3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0110  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0101  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0224  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0116  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0107  5d5/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0084  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0080  30  4s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9090  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3357  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3325  80  4s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0683  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9249  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9205  5s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9053  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3460  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3408  5s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0478  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9125  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9055  5p1/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0162  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0177  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0162  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2880  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3904  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3885  5p3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0437  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0119  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0136  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0578  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0990  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0961  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0172  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0075  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0069  5d3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0108  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0169  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0156  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0171  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0076  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0069  5d5/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0124  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0117  40  4s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8484  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3348  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3314  90  4s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3278  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3698  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3650  5s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8430  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3436  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3381  5s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='2952  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3406  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3329  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0262  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0612  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0596  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='4244  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5875  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='5855  5p3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0103  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0084  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0739  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1228  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1195  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0119  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0033  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0031  5d3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0173  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0236  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0220  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0119  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0036  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0032  5d5/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0117  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0165  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0155  50  4s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8280  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3813  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3777  100  4s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='7885  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1190  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1142  5s  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='8206  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3877  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='3820  5s  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='7322  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0568  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0496  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0733  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1131  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1115  5p1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='6515  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9319  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='9297  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0322  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0301  5p3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0889  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1478  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='1444  5d3/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0065  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0011  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0010  5d3/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0246  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0317  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0297  5d5/2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0068  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0006  5d5/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0158  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0206  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0193  21  TABLE VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The one-electron nuclear recoil correction beyond the Breit approximation for the  valence ns electron in neutral alkali metals in terms of the function F defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  labels CH and xα = 0, 1/3, 2/3, and 1 correspond to the different effective potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' See the text  for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Atom  Approach  CH  xα = 0  xα = 1/3  xα = 2/3  xα = 1  Na 3s  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0502  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0446  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0437  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0473  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0575  ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0581  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0516  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0508  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0553  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0676  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0561  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0499  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0496  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0544  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0671  K 4s  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0255  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0214  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0243  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0319  ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0313  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0263  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0302  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0400  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0311  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0261  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0263  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0302  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0401  Rb 5s  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0080  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0082  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0098  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0136  ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0141  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0112  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0116  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0195  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0113  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0117  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0141  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='0197  Cs 6s  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00645  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00504  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00525  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00642  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00927  ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01019  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00796  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00835  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01490  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01028  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00803  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00841  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01034  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01500  Fr 7s  ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00586  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00416  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00457  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00595  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00906  ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00999  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00708  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01022  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01563  Exact  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00712  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='00786  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01026  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01568  the operator exactly reproduces the QED recoil contributions for these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  For this  reason, the “predictive power” of the operator can be tested by applying it to evaluation  of the corresponding corrections for the states with higher values of the principal quantum  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In Table VI, the nuclear recoil contributions beyond the Breit approximation are  given for the 4s, 5s, 5p1/2, 5p3/2, 5d3/2, and 5d5/2 states in hydrogenlike ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The columns  labeled ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩ and ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩ contain the predictions obtained using the semilocal  part (29) of the model-QED operator and the total model-QED operator (28), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  These values are compared with the ab initio results taken from Tables I-V and shown in the  last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' As one can see, there is generally good agreement between the data, especially  for the s states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We stress the importance of the nonlocal part of the model-QED operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  For the np and nd states the functions Fnink change the sign for the middle values of Z  and, accordingly, have small absolute values there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' As a result, the relative accuracy of the  model-QED-operator predictions slightly decreases in these regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  22  To date, the QED contribution to the NMS in many-electron systems was usually eval-  uated within the independent-electron approximation by performing the calculations in the  extended Furry picture, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [4, 10, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' To demonstrate the performance of the  developed model-QED-operator approach, we apply it to evaluation of the nuclear recoil  effect on valence-electron energies in neutral alkali metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' First, we perform ab initio calcu-  lations of the one-electron contribution for the ns states by using a local effective potential  as V in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We use the core-Hartree (CH) and xα potentials, which are discussed in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The results of ab initio calculations of the QED contribution to the NMS are  presented in Table VII in rows labeled “Exact”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The model-QED-operator values are ob-  tained by averaging the operator (28) with the valence-electron wave functions determined  from the Dirac equation (5), in which the potential V is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Along with  the total model-operator predictions, ⟨ψv|˜hNMS  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' |ψv⟩, the results ⟨ψv|Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='|ψv⟩ for the semilocal  part (29) only are shown as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' As it is seen from Table VII, for all the alkali metals there  is good agreement between the approximate and exact values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Therefore, the model-QED  operator based on the calculations for hydrogenlike ions works also reasonably well in the  nonhydrogenic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  To conclude the discussion of alkali metals, let us note that, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [122], the strong  dependence of the final results on the choice of the potential for the initial approximation  takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The scatter of the results is related obviously with the approximate treatment  of the interelectronic-interaction effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We believe that the developed model-QED oper-  ator for the nuclear recoil effect merged with the standard methods to treat the electron  correlations will make it possible to perform much more accurate evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' However, such  calculations are out of the scope of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We reserve systematic calculations  with the model-QED operator for the nuclear recoil effect for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  CONCLUSION  In the present paper, we have worked out the model-QED-operator approach to treat the  nuclear recoil effect on binding energies in many-electron atomic systems beyond the Breit  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The approach is similar to the one proposed earlier for approximate calcula-  tions of the radiative corrections to energy levels [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The developed operator can be readily  included into any relativistic calculations based on the Dirac-Coulomb-Breit Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  23  The performance of the approach was demonstrated by comparing the model-QED-operator  predictions with the results of the rigorous QED calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  The work was supported by the Foundation for the Advancement of Theoretical Physics  and Mathematics “BASIS” and by RFBR and ROSATOM according to the research project  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 20-21-00098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The work of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' was supported by the German-Russian Interdisciplinary  Science Center (G-RISC) funded by the German Federal Foreign Office via the German  Academic Exchange Service (DAAD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Appendix A: One- and two-electron contributions to the nuclear recoil effect on  binding energies of quasi-degenerate levels  In the present Appendix, we derive the fully relativistic expressions for the contributions  of the nuclear recoil effect on binding energies within the two-time Green’s function (TTGF)  method [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Let us denote the unperturbed wave functions of the two states under consid-  eration as |ui⟩ and |uk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' During the derivation, we assume that the unperturbed energies  E(0)  i  and E(0)  k , which corresponds to these states, differ and, therefore, |ui⟩ ̸= |uk⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' However,  all the obtained expressions are valid for the coinciding energies and, of course, boil down to  the expressions (8) and (17) in the case of diagonal matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In order to derive the  formulas, we introduce a model subspace Ω, which is spanned by the states |ui⟩ and |uk⟩,  and construct for these states the QED perturbation theory as for quasi-degenerate levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The projector on Ω reads as  P (0) = |ui⟩⟨ui| + |uk⟩⟨uk| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A1)  The derivation procedure can be readily generalized for an arbitrary number of quasi-  degenerate levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  First, let us recall the basic ideas of the TTGF methods in the application to quasi-  degenerate levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The detailed description of the method can be found, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [96,  109, 126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The N-electron TTGF is defined as  G(t′, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , r′  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , rN) = ⟨0|Tψ(x′  1) · · ·ψ(x′  N) ¯ψ(xN) · · · ¯ψ(x1)|0⟩  �����t′0  1 =.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='=t′0  N≡t′  t0  1=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='=t0  N≡t  , (A2)  24  where ψ is the electron-positron field operator in the Heisenberg representation, ¯ψ = ψ†γ0,  x = (t0, r), and T is the time-ordering operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Turning to the mixed representation, one  obtains  G(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , r′  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , rN)δ(E − E′)  =  1  2πi  1  N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  � ∞  −∞  dtdt′ eiE′t′−iEt G(t′, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , r′  N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' , rN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A3)  Employing P (0), we can introduce the projection of the Green’s function (A3) on the sub-  space Ω,  g(E) = P (0)G(E)γ0  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' γ0  NP (0) ,  (A4)  and then determine the ˆK and ˆP operators:  ˆK ≡  1  2πi  �  Γ  dE Eg(E) ,  (A5)  ˆP ≡  1  2πi  �  Γ  dE g(E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A6)  The anticlockwise oriented contour Γ in the complex E plane surrounds the poles corre-  sponding to the quasi-degenerate levels and keeps outside all other singularities of g(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The investigated system is fully described by the effective operator ˆH defined as  ˆH = ˆP −1/2 ˆK ˆP −1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A7)  The perturbation theory for the Green’s function (A2) leads to the perturbation series for  the operator ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  To first order in m/M, the nuclear recoil effect is described by the one- and two-electron  Feynman diagrams depicted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The Feynman rules for these diagrams are  formulated, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [26], see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The first-order contribution to the operator ˆH  can be expressed as  ˆH(1) = ˆK(1) − 1  2  ˆP (1) ˆK(0) − 1  2  ˆK(0) ˆP (1) ,  (A8)  where the superscripts denote the orders in the expansion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The contributions of  the nuclear recoil effect are determined by the matrix elements of the operator (A8) in the  basis of the unperturbed functions |ui⟩ and |uk⟩:  H(1)  ik ≡ ⟨ui| ˆH(1)|uk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A9)  25  To zeroth order, the matrix of the operator ˆK is diagonal, K(0)  ik = E(0)  i δik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Therefore, one  can obtain  H(1)  ik = K(1)  ik − E(0)  i  + E(0)  k  2  P (1)  ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A10)  Let us now directly turn to the derivation of the desired nonperturbative (in αZ) formulas  for the nuclear recoil effect on binding energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' We start from the one-electron (NMS) con-  tribution corresponding to the diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this case, N = 1 and the unperturbed  wave functions |ui⟩, |uk⟩ and energies E(0)  i , E(0)  k  are given by the Dirac eigenfunctions and  eigenvalues, respectively, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The present derivation is similar to the one for the  self-energy diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Employing the Feynman rules [26], one can obtain the  following expression for the matrix element of the Green’s function ∆g(1)  NMS(E) projected on  the subspace Ω [96]:  ∆g(1)  NMS,ik(E) =  ⟨ψi|P(E)|ψk⟩  (E − εi)(E − εk) ,  (A11)  where the operator P is defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The corresponding contributions to the ˆK and  ˆP operators read as:  K(1)  NMS,ik =  1  2πi  �  Γ  dE E ∆g(1)  NMS,ik(E) =  εi  εi − εk  ⟨ψi|P(εi)|ψk⟩ +  εk  εk − εi  ⟨ψi|P(εk)|ψk⟩ , (A12)  P (1)  NMS,ik =  1  2πi  �  Γ  dE ∆g(1)  NMS,ik(E) =  1  εi − εk  ⟨ψi|P(εi)|ψk⟩ +  1  εk − εi  ⟨ψi|P(εk)|ψk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A13)  Substituting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A12) and (A13) into the formula (A10), one finally obtains the NMS  contribution  HNMS,ik = 1  2  �  ⟨ψi|P(εi)|ψk⟩ + ⟨ψi|P(εk)|ψk⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A14)  The derivation of the nonperturbative (in αZ) expression for the two-electron (SMS)  contribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 2 is also straightforward but more tedious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In this case, N = 2 and  we, for simplicity, assume that the unperturbed wave functions |ui⟩ and |uk⟩ are given  by the one-determinant wave functions Ψi1i2 and Ψk1k2, respectively, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  The  corresponding unperturbed energies are equal to E(0)  i  = εi1 + εi2 and E(0)  k  = εk1 + εk2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  derivation repeats the one for the one-photon-exchange diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The Green’s  26  function ∆g(1)  SMS(E) projected on the subspace Ω can be written as [26, 96]  ∆g(1)  SMS,ik(E) =  � i  2π  �2 �  dp1dp′  1  �  P  (−1)P ⟨ψP i1ψP i2|R(p1 − p′  1)|ψk1ψk2⟩  ×  1  (p′  1 − εP i1 + i0)(E − p′  1 − εP i2 + i0)  1  (p1 − εk1 + i0)(E − p1 − εk2 + i0) ,  (A15)  where the operator R is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Using the identity  1  (p′  1 − εP i1 + i0)(E − p′  1 − εP i2 + i0) =  1  E − E(0)  i  �  1  p′  1 − εP i1 + i0 +  1  E − p′  1 − εP i2 + i0  �  (A16)  and a similar one for the second pair of denominators in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A15), one can explicitly separate  the poles of the Green’s function located inside the contour Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' the contribution to  the ˆK operator is  K(1)  SMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='ik =  �  Γ  dE E ∆g(1)  SMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='ik(E)  =  � i  2π  �2 �  dp1dp′  1  �  P  (−1)P ⟨ψP i1ψP i2|R(p1 − p′  1)|ψk1ψk2⟩  ×  �  E(0)  i  E(0)  i  − E(0)  k  �2π  i δ(p′  1 − εP i1)  � �  1  p1 − εk1 + i0 +  1  E(0)  i  − p1 − εk2 + i0  �  +  E(0)  k  E(0)  k  − E(0)  i  �  1  p′  1 − εP i1 + i0 +  1  E(0)  k  − p′  1 − εP i2 + i0  � �2π  i δ(p1 − εk1)  � �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A17)  where the indentity  1  p − ε + i0 +  1  −p + ε + i0 = 2π  i δ(p − ε)  (A18)  was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Defining the coefficients A and B so that K(1)  SMS,ik ≡ AE(0)  i  +BE(0)  k , the contri-  bution to the operator ˆP can be expressed as P (1)  SMS,ik = A+B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Therefore, the formula (A10)  27  results in  HSMS,ik = 1  2  i  2π  �  dω  �  P  (−1)P  ×  �  ⟨ψP i1ψP i2|R(ω − εP i1)|ψk1ψk2⟩  �  1  ω − εk1 + i0 +  1  E(0)  i  − ω − εk2 + i0  �  + ⟨ψP i1ψP i2|R(εk1 − ω)|ψk1ψk2⟩  �  1  ω − εP i1 + i0 +  1  E(0)  k  − ω − εP i2 + i0  � �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (A19)  The integration over ω can be performed using the standard identity (ω1 < 0 < ω2):  ω2  �  ω1  dω f(ω)  ω ± i0 = ∓iπf(0) + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  ω2  �  ω1  dω f(ω)  ω  ,  (A20)  where P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='V means the principal-value integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Indeed, applying the formula (A20) to all  four terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A19) and taking into account that all the principal-value integrals vanish  due to the fact that R(ω) is the even function of ω, one finally obtains the SMS contribution  HSMS,ik = 1  2  �  P  (−1)P�  ⟨ψP i1ψP i2|R(∆1)|ψk1ψk2⟩ + ⟨ψP i1ψP i2|R(∆2)|ψk1ψk2⟩  �  ,  (A21)  where ∆1 = εP i1 − εk1 and ∆2 = εP i2 − εk2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Appendix B: Computational formulas for the one-electron contributions  In view of the definition (9), the expression (A14) obtained in Appendix A is given in a  form that allows one to use the finite-basis-set methods [120, 121, 127] for its calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Therefore, the main difficulty is related with the evaluation of the integral over ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' For large  real values of ω, the photon propagator (7) is a strongly oscillating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' It is convenient  to perform the Wick’s rotation to overcome this obstacle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In the present Appendix, we  discuss the related transformations for the contribution HNMS,ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Employing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (12), one can represent Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (A14) as the sum of the Coulomb, one-  transverse-photon, and two-transverse-photon contributions  HNMS,ik = Hc  NMS,ik + Htr1  NMS,ik + Htr2  NMS,ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (B1)  28  ���������  ��������  �  ���������  ��������  �  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The poles and the branch cuts of the integrand in the operator P(εx) for the one- and  two-transverse-photon contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The integration contour: the original one oriented along the  real axis (left panel);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' the rotated to the imaginary one (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  ���������  ��������  �  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The poles and the branch cuts of the integrand in the operator P(εx) for the one- and  two-transverse-photon contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The integration contour in the complex plane is chosen to  avoid all the singularities of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  In the Coulomb contribution, the integration over ω can be performed analytically by means  of the identity (A20)  Hc  NMS,ik = 1  2  � εn>0  �  n  ⟨ψiψn|Rc|ψnψk⟩ −  εn<0  �  n  ⟨ψiψn|Rc|ψnψk⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (B2)  The one- and two-transverse-photon contributions are handled identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In what follows  in this Appendix, we will refer to them together as the “transverse-photon” (tr) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In the  left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 6, the original integration contour oriented along the real axis is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The  poles and the branch cuts of the integrand are presented as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Upon the Wick’s rotation  to the imaginary axis, the bound-state poles of the electron Green’s function are picked up  as the residues, see the right panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The final formulas for the transverse-photon  29  contribution can be written as a sum of three terms  Htr  NMS,ik = Htr(a)  NMS,ik + Htr(b)  NMS,ik + Htr(c)  NMS,ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (B3)  The first term arises from the residues shown by the circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 6 and reads as  Htr(a)  NMS,ik = 1  2  �  x=i,k  −1<εn<εx  �  n  ⟨ψiψn|Rtr(εn − εx)|ψnψk⟩ ,  (B4)  where for brevity we have introduced the summation over x in order to take into account  the symmetric form of the expression with respect to the argument of P(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The second  term corresponds to the case of degeneracy between the states of the opposite parity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This  term originates from the poles located at ω = 0 and has the form  Htr(b)  NMS,ik = 1  4  �  x=i,k  εn=εx  �  n  ⟨ψiψn|Rtr(0)|ψnψk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (B5)  Finally, the third term corresponds to the integration over the imaginary axis  Htr(c)  NMS,ik = 1  2  �  x=i,k  εn̸=εx  �  n  1  π  � ∞  0  dy  εn − εx  y2 + (εn − εx)2⟨ψiψn|Rtr(iy)|ψnψk⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (B6)  The expressions similar to (B2), (B4), (B5), and (B6) were derived, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  However, only the case of the diagonal matrix elements was considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Moreover, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [32]  the formulas for the higher-order QED corrections beyond the Breit approximation were  given, while the present ones include the lowest-relativistic contributions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  As an additional crosscheck of the ω-integration routine, we perform it also by employ-  ing the contour schematically shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' This contour is chosen to bypass all the  singularities in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The matrix elements HNMS,ik have been evaluated for  both variants of the integration-contour rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The results are found to be in excellent  agreement with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Appendix C: Local effective potentials  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' IV, in order to test the performance of the model-QED operator for the nuclear  recoil effect, we need substitute the core-Hartree (CH) and xα potentials (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' [122])  instead of V in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' These potentials can be expressed via the charge densities of the  valence electron,  ρv(r) = g2  v(r) + f 2  v (r) ,  (C1)  30  and the core,  ρcore(r) =  �  c  (2jc + 1)  �  g2  c(r) + f 2  c (r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (C2)  Here the large, g, and small, f, components of the Dirac wave function in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (C1) and  (C2) are determined self-consistently in the local potential  V (r) = −αZeff(r)  r  ,  (C3)  where Zeff(r) is the effective charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The CH potential is given by  ZCH  eff (r) = Znucl(r) − r  ∞  �  0  dr′  ρcore(r′)  max{r, r′} ,  (C4)  where Znucl(r) accounts for the finite size of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Defining the total charge density,  ρtot = ρcore + ρv, the effective charge for the xα potentials can be written in the form  Zxα  eff (r) = Znucl(r) − r  ∞  �  0  dr′  ρtot(r′)  max{r, r′} + xα  � 81  32π2rρtot(r)  �1/3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  (C5)  We use the values of xα equal to 0, 1/3, 2/3, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' The choice xα = 0 leads to the Dirac-  Hartree potential, xα = 2/3 corresponds to the Kohn-Sham potential [123], while xα = 1 is  the Dirac-Slater potential [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' In order to restore the proper asymptotic behavior of the  xα potentials, we introduce the Latter correction [125].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Elliott, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beiersdorfer, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 76, 1031 (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 77, 4278(E)  (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Elliott, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beiersdorfer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Decaux, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Knapp, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C  57, 583 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schuch, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lindroth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Madzunkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fogle, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mohamed, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Indelicato, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 95, 183003 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [4] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soria Orts, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Crespo L´opez-Urrutia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bruhns,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mart´ınez, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jentschura, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lapierre, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mironov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tawara, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ullrich, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 97, 103002 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  31  [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brandau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhuharov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schippers, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bern-  hardt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B¨ohm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jacobi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schmidt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mokler, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bosch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kluge, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beckert, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beller, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nolden, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Steck, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gumberidze, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Reuschl, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Spillmann,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Currell, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jentschura, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wolf, and  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Stachura, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 100, 073201 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [6] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K¨ohler, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Block, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chenmarev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eliseev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gon-  charov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kracke, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nesterenko, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Novikov, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ramirez,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 7, 10246 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [7] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Maaß, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H¨uther, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K¨onig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kr¨amer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Krause, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lovato, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Puchalski, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Roth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S´anchez, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sommer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wiringa, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N¨ortersh¨auser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 122, 182501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [8] ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Koszor´us, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jiang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Novario, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Billowes, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Binnersley,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bissell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cocolios, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cooper, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' de Groote, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ekstr¨om, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flanagan,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Forss´en, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Franchoo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ruiz, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gustafsson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hagen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jansen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kanel-  lakopoulos, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kortelainen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nazarewicz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Neyens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Papenbrock, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Reinhard,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ricketts, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sahoo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Vernon, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wilkins, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 17, 439 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [9] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sailer, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Debierre, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Heiße, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K¨onig, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Morgner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, Nature 606, 479 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' King, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Spieß, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Micke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wilzewski, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Leopold, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Benkler, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lange,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Huntemann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Surzhykov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L´opez-Urrutia, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schmidt,  arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='13053 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='atom-ph] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Han, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Berengut, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Goriely,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Meng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zhang, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Research 4, 033049  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Frugiuele, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fuchs, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Perez, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schlaffer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' D 96, 015011 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [13] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Berengut, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Budker, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Delaunay, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flambaum, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Frugiuele, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fuchs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gro-  jean, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harnik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ozeri, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Perez, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soreq, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 120, 091801 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flambaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Geddes, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Viatkina, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 97, 032510 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [15] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Surzhykov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Micke, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schmidt, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  101, 012502 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  32  [16] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rehbehn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rosner, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bekker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Berengut, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schmidt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' King,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Micke, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Surzhykov, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L´opez-Urrutia, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  103, L040801 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [17] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Figueroa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Berengut, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flambaum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Budker, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Antypas,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 128, 073001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [18] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ono, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Saito, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ishiyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Higomoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Takano, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Takasu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yamamoto,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tanaka, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Takahashi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' X 12, 021033 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [19] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Debierre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, Physics Letters B 807, 135527 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [20] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Debierre and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Oreshkina, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 104, 032825 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [21] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Debierre, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman,  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01668 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='atom-ph] (2022),  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='01668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Debierre, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Oreshkina, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Valuev, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, rXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='04868  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='atom-ph] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [23] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 63, 588 (1985), [Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 63, 394 (1985)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Sov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 47, 69 (1988), [Yad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 47, 107 (1988)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Palmer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 20, 5987 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [26] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 57, 59 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [27] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Grotch, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 51, 1854 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [28] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yelkhovsky, e-print hep-th/9403095 (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 110, 431 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Adkins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Morrison, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sapirstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 76, 042508 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [30] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 115, 233002 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [31] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 93, 062514 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [32] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 52, 1884 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [33] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 57, 4235 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Popov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 51, 085001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [35] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Grant, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 19, 747 (1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [36] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Desclaux, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 9, 31 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [37] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bratzev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Deyneka, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' USSR, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 41, 173  (1977), [Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nauk SSSR, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 41, 2655 (1977)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  33  [38] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Indelicato and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lindroth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 46, 2426 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [39] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dzuba, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flambaum, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozlov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 54, 3948 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [40] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Safronova, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Johnson, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Derevianko, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 60, 4476 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [41] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Crespo L´opez-Urrutia, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dragani´c, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soria Orts, and  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ullrich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 68, 022511 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [42] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozlov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Porsev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Safronova, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  195, 199 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [43] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dzuba, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Berengut, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harabati, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Flambaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 95, 012503  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [44] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien,  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Methods Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B 408, 46 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [45] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Saue, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bast, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gomes, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jensen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Visscher, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Aucar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Di Remigio,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dyall, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eliav, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fasshauer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fleig, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Halbert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hedeg˚ard, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Helmich-Paris,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Iliaˇs, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Jacob, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Knecht, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Laerdahl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Vidal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Nayak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Olejniczak,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Olsen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pernpointner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Senjean, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shee, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sunaga, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' van Stralen,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 152, 204104 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [46] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 88, 012513 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [47] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Za-  ytsev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 106, 012806 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [48] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin,  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 189, 175  (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 223, 69 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [49] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozlov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Safronova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Dzuba, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 117, 253001 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [50] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Paˇsteka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eliav, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Borschevsky, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaldor, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schwerdtfeger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  118, 023002 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Machado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Szabo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Santos, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Amaro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Guerra, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gumberidze, Guojie Bian,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Isac, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Indelicato, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 97, 032517 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [52] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Si, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Guo, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brage, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hutton, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Froese Fischer, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  98, 012504 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [53] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lindroth, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Borovik, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=', P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hillenbrand, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Holste, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Indelicato,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kilcoyne, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Klumpp, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Martins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Viefhaus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wilhelm, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schippers, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  34  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 98, 033416 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [54] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zaytsev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Maltsev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 100, 052504  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [55] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Puchalski, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 102, 042816 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [56] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mironova, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 101, 052502 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [57] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bezborodov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mironova, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 128, 21 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [58] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Skripnikov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eliav, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Maly-  shev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Oleynichenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Titov, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zaitsevskii, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  104, 012819 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [59] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Skripnikov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 154, 201101 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [60] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Usov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eliav, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Oleyn-  ichenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Skripnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Titov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zaitsevskii,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 105, 062805 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [61] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H¨affner, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beier, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hermanspahn, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kluge, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Stahl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Verd´u, and  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 85, 5308 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [62] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Verd´u, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Djeki´c, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Stahl, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Valenzuela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Vogel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beier, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kluge, and  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 92, 093002 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [63] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wagner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schabinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zatorski, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kei-  tel, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 107, 023002 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [64] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wagner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K¨ohler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint,  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 110, 033003 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [65] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wagner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kretzschmar, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  87, 030501(R) (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [66] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' K¨ohler-Langes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Heiße, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quint,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rau, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Werth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 123, 173001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [67] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Arapoglou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Egl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H¨ocker, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sailer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Weigel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wolf, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cakir,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Oreshkina, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Agababaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zinenko,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Harman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Keitel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sturm, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blaum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 122, 253001  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  35  [68] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 64, 052104 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [69] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 88, 091801 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [70] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  119, 263001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [71] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, JETP Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 106, 765  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [72] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 98, 032512  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [73] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Aleksandrov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 98, 062521 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [74] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 101, 012513 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [75] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Aleksandrov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 128, 297 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [76] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Hughes and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Eckart, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 36, 694 (1930).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [77] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Korol and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozlov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 76, 022103 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [78] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 81, 042513 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [79] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaidamauskas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Naz´e, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rynkun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaigalas, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J¨onsson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 44, 175003 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [80] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Naz´e, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Verdebout, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rynkun, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaigalas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J¨onsson, At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Data  Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Data Tables 100, 1197 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [81] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Volotka, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brandau, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 90, 062512 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [82] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Naz´e, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Li, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 91, 032511 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [83] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Fischer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J¨onsson, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaigalas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 49, 182004 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [84] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brandau, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 93, 052502 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [85] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Filippin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Biero´n, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaigalas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J¨onsson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 96, 042502  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  36  [86] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A  98, 022517 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [87] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gamrath, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Palmeri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quinet, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Bouazza, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Radiat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Transf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 218, 38 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [88] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Anisimova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev,  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Brandau, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' St¨ohlker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 52, 185001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [89] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ekman, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J¨onsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Naz´e, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Gaigalas, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Biero´n, Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 235, 433 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [90] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Zubova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Popov, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Save-  lyev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 128, 1090 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [91] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M¨uller, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Surzhykov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 104, L020802  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [92] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Si, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schiffmann, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Godefroid, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 104, 012802  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [93] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schelfhout and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' McFerran, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 104, 022806 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [94] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Schelfhout and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' McFerran, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 105, 022805 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [95] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Furry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 81, 115 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [96] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 356, 119 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [97] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  28, 5201 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [98] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Artemyev, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Beier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' T80, 493 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [99] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Aleksandrov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shchepetnov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 48, 144004 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [100] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Grotch and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yennie, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 41, 350 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [101] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Borie and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rinker, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 54, 67 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [102] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='13381 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='atom-ph] (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [103] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Veitia and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 69, 042501 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [104] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Anisimova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mironova, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 100, 012510 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  37  [105] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Anisimova, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mironova, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plu-  nien, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 101, 052506 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [106] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mohr, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Taylor, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Newell, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 84, 1527 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [107] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sapirstein and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yennie,  Theory of hydrogenic bound states,  in Quantum  Electrodynamics, edited by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kinoshita, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 560, World Scientific, Singapore, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [108] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Pachucki and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Karshenboim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 28, L221 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [109] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' B: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 26, 4703 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [110] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 55, 166 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [111] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cheng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chen, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Johnson, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 77, 052504 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [112] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Surzhykov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 86, 042507 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [113] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kaygorodov, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kozhedub, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Malyshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Glazov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 99, 032505 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [114] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mohr, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=') 88, 26 (1974);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 88, 52 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [115] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Snyderman, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=') 211, 43 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [116] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Mohr, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 293, 227 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [117] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 60, 800 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [118] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Angeli and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Marinova, At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Data Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Data Tables 99, 69 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [119] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Data 44, 033103 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [120] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Johnson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Blundell, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sapirstein, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 37, 307 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [121] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Tupitsyn, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Yerokhin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Plunien, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Soff, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  93, 130405 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [122] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sapirstein and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Cheng, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 66, 042501 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [123] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Kohn and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Sham, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 140, A1133 (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [124] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Slater, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 81, 385 (1951).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [125] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Latter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' 99, 510 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [126] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Shabaev, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 50, 4521 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  [127] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Salomonson and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' ¨Oster, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content=' A 40, 5548 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
+page_content='  38' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE2T4oBgHgl3EQfzgi3/content/2301.04132v1.pdf'}
diff --git a/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/2301.05657v1.pdf.txt b/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/2301.05657v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..299c2529c0e2383165ca7212c2c74dc22856f3fc
--- /dev/null
+++ b/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/2301.05657v1.pdf.txt
@@ -0,0 +1,1793 @@
+1 
+ 
+Structural evolution of iodine on approach to the monatomic state 
+Elena Bykova1,3, Iskander G. Batyrev2, Maxim Bykov1,4, Eric Edmund1, Stella Chariton5, Vitali 
+B. Prakapenka5, and Alexander F. Goncharov1,4 
+1 Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA 
+2 U.S. Army Research Laboratory, RDRLWML-B, Aberdeen Proving Ground, Maryland 21005, 
+United States   
+3 Bayerisches Geoinstitut, University of Bayreuth, Universitätsstrasse 30, D-95447, Bayreuth, 
+Germany 
+4 Institute of Inorganic Chemistry, University of Cologne, Greinstrasse 6, 50939 Cologne, 
+Germany 
+5 Center for Advanced Radiation Sources, The University of Chicago, Chicago, Illinois 60637, 
+USA 
+We applied single-crystal X-ray diffraction and Raman spectroscopy in a diamond anvil cell 
+up to 36 GPa and first principles theoretical calculations to study the molecular dissociation 
+of solid iodine at high pressure. Unlike previously reported, we find that the familiar Cmce 
+molecular phase transforms to a Cmc21 molecular structure at 16 GPa, and then to an 
+incommensurate dynamically disordered Fmmm(00γ)s00 structure at 20 GPa, which can be 
+viewed as a stepwise formation of polymeric zigzag chains of three iodine atoms following by  
+the formation of the dynamically dissociated, incommensurately modulated i-Fmmm phase, 
+and the truly monatomic Immm phase at higher pressures.  
+Understanding pressure driven dissociation of simple diatomic molecules is important for a range 
+of topics including materials behavior under extremes and composition of planetary interiors. Such 
+transformations, which are commonly preceded or accompanied by metallization, occur at very 
+high pressures in the crystals made of light molecules (e.g., H2 and F2), where such investigations 
+are very challenging (1, 2). However, molecular dissociation and band gap closure occurs at much 
+lower pressures for heavier molecules. In fact, these phenomena in halogens Cl2, Br2, and I2 have 
+been thoroughly investigated revealing a common behavior (3-5), where molecular dissociation is 
+reported to occur in steps associated with a series of phase transitions.     
+Iodine is investigated most extensively largely because it transforms to a metallic state at 14-24 
+GPa and experiences molecular dissociation at above 21 GPa (6-11)—conditions, which are easily 
+accessible with the diamond anvil cell technique. At low pressures, iodine crystallizes in an 
+orthorhombic Cmce structure (3) (phase I), which consists of flat layers, formed as associations of 
+zigzag molecular chains (Fig. 1). Above 26 GPa, this structure transforms into a monatomic Immm 
+lattice (phase II), where iodine atoms occupy the corner sites forming a metallic body-centered 
+orthorhombic (BCO) single-atom unit cell (8). However, careful investigations (12) showed the 
+existence of an intermediate phase (V) at 24-26 GPa, which can coexist with the Immm structure 
+at 26-30 GPa. This intermediate structure can be viewed as a face-centered orthorhombic lattice 
+
+2 
+ 
+based on the main diffraction peaks, but the occurrence of satellite reflections from both sides of 
+the main peaks indicates the presence of an incommensurate modulation wave along the a-axis, 
+which shifts the atoms along the b-axis. In this structure, there are three- or four- atom chains in 
+the ab plane, with a continuous distribution of the nearest-neighbor interatomic distances covering 
+a range of approximately 0.26 Å. Raman spectroscopy demonstrated the presence of a specific 
+low-frequency soft mode for the modulated structures, called the amplitude mode (AM), which 
+corresponds to transverse atomic vibrations and preserves the symmetry of the modulation wave 
+(5). No Raman signal has been detected in phase II as expected from the Raman selection rules. 
+ 
+    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 1.  Fragments of crystal structures of Cmce (I), Cmc21 (VI), and Fmmm(00γ)s00 (VII) phases 
+determined in this work (Tables S1 and S2, of Supplemental Material (13)) projected to the bc(ac 
+for Fmmm) plane. The atoms occupying different crystallographic sites are shown in different 
+colors. The atoms are represented by the displacement ellipsoids (at 75% probability) as they occur 
+after the structural refinement. Arrows show the directions of shifts of the molecular layers that 
+occur at the Cmce to Cmc21 transition. Fmmm(00γ)s00 structure presentation is approximate (small 
+unit cell) (see detailed structural results in Table S2, Supplemental Material (13)).    
+In the stepwise molecular dissociation scenario presented above, it remains puzzling if molecular 
+phase I metallizes via band overlap and whether there is any indication of its instability toward the 
+transformation into modulated phase V. Indeed, theoretical calculations predict such instability 
+within Cmce structure due to a coupling of the bond charge density with a transverse optical 
+phonon (14). Raman measurements of phase I (5, 15) demonstrated the presence of 2Ag+2B3g in-
+plane stretching (S) and librational (L) modes, out of which Ag(L) mode turns over and becomes 
+a soft mode above 13 GPa. In the same pressure range, additional Raman modes appear that cannot 
+be explained within the Raman selection rules of Cmce structure. In addition, Mössbauer 
+experiments demonstrated an anomaly at 16 GPa suggesting a phase transformation (16), while X-
+ 
+3.08 Å
+3.08 Å
+2.725 Å
+Cmce
+2.97 Å
+3.14 Å
+2.74 Å
+Cmc21
+Fmmm
+2.84 Å
+2.80 Å
+3.19 Å
+2.82 Å
+2.826 Å
+3.21 Å
+3.208 Å
+3.224 Å
+
+a+ba+bcf3 
+ 
+ray absorption spectroscopy showed a hint of an increase in the interamolecular bond distance in 
+the same pressure range (4). Theoretical first-principles calculations (17) suggested that there is 
+another molecular phase (phase I), which has C2/m symmetry; it was proposed to coexist with 
+Cmce phase I (17). This phase has two types of I2 molecules in the unit cell with different 
+intramolecular lengths offering an explanation to the extra Raman bands, which appear above 14 
+GPa.  
+Here we present the results of concomitant synchrotron single-crystal (SC) X-ray diffraction 
+(XRD) and Raman measurements, which provide a detailed structural description of the multi-
+stage molecular dissociation of iodine. Our experiments unequivocally identify intermediate 
+phases - a Cmc21 molecular phase (hereafter I2-VI), and an incommensurate Fmmm(00γ)s00 phase 
+(hereafter I2-VII) characterized by the loss of well-defined stable molecules, which provide key 
+missing links between stable molecular I2-I and dynamically dissociated I2-V. Our first-principles 
+theoretical calculations show that Cmc21 I2-VI phase is energetically competitive, dynamically 
+stable and confirm the presence of the additional Raman bands as due the reduction in symmetry 
+upon transition to the Cmc21 phase.                    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 2.  Reconstructed reciprocal lattice planes of iodine at 15 GPa (upper panel) and at 17 GPa 
+(lower panel). The observed diffraction spots from the sample have been indexed and used to 
+determine the structure of Cmce (I2-I) and Cmc21 (I2-VI) phases, respectively (see details in Table 
+S1 of Supplementary Materials (13)). The X-ray extinctions rules for these phases are depicted.    
+ 
+ 
+
+Cmce
+hko:h,k=2n
+0
+2 01
+2-20
+Cmc21
+hk0:h+k=2n
+Q
+-510
+3.10
+00
+400
+510
+5810
+530
+3-304 
+ 
+Up to 15 GPa SC XRD patterns confirm the previously reported Cmce structure (Table S1 of 
+Supplementary Information (13)). However, SC XRD measurements at 17 GPa demonstrate a 
+difference in the X-ray extinction rules. Reflections (h k 0), with h=2n+1 and k = 2n+1 (e.g. (3 1 
+0), (5 1 0) and others) appear in the diffraction patterns (Fig. 2). These weaker reflections cannot 
+be observed in powder XRD patterns as they are much weaker than the main ones (<0.1%) and 
+interfere with stronger reflections, so this symmetry change was previously overlooked. A new 
+structure can be indexed with the Cmc21 space group, which represents a subgroup of the Cmce 
+group. Hereafter, we call this phase as I2-VI (cf. Ref. (18)). The structural distortion in the Cmc21 
+structure is very subtle (Fig. 1). It can be understood as a small shift of layers of collinear molecules 
+along the b-axis, which diminishes the lattice symmetry.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 3.  Reconstructed reciprocal lattice planes of iodine at 17, 19, and 20.8 GPa (from left to right). 
+Insets below show enlarged views of Bragg peaks marked by red rectangles. Splitting of the 
+zoomed in peak manifests the occurrence of incommensurate phase at 20.8 GPa. 
+ 
+On further compression, above 20 GPa, yet another subtle change in symmetry occurs (Figs. 1, 3). 
+At this point, a stable structural solution within Cmc21 space group is not possible. A sequence of 
+diffraction spots from (0kl) lattice planes (Fig. 3) show a splitting of several reflections along the 
+c* direction at 20.8 GPa, manifesting the occurrence of an incommensurate lattice with a 
+modulation vector of q=0.4837(2). An incommensurate phase has Fmmm(00γ)s00 structure, which 
+ 
+
+Okl
+Okl
+Okl
+p3
+p4
+p5
+Commensurate
+Commensurate
+Incommensurate
+Cmc2,
+Cmc2,
+Fmmm(00g)s005 
+ 
+is isosymmetrical with phase V at higher pressures. The existence of this phase is consistent with  
+Ref. (18), where this phase I2 was reported  at 16-23 GPa. In contrast, we find no incommensurate 
+structure below 20 GPa, where commensurate Cmc21 I2-VI is stable. Both commensurate Cmc21 
+and Cmce are the a×b×2c supercell subgroups of Fmmm(00γ)s00 with γ = 0.5 and various initial 
+phases of modulation t0 (1/8 for Cmce and general for Cmc21).  
+In incommensurate I2-VII phase, the nearest-neighbor interatomic distances are modulated (Fig. 
+S1 of Supplementary Information (13)) spreading a wide range, which covers typical intra-to-
+intermolecular I-I distances. This phase consists of a dynamic mixture of molecular I2 and 
+polymeric zigzag chains of three I atoms (see also Ref. (18)) in the ac-plane (Fig. S2 of 
+Supplementary Information (13)).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 4.  Raman spectroscopy data. Panel (a) shows a sequence of Raman spectra across the phase 
+transitions on pressure increase. The phase symmetries are labeled, and the traces are color coded. 
+Modes labeled as X and Y appear in Cmc21 phase and gain intensity. Panel (b) shows the measured 
+here pressure dependencies of the Raman frequencies and compared to previous investigations (5, 
+15) (left panel) as well as theoretically calculated (right panel) in this work.  
+Our concomitant Raman measurements below 24 GPa demonstrate smooth changes with pressure 
+(Fig. 4), which are generally consistent with previous observations (5, 15). Above 16 GPa, where 
+Cmc21 I2-VI is documented in XRD, new peaks appear, dubbed X and Y in the literature, with the 
+frequencies below Ag(L) and above B3g(L) modes, respectively. The appearance of these modes 
+can be understood by the Raman selection rule modification at the Cmce→Cmc21 phase transition 
+(Table S4 of Supplementary Information (13)); the most prominent difference in Cmc21 phase VI 
+is the relaxation of the rule of mutual exclusion for the Raman and IR active modes. According to 
+group-theory predictions, all optical modes are Raman-active in phase VI, out of which the most 
+intense 3A1+3B2 in-plane modes are expected to be observed in Raman spectra compared to 
+2Ag+2B3g modes of Cmce phase I. We note that X and Y modes are coupled to Ag(L) and B3g(L) 
+ 
+Raman spectra
+Raman Shift (cm-1)
+50
+100
+150
+200
+250
+Intensity (arb. units)
+13.8 GPa
+17.4 GPa
+19.0 GPa
+20.8 GPa
+21.8 GPa
+23.8 GPa
+32.5 GPa
+26.4 GPa
+24.0 GPa
+29.6 GPa
+Immm
+i-Fmmm (2)
+Cmc21
+Cmce
+X
+Y
+i-Fmmm (1)
+(a) 
+ 
+(b) 
+
+250
+Cmc2.Cmcm
+Cmca
+Cmc2,
+Cmce
+i-FmmmImmm
+A
+Ag(S)
+200
+B
+Bzboundary
+150
+B,
+B2
+100
+A1
+Thiswork
+50
+AM
+Olijnyketal,1994
+Kumeetal.,2005
+D
+0
+10
+20
+30
+10
+20
+Pressure(GPa)
+Pressure(GPa)6 
+ 
+originated modes, respectively. This is evident from their frequency curves vs. pressure (Fig. 4(b)), 
+which show characteristic avoided crossing dependencies and an exchange in intensities correlated 
+with the frequency approaching as in the case of Fermi resonance. This behavior shows that the 
+coupled modes are of the same symmetry, namely A1 and B1 modes of Cmc21 phase (Fig. 4(b)). 
+Thus, the appearance and behavior of the X and Y modes are well understood by the 
+Cmce→Cmc21 phase transformation; these modes originate from IR active B1u+B2u modes of 
+Cmce phase, which become Raman active in Cmc21 phase (Table S3 of Supplementary 
+Information (13)). These modes correspond to translational motions of I2 molecules along the b 
+and c axes, which couple with librational motions in phase VI.  
+Only subtle changes (mainly peak broadening) occur in the Raman spectra above 20 GPa, where 
+our XRD data indicate a transition into an incommensurate Fmmm(00γ)s00 I2-VII phase. The 
+presence of vibrational modes characteristic for molecular phases is likely because this phase still 
+consists of short lived I2 molecules demonstrating fluxional behavior, which is supported by MD 
+simulations of Ref. (18). This behavior is consistent with the incommensurate nature of this phase, 
+where molecular breakdown and recombination are associated with the modulation wave 
+propagating through the crystal. It is the instantaneous existence of molecules that leads to a Raman 
+response similar to that of Cmce and Cmc21 phases. At 23.8 GPa, the A1 mode originated from the 
+X-mode becomes a dominant mode in the spectrum; this mode is a soft one corresponding to 
+librational motions of I2 molecules with the eigenvector, which captures the pathway to a 
+monatomic lattice. The observed here mode behavior is qualitatively consistent with the first-
+principles theoretical calculations (Fig. 4 and Fig. S3 of Supplementary Information (13)). 
+Our first-principles theoretical calculations show that Cmc21 phase is energetically competitive 
+with respect to Cmce and C2/m in the pressure range above 15 GPa , where Cmce phase is reported 
+to transform in this work and Refs. (17, 18) (Fig. S4 of Supplementary Information (13)). However, 
+within GGA-PBE approximation, Cmc21 phase is substantially more stable than the experimental 
+Cmce ground-state phase below 15 GPa (see also Ref. (19)), which clearly contradicts the 
+experiments. This apparent inconsistency is removed if M06-L functional is used as the enthalpies 
+of Cmce and Cmc21 phases become nearly degenerate in the whole pressure range of interest. All 
+these phases are dynamically stable at 5-30 GPa as witnessed by calculations of the phonon 
+dispersion curves (Fig. S5 of Supplementary Information (13)). GGA-PBE calculations of the 
+electronic band structure show that the molecular phases determined here are semiconducting with 
+narrow band gaps decreasing with pressure, which are expected to close at 26 GPa (Fig. S6 of 
+Supplementary Information (13)).     
+At 24 GPa, XRD patterns and Raman spectra change abruptly. SC XRD performed in this work 
+find a modulated structure (Figs. S7, S8 and Table S3 of Supplementary Information (13)) as 
+reported in Ref. (12) based on powder diffraction data (phase V). Unlike Ref. (12), we find the 
+structure of phase V to be Fmmm(00γ)s00 (cf. Fmm2(a00)0s0 in Ref. (12)) in agreement with the 
+more recent analysis (20). We also find that the modulation vector value decreases with pressure 
+(Fig. S9) in agreement with Refs. (12, 20). However, we additionally find, based on SC XRD 
+measurements on high-quality laser annealed crystals (Fig. S7 of Supplementary Information 
+(13)), that there is a tiny atomic displacement along the c-axis (Table S3 of Supplementary 
+
+7 
+ 
+Information (13)). Overall, these results definitively establish the structure of a modulated phase 
+V proposed in Ref. (12). However, our data do not support the tetragonal 5D modulated structure 
+(I4/mmm(αα0)000s(−αα0)0000) proposed  in Ref. (18). As can be seen on the reconstructed 
+precession image (Fig. S7 of Supplementary Information (13)), there is only one modulation 
+vector, which can describe all the satellite reflections.   
+Raman spectra above 24 GPa do not show any modes of the lower pressure molecular phases I, 
+VI, and VII. Instead, a strong soft Raman mode appears at low frequencies, which has been 
+previously identified as an AM (5) (Fig. 4(b)). These observations confirm that the modulated 
+phase V is incommensurate, where the Brillouin zone center vibrational modes of the parent phase 
+are forbidden. Above 26 GPa, two new Raman excitations, a narrow and a broad at nearly 150 cm-
+1 appear and shift to higher frequencies with pressure (Fig. 4(b)). Monatomic (BCO) phase II 
+coexists with phase V in this regime, identified in powder XRD measurements. Above 30 GPa, 
+the AM mode disappears, and XRD measurements show a single-phase BCO structure. Based on 
+our theoretical calculations (Fig. S10), the Raman bands observed in this phase correspond well in 
+frequency and pressure shift to the Brillouin zone boundary transverse acoustic modes near the R, 
+W, and T points, where the corresponding dispersion curves are flat yielding sharp maxima in the 
+phonon density of states at 4 THz (134 cm-1). The mechanism of their Raman activity (e.g., defect 
+induced) is not clear at this stage.                              
+Our results show the presence of an intermediate molecular phase VI and fluxional mixed 
+molecular-zigzag incommensurate phase VII between a low-pressure molecular phase I and a 
+modulated and dynamically dissociated phase V. Phase VI has Cmc21 orthorhombic symmetry, 
+which can be described as a small distortion of Cmce low-pressure phase. This new phase can be 
+viewed as a slightly dissociated (with longer intramolecular distance (Figs. 1, 5) molecular phase, 
+where intermolecular coupling has been modified to approach zigzag chains of three iodine atoms. 
+Previous experiments and theoretical calculations proposed a monoclinic C2/m structure in this 
+regime (17), but our XRD experiments clearly rule out this phase. In a recent work, it has been 
+proposed that phase VI is a modulated incommensurate phase (18); however, our SC XRD data 
+clearly show that no satellite reflections are observed below 20 GPa. In a qualitative agreement 
+with the results of previous works (12, 18, 20), we find the structure of phase V at 24-30 GPa to 
+be modulated and incommensurate (Figs. S7-S9 of Supplementary Information (13)); however, 
+the symmetry and the dimensionality are different of that proposed in Ref. (18).   
+Our new phase identification is consistent with Mössbauer experiments (16), which proposed a 
+change in symmetry based on discontinuous changes in the electric field gradient (EFG) 
+parameters. We theoretically computed these parameters in Cmce and Cmc21 phases and found 
+discontinuous changes in EFG, which are qualitatively consistent with the Mössbauer observations 
+(Fig. S11 of Supplementary Information (13)).  
+The Fmmm(00γ)s00 incommensurate mixed molecular-zigzag phase VII differs from 
+incommensurate phase V in that the latter one can be viewed as totally dissociated, consisting of 
+I4 and I5 zigzag atomic chains (Fig. S1), while phase VII still has a large disparity between intra 
+and intermolecular bond lengths, which results in formation of zigzag I3 chains (cf. Ref. (18)), 
+while some I2 molecules still remain. The modulation vector of phase VII is substantially larger 
+
+8 
+ 
+than that of phase V (Fig. S9). The interatomic distances in both modulated phases V and VII 
+greatly vary (Fig. 5, Fig. S1), while the averaged distance goes down with pressure due to the 
+volume compression. The determined here phase sequence Commensurate (Cmce)→ 
+Commensurate (Cmc21)→ Incommensurate Fmmm(00γ)s00 (γ ~ 0.5) → Incommensurate 
+Fmmm(00γ)s00 (γ ~ 0.25) → Immm shows a seamless pathway for molecular dissociation of I2.   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 5.  Volume compression of iodine. The inset shows the nearest interatomic distance vs 
+pressure data. The results of this work (filled symbols, the uncertainty is smaller than the symbol 
+size) are compared to previously reported data (open symbols and crosses) from Refs. (4, 8, 21). 
+The solid line extrapolated to the transition into the modulated Fmmm phase is a Vinet fit to our 
+data (V0=340.9(2) Ǻ3, K0=8.1(5), and K0ʹ=7.0(3), where K0 and K0ʹ are the bulk modulus and its 
+pressure derivative at P=0). The vertical bars in inset show the spread of the nearest I-I distances 
+in incommensurate phases.   
+Our XRD data also address the volume change at metallization and molecular dissociation 
+transition (Fig. 5). There is no measurable volume change at the Cmce → Cmc21 (VI)→ 
+Fmmm(00γ)s00 (VII) phase transitions. However, there is a volume collapse of 1.8% that occurs 
+at the isosymmetrical VII-V transition (both have Fmmm(00γ)s00 symmetry) (cf. Ref. (12)). This 
+is smaller than that theoretically computed here using DFT calculations using GGA-PBE 
+functional (4.2%). It has been previously suggested that metallization occurs in a molecular state 
+thus preceding dissociation (4).  Based on our XRD results and theoretical calculations of the 
+electronic structure (Figs. S3, S4), we speculate that the Cmc21 → Fmmm(00γ)s00 (VII) phase 
+transitions at 20 GPa can be directly connected to metallization reported at 14-24 GPa (9). In this 
+case, there is no volume collapse associated with metallization. Molecular dissociation occurs 
+stepwise from truly molecular Cmce and Cmc21 to dynamically disordered Fmmm(00γ)s00 and 
+dynamically dissociated i-Fmmm phase. It would be of great interest to investigate if these results 
+are applicable to Br2 and Cl2, which were reported to have a similar structural behavior.        
+ 
+Pressure (GPa)
+0
+10
+20
+30
+Volume (Å3)
+200
+250
+300
+Cmce (I)
+Cmc21(VI)
+Fmmm-1 (V)
+Immm (II)
+0
+10
+20
+30
+Shortest Interatomic distance (Å) 
+2.70
+2.75
+2.80
+2.85
+2.90
+2.95
+3.00
+Cmce
+Cmc21
+Buontempo et al. 1998
+Shi et al., 2021
+This work
+Takemura et al., 1982
+This work, calculation
+This work, experiment
+Fmmm-2 (VII)
+Fmmm-2
+Fmmm-1
+Immm
+
+9 
+ 
+Parts of this research were carried out at the GeoSoilEnviroCARS (The University of Chicago, 
+Sector 13), Advanced Photon Source (Argonne National Laboratory). GeoSoilEnviroCARS is 
+supported by the National Science Foundation—Earth Sciences (No. EAR-1634415). Use of 
+the GSECARS Raman System was supported by the NSF MRI Proposal (No. EAR-1531583). The 
+Advanced Photon Source is a U.S. Department of Energy (DOE) Office of Science User Facility 
+operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DE-
+AC02-06CH11357. We acknowledge support by the Army Research Office accomplished under 
+the Cooperative Agreement No. W911NF-19-2-0172 and the Carnegie Institution of Washington. 
+M.B. acknowledges the support of Deutsche Forschungsgemeinschaft (DFG Emmy-Noether 
+project BY112/2-1). E.B. acknowledges financial support from the program ‘Promotion of Equal 
+Opportunities for Women in Research and Teaching’ funded by the Free State of Bavaria. 
+ 
+1. 
+A. Goncharov, Phase diagram of hydrogen at extreme pressures and temperatures; updated 
+through 2019 (Review article). Low Temperature Physics 46, 97-103 (2020). 
+2. 
+E. Gregoryanz et al., Everything you always wanted to know about metallic hydrogen but were 
+afraid to ask. Matter and Radiation at Extremes 5, 038101 (2020). 
+3. 
+H. Fujihisa, Y. Fujii, K. Takemura, O. Shimomura, Structural aspects of dense solid halogens 
+under high pressure studied by x-ray diffraction—Molecular dissociation and metallization. 
+Journal of Physics and Chemistry of Solids 56, 1439-1444 (1995). 
+4. 
+J. Shi et al., Halogen molecular modifications at high pressure: the case of iodine. Physical 
+Chemistry Chemical Physics 23, 3321-3326 (2021). 
+5. 
+T. Kume, T. Hiraoka, Y. Ohya, S. Sasaki, H. Shimizu, High Pressure Raman Study of Bromine and 
+Iodine: Soft Phonon in the Incommensurate Phase. Physical Review Letters 94, 065506 (2005). 
+6. 
+R. W. Lynch, H. G. Drickamer, Effect of Pressure on the Lattice Parameters of Iodine, Stannic 
+Iodide, and p‐Di‐iodobenzene. The Journal of Chemical Physics 45, 1020-1026 (1966). 
+7. 
+K. Takemura, S. Minomura, O. Shimomura, Y. Fujii, Observation of Molecular Dissociation of 
+Iodine at High Pressure by X-Ray Diffraction. Physical Review Letters 45, 1881-1884 (1980). 
+8. 
+K. Takemura, S. Minomura, O. Shimomura, Y. Fujii, J. D. Axe, Structural aspects of solid iodine 
+associated with metallization and molecular dissociation under high pressure. Physical Review B 
+26, 998-1004 (1982). 
+9. 
+T. Ishikawa et al., Metallization of solid iodine in phase I: X-ray diffraction measurements, 
+electrical resistance measurements, and ab initio calculations. High Pressure Research 33, 186-
+190 (2013). 
+10. 
+A. S. Balchan, H. G. Drickamer, Effect of Pressure on the Resistance of Iodine and Selenium. The 
+Journal of Chemical Physics 34, 1948-1949 (1961). 
+11. 
+N. Sakai, K.-i. Takemura, K. Tsuji, Electrical Properties of High-Pressure Metallic Modification of 
+Iodine. Journal of the Physical Society of Japan 51, 1811-1816 (1982). 
+12. 
+K. Takemura, S. Kyoko, F. Hiroshi, O. Mitsuko, Modulated structure of solid iodine during its 
+molecular dissociation under high pressure. Nature 423, 971-974 (2003). 
+13. 
+S. F. S.-S. See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevLett. 
+xxx This pdf file contains Materials and Methods, Tables S1-S4, and Bibliography which includes 
+Refs. [22–27]. 
+14. 
+T. Luty, J. C. Raich, Molecular to atomic transformation in solid iodine under high pressure. 
+Canadian Journal of Chemistry 66, 812-818 (1988). 
+
+10 
+ 
+15. 
+H. Olijnyk, W. Li, A. Wokaun, High-pressure studies of solid iodine by Raman spectroscopy. 
+Physical Review B 50, 712-716 (1994). 
+16. 
+M. Pasternak, J. N. Farrell, R. D. Taylor, Metallization and structural transformation of iodine 
+under pressure: A microscopic view. Physical Review Letters 58, 575-578 (1987). 
+17. 
+Q. Zeng et al., A new phase of solid iodine with different molecular covalent bonds. Proceedings 
+of the National Academy of Sciences 105, 4999-5001 (2008). 
+18. 
+H. Fujihisa, K. Takemura, M. Onoda, Y. Gotoh, Two intermediate incommensurate phases in the 
+molecular dissociation process of solid iodine under high pressure. Physical Review Research 3, 
+033174 (2021). 
+19. 
+J. George, C. Reimann, V. L. Deringer, T. Bredow, R. Dronskowski, On the DFT Ground State of 
+Crystalline Bromine and Iodine. ChemPhysChem 16, 728-732 (2015). 
+20. 
+K. Takemura, K. Sato, H. Fujihisa, M. Onoda, Structural phase transitions in iodine under high 
+pressure. Zeitschrift für Kristallographie - Crystalline Materials 219, 749-754 (2004). 
+21. 
+U. Buontempo et al., Anomalous Bond Length Expansion in Liquid Iodine at High Pressure. 
+Physical Review Letters 80, 1912-1915 (1998). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+11 
+ 
+Supplementary Information 
+Structural evolution of iodine on approach to the monatomic state 
+Elena Bykova1,3, Iskander G. Batyrev2, Maxim Bykov1,4, Eric Edmund1, Stella Chariton5, Vitali 
+B. Prakapenka5, and Alexander F. Goncharov1,4 
+1 Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA 
+2 U.S. Army Research Laboratory, RDRLWML-B, Aberdeen Proving Ground, Maryland 21005, 
+United States   
+3 Bayerisches Geoinstitut, University of Bayreuth, Universitätsstrasse 30, D-95447, Bayreuth, 
+Germany 
+4 Institute of Inorganic Chemistry, University of Cologne, Greinstrasse 6, 50939 Cologne, 
+Germany 
+5 Center for Advanced Radiation Sources, The University of Chicago, Chicago, Illinois 60637, 
+USA 
+ 
+This pdf file contains Materials and Methods, Supplemental Figures S1-S11, Tables S1-S4, and 
+Bibliography with References [22-27]. 
+ 
+Corresponding author: 
+agoncharov@carnegiescience.edu 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+12 
+ 
+Materials and methods  
+Experiments 
+The experimental procedure included concomitant SC XRD and Raman spectroscopy 
+measurements (see details in the Supplementary Information) at 10-36 GPa in the Sector 13 
+(GSECARS) of the Advanced Photon Source, Argonne National Laboratory. Samples in the form 
+of flakes were placed in a cavity of a diamond anvil cell formed in a central hole made in a 
+preindented rhenium gasket, sealed to avoid sublimation, and then loaded by compressing Ne gas 
+to 160 MPa at room temperature. Ruby served as an optical pressure gauge, while the equation of 
+state of Ne 22 has been used to cross check pressure during XRD measurements. Compressed Ne 
+serving as a pressure medium provided quasi-hydrostatic conditions in the high-pressure cavity, 
+as witnessed by SC XRD data, which were of sufficiently high quality up to at least 27 GPa. At 
+this pressure we performed laser annealing of the sample up to 1800 K to improve the quality of 
+SC XRD data as will be presented below. Additional Raman measurements have been collected at 
+Earth and Planets Laboratory of the Carnegie Institution for Science, before and after synchrotron 
+XRD experiments. 
+Theoretical calculations 
+First-principles theoretical calculations have been performed in Cmce, Cmc21, C2/m, Immm phases 
+at selected pressures between 0 and 30 GPa, where these structures were optimized using norm-
+conserving pseudopotentials, GGA-PBE functional, and Grimme2 dispersion corrections23. 
+Monkhorst-Pack grid size for k-points sampling of the Brillouin Zone (BZ) is 5x6x5 for all 
+structures except I2-II (Immm) at 30 GPa, where 6x7x6 sampling was used 24. The phonon 
+dispersion and phonon frequency calculations were performed using a finite displacement method 
+implemented in the CASTEP code 25.  The Raman spectra were calculated using the formalism 
+presented in Ref. 26. Since application of GGA-PBE functional incorrectly predicts that Cmce 
+phase is thermodynamically and dynamically unstable at 0 GPa 19, we performed calculations 
+using Minnesota 2006 local functional 27 (M06-L) in Cmce and Cmc21 phases up to 30 GPa. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+13 
+ 
+ 
+ 
+Fig. S1.  The I-I nearest-neighbor interatomic distances as a function of the modulation vector of 
+Fmmm(00γ)s00 I2-VII phase determined in this work. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+p05
+3.30
+I-I distances (A
+3.10
+2.90
+2.70
+2.50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+t14 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S2.  Fragments of crystal structures of Fmmm(00γ)s00 I2-VII and I2-V phases determined in 
+this work (Tables S2 and S3, respectively) projected to the ac plane. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+I2-VII 
+I2-V 
+ 
+d<2.9 Ǻ
+2.9 Ǻ <d<3.01 Ǻ
+d<2.953 Ǻ
+
+15 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S3.  Theoretically computed Raman frequencies vs pressure determined here from the first 
+principles. The irreducible representations are changed to be consistent with the axes name 
+convention used in experiment (Fig. 1). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Pressure (GPa)
+10
+12
+14
+16
+18
+20
+22
+Raman Frequency (cm-1)
+0
+50
+100
+150
+200
+Ag
+B3g
+B1
+Ag
+B3g
+A1
+A1
+B1
+B1
+A1
+Cmca
+Cmc21
+
+16 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S4.  Theoretically computed enthalpies of the candidate phases vs pressure determined here 
+from the first principles plotted with respect to results for Cmce phase. Solid lines are the results 
+computed using GGA-PBE functional, while a dashed line for Cmc21 phase is obtained using 
+Minnesota 2006 local functional (M06-L). The computed equilibrium phases are dynamically 
+stable in the presented here pressure range.   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Pressure (GPa)
+0
+5
+10
+15
+20
+25
+30
+Enthalpy / atom (meV)
+-10
+-5
+0
+5
+10
+Cmce
+C2/m
+Immm
+Cmc21
+Cmc21 (M06L)
+Cmcm
+
+17 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S5. Electronic band structure and density of electronic states of phases I (Cmce) and VI 
+(Cmc21) at 20 GPa.  
+ 
+ 
+ 
+ 
+ 
+ 
+Cmc21, 20 GPa 
+ 
+Cmce, 20 GPa 
+
+CASTEPBandStructure
+CASTEP Partial Density
+Band gap is 0.223 eV
+of States
+21
+21
+18
+18
+15
+15
+12
+12
+(eV)
+(eV)
+Energy (
+hergy
+-9
+-12
+-12
+-15
+-15
+5
+5
+X
+U
+R
+0
+2
++
+6
+8
+DensityofStates(electrons/ev）
+Band energy
+SumCASTEPPartial
+CASTEP Band Structure
+Density of
+Band gap is 0.221 eV
+States
+20
+20
+Energy (eV)
+Energy (eV)
+10
+10
+-10
+5
+Density of States(electr
+- Band energy
+Sum18 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S6. The electronic band gap calculated as a function of pressure in Cmce and Cmc21 
+structures.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Pressure (GPa)
+0
+5
+10
+15
+20
+25
+Bandgap (eV)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Cmc21 
+ Cmca 
+
+19 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+    
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S7. Reconstructed reciprocal lattice planes of iodine phase V at 26.4 GPa. The right panel 
+shows an enlarged view of a fine structure near the (402) reflection of a parent Fmmm phase. 
+Satellite reflections are indexed in terms of the modulation vector. The details of the structural 
+refinement are presented in Table S3 of Supplementary Materials). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+(hol)
+2g
+(402)
+-q
+-2q20 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S8. Reconstructed precession image of iodine phase V at 26.4 GPa. The right panel shows 
+an enlarged view of satellite reflections. Satellite reflections are indexed in terms of the 
+modulation vector. The details of the structural refinement are presented in Table S3 of 
+Supplementary Materials). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+3g
+29
+(hk-1) in 14/mmm setting21 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S9. The modulation vector value as a function of pressure determined here and Ref. 12. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Pressure (GPa)
+20
+22
+24
+26
+28
+Modulation vector value
+0.2
+0.3
+0.4
+0.5
+Takemura et 2003, Phase V 
+This work, Phase V 
+This work, Phase VII
+
+22 
+ 
+ 
+ 
+ 
+Fig. S10. Phonon dispersion curves and density of phonon states of phase II (Immm) at 30 GPa.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+CASTEP Density of Phonon
+CASTEP Phonon Dispersion
+States
+10 
+10
+8
+(ZHL) 
+(ZHL) 
+Frequency（
+Freguency
+-2
+2
+R
+X
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+Density of Phonon States (1/THz)
+- Phonon dispersion23 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. S11. Electric field gradient (EFG) parameters calculated for phases Cmce (phase I) and 
+Cmc21 (phase VI) compared to experimental data of Ref. 16. 
+ 
+ 
+ 
+Pressure (GPa)
+8
+10
+12
+14
+16
+18
+20
+22
+Anysotropy parameter ()
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Pressure (GPa)
+8
+10
+12
+14
+16
+18
+20
+22
+EFG
+-200
+-150
+-100
+-50
+0
+50
+100
+150
+Pasternak, LP
+Cmca
+Cmc21
+Cmc21
+Pasternak et al., HP1
+Pasternak et al., LP
+Pasternak et al., HP1
+Cmc21
+Cmca
+
+24 
+ 
+Table S1. Crystal structure details of commensurate phases of iodine  
+Details of crystal structure refinements for iodine phases at high pressures* 
+Pressure, GPa 
+9.90 
+15.0 
+17.0 
+19.0 
+32.0 
+Space group 
+Cmce 
+Cmce 
+Cmc21 
+Cmc21 
+Immm 
+Phase 
+iodine-I 
+iodine-I 
+iodine-VI 
+iodine-VI 
+iodine-II 
+Pressure step 
+p1 
+p2 
+p3 
+p4 
+p8 
+a (Å) 
+6.163(4) 
+5.978(4) 
+5.908(5) 
+5.829(6) 
+2.9077(6) 
+b (Å) 
+4.1435(3) 
+4.0396(4) 
+4.0055(3) 
+3.9773(6) 
+3.0333(12) 
+c (Å) 
+9.2864(8) 
+9.1487(9) 
+9.0992(11) 
+9.0739(13) 
+5.226(7) 
+Z 
+2 
+2 
+2 
+2 
+2 
+V (Å3) 
+237.15(16) 
+220.94(15) 
+215.3(2) 
+210.4(2) 
+46.10(7) 
+ρcalc (g/cm3) 
+7.109 
+7.63 
+7.83 
+8.014 
+9.143 
+μ/mm-1 
+13.804 
+20.519 
+15.204 
+15.563 
+17.755 
+2Θmin for data 
+collection (°) 
+2.624 
+3.549 
+3.158 
+2.575 
+6.508 
+2Θmax for data 
+collection (°) 
+14.627 
+15.055 
+15.001 
+14.776 
+13.102 
+Completeness to d = 
+0.8 Å 
+0.527 
+0.468 
+0.432 
+0.545 
+0.25 
+Reflections collected 
+259 
+165 
+243 
+327 
+21 
+Independent reflections 136 
+86 
+175 
+213 
+16 
+Independent reflections 
+[I > 2σ(I)] 
+132 
+80 
+171 
+213 
+15 
+Refined parameters 
+7 
+7 
+13 
+13 
+4 
+Rint(F2) 
+0.012 
+0.0131 
+0.0124 
+0.0087 
+0.083 
+R(σ) 
+0.0148 
+0.0097 
+0.0163 
+0.0126 
+0.0316 
+R1 [I > 2σ(I)] 
+0.0356 
+0.0246 
+0.031 
+0.0351 
+0.0836 
+wR2 [I > 2σ(I)] 
+0.1028 
+0.0606 
+0.0784 
+0.0922 
+0.1911 
+R1 
+0.0368 
+0.026 
+0.0315 
+0.0351 
+0.0826 
+wR2 
+0.1057 
+0.0626 
+0.0785 
+0.0922 
+0.1907 
+Goodness of fit on F2 
+1.201 
+1.139 
+1.1 
+1.164 
+1.308 
+Δρmax(e / Å3) 
+1.774 
+1.068 
+1.463 
+2.02 
+2.874 
+Δρmin(e / Å3) 
+-1.579 
+-0.825 
+-1.096 
+-1.807 
+-2.468 
+x(I 1) 
+0 
+0 
+0 
+0 
+0 
+y(I 1) 
+0.18376(9) 
+0.19087(6) 
+0.4299(2) 
+0.4306(3) 
+0 
+z(I 1) 
+0.62197(4) 
+0.62278(3) 
+0.26446(10) 0.26390(15) 
+0 
+Ueq(I 1) (Å2)* 
+0.0165(6) 
+0.0099(6) 
+0.0080(9) 
+0.0086(9) 
+0.011(5) 
+x(I 2) 
+ 
+ 
+0 
+0 
+ 
+
+25 
+ 
+y(I 2) 
+ 
+ 
+0.0387(3) 
+0.0291(6) 
+ 
+z(I 2) 
+ 
+ 
+0.51173(10) 0.51229(15) 
+ 
+Ueq(I 2) (Å2)* 
+ 
+ 
+0.0096(9) 
+0.0114(10) 
+ 
+*Ueq is defined as one 
+third of the trace of the 
+orthogonalized Uij 
+tensor. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+26 
+ 
+Table S2. Crystal structure details of incommensurate Fmmm(00γ)s00 phase VII of iodine. 
+Crystal data 
+Chemical formula 
+I 
+Mr 
+126.9 
+Crystal system, space group 
+Orthorhombic, Fmmm(00γ)s00† 
+Pressure step 
+P05 
+Pressure (GPa) 
+20.8 GPa 
+Wave vectors 
+q = 0.4837(2)c* 
+a, b, c (Å) 
+3.9536(14), 5.735(18), 4.530(2) 
+V (Å3) 
+102.7(3) 
+Z 
+4 
+Radiation type 
+X-ray, λ = 0.2952 Å 
+µ (mm−1) 
+16.191 
+Data collection 
+Diffractometer 
+GSECARS 13IDD 
+No. of main reflections: All/independent/observed independent 
+75/35/31 
+No. of satellite reflections m = 1: All/independent/observed independent 
+131/56/56 
+No. of satellite reflections m = 2: All/independent/observed independent 
+152/64/58 
+Rint 
+0.0138 
+(sin θ/λ)max (Å−1) 
+0.867 
+Refinement 
+R(obs)main + sattelites / wR (all) main + satellites 
+0.0360/0.1141 
+R(obs)main / wR (all) main 
+0.0344/0.0934 
+R(obs)1st order satellites / wR (all) 1st order satellites 
+0.0305/0.0812 
+R(obs)2nd order satellites / wR (all)2nd order satellites 
+0.0513/0.1538 
+No. of reflections 
+145 
+No. of parameters 
+6 
+Δρmax, Δρmin (e Å−3) 
+1.65, −1.13 
+Crystal structure 
+ 
+ 
+I (x,y,z) 
+(0, 0, 0) 
+𝐴𝑥
+1  
+0.0749(2) 
+𝐴𝑧
+2  
+-0.00286(9) 
+ 
+† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x4+1/2; (3) −x1, x2, −x3, −x4; (4) x1, −x2, −x3, −x4+1/2; (5) 
+−x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x4+1/2; (7) x1, −x2, x3, x4; (8) −x1, x2, x3, x4+1/2; (9) x1, x2+1/2, x3+1/2, x4; (10) −x1, 
+
+27 
+ 
+−x2+1/2, x3+1/2, x4+1/2; (11) −x1, x2+1/2, −x3+1/2, −x4; (12) x1, −x2+1/2, −x3+1/2, −x4+1/2; (13) −x1, −x2+1/2, 
+−x3+1/2, −x4; (14) x1, x2+1/2, −x3+1/2, −x4+1/2; (15) x1, −x2+1/2, x3+1/2, x4; (16) −x1, x2+1/2, x3+1/2, x4+1/2; (17) 
+x1+1/2, x2, x3+1/2, x4; (18) −x1+1/2, −x2, x3+1/2, x4+1/2; (19) −x1+1/2, x2, −x3+1/2, −x4; (20) x1+1/2, −x2, −x3+1/2, 
+−x4+1/2; (21) −x1+1/2, −x2, −x3+1/2, −x4; (22) x1+1/2, x2, −x3+1/2, −x4+1/2; (23) x1+1/2, −x2, x3+1/2, x4; (24) 
+−x1+1/2, x2, x3+1/2, x4+1/2; (25) x1+1/2, x2+1/2, x3, x4; (26) −x1+1/2, −x2+1/2, x3, x4+1/2; (27) −x1+1/2, x2+1/2, −x3, 
+−x4; (28) x1+1/2, −x2+1/2, −x3, −x4+1/2; (29) −x1+1/2, −x2+1/2, −x3, −x4; (30) x1+1/2, x2+1/2, −x3, −x4+1/2; (31) 
+x1+1/2, −x2+1/2, x3, x4; (32) −x1+1/2, x2+1/2, x3, x4+1/2.  
+Displacive modulations of I atom (𝑢𝑖(𝑥4
+̅̅̅) for i = x, y, z) are described by a truncated Fourier series, which 
+due to the symmetry restrictions take the following form: 𝑢𝑥(𝑥4
+̅̅̅) = 𝐴𝑥1 sin(2𝜋𝑥4
+̅̅̅), 𝑢𝑧(𝑥4
+̅̅̅)= 𝐴𝑧2 sin(4𝜋𝑥4
+̅̅̅) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+28 
+ 
+Table S3. Crystal structure details of incommensurate Fmmm(00γ)s00 phase V of iodine. 
+Crystal data 
+Chemical formula 
+I 
+I 
+Mr 
+126.9 
+126.9 
+Crystal system, space group 
+Orthorhombic, Fmmm(00γ)s00† 
+Orthorhombic, Fmmm(00γ)s00† 
+Pressure step 
+p6 
+p7 
+Pressure (GPa) 
+24.0  
+26.4 
+Wave vectors 
+q = 0.274(2)c* 
+q = 0.2381(6)c* 
+a, b, c (Å) 
+4.2098 (13), 5.539 (10), 4.2407 (13) 
+4.2018 (8), 5.434 (4), 4.2224 (8) 
+V (Å3) 
+98.88 (18) 
+96.41 (8) 
+Z 
+4 
+Radiation type 
+X-ray, λ = 0.2952 Å 
+µ (mm−1) 
+16.82 
+17.25 
+Data collection 
+Diffractometer 
+GSECARS 13IDD 
+GSECARS 13IDD 
+No. of main reflections 
+All/independent/observed independent 
+68/31/28 
+56/28/28 
+No. of satellite reflections m = 1  
+All/independent/observed independent 
+130/55/55 
+123/60/60 
+No. of satellite reflections m = 2  
+All/independent/observed independent 
+- 
+132/66/57 
+Rint 
+0.017 
+0.012 
+(sin θ/λ)max (Å−1) 
+0.842 
+0.876 
+Refinement 
+RF(obs)main + sattelites / wRF (all) main + satellites 
+0.0347/0.0419 
+0.0409/0.0515 
+RF(obs)main / wRF (all) main 
+0.0257/0.0291 
+0.0400/0.0518 
+RF(obs)1st order satellites / wRF (all) 1st order 
+satellites 
+0.0438/0.0499 
+0.0394/0.0486 
+RF(obs)2nd order satellites / wRF (all)2nd order 
+satellites 
+- 
+0.0468/0.0555 
+No. of reflections 
+86 
+154 
+No. of parameters 
+5 
+6 
+Δρmax, Δρmin (e Å−3) 
+1.84, −1.70 
+2.32, −3.05 
+Crystal structure 
+ 
+ 
+I (x,y,z) 
+(0, 0, 0) 
+(0, 0, 0) 
+𝐴𝑥
+1  
+0.0576(4) 
+0.0588(2) 
+𝐴𝑧
+2  
+- 
+0.00072(8) 
+
+29 
+ 
+ 
+† Symmetry operations: (1) x1, x2, x3, x4; (2) −x1, −x2, x3, x4+1/2; (3) −x1, x2, −x3, −x4; (4) x1, −x2, −x3, −x4+1/2; (5) 
+−x1, −x2, −x3, −x4; (6) x1, x2, −x3, −x4+1/2; (7) x1, −x2, x3, x4; (8) −x1, x2, x3, x4+1/2; (9) x1, x2+1/2, x3+1/2, x4; (10) −x1, 
+−x2+1/2, x3+1/2, x4+1/2; (11) −x1, x2+1/2, −x3+1/2, −x4; (12) x1, −x2+1/2, −x3+1/2, −x4+1/2; (13) −x1, −x2+1/2, 
+−x3+1/2, −x4; (14) x1, x2+1/2, −x3+1/2, −x4+1/2; (15) x1, −x2+1/2, x3+1/2, x4; (16) −x1, x2+1/2, x3+1/2, x4+1/2; (17) 
+x1+1/2, x2, x3+1/2, x4; (18) −x1+1/2, −x2, x3+1/2, x4+1/2; (19) −x1+1/2, x2, −x3+1/2, −x4; (20) x1+1/2, −x2, −x3+1/2, 
+−x4+1/2; (21) −x1+1/2, −x2, −x3+1/2, −x4; (22) x1+1/2, x2, −x3+1/2, −x4+1/2; (23) x1+1/2, −x2, x3+1/2, x4; (24) 
+−x1+1/2, x2, x3+1/2, x4+1/2; (25) x1+1/2, x2+1/2, x3, x4; (26) −x1+1/2, −x2+1/2, x3, x4+1/2; (27) −x1+1/2, x2+1/2, −x3, 
+−x4; (28) x1+1/2, −x2+1/2, −x3, −x4+1/2; (29) −x1+1/2, −x2+1/2, −x3, −x4; (30) x1+1/2, x2+1/2, −x3, −x4+1/2; (31) 
+x1+1/2, −x2+1/2, x3, x4; (32) −x1+1/2, x2+1/2, x3, x4+1/2.  
+Displacive modulations of I atom (𝑢𝑖(𝑥4
+̅̅̅) for i = x, y, z) are described by a truncated Fourier series, which 
+due to the symmetry restrictions take the following form: 𝑢𝑥(𝑥4
+̅̅̅) = 𝐴𝑥1 sin(2𝜋𝑥4
+̅̅̅), 𝑢𝑧(𝑥4
+̅̅̅)= 𝐴𝑧2 sin(4𝜋𝑥4
+̅̅̅) 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+30 
+ 
+Table S4. Vibrational modes and activity of Cmce and Cmc21 crystal structures 
+Space & 
+point 
+group 
+Cmce #64  (D2h) 
+Cmc21  #36 (C2v)  
+Cmcm #63 (D2h) 
+Site 
+symmetry 
+8f  (CS) 
+4a +4a  (CS) 
+4c (C2v) +   4a (C2h) 
+Acoustic 
+modes 
+B1u+B2u+B3u 
+A1+B1+B2 
+B1u+B2u+B3u 
+Optical 
+modes 
+Modes 
+Activity 
+Modes 
+Activity 
+Modes 
+Activity 
+ 
+Au 
+ 
+B1u+B2u+B3u 
+ 
+B1g+B2g 
+ 
+2Ag+2B3g 
+ 
+Silent 
+ 
+IR 
+ 
+Raman (a) 
+ 
+Raman (bc) 
+A2 
+ 
+A1+B2+B1 
+ 
+A2+B1 
+ 
+2A1+2B2 
+Raman 
+ 
+IR & Raman 
+ 
+Raman & IR 
+ 
+Raman & IR 
+Au 
+ 
+3B1u+3B2u+2B3u 
+(4ª) 
+none 
+(4c) 
+Ag+ B1g+B3g 
+ 
+Silent 
+ 
+IR 
+ 
+ 
+ 
+Raman 
+ 
+ 
+ 
+References 
+22. 
+Y. Fei, A. Ricolleau, M. Frank, K. Mibe, G. Shen and V. Prakapenka, Proceedings of the 
+National Academy of Sciences 104 (22), 9182-9186 (2007). 
+23. 
+D. R. Hamann, M. Schlüter and C. Chiang, Physical Review Letters 43 (20), 1494-1497 
+(1979). 
+24. 
+H. J. Monkhorst and J. D. Pack, Physical Review B 13 (12), 5188-5192 (1976). 
+25. 
+K. Refson, P. R. Tulip and S. J. Clark, Physical Review B 73 (15), 155114 (2006). 
+26. 
+D. Porezag and M. R. Pederson, Physical Review B 54 (11), 7830-7836 (1996). 
+27. 
+Y. Zhao and D. G. Truhlar, The Journal of Chemical Physics 125 (19), 194101 (2006). 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
diff --git a/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/load_file.txt b/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7259a37823495286eec12f8d0824f095ae1282a8
--- /dev/null
+++ b/ntE5T4oBgHgl3EQfjg_f/content/tmp_files/load_file.txt
@@ -0,0 +1,888 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf,len=887
+page_content='1  Structural evolution of iodine on approach to the monatomic state Elena Bykova1,3, Iskander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Batyrev2, Maxim Bykov1,4, Eric Edmund1, Stella Chariton5, Vitali B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Prakapenka5, and Alexander F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Goncharov1,4 1 Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA 2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Army Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' RDRLWML-B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Aberdeen Proving Ground,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Maryland 21005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' United States 3 Bayerisches Geoinstitut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' University of Bayreuth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Universitätsstrasse 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D-95447,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Bayreuth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Germany 4 Institute of Inorganic Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' University of Cologne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Greinstrasse 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 50939 Cologne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Germany 5 Center for Advanced Radiation Sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Illinois 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' USA We applied single-crystal X-ray diffraction and Raman spectroscopy in a diamond anvil cell up to 36 GPa and first principles theoretical calculations to study the molecular dissociation of solid iodine at high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Unlike previously reported, we find that the familiar Cmce molecular phase transforms to a Cmc21 molecular structure at 16 GPa, and then to an incommensurate dynamically disordered Fmmm(00γ)s00 structure at 20 GPa, which can be viewed as a stepwise formation of polymeric zigzag chains of three iodine atoms following by the formation of the dynamically dissociated, incommensurately modulated i-Fmmm phase, and the truly monatomic Immm phase at higher pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Understanding pressure driven dissociation of simple diatomic molecules is important for a range of topics including materials behavior under extremes and composition of planetary interiors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Such transformations, which are commonly preceded or accompanied by metallization, occur at very high pressures in the crystals made of light molecules (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', H2 and F2), where such investigations are very challenging (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, molecular dissociation and band gap closure occurs at much lower pressures for heavier molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In fact, these phenomena in halogens Cl2, Br2, and I2 have been thoroughly investigated revealing a common behavior (3-5), where molecular dissociation is reported to occur in steps associated with a series of phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Iodine is investigated most extensively largely because it transforms to a metallic state at 14-24 GPa and experiences molecular dissociation at above 21 GPa (6-11)—conditions, which are easily accessible with the diamond anvil cell technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' At low pressures, iodine crystallizes in an orthorhombic Cmce structure (3) (phase I), which consists of flat layers, formed as associations of zigzag molecular chains (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Above 26 GPa, this structure transforms into a monatomic Immm lattice (phase II), where iodine atoms occupy the corner sites forming a metallic body-centered orthorhombic (BCO) single-atom unit cell (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  However, careful investigations (12) showed the existence of an intermediate phase (V) at 24-26 GPa, which can coexist with the Immm structure at 26-30 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This intermediate structure can be viewed as a face-centered orthorhombic lattice  2  based on the main diffraction peaks, but the occurrence of satellite reflections from both sides of the main peaks indicates the presence of an incommensurate modulation wave along the a-axis, which shifts the atoms along the b-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In this structure, there are three- or four- atom chains in the ab plane, with a continuous distribution of the nearest-neighbor interatomic distances covering a range of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='26 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Raman spectroscopy demonstrated the presence of a specific low-frequency soft mode for the modulated structures, called the amplitude mode (AM), which corresponds to transverse atomic vibrations and preserves the symmetry of the modulation wave (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' No Raman signal has been detected in phase II as expected from the Raman selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fragments of crystal structures of Cmce (I), Cmc21 (VI), and Fmmm(00γ)s00 (VII) phases determined in this work (Tables S1 and S2, of Supplemental Material (13)) projected to the bc(ac for Fmmm) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The atoms occupying different crystallographic sites are shown in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The atoms are represented by the displacement ellipsoids (at 75% probability) as they occur after the structural refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Arrows show the directions of shifts of the molecular layers that occur at the Cmce to Cmc21 transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fmmm(00γ)s00 structure presentation is approximate (small unit cell) (see detailed structural results in Table S2, Supplemental Material (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In the stepwise molecular dissociation scenario presented above, it remains puzzling if molecular phase I metallizes via band overlap and whether there is any indication of its instability toward the transformation into modulated phase V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Indeed, theoretical calculations predict such instability within Cmce structure due to a coupling of the bond charge density with a transverse optical phonon (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Raman measurements of phase I (5, 15) demonstrated the presence of 2Ag+2B3g in- plane stretching (S) and librational (L) modes, out of which Ag(L) mode turns over and becomes a soft mode above 13 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In the same pressure range, additional Raman modes appear that cannot be explained within the Raman selection rules of Cmce structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  In addition, Mössbauer experiments demonstrated an anomaly at 16 GPa suggesting a phase transformation (16), while X-  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='08 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='08 Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='725 Å  Cmce  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='97 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='14 Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='74 Å  Cmc21  Fmmm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='84 Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='80 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='19 Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='82 Å  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='826 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='21 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='208 Å  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='224 Å  a+ba+bcf3  ray absorption spectroscopy showed a hint of an increase in the interamolecular bond distance in the same pressure range (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Theoretical first-principles calculations (17) suggested that there is another molecular phase (phase I\uf0a2), which has C2/m symmetry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' it was proposed to coexist with Cmce phase I (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This phase has two types of I2 molecules in the unit cell with different intramolecular lengths offering an explanation to the extra Raman bands, which appear above 14 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Here we present the results of concomitant synchrotron single-crystal (SC) X-ray diffraction (XRD) and Raman measurements, which provide a detailed structural description of the multi- stage molecular dissociation of iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our experiments unequivocally identify intermediate phases - a Cmc21 molecular phase (hereafter I2-VI), and an incommensurate Fmmm(00γ)s00 phase (hereafter I2-VII) characterized by the loss of well-defined stable molecules, which provide key missing links between stable molecular I2-I and dynamically dissociated I2-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our first-principles theoretical calculations show that Cmc21 I2-VI phase is energetically competitive, dynamically stable and confirm the presence of the additional Raman bands as due the reduction in symmetry upon transition to the Cmc21 phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reconstructed reciprocal lattice planes of iodine at 15 GPa (upper panel) and at 17 GPa (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The observed diffraction spots from the sample have been indexed and used to determine the structure of Cmce (I2-I) and Cmc21 (I2-VI) phases, respectively (see details in Table S1 of Supplementary Materials (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The X-ray extinctions rules for these phases are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Cmce  hko:h,k=2n  0  2 01  2 20  Cmc21  hk0:h+k=2n  Q 510  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='10  00  400  510  5810  530  3 304  Up to 15 GPa SC XRD patterns confirm the previously reported Cmce structure (Table S1 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, SC XRD measurements at 17 GPa demonstrate a difference in the X-ray extinction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reflections (h k 0), with h=2n+1 and k = 2n+1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (3 1 0), (5 1 0) and others) appear in the diffraction patterns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' These weaker reflections cannot be observed in powder XRD patterns as they are much weaker than the main ones (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1%) and interfere with stronger reflections, so this symmetry change was previously overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' A new structure can be indexed with the Cmc21 space group, which represents a subgroup of the Cmce group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Hereafter, we call this phase as I2-VI (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The structural distortion in the Cmc21 structure is very subtle (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' It can be understood as a small shift of layers of collinear molecules along the b-axis, which diminishes the lattice symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reconstructed reciprocal lattice planes of iodine at 17, 19, and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Insets below show enlarged views of Bragg peaks marked by red rectangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Splitting of the zoomed in peak manifests the occurrence of incommensurate phase at 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  On further compression, above 20 GPa, yet another subtle change in symmetry occurs (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' At this point, a stable structural solution within Cmc21 space group is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' A sequence of diffraction spots from (0kl) lattice planes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 3) show a splitting of several reflections along the c* direction at 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa, manifesting the occurrence of an incommensurate lattice with a modulation vector of q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4837(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' An incommensurate phase has Fmmm(00γ)s00 structure, which  Okl  Okl  Okl  p3  p4  p5  Commensurate  Commensurate  Incommensurate  Cmc2,  Cmc2,  Fmmm(00g)s005  is isosymmetrical with phase V at higher pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The existence of this phase is consistent with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18), where this phase I2 was reported  at 16-23 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In contrast, we find no incommensurate structure below 20 GPa, where commensurate Cmc21 I2-VI is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Both commensurate Cmc21 and Cmce are the a×b×2c supercell subgroups of Fmmm(00γ)s00 with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='5 and various initial phases of modulation t0 (1/8 for Cmce and general for Cmc21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In incommensurate I2-VII phase, the nearest-neighbor interatomic distances are modulated (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S1 of Supplementary Information (13)) spreading a wide range, which covers typical intra-to- intermolecular I-I distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This phase consists of a dynamic mixture of molecular I2 and polymeric zigzag chains of three I atoms (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18)) in the ac-plane (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S2 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Raman spectroscopy data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Panel (a) shows a sequence of Raman spectra across the phase transitions on pressure increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The phase symmetries are labeled, and the traces are color coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Modes labeled as X and Y appear in Cmc21 phase and gain intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Panel (b) shows the measured here pressure dependencies of the Raman frequencies and compared to previous investigations (5, 15) (left panel) as well as theoretically calculated (right panel) in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our concomitant Raman measurements below 24 GPa demonstrate smooth changes with pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4), which are generally consistent with previous observations (5, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Above 16 GPa, where Cmc21 I2-VI is documented in XRD, new peaks appear, dubbed X and Y in the literature, with the frequencies below Ag(L) and above B3g(L) modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The appearance of these modes can be understood by the Raman selection rule modification at the Cmce→Cmc21 phase transition (Table S4 of Supplementary Information (13));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' the most prominent difference in Cmc21 phase VI is the relaxation of the rule of mutual exclusion for the Raman and IR active modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' According to group-theory predictions, all optical modes are Raman-active in phase VI, out of which the most intense 3A1+3B2 in-plane modes are expected to be observed in Raman spectra compared to 2Ag+2B3g modes of Cmce phase I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' We note that X and Y modes are coupled to Ag(L) and B3g(L)  Raman spectra  Raman Shift (cm 1)  50  100  150  200  250  Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' units)  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 GPa  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 GPa  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='5 GPa  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 GPa  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 GPa  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='6 GPa  Immm  i Fmmm (2)  Cmc21  Cmce  X  Y  i Fmmm (1)  (a)  (b)  250  Cmc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Cmcm  Cmca  Cmc2,  Cmce  i FmmmImmm  A  Ag(S)  200  B  Bzboundary  150  B,  B2  100  A1  Thiswork  50  AM  Olijnyketal,1994  Kumeetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=',2005  D  0  10  20  30  10  20  Pressure(GPa)  Pressure(GPa)6  originated modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This is evident from their frequency curves vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4(b)), which show characteristic avoided crossing dependencies and an exchange in intensities correlated with the frequency approaching as in the case of Fermi resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This behavior shows that the coupled modes are of the same symmetry, namely A1 and B1 modes of Cmc21 phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Thus, the appearance and behavior of the X and Y modes are well understood by the Cmce→Cmc21 phase transformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' these modes originate from IR active B1u+B2u modes of Cmce phase, which become Raman active in Cmc21 phase (Table S3 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' These modes correspond to translational motions of I2 molecules along the b and c axes, which couple with librational motions in phase VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Only subtle changes (mainly peak broadening) occur in the Raman spectra above 20 GPa, where our XRD data indicate a transition into an incommensurate Fmmm(00γ)s00 I2-VII phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The presence of vibrational modes characteristic for molecular phases is likely because this phase still consists of short lived I2 molecules demonstrating fluxional behavior, which is supported by MD simulations of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This behavior is consistent with the incommensurate nature of this phase, where molecular breakdown and recombination are associated with the modulation wave propagating through the crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  It is the instantaneous existence of molecules that leads to a Raman response similar to that of Cmce and Cmc21 phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' At 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa, the A1 mode originated from the X-mode becomes a dominant mode in the spectrum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' this mode is a soft one corresponding to librational motions of I2 molecules with the eigenvector, which captures the pathway to a monatomic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The observed here mode behavior is qualitatively consistent with the first- principles theoretical calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S3 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our first-principles theoretical calculations show that Cmc21 phase is energetically competitive with respect to Cmce and C2/m in the pressure range above 15 GPa , where Cmce phase is reported to transform in this work and Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (17, 18) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S4 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, within GGA-PBE approximation, Cmc21 phase is substantially more stable than the experimental Cmce ground-state phase below 15 GPa (see also Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (19)), which clearly contradicts the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This apparent inconsistency is removed if M06-L functional is used as the enthalpies of Cmce and Cmc21 phases become nearly degenerate in the whole pressure range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' All these phases are dynamically stable at 5-30 GPa as witnessed by calculations of the phonon dispersion curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S5 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  GGA-PBE calculations of the electronic band structure show that the molecular phases determined here are semiconducting with narrow band gaps decreasing with pressure, which are expected to close at 26 GPa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S6 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' At 24 GPa, XRD patterns and Raman spectra change abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' SC XRD performed in this work find a modulated structure (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S7, S8 and Table S3 of Supplementary Information (13)) as reported in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12) based on powder diffraction data (phase V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Unlike Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12), we find the structure of phase V to be Fmmm(00γ)s00 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fmm2(a00)0s0 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12)) in agreement with the more recent analysis (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' We also find that the modulation vector value decreases with pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S9) in agreement with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12, 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, we additionally find, based on SC XRD measurements on high-quality laser annealed crystals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S7 of Supplementary Information (13)), that there is a tiny atomic displacement along the c-axis (Table S3 of Supplementary  7  Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Overall, these results definitively establish the structure of a modulated phase V proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, our data do not support the tetragonal 5D modulated structure (I4/mmm(αα0)000s(−αα0)0000) proposed  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' As can be seen on the reconstructed precession image (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S7 of Supplementary Information (13)), there is only one modulation vector, which can describe all the satellite reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Raman spectra above 24 GPa do not show any modes of the lower pressure molecular phases I, VI, and VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Instead, a strong soft Raman mode appears at low frequencies, which has been previously identified as an AM (5) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' These observations confirm that the modulated phase V is incommensurate, where the Brillouin zone center vibrational modes of the parent phase are forbidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Above 26 GPa, two new Raman excitations, a narrow and a broad at nearly 150 cm- 1 appear and shift to higher frequencies with pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Monatomic (BCO) phase II coexists with phase V in this regime, identified in powder XRD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Above 30 GPa, the AM mode disappears, and XRD measurements show a single-phase BCO structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Based on our theoretical calculations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  S10), the Raman bands observed in this phase correspond well in frequency and pressure shift to the Brillouin zone boundary transverse acoustic modes near the R, W, and T points, where the corresponding dispersion curves are flat yielding sharp maxima in the phonon density of states at 4 THz (134 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The mechanism of their Raman activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', defect induced) is not clear at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our results show the presence of an intermediate molecular phase VI and fluxional mixed molecular-zigzag incommensurate phase VII between a low-pressure molecular phase I and a modulated and dynamically dissociated phase V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Phase VI has Cmc21 orthorhombic symmetry, which can be described as a small distortion of Cmce low-pressure phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This new phase can be viewed as a slightly dissociated (with longer intramolecular distance (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1, 5) molecular phase, where intermolecular coupling has been modified to approach zigzag chains of three iodine atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Previous experiments and theoretical calculations proposed a monoclinic C2/m structure in this regime (17), but our XRD experiments clearly rule out this phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In a recent work, it has been proposed that phase VI is a modulated incommensurate phase (18);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' however, our SC XRD data clearly show that no satellite reflections are observed below 20 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' In a qualitative agreement with the results of previous works (12, 18, 20), we find the structure of phase V at 24-30 GPa to be modulated and incommensurate (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  S7-S9 of Supplementary Information (13));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' however, the symmetry and the dimensionality are different of that proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our new phase identification is consistent with Mössbauer experiments (16), which proposed a change in symmetry based on discontinuous changes in the electric field gradient (EFG) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' We theoretically computed these parameters in Cmce and Cmc21 phases and found discontinuous changes in EFG, which are qualitatively consistent with the Mössbauer observations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S11 of Supplementary Information (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The Fmmm(00γ)s00 incommensurate mixed molecular-zigzag phase VII differs from incommensurate phase V in that the latter one can be viewed as totally dissociated, consisting of I4 and I5 zigzag atomic chains (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S1), while phase VII still has a large disparity between intra and intermolecular bond lengths, which results in formation of zigzag I3 chains (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18)), while some I2 molecules still remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The modulation vector of phase VII is substantially larger  8  than that of phase V (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The interatomic distances in both modulated phases V and VII greatly vary (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 5, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S1), while the averaged distance goes down with pressure due to the volume compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The determined here phase sequence Commensurate (Cmce)→ Commensurate (Cmc21)→ Incommensurate Fmmm(00γ)s00 (γ ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='5) → Incommensurate Fmmm(00γ)s00 (γ ~ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='25) → Immm shows a seamless pathway for molecular dissociation of I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Volume compression of iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The inset shows the nearest interatomic distance vs pressure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The results of this work (filled symbols, the uncertainty is smaller than the symbol size) are compared to previously reported data (open symbols and crosses) from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (4, 8, 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The solid line extrapolated to the transition into the modulated Fmmm phase is a Vinet fit to our data (V0=340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9(2) Ǻ3, K0=8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1(5), and K0ʹ=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0(3), where K0 and K0ʹ are the bulk modulus and its pressure derivative at P=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The vertical bars in inset show the spread of the nearest I-I distances in incommensurate phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Our XRD data also address the volume change at metallization and molecular dissociation transition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' There is no measurable volume change at the Cmce → Cmc21 (VI)→ Fmmm(00γ)s00 (VII) phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' However, there is a volume collapse of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8% that occurs at the isosymmetrical VII-V transition (both have Fmmm(00γ)s00 symmetry) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' This is smaller than that theoretically computed here using DFT calculations using GGA-PBE functional (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' It has been previously suggested that metallization occurs in a molecular state thus preceding dissociation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Based on our XRD results and theoretical calculations of the electronic structure (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S3, S4), we speculate that the Cmc21 → Fmmm(00γ)s00 (VII) phase transitions at 20 GPa can be directly connected to metallization reported at 14-24 GPa (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  In this case, there is no volume collapse associated with metallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Molecular dissociation occurs stepwise from truly molecular Cmce and Cmc21 to dynamically disordered Fmmm(00γ)s00 and dynamically dissociated i-Fmmm phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' It would be of great interest to investigate if these results are applicable to Br2 and Cl2, which were reported to have a similar structural behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa)  0  10  20  30  Volume (Å3)  200  250  300  Cmce (I)  Cmc21(VI)  Fmmm 1 (V)  Immm (II)  0  10  20  30  Shortest Interatomic distance (Å)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='80  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='85  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='90  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='95  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='00  Cmce  Cmc21  Buontempo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1998  Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', 2021  This work  Takemura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', 1982  This work, calculation  This work, experiment  Fmmm 2 (VII)  Fmmm 2  Fmmm 1  Immm  9  Parts of this research were carried out at the GeoSoilEnviroCARS (The University of Chicago, Sector 13), Advanced Photon Source (Argonne National Laboratory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' GeoSoilEnviroCARS is supported by the National Science Foundation—Earth Sciences (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' EAR-1634415).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Use of the GSECARS Raman System was supported by the NSF MRI Proposal (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' EAR-1531583).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The Advanced Photon Source is a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' DE- AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' We acknowledge support by the Army Research Office accomplished under the Cooperative Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' W911NF-19-2-0172 and the Carnegie Institution of Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' acknowledges the support of Deutsche Forschungsgemeinschaft (DFG Emmy-Noether project BY112/2-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' acknowledges financial support from the program ‘Promotion of Equal Opportunities for Women in Research and Teaching’ funded by the Free State of Bavaria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Goncharov, Phase diagram of hydrogen at extreme pressures and temperatures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' updated through 2019 (Review article).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Low Temperature Physics 46, 97-103 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Gregoryanz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', Everything you always wanted to know about metallic hydrogen but were afraid to ask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Matter and Radiation at Extremes 5, 038101 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujihisa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujii, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shimomura, Structural aspects of dense solid halogens under high pressure studied by x-ray diffraction—Molecular dissociation and metallization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Journal of Physics and Chemistry of Solids 56, 1439-1444 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', Halogen molecular modifications at high pressure: the case of iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Chemistry Chemical Physics 23, 3321-3326 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Kume, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Hiraoka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ohya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Sasaki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shimizu, High Pressure Raman Study of Bromine and Iodine: Soft Phonon in the Incommensurate Phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review Letters 94, 065506 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Lynch, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Drickamer, Effect of Pressure on the Lattice Parameters of Iodine, Stannic Iodide, and p‐Di‐iodobenzene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The Journal of Chemical Physics 45, 1020-1026 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Minomura, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shimomura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujii, Observation of Molecular Dissociation of Iodine at High Pressure by X-Ray Diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review Letters 45, 1881-1884 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Minomura, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shimomura, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujii, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Axe, Structural aspects of solid iodine associated with metallization and molecular dissociation under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review B 26, 998-1004 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ishikawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', Metallization of solid iodine in phase I: X-ray diffraction measurements, electrical resistance measurements, and ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' High Pressure Research 33, 186- 190 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Balchan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Drickamer, Effect of Pressure on the Resistance of Iodine and Selenium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The Journal of Chemical Physics 34, 1948-1949 (1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Sakai, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Tsuji, Electrical Properties of High-Pressure Metallic Modification of Iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Journal of the Physical Society of Japan 51, 1811-1816 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Kyoko, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Hiroshi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Mitsuko, Modulated structure of solid iodine during its molecular dissociation under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Nature 423, 971-974 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' See Supplemental Material at http://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='org/supplemental/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' xxx This pdf file contains Materials and Methods, Tables S1-S4, and Bibliography which includes Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' [22–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Luty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Raich, Molecular to atomic transformation in solid iodine under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Canadian Journal of Chemistry 66, 812-818 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  10  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Olijnyk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Wokaun, High-pressure studies of solid iodine by Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review B 50, 712-716 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Pasternak, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Farrell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Taylor, Metallization and structural transformation of iodine under pressure: A microscopic view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review Letters 58, 575-578 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', A new phase of solid iodine with different molecular covalent bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Proceedings of the National Academy of Sciences 105, 4999-5001 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujihisa, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Onoda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Gotoh, Two intermediate incommensurate phases in the molecular dissociation process of solid iodine under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review Research 3, 033174 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' George, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reimann, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Deringer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Bredow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Dronskowski, On the DFT Ground State of Crystalline Bromine and Iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' ChemPhysChem 16, 728-732 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Takemura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Sato, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fujihisa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Onoda, Structural phase transitions in iodine under high pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Zeitschrift für Kristallographie - Crystalline Materials 219, 749-754 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Buontempo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', Anomalous Bond Length Expansion in Liquid Iodine at High Pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Physical Review Letters 80, 1912-1915 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  11  Supplementary Information Structural evolution of iodine on approach to the monatomic state Elena Bykova1,3, Iskander G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Batyrev2, Maxim Bykov1,4, Eric Edmund1, Stella Chariton5, Vitali B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Prakapenka5, and Alexander F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Goncharov1,4 1 Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015, USA 2 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Army Research Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' RDRLWML-B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Aberdeen Proving Ground,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Maryland 21005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' United States 3 Bayerisches Geoinstitut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' University of Bayreuth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Universitätsstrasse 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D-95447,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Bayreuth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Germany 4 Institute of Inorganic Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' University of Cologne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Greinstrasse 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 50939 Cologne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Germany 5 Center for Advanced Radiation Sources,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Illinois 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' USA  This pdf file contains Materials and Methods,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Supplemental Figures S1-S11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Tables S1-S4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' and Bibliography with References [22-27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Corresponding author:  agoncharov@carnegiescience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='edu  12  Materials and methods Experiments The experimental procedure included concomitant SC XRD and Raman spectroscopy measurements (see details in the Supplementary Information) at 10-36 GPa in the Sector 13 (GSECARS) of the Advanced Photon Source, Argonne National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Samples in the form of flakes were placed in a cavity of a diamond anvil cell formed in a central hole made in a preindented rhenium gasket, sealed to avoid sublimation, and then loaded by compressing Ne gas to 160 MPa at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ruby served as an optical pressure gauge, while the equation of state of Ne 22 has been used to cross check pressure during XRD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Compressed Ne serving as a pressure medium provided quasi-hydrostatic conditions in the high-pressure cavity, as witnessed by SC XRD data, which were of sufficiently high quality up to at least 27 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' At this pressure we performed laser annealing of the sample up to 1800 K to improve the quality of SC XRD data as will be presented below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Additional Raman measurements have been collected at Earth and Planets Laboratory of the Carnegie Institution for Science, before and after synchrotron XRD experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Theoretical calculations First-principles theoretical calculations have been performed in Cmce, Cmc21, C2/m, Immm phases at selected pressures between 0 and 30 GPa, where these structures were optimized using norm- conserving pseudopotentials, GGA-PBE functional, and Grimme2 dispersion corrections23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Monkhorst-Pack grid size for k-points sampling of the Brillouin Zone (BZ) is 5x6x5 for all structures except I2-II (Immm) at 30 GPa, where 6x7x6 sampling was used 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The phonon dispersion and phonon frequency calculations were performed using a finite displacement method implemented in the CASTEP code 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The Raman spectra were calculated using the formalism presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Since application of GGA-PBE functional incorrectly predicts that Cmce phase is thermodynamically and dynamically unstable at 0 GPa 19, we performed calculations using Minnesota 2006 local functional 27 (M06-L) in Cmce and Cmc21 phases up to 30 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The I-I nearest-neighbor interatomic distances as a function of the modulation vector of Fmmm(00γ)s00 I2-VII phase determined in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  p05  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='30  I I distances (A  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='90  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0  t14  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fragments of crystal structures of Fmmm(00γ)s00 I2-VII and I2-V phases determined in this work (Tables S2 and S3, respectively) projected to the ac plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  I2 VII  I2 V  d<2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9 Ǻ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9 Ǻ <d<3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='01 Ǻ  d<2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='953 Ǻ  15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Theoretically computed Raman frequencies vs pressure determined here from the first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The irreducible representations are changed to be consistent with the axes name convention used in experiment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa)  10  12  14  16  18  20  22  Raman Frequency (cm 1)  0  50  100  150  200  Ag  B3g  B1  Ag  B3g  A1  A1  B1  B1  A1  Cmca  Cmc21  16  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Theoretically computed enthalpies of the candidate phases vs pressure determined here from the first principles plotted with respect to results for Cmce phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Solid lines are the results computed using GGA-PBE functional, while a dashed line for Cmc21 phase is obtained using Minnesota 2006 local functional (M06-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The computed equilibrium phases are dynamically stable in the presented here pressure range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa)  0  5  10  15  20  25  30  Enthalpy / atom (meV) 10 5  0  5  10  Cmce  C2/m  Immm  Cmc21  Cmc21 (M06L)  Cmcm  17  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Electronic band structure and density of electronic states of phases I (Cmce) and VI (Cmc21) at 20 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Cmc21, 20 GPa  Cmce, 20 GPa  CASTEPBandStructure CASTEP Partial Density Band gap is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='223 eV of States 21 21 18 18 15 15 12 12 (eV) (eV) Energy ( hergy -9 -12 -12 -15 -15 5 5 X U R 0 2 + 6 8 DensityofStates(electrons/ev） Band energy SumCASTEPPartial CASTEP Band Structure Density of Band gap is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='221 eV States 20 20 Energy (eV) Energy (eV) 10 10 -10 5 Density of States(electr - Band energy Sum18  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The electronic band gap calculated as a function of pressure in Cmce and Cmc21 structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa)  0  5  10  15  20  25  Bandgap (eV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0  Cmc21  Cmca  19  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reconstructed reciprocal lattice planes of iodine phase V at 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The right panel shows an enlarged view of a fine structure near the (402) reflection of a parent Fmmm phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Satellite reflections are indexed in terms of the modulation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The details of the structural refinement are presented in Table S3 of Supplementary Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  (hol)  2g  (402) q 2q20  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Reconstructed precession image of iodine phase V at 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The right panel shows an enlarged view of satellite reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Satellite reflections are indexed in terms of the modulation vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The details of the structural refinement are presented in Table S3 of Supplementary Materials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  3g  29  (hk 1) in 14/mmm setting21  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' The modulation vector value as a function of pressure determined here and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa) 20 22 24 26 28 Modulation vector value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='5 Takemura et 2003, Phase V This work, Phase V This work, Phase VII  22  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Phonon dispersion curves and density of phonon states of phase II (Immm) at 30 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  CASTEP Density of Phonon CASTEP Phonon Dispersion States 10 10 8 (ZHL) (ZHL) Frequency（ Freguency -2 2 R X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='3 Density of Phonon States (1/THz) - Phonon dispersion23  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Electric field gradient (EFG) parameters calculated for phases Cmce (phase I) and Cmc21 (phase VI) compared to experimental data of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Pressure (GPa)  8  10  12  14  16  18  20  22  Anysotropy parameter (\uf068)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0  Pressure (GPa)  8  10  12  14  16  18  20  22  EFG 200 150 100 50  0  50  100  150  Pasternak, LP  Cmca  Cmc21  Cmc21  Pasternak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', HP1  Pasternak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', LP  Pasternak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=', HP1  Cmc21  Cmca  24  Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  Crystal structure details of commensurate phases of iodine Details of crystal structure refinements for iodine phases at high pressures* Pressure, GPa 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='90 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 Space group Cmce Cmce Cmc21 Cmc21 Immm Phase iodine-I iodine-I iodine-VI iodine-VI iodine-II Pressure step p1 p2 p3 p4 p8 a (Å) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='163(4) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='978(4) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='908(5) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='829(6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9077(6) b (Å) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1435(3) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0396(4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0055(3) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9773(6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0333(12) c (Å) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2864(8) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1487(9) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0992(11) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0739(13) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='226(7) Z 2 2 2 2 2 V (Å3) 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='15(16) 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='94(15) 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='3(2) 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4(2) 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='10(7) ρcalc (g/cm3) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='109 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='83 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='014 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='143 μ/mm-1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='804 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='519 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='204 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='563 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='755 2Θmin for data collection (°) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='624 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='549 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='158 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='575 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='508 2Θmax for data collection (°) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='627 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='055 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='001 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='776 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='102 Completeness to d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 Å 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='468  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='25 Reflections collected 259 165 243 327 21 Independent reflections 136 86 175 213 16 Independent reflections [I > 2σ(I)] 132 80 171 213 15 Refined parameters 7 7 13 13 4 Rint(F2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='083 R(σ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0316 R1 [I > 2σ(I)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0836 wR2 [I > 2σ(I)] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1911 R1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0826 wR2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1907 Goodness of fit on F2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='201 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='139 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='164 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='308 Δρmax(e / Å3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='774 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='068 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='463 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='874 Δρmin(e / Å3) -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='579 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='825 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='096 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='807 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='468 x(I 1) 0 0 0 0 0 y(I 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='18376(9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='19087(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4299(2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4306(3) 0 z(I 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='62197(4) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='62278(3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='26446(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='26390(15) 0 Ueq(I 1) (Å2)* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0165(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0099(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0080(9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0086(9) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='011(5) x(I 2)  0  0  25  y(I 2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0387(3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0291(6)  z(I 2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='51173(10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='51229(15)  Ueq(I 2) (Å2)*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0096(9)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0114(10)  Ueq is defined as one third of the trace of the orthogonalized Uij tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='  26  Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Crystal structure details of incommensurate Fmmm(00γ)s00 phase VII of iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Crystal data Chemical formula I Mr 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9 Crystal system, space group Orthorhombic, Fmmm(00γ)s00† Pressure step P05 Pressure (GPa) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='8 GPa Wave vectors q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4837(2)c* a, b, c (Å) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9536(14), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='735(18), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='530(2) V (Å3) 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='7(3) Z 4 Radiation type X-ray, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2952 Å µ (mm−1) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='191 Data collection Diffractometer GSECARS 13IDD No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of main reflections: All/independent/observed independent 75/35/31 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of satellite reflections m = 1: All/independent/observed independent 131/56/56 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of satellite reflections m = 2: All/independent/observed independent 152/64/58 Rint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0138 (sin θ/λ)max (Å−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='867 Refinement R(obs)main + sattelites / wR (all) main + satellites 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0360/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1141 R(obs)main / wR (all) main 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0344/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0934 R(obs)1st order satellites / wR (all) 1st order satellites 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0305/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0812 R(obs)2nd order satellites / wR (all)2nd order satellites 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0513/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='1538 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of reflections 145 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of parameters 6 Δρmax, Δρmin (e Å−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='65, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='13 Crystal structure  I (x,y,z)  (0, 0, 0)  𝐴𝑥  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0749(2)  𝐴𝑧  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='00286(9)  † Symmetry operations: (1) x1, x2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (2) −x1, −x2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (3) −x1, x2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (4) x1, −x2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (5) −x1, −x2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (6) x1, x2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (7) x1, −x2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (8) −x1, x2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (9) x1, x2+1/2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (10) −x1,  27  −x2+1/2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (11) −x1, x2+1/2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12) x1, −x2+1/2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (13) −x1, −x2+1/2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (14) x1, x2+1/2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (15) x1, −x2+1/2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (16) −x1, x2+1/2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (17) x1+1/2, x2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18) −x1+1/2, −x2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (19) −x1+1/2, x2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (20) x1+1/2, −x2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (21) −x1+1/2, −x2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (22) x1+1/2, x2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (23) x1+1/2, −x2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (24) −x1+1/2, x2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (25) x1+1/2, x2+1/2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (26) −x1+1/2, −x2+1/2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (27) −x1+1/2, x2+1/2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (28) x1+1/2, −x2+1/2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (29) −x1+1/2, −x2+1/2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (30) x1+1/2, x2+1/2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (31) x1+1/2, −x2+1/2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (32) −x1+1/2, x2+1/2, x3, x4+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Displacive modulations of I atom (𝑢𝑖(𝑥4 ̅̅̅) for i = x, y, z) are described by a truncated Fourier series, which due to the symmetry restrictions take the following form: 𝑢𝑥(𝑥4 ̅̅̅) = 𝐴𝑥1 sin(2𝜋𝑥4 ̅̅̅), 𝑢𝑧(𝑥4 ̅̅̅)= 𝐴𝑧2 sin(4𝜋𝑥4 ̅̅̅)  28  Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Crystal structure details of incommensurate Fmmm(00γ)s00 phase V of iodine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Crystal data Chemical formula I I Mr 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='9 Crystal system, space group Orthorhombic, Fmmm(00γ)s00† Orthorhombic, Fmmm(00γ)s00† Pressure step p6 p7 Pressure (GPa) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4 Wave vectors q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='274(2)c* q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2381(6)c* a, b, c (Å) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2098 (13), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='539 (10), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2407 (13) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2018 (8), 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='434 (4), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2224 (8) V (Å3) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='88 (18) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='41 (8) Z 4 Radiation type X-ray, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2952 Å µ (mm−1) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='82 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='25 Data collection Diffractometer GSECARS 13IDD GSECARS 13IDD No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of main reflections All/independent/observed independent 68/31/28 56/28/28 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of satellite reflections m = 1 All/independent/observed independent 130/55/55 123/60/60 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of satellite reflections m = 2 All/independent/observed independent - 132/66/57 Rint 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='012 (sin θ/λ)max (Å−1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='876 Refinement RF(obs)main + sattelites / wRF (all) main + satellites 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0347/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0409/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0515 RF(obs)main / wRF (all) main 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0257/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0400/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0518 RF(obs)1st order satellites / wRF (all) 1st order satellites 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0438/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0394/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0486 RF(obs)2nd order satellites / wRF (all)2nd order satellites - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0468/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0555 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of reflections 86 154 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' of parameters 5 6 Δρmax, Δρmin (e Å−3) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='84, −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='32, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='05 Crystal structure  I (x,y,z)  (0, 0, 0)  (0, 0, 0)  𝐴𝑥  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0576(4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='0588(2)  𝐴𝑧  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='00072(8)  29  † Symmetry operations: (1) x1, x2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (2) −x1, −x2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (3) −x1, x2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (4) x1, −x2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (5) −x1, −x2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (6) x1, x2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (7) x1, −x2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (8) −x1, x2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (9) x1, x2+1/2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (10) −x1, −x2+1/2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (11) −x1, x2+1/2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (12) x1, −x2+1/2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (13) −x1, −x2+1/2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (14) x1, x2+1/2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (15) x1, −x2+1/2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (16) −x1, x2+1/2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (17) x1+1/2, x2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (18) −x1+1/2, −x2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (19) −x1+1/2, x2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (20) x1+1/2, −x2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (21) −x1+1/2, −x2, −x3+1/2, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (22) x1+1/2, x2, −x3+1/2, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (23) x1+1/2, −x2, x3+1/2, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (24) −x1+1/2, x2, x3+1/2, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (25) x1+1/2, x2+1/2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (26) −x1+1/2, −x2+1/2, x3, x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (27) −x1+1/2, x2+1/2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (28) x1+1/2, −x2+1/2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (29) −x1+1/2, −x2+1/2, −x3, −x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (30) x1+1/2, x2+1/2, −x3, −x4+1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (31) x1+1/2, −x2+1/2, x3, x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' (32) −x1+1/2, x2+1/2, x3, x4+1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Displacive modulations of I atom (𝑢𝑖(𝑥4 ̅̅̅) for i = x, y, z) are described by a truncated Fourier series, which due to the symmetry restrictions take the following form: 𝑢𝑥(𝑥4 ̅̅̅) = 𝐴𝑥1 sin(2𝜋𝑥4 ̅̅̅), 𝑢𝑧(𝑥4 ̅̅̅)= 𝐴𝑧2 sin(4𝜋𝑥4 ̅̅̅)  30  Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Vibrational modes and activity of Cmce and Cmc21 crystal structures Space & point group Cmce #64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='(D2h) Cmc21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='#36 (C2v) Cmcm #63 (D2h) Site symmetry 8f  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='(CS) 4a +4a  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='(CS) 4c (C2v) +   ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='4a (C2h) Acoustic modes B1u+B2u+B3u A1+B1+B2 B1u+B2u+B3u Optical modes Modes Activity Modes Activity Modes Activity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Au  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='B1u+B2u+B3u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='B1g+B2g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2Ag+2B3g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Silent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='IR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman (a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman (bc)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='A2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='A1+B2+B1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='A2+B1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='2A1+2B2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='IR & Raman  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman & IR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman & IR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Au  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='3B1u+3B2u+2B3u  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='(4ª)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='none  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='(4c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Ag+ B1g+B3g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Silent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='IR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='Raman  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content='References 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Fei, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Ricolleau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Frank, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Mibe, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Shen and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Prakapenka, Proceedings of the National Academy of Sciences 104 (22), 9182-9186 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Hamann, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Schlüter and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Chiang, Physical Review Letters 43 (20), 1494-1497 (1979).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Monkhorst and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Pack, Physical Review B 13 (12), 5188-5192 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Refson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Tulip and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Clark, Physical Review B 73 (15), 155114 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Porezag and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Pederson, Physical Review B 54 (11), 7830-7836 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Zhao and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
+page_content=' Truhlar, The Journal of Chemical Physics 125 (19), 194101 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE5T4oBgHgl3EQfjg_f/content/2301.05657v1.pdf'}
diff --git a/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/2301.05669v1.pdf.txt b/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/2301.05669v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..90982f2f09594a4d5f36ed2b26f2ef8c8ab13268
--- /dev/null
+++ b/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/2301.05669v1.pdf.txt
@@ -0,0 +1,334 @@
+Mem. S.A.It. Vol. 75, 282
+© SAIt 2022
+Memorie della
+Mapping the Universe with slitless spectroscopy
+P. Monaco1,2,3,4, on behalf of the Euclid Consortium
+1 Universit`a di Trieste, Dipartimento di Fisica, Via Tiepolo 11, I-34131 Trieste, Italy
+2 Istituto Nazionale di Astrofisica – OATs, Via Tiepolo 11, I-34131 Trieste, Italy
+3 Istituto Nazionale di Fisica Nucleare Trieste, via Valerio 2, I-34127 Trieste
+4 Institute for the Fundamental Physics of the Universe, via Beirut 2, I-34151 Trieste
+e-mail: pierluigi.monaco@inaf.it
+Received: Day Month Year; Accepted: Day Month Year
+Abstract. Euclid will survey most of the accessible extragalactic sky with imaging and
+slitless spectroscopy observations, creating a unique spectroscopic catalog of galaxies with
+Hα line in emission that will map the Universe from z = 0.9 to 1.8. With low expected
+statistical errors, the error budget will likely be dominated by systematic errors related to
+uncertainties in the data and modelling. I will discuss the strategy that has been proposed
+to mitigate the expected systematic effects and propagate the uncertainty of mitigation to
+cosmological parameter errobars.
+Key words. Cosmology: observations – Large-scale structure of the Universe –
+Cosmological parameters
+1. Introduction
+Observations
+of
+the
+Cosmic
+Microwave
+Background
+(CMB,
+Planck
+Collaboration
+2018) have provided percent-accurate con-
+straints to cosmological parameters, strength-
+ening the case for a 6-parameter flat ΛCDM
+cosmological model; this is consistent with
+most available evidence on large scales, but
+at the cost of leaving 95% of the present
+mass-energy
+budget
+unexplained.
+In
+fact,
+most matter today is thought to be in the
+form of an unknown collisionless particle,
+and most energy today is thought to be in a
+dark energy component that is accelerating the
+Universe expansion, represented by a positive
+cosmological constant.
+To shed light on the dark sector, it is crucial
+to map the Universe at lower redshift, when
+dark energy becomes dominant. The European
+Space Agency has promoted the Euclid mis-
+sion (Laureijs et al. 2011), an optical and
+near-infrared space telescope that will survey
+the sky with a visual imager (VIS), optimised
+for galaxy lensing, and a near-infrared imager
+and spectrograph (NISP), optimised for galaxy
+clustering.
+2. Galaxy Clustering with slitless
+spectroscopy
+Spectroscopic observations from space are
+complicated by the impossibility to use screens
+with suitably pierced holes or slits, as custom-
+ary from the ground. While JWST has im-
+plemented a novel micro-mirror technology
+that provides a way to select what part of the
+image is to be dispersed by a grism, Euclid
+arXiv:2301.05669v1  [astro-ph.CO]  13 Jan 2023
+
+Monaco: Euclid’s slitless spectroscopy
+283
+will adopt a straight slitless spectroscopy strat-
+egy, already experimented with Hubble Space
+Telescope (Bagley et al. 2020).This means that
+each source will produce a straight track on
+the NISP detector, as showed in the simulation
+in Fig. 1. These tracks will be separated and
+used to produce 1D spectra with a resolution
+of R = 380. This observing strategy will allow
+us to detect emission-line galaxies (ELGs) and
+measure their redshift, provided that the emis-
+sion line is correctly recognised. The grism
+will be sensitive to wavelengths in the range
+λ ∈ [1.25, 1.85] µm, so the most abundant de-
+tectable ELGs are expected to be Hα emitters
+in the redshift range z ∈ [0.9, 1.8]. Simulations
+show that the probability of detecting an ELG
+drops for line fluxes below 2 × 10−16 erg s−1
+cm−2, the nominal line-flux limit of the Euclid
+Wide Survey.
+The obvious drawback of this strategy, the
+need to deblend all the sources in a field, is
+balanced by the ability to perform a matter-
+of-fact blind search of emission-line galaxies.
+Indeed, spectra will be extracted for any source
+detected in the photometric observations, that
+we may think as limited to HE < 24, and
+the number of Hα ELGs associated to fainter
+sources has been demonstrated to be negligible
+in Bagley et al. (2020). At the same time, al-
+though we expect the success rate of deblend-
+ing to be a function of surface density of all
+sources, the sample will be free of fiber colli-
+sion bias (e.g., Bianchi et al. 2018).
+3. Chasing systematic effects
+The Observational Systematics Work Package
+of the Galaxy Clustering Science Working
+Group has surveyed the whole pipeline, from
+raw data to the measure of galaxy clustering
+that is provided to the likelihood. Using the for-
+malism of Monaco, Di Dio & Sefusatti (2019),
+we have classified the possible systematic ef-
+fects as follows: (i) modulations of the effec-
+tive flux limit of the sample, due either to in-
+strumental issues (e.g. the tiling of the various
+dithers will produce an inhomogeneous expo-
+sure time map, while straylight from nearby
+bright stars will modulate the noise) or to astro-
+physical foregrounds (e.g. zodiacal light will
+add to the background noise, while Milky Way
+extinction will decrease the signal); (ii) redshift
+errors due to line misidentifications; (iii) red-
+shift errors due to noise fluctuations being in-
+terpreted as lines in an overall undetected spec-
+trum, thus creating ‘noise interlopers’.
+The first class of systematic effects will be
+mitigated by suitably constructing a random
+catalog. To get rid of the angular footprint of a
+survey, clustering estimators usually compare
+the density field measured with the data sam-
+ple with that from a random sample that covers
+the same area and is unclustered on the sky; its
+number density is usually taken to be 50 times
+that of the data sample (fitted by a model in
+order not to erase some radial modes), to min-
+imise the amount of extra shot noise introduced
+by the random. We will construct the random
+by forward-modeling the completeness and pu-
+rity of the spectroscopic sample, thus creating
+what we call a visibility mask. This will be
+done by taking profit of the Euclid Deep Field,
+where 50 deg2 of the sky will be surveyed ten
+times with various orientations of the grism,
+and with 40 more pointings with a ‘blue grism’
+that is sensitive in the range λ ∈ [0.92, 1.25]
+µm; from it, we will extract a bona fide sam-
+ple, pure at ∼ 99% level, of the ELGs that
+can be seen in the EWS. These galaxies will
+be used to create a parent random catalog that
+is unclustered on the sky, whose objects have
+the same physical properties of the target sam-
+ple; these random galaxies will be injected in
+the EWS NISP images, and processed to deter-
+mine their probability of detection. This way
+the space density of the selected random will
+be modulated on the sky by systematics in the
+same way as the data sample.
+While the third class of systematic effects,
+the noise interlopers, can be modeled by suit-
+ably adding a class of contaminants to the ran-
+dom catalog, the second class, line misidenti-
+fications, is more subtle to address (Addison
+et al. 2019). With a typically steep luminosity
+function of sources, most objects in a catalog
+are around the detection limit, where a single
+emission line is typically detected. If no further
+information is used, the contamination level
+is expected to be around ∼ 10–20 %, mostly
+coming from [Oiii] emitters at higher redshift.
+
+284
+Monaco: Euclid’s slitless spectroscopy
+Fig. 1. Simulations of a NISP exposure, where each source produces a stripe along the dispersion imprinted
+by the grism. Circles denote the positions of the Hα lines. Credit: Ben Granett, OU-SIM an OU-SIR teams.
+These galaxies will be moved from their red-
+shift to the one corresponding to Hα, carrying
+with them their rescaled clustering signal, so
+the measured two-point function ξ(r) will be
+the weighted sum of the target one (1− f)2ξtarget
+and the contaminant one f 2ξinterloper, where f
+is the fraction of contaminants in the sam-
+ple. This contamination can be mitigated at the
+likelihood level, comparing the measurements
+with a weighted sum of predictions relative to
+the target sample and to the significant contam-
+inants, with fi fractions treated as nuisance pa-
+rameters subject to a tight prior coming from
+measurement of the Deep Field.
+4. Propagating the uncertainty in the
+mitigation
+This mitigation strategy will anyway leave
+residuals that contaminate the sample. Every
+step in the modeling of the visibility mask of
+the EWS has an associated uncertainty that
+must be propagated to parameter errorbars.
+The most effective way to achieve this is to
+construct a set of simulated mock galaxy cat-
+alogs and process them in the same way as the
+parent random catalog described above: inject
+galaxies in the images and compute their prob-
+ability of being detected. However, this process
+should not be performed using our best knowl-
+edge of the visibility mask but a modulation of
+it, obtained by perturbing every single step in
+the pipeline, sampling its estimated error PDF.
+As an example, detection probability will de-
+pend on the measured noise level, and we will
+use the best-fit value of the noise to create the
+random, and a value drawn from its PDF for
+applying the visibility mask to mock galaxy
+catalogs.
+This approach will provide a brute-force
+numerical estimate of the covariance matrix
+that will include both cosmic covariance and
+the uncertainty in the mitigation of system-
+atic effects. Proper sampling of the matrix re-
+quires thousands of mocks; in this moment the
+Galaxy Clustering Science Working Group is
+preparing 3500 simulations of the Euclid sky.
+N-body codes are simply too expensive to ad-
+dress this massive production, so we will re-
+sort to approximate methods (Monaco 2016). I
+am presently working to prepare such a large
+set of simulations using the PINOCCHIO code
+(Monaco, Theuns & Taffoni 2002; Munari et
+al. 2017), based on Lagrangian Perturbation
+Theory. Fig. 2 shows one of these ligh-
+cones, reporting dark matter halos (with Mh >
+
+2000
+1750
+1500
+1250
+1000
+750
+500
+250
+0
+0
+250
+500
+750
+1000
+1250
+1500
+1750
+2000Monaco: Euclid’s slitless spectroscopy
+285
+Fig. 2. Lightcone generated with PINOCCHIO, see text for explanations.
+1012 M⊙, one in 500) in comoving coordinates,
+in a slice that cuts through the survey volume;
+the catalogs cover half of the sky, with the ex-
+ception of unobserved low Galactic latitudes,
+and start at z = 4. The red lines mark the
+box size (3380 h−1 Mpc), that is tiled to cover
+the survey volume. When ready, this will be
+the largest set of cosmological simulations ever
+produced.
+Acknowledgements. P.M. thanks L. Guzzo, W.
+Percival, Y. Wang, C. Scarlata, B. Granett, M.
+Moresco, S. De La Torre and the members of
+the Observational Systematics Work Package for
+many discussions. The simulation used for Fig.
+1 was produced by the Euclid Science Ground
+Segment, in particular Operational Units OU-SIM
+and OU-SIR. The Euclid Consortium acknowledges
+the European Space Agency and a number of
+agencies and institutes that have supported the
+development of Euclid, in particular the Academy
+of Finland, the Agenzia Spaziale Italiana, the
+Belgian
+Science
+Policy,
+the
+Canadian
+Euclid
+Consortium, the French Centre National d’Etudes
+Spatiales, the Deutsches Zentrum f¨ur Luft- und
+Raumfahrt, the Danish Space Research Institute,
+the Fundac¸˜ao para a Ciˆencia e a Tecnologia,
+the
+Ministerio
+de
+Ciencia
+e
+Innovaci´on,
+the
+National Aeronautics and Space Administration,
+the
+National
+Astronomical
+Observatory
+of
+Japan, the Netherlandse Onderzoekschool Voor
+Astronomie, the Norwegian Space Agency, the
+Romanian Space Agency, the State Secretariat
+for Education, Research and Innovation (SERI)
+at the Swiss Space Office (SSO), and the United
+Kingdom Space Agency. A complete and de-
+tailed list is available on the Euclid web site
+(http://www.euclid-ec.org).
+References
+Addison, G., et. al., 2019, ApJ, 879, Id.15
+Bagley, M., et al., 2020, ApJ, 897, Id.98
+Bianchi, D., et al., 2018, MNRAS, 481, 2338
+Laureijs, R., et al., 2011, arXiv:1110.3193
+Monaco, P., 2016, Galaxies, 4, 53
+Monaco, P., Di Dio, E. & Sefusatti, E., 2019,
+JCAP, 2019-4, Id.023
+Monaco, P., Theuns, T. & Taffoni, G., 2002,
+MNRAS, 331, 587
+Munari, E., et al., 2017, MNRAS, 465, 4658
+Planck Collaboration, Aghanim N., et al.,
+2018, A&A, 641, Id.A6
+
+M>1012M/hhalosinaslice(1/500)
+7000
+6000
+5000
+Mpc/h
+4000
+3000
+2000
+1000
+z=1;8
+之=0.9
+0
+-6000
+-4000
+-2000
+0
+2000
+4000
+6000
+Mpc/h
\ No newline at end of file
diff --git a/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/load_file.txt b/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5fb4df5e26d2bd9451b903c6497c6eca4b5ce7b2
--- /dev/null
+++ b/qtE5T4oBgHgl3EQflQ8e/content/tmp_files/load_file.txt
@@ -0,0 +1,161 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf,len=160
+page_content='Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 75, 282  © SAIt 2022  Memorie della  Mapping the Universe with slitless spectroscopy  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Monaco1,2,3,4, on behalf of the Euclid Consortium  1 Universit`a di Trieste, Dipartimento di Fisica, Via Tiepolo 11, I-34131 Trieste, Italy  2 Istituto Nazionale di Astrofisica – OATs, Via Tiepolo 11, I-34131 Trieste, Italy  3 Istituto Nazionale di Fisica Nucleare Trieste, via Valerio 2, I-34127 Trieste  4 Institute for the Fundamental Physics of the Universe, via Beirut 2, I-34151 Trieste  e-mail: pierluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='monaco@inaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='it  Received: Day Month Year;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Accepted: Day Month Year  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Euclid will survey most of the accessible extragalactic sky with imaging and  slitless spectroscopy observations, creating a unique spectroscopic catalog of galaxies with  Hα line in emission that will map the Universe from z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='9 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' With low expected  statistical errors, the error budget will likely be dominated by systematic errors related to  uncertainties in the data and modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' I will discuss the strategy that has been proposed  to mitigate the expected systematic effects and propagate the uncertainty of mitigation to  cosmological parameter errobars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Cosmology: observations – Large-scale structure of the Universe –  Cosmological parameters  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Introduction  Observations  of  the  Cosmic  Microwave  Background  (CMB,  Planck  Collaboration  2018) have provided percent-accurate con-  straints to cosmological parameters, strength-  ening the case for a 6-parameter flat ΛCDM  cosmological model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' this is consistent with  most available evidence on large scales, but  at the cost of leaving 95% of the present  mass-energy  budget  unexplained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  In  fact,  most matter today is thought to be in the  form of an unknown collisionless particle,  and most energy today is thought to be in a  dark energy component that is accelerating the  Universe expansion, represented by a positive  cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  To shed light on the dark sector, it is crucial  to map the Universe at lower redshift, when  dark energy becomes dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' The European  Space Agency has promoted the Euclid mis-  sion (Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2011), an optical and  near-infrared space telescope that will survey  the sky with a visual imager (VIS), optimised  for galaxy lensing, and a near-infrared imager  and spectrograph (NISP), optimised for galaxy  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Galaxy Clustering with slitless  spectroscopy  Spectroscopic observations from space are  complicated by the impossibility to use screens  with suitably pierced holes or slits, as custom-  ary from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' While JWST has im-  plemented a novel micro-mirror technology  that provides a way to select what part of the  image is to be dispersed by a grism, Euclid  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='05669v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='CO]  13 Jan 2023  Monaco: Euclid’s slitless spectroscopy  283  will adopt a straight slitless spectroscopy strat-  egy, already experimented with Hubble Space  Telescope (Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='This means that  each source will produce a straight track on  the NISP detector, as showed in the simulation  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' These tracks will be separated and  used to produce 1D spectra with a resolution  of R = 380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' This observing strategy will allow  us to detect emission-line galaxies (ELGs) and  measure their redshift, provided that the emis-  sion line is correctly recognised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' The grism  will be sensitive to wavelengths in the range  λ ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='85] µm, so the most abundant de-  tectable ELGs are expected to be Hα emitters  in the redshift range z ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Simulations  show that the probability of detecting an ELG  drops for line fluxes below 2 × 10−16 erg s−1  cm−2, the nominal line-flux limit of the Euclid  Wide Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  The obvious drawback of this strategy, the  need to deblend all the sources in a field, is  balanced by the ability to perform a matter-  of-fact blind search of emission-line galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  Indeed, spectra will be extracted for any source  detected in the photometric observations, that  we may think as limited to HE < 24, and  the number of Hα ELGs associated to fainter  sources has been demonstrated to be negligible  in Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' At the same time, al-  though we expect the success rate of deblend-  ing to be a function of surface density of all  sources, the sample will be free of fiber colli-  sion bias (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Chasing systematic effects  The Observational Systematics Work Package  of the Galaxy Clustering Science Working  Group has surveyed the whole pipeline, from  raw data to the measure of galaxy clustering  that is provided to the likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Using the for-  malism of Monaco, Di Dio & Sefusatti (2019),  we have classified the possible systematic ef-  fects as follows: (i) modulations of the effec-  tive flux limit of the sample, due either to in-  strumental issues (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the tiling of the various  dithers will produce an inhomogeneous expo-  sure time map, while straylight from nearby  bright stars will modulate the noise) or to astro-  physical foregrounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' zodiacal light will  add to the background noise, while Milky Way  extinction will decrease the signal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' (ii) redshift  errors due to line misidentifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' (iii) red-  shift errors due to noise fluctuations being in-  terpreted as lines in an overall undetected spec-  trum, thus creating ‘noise interlopers’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  The first class of systematic effects will be  mitigated by suitably constructing a random  catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' To get rid of the angular footprint of a  survey, clustering estimators usually compare  the density field measured with the data sam-  ple with that from a random sample that covers  the same area and is unclustered on the sky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' its  number density is usually taken to be 50 times  that of the data sample (fitted by a model in  order not to erase some radial modes), to min-  imise the amount of extra shot noise introduced  by the random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' We will construct the random  by forward-modeling the completeness and pu-  rity of the spectroscopic sample, thus creating  what we call a visibility mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' This will be  done by taking profit of the Euclid Deep Field,  where 50 deg2 of the sky will be surveyed ten  times with various orientations of the grism,  and with 40 more pointings with a ‘blue grism’  that is sensitive in the range λ ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='92, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='25]  µm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' from it, we will extract a bona fide sam-  ple, pure at ∼ 99% level, of the ELGs that  can be seen in the EWS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' These galaxies will  be used to create a parent random catalog that  is unclustered on the sky, whose objects have  the same physical properties of the target sam-  ple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' these random galaxies will be injected in  the EWS NISP images, and processed to deter-  mine their probability of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' This way  the space density of the selected random will  be modulated on the sky by systematics in the  same way as the data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  While the third class of systematic effects,  the noise interlopers, can be modeled by suit-  ably adding a class of contaminants to the ran-  dom catalog, the second class, line misidenti-  fications, is more subtle to address (Addison  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' With a typically steep luminosity  function of sources, most objects in a catalog  are around the detection limit, where a single  emission line is typically detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' If no further  information is used, the contamination level  is expected to be around ∼ 10–20 %, mostly  coming from [Oiii] emitters at higher redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  284  Monaco: Euclid’s slitless spectroscopy  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Simulations of a NISP exposure, where each source produces a stripe along the dispersion imprinted  by the grism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Circles denote the positions of the Hα lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Credit: Ben Granett, OU-SIM an OU-SIR teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  These galaxies will be moved from their red-  shift to the one corresponding to Hα, carrying  with them their rescaled clustering signal, so  the measured two-point function ξ(r) will be  the weighted sum of the target one (1− f)2ξtarget  and the contaminant one f 2ξinterloper, where f  is the fraction of contaminants in the sam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' This contamination can be mitigated at the  likelihood level, comparing the measurements  with a weighted sum of predictions relative to  the target sample and to the significant contam-  inants, with fi fractions treated as nuisance pa-  rameters subject to a tight prior coming from  measurement of the Deep Field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Propagating the uncertainty in the  mitigation  This mitigation strategy will anyway leave  residuals that contaminate the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Every  step in the modeling of the visibility mask of  the EWS has an associated uncertainty that  must be propagated to parameter errorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  The most effective way to achieve this is to  construct a set of simulated mock galaxy cat-  alogs and process them in the same way as the  parent random catalog described above: inject  galaxies in the images and compute their prob-  ability of being detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' However, this process  should not be performed using our best knowl-  edge of the visibility mask but a modulation of  it, obtained by perturbing every single step in  the pipeline, sampling its estimated error PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  As an example, detection probability will de-  pend on the measured noise level, and we will  use the best-fit value of the noise to create the  random, and a value drawn from its PDF for  applying the visibility mask to mock galaxy  catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  This approach will provide a brute-force  numerical estimate of the covariance matrix  that will include both cosmic covariance and  the uncertainty in the mitigation of system-  atic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Proper sampling of the matrix re-  quires thousands of mocks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' in this moment the  Galaxy Clustering Science Working Group is  preparing 3500 simulations of the Euclid sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  N-body codes are simply too expensive to ad-  dress this massive production, so we will re-  sort to approximate methods (Monaco 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' I  am presently working to prepare such a large  set of simulations using the PINOCCHIO code  (Monaco, Theuns & Taffoni 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Munari et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2017), based on Lagrangian Perturbation  Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2 shows one of these ligh-  cones, reporting dark matter halos (with Mh >  2000  1750  1500  1250  1000  750  500  250  0  0  250  500  750  1000  1250  1500  1750  2000Monaco: Euclid’s slitless spectroscopy  285  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Lightcone generated with PINOCCHIO, see text for explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  1012 M⊙, one in 500) in comoving coordinates,  in a slice that cuts through the survey volume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the catalogs cover half of the sky, with the ex-  ception of unobserved low Galactic latitudes,  and start at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' The red lines mark the  box size (3380 h−1 Mpc), that is tiled to cover  the survey volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' When ready, this will be  the largest set of cosmological simulations ever  produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' thanks L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Guzzo, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  Percival, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Scarlata, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Granett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  Moresco, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' De La Torre and the members of  the Observational Systematics Work Package for  many discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' The simulation used for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  1 was produced by the Euclid Science Ground  Segment, in particular Operational Units OU-SIM  and OU-SIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' The Euclid Consortium acknowledges  the European Space Agency and a number of  agencies and institutes that have supported the  development of Euclid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' in particular the Academy  of Finland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the Agenzia Spaziale Italiana,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the  Belgian  Science  Policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the  Canadian  Euclid  Consortium,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the French Centre National d’Etudes  Spatiales,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the Deutsches Zentrum f¨ur Luft- und  Raumfahrt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the Danish Space Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the Fundac¸˜ao para a Ciˆencia e a Tecnologia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the  Ministerio  de  Ciencia  e  Innovaci´on,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the  National Aeronautics and Space Administration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  the  National  Astronomical  Observatory  of  Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the Netherlandse Onderzoekschool Voor  Astronomie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the Norwegian Space Agency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the  Romanian Space Agency,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' the State Secretariat  for Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' Research and Innovation (SERI)  at the Swiss Space Office (SSO),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' and the United  Kingdom Space Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' A complete and de-  tailed list is available on the Euclid web site  (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='euclid-ec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='  References  Addison, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2019, ApJ, 879, Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='15  Bagley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2020, ApJ, 897, Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='98  Bianchi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2018, MNRAS, 481, 2338  Laureijs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2011, arXiv:1110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='3193  Monaco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2016, Galaxies, 4, 53  Monaco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', Di Dio, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' & Sefusatti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2019,  JCAP, 2019-4, Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='023  Monaco, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', Theuns, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=' & Taffoni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2002,  MNRAS, 331, 587  Munari, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', 2017, MNRAS, 465, 4658  Planck Collaboration, Aghanim N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content=',  2018, A&A, 641, Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='A6  M>1012M/hhalosinaslice(1/500)  7000  6000  5000  Mpc/h  4000  3000  2000  1000  z=1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='8  之=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
+page_content='9  0  6000  4000  2000  0  2000  4000  6000  Mpc/h' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qtE5T4oBgHgl3EQflQ8e/content/2301.05669v1.pdf'}
diff --git a/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/2301.04375v1.pdf.txt b/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/2301.04375v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..0f9f684f5ed15d1b1cbfdd856c1facc79cee9bc1
--- /dev/null
+++ b/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/2301.04375v1.pdf.txt
@@ -0,0 +1,4196 @@
+ 
+1 
+ 
+Defect engineering over anisotropic brookite towards substrate-specific photo-
+oxidation of alcohols 
+S. M. Hossein Hejazi1, Mahdi Shahrezaei1, Piotr Błoński1, Mattia Allieta2, Polina M. 
+Sheverdyaeva3, Paolo Moras3, Zdeněk Baďura1, Sergii Kalytchuk1, Elmira Mohammadi1, Radek 
+Zbořil1,4, Štěpán Kment1,4, Michal Otyepka1,5, Alberto Naldoni1*, Paolo Fornasiero6,7* 
+1Czech Advanced Technology and Research Institute, Regional Centre of Advanced 
+Technologies and Materials, Palacký University Olomouc, Křížkovského 511/8, 77900 
+Olomouc, Czech Republic 
+2Ronin Institute Montclair, NJ 07043 USA 
+3Istituto di Struttura della Materia-CNR (ISM-CNR), SS 14, Km 163,5, I-34149, Trieste, 
+Italy 
+4Nanotechnology Centre, Centre of Energy and Environmental Technologies, VŠB–
+Technical University of Ostrava, 17. listopadu 2172/15, 70800 Ostrava-Poruba, Czech 
+Republic 
+5IT4Innovations, VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 
+Ostrava-Poruba, Czech Republic 
+6Department of Chemical and Pharmaceutical Sciences, ICCOM-CNR Trieste 
+Research Unit, INSTM-Trieste, University of Trieste, Via L. Giorgieri 1, 34127 
+Trieste, Italy  
+7Center for Energy, Environment and Transport Giacomo Ciamician - University of Trieste, 
+Italy 
+*Corresponding authors: alberto.naldoni@upol.cz; pfornasiero@units.it 
+ 
+ 
+
+ 
+2 
+ 
+Summary 
+Generally adopted design strategies for enhancing the photocatalytic activity are aimed at tuning 
+properties such as the visible light response, the exposed crystal facets, and the nanocrystal shape. 
+Here, we present a different approach for designing efficient photocatalysts displaying a substrate-
+specific reactivity upon defect engineering. The defective anisotropic brookite TiO2 photocatalyst 
+functionalized with Pt nanocrystals are tested for alcohol photoreforming showing up to an 11-
+fold increase in methanol oxidation rate, compared to the unreduced one, whilst presenting much 
+lower ethanol or isopropanol specific oxidation rates. We demonstrate that the alcohol oxidation 
+and hydrogen evolution reactions are tightly related, and when the substrate-specific alcohol 
+oxidation ability is increased, the hydrogen evolution is significantly boosted. The reduced 
+anisotropic brookite shows up to twenty-six-fold higher specific photoactivity with respect to 
+anatase and brookite with isotropic nanocrystals, reflecting the different type of defective catalytic 
+sites formed depending on the TiO2 polymorph and its crystal shape. Advanced in-situ 
+characterizations and theoretical investigations reveal that controlled engineering over oxygen 
+vacancies and lattice strain produces large electron polarons hosting the substrate-specific active 
+sites for alcohol photo-oxidation.  
+ 
+Keywords: selective photocatalysis, oxygen vacancies, DFT calculations, photoreforming, black 
+TiO2 
+ 
+ 
+ 
+ 
+
+ 
+3 
+ 
+INTRODUCTION 
+The visionary idea of a world powered by solar light proposed by G. Ciamician more than a century 
+ago1 has become a reality, proving that complex organic synthesis2,3 and production of solar fuels 
+like hydrogen4–6 and ammonia7,8 can be performed more and more efficiently. However, these 
+intrinsically sustainable processes, before becoming industrially competitive with existing 
+polluting technologies, need further material design, fine tuning of light absorption properties, 
+charge carriers management, and surface engineering.9 Over the past decades, photocatalysis for 
+direct conversion of solar energy into molecular fuels has been focused on designing efficient 
+photocatalysts by improving their fundamental properties. The visible light photoactivity can be 
+enhanced by engineering heterojunction, introducing lattice defects into wide bandgap materials 
+like TiO2,10–12 or choosing semiconductors having small bandgap energy (e.g. Cu2O, ZnIn2S4, and 
+i.e. C3N4) and suitable bands position straddling the molecular redox levels of the investigated 
+chemical reaction.2,13–15 The use of inorganic nanocrystals with well-defined morphology, 
+determined crystal facets, or dimensional anisotropy have been also demonstrated to be beneficial 
+for the charge carrier separation.16–18 The kinetic competition between charge recombination and 
+surface catalysis is often overcome by the addition of proper metallic co-catalysts.19 This 
+playground has stimulated the exploration of countless options to prepare, mix, and engineer 
+semiconductor photocatalysts with enhanced opto-electronic properties with the aim of more 
+efficiently driving benchmark photocatalytic reactions such as hydrogen evolution from water 
+splitting and photoreforming of biomass. However, the majority of these studies involve 
+monitoring the products of the reductive cycle, i.e. the evolved hydrogen, while not analyzing the 
+oxidation products.20–22 When sacrificial biomass substrates are employed, i.e. alcohol 
+photoreforming, analyzing the oxidation pathway and reactivity becomes especially important not 
+only because they provide a kinetic gain, and therefore an improved hydrogen evolution compared 
+
+ 
+4 
+ 
+to the case of water oxidation, but also because the oxidized sacrificial agents often participate in 
+hydrogen evolution, thus directly regulating its kinetics.20,23 Furthermore, controlling the oxidation 
+process of sacrificial biomass is particularly relevant since it can lead to its partial oxidation and 
+the synthesis of added value products such as 2,5-furandicarboxylic acid (bioderived polymer that 
+may substitute PET) and diesel fuel precursors.3,24,25 
+We developed an approach to designing anisotropic defective brookite TiO2 nanocrystals that upon 
+high temperature reduction treatment expose precise defect site with substrate-specific photo-
+reactivity for methanol oxidation, compared to higher alcohols such as ethanol and isopropanol 
+(Fig. 1A). We show that the reaction rates for the photocatalytic alcohol oxidation and the parallel 
+hydrogen evolution reaction are tightly connected and that optimization of the methanol oxidation 
+leads to an increased production of hydrogen. Although the introduction of point defects and 
+structural deformation results in enhanced visible light absorption and reduced charge carrier 
+lifetime, we show that the selective affinity towards methanol oxidation of reduced brookite is the 
+main parameter enabling a higher apparent quantum yield (AQY) for hydrogen evolution. Using 
+a set of in-situ characterization aided by DFT calculations, we demonstrate that the substrate-
+specific activity is regulated by catalytic sites including sub-surface oxygen vacancies within a 
+locally strained lattice environment that generate shallow hole traps responsible for boosting the 
+first steps of methanol photo-oxidation. These photo-oxidation sites are crucial for the enhanced 
+substrate-specific photoreforming activity of reduced brookite, and they form preferentially within 
+anisotropic nanocrystals, which show twenty-six times higher specific hydrogen evolution rate, 
+compared to isotropic ones, where defect sites with different energy are formed.  
+ 
+ 
+ 
+
+ 
+5 
+ 
+RESULTS AND DISCUSSION 
+Synthesis and characterization of brookite nanorods 
+To engineer the photocatalytic sites for alcohol photoreforming at the atomic level, we selected 
+brookite TiO2 nanorods—a promising and still poorly investigated TiO2 polymorph—as a model 
+material and grew anisotropic nanostructures exposing the (210) surface on the lateral facets (Fig. 
+1B) by hydrothermal synthesis (see Supporting Information).26 We prepared various brookite 
+photocatalysts reduced under a H2 stream at different temperatures, along with reduced anatase 
+and commercial brookite reference samples (see Table S1). Elemental analysis revealed that the 
+obtained nanopowders did not contain significant quantity of non-metals coming from C, H, or N 
+incorporation (Table S2). Reduction of a pristine brookite TiO2 under pure hydrogen stream at 
+700°C for 1h created defective nanocrystals showing remarkable changes in their structural and 
+electronic properties alongside giving the best photocatalytic performance. This morphology 
+evolved from anisotropic nanostructures with well-defined shape and exposed crystal facets (Fig. 
+1B and S1A) into more isotropic nanoparticles that displayed irregular shape and aggregation 
+through twin boundaries formation (Fig. 1C and Fig. S1B). Notably, anatase and brookite with 
+isotropic crystal shapes (Fig. S2-S3) did not reveal any crystal reshaping upon high temperature 
+reduction treatment. However, they presented different color variations (Table S1), compared to 
+those observed for the anisotropic brookite, upon increasing the temperature of the hydrogen 
+treatment, thus suggesting a different reducibility behavior with respect to the TiO2 polymorph 
+and crystal shape, as confirmed by UV-vis reflectance spectroscopy, Raman spectroscopy, and 
+photolumiscence spectroscopy mapping (see below). In order to prepare highly active 
+photocatalysts for alcohol photoreforming, we functionalized the samples by photodepositing Pt 
+co-catalyst nanoparticles on their surface. Inductively coupled plasma mass spectrometry (ICP-
+MS) analysis detected similar Pt loading on both pristine (0.98 wt%) and reduced brookite samples 
+
+ 
+6 
+ 
+(0.90 wt%). HRTEM and STEM-HAADF micrographs as well as the elemental mapping showed 
+that Pt nanoparticles with an average diameter of 2.5 nm were homogeneously deposited on the 
+pristine brookite nanorods (Fig. 1D and Fig. S4A, S5, S6). Surprisingly, in the case of reduced 
+brookite nanocrystals, we detected the presence of small Pt nanoparticles with similar sizes as well 
+as larger Pt nanoparticle aggregates reaching 10–30 nm in size (Fig. 1E and Fig. S7–S10). This 
+result was confirmed by a HRTEM analysis of three different brookite batches. The larger Pt 
+conglomerates may be less reactive than the smaller ones, thus negatively affecting the 
+photocatalytic activity of the reduced brookite. Moreover, larger metal aggregates may screen the 
+incoming light during photocatalysis, decreasing the light harvesting efficiency. However, as 
+shown in the next section, in the present investigation these parameters did not reduced neither 
+one aspect nor the other for reduced brookite, suggesting that for the considered reaction the 
+defects in TiO2 played a more crucial role than that of Pt nanoparticles. The brookite treatment 
+under hydrogen was accompanied by a decrease in the BET (Brunauer–Emmett–Teller) specific 
+surface area from 67 to 47 m2 g-1 after reduction (Fig. S11 and Table S3). 
+The temperature (reduction)-dependent structural parameters extracted by the Rietveld refinement 
+of X-ray diffraction (XRD) patterns were in agreement with the described morphological evolution 
+(Fig. S12-S18 and Table S3–S6). Notably, the XRD analysis highlighted that the reduction 
+treatment introduced an anisotropic and preferential deformation of the brookite lattice along the 
+c-axis due to the creation of oxygen vacancies (Fig. S19). Their presence was further supported by 
+the blueshift in the main A1g vibrational mode detected by Raman spectroscopy measurements 
+(Fig. S20 and discussion in the Supporting Information) after reduction, which again pointed out 
+to a different reducing behavior dependent on the TiO2 polymorph and shape (Raman shift is 1.3 
+cm-1 for the reduced anisotropic brookite, 7.6 cm-1 for the reduced isotropic anatase, and no 
+observed shift for the reduced isotropic brookite). Having performed the bond valence sum 
+
+ 
+7 
+ 
+analysis, we observed an average depletion of ~0.5% of the Ti valence for the reduced brookite—
+a typical feature induced by the formation of oxygen vacancies.11,27 The moderate decrease in the 
+Ti valence upon a H2 treatment at high temperature is an indication of a low tendency toward 
+defects formation in brookite nanorods. This general feature was also reflected by the color 
+change—from white to grey—that brookite underwent after reduction at 700°C, as opposed to the 
+more reducible anatase phase that assumed a darker color already at lower temperatures (Table 
+S1). The reduced brookite nanorods showed an optical bandgap energy of ~3.38 eV, this making 
+no significant difference from the value retrieved for the as-synthesized sample (Fig. S21 and 
+Table S7). The same results were observed for both the isotropic brookite and anatase samples 
+(Fig. S22-S23 and Table S8-S9). Further analysis of the absorption spectra highlighted an 
+increased visible light absorption and an Urbach tail that ranged for the anisotropic brookite from 
+69 (for the as-synthesized brookite nanorods) to 115 meV (for nanorods reduced at 700°C). For 
+the isotropic anatase, the Urbach tail increased from 115 (for the as-synthesized anatase 
+nanoparticles) to 205 mV (for anatase reduced at 500°C). This supports the scenario of a phase- 
+and shape-dependent increase in the population of oxygen vacancies after the reduction treatment 
+(see Supporting Information for further discussion).  
+Alcohol photoreforming with reduced brookite 
+The photocatalytic activity of the platinized brookite nanorods was tested for methanol 
+photoreforming under a simulated AM 1.5G spectrum at one-sun intensity producing H2 and 
+oxidation products. As previously reported by others groups, the increased photocatalytic activity 
+of reduced TiO2 nanomaterials could be ascribed to the co-catalyst role in H2 evolution played by 
+oxygen vacancies.12,28 In contrast, in order to use the oxygen vacancies as catalytic sites in the 
+photocatalytic oxidation reaction, we photodeposited Pt nanoparticles over the optimized 
+photocatalysts. Following this procedure, H2 generation occurred on the Pt surface as the 
+
+ 
+8 
+ 
+photogenerated electrons were separated and stabilized into the Pt nanoparticles by the Schottky 
+barrier formation at the Pt–TiO2 interface, while photo-oxidation occurred on the TiO2 surface.19,29  
+In contrast to common practice, where only a H2 production rate is detected, we designed specific 
+experiments to follow the kinetics of methanol photo-oxidation using a solution including 
+deuterated methanol (i.e, CD3OD) and a small aliquot of methanol (i.e. CH3OH), whose 
+corresponding consumption was followed by an 1H-NMR analysis of the liquid phase (Fig. S24). 
+Table S10 reports the NMR signal integration of CH3OH relative to the adopted internal standard 
+(DMSO), which highlight no significant difference between the blank measurement containing no 
+photocatalyst and the one at time zero, i.e. after 30 min adsorption/desorption equilibrium in the 
+dark in the presence of the photocatalysts. From this data is clear that the methanol adsorption in 
+the dark did not affect the photocatalytic performance of the investigated photocatalysts. Fig. 1F 
+shows the amount of methanol oxidized over 24h of illumination illustrating the remarkable 
+oxidation activity of the reduced brookite over the pristine material. The corresponding specific 
+methanol consumption rates, computed by using the BET surface area of each sample, for the 
+platinized brookite nanorod samples (Fig. 1G, left) evidenced that the pristine brookite drove the 
+photo-oxidation reaction with a specific rate of 27 mol h-1 m-2, whereas for the reduced one, it 
+was 99 mol h-1 m-2. We obtained the photoactivity values by considering methanol consumption 
+after 24 h of reaction, which resulted in a 3.7-fold enhancement in favor of the reduced brookite. 
+Notably, if we consider kinetic data after 5 h (Fig. 1F), the reduced brookite performed methanol 
+photo-oxidation up to 11 times faster than the pristine sample. This suggests on the one hand that, 
+in the early stage of reaction, methanol was oxidized faster until the available surface reaction sites 
+were fully occupied and a steady state was reached, which ensured a higher methanol consumption 
+rate even after 24 h of reaction, as evidenced by the divergence of the reaction kinetics curves (Fig. 
+1F). On the other hand, this behavior can be due to a partial aggregation of the colloidal 
+
+ 
+9 
+ 
+photocatalysts after several hours of irradiation (see dynamic light scattering measurements in Fig. 
+S25), thus producing a reduced available surface for the methanol oxidation reaction to occur. 
+The amount of evolved hydrogen determined by gas chromatography analysis followed a linear 
+increase with time (Fig. S26) for both the pristine and the reduced brookite, corresponding to 
+optimized specific H2 production rates of 26 and 88 mol h-1 m-2, respectively, with reduced 
+brookite that evolved H2 3.4 times faster (Fig. 1G, right). The reduced brookite showed a 13% 
+decrease activity after 5 photocatalytic runs (Fig. S27). Notably, the activity decrease appeared 
+almost constant after each recycling test, thus suggesting that it can be due to the loss of catalyst 
+during the tests, which can happen during the centrifugation/re-suspension of the photocatalyst 
+and/or can be due to the loss of material attaching onto the reactor walls. Another aspect that can 
+produce this slight decrease in activity is the increased hydrodynamic diameter of the brookite 
+nanocrystals in suspension, as revealed by dynamic light scattering measurements after 24 h of 
+illumination (Fig. S25). Moreover, two more aspects may produce the observed photocatalytic 
+activity decrease after several recycling cycles. On the one hand, the catalyst surface may be 
+partially passivated by the presence of reaction intermediates, as we did not wash the catalyst 
+before subsequent tests. On the other hand, a partial modification of the surface population of 
+defects (as evidenced by the resonant PES valence band measurements on B700 after catalysis, 
+see Fig. S43) may induce a partial reactivity change. Interestingly, the H2 production rates followed 
+a reactivity trend closely resembling the one observed for the methanol photo-oxidation activity, 
+which suggests that hydrogen production is strictly regulated by the alcohol oxidation and it can 
+be used as reporter figures of merit for tracking the reactivity of the pristine and the reduced 
+brookite for alcohol photo-oxidation. Following this principle, we tested our platinized samples 
+for the photoreforming of ethanol and isopropanol and discovered that the reactivity of the reduced 
+brookite was markedly more pronounced and substrate-specific toward methanol in comparison 
+
+ 
+10 
+ 
+with the other tested alcohols (Fig. 1G, right). In the case of ethanol photoreforming, both reduced 
+and pristine anisotropic brookites showed a specific H2 production rate of 54 mol h-1 m-2, 
+suggesting that they have a similar affinity toward its photo-oxidation. On the other hand, in the 
+case of isopropanol photoreforming, the reduced anisotropic brookite presented a 1.7-fold higher 
+specific H2 production rate (41 mol h-1 m-2) when compared to the pristine sample, denoting a 
+higher photo-oxidation ability yet still much lower than that shown for methanol. This observation 
+confirmed that the H2 evolution activity of the reduced brookite was regulated by a substrate-
+specific reactivity toward alcohol photo-oxidation. Such a stark photo-reactivity toward methanol 
+oxidation was observed only for the brookite nanorods, while platinized reference samples made 
+by reduced spherical anatase nanocrystals or reduced isotropic brookite nanoparticles displayed 
+~1.8–1.9 times higher specific photocatalytic rates in comparison with the untreated materials (Fig. 
+S28). Notably, the reduced brookite nanorods loaded with Pt showed a remarkably higher specific 
+H2 evolution rate with respect to both the reduced anatase/Pt (19 times) and the reduced isotropic 
+brookite/Pt (26 times), as shown in Fig. S28. However, this difference is significantly reduced 
+when considering the photocatalytic activity per optimized mass (Fig. S29), with the reduced 
+brookite nanorods (B700/Pt) still showing more than two-times the hydrogen evolution rate 
+observed for the reduced anatase nanocrystals (A500/Pt). These observations suggest that the type 
+of produced defects/catalytic sites varies depending on the selected TiO2 polymorph as well as on 
+the crystal shape, which emphasizes how the exposure of different crystal facets having different 
+interfacial energy and therefore resistance to hydrogen treatment under high temperature regulates 
+the defects formation. This is demonstrated by the different degree of reducibility that each sample 
+exhibited, as supported by previously discussed absorption and Raman spectroscopy 
+measurements. 
+
+ 
+11 
+ 
+The apparent quantum yield (AQY) for hydrogen evolution from methanol photoreforming was 
+measured for a pristine and a reduced platinized brookite (Fig. 1H) using different monochromatic 
+light sources. The maximum AQY was reached at 334 nm and was 33.5% for the reduced brookite 
+and 22.2% for the pristine nanorods. These AQY values can be further increased by optimizing 
+the methanol concentration, metal loading, metal particle size, or photoreactor design, which 
+however goes beyond the scope of this study. Interestingly, despite its visible light absorption, the 
+reduced brookite did not show AQY in the visible region, with values of ~0.09 and 0.004% at 386 
+and 402 nm, respectively (AQY below the detection limit for the pristine brookite at both 
+wavelengths). This result was further confirmed by H2 evolution experiments under one-sun 
+illumination applying a longpass optical filter to cut off λ ≥ 380 nm, i.e. cutting optical excitation 
+above bandgap energy did not lead to detecting any H2 after 24 h of reaction. This result confirms 
+that oxygen vacancies introduced upon the hydrogen treatment at high temperature, and the related 
+optical transitions, did not produce visible light photocatalytic activity. We indeed suggest that the 
+introduced oxygen vacancies enhanced the reactivity toward the methanol oxidation by favoring 
+its activation, as discussed in further detail below. 
+
+ 
+12 
+ 
+ 
+Fig. 1. Morphology and photocatalytic activity. (A) Schematic representation of defect 
+engineering in reduced brookite showing an exemplary TiO2 surface during photocatalysis and a 
+zoomed view of the catalytic site containing oxygen vacancy (VO) and structural distortions. (B) 
+HAADF-STEM (left) and HRTEM (right) micrographs of a single brookite nanorod. (C) HRTEM 
+micrograph of isolated Pt nanoparticles deposited on pristine brookite. (D) HRTEM of brookite 
+nanorods reduced at 700°C. (E) HRTEM micrograph of aggregated Pt nanostructures deposited 
+on reduced brookite. (F) Methanol consumption in time for pristine (sky blue) and reduced 
+brookite (dark blue) nanorods. The points before zero time represent the methanol signal before 
+adding the photocatalysts, while time zero was measured once the adsorbtion/desorption 
+equilibrium in the dark was reached. (G) Specific methanol consumption rate (left) and specific 
+hydrogen evolution rate (right) during methanol, ethanol, and isopropanol photoreforming for 
+pristine (sky blue) and reduced (dark blue) brookite. Measurements were performed under a 
+simulated AM 1.5G spectrum at one-sun intensity for 24h using a 1:1 vol.% H2O:alcohol mixture. 
+(H) Apparent quantum yield for hydrogen evolution from methanol photoreforming for pristine 
+(sky blue) and reduced (dark blue) brookite. In all measurements both pristine and reduced 
+brookite were loaded with 1 wt.% Pt.  
+
+A
+H,O
+H
+CH.
+CO
+Catalytic sitewithoxygen vacancyV.
+structural
+deformation
+O
+B
+onm
+10nm
+F
+G
+100
+2500
+Light,
+1 m2),
+100
+m-2)
+2000
+CH,OH cons. rate (μmol h-1
+80
+[oum)
+80
+30
+rate
+1500
+60
+AQY %
+20
+CH,OH cons.
+1000
+40
+40
+10
+500
+20
+0
+n
+0
+5
+10
+15
+20
+25
+310
+330
+350
+370
+390
+410
+Time (h)
+Wavelength (nm) 
+13 
+ 
+In order to study in more detail the methanol photo-oxidation reaction on the reduced brookite, we 
+analyzed the reaction products by both the GC analysis of the gas phase and the NMR analysis of 
+the liquid phase after reaction. Carbon dioxide was the only detected reaction product. 
+Furthermore, we investigated the hydrogen production rate from different possible intermediates 
+of methanol oxidation (e.g. formaldehyde and formic acid) of as-synthesized and reduced brookite 
+nanorods loaded with 1 wt.% Pt. Interestingly, both samples showed similar specific photocatalytic 
+activity in the presence of formaldehyde and formic acid, presenting significant hydrogen 
+production rate of around 25–30 mol h-1 m-2 (Fig. S30). This result is far from being trivial, as it 
+has been reported that other TiO2 polymorphs usually oxidize methanol to formaldehyde, thus 
+stopping the methanol photo-oxidation after the first reaction step.30 Moreover, it repeatedly 
+emphasizes that a reduced brookite demonstrates a substrate-specific oxidation ability toward 
+methanol molecules. The blank test for photolysis of formaldehyde under AM 1.5G 1sun 
+illumination produced a very small hydrogen production rate, namely, ~85 nmol h-1 m-2.  
+Notably, the investigated TiO2 samples showed two order-lower alcohol photoreforming activity 
+without Pt loading; the data on the samples reduced at different temperatures are reported in Fig. 
+S31 and S32. These data are further supported by electron spin resonance spectroscopy (EPR) 
+investigations measured under dark and light conditions both for dried powders and in a 
+water/methanol medium (in situ conditions), demonstrating the increased reactivity of brookite 
+nanorods reduced at 700°C (Fig. S33–S35). For instance, in the case of EPR spectra for dried 
+powders of an anisotropic brookite reduced at different temperatures, the most efficient sample in 
+methanol photoreforming was B700, which indeed gave the highest differential EPR signal (light-
+dark) in comparison with samples with lower activity (e.g. a pristine brookite and B500). 
+Interestingly, the most active sample (B700) showed the weakest intensity in the EPR powder 
+spectrum among the series (Fig S33 and discussion in the Supporting Information). Therefore, the 
+
+ 
+14 
+ 
+number of spins recorded by EPR do not directly correlate with the system reactivity and its overall 
+efficiency in the photocatalytic process, all in agreement with previous reports.12,31  
+In-situ photoluminescence spectroscopy 
+To understand the nature of the methanol oxidation sites, we measured excitation-dependent 
+photoluminescence (PL) spectra at 80 K, obtaining energy-resolved two-dimensional maps of the 
+radiative recombinations occurring in the pristine and the reduced anisotropic brookite both under 
+inert gas atmosphere (N2) and in the presence of methanol (Fig. 2A). The PL maps in the presence 
+of the latter (i.e., a hole scavenger) showed drastic quenching of the signal, demonstrating that the 
+photogenerated holes trapped within the defect sites, i.e. oxygen vacancies, readily reacted with 
+the surface adsorbed methanol molecules. Moreover, this also provided evidence that such defect 
+sites must be located on either the surface or sub-surface of the brookite nanocrystals, from where 
+they can react with surface adsorbates.32 This is supported by the synchrotron-based photoemission 
+spectra for the Ti 2p region and valence band (see for details the next section). Comparing the two-
+dimensional PL maps measured under N2 gas atmosphere for the pristine and the anisotropic 
+brookite reduced at 500, 600, 700, and 800°C (Fig. 2A and Fig. S36), we observed a clear variation 
+in the energy position relative to the radiative recombination centers upon high temperature 
+treatment, eventually underlined by a subtle but significant re-organization of structural defects. 
+Notably, the reference samples (especially the isotropic anatase, which is more reducible than the 
+isotropic brookite) displayed a similar behavior (Fig. S37-S38 and discussion in the Supporting 
+Information).  
+Focusing on the reduced anisotropic brookite, we observed a significant blue shift in the PL peak 
+after reduction.  
+
+ 
+15 
+ 
+ 
+Fig. 2. Energy distribution of defects-related radiative recombinations and lifetime of 
+photogenerated charge carriers. (A) Excitation-emission color maps under N2 and in the 
+presence of methanol (MeOH) for pristine and reduced brookite. (B) PL spectra of pristine (blue 
+sky) and reduced brookite (dark blue) under excitation at 340 nm. (C) Time-resolved PL decay 
+curves collected at the corresponding emission maximum of pristine (blue sky) and reduced 
+brookite (dark blue) under excitation at 372 nm. 
+This is better highlighted in the PL spectra generated using a single excitation wavelength (340 
+nm) and by analyzing the weight of the deconvoluted components set at 2.75, 2.53, 2.27, and 2.0 
+eV for all the samples (Fig. 2B and Fig. S39). The dominant radiative recombinations for the as-
+synthesized anisotropic brookite (B-AS) localize at 2.27 and 2.0 eV, while the components at 
+higher energies are almost negligible. Notably, in the case of the reduced anisotropic brookite 
+(B700), the component at lower energy almost vanished, while the intensity of the radiative 
+recombinations with higher energies (2.75 and 2.53 eV) became dominant, denoting the formation 
+of shallower hole traps upon hydrogen reduction treatment at high temperature. Furthermore, we 
+investigated the lifetime of photogenerated charge carriers measured at PL maximum by time-
+resolved PL spectroscopy. The charge carriers’ lifetime (τ) decreased after the brookite’s 
+
+B
+2.5
+3.4
+2.8
+2.4
+2.1
+1.8
+1.6
+B-AS-N2
+B-AS-MeOH
+(eV)
+ intensity (a.u.)
+B-AS
+2.7
+B700
+Excitation energy
+3.1
+2.8x104
+3.5
+2.5x104
+PLi
+2.2x104
+4.0
+2.0x104
+1.7x104
+4.8-
+1.4x104
+360
+440
+520
+600
+680
+760
+2.5
+1.1x104
+Wavelength (nm)
+B700-N2
+C
+B700-MeOH
+8.4x103
+(eV)
+2.7
+5.6x103
+10°
+B-AS
+Excitation energy
+2.8x103
+(a.u.)
+B700
+3.1
+Intensity
+10-1
+3.5
+=5.3ns
+4.0
+C=2.0ns
+10-2
+4.8-
+3.4
+2.8
+2.4
+2.1
+1.8
+1.7 3.4
+2.8
+1.8
+1.7
+0
+5
+10
+15
+20
+Emissionenergy (eV)
+Emission energy (eV)
+Time (ns) 
+16 
+ 
+reduction, similarly to the other TiO2 reference samples, from 5.3 ns to 2.0 ns (Fig. 3C, Fig. S40 
+and Table S11), suggesting that the enhanced photocatalytic activity of the reduced anisotropic 
+brookite is not related to the enhanced charge separation. 
+Synchrotron resonant photoemission spectroscopy 
+The investigation by conventional lab scale X-ray photoelectron spectroscopy (XPS) analysis 
+provided similar results for pristine and reduced TiO2 samples (Fig. S41). Therefore, we 
+investigated in more detail the electronic structure of our brookite samples at the VUV-
+Photoemission beamline (Elettra, Trieste) by synchrotron-based photoemission spectroscopy 
+(PES) for the Ti 2p (Fig. S42), O 1s (Fig. S43, see discussion in the next section), and the valence 
+band (VB) regions. The Ti 2p spectra of both brookite samples contained two components 
+corresponding to the presence of Ti4+, due to the coordination of Ti into the stoichiometric lattice, 
+and Ti3+ species introduced by the oxygen vacancy formation near or at the TiO2 surface. The 
+presence of low valence Ti ions in the pristine brookite is a common observation, especially in 
+nanocrystals obtained through hydrothermal synthesis and not subjected to a following heat 
+treatment like in the present case. Next, by using soft X-ray photons with energy that is resonant 
+to the Ti absorption edge, it is possible to highlight electronic states even in samples containing a 
+low amount of defects.33 Fig. 3A shows the VB PES spectra for the pristine and the reduced 
+anisotropic brookite. The main VB edge did not significantly shift upon reduction, while the 
+density of states (DOS) within the bandgap showed a stark difference. The pristine brookite 
+displayed localized mid-gap states peaking at around 1 eV below the Fermi energy. In contrast, 
+the reduced brookite revealed an increased electron density showing an intense VB tailing. We 
+also investigated the reduced brookite after 24 h of photocatalytic reaction B700-AR and compared 
+the result with the spectra of the pristine brookite B-AS and the reduced brookite before reaction 
+B700-BR (Fig. S44). The post-catalytic characterization evidence a spectrum, which features a 
+
+ 
+17 
+ 
+localize state at around 1 eV below the Fermi level (similarly to B-AS) and an increase density of 
+states –band tailing - at energies closer to the valence band maximum (similarly to B700-BR). The 
+slight modification of the spectrum can be due both to a partial passivation of the surface defects 
+in B700 during photocatalysis and to the adsorption of methoxy groups / reaction intermediates, 
+i.e. the sample was not regenerated after reaction. 
+Density functional theory calculations of the photocatalytic sites 
+In order to understand the origin of the VB tailing in the electronic structure of the reduced 
+brookite, we calculated the energy band structure using ab initio density functional theory (DFT) 
+calculations. Driven by the XRD results, we focused on the (210) surface of the brookite TiO2 
+introducing two types of structural defects: (1) oxygen vacancies located at different distance from 
+the surface (denoted by V1–V8 in Fig. 3B and Fig. S45), and (2) distortion of TiO6 octahedra (for 
+methods see supplementary materials) by modifying up to ±0.1 Å either the axial or randomly 
+chosen Ti-O distances.  
+ 
+
+ 
+18 
+ 
+  
+Fig. 3. Experimental and theoretical determination of the electronic structure of the 
+photocatalytic sites. (A) Synchrotron-based photoemission spectra around the valence band (VB) 
+region for the pristine (light blue) and the reduced (dark blue) brookite. Inset: zoom of the VB 
+region around the Fermi energy. (B) Brookite TiO2 supercell employed for the calculations 
+exposing the (210) surface: Ti atoms plotted in grey, O atoms in red, oxygen vacancies in orange. 
+The middle part of the slab corresponds to the bulk region of TiO2 enclosed by green planes, while 
+the supercell’s boundaries are marked by the dotted lines. (C) Calculated total DOS of the ideal 
+(210) brookite surface, of various defective brookite surfaces with an oxygen vacancy (V in the 
+figure) placed at different locations in the lattice, two distorted brookite surfaces (rdm. and ax. 
+stand for random and axial distortions, respectively). The energy of the VB maximum of the ideal 
+(210) surface is taken to be zero. Spin up/down derived DOS are shown by solid/dashed lines. (D) 
+Excess electron density donated by introducing V3 and V5 oxygen vacancies (yellow iso-surface). 
+The presence of oxygen vacancies introduces localized mid-gap states deriving from the 
+hybridization of O 2p and Ti 3d orbitals (Fig. 3C and Fig. S46) with their energy position that 
+
+A
+B
+/3
+V4
+V5
+Intensity (a.u.)
+-3
+-2
+V6
+-1
+0
+V7
+E-E (eV)
+V8
+-10
+-8
+9-
+-4
+2
+0
+E-E (eV)
+D
+(210)ax.dist.
+V3
+DOs (a.u.)
+(210) rdm.dist.
+V
+V5
+V
+(210)
+A
+-2
+0
+-2
+-4
+Energy (eV) 
+19 
+ 
+varies with respect to the defect’s distance from the surface. In contrast, the primary effect of the 
+expansion of Ti-O axial distances is to produce strong band tailing near the VB edge (Fig. 3C) 
+entering the band gap by ~0.4 eV. When we introduced a lattice disorder by random displacements 
+of both Ti and O atoms from their equilibrium positions, both mid-gap states and VB tailing were 
+seen in the DOS (Fig. 3C). The computational results confirmed that the DOS envelope of the 
+reduced brookite was formed by mid-gap states and VB band tailing due to the combined effect of 
+oxygen vacancies and lattice distortions. The location of oxygen vacancies in the real samples is 
+represented by a statistical distribution of lattice positions. Each of such defect populations 
+produce different DOS and their convolution, alongside the effect from lattice distortions, results 
+in the formation of the VB tailing. 
+We also examined the differential electron densities due to the introduction of an oxygen vacancy 
+at two different positions in the slab, namely, surface/near-surface (V3) or sub-surface (V5) 
+positions (Fig. 3B and 3D). In both cases, the excess of charge was spread over many lattice sites 
+and accompanied by the relaxation of the lattice atoms by up to 2–4% of the equilibrium Ti-O 
+bond length, thus denoting the generation of a large electron polaron around the oxygen vacancies.  
+We propose that these kind of bound states between oxygen vacancies and large electron polarons 
+represent the substrate-specific photocatalytic active sites for methanol oxidation. The size of this 
+photocatalytic active site and the charge distribution around it make it a well-defined reactive 
+pocket with high specificity toward the methanol molecule rather than to higher alcohols. The 
+finding of Zhang and co-workers support our results, as they recently observed a similar reactivity 
+pattern in Cu-doped TiO2 nanosheets, where the oxygen vacancies within a strained environment 
+enabled strong chemisorption and activation of molecular N2 and water, resulting in high 
+photocatalytic NH3 evolution under visible-light irradiation.8 Diebold and co-workers recently 
+reported the photo-oxidation mechanism of methanol at the surface of anatase TiO2.34 Values 
+
+ 
+20 
+ 
+obtained from DFT calculations, scanning tunneling microscopy, and temperature programmed 
+desorption aided by XPS showed the existence of two different, more favorable pathways for 
+activating methanol adsorbed on TiO2. The methanol molecules are first adsorbed onto the surface 
+Ti5c atoms dissociating into methoxy groups and hydrogen atoms, which are then oxidized to 
+formaldehyde (and eventually to formic acid and carbon dioxide) and molecular hydrogen. 
+Methanol molecules must first dissociate into methoxy groups, and after this step, the hole transfer 
+from TiO2 becomes energetically favorable. Methanol can be activated via two pathways (Fig. 
+S47): (A) by reaction with dissociated H2O forming terminal OH– species bound to surface Ti5c 
+atoms, and (B) by reaction with activated adsorbed O2.Error! Reference source not found. M
+echanism (A) begins with the spontaneous dissociative adsorption of water enabled by the extra 
+charge density due to oxygen vacancies and reflected by the formation of hydroxyl ions.34,35 
+Interestingly, brookite TiO2(210) (the same crystallographic direction expressed on the lateral 
+facets of our brookite nanorods) has the same structural building block of anatase TiO2(101), but 
+interatomic distances are slightly shorter and the blocks are arranged in a different way. Selloni 
+and co-workers found that these differences significantly change the reactivity toward adsorption 
+of water (and formic acid), making its dissociation more possible to occur on the brookite surface 
+rather than on the anatase.36 This may underlie the enhanced specific photocatalytic activity that 
+we observed for the anisotropic brookite over the anatase during methanol photoreforming. This 
+scenario is corroborated by our synchrotron XPS PES of the O 1s region that shows a significant, 
+24% increase in the OH– species adsorbed on the reduced surface in comparison with the pristine 
+brookite. These species may be derived from the dissociative adsorption of water, which is more 
+favored on the reduced brookite due to the extra electrons provided by the subsurface oxygen 
+vacancies.32 Mechanism (B) is less probable, as our experiments are performed in the absence of 
+oxygen (under Ar atmosphere). However, it should be noted that some traces of peroxide species 
+
+ 
+21 
+ 
+were detected by EPR,12,37 suggesting that mechanism B may occur even at a lower extent than 
+mechanism A. This could also point to a faster decomposition of hydrogen peroxide to water and 
+bridging oxygen dimer (step (iii) → (iv) in mechanism B) in the most photoactive sample (B700) 
+after illumination. Finally, besides the pure A and B mechanisms, an intermediate case can be also 
+considered, in which the OH– formation results from the reaction of coadsorbed O2 and H2O.38 
+CONCLUSIONS 
+In summary, we demonstrated the concept of enhancing the photocatalytic activity during alcohol 
+photoreforming by engineering the defect sites in an anisotropic brookite in a way that enables 
+substrate-specific oxidation photo-activity. Synchrotron photoemission spectroscopy and in situ 
+photoluminescence investigations aided by DFT calculations showed that creating a low amount 
+of defects (i.e. oxygen vacancies) in well-defined lattice positions produces a kind of bound states 
+between oxygen vacancies and large electron polarons hosting the photocatalytic active sites, 
+which act as shallow hole traps during alcohol photoreforming. Our results also demonstrate that 
+the nature of the produced defects/photocatalytic sites varies with respect to the selected TiO2 
+polymorph and on its crystal shape. This work highlights the value of analyzing the reaction 
+products of both the reductive and the oxidative pathways during photocatalytic reactions 
+alongside opening new avenues for substrate-selective photocatalytic biomass conversion through 
+the atomic design of the active sites. 
+ 
+ 
+ 
+
+ 
+22 
+ 
+Acknowledgments: A.N. and R.Z. gratefully acknowledge the support of the Czech Science 
+Foundation (GACR) through the projects no. 20-17636S and 19-27454X. The authors gratefully 
+acknowledge the support by the Operational Programme Research, Development and Education - 
+European Regional Development Fund, project no. CZ.02.1.01/0.0/0.0/15_003/0000416 of the 
+Ministry of Education, Youth and Sports of the Czech Republic. P.F. acknowledge financial 
+support from the European Community (projects H2020 – RIA-CE-NMBP-25 Program – Grant 
+No. 862030 – and H2020-LC-SC3-2019-NZE-RES-CC – Grant No. 884444), INSTM consortium 
+and ICCOM-CNR. P.M.S. and P.M. gratefully acknowledge financial support through the project 
+EUROFEL-ROADMAP ESFRI. The authors gratefully acknowledged G. Zoppellaro and O. 
+Tomanec for EPR discussion and TEM measurements, respectively.  
+Author contributions:  
+Conceptualization: PF, AN 
+Methodology: SMHH, MA, EM, PB 
+Investigation: SMHH, MS, PMS, PM, ZB, SK, EM, PB 
+Visualization: SMHH, PB, AN 
+Funding acquisition: RZ, SK, MO, PF, AN  
+Project administration: AN 
+Supervision: PF, AN 
+Writing – original draft: SMHH, AN 
+Writing – review & editing: SMHH, MA, PM, PB, RZ, MO, PF, AN 
+Competing interests: Authors declare that they have no competing interests. 
+Data and materials availability: All data are available in the main text or the supplementary 
+materials. 
+ 
+
+ 
+23 
+ 
+REFERENCES  
+1. Ciamician, G. (1912). The Photochemistry of the Future. Science 36, 385–394. 
+2. Ghosh, I., Khamrai, J., Savateev, A., Shlapakov, N., Antonietti, M., and König, B. (2019). 
+Organic semiconductor photocatalyst can bifunctionalize arenes and heteroarenes. Science 
+365, 360–366. 
+3. Luo, N., Montini, T., Zhang, J., Fornasiero, P., Fonda, E., Hou, T., Nie, W., Lu, J., Liu, J., 
+Heggen, M., et al. (2019). Visible-light-driven coproduction of diesel precursors and 
+hydrogen from lignocellulose-derived methylfurans. Nat Energy 4, 575–584. 
+4. Wang, Q., Hisatomi, T., Jia, Q., Tokudome, H., Zhong, M., Wang, C., Pan, Z., Takata, T., 
+Nakabayashi, M., Shibata, N., et al. (2016). Scalable water splitting on particulate 
+photocatalyst sheets with a solar-to-hydrogen energy conversion efficiency exceeding 1%. 
+Nature Mater 15, 611–615. 
+5. Kosco, J., Bidwell, M., Cha, H., Martin, T., Howells, C.T., Sachs, M., Anjum, D.H., 
+Gonzalez Lopez, S., Zou, L., Wadsworth, A., et al. (2020). Enhanced photocatalytic 
+hydrogen evolution from organic semiconductor heterojunction nanoparticles. Nat. Mater. 
+19, 559–565. 
+6. Wang, Z., Li, C., and Domen, K. (2019). Recent developments in heterogeneous 
+photocatalysts for solar-driven overall water splitting. Chem. Soc. Rev. 48, 2109–2125. 
+7. Zhang, N., Jalil, A., Wu, D., Chen, S., Liu, Y., Gao, C., Ye, W., Qi, Z., Ju, H., Wang, C., et 
+al. (2018). Refining Defect States in W 18 O 49 by Mo Doping: A Strategy for Tuning N 2 
+Activation towards Solar-Driven Nitrogen Fixation. J. Am. Chem. Soc. 140, 9434–9443. 
+8. Zhao, Y., Zhao, Y., Shi, R., Wang, B., Waterhouse, G.I.N., Wu, L., Tung, C., and Zhang, T. 
+(2019). Tuning Oxygen Vacancies in Ultrathin TiO 2 Nanosheets to Boost Photocatalytic 
+Nitrogen Fixation up to 700 nm. Adv. Mater. 31, 1806482. 
+9. Takanabe, K. (2017). Photocatalytic Water Splitting: Quantitative Approaches toward 
+Photocatalyst by Design. ACS Catal. 7, 8006–8022. 
+10. Chen, X., Liu, L., Yu, P.Y., and Mao, S.S. (2011). Increasing Solar Absorption for 
+Photocatalysis with Black Hydrogenated Titanium Dioxide Nanocrystals. Science 331, 746–
+750. 
+11. Naldoni, A., Allieta, M., Santangelo, S., Marelli, M., Fabbri, F., Cappelli, S., Bianchi, C.L., 
+Psaro, R., and Dal Santo, V. (2012). Effect of Nature and Location of Defects on Bandgap 
+Narrowing in Black TiO2 Nanoparticles. J. Am. Chem. Soc. 134, 7600–7603. 
+12. Naldoni, A., Altomare, M., Zoppellaro, G., Liu, N., Kment, Š., Zbořil, R., and Schmuki, P. 
+(2019). Photocatalysis with Reduced TiO 2 : From Black TiO 2 to Cocatalyst-Free Hydrogen 
+Production. ACS Catal. 9, 345–364. 
+
+ 
+24 
+ 
+13. Wang, X., Maeda, K., Thomas, A., Takanabe, K., Xin, G., Carlsson, J.M., Domen, K., and 
+Antonietti, M. (2009). A metal-free polymeric photocatalyst for hydrogen production from 
+water under visible light. Nature Materials 8, 76–80. 
+14. Filippini, G., Longobardo, F., Forster, L., Criado, A., Carmine, G.D., Nasi, L., D’Agostino, 
+C., Melchionna, M., Fornasiero, P., and Prato, M. (2020). Light-driven, heterogeneous 
+organocatalysts for C–C bond formation toward valuable perfluoroalkylated intermediates. 
+Science Advances 6, eabc9923. 
+15. Li, H., Chen, S., Shang, H., Wang, X., Yang, Z., Ai, Z., and Zhang, L. (2020). Surface 
+hydrogen bond network spatially confined BiOCl oxygen vacancy for photocatalysis. 
+Science Bulletin 65, 1916–1923. 
+16. Wu, Y.A., McNulty, I., Liu, C., Lau, K.C., Liu, Q., Paulikas, A.P., Sun, C.-J., Cai, Z., Guest, 
+J.R., Ren, Y., et al. (2019). Facet-dependent active sites of a single Cu 2 O particle 
+photocatalyst for CO 2 reduction to methanol. Nature Energy 4, 957–968. 
+17. Cargnello, M., Montini, T., Smolin, S.Y., Priebe, J.B., Jaén, J.J.D., Doan-Nguyen, V.V.T., 
+McKay, I.S., Schwalbe, J.A., Pohl, M.-M., Gordon, T.R., et al. (2016). Engineering titania 
+nanostructure to tune and improve its photocatalytic activity. PNAS 113, 3966–3971. 
+18. Li, R., Zhang, F., Wang, D., Yang, J., Li, M., Zhu, J., Zhou, X., Han, H., and Li, C. (2013). 
+Spatial separation of photogenerated electrons and holes among {010} and {110} crystal 
+facets of BiVO 4. Nature Communications 4, 1432. 
+19. Chen, X., Shen, S., Guo, L., and Mao, S.S. (2010). Semiconductor-based Photocatalytic 
+Hydrogen Generation. Chem. Rev. 110, 6503–6570. 
+20. Kamat, P.V., and Jin, S. (2018). Semiconductor Photocatalysis: “ Tell Us the Complete 
+Story! .” ACS Energy Lett. 3, 622–623. 
+21. Schneider, J., and Bahnemann, D.W. (2013). Undesired Role of Sacrificial Reagents in 
+Photocatalysis. J. Phys. Chem. Lett. 4, 3479–3483. 
+22. Hainer, A.S., Hodgins, J.S., Sandre, V., Vallieres, M., Lanterna, A.E., and Scaiano, J.C. 
+(2018). Photocatalytic Hydrogen Generation Using Metal-Decorated TiO 2 : Sacrificial 
+Donors vs True Water Splitting. ACS Energy Lett. 3, 542–545. 
+23. Belhadj, H., Hamid, S., Robertson, P.K.J., and Bahnemann, D.W. (2017). Mechanisms of 
+Simultaneous Hydrogen Production and Formaldehyde Oxidation in H 2 O and D 2 O over 
+Platinized TiO 2. ACS Catal. 7, 4753–4758. 
+24. Xu, C., Paone, E., Rodríguez-Padrón, D., Luque, R., and Mauriello, F. (2020). Recent 
+catalytic routes for the preparation and the upgrading of biomass derived furfural and 5-
+hydroxymethylfurfural. Chem. Soc. Rev. 49, 4273–4306. 
+25. Wu, X., Li, J., Xie, S., Duan, P., Zhang, H., Feng, J., Zhang, Q., Cheng, J., and Wang, Y. 
+(2020). Selectivity Control in Photocatalytic Valorization of Biomass-Derived Platform 
+Compounds by Surface Engineering of Titanium Oxide. Chem 6, 3038–3053. 
+
+ 
+25 
+ 
+26. Naldoni, A., Montini, T., Malara, F., Mróz, M.M., Beltram, A., Virgili, T., Boldrini, C.L., 
+Marelli, M., Romero-Ocaña, I., Delgado, J.J., et al. (2017). Hot Electron Collection on 
+Brookite Nanorods Lateral Facets for Plasmon-Enhanced Water Oxidation. ACS Catal. 7, 
+1270–1278. 
+27. Di Valentin, C., Pacchioni, G., and Selloni, A. (2009). Reduced and n-Type Doped TiO2: 
+Nature of Ti3+ Species. J. Phys. Chem. C 113, 20543–20552. 
+28. Wierzbicka, E., Altomare, M., Wu, M., Liu, N., Yokosawa, T., Fehn, D., Qin, S., Meyer, K., 
+Unruh, T., Spiecker, E., et al. (2021). Reduced grey brookite for noble metal free 
+photocatalytic H2 evolution. J. Mater. Chem. A 9, 1168–1179. 
+29. Wang, H., Zhang, L., Chen, Z., Hu, J., Li, S., Wang, Z., Liu, J., and Wang, X. (2014). 
+Semiconductor heterojunction photocatalysts: design, construction, and photocatalytic 
+performances. Chem. Soc. Rev. 43, 5234–5244. 
+30. Huang, H., Feng, J., Zhang, S., Zhang, H., Wang, X., Yu, T., Chen, C., Yi, Z., Ye, J., Li, Z., 
+et al. (2020). Molecular-level understanding of the deactivation pathways during methanol 
+photo-reforming on Pt-decorated TiO2. Applied Catalysis B: Environmental 272, 118980. 
+31. Bad’ura, Z., Naldoni, A., Qin, S., Bakandritsos, A., Kment, Š., Schmuki, P., and Zoppellaro, 
+G. Light-Induced Migration of Spin Defects in TiO2 Nanosystems and their Contribution to 
+the H2 Evolution Catalysis from Water. ChemSusChem n/a. 
+32. Setvín, M., Aschauer, U., Scheiber, P., Li, Y.-F., Hou, W., Schmid, M., Selloni, A., and 
+Diebold, U. (2013). Reaction of O2 with Subsurface Oxygen Vacancies on TiO2 Anatase 
+(101). Science 341, 988–991. 
+33. Naldoni, A., Riboni, F., Marelli, M., Bossola, F., Ulisse, G., Carlo, A.D., Píš, I., Nappini, S., 
+Malvestuto, M., Dozzi, M.V., et al. (2016). Influence of TiO2 electronic structure and strong 
+metal–support interaction on plasmonic Au photocatalytic oxidations. Catal. Sci. Technol. 6, 
+3220–3229. 
+34. Setvin, M., Shi, X., Hulva, J., Simschitz, T., Parkinson, G.S., Schmid, M., Di Valentin, C., 
+Selloni, A., and Diebold, U. (2017). Methanol on Anatase TiO2 (101): Mechanistic Insights 
+into Photocatalysis. ACS Catal. 7, 7081–7091. 
+35. Selcuk, S., and Selloni, A. (2016). Facet-dependent trapping and dynamics of excess 
+electrons at anatase TiO2 surfaces and aqueous interfaces. Nature Mater 15, 1107–1112. 
+36. Li, W.-K., Gong, X.-Q., Lu, G., and Selloni, A. (2008). Different Reactivities of TiO 2 
+Polymorphs: Comparative DFT Calculations of Water and Formic Acid Adsorption at 
+Anatase and Brookite TiO 2 Surfaces. J. Phys. Chem. C 112, 6594–6596. 
+37. Naldoni, A., D’Arienzo, M., Altomare, M., Marelli, M., Scotti, R., Morazzoni, F., Selli, E., 
+and Dal Santo, V. (2013). Pt and Au/TiO2 photocatalysts for methanol reforming: Role of 
+metal nanoparticles in tuning charge trapping properties and photoefficiency. Applied 
+Catalysis B: Environmental 130–131, 239–248. 
+
+ 
+26 
+ 
+38. Setvin, M., Aschauer, U., Hulva, J., Simschitz, T., Daniel, B., Schmid, M., Selloni, A., and 
+Diebold, U. (2016). Following the Reduction of Oxygen on TiO2 Anatase (101) Step by 
+Step. Journal of the American Chemical Society 138, 9565–9571. 
+ 
+
+S1 
+ 
+ 
+Supporting Information  
+ 
+Defect engineering over anisotropic brookite towards substrate-specific photo-
+oxidation of alcohols 
+ 
+S. M. Hossein Hejazi1, Mahdi Shahrezaei1, Piotr Błoński1, Mattia Allieta2, Polina M. 
+Sheverdyaeva3, Paolo Moras3, Zdeněk Baďura1, Sergii Kalytchuk1, Elmira Mohammadi1, Radek 
+Zbořil1,4, Štěpán Kment1,4, Michal Otyepka1,5, Alberto Naldoni1*, Paolo Fornasiero6,7*,  
+ 
+1Czech Advanced Technology and Research Institute, Regional Centre of Advanced 
+Technologies and Materials, Palacký University Olomouc, Křížkovského 511/8, 77900 
+Olomouc, Czech Republic 
+2Ronin Institute Montclair, NJ 07043 USA 
+3Istituto di Struttura della Materia-CNR (ISM-CNR), SS 14, Km 163,5, I-34149, Trieste, Italy 
+4Nanotechnology Centre, Centre of Energy and Environmental Technologies, VŠB–Technical 
+University of Ostrava, 17. listopadu 2172/15, 70800 Ostrava-Poruba, Czech Republic 
+5IT4Innovations, VSB – Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava-
+Poruba, Czech Republic 
+6Department of Chemical and Pharmaceutical Sciences, ICCOM-CNR Trieste 
+Research Unit, INSTM-Trieste, University of Trieste, Via L. Giorgieri 1, 34127 
+Trieste, Italy  
+7Center for Energy, Environment and Transport Giacomo Ciamician - University of Trieste, Italy 
+ 
+*Corresponding authors: alberto.naldoni@upol.cz; pfornasiero@units.it 
+ 
+ 
+ 
+
+S2 
+ 
+Preparation of TiO2 photocatalysts 
+Titanium (IV) bis (ammonium lactate) dihydroxide Ti(NH4C3H4O3)2(OH)2 aqueous solution (50 wt.%, 
+Sigma–Aldrich) (TALH) and urea (ACS reagent, Sigma–Aldrich), were used as precursors for the synthesis 
+of TiO2 nanocrystals using a hydrothermal method, according to the procedure previously reported with 
+some modifications 1–4. A solution containing 45 mL of urea in deionized (DI) water and 5 mL of TALH 
+was stirred until a clear solution was obtained. The solution was afterwards transferred to a 125 mL Teflon 
+lined autoclave and placed in an oil bath at 180°C and stirred at this temperature with 800 rpm for 20 days. 
+The autoclave was then cooled down in air and the precipitate was centrifuged and dispersed by sonication 
+in DI water for several times until the pH of supernatant water became ~7. Finally, the precipitate was dried 
+at 80°C for 12 h. To prepare pure brookite and pure anatase samples, 0.15M and 11.5M urea solution in DI 
+water were used, respectively. Commercial TiO2 brookite was purchased from Sigma-Aldrich (99.99 wt. 
+% purity). To prepare the reduced powders, 20 mg of TiO2 nanopowders were placed in a crucible within 
+a quartz chamber in a tubular furnace (10 °C min-1 heating/cooling ramp in N2 flow rate 10 mL min-1, 1 h 
+dwell in H2 flow rate 10 mL min-1 at predefined temperature. Before starting the heat treatment, the tube 
+furnace was cleaned up increasing the temperature up to 1000°C in air. 1 wt.% platinum nanoparticles were 
+loaded on TiO2 by via photodeposition method. Briefly, 50 mg of TiO2 powder suspended in 25 mL of 
+methanol (ACS reagent, Sigma–Aldrich) and bubbled with Ar for 30 min. Then, a solution of H2PtCl6.6H2O 
+(ACS reagent, Sigma–Aldrich) was added and stirred for 20 min in the dark to favor Pt adsorption of the 
+TiO2 surface. Then the solution was illuminated for 1h using a solar simulator equipped with a 150 W Xe 
+arc lamp and an AM 1.5G filter and calibrated to deliver a power of 100 mW cm-2 (1 Sun). 
+ 
+Characterization 
+The morphological analyses of the samples were performed by transmission electron microscopy (TEM) 
+JEM-2100 (JEOL, Tokyo, Japan) at 200 kV of accelerating voltage. For TEM measurements, the samples 
+were dispersed in ethanol by sonication for 5 minutes and then the suspensions were dropped on the copper 
+grid with holey carbon film and dried upon air exposure. The average particle size of brookite nanorods 
+were assessed by analyzing TEM micrographs and by considering at least 100 nanorods. The high 
+resolution transmission electron microscopy (HRTEM) analysis were performed using a HRTEM Titan G2 
+(FEI) with image corrector on accelerating voltage 300 kV. Images were taken with BM UltraScan CCD 
+camera (Gatan). 
+X-ray diffraction (XRD) patterns were recorded at room temperature with an Empyrean (PANalytical, 
+Almelo, The Netherlands) diffractometer in the Bragg-Brentano geometry and using Co-Kα radiation (40 
+kV, 30 mA, λ = 0.1789 nm). The diffractometer was equipped with a PIXcel3D detector and programmable 
+divergence and diffracted beam anti-scatter slits. The same amount of powders was placed on a zero-
+background Si slide. The measurement range was 2θ = 10° - 100°, with a step size of 0.0167° and acquisition 
+time of 4 s per step. Standards SRM640 (Si) and SRM660 (LaB6) were used to evaluate the line position 
+and the instrumental line broadening, respectively. The identification of crystalline phases was performed 
+using the High Score Plus software that includes the PDF-4+ and ICSD databases. Rietveld analysis was 
+performed through the GSAS program 5. We use the brookite orthorhombic model of Pbca space group 
+with Ti and two O, namely O1, O2, in 8c position (x,y,z) 6. During the refinement, the background was 
+subtracted using shifted Chebyshev polynomials and the diffraction peak profiles were fitted with a 
+modified pseudo-Voigt function. In the last calculation cycles all the parameters were refined: cell 
+parameters, atomic positional degrees of freedom, isotropic thermal parameters, anisotropic microstrain 
+broadening parameters, background, diffractometer zero point. To evaluate annealing T evolution ionic 
+charge of Ti from the experimental dTi-O1, dTi-O2 of brookite, we calculated Bond Valence Sum (BVS) 
+by using the tabulated parameters 7. Crystallite size of synthesized brookite samples was estimated through 
+the Williamson-Hall (WH) method 8 employing at least 15 reflections for each calculation. Single peak 
+fitting to extract peak positions and profile parameters was performed through the WinPLOTR Software 9. 
+The crystallite size of as received and reduced commercial brookite and anatase was calculated from XRD 
+patterns according to the Scherrer equation as follows: 
+
+S3 
+ 
+𝐷 =
+𝐾 × 𝜆
+𝛽 × 𝑐𝑜𝑠𝜃 
+where, D is the mean size of crystallite, K is the dimensionless shape factor, λ is the x-ray wavelength, β is 
+the full width half maximum intensity (FWHM), and θ is the Bragg angle and considering K=0.9, λ=1.79 
+Å (Co). 
+Raman spectra were collected using a DXR Raman spectrometer (Thermo Scientific, Massachusetts, USA). 
+The excitation laser operated at the wavelength of 455 nm. The samples were deposited on a silicon wafer 
+and the laser was focused on its surface and tuned to maximize the signal. The laser power on the sample 
+was set to 0.1 mW cm-2 and exposure time was 3 s. The reported Raman spectra were averaged over 512 
+experimental microscans. 
+The surface area and pore size analyses were performed by means of N2 adsorption/desorption 
+measurements at 77 K on a volumetric gas adsorption analyzer 3 Flex (Micromeritics, Georgia, USA) up 
+to 0.965 P/P0. Prior the analysis, the sample was degassed under high vacuum (10-4 Pa) at 130°C for 12 
+hours, while high purity (99.999 %) N2 and He gases were used for the measurements. The Brunauer–
+Emmett–Teller area (BET) was determined with respect to Rouquerol criteria 10 for N2 isotherm. 
+The ultraviolet-visible diffuse reflectance spectra (UV-Vis DRS) of the fabricated samples were obtained 
+by Specord 250 plus (Analytik Jena, Jena, Germany) spectrophotometer. An integrating sphere was used 
+to collect the spectrum and a Spectralon reference sample was used to measure the background. 
+X-ray photoelectron spectroscopy (XPS) was carried out with a PHI 5000 VersaProbe II (Physical 
+Electronics, Chanhassen, USA) spectrometer using an Al Kα source (15 kV, 50 W). The obtained data were 
+evaluated with the MultiPak software package (Ulvac-PHI Inc., Chigasaki, Japan). High-resolution spectra 
+of C1s peaks were acquired by setting the pass energy to 23.500 eV and step size to 0.200 eV. The binding 
+energy values were corrected considering the C1s peak at 284.8 eV as a reference. The spectral analysis 
+included Shirley background subtraction and peak deconvolution using Gaussian functions. 
+Photoluminescence spectroscopy (PL) was performed on an FLS980 fluorescence spectrometer (Edinburgh 
+Instruments, Livingston, United Kingdom) equipped with a R928P photomultiplier (Hamamatsu, Japan), 
+with a 450 W xenon arc lamp as the excitation source for steady-state spectra and an EPL-375 picosecond 
+pulsed diode laser (λem= 372 nm with a pulse width of 66.5 ps, a repetition rate of 10 MHz and an average 
+power of 75 μW, Edinburgh Instruments) in conjunction with a time-correlated single-photon counting 
+system for time-resolved photoluminescence measurements. Spectral correction curves were provided by 
+Edinburgh Instruments. The emission of TRPL spectra were detected at 450 nm. PL decay curves were 
+fitted using a multi-exponential function:  
+𝐼(𝑡) = ∑
+𝐵𝑖 exp (−
+𝑡
+𝜏𝑖)
+𝑛
+𝑖=1
+, ∑
+𝐵𝑖 = 1
+𝑛
+𝑖=1
+, 
+Where, the fit parameter τi represents the decay time constant, Bi represents the normalized amplitude of 
+each component, n is the number of decay times. 
+The amplitude weighted average decay lifetime τave of the entire PL decay process reads as: 
+𝜏𝑎𝑣𝑒 =
+∑ 𝐵𝑖𝜏𝑖
+2
+∑ 𝐵𝑖𝜏𝑖
+ 
+A nitrogen bath cryostat holder Optistat/DNV (Oxford instruments, Abingdon, United Kingdom) was used 
+to control the temperature of sample during measurements. Since the PL emission of TiO2 is weak at room 
+temperature and due to highly scattering nature of a typical nano-TiO2 sample, the stray excitation light 
+could be wrongly assigned as PL signal 11. To avoid this, the PL spectra were measured at 80 K. Using low 
+temperature condition results in slower non-radiative decay and brighter PL 11, which decreases the effect 
+of stray light scattered by the sample. The powders in solid were pressed between two flat quartz and put 
+into the chamber. N2 and methanol were used to investigate the PL behavior of the TiO2 samples in contact 
+with different environments. The methanol was degassed with argon bubbling for 10 minutes before wetting 
+the sample.  
+An in-depth analysis of the electronic structure of the samples was carried out at the VUV-Photoemission 
+synchrotron beamline (Elettra, Trieste) at room temperature with a Scienta R-4000 electron spectrometer. 
+The O1s and Ti2p core levels were measured with photon b 650 eV with an instrumental energy resolution 
+
+S4 
+ 
+of 0.2 eV. The valence band was probed in near-resonant conditions to the Ti L2,3 edge, in order to enhance 
+the signal of Ti-related states. A photon energy of 468 eV (energy resolution 0.14 eV) was used to avoid 
+the appearance of spurious Ti2p signal in the region of interest (from 4 eV binding energy up to the Fermi 
+level), due to high harmonics contribution from the beamline. 
+Electron paramagnetic resonance (EPR) spectra were recorded using a continuous wave X-band JEOL JES-
+X-320 spectrometer operating at 9.1 GHz. The EPR spectrometer is equipped with a variable temperature 
+control ES 13060DVT5 apparatus. The cavity Q quality factor was kept above 6000 in all measurements. 
+Highly pure quartz tubes were used (Suprasil, Wilmad, ≤ 0.5 OD) and accuracy on g-values was obtained 
+against a Mn2+/MgO standard (JEOL standard). For all experiments the same acquisition conditions were 
+kept. The microwave power was set to 1.5 mW, therefore, no power saturation effects was occurring in the 
+EPR traces. The modulation width of 0.7 mT and modulation frequency 100 Hz were used. Experimental 
+temperature was set to 78 K. All spectra were recorded with 30 ms time constant and 2 minutes sweep time 
+with 3 accumulations, to improve signal to noise ratio. HeCd (200 mW) laser with 325 nm wavelength was 
+employed as the UV light source during EPR experiments. EPR envelopes were simulated in Matlab 
+software where the spin-Hamiltonian EasySpin simulation package 12 is implemented. During the 
+measuring the samples in contact with DI water + methanol solution at 80K, the head space of EPR tube 
+was purged by nitrogen to avoid any parasitic oxygen signals coming from the air. The EPR cavity was 
+kept in constant flow of nitrogen to eliminate the formation of ice. The samples were measured in 
+suspension form. For each experiment 10 mg of the powder and 100 µl of DI water/methanol mixture were 
+used. In all experiments, to make spectra more comparable, the position of the EPR tube inside the cavity 
+was kept the same. 
+1H-NMR (proton nuclear magnetic resonance) spectra were obtained on a JEOL 400 MHz spectrometer in 
+CD3OD, using dimethyl sulfoxide (DMSO) as the internal standard. All the measurements were collected 
+at ambient temperature with a spectral width of 20 ppm, a pulse width of 5.7 μs (90 °), a relaxation delay 
+of 60 s, and 8 scans. Chemical shifts (δ) are expressed in ppm. 
+C, H and N elemental analyses was performed on a Flash EA 1112 instrument (Thermo Finnigan, North 
+Carolina, USA). 
+Pt loading on TiO2 was determined by ICP-MS (Agilent 7700x, Agilent, USA) at isotope 105 using He 
+mode and an external calibration. Calibration solutions were prepared from a certified reference material 
+with Pt concentration 100,0 +/- 0,2 mg/L (Analytika Ltd., Czech Republic). A mixture of nitric acid (ACS 
+reagent, 70% , Sigma–Aldrich) and hydrochloric acid (ACS reagent, 37% , Sigma–Aldrich) in a molar ratio 
+of 1:3, was used to digest the Pt. The Pt loading in pristine and reduced brookite nanorods was 0.98 and 
+0.90%, respectively. 
+The hydrodynamic diameter of the brookite B700 was measured by Dynamic Light Scattering (DLS) at 
+23°C, using a Malvern Nano-ZS instrument (Malvern Ltd., Leamington Spa, UK). The light source was a 
+laser 633 nm, 4 mW. The measurement was performed at the beginning (time = 0 h) and after specific times 
+of reaction. The sample was taken from 10 mL of DI water and methanol (1:1 vol) solution with 2mg of 
+dispersed B700-Pt. At time = 0h the solution was ultrasonically dispersed for 10 min.  
+ 
+Photocatalytic experiments 
+Photocatalytic activity was measured in a quartz reactor with 10 mL solution of DI water and methanol 
+(volume ratio 1:1). The same conditions and volume ratio were applied for the photocatalytic reactions with 
+ethanol (99.8 %, BC Chemservic), isopropanol (ACS reagent, 99.8%, Sigma–Aldrich), formaldehyde (36-
+38%, Penta) and formic acid (99%, Penta). After sealing the reactor with a rubber septum, the photocatalysts 
+were sonicated for 10 min to create a homogeneous and dispersed suspension. Afterwards, the suspension 
+was bubbled with argon for 30 min to remove the unwanted gasses and dissolved oxygen. The samples 
+were irradiated using a solar simulator equipped with a 150 W Xe arc lamp and an AM 1.5G filter and 
+calibrated to deliver a power of 100 mW cm-2 (1 sun). A calibrated reference solar cell (Newport, California, 
+USA) was used before and after reaction to check the power of irradiation (1 Sun). Each sample was 
+irradiated for 24 h under continuous stirring before measuring the amount of evolved hydrogen. The test 
+was repeated three times and the average amount of measured hydrogen was reported. The photocatalytic 
+
+S5 
+ 
+hydrogen was detected with a gas chromatograph GCMS-QP2010 SE (Shimadzu, Kyoto, Japan ) and a 
+TCD (Thermal conductivity detector), using Ar as carrier gas. The temperature of the reaction suspension 
+was measured with a thermocouple and was 23°C both before and after 24 h of irradiation under 1 sun 
+illumination. 
+To identify the wavelength dependence of AQY, the reactor was illuminated with the wavelengths 316 (1.6 
+mW cm-2), 334 (1.2 mW cm-2), 360 (2.8 mW cm-2), 369 (8.7 mW cm-2), 386 (5 mW cm-2) and 402 nm (17.7 
+mW cm-2) using a tunable diode light source of Zahner CIMPS PP201 system. The illuminated surface area 
+was 0.94 cm2 and the power of each wavelength was measured using an external digital power meter 
+Thorlabs PM100D. The sample was illuminated for 1h and the reacting medium was a 1:1 vol/vol% 
+water:methanol mixture. The apparent quantum yield (AQY) was calculated according to the following 
+equation: 
+ 
+𝐴𝑄𝑌 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 × 100 = 2 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑣𝑜𝑙𝑣𝑒𝑑 𝐻2 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠
+𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝ℎ𝑜𝑡𝑜𝑛𝑠
+ 
+ 
+The qNMR (quantitative nuclear magnetic resonance) analysis was used to investigate the rate of methanol 
+consumption during the photocatalytic hydrogen evolution. For this purpose, a 10 mL mixture of DI water: 
+CD3OD (volume ratio 1:1) was prepared. To have a detectable signal, 50 μL of non-deuterated methanol 
+(i.e. CH3OH) was added to the above mixture. The photocatalytic experiment was performed following the 
+same procedure explained above for hydrogen evolution measurements. At the specified times, 600 μL of 
+solution was taken by syringe and centrifuged at 15000 rpm for 30min to separate the catalyst. Then the 
+clear and transparent solution was transferred to a quartz NMR tube and 0.2 µL DMSO was added as 
+internal standard. The NMR spectra of the samples were recorded and the peak area of methanol at 3.32 
+ppm was compared to the peak area of internal standard to study the rate of photocatalytic methanol 
+consumption over time. 
+ 
+DFT calculations 
+Calculations were carried out by using the DFT-based Vienna Ab Initio Simulation Package (VASP) 13,14. 
+The projector-augmented-wave (PAW) formalism 15,16 was used to treat the electron–ionic core interactions. 
+A plane-wave basis with a 400 eV energy cutoff was used. Test calculations were also performed with 
+energy cutoff increased to 600 eV. Exchange and correlation effects were treated within a generalized-
+gradient approximation (GGA) by using Perdew-Burke-Ernzerhof (PBE) functional 17,18. To counteract the 
+problems of standard density functionals associated with the self-interaction error (SIE) we applied on-site 
+Hubbard corrections 19 to both Ti-d and O-p states 20 with an effective U parameter of 6 eV. All 
+computations were performed in spin unrestricted manner. Brillouin zone samplings were kept restricted to 
+Gamma point only due to the supercell dimensions being sufficiently large. We modeled the (210) surface 
+of brookite by a bulk-terminated slab of 12 Ti-layers in thickness and 14.38 Å × 15.41 Å of surface area 
+and containing 432 atoms (see the structure shown in Figure S41) 21, and with a vacuum layer of length 
+∼20 Å deployed along the off-planar direction to ward off spurious interactions with the periodic images. 
+Except atoms in the middle part of the slab (area enclosed by green planes in the structure displayed in 
+Figure S41), all other atoms were relaxed until all forces were reduced below 0.025 eV/Å and the change 
+in total energy between successive iteration steps became smaller than 10−6 eV. 
+Several possible sites for oxygen vacancy formation were considered (and denoted by V1–V8 in the 
+structure in Figure S41) and the system was re-optimized. For oxygen vacancy defects present in the bulk 
+region of the slab, atoms in the immediate vicinity of the vacancy were allowed to relax too.  
+The effect of distortion of TiO6 octahedra on the electronic structure of brookite was also considered. Two 
+models were considered: (i) Several Ti-O axial distances were modified by up to ±0.1 Å and the change in 
+densities of states with respect to an ideal (210) surface of brookite was monitored. (ii) Randomly chosen 
+Ti-O distances were modified within ±0.1 Å and accompanied by a displacement of Ti atoms from its 
+equilibrium positions. 
+
+S6 
+ 
+Supplementary Text 
+ 
+Digital pictures of TiO2 photocatalysts 
+Upon reduction under hydrogen atmosphere at different temperatures, the color of synthesized brookite and 
+anatase samples changed from white to gray and black, while the commercial brookite showed a relatively 
+small color change (Table S1), suggesting its non-reducibility, as also confirmed by the other 
+characterizations reported below. 
+ 
+Table S1. Digital photographs of as synthesized brookite (B-AS), as-received commercial brookite (CB-
+AR), as-synthesized anatase (A-AS) and reduced samples at different temperatures. The numbers after 
+abbreviations stand for reducing temperature in °C. 
+Sample 
+ 
+Sample 
+ 
+Sample 
+ 
+B-AS 
+ 
+CB-AR 
+ 
+A-AS 
+ 
+B500 
+ 
+CB400 
+ 
+A400 
+ 
+B600 
+ 
+CB500 
+ 
+A500 
+ 
+B700 
+ 
+CB600 
+ 
+A600 
+ 
+B800 
+ 
+CB700 
+ 
+A700 
+ 
+B900 
+ 
+CB800 
+ 
+ 
+ 
+B1000 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+CHN elemental analysis 
+To exclude the contribution of carbon, nitrogen, and hydrogen impurities in the photocatalytic activity, 
+CHN analysis of TiO2 samples were carried out before and after reduction under hydrogen atmosphere for 
+the most active photocatalysts. As Table S2 shows, there is no significant difference between carbon, 
+nitrogen, and hydrogen concentration in the samples before and after the thermal treatment in hydrogen. 
+ 
+
+S7 
+ 
+Table S2. CHN analysis of the pristine and the most photoactive sample for brookite, commercial brookite 
+and anatase. 
+Sample 
+C (wt. %) 
+H (wt. %) 
+N (wt. %) 
+B-AS 
+0.25 ± 0.08 
+0.27 ± 0.03 
+0.12 ± 0.03 
+B700 
+0.19 ± 0.02 
+0.07 ± 0.01 
+0.03 ± 0.01 
+CB-AR 
+1.22 ± 0.04 
+0.83 ± 0.02 
+0.07 ± 0.01 
+CB600 
+0.73 ± 0.05 
+0.08 ± 0.01 
+0.09 ± 0.03 
+A-AS 
+1.03 ± 0.06 
+1.43 ± 0.09 
+0.11 ± 0.02 
+A-500 
+0.39 ± 0.06 
+0.16 ± 0.05 
+0.06 ± 0.02 
+ 
+Morphology of TiO2 photocatalysts 
+Figure S1A and Figure S1B show the TEM images of B-AS and B700, respectively. The as-synthesized 
+brookite nanorods have an average length of 90 ± 35 nm that after reduction under hydrogen atmosphere at 
+700°C decreased to 82 ± 28 nm. However, the reduction in high temperature resulted in aggregation of 
+particles due to sintering as well as losing their well-defined facets. Figure S2A and Figure S2B present the 
+TEM images of CB-AR and CB600, respectively. The particle shape of commercial brookite is rounded 
+(average diameter 29 ± 10 nm) in comparison with the synthesized brookite and showed almost no change 
+in shape and a slightly increase in size after reduction (average diameter 32 ± 7 nm). The as synthesized 
+anatase is spherical in shape with average diameter of 6 ± 1 nm (Figure S3) and its shape remained 
+unchanged with slighty increase in diameter after reduction (average diameter 8 ± 2 nm). 
+
+S8 
+ 
+ 
+Figure S1. TEM images of (A) B-AS and (B) B700 with associated histograms of size distributions 
+(bottom) based on 100 measurements of nanorods length.  
+ 
+Figure S2. TEM images of (A) CB-AR and (B) CB600 with associated histograms of size distributions 
+(bottom) based on 100 measurements of particle diameter. 
+
+A
+B
+100nm
+100nm
+45
+45
+B-AS
+B700
+Counts
+30
+30
+15
+15
+0
+0
+20
+50
+80
+110
+140
+170
+20
+50
+80
+110
+140
+170
+Size (nm)
+Size (nm)A
+B
+50 nm
+50nm
+45
+CB-AR
+CB600
+45
+Counts
+30
+30
+15
+15
+0
+0
+10
+20
+30
+40
+50
+60
+10
+20
+30
+40
+50
+60
+Size (nm)
+Size (nm)S9 
+ 
+ 
+Figure S3. TEM images of (A) A-AS and (B) A500 with associated histograms of size distributions 
+(bottom) based on 100 measurements of particle diameter. 
+
+A
+B
+40 nm
+40nm
+50
+50
+A-AS
+A500
+Counts
+35
+35
+20
+20
+5
+5
+2
+4
+6
+8
+10
+12
+14
+16
+2
+4
+6
+8
+10
+12
+14
+16
+Size (nm)
+Size (nm)S10 
+ 
+ 
+Figure S4. TEM images of (A) platinized pristine brookite and (B) platinized reduced brookite at 700°C 
+with associated histograms of size distributions of Pt particles (bottom) based on 100 measurements of 
+particle diameter. The loaded reduced brookite present large Pt aggregates and therefore the size distribution 
+refers to the isolated Pt nanoparticles found in the sample. The photodeposition of Pt results clearly in 
+different results: pristine brookite present isolated homogeneously dispersed Pt nanoparticles, while 
+reduced brookite show mainly large Pt aggregates and some isolated Pt nanoparticles. 
+ 
+ 
+ 
+
+A
+B
+50 nm
+50 nm
+B-AS/Pt
+B700/Pt
+39
+39
+Counts
+Counts
+26
+26
+13
+13
+0
+0
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+Size (nm)
+Size (nm)S11 
+ 
+ 
+ 
+Figure S5. (A,B) TEM and HR-TEM (C-F) micrographs of pristine brookite nanorods loaded with 1 wt% 
+Pt showing their homogeneous distribution over the TiO2 matrix. The darker dots are the Pt nanoparticles. 
+ 
+ 
+ 
+Figure S6. (A-C) STEM-HAADF images and (D) elemental EDS mapping of (C) for pristine brookite 
+nanorods loaded with 1 wt% Pt showing their homogeneous distribution over the TiO2 matrix. 
+ 
+
+B
+C
+200
+200 nm
+nm
+D
+nmA
+B
+HAADF
+50 nm
+50nm
+60 nm
+Ti
+Pt
+Tilo
+Pt
+60nm
+60 nm
+60nm
+60 nmS12 
+ 
+ 
+Figure S7. (A-C) TEM and HR-TEM (D) of micrographs reduced brookite nanorods (at 700°C) loaded 
+with 1 wt% Pt showing their aggregation over the TiO2 matrix. The darker dots and nanostructured 
+aggregates are made by Pt. 
+ 
+ 
+ 
+ 
+Figure S8. (A-C) TEM and HR-TEM (D) of micrographs reduced brookite nanorods (at 700°C) loaded 
+with 1 wt% Pt showing their aggregation over the TiO2 matrix. The Pt nanocrystals are the brighter spots. 
+ 
+ 
+ 
+ 
+ 
+ 
+
+A
+B
+200nm
+200nm
+D
+50
+nm
+10nmA
+B
+50 nm
+20 nmS13 
+ 
+ 
+Figure S9. STEM-HAADF and elemental mapping images of reduced brookite nanorods (at 700°C) loaded 
+with 1 wt% Pt showing their aggregation over the TiO2 matrix.  
+ 
+ 
+ 
+Figure S10. STEM-HAADF and elemental mapping images of reduced brookite nanorods (at 700°C) 
+loaded with 1 wt% Pt showing their aggregation over the TiO2 matrix.  
+ 
+ 
+ 
+
+HAADF
+Ti
+90 nm
+90nm
+90 nm
+Pt
+Tio
+Pt
+90 nm
+90nmHAADF
+Ti
+60nm
+60nm
+60 nm
+Pt
+Tio
+Pt
+60nm
+60nmS14 
+ 
+Specific surface area measurement 
+Figure S11 shows the nitrogen sorption isotherm profiles of the samples. In all cases, the BET surface area 
+decreases after reduction under hydrogen atmosphere at high temperature (Table S3). 
+ 
+Figure S11. N2 adsorption/desorption type IV isotherms (mesoporous solids) at 77K for (A) B-AS and 
+B700, (B) CB-AR and CB600, and (C) A-AS and A500. 
+Table S3. Brunauer−Emmett−Teller (BET) specific surface area for the pristine and the most photoactive 
+sample of each studied phases of TiO2. 
+Sample 
+BET surface area 
+(m2 g-1) 
+anataseB-
+AS 
+67 
+B700 
+47 
+CB-AR 
+50 
+CB600 
+48 
+A-AS 
+273 
+A-500 
+189 
+
+A
+180
+B
+STP)
+STP)
+10
+8
+135
+-B-AS Adsorption
+CB-AR Adsorption
+-B-AS Desoption
+6
+CB-AR Desoption
+90
+(cm3
+4
+45
+2
+Quantity Adsorbed
+0
+0
+8
+135
+-B700 Adsorption
+CB600 Adsorption
+B700Desorption
+6
+CB600 Desorption
+90
+4
+45
+2
+0
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Pressure (p/p°)
+Relative Pressure (p/p°)
+C
+P180
+90
+45
+-A-As Adsorption
+0
+-A-AS Desoption
+135
+A500Adsorption
+A500 Desorption
+90
+45
+0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Relative Pressure (p/p°)S15 
+ 
+Structural characterization 
+X-ray diffraction  
+XRD analysis (Figure S12 and Table S4) revealed that the synthesized brookite is crystalline and stable up 
+to 700°C. The sample treated at 800°C, instead, presented around 91 wt. % of rutile phase, while at 900°C 
+almost all the rutile is transformed to Magnéli phase Ti9O17. At 1000°C, the other Magnéli phases Ti4O7 in 
+94 wt. % and Ti5O9 in 6 wt. % were generated. 
+The phase stability of commercial brookite is up to 700°C (the same as synthesized brookite) and after that 
+reduction at 800°C induced the complete conversion to rutile (Figure S13 and Table S5). The XRD pattern 
+of as synthesized and reduced anatase (Figure S14) samples indicates that the as synthesized anatase 
+photocatalysts are stable at up to 500°C , while almost full conversion to rutile is obtained at 700°C (Table 
+S6). The average particle sizes obtained using the Scherrer method for commercial brookite and anatase 
+samples are in good agreement with those retrieved from TEM micrographs analysis. 
+ 
+Figure S12. (A) XRD patterns of brookite samples and reference patterns for brookite (bottom) and rutile 
+(top). (B) XRD patterns for reduced brookite at 900 and 1000 °C and reference patterns (bottom) for rutile 
+and various magnéli phases (TixO2x-1). 
+ 
+Figure S13. XRD patterns for commercial brookite samples and reference patterns for brookite (bottom) 
+and rutile (top). 
+
+A
+B
+41-008-7847
+Rutile
+B1000
+Intensity (a.u.)
+B800
+B700
+B900
+B600
+(Ti,O1z) 00-050-0791
+B500
+(Ti,Og)04-005-4465
+B-AS
+(Ti,07) 04-005-4521
+01-071-6410
+(Rutile) 01-071-6410
+Brookite
+20
+40
+60
+80
+20
+40
+60
+80
+20 (°), Co-kα
+20 (°), Co-kα(Rutile) 41-008-7847
+CB800
+Intensity (a.u.)
+CB700
+CB600
+CB500
+CB400
+CB-AR
+(Brookite) 01-071-641Q
+20
+40
+60
+80
+20 (°), Co-kαS16 
+ 
+ 
+Figure S14. XRD patterns for anatase samples and reference patterns for anatase (bottom) and rutile (top). 
+ 
+Table S4. Phase composition of brookite samples retrieved from Rietveld refinement of XRD patterns. 
+Sample 
+Brookite 
+(wt.%) 
+Rutile 
+(wt.%) 
+Ti4O7 
+(wt.%) 
+Ti5O9 
+(wt.%) 
+Ti9O17 
+(wt.%) 
+B-AS 
+100 
+- 
+- 
+- 
+- 
+B500 
+100 
+- 
+- 
+- 
+- 
+B600 
+100 
+- 
+- 
+- 
+- 
+B700 
+100 
+- 
+- 
+- 
+- 
+B800 
+9 
+91 
+- 
+- 
+- 
+B900 
+- 
+9 
+- 
+- 
+91 
+B1000 
+- 
+- 
+94 
+6 
+- 
+ 
+Table S5. Rietveld quantitative phase analysis of commercial brookite samples and crystallite size 
+calculated according to the Scherrer method. 
+Sample 
+Brookite 
+(wt.%) 
+Anatase 
+(wt.%) 
+Rutile 
+(wt.%) 
+Crystallite size 
+(nm) 
+CB-AR 
+100 
+- 
+- 
+27 
+CB400 
+100 
+- 
+- 
+27 
+CB500 
+100 
+- 
+- 
+28 
+CB600 
+100 
+- 
+- 
+31 
+CB700 
+100 
+- 
+- 
+41 
+CB800 
+- 
+- 
+100 
+69 
+ 
+ 
+ 
+
+41-008-7847
+Rutile
+A700
+Intensity (a.u.)
+A600
+A500
+A400
+A-AS
+01-071-6410
+Anatase
+20
+40
+60
+80
+20 (°), Co-kαS17 
+ 
+Table S6. Rietveld quantitative phase analysis of anatase samples and crystallite size calculated according 
+to the Scherrer method. 
+Sample 
+Anatase 
+(wt.%) 
+Rutile 
+(wt.%) 
+Crystallite size 
+(nm) 
+A-AS 
+100 
+- 
+8 
+A400 
+100 
+- 
+8 
+A500 
+100 
+- 
+13 
+A600 
+68 
+32 
+24 
+A700 
+- 
+100 
+28 
+To investigate more in depth the structural modifications induced by the reduction treatment, we performed 
+detailed Rietveld refinements for the samples treated up to 700°C (Figure S15). All of the patterns are well 
+described by single phase of brookite TiO2 and, within the resolution of our measurements, we did not 
+detect any spurious crystalline phases. 
+Figure S1A and B shows the reduction temperature (T) dependence of the average particle size, D (nm), 
+and the average strain as obtained using the Williamson-Hall method. We observed a clear decrease of D 
+for T = 400°C followed by an increase of average particle size above this T. On the other hand, the average 
+particle strain weakly decreases at T=400°C keeping roughly the same value at higher T. This phenomenon 
+can be associated to particle sintering since the sudden aggregation of particle can result in an increase of 
+D owing to relieve of intraparticle tension by decreasing the overall particle strain. This is confirmed by the 
+annealing temperature evolution of component of empirical extension of anisotropic microstrain 
+broadening tensor refined by Rietveld refinements (Figure S1C). Indeed, we note a progressive convergence 
+of these parameters to the same values indicating that the evolution of Williamson-Hall strain is explained 
+by the tendency to form more isotropic particle aggregates at high annealing temperature. This 
+morphological observation is in agreement with the TEM micrographs of reduced brookite B700 (Figure 
+S1B), which confirm that brookite nanorods tends to aggregate in isotropic agglomerate upon reduction. 
+The tendency toward powder aggregation can also explain the disagreement between the average nanorod 
+lengths detected by TEM, both before and after reduction, and the average particle size retrieved by Rietveld 
+refinements. 
+Figure S1D shows the annealing temperature evolution of the refined lattice parameters. Orthorhombic 
+strain is defined as: 
+𝜂 = 2(𝑎 − 𝑏)
+(𝑎 + 𝑏)  
+ 
+where a and b are the unit cell axes. We observed that  remains almost unchanged for all the investigated 
+annealing temperature whereas the c-axis shows clear decrease. In other words, this indicates that the 
+annealing temperature does not affect the ab plane but induces distortion along the c-axis only. This 
+anisotropic evolution of structural parameters upon reduction is even better outlined by the interatomic 
+distances related to the TiO6 octahedra. TiO2 structure is composed of TiO6 octahedra, each with a titanium 
+atom at its center and oxygen atoms at its corners (Figure S1A). In brookite TiO6 are distorted oxygen atoms 
+in two different positions, namely O1, O2. This results in two groups of distances namely dTi-O1, dTi-O2 
+(Figure S1B, C) which can be regrouped into two axial and four equatorial different interatomic distances 
+which have been averaged out and shown in Figure S1. Averaged axial distance expand upon increasing 
+temperature, whereas the average equatorial distances show a weak contraction. Both distances tend to the 
+same value and this may indicate that the annealing process induces the TiO6 to be more regular, i.e. 
+isotropic, and less distorted at high temperature. To figure out the effect of reduction treatment on the 
+brookite structure we argue that the reduction of TiO2 at high temperature creates oxygen vacancies (𝑉𝑂,  
+
+S18 
+ 
+in Kröger–Vink notation) 22,23 and induces Ti3+ ions electron trapped in Ti4+ lattice sites ( 𝑇𝑖𝑇𝑖
+′  ) according 
+to the following relation: 
+ 
+𝑂𝑂
+𝑥 + 2𝑇𝑖𝑇𝑖
+𝑥 + 𝐻2  → 𝑉𝑂
+•• + 2𝑇𝑖𝑇𝑖
+′ + 𝐻2𝑂 
+ 
+Two negatively charged 𝑇𝑖𝑇𝑖
+′  species can then couple with one double positive charged 𝑉𝑂
+•• promoting the 
+formation < 𝑇𝑖𝑇𝑖
+′ − 𝑉𝑂
+•• − 𝑇𝑖𝑇𝑖
+′ > defects. The mutual attraction between them can produce a contraction 
+of the structure to account for electron neutrality, which is fully in agreement with the observed c-axis 
+contraction upon reduction at high temperature. The reduction of Ti valence to produce 𝑇𝑖𝑇𝑖
+′  species, is 
+compatible with a depletion of ≈ - 0.5% as evidenced by bond valence sum (BVS) analysis, see for B700 
+(Figure S1A). Further, the correlation between the BVS and c-axis contraction is shown in Figure S1B, 
+showing a nearly linear relationship between these two parameters and thus indicating that more 𝑉𝑂𝑠 are 
+induced by the reduction and more the c-axis resulted to be contracted. This highlights that reduction 
+treatment introduced an anisotropic and preferential deformation of the brookite lattice along the c-axis. 
+ 
+Figure S15. Rietveld refinements of brookite samples reduced at different temperatures. Dots are 
+experimental data; continuous lines are the calculated profiles; Rietveld agreement factors [R (F2)] between 
+observed and calculated patterns ranged from 0.06 to 0.08. 
+
+Intensity (a.u.)
+B-AS
+Experimental
+Calculated
+Residual
+Intensity (a.u.)
+B400
+Intensity (a.u.)
+B500
+Intensity (a.u.)
+B600
+Intensity (a.u.)
+B700
+20
+30
+40
+50
+60
+70
+80
+90
+100
+20 (), Co-kaαS19 
+ 
+ 
+Figure S16. Structural parameters retrieved from Rietveld refinements of XRD patterns for as synthesized 
+brookite and brookite reduced at different temperatures (i.e. annealing T in x axis). (A) Reducing 
+temperature dependence of average particle size (D), and (B) crystal structure strain as obtained by 
+Williamson- Hall (WH) method. (C) Annealing temperature evolution of components of empirical 
+extension of anisotropic microstrain broadening tensor (L) refined by Rietveld refinements. Indexes are 
+referring to the following expression: L = L11h2+L22k2+L33l2+2L12hk+2L13hl+2L23kl. (D) Annealing 
+temperature dependence of orthorhombic strain (see text) and c-axis (inset). 
+ 
+ 
+
+A
+390
+B
+0.0014
+360
+0.0012
+330
+ Strain
+0.0010
+300
+D
+WH
+0.0008
+270
+0.0006
+240
+0.0004
+210
+0
+100
+200300400500
+600
+700
+800
+0
+100
+200300400500
+600
+700
+800
+Annealing T (°C)
+Annealing T (°C)
+C
+D
+0.4
+0.510
+0.3
+33
+E
+0.509
+Orthorombic strain (
+0.2 
+13
+23
+0.508
+Microstrain broadening 
+0.1
+0.507
+5.135
+0.0
+0.506
+5.134
+100200300
+400
+500600700800
+0.1
+Annealing T (°C)
+0.505
+0
+100
+200
+300
+400
+500
+600
+700
+800
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Annealing T (°C)
+Annealing T (°C)S20 
+ 
+ 
+Figure S17. (A) Representation of octahedra (TiO6) in TiO2 brookite unit cell (Pbca) showing the 
+arrangement of distorted TiO6 resulting in two groups of distances, namely dTi-O1 and dTi-O2, composed by 
+two axial and four equatorial different interatomic distances. Values refer to structure refined at room 
+temperature. (B) Evolution of axial and (C) equatorial interatomic distances retrieved from Rietveld 
+refinements of XRD patterns for as synthesized brookite and brookite reduced at different temperatures (i.e. 
+annealing T in x axis). Solid lines are guide to the eye. 
+
+A
+dT-011.913 A
+T-02 1.918 A
+Axial
+.Equatorial
+dT-01=1.943 A
+Ti-021.918A
+dt-011.955 A
+2.082A
+B
+2.02
+Axial
+Lo-!lp
+dTi-02
+2.00
+1.98
+1.96
+1.94
+1.92
+1.90
+0
+200
+400
+600
+800
+C
+Annealing T (°C)
+2.10
+Equatorial
+2.05
+dti-01
+dti-02
+di-01
+2.00
+1.95
+1.90
+1.85
+1.80
+0
+200
+400
+600
+800
+Annealing T (°C)S21 
+ 
+ 
+Figure S18. Annealing Temperature dependence of averaged axial and equatorial interatomic retrieved 
+from Rietveld refinements of XRD patterns for as synthesized brookite and brookite reduced at different 
+temperatures (i.e. annealing T in x axis). 
+ 
+ 
+Figure S19. (A) Annealing Temperature dependence of Ti Bond Valence Sum (BVS) and (B) correlation 
+between Ti BVS and c-axis contraction retrieved from Rietveld refinements of XRD patterns for as 
+synthesized brookite and brookite reduced at different temperatures (i.e. annealing T in x axis). 
+ 
+ 
+
+1.97
+1.96
+1.96
+1.95
+1.95
+1.94
+1.94
+Aaverage axial dti-o
+1.93
+Aaverage equatorial dti-o
+1.93
+0
+100
+200
+300
+400
+500
+600
+700
+800
+Annealing T (°C)A
+4.17
+B
+4.17
+4.16-
+[B-AS
+4.16.
+4.15
+4.15
+TB600
+TB400
+S
+ 4.14 .
+B700
+B500
+4.13
+4.13-
+4.12
+4.12
+4.11
+4.11-
+0
+100
+200
+300
+400
+500
+600
+700
+800
+5.134
+5.135
+5.136
+5.137
+5.138
+Annealing T (°C)
+c-axis (A)S22 
+ 
+Raman spectroscopy 
+The crystal system of brookite is orthorhombic and has eight formula units per unit cell and thirty-six 
+Raman active modes (9A1g+9B1g+9B2g+9B3g) 24. The crystal system of anatase is tetragonal and has two 
+formula units per unit cell and six Raman active modes (A1g+2B1g+3Eg) 24. The crystal system of rutile is 
+tetragonal and has two formula units per unit cell and four Raman active modes (A1g+B1g+B2g+Eg) 24. 
+Raman spectra of synthesized and reduced brookite samples (Figure S20A) confirmed that the synthesized 
+brookite is stable up to 700°C and after that it transforms to rutile, as evidenced by the disappearance of 
+brookite peaks and appearance of rutile peaks in Raman spectra of B700 and B800. The same Raman modes 
+of rutile remained evident also for B900 and B1000, where an additional phase transition to Magnéli phases 
+is completed. The main Raman peak of as synthesized brookite at 147.3 cm-1 is blue shifted to 146.0 cm-1 
+after reduction at 700°C. Moreover, there is a peak narrowing in the main peak due to the reduction 
+(FWHMB-AS = 15.82 cm-1, FWHMB700 = 12.81 cm-1). 
+Raman spectroscopy measurements also revealed that the commercial brookite sample is stable under 
+reducing conditions up to 600°C, after which, rutile Raman fingerprint developed. There is almost no peak 
+shift and no change in FWHM of the main peak (FWHMCB-AR = 16.41 cm-1, FWHMCB600 = 16.64 cm-1) 
+(Figure S20B), suggesting that no significant modifications occurred in the commercial brookite lattice 
+upon reduction at high temperature. 
+Figure S20C shows, instead, that anatase is stable up to 500°C in our reduction conditions showing also a 
+significant blueshift of the main Raman mode from 142.2 cm-1 for as-synthesized anatase to 134.6 cm-1 for 
+anatase reduced at 500°C. The same peak narrowed significantly from FWHMA-AS = 30.97 cm-1 to 
+FWHMA500 = 13.80 cm-1. 
+Several phenomena can give rise to TiO2 Raman peak shifting and broadening (or narrowing) 24–28 as 
+follows: (i) the lattice strain 26, (ii) the crystal size that could regulate the phonon confinement and the 
+Raman scattering (increasing in crystal size results in redshift and peak narrowing) 24, and (iii) the oxygen 
+stoichiometry, depending on the phase of TiO2, could affect the position of Raman peaks and also could 
+modify the FWHM 25. In the case of synthesized brookite and anatase, we observed a blueshift and peak 
+narrowing of the main Raman peak after reduction. Upon reduction, we detected a reduction of lattice 
+microstrain in brookite nanorods (Figure S20C) as well as an increase in the overall crystal size due to 
+nanorods aggregation. Therefore, we propose that the change in oxygen stoichiometry (as also evidenced 
+by the other characterizations reported) underlies the blue shift and the peak narrowing witnessed for both 
+brookite and anatase 25. In contrast, in case of commercial brookite neither peak shifting nor peak narrowing 
+(or broadening) were observed. This result confirms that the commercial brookite was not reduced under 
+reduction conditions as already suggested by UV-vis diffuse reflectance spectra analysis, TPR-MS 
+measurements, and also by the color of the samples, which did not change upon reduction treatment (Table 
+S1). 
+
+S23 
+ 
+ 
+Figure S20. Raman spectra of (A) as-synthesized and reduced anisotropic brookite samples, (B) as-received 
+and reduced commercial brookite samples, and (C) as-synthesized and reduced anatase samples. 
+Measurements performed with a 455 nm laser at a power density of 0.1 mW cm-2. The left part of each 
+panel is the zoomed view of the main Raman peaks. 
+ 
+
+B
+Intensity (a.u.)
+Intensity (a.u.)
+3900
+CB800
+8800
+CB70C
+B700
+CB600
+B600
+CB500
+B500
+B400
+CB400
+B-AS
+CB-AR
+130153100
+200
+300
+400
+500
+600
+700
+800
+130153100
+200
+300
+400
+500
+600
+700
+800
+Raman shift (cm-1)
+Raman shift (cm-1)
+C
+Intensity (a.u.)
+A700
+A600
+A500
+A400
+A-AS
+120148100
+200
+300
+400
+500
+600
+700
+800
+Raman shift (cm-1)S24 
+ 
+Diffuse reflectance spectroscopy 
+The Tauc method 29 is one of the most common procedures for determining the bandgap of materials. This 
+method makes a relationship between absorption coefficient and optical bandgap of material base on the 
+following formula: 
+(𝛼ℎ𝜈)𝑛 = (ℎ𝜈 − 𝐸𝑔) 
+where, α is the absorption coefficient, h is the Plank constant, ν is the frequency of radiation and Eg is the 
+bandgap of material. The n factor depends on the nature of the electronic transitions and is equal to 2 for 
+TiO2 as it is an indirect band gap semiconductor 30,31. This method assumes that the scattering component 
+of the reflected irradiation is zero. However, in case of nanopowders with high amount of scattering, this 
+component could not be neglected. Therefore, the Kubelka-Munk theory is used to make an estimation of 
+absorption from reflectance according to the following formula 32: 
+𝐹(𝑅) = 𝛼 = 𝐾
+𝑆 = (1 − 𝑅)2
+2𝑅
+ 
+where, R is the reflectance of the sample with infinite thickness to avoid any contribution of substrate, K 
+and S are the absorption and scattering coefficients, respectively. The bandgap of TiO2 can be retrieved 
+using the Tauc method as follows: 
+(𝐹(𝑅)ℎ𝜈)1/2 = (ℎ𝜈 − 𝐸𝑔) 
+From the Tauc plot, the x-axis intersection point of the tangent to the linear increase of light absorption in 
+the Tauc plot gives the band gap energy. This method is accurate and used here for determining the bandgap 
+energy of pristine TiO2 materials, while for reduced TiO2 powders the baseline method was employed to 
+calculate the bandgap 32. 
+Furthermore, we analyzed the Urbach tail in absorption spectra, as it originates optical transitions involving 
+intragap states related to defects 33–35. The Urbach energy can be calculated as follow 36,37: 
+𝛼 = 𝛼0 + exp ( 𝐸
+𝐸𝑢
+) 
+where α is the absorption coefficient, E is the photon energy equal to hν and Eu is the Urbach energy, which 
+can be retrieved by the reciprocal of the slope of the linear part of the curve. 
+Figure S21 shows the absorption spectra of as synthesized and reduced anisotropic brookite samples. As 
+expected, most of the UV light is absorbed owing to the wide band gap of brookite, which was found to be 
+comprised between 3.32 and 3.36 eV (Table S7) for all samples containing only the brookite phase (B-AS, 
+B500, B600, B700). A large shift in the absorption edge of B800 and B900 was observed due to the phase 
+change to rutile, with a corresponding decrease of bandgap values around 3.1 eV. The sample B1000 
+absorbs almost the entire spectrum and it is not possible to calculate any bandgap. This can be ascribed to 
+a phase composition including several semimetallic (Magnéli) phases 38. The absorption of anisotropic 
+brookite samples in the visible region increases by increasing the temperature of reduction, which is 
+predictable from the color of powders, and it can be assigned to the introduction of oxygen vacancies and 
+Ti3+ electronic states. However, visible light has no influence in the photocatalytic activity of the reduced 
+brookite, as discussed in photocatalysis section reported above. Furthermore, in all of the samples, a change 
+in the Urbach tail due to the reduction at different temperatures was observed. It varied from 69 to 115 meV 
+passing from B-AS to B700, suggesting an increased population of point defects in TiO2. 
+Reflectance spectra for commercial brookite and synthesized anatase (with isotropic crystal shape) are 
+reported in Figure S22 and Figure S23. In addition, Table S8 and Table S9 provide the results obtained 
+from analysis of bandgap and Urbach energy of commercial brookite and anatase, respectively. It is 
+apparent from the results that in all cases the bandgap underwent to only a slightly modification after 
+reduction, remaining around 3.3 and 3.2, for commercial brookite and anatase, respectively. After phase 
+transformation to rutile (at 800°C in commercial brookite and at 600°C in anatase), bandgap values 
+decreased and Urbach energy increased. Furthermore, no change in color of the commercial brookite 
+samples was observed after reduction at different temperatures (Table S1), while in anatase samples the 
+color change from white to gray and black. The variations of Urbach energy is in agreement with this 
+observation: for commercial brookite it remained at a stable value of ~55 meV before and after reduction, 
+
+S25 
+ 
+while for anatase Urbach energy increased from 115 (A-AS) to 205 meV (A500). These results further 
+confirmed that commercial brookite was hardly reducible, while defects could be introduced in the 
+synthesized anatase. 
+ 
+Figure S21. (A) Diffusive reflectance spectra and (B) Tauc plots of as synthesized and reduced anisotropic 
+brookite samples. 
+ 
+ 
+ 
+Figure S22. (A) Diffusive reflectance spectra and (B) Tauc plots of as received and reduced commercial 
+brookite samples. 
+ 
+
+A 
+100
+B
+B-AS
+B800
+80
+B500
+B900
+Reflectance (%)
+B600
+B1000
+B700
+3.13eV
+60
+40
+B-AS
+B500
+B600
+20
+3.38ev
+B700
+3.38eV
+B800
+3.37 eV
+B900
+3.32 eV
+B1000
+0
+300
+500
+700
+900
+1100
+2
+3
+4
+5
+Wavelength (nm)
+hu (eV)A
+100
+B
+CB-RT
+CB400
+80
+CB500
+Reflectance (%)
+(a.u.)
+CB600
+CB700
+CB800
+60
+CB-RT
+[F(R)hu]1/2 (
+CB400
+CB500
+40
+CB600
+3.04 eV
+CB700
+CB800
+3.34 eV
+20
+3.33 eV
+3.34eV7
+3.32eV7
+3.32eV
+0
+300
+500
+700
+900
+1100
+2
+3
+4
+5
+Wavelength (nm)
+hu (eV)S26 
+ 
+ 
+Figure S23. (A) Diffusive reflectance spectra and (B) Tauc plots of as synthesized and reduced anatase 
+samples. 
+ 
+Table S7. Optical bandgap and Urbach energy of anisotropic brookite samples. 
+Sample 
+Optical bandgap (eV) 
+Urbach energy (meV) 
+B-AS 
+3.32 
+69 
+B500 
+3.37 
+67 
+B600 
+3.38 
+104 
+B700 
+3.38 
+115 
+B800 
+3.13 
+155 
+B900 
+3.13 
+- 
+B1000 
+- 
+- 
+ 
+ 
+Table S8. Optical bandgap and Urbach energy of commercial brookite samples. 
+Sample 
+Optical bandgap (eV) 
+Urbach energy (meV) 
+CB-AR 
+3.32 
+58 
+CB400 
+3.32 
+55 
+CB500 
+3.34 
+54 
+CB600 
+3.32 
+56 
+CB700 
+3.34 
+39 
+CB800 
+3.04 
+36 
+
+A 
+100
+B
+A-AS
+A-AS
+A400
+A400
+A500
+80
+Reflectance (%)
+A500
+A600
+A600
+A700
+A700
+60
+3.09
+40
+3.09eV
+20
+3.29
+3.20 eV
+0
+300
+500
+700
+900
+1100
+2
+3
+4
+5
+Wavelength (nm)
+hu (eV)S27 
+ 
+Table S9. Optical bandgap and Urbach energy of anatase samples. 
+Sample 
+Optical bandgap (eV) 
+Urbach energy (meV) 
+A-AS 
+3.20 
+115 
+A400 
+3.29 
+154 
+A500 
+3.26 
+205 
+A600 
+3.09 
+252 
+A700 
+3.09 
+100 
+ 
+ 
+ 
+ 
+ 
+ 
+
+S28 
+ 
+Methanol photoreforming 
+In order to demonstrate the better photo-oxidation acitivty toward methanol of reduced brookite, we 
+performed a series of photocatalytic experiments detecting (1) the methanol consumption rate through 1H-
+NMR spectroscopy and (2) the hydrogen evolution rate with GC for platinized B-AS and B700 (Figure 
+SS24-S25 and Figure 1 of the main text). 
+We did not detect any trace of formaldehyde nor formic acid in both liquid phase (through NMR analysis) 
+and gas phase (through GC analysis), suggesting that methanol oxidation proceeded toward CO2, as 
+confimed by the high reactivity of brookite samples toward the oxidation of formaldehyde and formic acid 
+solutions (Figure S27). 
+ 
+ 
+ 
+Figure S24. 1H-NMR spectra of the solutions after photocatalysis at representative reaction times for B-
+AS/Pt and B700/Pt samples. The blank samples were recorded by using solutions containing CD3OD, DI 
+water, and DMSO ((CH3)2SO) as the internal standard. In the other samples, an aliquot of 50 μL of CH3OH 
+was added to the solution and its photocatalytic consumption rate was measured by assessing the area of its 
+characteristic NMR peak. CHD2OH is the impurity presents in the deuterated methanol. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+F
+AF
+AF
+B-AS/Pt 4h
+B700/Pt 4h
+0.05 
+H
+0
+0
+H-CO-H
+H-C-O-H
+工
+HsC-0~CH3
+0.03
+工 ·
+-
+-
+-
+0.02
+D
+ H-C-O-H
+-
+H-C-O-H
+-
+1
+1
+0.00
+-
+1
+-
+1
+B-AS/Pt Oh
+1
+1
+1
+B700/Pt0h
+(a.u.
+0.05
+1
+1
+1
+1
+-
+-
+-
+-
+1
+-
+-
+-
+-
+1
+-
+0.03
+-
+-
+-
+-
+-
+-
+-
+-
+-
+0.02
+1
+1
+-
+-
+-
+/1
+1
+1
+0.00
+-
+Blank
+-
+1
+Blank
+0.05
+-
+-
+-
+-
+-
+-
+-
+0.03
+-
+-
+-
+0.02 
+-
+-
+1
+-
+-
+1
+0.00
+1
+1
+AF
+AF
+3.40
+3.30
+3.20
+2.70
+2.60
+3.40
+3.30
+3.20
+2.70
+2.60
+ppm
+wddS29 
+ 
+Table S10. Relative NMR integration values of MeOH signal to internal standard (DMSO) at different 
+reaction times for the pristine (B-AS/Pt) and reduced brookite (B700/Pt) samples. 
+Time (h) 
+B-AS/Pt 
+B700/Pt 
+-a 
+1.04a 
+0.92a 
+0b 
+1.03b 
+0.92b 
+2 
+1.03 
+0.88 
+4 
+1.02 
+0.85 
+6 
+1.02 
+0.83 
+8 
+1.01 
+0.83 
+10 
+1.02 
+0.82 
+16 
+0.99 
+0.78 
+24 
+0.96 
+0.75 
+aBlank measurements with no photocatalyst, but in the presence of the reagents d4-MeOH (5 mL), H2O (5 
+mL), MeOH (50 μL), and DMSO (2 μL). 
+bResults were obtained after addition of the photocatalysts (B-AS-Pt and B700-Pt) to the solutions and 
+stirring for 30 min in the absence of light to allow for reaching the adsorption/desorption equilibrium. 
+ 
+ 
+ 
+ 
+Figure S25. Dynamic Light Scattering (DLS) measurements analysis showing the hydrodynamic diameter 
+evolution of the B700/Pt catalysts during the reaction. 
+ 
+200
+400
+600
+800
+0
+14
+28
+42
+0
+14
+28
+42
+0
+14
+28
+42
+0
+14
+28
+42
+ 
+diameter (nm) 
+ 0h
+ 
+Intensity (%)
+ 4h
+ 
+ 16h
+ 
+ 
+ 24h
+
+S30 
+ 
+ 
+Figure S26. Hydrogen evolution kinetics for B-AS/Pt and B700/Pt under one sun illumination. 
+ 
+ 
+ 
+ 
+ 
+Figure S27. Hydrogen amount evolved from five consecutive photocatalytic cycles for B700/Pt during 
+methanol photoreforming under 1 Sun illumination.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+0
+5
+10
+15
+20
+25
+0
+500
+1000
+1500
+2000
+2500
+Amount of H2 (mmol m-2)
+Time (h)
+ B700/Pt
+ B-AS/Pt
+0
+24
+48
+72
+96
+120
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Amount of evolved H2 (mmol m-2)
+Time (h)
+
+S31 
+ 
+ 
+Figure S28. Specific hydrogen evolution rate obtained for samples loaded with 1 wt.% Pt by normalizing 
+the photocatalytic rates by BET specific surface area for the as-received (CB-AR/Pt) and the most active 
+reduced commercial brookite (CB600/Pt), and as-synthesized (A-AS/Pt) and the most active reduced 
+anatase (A500/Pt).  
+ 
+ 
+ 
+ 
+ 
+Figure S29. Photocatalytic hydrogen evolution rate optimized per mass of used photocatalyst for (A) for 
+B-AS/Pt and B700/Pt; (B) A-AS/Pt and A500/Pt under one sun illumination. 
+ 
+
+9
+5
+4
+3
+2
+工
+0
+CB-AR/Pt
+CB600/Pt
+A-AS/Pt
+A500/PtA
+B
+10
+10
+B-AS-Pt
+A-AS-Pt
+B700-Pt
+A500-Pt
+8
+(μmol h-1)
+8.
+6
+evolution rate
+4
+?
+2
+I/
+0
+0
+2
+4
+6
+8
+10
+0
+2
+4
+6
+8
+10
+Amount of photocatalyst (mg)
+Amount of photocatalyst (mg)S32 
+ 
+ 
+Figure S30. (A) Hydrogen production rate from different intermediates of methanol oxidation 
+(formaldehyde and formic acid) of as-synthesized (light blue) and reduced at 700°C (dark blue) brookite 
+samples loaded with 1 wt.% Pt. These experiments confirmed GC analysis of the gas phase (i.e., we detected 
+only carbon dioxide) and NMR analysis of the liquid phase after reaction (i.e. we did not detect any signal 
+related to formaldehyde nor formic acid) indicating that the methanol photo-oxidation proceeded to carbon 
+dioxide, due to reactivity of brookite nanorods toward oxidation of the intermediates (formaldehyde and 
+formic acid) of methanol oxidation. Photolysis of formaldehyde under AM 1.5G 1sun illumination 
+produced a very small hydrogen production rate, namely, ~85 nmol m-2 h-1. 
+ 
+
+45
+40
+2 prod. rate (μmol h-1 m*2
+35
+30
+25
+20
+15
+H2
+10
+5
+0
+Formaldehyde
+Formic acidS33 
+ 
+ 
+Figure S31. Co-catalyst free rate of hydrogen evolution of TiO2 samples tested under 1 Sun illumination 
+for 24 h and reduced under pure hydrogen for 1 h: (A) brookite nanorods, (B) brookite nanorods reduced 
+at 700°C for different times, (C) commercial brookite, and (D) synthesized anatase. We report the specific 
+H2 evolution rate only for the most performing sample for each series in Figure S29. 
+ 
+ 
+
+A
+50
+B
+50
+40
+40
+30
+30
+20
+20
+10
+10
+H2
+0
+B1000
+0
+B-AS
+B500
+B600
+B700
+B800
+B900
+0.5h
+1 h
+2 h
+Timeofreductionat7oo°c
+C
+D
+40
+12
+10
+26
+30
+jowm)
+8
+evolutionrate
+20
+6
+4
+10
+2
+0
+0
+CB-RT
+CB400
+CB500
+CB600
+CB700
+CB800
+A-AS
+A400
+A500
+A600
+A700S34 
+ 
+  
+Figure S32. Specific hydrogen evolution rate obtained by normalizing the photocatalytic rates by BET 
+specific surface area for as-synthesized (B-AS) and the most active reduced brookite (B700), as-received 
+(CB-AR) and the most active reduced commercial brookite (CB600), and as-synthesized (A-AS) and the 
+most active reduced anatase (A500). 
+The specific photocatalytic H2 evolution rates expressed in µmol m-2 h-1 shown in Figure S29 underly that 
+nanorods of reduced brookite (B700) have a specific photocatalytic activity that is 4.4-, 5.6-, and 5.7-fold 
+the ones of as synthesized brookite (B-AS), reduced commercial brookite (CB600), and reduced anatase 
+(A500), respectively. These results demonstrate that higly active sites for photocatalysis are formed in B700 
+upon reduction under hydrogen atmosphere. Interestingly, when we performed H2 evolution experiments 
+with B700 in H2O/methanol under 1 sun light irradiation and applying a longpass optical filter to cutoff λ 
+≥ 380 nm, i.e. cutting optical excitation above bandgap energy, we did not detect any H2 after 24 h of 
+reaction. This finding suggests that the intragap electronic states due to the introduction of defects in TiO2 
+upon reduction (see materials characterization and DFT calculations below) do not directly participate to 
+the photocatalytic activity in the H2 evolution reaction. Further, experiments performed under full 1 sun 
+illumination and using Na2S (0.1 M) instead of methanol as a hole scavanger, did not produce any H2 
+suggesting that the H2 evolution activity of our reduced brookite is regulated by its selective methanol 
+photo-oxidation ability. This is supported by the photocatalytic experiments carried out using different 
+alcohols (Figure 1 main text).  
+ 
+ 
+ 
+
+1.0 -
+m
+h-1
+0.8-
+(μmol I
+0.6-
+0.4-
+0.2
+0.0
+s
+CB-AR
+CB600
+S
+A
+1
+B7
+BS35 
+ 
+Electron paramagnetic resonance spectroscopy 
+In order to identify the nature of spin containing sites and their contribution to photocatalysis we used 
+electron paramagnetic resonance (EPR) spectroscopy. Three samples have been tested, which differ in the 
+annealing temperature in hydrogen atmosphere, B-AS, B500, and B700. We investigated these samples in 
+powder form and in situ under photocatalytic conditions by using a water/methanol (MeOH) matrix. 
+ 
+Figure S33. X-band EPR spectra (9.1 GHz, T = 77 K) of (A) B-AS, (B) B500, and (C) B-700 recorded in 
+powder form and measured in dark (blue lines) and under UV irradiation  (red lines). 
+The EPR spectrum of B-AS in powder form shown in Figure SA exhibits a broad resonance, centered at g 
+ 2.0. This signal arises from delocalized defects and presence of significant strains in the crystalline lattice, 
+due to the synthetic procedure pursued. Following irradiation, a new resonant line appears in the spectrum, 
+around g = 2.017. This signal originates from holes/oxygen-based radicals. From the double integration of 
+the EPR signal recorded in light and dark conditions, we calculated an increase of about 25.6% in the total 
+number of spin contacting sites after illumination. The observed increase in spin concentration indicates 
+that fast electron/hole recombination processes are here hindered; thus, the photogenerated states are rather 
+stable. 
+The EPR trace of B500 in powder form, on the contrary, is very strong and it is dominated by a sharp 
+resonant line at g=1.997 (Figure S30B). This resonance arises from localized lattice embedded Ti3+ sites in 
+an octahedral field. Upon the UV irradiation, we witness an increase of the signal in the region of the 
+
+A
+B
+B-AS
+B500
+UV-On
+UV-On
+UV-Off
+UV-Off
+2.2
+2.1
+2.0
+1.9
+1.8
+2.2
+2.1
+2.0
+1.9
+1.8
+g-value
+g-value
+B700
+"/dB (a.u.)
+UV-On
+UV-Off
+2.2
+2.1
+2.0
+1.9
+1.8
+g-valueS36 
+ 
+spectrum associated to oxygen-based radicals at g=2.017. In this case, the double integrated EPR signal 
+shows after UV light exposure and increase in the total number of spins of 13.7%. Therefore, the 
+photoexcited states are here less stable than in B-RT and fast e-/h+ recombination processes occur in this 
+sample.  
+ 
+Figure S34. X-band EPR spectra (9.1 GHz, T = 77 K) of three samples studied (A) B-AS, (B) B500, and 
+(C) B700 dispersed in the frozen solution of DI water and methanol (1:1, volume ratio) recorded in dark 
+(blue lines) and under UV irradiation (red lines). 
+ 
+The sample B700 in powder form in the dark (Figure SC) gives a weaker EPR signal compared to B500, 
+although being qualitatively similar. The major difference between B700 and B500 appears, in the former, 
+to be linked with a larger distribution in crystal field strain (octahedral to rhombic) associated to the Ti3+ 
+spins due to structural defects. Upon UV irradiation, an increase of 56.6% in spin concentration was 
+observed. This behavior is interpreted as due to the high stability of photoexcited states and slow 
+recombination processes in the material, i.e. in B700 in powder form we observed the highest increase of 
+paramagnetic species accumulating under irradiation. 
+It is worthy to point out that the most efficient sample in hydrogen production and methanol consumption 
+is B700, which gives indeed the weakest intensity in the EPR powder spectrum among the series here 
+
+A
+B
+B-AS
+B500
+UV-On
+UV-On
+UV-Off
+UV-Off
+2.2
+2.1
+2.0
+1.9
+1.8
+2.2
+2.1
+2.0
+1.9
+1.8
+g-value
+g-value
+C
+B700
+UV-On
+I/dB (a.u.)
+UV-Off
+2.2
+2.1
+2.0
+1.9
+1.8
+g-valueS37 
+ 
+shown. Therefore, the number of spins recorded by EPR do not directly correlate with the system reactivity 
+and its overall efficiency in the catalytic process.  
+To unveil in more detail the reasons underlying the different efficiency in the hydrogen production recorded 
+in this series of tested materials, we performed a set of experiments under in situ photocatalytic conditions. 
+In our experiments, 10 mg of materials were dispersed in 100 µL of solution of deionized (DI) water and 
+MeOH (50:50). Under dark conditions the EPR fingerprints in frozen solutions of B-AS, B500, and B700 
+(Figure S32A-E) appear very similar to those observed in their correspondent powder forms in contact with 
+N2. However, upon irradiation, significant differences in the ability of MeOH molecules to quench the 
+photogenerated holes emerged. In particular, while in B-AS and B700 the interaction of MeOH molecules 
+with photogenerated h+ seems more effective, in B500 is not, as validated from the appearance of a strong 
+peak at g=2.017 associated to accumulation of holes. Therefore, the decreased catalytic efficiency of B500 
+compared to B-AS and B700 arises from combination of two factors, i) fast electron/hole quenching during 
+photoexcitation and ii) when holes are formed they do not tend to react with MeOH molecules, which 
+translates into a lower probability of successful delivery of electrons for hydrogen production. From time-
+resolved PL measurements, however, the average lifetime of charge carriers is similar for both B500 (1.9 
+ns) and B700 (2.0 ns), suggesting how the reaction between holes and methanol is a determining factor 
+addressing the photocatalytic activity trend.  
+ 
+ 
+Figure S35. (A) an illustrative EPR envelope of B700 in contact with N2 at 80K (grey spectrum) together 
+with individual components (C1-C6) derived from simulation and discussed below, (B-E) comparisons of 
+EPR spectra of B700 recorded in different conditions together with theoretically proposed model.  
+ 
+The simulation and deconvolution of EPR spectra for samples B700 in light and dark conditions in contact 
+with N2 (in powder form) and DI water: MeOH (1:1, volume ratio) at 80K is shown in Figure S32. By 
+deconvolution of EPR spectra we simulated the possible model of paramagnetic species present in the 
+sample B700, which can be applied on all related spectra just by varying the proportions of individual 
+components. We assigned five main species commented bellow. 
+ 
+Component C1: g=2.059 related to peroxyl radicals (Ti-O-O•) on the surface. 
+Component C2: S=1/2 system associated with a trapped hole (O-centre) in the lattice with gx=2.038, 
+gy=2.017 and gz=1.999. 
+
+A
+B
+Experiment
+D
+Experiment
+C5
+Simulation
+Simulation
+C5
+C4
+C3
+B700/N2
+B700/DI/MeOH
+[a.u]
+Uv off
+Uv off
+/dB
+c
+Experiment
+E
+Experiment
+Simulation
+Simulation
+C2
+C1
+B700/N2
+B700/DI/MeOH
+Uv On
+UV On
+280
+300
+320
+340
+360
+B [mT]S38 
+ 
+Component C3: surface exposed Ti3+ centres which are highly disordered, and which give wide resonant 
+line at g=1.935. 
+Component C4: sharp isotropic signal at g=1.992, we assume that this signal is associated with interstitial 
+Ti3+ sides in pseudo octahedral positions.  
+Component C5: typical axial signal arising from Ti3+ in regular lattice positions with gx,y=1.978 and 
+gz=1.952.  
+Component C6:  Ti3+ in the lattice located in some sublayer under the surface, where are lattice parameters 
+slightly shortened and therefore g value is smaller comparing to C5, to be precise gx,y=1.963 and gz=1.944. 
+ 
+ 
+ 
+
+S39 
+ 
+Photoluminescence (PL) and time-resolved photoluminescence (TRPL) spectroscopy 
+Photoluminescence spectroscopy is a powerful technique for studying the radiative recombination of 
+electrons and holes that populate intragap energy positions and that are due to the presence of structural 
+defects in semiconducting materials 39. Photoluminescence spectroscopy is often used in the literature to 
+reconcile charge carrier recombination phenomena occurring in TiO2 and its photocatalytic activity 40–42. 
+At first approximation, an higher PL intensity and lower PL lifetime underlie a higher electron and hole 
+recombination rate and consequently a lower photocatalytic activity. However, the way radiative 
+recombination processes may bring insight to photocatalytic activity also relates to: (i) the position of 
+structural defects into the TiO2 lattice (i.e. surface, subsurface, bulk) that, in turn, influences the energy 
+distribution of electronic states due to defects (and therefore also the energy distribution of PL spectra), and 
+(ii) how PL intensity and energy distribution change upon exposure to molecules employed in the 
+photocatalytic tests as a hole scavenger (i.e. methanol). 
+To examine the luminescence behavior of our TiO2 samples, we measured PL map spectra using several 
+excitation wavelengths from 258 to 590 nm (4.8 eV down to 2.1 eV) and monitoring PL emission in the 
+energy range from 364 to 730 nm (3.4 eV down to 1.7 eV). The samples were measured initially in the solid 
+state at the temperature 80 K in contact with N2 atmosphere. The PL maps for as synthesized brookite and 
+B700 (the most active photocatalyst) are reported in the main text (Figure 2), while PL maps for B500, 
+B600 and B800 are illustrated in Figure S36. From these PL maps, it is evident that PL intensity and energy 
+distribution changed in each sample depending on the reduction temperature, suggesting that recombination 
+centers (i.e. defects) re-organized and moved within the brookite lattice upon reduction at increasing 
+temperatures 23,43. A similar phenomenon was observed for each set of samples investigated, namely 
+commercial brookite (Figure S37) and anatase (Figure S38). Within the synthesized brookite series, the 
+most intense PL signal was recorded for B700 ≈ B-AS > B500 > B600 > B800. According to the XRD 
+results, the sample B800 is already transformed to rutile phase and for this reason show a very weak PL in 
+the investigated emission energy range. It is interesting to note that B-AS presents PL signal only when 
+excited with above-bandgap photons, while B700 highlights a complex PL signal even for below-bandgap 
+excitations, demonstrating that reduction modified the electronic structure of reduced brookite. In order to 
+investigate more in depth this aspect, we deconvoluted the PL spectra of synthesized and reduced brookite 
+(Figure S39) obtained using an excitation wavelength 340 nm (3.6 eV, above-bandgap). The spectra were 
+fitted by 4 components peaking at 450, 490, 545 and 620 nm (2.75, 2.53, 2.27 and 2.0 eV, respectively). In 
+each case, the peak position and the full width at half maximum (FWHM) of the deconvoluted peaks were 
+kept the same and just the intensity of peaks was free to change.  
+Generally, it has been shown that anatase and brookite exhibit distinct PL bands in the visible region of the 
+electromagnetic spectrum 44. The well-accepted interpretative model 39 for these emissions include the 
+presence of the main visible emission is composed of a green component (“type 1 PL” or “green PL”), at 
+higher energies, due to recombinations between shallowly trapped electrons (or conduction band electrons) 
+and deeply trapped holes, and a red component (“type 2 PL” or “red PL”), at lower energies, due to 
+recombinations between valence band holes and deeply trapped electrons 45–49. Spectra deconvolution 
+reported in Figure S39 show that the weight of different PL components in our brookite samples 
+significantly changes upon treatment in hydrogen at increasing temperatures. Importantly, the most intense 
+PL components for B-AS are those at 2 and 2.27 eV, while for B700 (i.e. the most active photocatalyst), 
+the higher energies PL components become dominant. The experimentally determined valence band 
+photoemission spectra recorded at synchrotron using photon energy in resonance with the titanium x-ray 
+absorption edge (Figure 3A in the main text) show that in case of B-AS, the native oxygen vacancies present 
+in the brookite induced a distribution of intragap states acting as deep electronic traps, while for B700 a 
+more defective structure induced a valence band tailing, offering therefore electronic states that may host 
+deeply/shallowly trapped holes. These findings demonstrates that the PL emissions observed for our 
+samples are in agreement with the proposed mechanism about “green and red PL”. The different PL 
+behavior in B-AS and B700 therefore remarks that the reduction treatment introduce different defects that 
+have a different PL signatures and influence in different way the photocatalytic activity of brookite samples. 
+These results reflect those of Vequizo et al. 50 who also found that the presence of an appropriate depth of 
+
+S40 
+ 
+the traps can effectively contribute to enhance the overall photocatalytic activity of TiO2. They found that 
+in case of as synthesized brookite, the moderate depth of electron traps in comparison with the anatase and 
+the rutile phase with shallow and deep electron traps, respectively, help brookite to be active for the both 
+oxidation and reduction reactions. In our case, instead, we provide evidences that recombinations centers 
+in reduced brookite are those defects that regulate the photo-oxidation reaction during H2 evolution from 
+methanol/H2O photoreforming. Further, PL maps recorded in the presence of methanol (Figure 2 in the 
+main text) show for both B-AS and B700 that the radiative recombinations are almost completely quenched 
+in these conditions, confirming the proposed PL mechanism and the fact that during photocatalysis holes 
+are mainly employed for the photo-oxidation reaction. 
+Figure S40 shows the TRPL of as-synthesized (received) and reduced samples, while Table S10 shows the 
+fitted parameters employed to calculating the average electron life time of each sample. In all cases, the 
+amplitude of the slow component (B1) is higher than the amplitude of the fast component (B2). The average 
+electron lifetime (τave) of reduced samples, in all cases, is less than that of the corresponding pristine 
+samples. This behavior of reduced TiO2 samples suggests that after reduction, the defective centers are 
+responsible for a faster charge carrier recombination 51. Importantly, the results from TPRL measurements 
+demonstrates that the improved photocatalytic activity of our reduced TiO2 (B700, A500, CB600) is not 
+due to an improved photo-induced charge separation, as shown previously in other TiO2 systems 52,53. In 
+contrast, in this case it is regulated by other parameters such as the photo-reactivity of the heterogeneous 
+catalytic sites formed around oxygen vacancies and comprised of several Ti atoms sharing extra charge (see 
+DFT calculations below) and lattice distortions (see XRD analysis). 
+ 
+Figure S36. PL maps for (A) B500, (B) B600, and (C) B800. The measurement temperature was 80 K 
+under N2 atmosphere. 
+
+A
+B
+2.1
+2.8x104
+2.1
+2.8x104
+B500-N2
+2.5x104
+B600-N2
+2.5x104
+2.3
+2.2x104
+2.3
+2.2x104
+(a.u.)
+1.u.
+2.0x104
+2.0x104
+2.7
+2.7
+1.7x104
+intensity
+1.7x104
+intensity
+1.4x104
+1.4x104
+3.1
+1.1x104
+3.1
+1.1x104
+8.4x103
+8.4x103
+PL
+3.8
+P
+3.8
+5.6x103
+5.6x103
+2.8x103
+2.8x103
+4.8
+0.0
+4.8
+0.0
+3.4
+2.9
+2.4
+2.1
+1.9
+1.7
+3.4
+2.9
+2.4
+2.1
+1.9
+1.7
+Emissionenergy(eV)
+Emission energy (eV)
+C
+2.1
+2.8x104
+B800-N2
+2.5x104
+2.3
+2.2×104
+PL intensity (a.u.)
+2.0x104
+2.7
+1.7x104
+1.4x104
+3.1
+1.1x104
+8.4x103
+3.8
+5.6x103
+2.8x103
+4.8-
+0.0
+3.4
+2.9
+2.4
+2.1
+1.9
+1.7
+Emission energy (eV)S41 
+ 
+ 
+Figure S37. PL maps for (A) as-received commercial brookite and (B) reduced commercial brookite at 
+600°C. The measurement temperature was 80 K under N2 atmosphere. 
+ 
+
+A
+B
+2.1
+2.1
+1.3x105
+1.3x105
+(eV)
+(eV)
+CB-AR
+1.2x105
+CB600
+1.2x105
+2.3
+1.0x105
+2.3
+1.0x105
+Tenergy
+intensitv(a.u.)
+lenergy
+(a.u.)
+9.0x104
+9.0x104
+2.7
+7.7x10
+2.7
+7.7x104
+intensity
+6.4x104
+6.4x104
+Excitation 
+Excitation
+3.2
+5.1x104
+3.2
+5.1x104
+3.9x104
+PL
+3.9x104
+P
+3.9
+2.6x104
+3.9
+2.6x104
+1.3x104
+1.3x104
+5.0
+0.0
+5.0
+0.0
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+Emissionenergy(eV)
+Emissionenergy(eV)S42 
+ 
+ 
+Figure S38. PL maps for (A) as-synthesized anatase, and reduced anatase at (B) 400°C, (B) 500°C, (B) 
+600°C, and (B) 700°C. The measurement temperature was 80 K under N2 atmosphere. 
+ 
+ 
+
+A
+1.8
+B
+1.8
+7x104
+7x104
+(eV)
+6x104
+(eV)
+A-AS
+A400
+6x104
+Excitation energy (
+2.0
+5x104
+2.0
+intensity (a.u.)
+5x104
+PL intensity (a.u.)
+5x104
+2.4
+4x104
+2.4
+3x104
+2.9
+3x104
+2.9
+2x104
+2x104
+3.6
+1x104
+3.6
+1x104
+7x103
+7x103
+5.0 -
+5.0
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+Emission energy (eV)
+Emission energy (eV)
+C
+1.8
+D
+7x104
+1.8
+7x104
+(eV)
+(eV)
+A500
+6x104
+A600
+6x104
+2.0
+5x104
+2.0
+Excitation energy (
+5x104
+intensity (a.u.)
+Excitation energy 
+PL intensity (a.u.)
+5x10*
+5x104
+2.4
+4x104
+2.4
+3x104
+3x104
+2.9
+3x104
+2.9
+3x104
+2x104
+2x104
+3.6
+1x104
+3.6
+1x104
+7x103
+7x103
+5.0
+5.0
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+Emissionenergy(eV)
+Emissionenergy(eV)
+E
+1.8
+A700
+0.9
+Normalized PL intensity
+2.0
+2.4
+0.6
+0.5
+2.9
+0.3 
+3.6
+0.2
+0.1
+5.0
+3.4
+2.7
+2.2
+1.9
+1.6
+1.5
+Emission energy (eV)S43 
+ 
+ 
+Figure S39. (A) Deconvoluted PL spectra for as-synthesized and reduced brookite samples, using an 
+excitation energy 3.6 eV. (B) PL peak area related to each component retrieved from deconvolution and 
+centered at 2, 2.27, 2.53, 2.75 eV. Both peak energy and width of each component were kept constant in 
+each sample.  
+
+A
+EmissionEnergy(eV)
+B
+B-AS
+3.10
+2.76
+2.48
+2.25
+2.07
+1.91
+1.77
+1.65
+B500
+B-AS
+B600
+13
+B700
+(a.u.)
+PL intensity (a.u.)
+13
+B500
+area
+13
+B600
+peak
+B700
+Experimentaldata
+Fitteddata
+Component4
+Component3
+Component2
+Component1
+400450500550600
+650700
+750
+Emissionwavelength(nm)
+2.75 eV
+2.53 eV
+2.27 eV
+2.0eVS44 
+ 
+ 
+Figure S40. Time–resolved photoluminescence spectra of (A) brookite reduced at 500°C and 600°C, (B) 
+as-received and reduced commercial brookite at 600°C, and (C) as-synthesized and reduced anatase at 400, 
+500, and 600°C. The dots are experimental data and the solid lines are the fitted curve. The τave is the average 
+electron lifetime extracted from the fitting. 
+ 
+ 
+
+A
+B
+100
+B500
+Tave = 1.9 ns
+100
+CB-AR Tave = 3.1 ns
+B600
+CB600 tave = 2.6 ns
+Tave = 1.8 ns
+PL Intensity (a.u.)
+PL Intensity (a.u.)
+10-1
+10-1
+10-2
+10-2
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+Time (ns)
+Time (ns)
+c
+100
+A-AS
+Tave = 17.7 ns
+A400
+Tave = 2.3 ns
+PL Intensity (a.u.)
+A500
+tave = 2.3 ns
+A600
+tave = 4.3 ns
+10-1
+10-2
+0
+5
+10
+15
+20
+Time (ns)S45 
+ 
+Table S10. Time–resolved photoluminescence fitted parameters for as-synthesized (as-received) and 
+reduced samples at different temperatures. 
+Sample 
+B1, % 
+τ1, ns 
+B2, % 
+τ2, ns 
+τave, ns 
+B-AS 
+85 
+1.2 
+15 
+8.6 
+5.3 
+B500 
+87 
+0.7 
+13 
+3.6 
+1.9 
+B600 
+91 
+0.6 
+9 
+3.7 
+1.8 
+B700 
+90 
+0.8 
+10 
+4.0 
+2.0 
+A-AS 
+80.2 
+1.8 
+19.8 
+22.7 
+17.7 
+A400 
+85.1 
+1.1 
+14.9 
+4.1 
+2.3 
+A500 
+84.7 
+0.9 
+15.3 
+4.1 
+2.3 
+A600 
+95.4 
+1.0 
+4.6 
+10.8 
+4.3 
+CB-AR 
+100.0 
+3.1 
+- 
+- 
+3.1 
+CB600 
+77.1 
+0.9 
+22.9 
+3.9 
+2.6 
+ 
+ 
+
+S46 
+ 
+Electronic characterization 
+ 
+Photoemission spectroscopy 
+ 
+We examined the electronic state of elements at the surface of the as-synthesized (received) and the best 
+reduced samples using an XPS laboratory source. Figure S compares the XPS survey spectra of samples 
+before and after reduction treatment. An important finding from these survey spectra is that the samples are 
+not contaminated during the reduction process, as already confirmed by CHN analysis (Table S2), since 
+there is no difference between total XPS survey before and after process. This observation supports our 
+hypothesis that all of the changes in photoactivity of the samples are due to the reduction process and not 
+due to materials contamination. The C1s peak at 284.8 eV was used as a reference for the energy scale to 
+compensate the charging effect of the samples (all spectra were shifted according to this reference). 
+However, the observed difference between XPS spectra of Ti2p and O1s for pristine and reduced samples 
+were not significant. A possible explanation for this might be that the amount of changes in the lattice of 
+TiO2 induced through reduction are below the detection limit of XPS 38 or the reduced species like Ti3+ can 
+be easily oxidized by exposure to the ambient air 54. Since, no difference was detected in XPS analysis 
+between the samples before and after reduction, we used synchrotron-based XPS (VUV-Photoemission 
+beamline, Elettra) to study them in more detail. Figure S39 shows the synchrotron-based XPS Ti2p spectra 
+of the pristine and the most photoactive sample of brookite. There is no evidence of an increase in Ti3+ 
+species after reduction. Additionally, synchrotron-based XPS study of O1s orbital of the pristine and the 
+most photoactive sample of brookite (Figure S40) reveals about 18% increase in OH groups on the surface 
+of TiO2 after reduction. This supports the water dissociation on superficial defects created upon reduction 
+treatment, see more details in the mechanism part below. 
+ 
+ 
+
+S47 
+ 
+ 
+Figure S41. (A) XPS survey spectra of the pristine and the most photoactive sample of brookite (left), 
+commercial brookite (middle) and anatase(right) (B) High resolution XPS spectra of the pristine and the 
+most photoactive sample of brookite (left), commercial brookite (middle) and anatase.(right) of (B) C1s, 
+(C) Ti2p and (D) O1s orbitals. 
+
+A
+B-AS
+01s
+CB-AR
+01s
+A-AS
+01s
+B700
+CB600
+A500
+(a.u.)
+dz !↓
+Intensity (a.u.)
+Intensity (a.u.)
+dz !!
+2
+AURL.
+dz !
+S Ti LMM-2
+Ti2s
+STi LMM-2
+Intensity (
+C 1s
+2.3
+1000
+800
+600
+400
+200
+0
+1000
+800
+600
+400
+200
+0
+1000
+800
+600
+400
+200
+0
+Binding energy (eV)
+Binding energy (eV)
+Binding energy (eV)
+B
+(a.u.)
+B-AS
+C 1s
+Normalized intensity (a.u.)
+CB-AR
+C 1s
+Normalized intensity (a.u.)
+ A-AS
+C 1s
+B700
+CB600
+A500
+Normalized intensity 
+290
+288
+286
+284
+282
+281
+292
+290
+288
+286
+284
+282
+28
+290
+288
+286
+284
+282
+280
+Binding energy (eV)
+Binding energy (eV)
+Binding energy (eV)
+c
+(a.u.)
+B-AS
+Normalized intensity (a.u.)
+CB-AR
+Ti 2p
+Ti 2p
+A-AS
+Ti 2p
+B700
+CB600
+A500
+Normalized intensity 
+Normalized intensity (
+468
+464
+460
+456
+45
+468
+464
+460
+456
+45
+468
+464
+460
+456
+452
+Binding energy (eV)
+Binding energy (eV)
+D
+Binding energy (eV)
+(a.u.)
+B-AS
+Normalized intensity (a.u.)
+CB-AR
+(a.u.)
+A-AS
+0 1s
+0 1s
+0 1s
+B700
+CB600
+A500
+Normalized intensity 
+Normalized intensity (
+536
+534
+532
+530
+528
+526
+536
+534
+532
+530
+528
+52
+536
+534
+532
+530
+528
+526
+Binding energy (eV)
+Binding energy (eV)
+Binding energy (eV)S48 
+ 
+ 
+Figure S42. Synchrotron-based XPS spectra in the Ti 2p region of pristine brookite (B-AS), the most 
+photoactive brookite sample (B700). 
+ 
+ 
+Figure S43. Synchrotron-based XPS spectra of the pristine (top) and the most photoactive sample of 
+brookite (bottom), O1s orbital. 
+468
+466
+464
+462
+460
+458
+456
+ 
+Binding energy (eV)
+ B-AS
+ Fit
+ Ti4+
+ Ti3+
+2p3/2
+2p3/2
+2p1/2
+ B700
+ Fit
+ Ti4+
+ Ti3+
+ 
+Intensity (a.u.)
+2p1/2
+
+B-AS
+Fitted
+O 1s (76%)
+Intensity (a.u.)
+OH (24%)
+B700
+Fitted
+O 1s (58%)
+Intensity (a.u.)
+OH (42%)
+536
+535
+534533532531530529
+528
+527
+Binding energy (eV)S49 
+ 
+ 
+Figure S44. Synchrotron-based photoemission spectra around the valence band (VB) region for the pristine 
+brookite B-AS (light blue, full circles), the reduced brookite before reaction B700-BR (dark blue, full 
+circles), and the reduced brookite after 24 h of photocatalytic reaction B700-AR (dark blue, empty circles). 
+ 
+ 
+ 
+
+0.04
+-B-AS
+B700-BR
+0.03
+-O—B700-AR
+Intensity (a.u.)
+0.02
+0.01 
+0.00 -
+-3
+-2
+-1
+0
+E-E (eV)S50 
+ 
+Density functional theory calculations 
+ 
+Figure S45. Spin-resolved density of states (DOS) plots of brookite TiO2 supercell exposing the (210) 
+surface. The simulated TiO2 structure is shown on the right side of the panel with Ti atoms plotted in grey, 
+O in red, and O vacancies indicated in orange. The middle part of the slab corresponds to the bulk region 
+of TiO2 enclosed by green planes, and the supercell’s boundaries are indicated by the dotted lines. (A) DOS 
+for a perfect slab, (B–I) is for the same slab including one O vacancy denoted by V1–V8 in the structure on 
+the right. All plots are zeroth to the Fermi level. DOS of V2 and V4 were calculated for a fixed geometry 
+to prevent the migration of the vacancy to the on-surface V1 position. The DOS plots clearly show that 
+varying the lattice position (surface, subsurface, bulk) of an oxygen vacancy results in the formation of 
+intragap electronic states with different energy and features. 
+ 
+ 
+
+200
+spin ↑
+0
+-200
+Spin
+ev)
+200
+DOS (states /
+V6
+V7
+0
+V8
+200
+200
+0
+-200
+-4
+-2
+0
+2
+4
+.4
+-2
+0
+2
+4
+-4
+-2
+0
+2
+4
+Energy (eV)S51 
+ 
+ 
+Figure S46. (A) DOS of the surface Ti bilayer and (B) the corresponding O atoms with the missing O atom 
+at V3 position. (C) DOS of subsurface Ti bilayer and (D) the corresponding O atoms. (E–J) Orbital resolved 
+partial DOS for the surface Ti bilayer (E, G, I) and O atoms (F, H, J). All plots are zeroth to EF. It is 
+interesting to note how intragap states are composed by hybridized electronic contributions belonging both 
+to Ti and O atoms. 
+ 
+ 
+ 
+
+60
+spd
+spin 
+sp
+0
+-60
+spin 
+20
+L
+6
+0
+ (states / eV)
+6
+5
+Pz
+DOS
+5
+0
+VZ
+-5
+5
+p
+0
+-4
+-2
+0
+2
+4
+-2
+0
+2
+4
+Energy (eV)
+Energy (eV)S52 
+ 
+Mechanism of methanol photo-oxidation 
+  
+ 
+Figure S47. Methanol oxidation mechanism at the TiO2surface. (A) Methanol activation by terminal OH–. 
+In step (i) → (ii) the OH reacts with methanol producing H2O and the methoxy group. In (ii) → (iii) the 
+methoxy group accepts a hole from TiO2 substrate (or donates an electron to TiO2) and converts to methoxy 
+radical, while the water molecule is released from the TiO2 surface. In (iii) → (iv) the methoxy radical 
+decomposes to adsorbed formaldehyde and proton ion. (B) Methanol activation by coadsorbed O2. In step 
+(i) → (ii) The superoxide activates the methanol and to methoxy radical and produces the peroxide radical. 
+Then in step (ii) → (iii) such radicals are converted to adsorbed formaldehyde and hydrogen peroxide, 
+respectively. In step (iii) → (iv) the hydrogen peroxide decomposes to water (then released from the TiO2 
+surface) and a bridging oxygen dimer 55. 
+ 
+ 
+ 
+
+A ()
+(ii)
+(ili)
+(iv)
+CH2
+CH3
+CH3
+H
+H
+CH
+H
+H
+0
+0
+0
+0
+O
+Thermally activated
+Tisc
+Tisc
+Tisc
+Tisc
+Ti3+
+Tise
+Tisc
+hx
+B (i)
+(ii)
+(ili)
+(iv)
+CH2
+CH2
+CH,
+H
+CH
+H-0
+H-0
+H
+H
+02
+0
+0
+0.
+0
+0
+0
+0
+Thermally activated
+Tisc
+Tisc
+Tisc
+Tisc
+Tisc
+Tisc
+Tisc
+Tisc
+h+S53 
+ 
+References 
+1. Kandiel, T.A., Feldhoff, A., Robben, L., Dillert, R., and Bahnemann, D.W. (2010). Tailored titanium 
+dioxide nanomaterials: Anatase nanoparticles and brookite nanorods as highly active photocatalysts. 
+Chem. Mater. 22, 2050–2060. 
+2. Zhao, H., Liu, L., Andino, J.M., and Li, Y. (2013). Bicrystalline TiO2 with controllable anatase-
+brookite phase content for enhanced CO2 photoreduction to fuels. J. Mater. Chem. 1, 8209–8216. 
+3. Beltram, A., Romero-Ocaña, I., Josè Delgado Jaen, J., Montini, T., and Fornasiero, P. (2016). 
+Photocatalytic valorization of ethanol and glycerol over TiO2 polymorphs for sustainable hydrogen 
+production. Appl. Catal. A: Gen 518, 167–175. 
+4. Naldoni, A., Montini, T., Malara, F., Mróz, M.M., Beltram, A., Virgili, T., Boldrini, C.L., Marelli, M., 
+Romero-Ocaña, I., Delgado, J.J., et al. (2017). Hot electron collection on brookite nanorods lateral 
+facets for plasmon-enhanced water oxidation. ACS Catal. 7, 1270–1278. 
+5. Larson, A.C., and Von Dreele, A.C. (2004). General structure analysis system (GSAS). 
+6. Weyl, R. (1977). Zeitschrift fuer Kristallographie. 
+7. http://www.iucr.org/resources/data/datasets/bond-valenceparameters. 
+8. Williamson, G.K., and Hall, W.H. (1953). X-ray line broadening from filed aluminium and wolfram. 
+Acta metall. 1, 22–31. 
+9. Roisnel, T., and Rodríguez-Carvajal, J. (2001). WinPLOTR: A windows tool for powder diffraction 
+pattern analysis. Mater. Sci. Forum 378–381, 118–123. 
+10. Rouquerol, F., Rouquerol, J., Llewellyn, P., Maurin, G., Sing, K., (1999). Adsorption by powders and 
+porous solids (Elsevier). 
+11. Rex, R.E., Knorr, F. J., and McHale, J.L. (2013). Comment on "Characterization of oxygen vacancy 
+associates within hydrogenated TiO2: A positron annihilation study". J. Phys. Chem 117, 7949–7951. 
+12. Stoll, S., and Schweiger, A. (2006). EasySpin, a comprehensive software package for spectral 
+simulation and analysis in EPR. J. Magn. Reson. 178, 42–55. 
+13. Kresse, G., and Furthmuller, J. (1996). Efficient iterative schemes for ab initio total-energy calculations 
+using a plane-wave basis set. Phys. Rev. B Condens. Matter Mater. Phys. 54, 11169–11186. 
+14. Kresse, G., and Furthmüller, J. (1996). Efficiency of ab-initio total energy calculations for metals and 
+semiconductors using a plane-wave basis set. Comput. Mater. Sci. 6, 15–50. 
+15. Blöchl, P.E. (1994). Projector augmented-wave method. Phys. Rev. B Condens. Matter Mater. Phys. 
+50, 17953–17979. 
+16. Kresse, G., and Joubert, D. (1999). From ultrasoft pseudopotentials to the projector augmented-wave 
+method. Phys. Rev. B Condens. Matter Mater. Phys. 59, 1758–1775. 
+17. Perdew, J.P., Burke, K., and Ernzerhof, M. (1996). Generalized gradient approximation made simple. 
+Phys. Rev. Lett. 77, 3865–3868. 
+
+S54 
+ 
+18. Perdew, J.P., Burke, K., and Ernzerhof, M. (1997). Generalized gradient approximation made simple 
+(Errata). Phys. Rev. Lett. 78, 1396–1396. 
+19. Dudarev, S.L., Botton, G.A., Savrasov, S.Y., Humphreys, C.J., and Sutton, A.P. (1998). Electron-
+energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study. Phys. Rev. B 57, 
+1505–1509. 
+20. Morgan, B.J., and Watson, G.W. (2009). Polaronic trapping of electrons and holes by native defects in 
+anatase TiO2. Phys. Rev. B 80, 2331021–2331024. 
+21. Holmström, E., Ghan, S., Asakawa, H., Fujita, Y., Fukuma, T., Kamimura, S., Ohno, T., and Foster, 
+A.S. (2017). Hydration structure of brookite TiO2 (210). J. Phys. Chem 121, 20790–20801. 
+22. Naldoni, A., Allieta, M., Santangelo, S., Marelli, M., Fabbri, F., Cappelli, S., Bianchi, C.L., Psaro, R., 
+and Dal Santo, V. (2012). Effect of nature and location of defects on bandgap narrowing in black TiO2 
+nanoparticles. J. Am. Chem. Soc. 134, 7600–7603. 
+23. Naldoni, A., Altomare, M., Zoppellaro, G., and Liu, N. (2019). Photocatalysis with reduced TiO2 : from 
+black TiO2 to cocatalyst- free hydrogen production. ACS Catal. 9, 345–364. 
+24. Bersani, D., Lottici, P.P., and Ding, X. (1998). Phonon confinement effects in the Raman scattering by 
+TiO2 nanocrystals. Appl. Phys. Lett. 72, 73–75. 
+25. Parker, J.C., and Siegel, R.W. (1990). Calibration of the Raman spectrum to the oxygen stoichiometry 
+of nanophase TiO2. Appl. Phys. Lett. 57. 
+26. Fauchet, I.H.C. and P.M. (1986). The effects of microcrystal size and shape of the one phonon raman 
+spectra of crystalline semiconductor. Solid State Commun. 58, 739–741. 
+27. Venkatasubbu, G.D., Ramakrishnan, V., Sasirekha, V., Ramasamy, S., and Kumar, J. (2014). Influence 
+of particle size on the phonon confinement of TiO2 nanoparticles. J. Exp. Nanosci. 9, 661–668. 
+28. Fisica, D., Infm, C., Chimica, S., Parma, U., and Scienze, V. (1993). Raman scattering characterization 
+of gel-derived titania glass. J. Mater. Sci. 28, 177–183. 
+29. Tauc, J., Grigorovici, R., Vancu, A. (1966). Optical properties and electronic structure of amorphous 
+germanium. Phys. Stat. Sol. 627, 627–637. 
+30. Escobedo-Morales, A., Ruiz-López, I.I., Ruiz-Peralta, M.deL., Tepech-Carrillo, L., Sánchez-Cantú, 
+M., and Moreno-Orea, J.E. (2019). Automated method for the determination of the band gap energy of 
+pure and mixed powder samples using diffuse reflectance spectroscopy. Heliyon 5, 1–19. 
+31. Mohajernia, S., Andryskova, P., Zoppellaro, G., Hejazi, S., Kment, S., Zboril, R., Schmidt, J., and 
+Schmuki, P. (2020). Influence of Ti3+ defect-type on heterogeneous photocatalytic H2 evolution activity 
+of TiO2. J. Mater. Chem. A 8, 1432–1442. 
+32. Makuła, P., Pacia, M., and Macyk, W. (2018). How to correctly determine the band gap energy of 
+modified semiconductor photocatalysts based on UV-Vis spectra. J. Phys. Chem. Lett. 9, 6814–6817. 
+33. Urbach, T. (1953). The long-wavelength edge of photographic sensitivity and of the electronic 
+absorption of solids. Phys. Rev. 92, 1324. 
+
+S55 
+ 
+34. Rai, R.C. (2013). Analysis of the Urbach tails in absorption spectra of undoped ZnO thin films. J. Appl. 
+Phys. 113, 153508. 
+35. John, S., Soukoulis, C., Cohen, M.H., and Economou, E.N. (1986). Theory of electron band tails and 
+the Urbach optical-absorption edge. Phys. Rev. Lett. 57, 1777–1780. 
+36. Akshay, V.R., Arun, B., Mandal, G., and Vasundhara, M. (2019). Visible range optical absorption, 
+Urbach energy estimation and paramagnetic response in Cr-doped TiO2 nanocrystals derived by a sol–
+gel method. Phys. Chem. Chem. Phys. 21, 12991–13004. 
+37. H. Tang, F. Levy, H. Berger, P.E.S. (1995). Urbach tail of anatase TiO2. Phys. Rev. B 52, 7771–7774. 
+38. Mohajernia, S., Andryskova, P., Zoppellaro, G., Hejazi, S., Kment, S., Zboril, R., Schmidt, J., and 
+Schmuki, P. (2020). Influence of Ti3+ defect-type on heterogeneous photocatalytic H2 evolution activity 
+of TiO2. Journal of Materials Chemistry A 8, 1432–1442. 
+39. Mascaretti, L., Russo, V., Zoppellaro, G., Lucotti, A., Casari, C.S., Kment, Š., Naldoni, A., and Li 
+Bassi, A. (2019). Excitation wavelength- and medium-dependent photoluminescence of reduced 
+nanostructured TiO2 films. J. Phys. Chem. C 123, 11292–11303. 
+40. Liqiang, J., Yichun, Q., Baiqi, W., Shudan, L., Baojiang, J., Libin, Y., Wei, F., Honggang, F., and 
+Jiazhong, S. (2006). Review of photoluminescence performance of nano-sized semiconductor materials 
+and its relationships with photocatalytic activity. Sol. Energy Mater Sol. Cells 90, 1773–1787. 
+41. Gerosa, M., Bottani, C.E., Caramella, L., Onida, G., Di Valentin, C., and Pacchioni, G. (2015). Defect 
+calculations in semiconductors through a dielectric-dependent hybrid DFT functional: The case of 
+oxygen vacancies in metal oxides. J. Chem. Phys. 143. 
+42. Shi, J., Chen, J., Feng, Z., Chen, T., Lian, Y., Wang, X., and Li, C. (2007). Photoluminescence 
+characteristics of TiO2 and their relationship to the photoassisted reaction of water/methanol mixture. 
+J. Phys. Chem. C 111, 693–699. 
+43. Mercado, C.C., Knorr, F.J., McHale, J.L., Usmani, S.M., Ichimura, A.S., and Saraf, L. V. (2012). 
+Location of hole and electron traps on nanocrystalline anatase TiO2. J. Phys. Chem. C 116, 10796–
+10804. 
+44. Vequizo, J.J.M., Kamimura, S., Ohno, T., and Yamakata, A. (2018). Oxygen induced enhancement of 
+NIR emission in brookite TiO2 powders: Comparison with rutile and anatase TiO2 powders. Phys. 
+Chem. Chem. Phys. 20, 3241–3248. 
+45. Pallotti, D.K., Passoni, L., Maddalena, P., Di Fonzo, F., and Lettieri, S. (2017). Photoluminescence 
+mechanisms in anatase and rutile TiO2. J. Phys. Chem. C 121, 9011–9021. 
+46. Mercado, C., Seeley, Z., Bandyopadhyay, A., Bose, S., and McHale, J.L. (2011). Photoluminescence 
+of dense nanocrystalline titanium dioxide thin films: Effect of doping and thickness and relation to gas 
+sensing. ACS Appl. Mater. Interfaces 3, 2281–2288. 
+47. Knorr, F.J., and McHale, J.L. (2013). Spectroelectrochemical photoluminescence of trap states of 
+nanocrystalline TiO2 in aqueous media. J. Phys. Chem. C 117, 13654–13662. 
+
+S56 
+ 
+48. Knorr, F.J., Mercado, C.C., and McHale, J.L. (2008). Trap-state distributions and carrier transport in 
+pure and mixed-phase TiO2: Influence of contacting solvent and interphasial electron transfer. J. Phys. 
+Chem. C 112, 12786–12794. 
+49. Mercado, C.C., McHale, J.L. Defect photoluminescence of TiO2 nanotubes. (2010). MRS Online 
+Proceedings Library 1268, 310. 
+50. Vequizo, J.J.M., Matsunaga, H., Ishiku, T., Kamimura, S., Ohno, T., and Yamakata, A. (2017). 
+Trapping-induced enhancement of photocatalytic activity on brookite TiO2 powders: Comparison with 
+anatase and rutile TiO2 powders. ACS Catal. 7, 2644–2651. 
+51. Yan, Y., Han, M., Konkin, A., Koppe, T., Wang, D., Andreu, T., Chen, G., Vetter, U., Morante, J.R., 
+and Schaaf, P. (2014). Slightly hydrogenated TiO2 with enhanced photocatalytic performance. J. Mater. 
+Chem. A 2, 12708–12716. 
+52. Wu, Z., Cao, S., Zhang, C., and Piao, L. (2017). Effects of bulk and surface defects on the 
+photocatalytic performance of size-controlled TiO2 nanoparticles. Nanotechnology 28. 
+53. Alsalka, Y., Hakki, A., Schneider, J., and Bahnemann, D.W. (2018). Co-catalyst-free photocatalytic 
+hydrogen evolution on TiO2: Synthesis of optimized photocatalyst through statistical material science. 
+Appl. Catal. B 238, 422–433. 
+54. Mohajernia, S., Hejazi, S., Mazare, A., Nguyen, N.T., and Schmuki, P. (2017). Photoelectrochemical 
+H2 generation from suboxide TiO2 nanotubes: visible-light absorption versus conductivity.  
+Chem. 
+Eur. J. 23, 12406–12411. 
+55. Setviń, M., Aschauer, U., Scheiber, P., Li, Y.F., Hou, W., Schmid, M., Selloni, A., and Diebold, U. 
+(2013). Reaction of O2 with subsurface oxygen vacancies on TiO2 anatase (101). Science 341, 988–
+991. 
+
diff --git a/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/load_file.txt b/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..7d54a1d145a02f4dca2fbd40e60ce8b545125a7a
--- /dev/null
+++ b/r9E3T4oBgHgl3EQfMwkD/content/tmp_files/load_file.txt
@@ -0,0 +1,2856 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf,len=2855
+page_content='1  Defect engineering over anisotropic brookite towards substrate-specific photo- oxidation of alcohols S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hossein Hejazi1, Mahdi Shahrezaei1, Piotr Błoński1, Mattia Allieta2, Polina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sheverdyaeva3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Paolo Moras3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zdeněk Baďura1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sergii Kalytchuk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Elmira Mohammadi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Radek Zbořil1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Štěpán Kment1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Michal Otyepka1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Alberto Naldoni1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Paolo Fornasiero6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7* 1Czech Advanced Technology and Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Regional Centre of Advanced Technologies and Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Palacký University Olomouc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Křížkovského 511/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 77900 Olomouc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Czech Republic 2Ronin Institute Montclair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' NJ 07043 USA 3Istituto di Struttura della Materia-CNR (ISM-CNR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' SS 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Km 163,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' I-34149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Italy 4Nanotechnology Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Centre of Energy and Environmental Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' VŠB– Technical University of Ostrava,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' listopadu 2172/15, 70800 Ostrava-Poruba, Czech Republic 5IT4Innovations, VSB – Technical University of Ostrava, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' listopadu 2172/15, 708 00 Ostrava-Poruba, Czech Republic 6Department of Chemical and Pharmaceutical Sciences, ICCOM-CNR Trieste Research Unit, INSTM-Trieste, University of Trieste, Via L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Giorgieri 1, 34127 Trieste, Italy 7Center for Energy, Environment and Transport Giacomo Ciamician - University of Trieste, Italy *Corresponding authors: alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='naldoni@upol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='cz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' pfornasiero@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='it  2  Summary Generally adopted design strategies for enhancing the photocatalytic activity are aimed at tuning properties such as the visible light response, the exposed crystal facets, and the nanocrystal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Here, we present a different approach for designing efficient photocatalysts displaying a substrate- specific reactivity upon defect engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The defective anisotropic brookite TiO2 photocatalyst functionalized with Pt nanocrystals are tested for alcohol photoreforming showing up to an 11- fold increase in methanol oxidation rate, compared to the unreduced one, whilst presenting much lower ethanol or isopropanol specific oxidation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We demonstrate that the alcohol oxidation and hydrogen evolution reactions are tightly related, and when the substrate-specific alcohol oxidation ability is increased, the hydrogen evolution is significantly boosted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The reduced anisotropic brookite shows up to twenty-six-fold higher specific photoactivity with respect to anatase and brookite with isotropic nanocrystals, reflecting the different type of defective catalytic sites formed depending on the TiO2 polymorph and its crystal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Advanced in-situ characterizations and theoretical investigations reveal that controlled engineering over oxygen vacancies and lattice strain produces large electron polarons hosting the substrate-specific active sites for alcohol photo-oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Keywords: selective photocatalysis, oxygen vacancies, DFT calculations, photoreforming, black TiO2  3  INTRODUCTION The visionary idea of a world powered by solar light proposed by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Ciamician more than a century ago1 has become a reality, proving that complex organic synthesis2,3 and production of solar fuels like hydrogen4–6 and ammonia7,8 can be performed more and more efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, these intrinsically sustainable processes, before becoming industrially competitive with existing polluting technologies, need further material design, fine tuning of light absorption properties, charge carriers management, and surface engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 Over the past decades, photocatalysis for direct conversion of solar energy into molecular fuels has been focused on designing efficient photocatalysts by improving their fundamental properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The visible light photoactivity can be enhanced by engineering heterojunction, introducing lattice defects into wide bandgap materials like TiO2,10–12 or choosing semiconductors having small bandgap energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Cu2O, ZnIn2S4, and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  C3N4) and suitable bands position straddling the molecular redox levels of the investigated chemical reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2,13–15 The use of inorganic nanocrystals with well-defined morphology, determined crystal facets, or dimensional anisotropy have been also demonstrated to be beneficial for the charge carrier separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='16–18 The kinetic competition between charge recombination and surface catalysis is often overcome by the addition of proper metallic co-catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='19 This playground has stimulated the exploration of countless options to prepare, mix, and engineer semiconductor photocatalysts with enhanced opto-electronic properties with the aim of more efficiently driving benchmark photocatalytic reactions such as hydrogen evolution from water splitting and photoreforming of biomass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, the majority of these studies involve monitoring the products of the reductive cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' the evolved hydrogen, while not analyzing the oxidation products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20–22 When sacrificial biomass substrates are employed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' alcohol photoreforming, analyzing the oxidation pathway and reactivity becomes especially important not only because they provide a kinetic gain, and therefore an improved hydrogen evolution compared  4  to the case of water oxidation, but also because the oxidized sacrificial agents often participate in hydrogen evolution, thus directly regulating its kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20,23 Furthermore, controlling the oxidation process of sacrificial biomass is particularly relevant since it can lead to its partial oxidation and the synthesis of added value products such as 2,5-furandicarboxylic acid (bioderived polymer that may substitute PET) and diesel fuel precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3,24,25 We developed an approach to designing anisotropic defective brookite TiO2 nanocrystals that upon high temperature reduction treatment expose precise defect site with substrate-specific photo- reactivity for methanol oxidation, compared to higher alcohols such as ethanol and isopropanol (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We show that the reaction rates for the photocatalytic alcohol oxidation and the parallel hydrogen evolution reaction are tightly connected and that optimization of the methanol oxidation leads to an increased production of hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Although the introduction of point defects and structural deformation results in enhanced visible light absorption and reduced charge carrier lifetime, we show that the selective affinity towards methanol oxidation of reduced brookite is the main parameter enabling a higher apparent quantum yield (AQY) for hydrogen evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Using a set of in-situ characterization aided by DFT calculations, we demonstrate that the substrate- specific activity is regulated by catalytic sites including sub-surface oxygen vacancies within a locally strained lattice environment that generate shallow hole traps responsible for boosting the first steps of methanol photo-oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These photo-oxidation sites are crucial for the enhanced substrate-specific photoreforming activity of reduced brookite, and they form preferentially within anisotropic nanocrystals, which show twenty-six times higher specific hydrogen evolution rate, compared to isotropic ones, where defect sites with different energy are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  5  RESULTS AND DISCUSSION Synthesis and characterization of brookite nanorods To engineer the photocatalytic sites for alcohol photoreforming at the atomic level, we selected brookite TiO2 nanorods—a promising and still poorly investigated TiO2 polymorph—as a model material and grew anisotropic nanostructures exposing the (210) surface on the lateral facets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1B) by hydrothermal synthesis (see Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='26 We prepared various brookite photocatalysts reduced under a H2 stream at different temperatures, along with reduced anatase and commercial brookite reference samples (see Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Elemental analysis revealed that the obtained nanopowders did not contain significant quantity of non-metals coming from C, H, or N incorporation (Table S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reduction of a pristine brookite TiO2 under pure hydrogen stream at 700°C for 1h created defective nanocrystals showing remarkable changes in their structural and electronic properties alongside giving the best photocatalytic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This morphology evolved from anisotropic nanostructures with well-defined shape and exposed crystal facets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1B and S1A) into more isotropic nanoparticles that displayed irregular shape and aggregation through twin boundaries formation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1C and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, anatase and brookite with isotropic crystal shapes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S2-S3) did not reveal any crystal reshaping upon high temperature reduction treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  However, they presented different color variations (Table S1), compared to those observed for the anisotropic brookite, upon increasing the temperature of the hydrogen treatment, thus suggesting a different reducibility behavior with respect to the TiO2 polymorph and crystal shape, as confirmed by UV-vis reflectance spectroscopy, Raman spectroscopy, and photolumiscence spectroscopy mapping (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In order to prepare highly active photocatalysts for alcohol photoreforming, we functionalized the samples by photodepositing Pt co-catalyst nanoparticles on their surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Inductively coupled plasma mass spectrometry (ICP- MS) analysis detected similar Pt loading on both pristine (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='98 wt%) and reduced brookite samples  6  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='90 wt%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' HRTEM and STEM-HAADF micrographs as well as the elemental mapping showed that Pt nanoparticles with an average diameter of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 nm were homogeneously deposited on the pristine brookite nanorods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1D and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S4A, S5, S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Surprisingly, in the case of reduced brookite nanocrystals, we detected the presence of small Pt nanoparticles with similar sizes as well as larger Pt nanoparticle aggregates reaching 10–30 nm in size (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1E and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S7–S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This result was confirmed by a HRTEM analysis of three different brookite batches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The larger Pt conglomerates may be less reactive than the smaller ones, thus negatively affecting the photocatalytic activity of the reduced brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Moreover, larger metal aggregates may screen the incoming light during photocatalysis, decreasing the light harvesting efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, as shown in the next section, in the present investigation these parameters did not reduced neither one aspect nor the other for reduced brookite, suggesting that for the considered reaction the defects in TiO2 played a more crucial role than that of Pt nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The brookite treatment under hydrogen was accompanied by a decrease in the BET (Brunauer–Emmett–Teller) specific surface area from 67 to 47 m2 g-1 after reduction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S11 and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The temperature (reduction)-dependent structural parameters extracted by the Rietveld refinement of X-ray diffraction (XRD) patterns were in agreement with the described morphological evolution (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S12-S18 and Table S3–S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, the XRD analysis highlighted that the reduction treatment introduced an anisotropic and preferential deformation of the brookite lattice along the c-axis due to the creation of oxygen vacancies (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Their presence was further supported by the blueshift in the main A1g vibrational mode detected by Raman spectroscopy measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S20 and discussion in the Supporting Information) after reduction, which again pointed out to a different reducing behavior dependent on the TiO2 polymorph and shape (Raman shift is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 cm-1 for the reduced anisotropic brookite, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 cm-1 for the reduced isotropic anatase, and no observed shift for the reduced isotropic brookite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Having performed the bond valence sum  7  analysis, we observed an average depletion of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5% of the Ti valence for the reduced brookite— a typical feature induced by the formation of oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='11,27 The moderate decrease in the Ti valence upon a H2 treatment at high temperature is an indication of a low tendency toward defects formation in brookite nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This general feature was also reflected by the color change—from white to grey—that brookite underwent after reduction at 700°C, as opposed to the more reducible anatase phase that assumed a darker color already at lower temperatures (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The reduced brookite nanorods showed an optical bandgap energy of ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38 eV, this making no significant difference from the value retrieved for the as-synthesized sample (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S21 and Table S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The same results were observed for both the isotropic brookite and anatase samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S22-S23 and Table S8-S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Further analysis of the absorption spectra highlighted an increased visible light absorption and an Urbach tail that ranged for the anisotropic brookite from 69 (for the as-synthesized brookite nanorods) to 115 meV (for nanorods reduced at 700°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For the isotropic anatase, the Urbach tail increased from 115 (for the as-synthesized anatase nanoparticles) to 205 mV (for anatase reduced at 500°C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This supports the scenario of a phase- and shape-dependent increase in the population of oxygen vacancies after the reduction treatment (see Supporting Information for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Alcohol photoreforming with reduced brookite The photocatalytic activity of the platinized brookite nanorods was tested for methanol photoreforming under a simulated AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G spectrum at one-sun intensity producing H2 and oxidation products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' As previously reported by others groups, the increased photocatalytic activity of reduced TiO2 nanomaterials could be ascribed to the co-catalyst role in H2 evolution played by oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='12,28 In contrast, in order to use the oxygen vacancies as catalytic sites in the photocatalytic oxidation reaction, we photodeposited Pt nanoparticles over the optimized photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Following this procedure, H2 generation occurred on the Pt surface as the  8  photogenerated electrons were separated and stabilized into the Pt nanoparticles by the Schottky barrier formation at the Pt–TiO2 interface, while photo-oxidation occurred on the TiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='19,29 In contrast to common practice, where only a H2 production rate is detected, we designed specific experiments to follow the kinetics of methanol photo-oxidation using a solution including deuterated methanol (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e, CD3OD) and a small aliquot of methanol (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CH3OH), whose corresponding consumption was followed by an 1H-NMR analysis of the liquid phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Table S10 reports the NMR signal integration of CH3OH relative to the adopted internal standard (DMSO), which highlight no significant difference between the blank measurement containing no photocatalyst and the one at time zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' after 30 min adsorption/desorption equilibrium in the dark in the presence of the photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' From this data is clear that the methanol adsorption in the dark did not affect the photocatalytic performance of the investigated photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1F shows the amount of methanol oxidized over 24h of illumination illustrating the remarkable oxidation activity of the reduced brookite over the pristine material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The corresponding specific methanol consumption rates, computed by using the BET surface area of each sample, for the platinized brookite nanorod samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  1G, left) evidenced that the pristine brookite drove the photo-oxidation reaction with a specific rate of 27 \uf06dmol h-1 m-2, whereas for the reduced one, it was 99 \uf06dmol h-1 m-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We obtained the photoactivity values by considering methanol consumption after 24 h of reaction, which resulted in a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7-fold enhancement in favor of the reduced brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, if we consider kinetic data after 5 h (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1F), the reduced brookite performed methanol photo-oxidation up to 11 times faster than the pristine sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This suggests on the one hand that, in the early stage of reaction, methanol was oxidized faster until the available surface reaction sites were fully occupied and a steady state was reached, which ensured a higher methanol consumption rate even after 24 h of reaction, as evidenced by the divergence of the reaction kinetics curves (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' On the other hand, this behavior can be due to a partial aggregation of the colloidal  9  photocatalysts after several hours of irradiation (see dynamic light scattering measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S25), thus producing a reduced available surface for the methanol oxidation reaction to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The amount of evolved hydrogen determined by gas chromatography analysis followed a linear increase with time (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S26) for both the pristine and the reduced brookite, corresponding to optimized specific H2 production rates of 26 and 88 \uf06dmol h-1 m-2, respectively, with reduced brookite that evolved H2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 times faster (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1G, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The reduced brookite showed a 13% decrease activity after 5 photocatalytic runs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, the activity decrease appeared almost constant after each recycling test, thus suggesting that it can be due to the loss of catalyst during the tests, which can happen during the centrifugation/re-suspension of the photocatalyst and/or can be due to the loss of material attaching onto the reactor walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Another aspect that can produce this slight decrease in activity is the increased hydrodynamic diameter of the brookite nanocrystals in suspension, as revealed by dynamic light scattering measurements after 24 h of illumination (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Moreover, two more aspects may produce the observed photocatalytic activity decrease after several recycling cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' On the one hand, the catalyst surface may be partially passivated by the presence of reaction intermediates, as we did not wash the catalyst before subsequent tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  On the other hand, a partial modification of the surface population of defects (as evidenced by the resonant PES valence band measurements on B700 after catalysis, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S43) may induce a partial reactivity change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interestingly, the H2 production rates followed a reactivity trend closely resembling the one observed for the methanol photo-oxidation activity, which suggests that hydrogen production is strictly regulated by the alcohol oxidation and it can be used as reporter figures of merit for tracking the reactivity of the pristine and the reduced brookite for alcohol photo-oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Following this principle, we tested our platinized samples for the photoreforming of ethanol and isopropanol and discovered that the reactivity of the reduced brookite was markedly more pronounced and substrate-specific toward methanol in comparison  10  with the other tested alcohols (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1G, right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In the case of ethanol photoreforming, both reduced and pristine anisotropic brookites showed a specific H2 production rate of 54 \uf06dmol h-1 m-2, suggesting that they have a similar affinity toward its photo-oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' On the other hand, in the case of isopropanol photoreforming, the reduced anisotropic brookite presented a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7-fold higher specific H2 production rate (41 \uf06dmol h-1 m-2) when compared to the pristine sample, denoting a higher photo-oxidation ability yet still much lower than that shown for methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This observation confirmed that the H2 evolution activity of the reduced brookite was regulated by a substrate- specific reactivity toward alcohol photo-oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Such a stark photo-reactivity toward methanol oxidation was observed only for the brookite nanorods, while platinized reference samples made by reduced spherical anatase nanocrystals or reduced isotropic brookite nanoparticles displayed ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 times higher specific photocatalytic rates in comparison with the untreated materials (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, the reduced brookite nanorods loaded with Pt showed a remarkably higher specific H2 evolution rate with respect to both the reduced anatase/Pt (19 times) and the reduced isotropic brookite/Pt (26 times), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, this difference is significantly reduced when considering the photocatalytic activity per optimized mass (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S29), with the reduced brookite nanorods (B700/Pt) still showing more than two-times the hydrogen evolution rate observed for the reduced anatase nanocrystals (A500/Pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These observations suggest that the type of produced defects/catalytic sites varies depending on the selected TiO2 polymorph as well as on the crystal shape, which emphasizes how the exposure of different crystal facets having different interfacial energy and therefore resistance to hydrogen treatment under high temperature regulates the defects formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This is demonstrated by the different degree of reducibility that each sample exhibited, as supported by previously discussed absorption and Raman spectroscopy measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  11  The apparent quantum yield (AQY) for hydrogen evolution from methanol photoreforming was measured for a pristine and a reduced platinized brookite (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1H) using different monochromatic light sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The maximum AQY was reached at 334 nm and was 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5% for the reduced brookite and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2% for the pristine nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These AQY values can be further increased by optimizing the methanol concentration, metal loading, metal particle size, or photoreactor design, which however goes beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interestingly, despite its visible light absorption, the reduced brookite did not show AQY in the visible region, with values of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='004% at 386 and 402 nm, respectively (AQY below the detection limit for the pristine brookite at both wavelengths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This result was further confirmed by H2 evolution experiments under one-sun illumination applying a longpass optical filter to cut off λ ≥ 380 nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' cutting optical excitation above bandgap energy did not lead to detecting any H2 after 24 h of reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This result confirms that oxygen vacancies introduced upon the hydrogen treatment at high temperature, and the related optical transitions, did not produce visible light photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We indeed suggest that the introduced oxygen vacancies enhanced the reactivity toward the methanol oxidation by favoring its activation, as discussed in further detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  12  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Morphology and photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Schematic representation of defect engineering in reduced brookite showing an exemplary TiO2 surface during photocatalysis and a zoomed view of the catalytic site containing oxygen vacancy (VO) and structural distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) HAADF-STEM (left) and HRTEM (right) micrographs of a single brookite nanorod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (C) HRTEM micrograph of isolated Pt nanoparticles deposited on pristine brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (D) HRTEM of brookite nanorods reduced at 700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (E) HRTEM micrograph of aggregated Pt nanostructures deposited on reduced brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (F) Methanol consumption in time for pristine (sky blue) and reduced brookite (dark blue) nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The points before zero time represent the methanol signal before adding the photocatalysts, while time zero was measured once the adsorbtion/desorption equilibrium in the dark was reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (G) Specific methanol consumption rate (left) and specific hydrogen evolution rate (right) during methanol, ethanol, and isopropanol photoreforming for pristine (sky blue) and reduced (dark blue) brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Measurements were performed under a simulated AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G spectrum at one-sun intensity for 24h using a 1:1 vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% H2O:alcohol mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (H) Apparent quantum yield for hydrogen evolution from methanol photoreforming for pristine (sky blue) and reduced (dark blue) brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In all measurements both pristine and reduced brookite were loaded with 1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A H,O H CH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CO Catalytic sitewithoxygen vacancyV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' structural deformation O B onm 10nm F G 100 2500 Light, 1 m2), 100 m-2) 2000 CH,OH cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' rate (μmol h-1 80 [oum) 80 30 rate 1500 60 AQY % 20 CH,OH cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1000 40 40 10 500 20 0 n 0 5 10 15 20 25 310 330 350 370 390 410 Time (h) Wavelength (nm) 13  In order to study in more detail the methanol photo-oxidation reaction on the reduced brookite, we analyzed the reaction products by both the GC analysis of the gas phase and the NMR analysis of the liquid phase after reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Carbon dioxide was the only detected reaction product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Furthermore, we investigated the hydrogen production rate from different possible intermediates of methanol oxidation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' formaldehyde and formic acid) of as-synthesized and reduced brookite nanorods loaded with 1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interestingly, both samples showed similar specific photocatalytic activity in the presence of formaldehyde and formic acid, presenting significant hydrogen production rate of around 25–30 \uf06dmol h-1 m-2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This result is far from being trivial, as it has been reported that other TiO2 polymorphs usually oxidize methanol to formaldehyde, thus stopping the methanol photo-oxidation after the first reaction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='30 Moreover, it repeatedly emphasizes that a reduced brookite demonstrates a substrate-specific oxidation ability toward methanol molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The blank test for photolysis of formaldehyde under AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G 1sun illumination produced a very small hydrogen production rate, namely, ~85 nmol h-1 m-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, the investigated TiO2 samples showed two order-lower alcohol photoreforming activity without Pt loading;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' the data on the samples reduced at different temperatures are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S31 and S32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  These data are further supported by electron spin resonance spectroscopy (EPR) investigations measured under dark and light conditions both for dried powders and in a water/methanol medium (in situ conditions), demonstrating the increased reactivity of brookite nanorods reduced at 700°C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S33–S35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For instance, in the case of EPR spectra for dried powders of an anisotropic brookite reduced at different temperatures, the most efficient sample in methanol photoreforming was B700, which indeed gave the highest differential EPR signal (light- dark) in comparison with samples with lower activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' a pristine brookite and B500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interestingly, the most active sample (B700) showed the weakest intensity in the EPR powder spectrum among the series (Fig S33 and discussion in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, the  14  number of spins recorded by EPR do not directly correlate with the system reactivity and its overall efficiency in the photocatalytic process, all in agreement with previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='12,31 In-situ photoluminescence spectroscopy To understand the nature of the methanol oxidation sites, we measured excitation-dependent photoluminescence (PL) spectra at 80 K, obtaining energy-resolved two-dimensional maps of the radiative recombinations occurring in the pristine and the reduced anisotropic brookite both under inert gas atmosphere (N2) and in the presence of methanol (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The PL maps in the presence of the latter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', a hole scavenger) showed drastic quenching of the signal, demonstrating that the photogenerated holes trapped within the defect sites, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' oxygen vacancies, readily reacted with the surface adsorbed methanol molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Moreover, this also provided evidence that such defect sites must be located on either the surface or sub-surface of the brookite nanocrystals, from where they can react with surface adsorbates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 This is supported by the synchrotron-based photoemission spectra for the Ti 2p region and valence band (see for details the next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Comparing the two- dimensional PL maps measured under N2 gas atmosphere for the pristine and the anisotropic brookite reduced at 500, 600, 700, and 800°C (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2A and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S36), we observed a clear variation in the energy position relative to the radiative recombination centers upon high temperature treatment, eventually underlined by a subtle but significant re-organization of structural defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, the reference samples (especially the isotropic anatase, which is more reducible than the isotropic brookite) displayed a similar behavior (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S37-S38 and discussion in the Supporting Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Focusing on the reduced anisotropic brookite, we observed a significant blue shift in the PL peak after reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  15  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Energy distribution of defects-related radiative recombinations and lifetime of photogenerated charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Excitation-emission color maps under N2 and in the presence of methanol (MeOH) for pristine and reduced brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) PL spectra of pristine (blue sky) and reduced brookite (dark blue) under excitation at 340 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (C) Time-resolved PL decay curves collected at the corresponding emission maximum of pristine (blue sky) and reduced brookite (dark blue) under excitation at 372 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This is better highlighted in the PL spectra generated using a single excitation wavelength (340 nm) and by analyzing the weight of the deconvoluted components set at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='53, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 eV for all the samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The dominant radiative recombinations for the as- synthesized anisotropic brookite (B-AS) localize at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 eV, while the components at higher energies are almost negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Notably, in the case of the reduced anisotropic brookite (B700), the component at lower energy almost vanished, while the intensity of the radiative recombinations with higher energies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='53 eV) became dominant, denoting the formation of shallower hole traps upon hydrogen reduction treatment at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Furthermore, we investigated the lifetime of photogenerated charge carriers measured at PL maximum by time- resolved PL spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The charge carriers’ lifetime (τ) decreased after the brookite’s  B  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  B AS N2  B AS MeOH  (eV)  intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B AS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  B700  Excitation energy  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5x104  PLi  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2x104  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x104  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  360  440  520  600  680  760  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  Wavelength (nm)  B700 N2  C  B700 MeOH  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x103  (eV)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x103  10°  B AS  Excitation energy  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x103  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B700  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  Intensity  10 1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3ns  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  C=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0ns  10 2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  0  5  10  15  20  Emissionenergy (eV)  Emission energy (eV)  Time (ns)  16  reduction, similarly to the other TiO2 reference samples, from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 ns to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 ns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3C, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S40 and Table S11), suggesting that the enhanced photocatalytic activity of the reduced anisotropic brookite is not related to the enhanced charge separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Synchrotron resonant photoemission spectroscopy The investigation by conventional lab scale X-ray photoelectron spectroscopy (XPS) analysis provided similar results for pristine and reduced TiO2 samples (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, we investigated in more detail the electronic structure of our brookite samples at the VUV- Photoemission beamline (Elettra, Trieste) by synchrotron-based photoemission spectroscopy (PES) for the Ti 2p (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S42), O 1s (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S43, see discussion in the next section), and the valence band (VB) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The Ti 2p spectra of both brookite samples contained two components corresponding to the presence of Ti4+, due to the coordination of Ti into the stoichiometric lattice, and Ti3+ species introduced by the oxygen vacancy formation near or at the TiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The presence of low valence Ti ions in the pristine brookite is a common observation, especially in nanocrystals obtained through hydrothermal synthesis and not subjected to a following heat treatment like in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Next, by using soft X-ray photons with energy that is resonant to the Ti absorption edge, it is possible to highlight electronic states even in samples containing a low amount of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='33 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  3A shows the VB PES spectra for the pristine and the reduced anisotropic brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The main VB edge did not significantly shift upon reduction, while the density of states (DOS) within the bandgap showed a stark difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The pristine brookite displayed localized mid-gap states peaking at around 1 eV below the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In contrast, the reduced brookite revealed an increased electron density showing an intense VB tailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We also investigated the reduced brookite after 24 h of photocatalytic reaction B700-AR and compared the result with the spectra of the pristine brookite B-AS and the reduced brookite before reaction B700-BR (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The post-catalytic characterization evidence a spectrum, which features a  17  localize state at around 1 eV below the Fermi level (similarly to B-AS) and an increase density of states –band tailing - at energies closer to the valence band maximum (similarly to B700-BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The slight modification of the spectrum can be due both to a partial passivation of the surface defects in B700 during photocatalysis and to the adsorption of methoxy groups / reaction intermediates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' the sample was not regenerated after reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Density functional theory calculations of the photocatalytic sites In order to understand the origin of the VB tailing in the electronic structure of the reduced brookite, we calculated the energy band structure using ab initio density functional theory (DFT) calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Driven by the XRD results, we focused on the (210) surface of the brookite TiO2 introducing two types of structural defects: (1) oxygen vacancies located at different distance from the surface (denoted by V1–V8 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3B and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S45), and (2) distortion of TiO6 octahedra (for methods see supplementary materials) by modifying up to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 Å either the axial or randomly chosen Ti-O distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  18  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Experimental and theoretical determination of the electronic structure of the photocatalytic sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Synchrotron-based photoemission spectra around the valence band (VB) region for the pristine (light blue) and the reduced (dark blue) brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Inset: zoom of the VB region around the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) Brookite TiO2 supercell employed for the calculations exposing the (210) surface: Ti atoms plotted in grey, O atoms in red, oxygen vacancies in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The middle part of the slab corresponds to the bulk region of TiO2 enclosed by green planes, while the supercell’s boundaries are marked by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (C) Calculated total DOS of the ideal (210) brookite surface, of various defective brookite surfaces with an oxygen vacancy (V in the figure) placed at different locations in the lattice, two distorted brookite surfaces (rdm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' and ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' stand for random and axial distortions, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The energy of the VB maximum of the ideal (210) surface is taken to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spin up/down derived DOS are shown by solid/dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (D) Excess electron density donated by introducing V3 and V5 oxygen vacancies (yellow iso-surface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The presence of oxygen vacancies introduces localized mid-gap states deriving from the hybridization of O 2p and Ti 3d orbitals (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3C and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S46) with their energy position that  A  B  /3  V4  V5  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=') 3 2  V6 1  0  V7  E E (eV)  V8 10 8  9 4  2  0  E E (eV)  D  (210)ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  V3  DOs (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  (210) rdm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  V  V5  V  (210)  A 2  0 2 4  Energy (eV)  19  varies with respect to the defect’s distance from the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In contrast, the primary effect of the expansion of Ti-O axial distances is to produce strong band tailing near the VB edge (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3C) entering the band gap by ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' When we introduced a lattice disorder by random displacements of both Ti and O atoms from their equilibrium positions, both mid-gap states and VB tailing were seen in the DOS (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The computational results confirmed that the DOS envelope of the reduced brookite was formed by mid-gap states and VB band tailing due to the combined effect of oxygen vacancies and lattice distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The location of oxygen vacancies in the real samples is represented by a statistical distribution of lattice positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Each of such defect populations produce different DOS and their convolution, alongside the effect from lattice distortions, results in the formation of the VB tailing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We also examined the differential electron densities due to the introduction of an oxygen vacancy at two different positions in the slab, namely, surface/near-surface (V3) or sub-surface (V5) positions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3B and 3D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In both cases, the excess of charge was spread over many lattice sites and accompanied by the relaxation of the lattice atoms by up to 2–4% of the equilibrium Ti-O bond length, thus denoting the generation of a large electron polaron around the oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  We propose that these kind of bound states between oxygen vacancies and large electron polarons represent the substrate-specific photocatalytic active sites for methanol oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The size of this photocatalytic active site and the charge distribution around it make it a well-defined reactive pocket with high specificity toward the methanol molecule rather than to higher alcohols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The finding of Zhang and co-workers support our results, as they recently observed a similar reactivity pattern in Cu-doped TiO2 nanosheets, where the oxygen vacancies within a strained environment enabled strong chemisorption and activation of molecular N2 and water, resulting in high photocatalytic NH3 evolution under visible-light irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 Diebold and co-workers recently reported the photo-oxidation mechanism of methanol at the surface of anatase TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34 Values  20  obtained from DFT calculations, scanning tunneling microscopy, and temperature programmed desorption aided by XPS showed the existence of two different, more favorable pathways for activating methanol adsorbed on TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The methanol molecules are first adsorbed onto the surface Ti5c atoms dissociating into methoxy groups and hydrogen atoms, which are then oxidized to formaldehyde (and eventually to formic acid and carbon dioxide) and molecular hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Methanol molecules must first dissociate into methoxy groups, and after this step, the hole transfer from TiO2 becomes energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Methanol can be activated via two pathways (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' S47): (A) by reaction with dissociated H2O forming terminal OH– species bound to surface Ti5c atoms, and (B) by reaction with activated adsorbed O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Error!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reference source not found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' M echanism (A) begins with the spontaneous dissociative adsorption of water enabled by the extra charge density due to oxygen vacancies and reflected by the formation of hydroxyl ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34,35 Interestingly, brookite TiO2(210) (the same crystallographic direction expressed on the lateral facets of our brookite nanorods) has the same structural building block of anatase TiO2(101), but interatomic distances are slightly shorter and the blocks are arranged in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Selloni and co-workers found that these differences significantly change the reactivity toward adsorption of water (and formic acid), making its dissociation more possible to occur on the brookite surface rather than on the anatase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='36 This may underlie the enhanced specific photocatalytic activity that we observed for the anisotropic brookite over the anatase during methanol photoreforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This scenario is corroborated by our synchrotron XPS PES of the O 1s region that shows a significant, 24% increase in the OH– species adsorbed on the reduced surface in comparison with the pristine brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These species may be derived from the dissociative adsorption of water, which is more favored on the reduced brookite due to the extra electrons provided by the subsurface oxygen vacancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 Mechanism (B) is less probable, as our experiments are performed in the absence of oxygen (under Ar atmosphere).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, it should be noted that some traces of peroxide species  21  were detected by EPR,12,37 suggesting that mechanism B may occur even at a lower extent than mechanism A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This could also point to a faster decomposition of hydrogen peroxide to water and bridging oxygen dimer (step (iii) → (iv) in mechanism B) in the most photoactive sample (B700) after illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Finally, besides the pure A and B mechanisms, an intermediate case can be also considered, in which the OH– formation results from the reaction of coadsorbed O2 and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38 CONCLUSIONS In summary, we demonstrated the concept of enhancing the photocatalytic activity during alcohol photoreforming by engineering the defect sites in an anisotropic brookite in a way that enables substrate-specific oxidation photo-activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Synchrotron photoemission spectroscopy and in situ photoluminescence investigations aided by DFT calculations showed that creating a low amount of defects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' oxygen vacancies) in well-defined lattice positions produces a kind of bound states between oxygen vacancies and large electron polarons hosting the photocatalytic active sites, which act as shallow hole traps during alcohol photoreforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Our results also demonstrate that the nature of the produced defects/photocatalytic sites varies with respect to the selected TiO2 polymorph and on its crystal shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  This work highlights the value of analyzing the reaction products of both the reductive and the oxidative pathways during photocatalytic reactions alongside opening new avenues for substrate-selective photocatalytic biomass conversion through the atomic design of the active sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  22  Acknowledgments: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' gratefully acknowledge the support of the Czech Science Foundation (GACR) through the projects no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 20-17636S and 19-27454X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The authors gratefully acknowledge the support by the Operational Programme Research, Development and Education - European Regional Development Fund, project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0/15_003/0000416 of the Ministry of Education, Youth and Sports of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' acknowledge financial support from the European Community (projects H2020 – RIA-CE-NMBP-25 Program – Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 862030 – and H2020-LC-SC3-2019-NZE-RES-CC – Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 884444), INSTM consortium and ICCOM-CNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' gratefully acknowledge financial support through the project EUROFEL-ROADMAP ESFRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The authors gratefully acknowledged G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zoppellaro and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Tomanec for EPR discussion and TEM measurements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Author contributions: Conceptualization: PF, AN Methodology: SMHH, MA, EM, PB Investigation: SMHH, MS, PMS, PM, ZB, SK, EM, PB Visualization: SMHH, PB, AN Funding acquisition: RZ, SK, MO, PF, AN Project administration: AN Supervision: PF, AN Writing – original draft: SMHH, AN Writing – review & editing: SMHH, MA, PM, PB, RZ, MO, PF, AN Competing interests: Authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Data and materials availability: All data are available in the main text or the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  23  REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Ciamician, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1912).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The Photochemistry of the Future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science 36, 385–394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Ghosh, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Khamrai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Savateev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shlapakov, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Antonietti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and König, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Organic semiconductor photocatalyst can bifunctionalize arenes and heteroarenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science 365, 360–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Luo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Montini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fornasiero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fonda, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hou, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Nie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Heggen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Visible-light-driven coproduction of diesel precursors and hydrogen from lignocellulose-derived methylfurans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nat Energy 4, 575–584.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hisatomi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Jia, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Tokudome, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhong, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Pan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Takata, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Nakabayashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shibata, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Scalable water splitting on particulate photocatalyst sheets with a solar-to-hydrogen energy conversion efficiency exceeding 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nature Mater 15, 611–615.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kosco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bidwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cha, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Martin, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Howells, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sachs, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Anjum, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Gonzalez Lopez, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wadsworth, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Enhanced photocatalytic hydrogen evolution from organic semiconductor heterojunction nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 19, 559–565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Domen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Recent developments in heterogeneous photocatalysts for solar-driven overall water splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 48, 2109–2125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zhang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Jalil, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Gao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ye, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ju, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Refining Defect States in W 18 O 49 by Mo Doping: A Strategy for Tuning N 2 Activation towards Solar-Driven Nitrogen Fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 140, 9434–9443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Waterhouse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Tung, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Tuning Oxygen Vacancies in Ultrathin TiO 2 Nanosheets to Boost Photocatalytic Nitrogen Fixation up to 700 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 31, 1806482.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Takanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalytic Water Splitting: Quantitative Approaches toward Photocatalyst by Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 8006–8022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Mao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Increasing Solar Absorption for Photocatalysis with Black Hydrogenated Titanium Dioxide Nanocrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science 331, 746– 750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Allieta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Santangelo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fabbri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cappelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bianchi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Psaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Dal Santo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Effect of Nature and Location of Defects on Bandgap Narrowing in Black TiO2 Nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 134, 7600–7603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kment, Š.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zbořil, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schmuki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalysis with Reduced TiO 2 : From Black TiO 2 to Cocatalyst-Free Hydrogen Production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9, 345–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  24  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Maeda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Thomas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Takanabe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Xin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Carlsson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Domen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Antonietti, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A metal-free polymeric photocatalyst for hydrogen production from water under visible light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nature Materials 8, 76–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Filippini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Longobardo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Forster, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Criado, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Carmine, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Nasi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', D’Agostino, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Melchionna, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fornasiero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Prato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Light-driven, heterogeneous organocatalysts for C–C bond formation toward valuable perfluoroalkylated intermediates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science Advances 6, eabc9923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Surface hydrogen bond network spatially confined BiOCl oxygen vacancy for photocatalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science Bulletin 65, 1916–1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', McNulty, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lau, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Paulikas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Guest, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ren, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Facet-dependent active sites of a single Cu 2 O particle photocatalyst for CO 2 reduction to methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nature Energy 4, 957–968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Cargnello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Montini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Smolin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Priebe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Jaén, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Doan-Nguyen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', McKay, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schwalbe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Pohl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Gordon, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Engineering titania nanostructure to tune and improve its photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' PNAS 113, 3966–3971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhou, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Han, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spatial separation of photogenerated electrons and holes among {010} and {110} crystal facets of BiVO 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Nature Communications 4, 1432.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Guo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Mao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Semiconductor-based Photocatalytic Hydrogen Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 110, 6503–6570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kamat, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Jin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Semiconductor Photocatalysis: “ Tell Us the Complete Story!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' .” ACS Energy Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3, 622–623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Schneider, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Bahnemann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Undesired Role of Sacrificial Reagents in Photocatalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 4, 3479–3483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hainer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hodgins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sandre, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Vallieres, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lanterna, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Scaiano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalytic Hydrogen Generation Using Metal-Decorated TiO 2 : Sacrificial Donors vs True Water Splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Energy Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3, 542–545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Belhadj, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hamid, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Robertson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Bahnemann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mechanisms of Simultaneous Hydrogen Production and Formaldehyde Oxidation in H 2 O and D 2 O over Platinized TiO 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 4753–4758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Paone, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Rodríguez-Padrón, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Luque, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Mauriello, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Recent catalytic routes for the preparation and the upgrading of biomass derived furfural and 5- hydroxymethylfurfural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 49, 4273–4306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Xie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Duan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Selectivity Control in Photocatalytic Valorization of Biomass-Derived Platform Compounds by Surface Engineering of Titanium Oxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem 6, 3038–3053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  25  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Montini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Malara, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Mróz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Beltram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Virgili, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Boldrini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Romero-Ocaña, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Delgado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hot Electron Collection on Brookite Nanorods Lateral Facets for Plasmon-Enhanced Water Oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 1270–1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Di Valentin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Pacchioni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reduced and n-Type Doped TiO2: Nature of Ti3+ Species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 113, 20543–20552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wierzbicka, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yokosawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fehn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Qin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Meyer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Unruh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Spiecker, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reduced grey brookite for noble metal free photocatalytic H2 evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A 9, 1168–1179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Semiconductor heterojunction photocatalysts: design, construction, and photocatalytic performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 43, 5234–5244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Feng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ye, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Molecular-level understanding of the deactivation pathways during methanol photo-reforming on Pt-decorated TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Applied Catalysis B: Environmental 272, 118980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Bad’ura, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Qin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bakandritsos, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kment, Š.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmuki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Light-Induced Migration of Spin Defects in TiO2 Nanosystems and their Contribution to the H2 Evolution Catalysis from Water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ChemSusChem n/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Setvín, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Aschauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Scheiber, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Diebold, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reaction of O2 with Subsurface Oxygen Vacancies on TiO2 Anatase (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science 341, 988–991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Riboni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bossola, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ulisse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Carlo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Píš, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Nappini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Malvestuto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Dozzi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Influence of TiO2 electronic structure and strong metal–support interaction on plasmonic Au photocatalytic oxidations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 6, 3220–3229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Setvin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hulva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Simschitz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Parkinson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Di Valentin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Diebold, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Methanol on Anatase TiO2 (101): Mechanistic Insights into Photocatalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 7081–7091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Selcuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Facet-dependent trapping and dynamics of excess electrons at anatase TiO2 surfaces and aqueous interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nature Mater 15, 1107–1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Gong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Different Reactivities of TiO 2 Polymorphs: Comparative DFT Calculations of Water and Formic Acid Adsorption at Anatase and Brookite TiO 2 Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 112, 6594–6596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', D’Arienzo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Scotti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Morazzoni, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Selli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Dal Santo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Pt and Au/TiO2 photocatalysts for methanol reforming: Role of metal nanoparticles in tuning charge trapping properties and photoefficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Applied Catalysis B: Environmental 130–131, 239–248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  26  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Setvin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Aschauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hulva, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Simschitz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Daniel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Diebold, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Following the Reduction of Oxygen on TiO2 Anatase (101) Step by Step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Journal of the American Chemical Society 138, 9565–9571.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S1  Supporting Information  Defect engineering over anisotropic brookite towards substrate-specific photo- oxidation of alcohols  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hossein Hejazi1, Mahdi Shahrezaei1, Piotr Błoński1, Mattia Allieta2, Polina M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sheverdyaeva3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Paolo Moras3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zdeněk Baďura1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sergii Kalytchuk1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Elmira Mohammadi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Radek Zbořil1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Štěpán Kment1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Michal Otyepka1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Alberto Naldoni1*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Paolo Fornasiero6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  1Czech Advanced Technology and Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Regional Centre of Advanced Technologies and Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Palacký University Olomouc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Křížkovského 511/8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 77900 Olomouc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Czech Republic 2Ronin Institute Montclair,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' NJ 07043 USA 3Istituto di Struttura della Materia-CNR (ISM-CNR),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' SS 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Km 163,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' I-34149,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Trieste,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Italy 4Nanotechnology Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Centre of Energy and Environmental Technologies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' VŠB–Technical University of Ostrava,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' listopadu 2172/15, 70800 Ostrava-Poruba, Czech Republic 5IT4Innovations, VSB – Technical University of Ostrava, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' listopadu 2172/15, 708 00 Ostrava- Poruba, Czech Republic 6Department of Chemical and Pharmaceutical Sciences, ICCOM-CNR Trieste Research Unit, INSTM-Trieste, University of Trieste, Via L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Giorgieri 1, 34127 Trieste, Italy 7Center for Energy, Environment and Transport Giacomo Ciamician - University of Trieste, Italy  Corresponding authors: alberto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='naldoni@upol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='cz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' pfornasiero@units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='it  S2  Preparation of TiO2 photocatalysts Titanium (IV) bis (ammonium lactate) dihydroxide Ti(NH4C3H4O3)2(OH)2 aqueous solution (50 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%, Sigma–Aldrich) (TALH) and urea (ACS reagent, Sigma–Aldrich), were used as precursors for the synthesis of TiO2 nanocrystals using a hydrothermal method, according to the procedure previously reported with some modifications 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A solution containing 45 mL of urea in deionized (DI) water and 5 mL of TALH was stirred until a clear solution was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The solution was afterwards transferred to a 125 mL Teflon lined autoclave and placed in an oil bath at 180°C and stirred at this temperature with 800 rpm for 20 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The autoclave was then cooled down in air and the precipitate was centrifuged and dispersed by sonication in DI water for several times until the pH of supernatant water became ~7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Finally, the precipitate was dried at 80°C for 12 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To prepare pure brookite and pure anatase samples, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='15M and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5M urea solution in DI water were used, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Commercial TiO2 brookite was purchased from Sigma-Aldrich (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='99 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' % purity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To prepare the reduced powders, 20 mg of TiO2 nanopowders were placed in a crucible within a quartz chamber in a tubular furnace (10 °C min-1 heating/cooling ramp in N2 flow rate 10 mL min-1, 1 h dwell in H2 flow rate 10 mL min-1 at predefined temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Before starting the heat treatment, the tube furnace was cleaned up increasing the temperature up to 1000°C in air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% platinum nanoparticles were loaded on TiO2 by via photodeposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Briefly, 50 mg of TiO2 powder suspended in 25 mL of methanol (ACS reagent, Sigma–Aldrich) and bubbled with Ar for 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Then, a solution of H2PtCl6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6H2O (ACS reagent, Sigma–Aldrich) was added and stirred for 20 min in the dark to favor Pt adsorption of the TiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Then the solution was illuminated for 1h using a solar simulator equipped with a 150 W Xe arc lamp and an AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G filter and calibrated to deliver a power of 100 mW cm-2 (1 Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Characterization The morphological analyses of the samples were performed by transmission electron microscopy (TEM) JEM-2100 (JEOL, Tokyo, Japan) at 200 kV of accelerating voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For TEM measurements, the samples were dispersed in ethanol by sonication for 5 minutes and then the suspensions were dropped on the copper grid with holey carbon film and dried upon air exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The average particle size of brookite nanorods were assessed by analyzing TEM micrographs and by considering at least 100 nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The high resolution transmission electron microscopy (HRTEM) analysis were performed using a HRTEM Titan G2 (FEI) with image corrector on accelerating voltage 300 kV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Images were taken with BM UltraScan CCD camera (Gatan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' X-ray diffraction (XRD) patterns were recorded at room temperature with an Empyrean (PANalytical, Almelo, The Netherlands) diffractometer in the Bragg-Brentano geometry and using Co-Kα radiation (40 kV, 30 mA, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1789 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The diffractometer was equipped with a PIXcel3D detector and programmable divergence and diffracted beam anti-scatter slits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The same amount of powders was placed on a zero- background Si slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The measurement range was 2θ = 10° - 100°, with a step size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0167° and acquisition time of 4 s per step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Standards SRM640 (Si) and SRM660 (LaB6) were used to evaluate the line position and the instrumental line broadening, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The identification of crystalline phases was performed using the High Score Plus software that includes the PDF-4+ and ICSD databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rietveld analysis was performed through the GSAS program 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We use the brookite orthorhombic model of Pbca space group with Ti and two O, namely O1, O2, in 8c position (x,y,z) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' During the refinement, the background was subtracted using shifted Chebyshev polynomials and the diffraction peak profiles were fitted with a modified pseudo-Voigt function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In the last calculation cycles all the parameters were refined: cell parameters, atomic positional degrees of freedom, isotropic thermal parameters, anisotropic microstrain broadening parameters, background, diffractometer zero point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To evaluate annealing T evolution ionic charge of Ti from the experimental dTi-O1, dTi-O2 of brookite, we calculated Bond Valence Sum (BVS) by using the tabulated parameters 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Crystallite size of synthesized brookite samples was estimated through the Williamson-Hall (WH) method 8 employing at least 15 reflections for each calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Single peak fitting to extract peak positions and profile parameters was performed through the WinPLOTR Software 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The crystallite size of as received and reduced commercial brookite and anatase was calculated from XRD patterns according to the Scherrer equation as follows:  S3  𝐷 = 𝐾 × 𝜆 𝛽 × 𝑐𝑜𝑠𝜃 where, D is the mean size of crystallite, K is the dimensionless shape factor, λ is the x-ray wavelength, β is the full width half maximum intensity (FWHM), and θ is the Bragg angle and considering K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9, λ=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='79 Å (Co).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Raman spectra were collected using a DXR Raman spectrometer (Thermo Scientific, Massachusetts, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The excitation laser operated at the wavelength of 455 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The samples were deposited on a silicon wafer and the laser was focused on its surface and tuned to maximize the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The laser power on the sample was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 mW cm-2 and exposure time was 3 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The reported Raman spectra were averaged over 512 experimental microscans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The surface area and pore size analyses were performed by means of N2 adsorption/desorption measurements at 77 K on a volumetric gas adsorption analyzer 3 Flex (Micromeritics, Georgia, USA) up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='965 P/P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Prior the analysis, the sample was degassed under high vacuum (10-4 Pa) at 130°C for 12 hours, while high purity (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='999 %) N2 and He gases were used for the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The Brunauer– Emmett–Teller area (BET) was determined with respect to Rouquerol criteria 10 for N2 isotherm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The ultraviolet-visible diffuse reflectance spectra (UV-Vis DRS) of the fabricated samples were obtained by Specord 250 plus (Analytik Jena, Jena, Germany) spectrophotometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' An integrating sphere was used to collect the spectrum and a Spectralon reference sample was used to measure the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  X-ray photoelectron spectroscopy (XPS) was carried out with a PHI 5000 VersaProbe II (Physical Electronics, Chanhassen, USA) spectrometer using an Al Kα source (15 kV, 50 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The obtained data were evaluated with the MultiPak software package (Ulvac-PHI Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chigasaki, Japan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' High-resolution spectra of C1s peaks were acquired by setting the pass energy to 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='500 eV and step size to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='200 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The binding energy values were corrected considering the C1s peak at 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 eV as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The spectral analysis included Shirley background subtraction and peak deconvolution using Gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoluminescence spectroscopy (PL) was performed on an FLS980 fluorescence spectrometer (Edinburgh Instruments, Livingston, United Kingdom) equipped with a R928P photomultiplier (Hamamatsu, Japan), with a 450 W xenon arc lamp as the excitation source for steady-state spectra and an EPL-375 picosecond pulsed diode laser (λem= 372 nm with a pulse width of 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 ps, a repetition rate of 10 MHz and an average power of 75 μW, Edinburgh Instruments) in conjunction with a time-correlated single-photon counting system for time-resolved photoluminescence measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spectral correction curves were provided by Edinburgh Instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The emission of TRPL spectra were detected at 450 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  PL decay curves were fitted using a multi-exponential function: 𝐼(𝑡) = ∑ 𝐵𝑖 exp (− 𝑡 𝜏𝑖) 𝑛 𝑖=1 , ∑ 𝐵𝑖 = 1 𝑛 𝑖=1 , Where, the fit parameter τi represents the decay time constant, Bi represents the normalized amplitude of each component, n is the number of decay times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The amplitude weighted average decay lifetime τave of the entire PL decay process reads as: 𝜏𝑎𝑣𝑒 = ∑ 𝐵𝑖𝜏𝑖 2 ∑ 𝐵𝑖𝜏𝑖  A nitrogen bath cryostat holder Optistat/DNV (Oxford instruments, Abingdon, United Kingdom) was used to control the temperature of sample during measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Since the PL emission of TiO2 is weak at room temperature and due to highly scattering nature of a typical nano-TiO2 sample, the stray excitation light could be wrongly assigned as PL signal 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To avoid this, the PL spectra were measured at 80 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Using low temperature condition results in slower non-radiative decay and brighter PL 11, which decreases the effect of stray light scattered by the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The powders in solid were pressed between two flat quartz and put into the chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' N2 and methanol were used to investigate the PL behavior of the TiO2 samples in contact with different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The methanol was degassed with argon bubbling for 10 minutes before wetting the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' An in-depth analysis of the electronic structure of the samples was carried out at the VUV-Photoemission synchrotron beamline (Elettra, Trieste) at room temperature with a Scienta R-4000 electron spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The O1s and Ti2p core levels were measured with photon b 650 eV with an instrumental energy resolution  S4  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The valence band was probed in near-resonant conditions to the Ti L2,3 edge, in order to enhance the signal of Ti-related states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A photon energy of 468 eV (energy resolution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='14 eV) was used to avoid the appearance of spurious Ti2p signal in the region of interest (from 4 eV binding energy up to the Fermi level), due to high harmonics contribution from the beamline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Electron paramagnetic resonance (EPR) spectra were recorded using a continuous wave X-band JEOL JES- X-320 spectrometer operating at \uf07e9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The EPR spectrometer is equipped with a variable temperature control ES 13060DVT5 apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The cavity Q quality factor was kept above 6000 in all measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Highly pure quartz tubes were used (Suprasil, Wilmad, ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 OD) and accuracy on g-values was obtained against a Mn2+/MgO standard (JEOL standard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For all experiments the same acquisition conditions were kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The microwave power was set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 mW, therefore, no power saturation effects was occurring in the EPR traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The modulation width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 mT and modulation frequency 100 Hz were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Experimental temperature was set to 78 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All spectra were recorded with 30 ms time constant and 2 minutes sweep time with 3 accumulations, to improve signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' HeCd (200 mW) laser with 325 nm wavelength was employed as the UV light source during EPR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' EPR envelopes were simulated in Matlab software where the spin-Hamiltonian EasySpin simulation package 12 is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  During the measuring the samples in contact with DI water + methanol solution at 80K, the head space of EPR tube was purged by nitrogen to avoid any parasitic oxygen signals coming from the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The EPR cavity was kept in constant flow of nitrogen to eliminate the formation of ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The samples were measured in suspension form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For each experiment 10 mg of the powder and 100 µl of DI water/methanol mixture were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In all experiments, to make spectra more comparable, the position of the EPR tube inside the cavity was kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1H-NMR (proton nuclear magnetic resonance) spectra were obtained on a JEOL 400 MHz spectrometer in CD3OD, using dimethyl sulfoxide (DMSO) as the internal standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All the measurements were collected at ambient temperature with a spectral width of 20 ppm, a pulse width of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 μs (90 °), a relaxation delay of 60 s, and 8 scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chemical shifts (δ) are expressed in ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C, H and N elemental analyses was performed on a Flash EA 1112 instrument (Thermo Finnigan, North Carolina, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Pt loading on TiO2 was determined by ICP-MS (Agilent 7700x, Agilent, USA) at isotope 105 using He mode and an external calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Calibration solutions were prepared from a certified reference material with Pt concentration 100,0 +/- 0,2 mg/L (Analytika Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Czech Republic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A mixture of nitric acid (ACS reagent, 70% , Sigma–Aldrich) and hydrochloric acid (ACS reagent, 37% , Sigma–Aldrich) in a molar ratio of 1:3, was used to digest the Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The Pt loading in pristine and reduced brookite nanorods was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='98 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='90%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The hydrodynamic diameter of the brookite B700 was measured by Dynamic Light Scattering (DLS) at 23°C, using a Malvern Nano-ZS instrument (Malvern Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Leamington Spa, UK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The light source was a laser 633 nm, 4 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The measurement was performed at the beginning (time = 0 h) and after specific times of reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The sample was taken from 10 mL of DI water and methanol (1:1 vol) solution with 2mg of dispersed B700-Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' At time = 0h the solution was ultrasonically dispersed for 10 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Photocatalytic experiments Photocatalytic activity was measured in a quartz reactor with 10 mL solution of DI water and methanol (volume ratio 1:1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The same conditions and volume ratio were applied for the photocatalytic reactions with ethanol (99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 %, BC Chemservic), isopropanol (ACS reagent, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8%, Sigma–Aldrich), formaldehyde (36- 38%, Penta) and formic acid (99%, Penta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' After sealing the reactor with a rubber septum, the photocatalysts were sonicated for 10 min to create a homogeneous and dispersed suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Afterwards, the suspension was bubbled with argon for 30 min to remove the unwanted gasses and dissolved oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The samples were irradiated using a solar simulator equipped with a 150 W Xe arc lamp and an AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G filter and calibrated to deliver a power of 100 mW cm-2 (1 sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A calibrated reference solar cell (Newport, California, USA) was used before and after reaction to check the power of irradiation (1 Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Each sample was irradiated for 24 h under continuous stirring before measuring the amount of evolved hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The test was repeated three times and the average amount of measured hydrogen was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The photocatalytic  S5  hydrogen was detected with a gas chromatograph GCMS-QP2010 SE (Shimadzu, Kyoto, Japan ) and a TCD (Thermal conductivity detector), using Ar as carrier gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The temperature of the reaction suspension was measured with a thermocouple and was 23°C both before and after 24 h of irradiation under 1 sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To identify the wavelength dependence of AQY, the reactor was illuminated with the wavelengths 316 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 mW cm-2), 334 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 mW cm-2), 360 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 mW cm-2), 369 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 mW cm-2), 386 (5 mW cm-2) and 402 nm (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 mW cm-2) using a tunable diode light source of Zahner CIMPS PP201 system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The illuminated surface area was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='94 cm2 and the power of each wavelength was measured using an external digital power meter Thorlabs PM100D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The sample was illuminated for 1h and the reacting medium was a 1:1 vol/vol% water:methanol mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The apparent quantum yield (AQY) was calculated according to the following equation:  𝐴𝑄𝑌 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑎𝑐𝑡𝑒𝑑 𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝ℎ𝑜𝑡𝑜𝑛𝑠 × 100 = 2 × 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑣𝑜𝑙𝑣𝑒𝑑 𝐻2 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑖𝑛𝑐𝑖𝑑𝑒𝑛𝑡 𝑝ℎ𝑜𝑡𝑜𝑛𝑠  The qNMR (quantitative nuclear magnetic resonance) analysis was used to investigate the rate of methanol consumption during the photocatalytic hydrogen evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For this purpose, a 10 mL mixture of DI water: CD3OD (volume ratio 1:1) was prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To have a detectable signal, 50 μL of non-deuterated methanol (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CH3OH) was added to the above mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The photocatalytic experiment was performed following the same procedure explained above for hydrogen evolution measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' At the specified times, 600 μL of solution was taken by syringe and centrifuged at 15000 rpm for 30min to separate the catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Then the clear and transparent solution was transferred to a quartz NMR tube and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 µL DMSO was added as internal standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The NMR spectra of the samples were recorded and the peak area of methanol at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 ppm was compared to the peak area of internal standard to study the rate of photocatalytic methanol consumption over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  DFT calculations Calculations were carried out by using the DFT-based Vienna Ab Initio Simulation Package (VASP) 13,14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The projector-augmented-wave (PAW) formalism 15,16 was used to treat the electron–ionic core interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A plane-wave basis with a 400 eV energy cutoff was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Test calculations were also performed with energy cutoff increased to 600 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Exchange and correlation effects were treated within a generalized- gradient approximation (GGA) by using Perdew-Burke-Ernzerhof (PBE) functional 17,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To counteract the problems of standard density functionals associated with the self-interaction error (SIE) we applied on-site Hubbard corrections 19 to both Ti-d and O-p states 20 with an effective U parameter of 6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All computations were performed in spin unrestricted manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Brillouin zone samplings were kept restricted to Gamma point only due to the supercell dimensions being sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We modeled the (210) surface of brookite by a bulk-terminated slab of 12 Ti-layers in thickness and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38 Å × 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='41 Å of surface area and containing 432 atoms (see the structure shown in Figure S41) 21, and with a vacuum layer of length ∼20 Å deployed along the off-planar direction to ward off spurious interactions with the periodic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Except atoms in the middle part of the slab (area enclosed by green planes in the structure displayed in Figure S41), all other atoms were relaxed until all forces were reduced below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='025 eV/Å and the change in total energy between successive iteration steps became smaller than 10−6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Several possible sites for oxygen vacancy formation were considered (and denoted by V1–V8 in the structure in Figure S41) and the system was re-optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' For oxygen vacancy defects present in the bulk region of the slab, atoms in the immediate vicinity of the vacancy were allowed to relax too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The effect of distortion of TiO6 octahedra on the electronic structure of brookite was also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Two models were considered: (i) Several Ti-O axial distances were modified by up to ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 Å and the change in densities of states with respect to an ideal (210) surface of brookite was monitored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (ii) Randomly chosen Ti-O distances were modified within ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 Å and accompanied by a displacement of Ti atoms from its equilibrium positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S6  Supplementary Text  Digital pictures of TiO2 photocatalysts Upon reduction under hydrogen atmosphere at different temperatures, the color of synthesized brookite and anatase samples changed from white to gray and black, while the commercial brookite showed a relatively small color change (Table S1), suggesting its non-reducibility, as also confirmed by the other characterizations reported below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Digital photographs of as synthesized brookite (B-AS), as-received commercial brookite (CB- AR), as-synthesized anatase (A-AS) and reduced samples at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The numbers after abbreviations stand for reducing temperature in °C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample  Sample  Sample  B AS  CB AR  A AS  B500  CB400  A400  B600  CB500  A500  B700  CB600  A600  B800  CB700  A700  B900  CB800  B1000  CHN elemental analysis To exclude the contribution of carbon, nitrogen, and hydrogen impurities in the photocatalytic activity, CHN analysis of TiO2 samples were carried out before and after reduction under hydrogen atmosphere for the most active photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' As Table S2 shows, there is no significant difference between carbon, nitrogen, and hydrogen concentration in the samples before and after the thermal treatment in hydrogen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S7  Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CHN analysis of the pristine and the most photoactive sample for brookite, commercial brookite and anatase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample C (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' %) H (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' %) N (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' %) B-AS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 B700 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01 CB-AR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01 CB600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 A-AS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 A-500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02  Morphology of TiO2 photocatalysts Figure S1A and Figure S1B show the TEM images of B-AS and B700, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The as-synthesized brookite nanorods have an average length of 90 ± 35 nm that after reduction under hydrogen atmosphere at 700°C decreased to 82 ± 28 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, the reduction in high temperature resulted in aggregation of particles due to sintering as well as losing their well-defined facets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S2A and Figure S2B present the TEM images of CB-AR and CB600, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The particle shape of commercial brookite is rounded (average diameter 29 ± 10 nm) in comparison with the synthesized brookite and showed almost no change in shape and a slightly increase in size after reduction (average diameter 32 ± 7 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The as synthesized anatase is spherical in shape with average diameter of 6 ± 1 nm (Figure S3) and its shape remained unchanged with slighty increase in diameter after reduction (average diameter 8 ± 2 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S8  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' TEM images of (A) B-AS and (B) B700 with associated histograms of size distributions (bottom) based on 100 measurements of nanorods length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' TEM images of (A) CB-AR and (B) CB600 with associated histograms of size distributions (bottom) based on 100 measurements of particle diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  100nm  100nm  45  45  B AS  B700  Counts  30  30  15  15  0  0  20  50  80  110  140  170  20  50  80  110  140  170  Size (nm)  Size (nm)A  B  50 nm  50nm  45  CB AR  CB600  45  Counts  30  30  15  15  0  0  10  20  30  40  50  60  10  20  30  40  50  60  Size (nm)  Size (nm)S9  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' TEM images of (A) A-AS and (B) A500 with associated histograms of size distributions (bottom) based on 100 measurements of particle diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  40 nm  40nm  50  50  A AS  A500  Counts  35  35  20  20  5  5  2  4  6  8  10  12  14  16  2  4  6  8  10  12  14  16  Size (nm)  Size (nm)S10  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' TEM images of (A) platinized pristine brookite and (B) platinized reduced brookite at 700°C with associated histograms of size distributions of Pt particles (bottom) based on 100 measurements of particle diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The loaded reduced brookite present large Pt aggregates and therefore the size distribution refers to the isolated Pt nanoparticles found in the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The photodeposition of Pt results clearly in different results: pristine brookite present isolated homogeneously dispersed Pt nanoparticles, while reduced brookite show mainly large Pt aggregates and some isolated Pt nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  50 nm  50 nm  B AS/Pt  B700/Pt  39  39  Counts  Counts  26  26  13  13  0  0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  Size (nm)  Size (nm)S11  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A,B) TEM and HR-TEM (C-F) micrographs of pristine brookite nanorods loaded with 1 wt% Pt showing their homogeneous distribution over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The darker dots are the Pt nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A-C) STEM-HAADF images and (D) elemental EDS mapping of (C) for pristine brookite nanorods loaded with 1 wt% Pt showing their homogeneous distribution over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  B  C  200  200 nm  nm  D  nmA  B  HAADF  50 nm  50nm  60 nm  Ti  Pt  Tilo  Pt  60nm  60 nm  60nm  60 nmS12  Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A-C) TEM and HR-TEM (D) of micrographs reduced brookite nanorods (at 700°C) loaded with 1 wt% Pt showing their aggregation over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The darker dots and nanostructured aggregates are made by Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A-C) TEM and HR-TEM (D) of micrographs reduced brookite nanorods (at 700°C) loaded with 1 wt% Pt showing their aggregation over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The Pt nanocrystals are the brighter spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  200nm  200nm  D  50  nm  10nmA  B  50 nm  20 nmS13  Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' STEM-HAADF and elemental mapping images of reduced brookite nanorods (at 700°C) loaded with 1 wt% Pt showing their aggregation over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' STEM-HAADF and elemental mapping images of reduced brookite nanorods (at 700°C) loaded with 1 wt% Pt showing their aggregation over the TiO2 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  HAADF  Ti  90 nm  90nm  90 nm  Pt  Tio  Pt  90 nm  90nmHAADF  Ti  60nm  60nm  60 nm  Pt  Tio  Pt  60nm  60nmS14  Specific surface area measurement Figure S11 shows the nitrogen sorption isotherm profiles of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In all cases, the BET surface area decreases after reduction under hydrogen atmosphere at high temperature (Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' N2 adsorption/desorption type IV isotherms (mesoporous solids) at 77K for (A) B-AS and B700, (B) CB-AR and CB600, and (C) A-AS and A500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Brunauer−Emmett−Teller (BET) specific surface area for the pristine and the most photoactive sample of each studied phases of TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample BET surface area (m2 g-1) anataseB- AS 67 B700 47 CB-AR 50 CB600 48 A-AS 273 A-500 189  A  180  B  STP)  STP)  10  8  135 B AS Adsorption  CB AR Adsorption B AS Desoption  6  CB AR Desoption  90  (cm3  4  45  2  Quantity Adsorbed  0  0  8  135 B700 Adsorption  CB600 Adsorption  B700Desorption  6  CB600 Desorption  90  4  45  2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  Relative Pressure (p/p°)  Relative Pressure (p/p°)  C  P180  90  45 A As Adsorption  0 A AS Desoption  135  A500Adsorption  A500 Desorption  90  45  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  Relative Pressure (p/p°)S15  Structural characterization X-ray diffraction XRD analysis (Figure S12 and Table S4) revealed that the synthesized brookite is crystalline and stable up to 700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The sample treated at 800°C, instead, presented around 91 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' % of rutile phase, while at 900°C almost all the rutile is transformed to Magnéli phase Ti9O17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' At 1000°C, the other Magnéli phases Ti4O7 in 94 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' % and Ti5O9 in 6 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' % were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The phase stability of commercial brookite is up to 700°C (the same as synthesized brookite) and after that reduction at 800°C induced the complete conversion to rutile (Figure S13 and Table S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The XRD pattern of as synthesized and reduced anatase (Figure S14) samples indicates that the as synthesized anatase photocatalysts are stable at up to 500°C , while almost full conversion to rutile is obtained at 700°C (Table S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The average particle sizes obtained using the Scherrer method for commercial brookite and anatase samples are in good agreement with those retrieved from TEM micrographs analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) XRD patterns of brookite samples and reference patterns for brookite (bottom) and rutile (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) XRD patterns for reduced brookite at 900 and 1000 °C and reference patterns (bottom) for rutile and various magnéli phases (TixO2x-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' XRD patterns for commercial brookite samples and reference patterns for brookite (bottom) and rutile (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  41 008 7847  Rutile  B1000  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B800  B700  B900  B600  (Ti,O1z) 00 050 0791  B500  (Ti,Og)04 005 4465  B AS  (Ti,07) 04 005 4521  01 071 6410  (Rutile) 01 071 6410  Brookite  20  40  60  80  20  40  60  80  20 (°), Co kα  20 (°), Co kα(Rutile) 41 008 7847  CB800  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  CB700  CB600  CB500  CB400  CB AR  (Brookite) 01 071 641Q  20  40  60  80  20 (°), Co kαS16  Figure S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' XRD patterns for anatase samples and reference patterns for anatase (bottom) and rutile (top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phase composition of brookite samples retrieved from Rietveld refinement of XRD patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Brookite (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Rutile (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Ti4O7 (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Ti5O9 (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Ti9O17 (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) B-AS 100 - - - - B500 100 - - - - B600 100 - - - - B700 100 - - - - B800 9 91 - - - B900 - 9 - - 91 B1000 - - 94 6 -  Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rietveld quantitative phase analysis of commercial brookite samples and crystallite size calculated according to the Scherrer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Brookite (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Anatase (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Rutile (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Crystallite size (nm) CB-AR 100 - - 27 CB400 100 - - 27 CB500 100 - - 28 CB600 100 - - 31 CB700 100 - - 41 CB800 - - 100 69  41 008 7847  Rutile  A700  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  A600  A500  A400  A AS  01 071 6410  Anatase  20  40  60  80  20 (°), Co kαS17  Table S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rietveld quantitative phase analysis of anatase samples and crystallite size calculated according to the Scherrer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Anatase (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Rutile (wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='%) Crystallite size (nm) A-AS 100 - 8 A400 100 - 8 A500 100 - 13 A600 68 32 24 A700 - 100 28 To investigate more in depth the structural modifications induced by the reduction treatment, we performed detailed Rietveld refinements for the samples treated up to 700°C (Figure S15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All of the patterns are well described by single phase of brookite TiO2 and, within the resolution of our measurements, we did not detect any spurious crystalline phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S1A and B shows the reduction temperature (T) dependence of the average particle size, D (nm), and the average strain as obtained using the Williamson-Hall method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We observed a clear decrease of D for T = 400°C followed by an increase of average particle size above this T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' On the other hand, the average particle strain weakly decreases at T=400°C keeping roughly the same value at higher T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This phenomenon can be associated to particle sintering since the sudden aggregation of particle can result in an increase of D owing to relieve of intraparticle tension by decreasing the overall particle strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This is confirmed by the annealing temperature evolution of component of empirical extension of anisotropic microstrain broadening tensor refined by Rietveld refinements (Figure S1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Indeed, we note a progressive convergence of these parameters to the same values indicating that the evolution of Williamson-Hall strain is explained by the tendency to form more isotropic particle aggregates at high annealing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This morphological observation is in agreement with the TEM micrographs of reduced brookite B700 (Figure S1B), which confirm that brookite nanorods tends to aggregate in isotropic agglomerate upon reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The tendency toward powder aggregation can also explain the disagreement between the average nanorod lengths detected by TEM, both before and after reduction, and the average particle size retrieved by Rietveld refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S1D shows the annealing temperature evolution of the refined lattice parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Orthorhombic strain is defined as: 𝜂 = 2(𝑎 − 𝑏) (𝑎 + 𝑏)  where a and b are the unit cell axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We observed that \uf068 remains almost unchanged for all the investigated annealing temperature whereas the c-axis shows clear decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In other words, this indicates that the annealing temperature does not affect the ab plane but induces distortion along the c-axis only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This anisotropic evolution of structural parameters upon reduction is even better outlined by the interatomic distances related to the TiO6 octahedra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' TiO2 structure is composed of TiO6 octahedra, each with a titanium atom at its center and oxygen atoms at its corners (Figure S1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In brookite TiO6 are distorted oxygen atoms in two different positions, namely O1, O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This results in two groups of distances namely dTi-O1, dTi-O2 (Figure S1B, C) which can be regrouped into two axial and four equatorial different interatomic distances which have been averaged out and shown in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Averaged axial distance expand upon increasing temperature, whereas the average equatorial distances show a weak contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Both distances tend to the same value and this may indicate that the annealing process induces the TiO6 to be more regular, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' isotropic, and less distorted at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To figure out the effect of reduction treatment on the brookite structure we argue that the reduction of TiO2 at high temperature creates oxygen vacancies (𝑉𝑂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S18  in Kröger–Vink notation) 22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='23 and induces Ti3+ ions electron trapped in Ti4+ lattice sites ( 𝑇𝑖𝑇𝑖 ′  ) according to the following relation:  𝑂𝑂 𝑥 + 2𝑇𝑖𝑇𝑖 𝑥 + 𝐻2  → 𝑉𝑂 •• + 2𝑇𝑖𝑇𝑖 ′ + 𝐻2𝑂  Two negatively charged 𝑇𝑖𝑇𝑖 ′  species can then couple with one double positive charged 𝑉𝑂 •• promoting the formation < 𝑇𝑖𝑇𝑖 ′ − 𝑉𝑂 •• − 𝑇𝑖𝑇𝑖 ′ > defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The mutual attraction between them can produce a contraction of the structure to account for electron neutrality, which is fully in agreement with the observed c-axis contraction upon reduction at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The reduction of Ti valence to produce 𝑇𝑖𝑇𝑖 ′  species, is compatible with a depletion of ≈ - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5% as evidenced by bond valence sum (BVS) analysis, see for B700 (Figure S1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Further, the correlation between the BVS and c-axis contraction is shown in Figure S1B, showing a nearly linear relationship between these two parameters and thus indicating that more 𝑉𝑂𝑠 are induced by the reduction and more the c-axis resulted to be contracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This highlights that reduction treatment introduced an anisotropic and preferential deformation of the brookite lattice along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rietveld refinements of brookite samples reduced at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Dots are experimental data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' continuous lines are the calculated profiles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rietveld agreement factors [R (F2)] between observed and calculated patterns ranged from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='06 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B AS  Experimental  Calculated  Residual  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B400  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B500  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B600  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B700  20  30  40  50  60  70  80  90  100  20 (), Co kaαS19  Figure S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Structural parameters retrieved from Rietveld refinements of XRD patterns for as synthesized brookite and brookite reduced at different temperatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' annealing T in x axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Reducing temperature dependence of average particle size (D), and (B) crystal structure strain as obtained by Williamson- Hall (WH) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (C) Annealing temperature evolution of components of empirical extension of anisotropic microstrain broadening tensor (L) refined by Rietveld refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Indexes are referring to the following expression: L = L11h2+L22k2+L33l2+2L12hk+2L13hl+2L23kl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (D) Annealing temperature dependence of orthorhombic strain (see text) and c-axis (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  390  B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0014  360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0012  330  Strain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0010  300  D  WH  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0008  270  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0006  240  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0004  210  0  100  200300400500  600  700  800  0  100  200300400500  600  700  800  Annealing T (°C)  Annealing T (°C)  C  D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='510  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  33  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='509  Orthorombic strain (  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  13  23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='508  Microstrain broadening  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='507  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='135  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='506  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='134  100200300  400  500600700800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  Annealing T (°C)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='505  0  100  200  300  400  500  600  700  800  0  100  200  300  400  500  600  700  800  Annealing T (°C)  Annealing T (°C)S20  Figure S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Representation of octahedra (TiO6) in TiO2 brookite unit cell (Pbca) showing the arrangement of distorted TiO6 resulting in two groups of distances, namely dTi-O1 and dTi-O2, composed by two axial and four equatorial different interatomic distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Values refer to structure refined at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) Evolution of axial and (C) equatorial interatomic distances retrieved from Rietveld refinements of XRD patterns for as synthesized brookite and brookite reduced at different temperatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' annealing T in x axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Solid lines are guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  dT 011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='913 A  T 02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='918 A  Axial  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Equatorial  dT 01=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='943 A  Ti 021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='918A  dt 011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='955 A  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='082A  B  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02  Axial  Lo !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='lp  dTi 02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='98  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='92  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='90  0  200  400  600  800  C  Annealing T (°C)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='10  Equatorial  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05  dti 01  dti 02  di 01  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='90  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='85  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='80  0  200  400  600  800  Annealing T (°C)S21  Figure S18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Annealing Temperature dependence of averaged axial and equatorial interatomic retrieved from Rietveld refinements of XRD patterns for as synthesized brookite and brookite reduced at different temperatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' annealing T in x axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Annealing Temperature dependence of Ti Bond Valence Sum (BVS) and (B) correlation between Ti BVS and c-axis contraction retrieved from Rietveld refinements of XRD patterns for as synthesized brookite and brookite reduced at different temperatures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' annealing T in x axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='97  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='96  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='94  Aaverage axial dti o  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='93  Aaverage equatorial dti o  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='93  0  100  200  300  400  500  600  700  800  Annealing T (°C)A  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='17  B  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='16 [B AS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='15  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='15  TB600  TB400  S  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  B700  B500  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='13  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='11 0  100  200  300  400  500  600  700  800  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='134  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='135  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='136  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='137  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='138  Annealing T (°C)  c axis (A)S22  Raman spectroscopy The crystal system of brookite is orthorhombic and has eight formula units per unit cell and thirty-six Raman active modes (9A1g+9B1g+9B2g+9B3g) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The crystal system of anatase is tetragonal and has two formula units per unit cell and six Raman active modes (A1g+2B1g+3Eg) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The crystal system of rutile is tetragonal and has two formula units per unit cell and four Raman active modes (A1g+B1g+B2g+Eg) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Raman spectra of synthesized and reduced brookite samples (Figure S20A) confirmed that the synthesized brookite is stable up to 700°C and after that it transforms to rutile, as evidenced by the disappearance of brookite peaks and appearance of rutile peaks in Raman spectra of B700 and B800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The same Raman modes of rutile remained evident also for B900 and B1000, where an additional phase transition to Magnéli phases is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The main Raman peak of as synthesized brookite at 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 cm-1 is blue shifted to 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 cm-1 after reduction at 700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Moreover, there is a peak narrowing in the main peak due to the reduction (FWHMB-AS = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='82 cm-1, FWHMB700 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='81 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Raman spectroscopy measurements also revealed that the commercial brookite sample is stable under reducing conditions up to 600°C, after which, rutile Raman fingerprint developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  There is almost no peak shift and no change in FWHM of the main peak (FWHMCB-AR = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='41 cm-1, FWHMCB600 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='64 cm-1) (Figure S20B), suggesting that no significant modifications occurred in the commercial brookite lattice upon reduction at high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S20C shows, instead, that anatase is stable up to 500°C in our reduction conditions showing also a significant blueshift of the main Raman mode from 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 cm-1 for as-synthesized anatase to 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 cm-1 for anatase reduced at 500°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The same peak narrowed significantly from FWHMA-AS = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='97 cm-1 to FWHMA500 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='80 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Several phenomena can give rise to TiO2 Raman peak shifting and broadening (or narrowing) 24–28 as follows: (i) the lattice strain 26, (ii) the crystal size that could regulate the phonon confinement and the Raman scattering (increasing in crystal size results in redshift and peak narrowing) 24, and (iii) the oxygen stoichiometry, depending on the phase of TiO2, could affect the position of Raman peaks and also could modify the FWHM 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In the case of synthesized brookite and anatase, we observed a blueshift and peak narrowing of the main Raman peak after reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Upon reduction, we detected a reduction of lattice microstrain in brookite nanorods (Figure S20C) as well as an increase in the overall crystal size due to nanorods aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Therefore, we propose that the change in oxygen stoichiometry (as also evidenced by the other characterizations reported) underlies the blue shift and the peak narrowing witnessed for both brookite and anatase 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In contrast, in case of commercial brookite neither peak shifting nor peak narrowing (or broadening) were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This result confirms that the commercial brookite was not reduced under reduction conditions as already suggested by UV-vis diffuse reflectance spectra analysis, TPR-MS measurements, and also by the color of the samples, which did not change upon reduction treatment (Table S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S23  Figure S20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Raman spectra of (A) as-synthesized and reduced anisotropic brookite samples, (B) as-received and reduced commercial brookite samples, and (C) as-synthesized and reduced anatase samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Measurements performed with a 455 nm laser at a power density of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 mW cm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The left part of each panel is the zoomed view of the main Raman peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  B  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  3900  CB800  8800  CB70C  B700  CB600  B600  CB500  B500  B400  CB400  B AS  CB AR  130153100  200  300  400  500  600  700  800  130153100  200  300  400  500  600  700  800  Raman shift (cm 1)  Raman shift (cm 1)  C  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  A700  A600  A500  A400  A AS  120148100  200  300  400  500  600  700  800  Raman shift (cm 1)S24  Diffuse reflectance spectroscopy The Tauc method 29 is one of the most common procedures for determining the bandgap of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This method makes a relationship between absorption coefficient and optical bandgap of material base on the following formula: (𝛼ℎ𝜈)𝑛 = (ℎ𝜈 − 𝐸𝑔) where, α is the absorption coefficient, h is the Plank constant, ν is the frequency of radiation and Eg is the bandgap of material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The n factor depends on the nature of the electronic transitions and is equal to 2 for TiO2 as it is an indirect band gap semiconductor 30,31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This method assumes that the scattering component of the reflected irradiation is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, in case of nanopowders with high amount of scattering, this component could not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, the Kubelka-Munk theory is used to make an estimation of absorption from reflectance according to the following formula 32: 𝐹(𝑅) = 𝛼 = 𝐾 𝑆 = (1 − 𝑅)2 2𝑅  where, R is the reflectance of the sample with infinite thickness to avoid any contribution of substrate, K and S are the absorption and scattering coefficients, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The bandgap of TiO2 can be retrieved using the Tauc method as follows: (𝐹(𝑅)ℎ𝜈)1/2 = (ℎ𝜈 − 𝐸𝑔) From the Tauc plot, the x-axis intersection point of the tangent to the linear increase of light absorption in the Tauc plot gives the band gap energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This method is accurate and used here for determining the bandgap energy of pristine TiO2 materials, while for reduced TiO2 powders the baseline method was employed to calculate the bandgap 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Furthermore, we analyzed the Urbach tail in absorption spectra, as it originates optical transitions involving intragap states related to defects 33–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The Urbach energy can be calculated as follow 36,37: 𝛼 = 𝛼0 + exp ( 𝐸 𝐸𝑢 ) where α is the absorption coefficient, E is the photon energy equal to hν and Eu is the Urbach energy, which can be retrieved by the reciprocal of the slope of the linear part of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S21 shows the absorption spectra of as synthesized and reduced anisotropic brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' As expected, most of the UV light is absorbed owing to the wide band gap of brookite, which was found to be comprised between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='36 eV (Table S7) for all samples containing only the brookite phase (B-AS, B500, B600, B700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A large shift in the absorption edge of B800 and B900 was observed due to the phase change to rutile, with a corresponding decrease of bandgap values around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The sample B1000 absorbs almost the entire spectrum and it is not possible to calculate any bandgap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This can be ascribed to a phase composition including several semimetallic (Magnéli) phases 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The absorption of anisotropic brookite samples in the visible region increases by increasing the temperature of reduction, which is predictable from the color of powders, and it can be assigned to the introduction of oxygen vacancies and Ti3+ electronic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, visible light has no influence in the photocatalytic activity of the reduced brookite, as discussed in photocatalysis section reported above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Furthermore, in all of the samples, a change in the Urbach tail due to the reduction at different temperatures was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' It varied from 69 to 115 meV passing from B-AS to B700, suggesting an increased population of point defects in TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reflectance spectra for commercial brookite and synthesized anatase (with isotropic crystal shape) are reported in Figure S22 and Figure S23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In addition, Table S8 and Table S9 provide the results obtained from analysis of bandgap and Urbach energy of commercial brookite and anatase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  It is apparent from the results that in all cases the bandgap underwent to only a slightly modification after reduction, remaining around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2, for commercial brookite and anatase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' After phase transformation to rutile (at 800°C in commercial brookite and at 600°C in anatase), bandgap values decreased and Urbach energy increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Furthermore, no change in color of the commercial brookite samples was observed after reduction at different temperatures (Table S1), while in anatase samples the color change from white to gray and black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The variations of Urbach energy is in agreement with this observation: for commercial brookite it remained at a stable value of ~55 meV before and after reduction,  S25  while for anatase Urbach energy increased from 115 (A-AS) to 205 meV (A500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These results further confirmed that commercial brookite was hardly reducible, while defects could be introduced in the synthesized anatase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Diffusive reflectance spectra and (B) Tauc plots of as synthesized and reduced anisotropic brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Diffusive reflectance spectra and (B) Tauc plots of as received and reduced commercial brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  100  B  B AS  B800  80  B500  B900  Reflectance (%)  B600  B1000  B700  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='13eV  60  40  B AS  B500  B600  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38ev  B700  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38eV  B800  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='37 eV  B900  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 eV  B1000  0  300  500  700  900  1100  2  3  4  5  Wavelength (nm)  hu (eV)A  100  B  CB RT  CB400  80  CB500  Reflectance (%)  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  CB600  CB700  CB800  60  CB RT  [F(R)hu]1/2 (  CB400  CB500  40  CB600  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='04 eV  CB700  CB800  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34 eV  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='33 eV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34eV7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32eV7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32eV  0  300  500  700  900  1100  2  3  4  5  Wavelength (nm)  hu (eV)S26  Figure S23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Diffusive reflectance spectra and (B) Tauc plots of as synthesized and reduced anatase samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Table S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Optical bandgap and Urbach energy of anisotropic brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Optical bandgap (eV) Urbach energy (meV) B-AS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 69 B500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='37 67 B600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38 104 B700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='38 115 B800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='13 155 B900 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='13 - B1000 - -  Table S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Optical bandgap and Urbach energy of commercial brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Optical bandgap (eV) Urbach energy (meV) CB-AR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 58 CB400 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 55 CB500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34 54 CB600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='32 56 CB700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='34 39 CB800 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='04 36  A  100  B  A AS  A AS  A400  A400  A500  80  Reflectance (%)  A500  A600  A600  A700  A700  60  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09  40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09eV  20  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='29  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20 eV  0  300  500  700  900  1100  2  3  4  5  Wavelength (nm)  hu (eV)S27  Table S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Optical bandgap and Urbach energy of anatase samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample Optical bandgap (eV) Urbach energy (meV) A-AS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20 115 A400 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='29 154 A500 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='26 205 A600 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09 252 A700 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='09 100  S28  Methanol photoreforming In order to demonstrate the better photo-oxidation acitivty toward methanol of reduced brookite, we performed a series of photocatalytic experiments detecting (1) the methanol consumption rate through 1H- NMR spectroscopy and (2) the hydrogen evolution rate with GC for platinized B-AS and B700 (Figure SS24-S25 and Figure 1 of the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We did not detect any trace of formaldehyde nor formic acid in both liquid phase (through NMR analysis) and gas phase (through GC analysis), suggesting that methanol oxidation proceeded toward CO2, as confimed by the high reactivity of brookite samples toward the oxidation of formaldehyde and formic acid solutions (Figure S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1H-NMR spectra of the solutions after photocatalysis at representative reaction times for B- AS/Pt and B700/Pt samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The blank samples were recorded by using solutions containing CD3OD, DI water, and DMSO ((CH3)2SO) as the internal standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In the other samples, an aliquot of 50 μL of CH3OH was added to the solution and its photocatalytic consumption rate was measured by assessing the area of its characteristic NMR peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' CHD2OH is the impurity presents in the deuterated methanol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  F  AF  AF  B AS/Pt 4h  B700/Pt 4h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05  H  0  0  H CO H  H C O H  工  HsC 0~CH3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03  工 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02  D  H C O H H C O H 1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00 1 1  B AS/Pt Oh  1  1  1  B700/Pt0h  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05  1  1  1  1 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02  1  1 /1  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00 Blank 1  Blank  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 1 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00  1  1  AF  AF  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='60  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='30  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='70  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='60  ppm  wddS29  Table S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Relative NMR integration values of MeOH signal to internal standard (DMSO) at different reaction times for the pristine (B-AS/Pt) and reduced brookite (B700/Pt) samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Time (h) B-AS/Pt B700/Pt -a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='04a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='92a 0b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='92b 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='88 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='85 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='83 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='83 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='82 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='78 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75 aBlank measurements with no photocatalyst, but in the presence of the reagents d4-MeOH (5 mL), H2O (5 mL), MeOH (50 μL), and DMSO (2 μL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' bResults were obtained after addition of the photocatalysts (B-AS-Pt and B700-Pt) to the solutions and stirring for 30 min in the absence of light to allow for reaching the adsorption/desorption equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Dynamic Light Scattering (DLS) measurements analysis showing the hydrodynamic diameter evolution of the B700/Pt catalysts during the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  200  400  600  800  0  14  28  42  0  14  28  42  0  14  28  42  0  14  28  42  diameter (nm)  0h  Intensity (%)  4h  16h  24h  S30  Figure S26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hydrogen evolution kinetics for B-AS/Pt and B700/Pt under one sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hydrogen amount evolved from five consecutive photocatalytic cycles for B700/Pt during methanol photoreforming under 1 Sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  0 5 10 15 20 25 0 500 1000 1500 2000 2500 Amount of H2 (mmol m-2) Time (h) B700/Pt B-AS/Pt 0 24 48 72 96 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 Amount of evolved H2 (mmol m-2) Time (h)  S31  Figure S28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Specific hydrogen evolution rate obtained for samples loaded with 1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% Pt by normalizing the photocatalytic rates by BET specific surface area for the as-received (CB-AR/Pt) and the most active reduced commercial brookite (CB600/Pt), and as-synthesized (A-AS/Pt) and the most active reduced anatase (A500/Pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalytic hydrogen evolution rate optimized per mass of used photocatalyst for (A) for B-AS/Pt and B700/Pt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) A-AS/Pt and A500/Pt under one sun illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  9  5  4  3  2  工  0  CB AR/Pt  CB600/Pt  A AS/Pt  A500/PtA  B  10  10  B AS Pt  A AS Pt  B700 Pt  A500 Pt  8  (μmol h 1)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  6  evolution rate  4  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  2  I/  0  0  2  4  6  8  10  0  2  4  6  8  10  Amount of photocatalyst (mg)  Amount of photocatalyst (mg)S32  Figure S30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Hydrogen production rate from different intermediates of methanol oxidation (formaldehyde and formic acid) of as-synthesized (light blue) and reduced at 700°C (dark blue) brookite samples loaded with 1 wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='% Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These experiments confirmed GC analysis of the gas phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', we detected only carbon dioxide) and NMR analysis of the liquid phase after reaction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' we did not detect any signal related to formaldehyde nor formic acid) indicating that the methanol photo-oxidation proceeded to carbon dioxide, due to reactivity of brookite nanorods toward oxidation of the intermediates (formaldehyde and formic acid) of methanol oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photolysis of formaldehyde under AM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5G 1sun illumination produced a very small hydrogen production rate, namely, ~85 nmol m-2 h-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  45 40 2 prod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' rate (μmol h-1 m*2 35 30 25 20 15 H2 10 5 0 Formaldehyde Formic acidS33  Figure S31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Co-catalyst free rate of hydrogen evolution of TiO2 samples tested under 1 Sun illumination for 24 h and reduced under pure hydrogen for 1 h: (A) brookite nanorods, (B) brookite nanorods reduced at 700°C for different times, (C) commercial brookite, and (D) synthesized anatase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We report the specific H2 evolution rate only for the most performing sample for each series in Figure S29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  50  B  50  40  40  30  30  20  20  10  10  H2  0  B1000  0  B AS  B500  B600  B700  B800  B900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5h  1 h  2 h  Timeofreductionat7oo°c  C  D  40  12  10  26  30  jowm)  8  evolutionrate  20  6  4  10  2  0  0  CB RT  CB400  CB500  CB600  CB700  CB800  A AS  A400  A500  A600  A700S34  Figure S32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Specific hydrogen evolution rate obtained by normalizing the photocatalytic rates by BET specific surface area for as-synthesized (B-AS) and the most active reduced brookite (B700), as-received (CB-AR) and the most active reduced commercial brookite (CB600), and as-synthesized (A-AS) and the most active reduced anatase (A500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The specific photocatalytic H2 evolution rates expressed in µmol m-2 h-1 shown in Figure S29 underly that nanorods of reduced brookite (B700) have a specific photocatalytic activity that is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4-, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6-, and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7-fold the ones of as synthesized brookite (B-AS), reduced commercial brookite (CB600), and reduced anatase (A500), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These results demonstrate that higly active sites for photocatalysis are formed in B700 upon reduction under hydrogen atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interestingly, when we performed H2 evolution experiments with B700 in H2O/methanol under 1 sun light irradiation and applying a longpass optical filter to cutoff λ ≥ 380 nm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' cutting optical excitation above bandgap energy, we did not detect any H2 after 24 h of reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This finding suggests that the intragap electronic states due to the introduction of defects in TiO2 upon reduction (see materials characterization and DFT calculations below) do not directly participate to the photocatalytic activity in the H2 evolution reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Further, experiments performed under full 1 sun illumination and using Na2S (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 M) instead of methanol as a hole scavanger, did not produce any H2 suggesting that the H2 evolution activity of our reduced brookite is regulated by its selective methanol photo-oxidation ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This is supported by the photocatalytic experiments carried out using different alcohols (Figure 1 main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 m  h 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 (μmol I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  s  CB AR  CB600  S  A  1  B7  BS35  Electron paramagnetic resonance spectroscopy In order to identify the nature of spin containing sites and their contribution to photocatalysis we used electron paramagnetic resonance (EPR) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Three samples have been tested, which differ in the annealing temperature in hydrogen atmosphere, B-AS, B500, and B700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We investigated these samples in powder form and in situ under photocatalytic conditions by using a water/methanol (MeOH) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' X-band EPR spectra (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 GHz, T = 77 K) of (A) B-AS, (B) B500, and (C) B-700 recorded in powder form and measured in dark (blue lines) and under UV irradiation  (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The EPR spectrum of B-AS in powder form shown in Figure SA exhibits a broad resonance, centered at g \uf07e 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This signal arises from delocalized defects and presence of significant strains in the crystalline lattice, due to the synthetic procedure pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Following irradiation, a new resonant line appears in the spectrum, around g = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This signal originates from holes/oxygen-based radicals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' From the double integration of the EPR signal recorded in light and dark conditions, we calculated an increase of about 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6% in the total number of spin contacting sites after illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The observed increase in spin concentration indicates that fast electron/hole recombination processes are here hindered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' thus, the photogenerated states are rather stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The EPR trace of B500 in powder form, on the contrary, is very strong and it is dominated by a sharp resonant line at g=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='997 (Figure S30B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This resonance arises from localized lattice embedded Ti3+ sites in an octahedral field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Upon the UV irradiation, we witness an increase of the signal in the region of the  A  B  B AS  B500  UV On  UV On  UV Off  UV Off  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  g value  g value  B700  "/dB (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  UV On  UV Off  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  g valueS36  spectrum associated to oxygen-based radicals at g=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In this case, the double integrated EPR signal shows after UV light exposure and increase in the total number of spins of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, the photoexcited states are here less stable than in B-RT and fast e-/h+ recombination processes occur in this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' X-band EPR spectra (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 GHz, T = 77 K) of three samples studied (A) B-AS, (B) B500, and (C) B700 dispersed in the frozen solution of DI water and methanol (1:1, volume ratio) recorded in dark (blue lines) and under UV irradiation (red lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The sample B700 in powder form in the dark (Figure SC) gives a weaker EPR signal compared to B500, although being qualitatively similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The major difference between B700 and B500 appears, in the former, to be linked with a larger distribution in crystal field strain (octahedral to rhombic) associated to the Ti3+ spins due to structural defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Upon UV irradiation, an increase of 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6% in spin concentration was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This behavior is interpreted as due to the high stability of photoexcited states and slow recombination processes in the material, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' in B700 in powder form we observed the highest increase of paramagnetic species accumulating under irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' It is worthy to point out that the most efficient sample in hydrogen production and methanol consumption is B700, which gives indeed the weakest intensity in the EPR powder spectrum among the series here  A  B  B AS  B500  UV On  UV On  UV Off  UV Off  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  g value  g value  C  B700  UV On  I/dB (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  UV Off  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  g valueS37  shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, the number of spins recorded by EPR do not directly correlate with the system reactivity and its overall efficiency in the catalytic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To unveil in more detail the reasons underlying the different efficiency in the hydrogen production recorded in this series of tested materials, we performed a set of experiments under in situ photocatalytic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In our experiments, 10 mg of materials were dispersed in 100 µL of solution of deionized (DI) water and MeOH (50:50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Under dark conditions the EPR fingerprints in frozen solutions of B-AS, B500, and B700 (Figure S32A-E) appear very similar to those observed in their correspondent powder forms in contact with N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, upon irradiation, significant differences in the ability of MeOH molecules to quench the photogenerated holes emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In particular, while in B-AS and B700 the interaction of MeOH molecules with photogenerated h+ seems more effective, in B500 is not, as validated from the appearance of a strong peak at g=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='017 associated to accumulation of holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Therefore, the decreased catalytic efficiency of B500 compared to B-AS and B700 arises from combination of two factors, i) fast electron/hole quenching during photoexcitation and ii) when holes are formed they do not tend to react with MeOH molecules, which translates into a lower probability of successful delivery of electrons for hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  From time- resolved PL measurements, however, the average lifetime of charge carriers is similar for both B500 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 ns) and B700 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 ns), suggesting how the reaction between holes and methanol is a determining factor addressing the photocatalytic activity trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) an illustrative EPR envelope of B700 in contact with N2 at 80K (grey spectrum) together with individual components (C1-C6) derived from simulation and discussed below, (B-E) comparisons of EPR spectra of B700 recorded in different conditions together with theoretically proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The simulation and deconvolution of EPR spectra for samples B700 in light and dark conditions in contact with N2 (in powder form) and DI water: MeOH (1:1, volume ratio) at 80K is shown in Figure S32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' By deconvolution of EPR spectra we simulated the possible model of paramagnetic species present in the sample B700, which can be applied on all related spectra just by varying the proportions of individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' We assigned five main species commented bellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Component C1: g=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='059 related to peroxyl radicals (Ti-O-O•) on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Component C2: S=1/2 system associated with a trapped hole (O-centre) in the lattice with gx=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='038, gy=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='017 and gz=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  Experiment  D  Experiment  C5  Simulation  Simulation  C5  C4  C3  B700/N2  B700/DI/MeOH  [a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u]  Uv off  Uv off  /dB  c  Experiment  E  Experiment  Simulation  Simulation  C2  C1  B700/N2  B700/DI/MeOH  Uv On  UV On  280  300  320  340  360  B [mT]S38  Component C3: surface exposed Ti3+ centres which are highly disordered, and which give wide resonant line at g=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Component C4: sharp isotropic signal at g=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='992, we assume that this signal is associated with interstitial Ti3+ sides in pseudo octahedral positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Component C5: typical axial signal arising from Ti3+ in regular lattice positions with gx,y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='978 and gz=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Component C6:  Ti3+ in the lattice located in some sublayer under the surface, where are lattice parameters slightly shortened and therefore g value is smaller comparing to C5, to be precise gx,y=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='963 and gz=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='944.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S39  Photoluminescence (PL) and time-resolved photoluminescence (TRPL) spectroscopy Photoluminescence spectroscopy is a powerful technique for studying the radiative recombination of electrons and holes that populate intragap energy positions and that are due to the presence of structural defects in semiconducting materials 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoluminescence spectroscopy is often used in the literature to reconcile charge carrier recombination phenomena occurring in TiO2 and its photocatalytic activity 40–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' At first approximation, an higher PL intensity and lower PL lifetime underlie a higher electron and hole recombination rate and consequently a lower photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, the way radiative recombination processes may bring insight to photocatalytic activity also relates to: (i) the position of structural defects into the TiO2 lattice (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' surface, subsurface, bulk) that, in turn, influences the energy distribution of electronic states due to defects (and therefore also the energy distribution of PL spectra), and (ii) how PL intensity and energy distribution change upon exposure to molecules employed in the photocatalytic tests as a hole scavenger (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' methanol).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' To examine the luminescence behavior of our TiO2 samples, we measured PL map spectra using several excitation wavelengths from 258 to 590 nm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 eV down to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 eV) and monitoring PL emission in the energy range from 364 to 730 nm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 eV down to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The samples were measured initially in the solid state at the temperature 80 K in contact with N2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The PL maps for as synthesized brookite and B700 (the most active photocatalyst) are reported in the main text (Figure 2), while PL maps for B500, B600 and B800 are illustrated in Figure S36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' From these PL maps, it is evident that PL intensity and energy distribution changed in each sample depending on the reduction temperature, suggesting that recombination centers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' defects) re-organized and moved within the brookite lattice upon reduction at increasing temperatures 23,43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A similar phenomenon was observed for each set of samples investigated, namely commercial brookite (Figure S37) and anatase (Figure S38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Within the synthesized brookite series, the most intense PL signal was recorded for B700 ≈ B-AS > B500 > B600 > B800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' According to the XRD results, the sample B800 is already transformed to rutile phase and for this reason show a very weak PL in the investigated emission energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' It is interesting to note that B-AS presents PL signal only when excited with above-bandgap photons, while B700 highlights a complex PL signal even for below-bandgap excitations, demonstrating that reduction modified the electronic structure of reduced brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In order to investigate more in depth this aspect, we deconvoluted the PL spectra of synthesized and reduced brookite (Figure S39) obtained using an excitation wavelength 340 nm (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 eV, above-bandgap).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The spectra were fitted by 4 components peaking at 450, 490, 545 and 620 nm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='53, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 eV, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In each case, the peak position and the full width at half maximum (FWHM) of the deconvoluted peaks were kept the same and just the intensity of peaks was free to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Generally, it has been shown that anatase and brookite exhibit distinct PL bands in the visible region of the electromagnetic spectrum 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The well-accepted interpretative model 39 for these emissions include the presence of the main visible emission is composed of a green component (“type 1 PL” or “green PL”), at higher energies, due to recombinations between shallowly trapped electrons (or conduction band electrons) and deeply trapped holes, and a red component (“type 2 PL” or “red PL”), at lower energies, due to recombinations between valence band holes and deeply trapped electrons 45–49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spectra deconvolution reported in Figure S39 show that the weight of different PL components in our brookite samples significantly changes upon treatment in hydrogen at increasing temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Importantly, the most intense PL components for B-AS are those at 2 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27 eV, while for B700 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' the most active photocatalyst), the higher energies PL components become dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The experimentally determined valence band photoemission spectra recorded at synchrotron using photon energy in resonance with the titanium x-ray absorption edge (Figure 3A in the main text) show that in case of B-AS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' the native oxygen vacancies present in the brookite induced a distribution of intragap states acting as deep electronic traps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' while for B700 a more defective structure induced a valence band tailing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' offering therefore electronic states that may host deeply/shallowly trapped holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These findings demonstrates that the PL emissions observed for our samples are in agreement with the proposed mechanism about “green and red PL”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The different PL behavior in B-AS and B700 therefore remarks that the reduction treatment introduce different defects that have a different PL signatures and influence in different way the photocatalytic activity of brookite samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' These results reflect those of Vequizo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 50 who also found that the presence of an appropriate depth of  S40  the traps can effectively contribute to enhance the overall photocatalytic activity of TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' They found that in case of as synthesized brookite, the moderate depth of electron traps in comparison with the anatase and the rutile phase with shallow and deep electron traps, respectively, help brookite to be active for the both oxidation and reduction reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In our case, instead, we provide evidences that recombinations centers in reduced brookite are those defects that regulate the photo-oxidation reaction during H2 evolution from methanol/H2O photoreforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Further, PL maps recorded in the presence of methanol (Figure 2 in the main text) show for both B-AS and B700 that the radiative recombinations are almost completely quenched in these conditions, confirming the proposed PL mechanism and the fact that during photocatalysis holes are mainly employed for the photo-oxidation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S40 shows the TRPL of as-synthesized (received) and reduced samples, while Table S10 shows the fitted parameters employed to calculating the average electron life time of each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In all cases, the amplitude of the slow component (B1) is higher than the amplitude of the fast component (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The average electron lifetime (τave) of reduced samples, in all cases, is less than that of the corresponding pristine samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This behavior of reduced TiO2 samples suggests that after reduction, the defective centers are responsible for a faster charge carrier recombination 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Importantly, the results from TPRL measurements demonstrates that the improved photocatalytic activity of our reduced TiO2 (B700, A500, CB600) is not due to an improved photo-induced charge separation, as shown previously in other TiO2 systems 52,53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In contrast, in this case it is regulated by other parameters such as the photo-reactivity of the heterogeneous catalytic sites formed around oxygen vacancies and comprised of several Ti atoms sharing extra charge (see DFT calculations below) and lattice distortions (see XRD analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' PL maps for (A) B500, (B) B600, and (C) B800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The measurement temperature was 80 K under N2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x104  B500 N2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5x104  B600 N2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2x104  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x104  intensity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x104  intensity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x103  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x103  PL  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  P  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x103  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x103  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x103  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x103  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  Emissionenergy(eV)  Emission energy (eV)  C  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x104  B800 N2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2×104  PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x103  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x103  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8x103  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  Emission energy (eV)S41  Figure S37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' PL maps for (A) as-received commercial brookite and (B) reduced commercial brookite at 600°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The measurement temperature was 80 K under N2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3x105  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3x105  (eV)  (eV)  CB AR  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2x105  CB600  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2x105  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x105  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x105  Tenergy  intensitv(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  lenergy  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7x104  intensity  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4x104  Excitation  Excitation  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9x104  PL  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9x104  P  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3x104  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  Emissionenergy(eV)  Emissionenergy(eV)S42  Figure S38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' PL maps for (A) as-synthesized anatase, and reduced anatase at (B) 400°C, (B) 500°C, (B) 600°C, and (B) 700°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The measurement temperature was 80 K under N2 atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  B  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  7x104  7x104  (eV)  6x104  (eV)  A AS  A400  6x104  Excitation energy (  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  5x104  PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  4x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  3x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  3x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  2x104  2x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1x104  7x103  7x103  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  Emission energy (eV)  Emission energy (eV)  C  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  D  7x104  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  7x104  (eV)  (eV)  A500  6x104  A600  6x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  Excitation energy (  5x104  intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  Excitation energy  PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  5x10 5x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  4x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  3x104  3x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  3x104  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  3x104  2x104  2x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1x104  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1x104  7x103  7x103  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  Emissionenergy(eV)  Emissionenergy(eV)  E  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8  A700  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  Normalized PL intensity  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='5  Emission energy (eV)S43  Figure S39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Deconvoluted PL spectra for as-synthesized and reduced brookite samples, using an excitation energy 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) PL peak area related to each component retrieved from deconvolution and centered at 2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='53, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Both peak energy and width of each component were kept constant in each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  EmissionEnergy(eV)  B  B AS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='10  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='76  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='48  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='07  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='91  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='77  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='65  B500  B AS  B600  13  B700  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  PL intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  13  B500  area  13  B600  peak  B700  Experimentaldata  Fitteddata  Component4  Component3  Component2  Component1  400450500550600  650700  750  Emissionwavelength(nm)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='75 eV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='53 eV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='27 eV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0eVS44  Figure S40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Time–resolved photoluminescence spectra of (A) brookite reduced at 500°C and 600°C, (B) as-received and reduced commercial brookite at 600°C, and (C) as-synthesized and reduced anatase at 400, 500, and 600°C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The dots are experimental data and the solid lines are the fitted curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The τave is the average electron lifetime extracted from the fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A B 100 B500 Tave = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 ns 100 CB-AR Tave = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 ns B600 CB600 tave = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 ns Tave = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 ns PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=') PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=') 10-1 10-1 10-2 10-2 0 5 10 15 20 0 5 10 15 20 Time (ns) Time (ns) c 100 A-AS Tave = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 ns A400 Tave = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 ns PL Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=') A500 tave = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 ns A600 tave = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 ns 10-1 10-2 0 5 10 15 20 Time (ns)S45  Table S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Time–resolved photoluminescence fitted parameters for as-synthesized (as-received) and reduced samples at different temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sample B1, % τ1, ns B2, % τ2, ns τave, ns B-AS 85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 B500 87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 B600 91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 B700 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 A-AS 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 A400 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 A500 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 A600 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3 CB-AR 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 - - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 CB600 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='6  S46  Electronic characterization  Photoemission spectroscopy  We examined the electronic state of elements at the surface of the as-synthesized (received) and the best reduced samples using an XPS laboratory source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Figure S compares the XPS survey spectra of samples before and after reduction treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' An important finding from these survey spectra is that the samples are not contaminated during the reduction process, as already confirmed by CHN analysis (Table S2), since there is no difference between total XPS survey before and after process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This observation supports our hypothesis that all of the changes in photoactivity of the samples are due to the reduction process and not due to materials contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The C1s peak at 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='8 eV was used as a reference for the energy scale to compensate the charging effect of the samples (all spectra were shifted according to this reference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' However, the observed difference between XPS spectra of Ti2p and O1s for pristine and reduced samples were not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A possible explanation for this might be that the amount of changes in the lattice of TiO2 induced through reduction are below the detection limit of XPS 38 or the reduced species like Ti3+ can be easily oxidized by exposure to the ambient air 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Since, no difference was detected in XPS analysis between the samples before and after reduction, we used synchrotron-based XPS (VUV-Photoemission beamline, Elettra) to study them in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S39 shows the synchrotron-based XPS Ti2p spectra of the pristine and the most photoactive sample of brookite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' There is no evidence of an increase in Ti3+ species after reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Additionally, synchrotron-based XPS study of O1s orbital of the pristine and the most photoactive sample of brookite (Figure S40) reveals about 18% increase in OH groups on the surface of TiO2 after reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' This supports the water dissociation on superficial defects created upon reduction treatment, see more details in the mechanism part below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S47  Figure S41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) XPS survey spectra of the pristine and the most photoactive sample of brookite (left), commercial brookite (middle) and anatase(right) (B) High resolution XPS spectra of the pristine and the most photoactive sample of brookite (left), commercial brookite (middle) and anatase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (right) of (B) C1s, (C) Ti2p and (D) O1s orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A  B AS  01s  CB AR  01s  A AS  01s  B700  CB600  A500  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  dz !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='↓  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  dz !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  2  AURL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  dz !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S Ti LMM 2  Ti2s  STi LMM 2  Intensity (  C 1s  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='3  1000  800  600  400  200  0  1000  800  600  400  200  0  1000  800  600  400  200  0  Binding energy (eV)  Binding energy (eV)  Binding energy (eV)  B  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B AS  C 1s  Normalized intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  CB AR  C 1s  Normalized intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  A AS  C 1s  B700  CB600  A500  Normalized intensity  290  288  286  284  282  281  292  290  288  286  284  282  28  290  288  286  284  282  280  Binding energy (eV)  Binding energy (eV)  Binding energy (eV)  c  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B AS  Normalized intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  CB AR  Ti 2p  Ti 2p  A AS  Ti 2p  B700  CB600  A500  Normalized intensity  Normalized intensity (  468  464  460  456  45  468  464  460  456  45  468  464  460  456  452  Binding energy (eV)  Binding energy (eV)  D  Binding energy (eV)  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  B AS  Normalized intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  CB AR  (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  A AS  0 1s  0 1s  0 1s  B700  CB600  A500  Normalized intensity  Normalized intensity (  536  534  532  530  528  526  536  534  532  530  528  52  536  534  532  530  528  526  Binding energy (eV)  Binding energy (eV)  Binding energy (eV)S48  Figure S42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Synchrotron-based XPS spectra in the Ti 2p region of pristine brookite (B-AS), the most photoactive brookite sample (B700).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Figure S43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Synchrotron-based XPS spectra of the pristine (top) and the most photoactive sample of brookite (bottom), O1s orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 468 466 464 462 460 458 456  Binding energy (eV)  B AS  Fit  Ti4+  Ti3+  2p3/2  2p3/2  2p1/2  B700  Fit  Ti4+  Ti3+  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  2p1/2  B AS  Fitted  O 1s (76%)  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  OH (24%)  B700  Fitted  O 1s (58%)  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  OH (42%)  536  535  534533532531530529  528  527  Binding energy (eV)S49  Figure S44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Synchrotron-based photoemission spectra around the valence band (VB) region for the pristine brookite B-AS (light blue, full circles), the reduced brookite before reaction B700-BR (dark blue, full circles), and the reduced brookite after 24 h of photocatalytic reaction B700-AR (dark blue, empty circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='04 B AS  B700 BR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='03 O—B700 AR  Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='00 3 2 1  0  E E (eV)S50  Density functional theory calculations  Figure S45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spin-resolved density of states (DOS) plots of brookite TiO2 supercell exposing the (210) surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The simulated TiO2 structure is shown on the right side of the panel with Ti atoms plotted in grey, O in red, and O vacancies indicated in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The middle part of the slab corresponds to the bulk region of TiO2 enclosed by green planes, and the supercell’s boundaries are indicated by the dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) DOS for a perfect slab, (B–I) is for the same slab including one O vacancy denoted by V1–V8 in the structure on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All plots are zeroth to the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' DOS of V2 and V4 were calculated for a fixed geometry to prevent the migration of the vacancy to the on-surface V1 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' The DOS plots clearly show that varying the lattice position (surface, subsurface, bulk) of an oxygen vacancy results in the formation of intragap electronic states with different energy and features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  200  spin ↑  0 200  Spin  ev)  200  DOS (states /  V6  V7  0  V8  200  200  0 200 4 2  0  2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='4 2  0  2  4 4 2  0  2  4  Energy (eV)S51  Figure S46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) DOS of the surface Ti bilayer and (B) the corresponding O atoms with the missing O atom at V3 position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (C) DOS of subsurface Ti bilayer and (D) the corresponding O atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (E–J) Orbital resolved partial DOS for the surface Ti bilayer (E, G, I) and O atoms (F, H, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' All plots are zeroth to EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' It is interesting to note how intragap states are composed by hybridized electronic contributions belonging both to Ti and O atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  60  spd  spin  sp  0 60  spin  20  L  6  0  (states / eV)  6  5  Pz  DOS  5  0  VZ 5  5  p  0 4 2  0  2  4 2  0  2  4  Energy (eV)  Energy (eV)S52  Mechanism of methanol photo oxidation  Figure S47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Methanol oxidation mechanism at the TiO2surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (A) Methanol activation by terminal OH–.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In step (i) → (ii) the OH reacts with methanol producing H2O and the methoxy group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In (ii) → (iii) the methoxy group accepts a hole from TiO2 substrate (or donates an electron to TiO2) and converts to methoxy radical, while the water molecule is released from the TiO2 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In (iii) → (iv) the methoxy radical decomposes to adsorbed formaldehyde and proton ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (B) Methanol activation by coadsorbed O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In step (i) → (ii) The superoxide activates the methanol and to methoxy radical and produces the peroxide radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Then in step (ii) → (iii) such radicals are converted to adsorbed formaldehyde and hydrogen peroxide, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' In step (iii) → (iv) the hydrogen peroxide decomposes to water (then released from the TiO2 surface) and a bridging oxygen dimer 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  A ()  (ii)  (ili)  (iv)  CH2  CH3  CH3  H  H  CH  H  H  0  0  0  0  O  Thermally activated  Tisc  Tisc  Tisc  Tisc  Ti3+  Tise  Tisc  hx  B (i)  (ii)  (ili)  (iv)  CH2  CH2  CH,  H  CH  H 0  H 0  H  H  02  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  0  0  0  0  Thermally activated  Tisc  Tisc  Tisc  Tisc  Tisc  Tisc  Tisc  Tisc  h+S53  References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kandiel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Feldhoff, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Robben, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Dillert, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Bahnemann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Tailored titanium dioxide nanomaterials: Anatase nanoparticles and brookite nanorods as highly active photocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 22, 2050–2060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Andino, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Bicrystalline TiO2 with controllable anatase- brookite phase content for enhanced CO2 photoreduction to fuels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1, 8209–8216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Beltram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Romero-Ocaña, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Josè Delgado Jaen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Montini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Fornasiero, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalytic valorization of ethanol and glycerol over TiO2 polymorphs for sustainable hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A: Gen 518, 167–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Montini, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Malara, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Mróz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Beltram, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Virgili, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Boldrini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Romero-Ocaña, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Delgado, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hot electron collection on brookite nanorods lateral facets for plasmon-enhanced water oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 1270–1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Larson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Von Dreele, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' General structure analysis system (GSAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Weyl, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Zeitschrift fuer Kristallographie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='iucr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='org/resources/data/datasets/bond-valenceparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Williamson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Hall, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' X-ray line broadening from filed aluminium and wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Acta metall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 1, 22–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Roisnel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Rodríguez-Carvajal, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' WinPLOTR: A windows tool for powder diffraction pattern analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Forum 378–381, 118–123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Rouquerol, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Rouquerol, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Llewellyn, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Maurin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sing, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Adsorption by powders and porous solids (Elsevier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rex, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Knorr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Comment on "Characterization of oxygen vacancy associates within hydrogenated TiO2: A positron annihilation study".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem 117, 7949–7951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Stoll, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schweiger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' EasySpin, a comprehensive software package for spectral simulation and analysis in EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 178, 42–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Furthmuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 54, 11169–11186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Furthmüller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 6, 15–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Blöchl, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Projector augmented-wave method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 50, 17953–17979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Joubert, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' From ultrasoft pseudopotentials to the projector augmented-wave method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 59, 1758–1775.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Generalized gradient approximation made simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 77, 3865–3868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S54  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Generalized gradient approximation made simple (Errata).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 78, 1396–1396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Dudarev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Botton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Savrasov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Humphreys, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Sutton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Electron- energy-loss spectra and the structural stability of nickel oxide: An LSDA+U study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B 57, 1505–1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Morgan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Watson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Polaronic trapping of electrons and holes by native defects in anatase TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B 80, 2331021–2331024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Holmström, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ghan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Asakawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fujita, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fukuma, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kamimura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ohno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Hydration structure of brookite TiO2 (210).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem 121, 20790–20801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Allieta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Santangelo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Marelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Fabbri, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cappelli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bianchi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Psaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Dal Santo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Effect of nature and location of defects on bandgap narrowing in black TiO2 nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 134, 7600–7603.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Altomare, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photocatalysis with reduced TiO2 : from black TiO2 to cocatalyst- free hydrogen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9, 345–364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Bersani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lottici, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Ding, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phonon confinement effects in the Raman scattering by TiO2 nanocrystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 72, 73–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Parker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Siegel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Calibration of the Raman spectrum to the oxygen stoichiometry of nanophase TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Fauchet, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The effects of microcrystal size and shape of the one phonon raman spectra of crystalline semiconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Solid State Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 58, 739–741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Venkatasubbu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ramakrishnan, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sasirekha, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ramasamy, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Influence of particle size on the phonon confinement of TiO2 nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nanosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9, 661–668.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Fisica, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Infm, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chimica, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Parma, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Scienze, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Raman scattering characterization of gel-derived titania glass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 28, 177–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Tauc, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Grigorovici, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Vancu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Optical properties and electronic structure of amorphous germanium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 627, 627–637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Escobedo-Morales, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ruiz-López, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ruiz-Peralta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='deL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Tepech-Carrillo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Sánchez-Cantú, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Moreno-Orea, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Automated method for the determination of the band gap energy of pure and mixed powder samples using diffuse reflectance spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Heliyon 5, 1–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mohajernia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Andryskova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hejazi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kment, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zboril, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schmuki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Influence of Ti3+ defect-type on heterogeneous photocatalytic H2 evolution activity of TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A 8, 1432–1442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Makuła, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Pacia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Macyk, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' How to correctly determine the band gap energy of modified semiconductor photocatalysts based on UV-Vis spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 9, 6814–6817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Urbach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  The long-wavelength edge of photographic sensitivity and of the electronic absorption of solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 92, 1324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S55  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rai, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Analysis of the Urbach tails in absorption spectra of undoped ZnO thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 113, 153508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' John, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Soukoulis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cohen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Economou, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Theory of electron band tails and the Urbach optical-absorption edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 57, 1777–1780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Akshay, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Arun, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Mandal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Vasundhara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Visible range optical absorption, Urbach energy estimation and paramagnetic response in Cr-doped TiO2 nanocrystals derived by a sol– gel method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 21, 12991–13004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Tang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Levy, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Berger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Urbach tail of anatase TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B 52, 7771–7774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mohajernia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Andryskova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hejazi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kment, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zboril, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmidt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schmuki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Influence of Ti3+ defect-type on heterogeneous photocatalytic H2 evolution activity of TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Journal of Materials Chemistry A 8, 1432–1442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mascaretti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Russo, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zoppellaro, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lucotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Casari, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kment, Š.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Naldoni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Li Bassi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Excitation wavelength- and medium-dependent photoluminescence of reduced nanostructured TiO2 films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 123, 11292–11303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Liqiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Yichun, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Baiqi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Shudan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Baojiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Libin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wei, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Honggang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Jiazhong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Review of photoluminescence performance of nano-sized semiconductor materials and its relationships with photocatalytic activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Energy Mater Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Cells 90, 1773–1787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Gerosa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bottani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Caramella, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Onida, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Di Valentin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Pacchioni, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Defect calculations in semiconductors through a dielectric-dependent hybrid DFT functional: The case of oxygen vacancies in metal oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Shi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Feng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Lian, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoluminescence characteristics of TiO2 and their relationship to the photoassisted reaction of water/methanol mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 111, 693–699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mercado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Knorr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Usmani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ichimura, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Saraf, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Location of hole and electron traps on nanocrystalline anatase TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 116, 10796– 10804.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Vequizo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kamimura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ohno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Yamakata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Oxygen induced enhancement of NIR emission in brookite TiO2 powders: Comparison with rutile and anatase TiO2 powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 20, 3241–3248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Pallotti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Passoni, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Maddalena, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Di Fonzo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Lettieri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoluminescence mechanisms in anatase and rutile TiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 121, 9011–9021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mercado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Seeley, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bandyopadhyay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Bose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoluminescence of dense nanocrystalline titanium dioxide thin films: Effect of doping and thickness and relation to gas sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Interfaces 3, 2281–2288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Knorr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Spectroelectrochemical photoluminescence of trap states of nanocrystalline TiO2 in aqueous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 117, 13654–13662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  S56  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Knorr, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Mercado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Trap-state distributions and carrier transport in pure and mixed-phase TiO2: Influence of contacting solvent and interphasial electron transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' C 112, 12786–12794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mercado, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', McHale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Defect photoluminescence of TiO2 nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' MRS Online Proceedings Library 1268, 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Vequizo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Matsunaga, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ishiku, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Kamimura, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Ohno, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Yamakata, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Trapping-induced enhancement of photocatalytic activity on brookite TiO2 powders: Comparison with anatase and rutile TiO2 powders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 7, 2644–2651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Han, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Konkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Koppe, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Andreu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Chen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Vetter, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Morante, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schaaf, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Slightly hydrogenated TiO2 with enhanced photocatalytic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' A 2, 12708–12716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Wu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Cao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Piao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Effects of bulk and surface defects on the photocatalytic performance of size-controlled TiO2 nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Nanotechnology 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Alsalka, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hakki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schneider, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Bahnemann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Co-catalyst-free photocatalytic hydrogen evolution on TiO2: Synthesis of optimized photocatalyst through statistical material science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' B 238, 422–433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Mohajernia, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hejazi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Mazare, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Nguyen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Schmuki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Photoelectrochemical H2 generation from suboxide TiO2 nanotubes: visible-light absorption versus conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 23, 12406–12411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='  Setviń, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Aschauer, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Scheiber, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Hou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Schmid, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', Selloni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=', and Diebold, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Reaction of O2 with subsurface oxygen vacancies on TiO2 anatase (101).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
+page_content=' Science 341, 988– 991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/r9E3T4oBgHgl3EQfMwkD/content/2301.04375v1.pdf'}
diff --git a/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/2301.02810v1.pdf.txt b/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/2301.02810v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cf8ffec4d82d832695d6c2e92bc184f6af4d7b55
--- /dev/null
+++ b/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/2301.02810v1.pdf.txt
@@ -0,0 +1,6669 @@
+ 
+This work is licensed under the Creative Commons Attribution-NonCommercial-
+NoDerivatives 4.0 International License. To view a copy of this license, visit 
+http://creativecommons.org/licenses/by-nc-nd/4.0/ or send a letter to Creative 
+Commons, PO Box 1866, Mountain View, CA 94042, USA. 
+ 
+Coping with geometric discontinuities in porous shallow water models 
+Giada Varra1, Renata Della Morte2, Luigi Cimorelli3, Luca Cozzolino4 
+ 
+Abstract 
+Porosity-based models are a viable alternative to classical two-dimensional (2-d) Shallow water 
+Equations (SWE) when the interaction of shallow flows with obstacles is modelled. The exact solution 
+of the Single Porosity (SP) Riemann problem, which is the building block of numerous porosity 
+models solved with the Finite Volume method, exhibits an interesting feature, namely the multiplicity 
+of solutions when a supercritical flow impinges on a sudden porosity reduction. In the present paper, 
+this ambiguity is overcome by systematically comparing the solution of the one-dimensional (1-d) SP 
+Riemann problem with the corresponding 2-d SWE numerical solutions at local porosity 
+discontinuities. An additional result of this comparison is that the SP Riemann problem should 
+incorporate an adequate amount of head loss through porosity discontinuities when strongly 
+supercritical flows are considered. An approximate Riemann solver, able to pick the physically 
+congruent solution among the alternatives and equipped with the required head loss amount, shows 
+promising results when implemented in a 1-d Single Porosity Finite Volume scheme. 
+ 
+1 Res., Ph. D., Dept. of Engrg., Parthenope Univ., Centro Direzionale di Napoli – Is. C4, 80143 Napoli, Italy. E-mail: 
+giada.varra@uniparthenope.it 
+2 Full Prof., Ph. D., Dept. of Engrg., Parthenope Univ., Centro Direzionale di Napoli – Is. C4, 80143 Napoli, Italy. E-
+mail: renata.dellamorte@uniparthenope.it 
+3 Ass Prof., DICEA, Federico II Univ., Via Claudio 21, 80125 Napoli, Italy. E-mail: luigi.cimorelli@unina.it 
+4 Ass. Prof., Ph. D., Dept. of Engrg., Parthenope Univ., Centro Direzionale di Napoli – Is. C4, 80143 Napoli, Italy. E-
+mail: luca.cozzolino@uniparthenope.it 
+ 
+ 
+ 
+
+CC
+S
+BY
+NC
+ND 
+Subject headings: Shallow water Equations; Porous Shallow water Equations; Single Porosity 
+Shallow water model; urban flooding; Differential equations; Riemann problem. 
+Corresponding author: Giada Varra 
+E-mail address: giada.varra@uniparthenope.it 
+Address: Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Isola C4, 80143 Napoli (Italy) 
+
+1. Introduction 
+Porosity-based shallow water models have been developed over the last two decades to provide a 
+viable alternative to classical two-dimensional (2-d) Shallow water Equations (SWE) for partially dry 
+areas and transitional environments (Defina 2000), large‐scale urban flood modelling (Guinot and 
+Soares-Frazão 2006, Sanders et al. 2008) and runoff simulation on vegetated hillslopes (Ion et al. 
+2022). 
+The available porosity shallow water models differ from each other by the conceptual 
+formulation and the underlying physical assumptions. However, Varra et al. (2020) have 
+demonstrated that the Single Porosity (SP) model (Guinot and Soares-Frazão 2006), the Binary Single 
+Porosity model (BSP, Varra et al. 2020), and the integral formulation of the SWE with obstacles by 
+Sanders et al. (2008), constitute a family of models which share the same mathematical structure and 
+features such as hyperbolicity, presence of non-conservative products, and small disturbance 
+celerities coinciding with those exhibited by the 2-d SWE model. Not surprisingly, the observation 
+that a common Finite Volume numerical framework can be used for their approximate solution 
+confirms their common mathematical structure. The Integral Porosity model (IP, Sanders et al. 2008), 
+the Dual Integral Porosity model (DIP, Guinot et al. 2017), and the numerical schemes by Cozzolino 
+et al. (2018b) and Cea and Vázquez-Cendón (2010), are all examples of numerical schemes falling in 
+this framework. 
+Due to the rapid variations of urban fabric density and flow characteristics through the urban 
+environment, the numerical fluxes over 2-d Finite Volume cell edges are usually calculated by solving 
+a local plane SP Riemann problem (Guinot and Soares-Frazão 2006, Sanders et al. 2008, Finaud-
+Guyot et al. 2010, Cea and Vázquez-Cendón 2010, Cozzolino et al. 2018b, Jung 2022). The presence 
+of porosity discontinuities requires the definition of appropriate generalized Rankine-Hugoniot 
+conditions (LeFloch 1989, Dal Maso et al. 1995), i.e., relationships between the flow variables at the 
+two sides of the porosity discontinuity that have a strong influence on the Riemann solution. 
+
+Numerical methods should incorporate the Rankine-Hugoniot conditions to reproduce the 
+corresponding Riemann exact solutions at porosity discontinuities.  
+Regarding these solutions, previous studies (Cozzolino et al. 2018a, Varra et al. 2020, 2021) 
+have shown that they present a fundamental ambiguity consisting in the appearance of multiple exact 
+solutions for certain initial conditions characterized by a supercritical flow impacting on a porosity 
+reduction. At this point, two problems arise, namely the need i) to solve this ambiguity by finding the 
+unique physically congruent solution among the alternatives and ii) to construct a numerical scheme 
+able to reproduce the corresponding relevant solution. The one-dimensional (1-d) SP model formally 
+coincides with the 1-d SWE in rectangular channels with variable width, where the porosity symbol 
+substitutes the width symbol (Guinot and Soares-Frazão 2006, Sanders et al. 2008). This physical 
+analogy is exploited in the present paper to disambiguate the multiple 1-d SP Riemann exact solutions 
+by means of a systematic comparison with the corresponding 2-d SWE numerical solutions at local 
+geometric discontinuities, because the 1-d variable-width SWE model is nothing but a crude 
+simplification of the 2-d SWE model in a rectangular channel. 
+Besides the effects of friction (Guinot et al. 2018), shallow flows in urban environments may 
+dissipate energy by means of different mechanisms, such as the drag induced by obstacles (Sanders 
+et al. 2008), the propagation of bores reflected by buildings (Guinot et al. 2017, 2018), and local 
+effects at geometric discontinuities (Guinot and Soares-Frazão 2006, Varra et al. 2020). In porosity 
+models, the adoption of computational cells of greater size than that usually adopted in shallow water 
+models causes the loss of geometrical and hydraulic information, which in turn causes the 
+underestimation of the energy dissipated by the flow propagating through the urban fabric (Guinot et 
+al. 2017, 2018, Varra et al. 2020). To reproduce missing dissipative effects, structural changes have 
+often been introduced in the original porosity shallow water models, for example altering the physical 
+momentum fluxes via reduction coefficients (Guinot et al. 2017, 2018). 
+Laboratory (Akers and Bokhove 2008, Defina and Viero 2010) and 2-d SWE numerical 
+experiments (Varra et al. 2020) show that supercritical flows suffer intense head loss across channel 
+
+contractions, implying that a corresponding energy dissipation must be experienced through rapid 
+porosity reductions. In the present work, this energy dissipation is considered by appropriately 
+reformulating the generalized Rankine-Hugoniot conditions in a head-balance form. The introduction 
+of an interface head loss through the definition itself of porosity discontinuity has the advantage of 
+leaving the structure of the mathematical model unchanged and it can be very naturally used to take 
+into account, at least partly, the drag forces through urban fabrics. However, the amount of head loss 
+to be introduced across the discontinuity needs to be evaluated in a proper way, depending on the 
+flow characteristics across the geometric transition. Also in this case, a systematic study of 2-d SWE 
+numerical results at isolated geometric discontinuities is conducted to supply the general conditions 
+under which this energy dissipation is present and how it can be evaluated. 
+With the aim of reproducing the effects that in 2-d shallow water models are caused by the 
+flow interaction with isolated geometric discontinuities, the present work proposes a novel 
+approximate Riemann solver that discriminates the existence of multiple solutions and considers 
+adequate head loss in case of supercritical flows at porosity discontinuities. This solver is 
+implemented in a 1-d Finite Volume scheme adopting the Single Porosity formulation of SWE 
+(Guinot and Soares-Frazão 2006). The capability of the 1-d numerical model with porosity of 
+reproducing the effects that in 2-d models are caused by the interaction between the flow and a 
+geometric transition is assessed against several Riemann problems by comparing the corresponding 
+results with the ones provided by a reference 2-d SWE numerical model. 
+The present paper is organized as follows. The structure of the SP Riemann problem solution, 
+where the definition by Cozzolino et al. (2018b) is used for generalized Rankine-Hugoniot conditions, 
+is discussed in Section 2. This solution is validated in Section 3 using 2-d SWE numerical 
+experiments, and a novel definition of porosity discontinuity is given in Section 4 to better reproduce 
+the 2-d SWE numerical results. In Section 5, it is shown how it is possible to construct a numerical 
+model able to discriminate multiple solutions and introduce the requested head loss amount. These 
+findings are discussed in Section 6. Finally, the paper is closed by a Conclusions section. 
+
+ 
+2. Mathematical model 
+In the present Section, the plane Riemann problem for the SP model is reviewed, showing that it 
+reduces to a 1-d SP Riemann problem. The corresponding solution requires the definition of 
+generalized Rankine-Hugoniot conditions to be used through porosity discontinuities. Exploiting the 
+analogy between the 1-d SP model and the 1-d SWE in rectangular channels with variable width 
+(Guinot and Soares-Frazão 2006, Sanders et al. 2008), we introduce and discuss a head-balance form 
+defining this relationship. 
+ 
+2.1 The 1-d SP model 
+The plane SP model considered here is an augmented 1-d system obtained from the 2-d SP model 
+(Guinot and Soares-Frazão 2006) by setting to zero the derivatives with respect to the y-axis and 
+neglecting the flow resistance components (Ferrari et al. 2017, Varra et al. 2020): 
+ 
+(1) 
+2
+2
+2
+0
+0
+2
+2
+0
+h
+hu
+t
+x
+hu
+gh
+gh
+hu
+t
+x
+x
+hv
+huv
+t
+x
+
+
+
+
+
+
+
+
+
+
+
++
+=
+ 
+
+
+
+
+
+
+
+
++
++
+−
+=
+
+
+
+
+
+
+
+
+
+
+
++
+=
+ 
+
+
+. 
+ 
+ 
+ 
+ 
+ 
+     
+The solution of the corresponding Riemann problem is the building block for the computation 
+of interface numerical fluxes in shock capturing Finite Volume schemes (Godlewski and Raviart 
+1996). In Eq. (1), the symbols have the following meaning: x and y are the space independent variables 
+of the inertial reference frame Oxy, while t is the time variable; h(x, y, t) is the flow depth; u(x, y, t) 
+and v(x, y, t) are the vertically averaged components of the flow velocity along x and y, respectively; 
+g is the gravity acceleration; and the porosity (x, y)  [0, 1] represents the fraction of urban area not 
+occupied by buildings and obstacles (storage porosity). In the following, the dependence of the 
+
+variables on y will be omitted because all the quantities should be considered constant along y (plane 
+problem). The effects of variable bed elevation are neglected here because the focus of the present 
+work is on obstacle modelling.  
+Despite a distinction is often made in the urban hydrology literature between storage porosity 
+ and conveyance porosity  (Lhomme 2006, Guinot and Delenne 2014, Guinot et al. 2017), the last 
+being related to the mass and momentum transport (Dewals et al. 2021), the two definitions are 
+genuinely different only in porosity models written in integral form while they coincide in differential 
+models (Varra et al. 2020). This result, which derives from a classical proof developed in the theory 
+of fluid motion in porous media (Whitaker 1969), states that the geometric parameter  in Eq. (1) 
+should always be interpreted not only as a storage but also as a conveyance porosity. 
+In Finite Volume schemes for the approximate solution of the 2-d SP model, the system of 
+Eq. (1), where x is a local reference normal to the cell interface, is solved using initial discontinuous 
+conditions (Guinot and Soares-Frazão 2006, Soares-Frazão et al. 2008, Sanders et al. 2008, Cea and 
+Vázquez-Cendón 2010, Finaud-Guyot et al. 2010, Özgen et al. 2016b, Özgen et al. 2017, Guinot et 
+al. 2017). The presence of the non-conservative product 
+2
+0.5
+
+
+
+gh
+x , which models the force per 
+unit-width exerted by the obstacles on the flow through the cell interface, requires careful 
+mathematical and numerical treatment because it cannot be recast in divergence form (Cozzolino et 
+al. 2018b). This point is central to the present discussion, and it will be further clarified in the 
+following. 
+The first two relations of Eq. (1) do not contain the conserved variable hv and can be decoupled 
+from the third (Varra et al. 2021), leading to the 1-d SP model (Sanders et al. 2008, Cozzolino et al. 
+2018b) 
+ 
+(2) 
+( )
+( )
+0
+
+
+
+
+
+
++
++
+=
+
+
+
+t
+x
+x
+f u
+u
+h u
+. 
+ 
+
+In Eq. (2), the meaning of the symbols is as follows: 
+(
+)
+=
+T
+h
+hu
+u
+ and 
+( ) (
+)
+2
+2
+0.5
+=
++
+T
+hu
+gh
+hu
+f u
+ are the vectors of the conserved variables and fluxes, respectively, of 
+the 1-d SWE model; T is the matrix transpose symbols; 
+( ) (
+)
+2
+0
+0.5
+=
+−
+T
+gh
+h u
+ is a vector 
+representing the hydrostatic thrust per unit-width exerted by obstacles on the flow. 
+The 1-d system of Eq. (2) is at the core of the solution to the plane Riemann problem for Eq. 
+(1). In fact, once that h and hu are known from Eq. (2), hv in Eq. (1) is readily computed with the 
+passive tracer equation (Varra et al. 2021): 
+ 
+(3) 
+0
+
+
++
+=
+
+
+v
+v
+u
+t
+x
+. 
+ 
+2.1.1 Porosity Riemann problem 
+In the Riemann problem of the 1-d SP model, Eq. (2) is solved under the following discontinuous 
+flow initial conditions and porosity 
+ 
+(4) (
+)
+,
+0
+,0
+,
+0
+
+
+= 
+
+
+L
+R
+x
+x
+x
+u
+u
+u
+, 
+( )
+,
+0
+,
+0
+
+
+
+
+
+= 
+
+
+L
+R
+x
+x
+x
+, 
+ 
+where 
+(
+)
+=
+T
+L
+L
+L
+L
+h
+h u
+u
+ and 
+(
+)
+=
+T
+R
+R
+R
+R
+h
+h u
+u
+ are the states initially to the left and right of the 
+geometric discontinuity in x = 0, respectively, while L  and R  are the corresponding porosities. The 
+solution of Eq. (2) with the initial conditions of Eq. (4) is self-similar and consists of a sequence of 
+constant states, the leftmost and rightmost of which are 
+L
+u  and 
+R
+u . All these states are in turn 
+connected by standing or moving waves (Varra et al. 2021). Being the solution self-similar, it exists 
+a vector function 
+( )
+
+w
+ of the scalar parameter  such that the Riemann problem solution can be 
+
+expressed as 
+(
+)
+(
+)
+,
+=
+x t
+x t
+u
+w
+. This implies that the states 
+(
+)
+1
+1
+1
+1
+=
+T
+h
+h u
+u
+ and 
+(
+)
+2
+2
+2
+2
+=
+T
+h
+h u
+u
+ 
+immediately to the left and right of the geometric discontinuity, respectively,  are constant in time 
+because they can be expressed as 
+(
+)
+( )
+1
+0 ,
+0
+−
+−
+=
+=
+t
+u
+u
+w
+ and 
+(
+)
+( )
+2
+0 ,
+0
++
++
+=
+=
+t
+u
+u
+w
+. 
+Based on the initial conditions of Eq. (4), the porosity is uniform to the left and right of the 
+geometric discontinuity in x = 0, implying that the non-conservative product 
+2
+0.5
+
+
+
+gh
+x  is null 
+and the system of Eq. (2) is conservative for x < 0 and x > 0. It follows that the moving waves (shock 
+or rarefactions) coincide with those of the classic 1-d SWE model and that the shocks are defined by 
+the classic Rankine-Hugoniot conditions (Varra et al. 2021). Vice versa, the non-conservative product 
+2
+0.5
+
+
+
+gh
+x  is active through x = 0, implying that the classic Rankine-Hugoniot conditions cannot 
+be used at porosity discontinuities. For this reason, the generalized Rankine-Hugoniot conditions 
+introduced by LeFloch (1989) and Dal Maso et al. (1995) must be used to define an appropriate 
+relationship between u1 and u2. In the SP Riemann problem, the self-similarity of the solution implies 
+that this relationship is constant in time because u1 and u2 are constant in time.  
+ 
+2.1.2 Generalized Rankine-Hugoniot conditions at porosity discontinuities 
+Following the definition introduced by Dal Maso et al. (1995) for hyperbolic systems of differential 
+equations with non-conservative products, the generalized Rankine-Hugoniot conditions across the 
+porosity discontinuity in Eq. (2) reduce to (Cozzolino et al. 2018b, Varra et al. 2020) 
+ 
+(5.a) 
+2
+2
+1 1
+0
+
+
+−
+=
+R
+L
+h u
+hu
+, 
+(5.b) 
+(
+)
+2
+2
+2
+2
+2
+2
+2
+1
+1 1
+1
+2
+,
+,
+,
+2
+2
+
+
+ 
+
+
+
+
+
++
+−
++
+=
+
+
+
+
+
+
+
+
+R
+L
+L
+R
+g
+g
+h
+h u
+h
+hu
+S
+u u
+, 
+ 
+where
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+ is the force exerted on the flow by the obstacles across the unit-width porosity 
+discontinuity in x = 0. Eq. (5.a) states that the unit-width discharge 
+
+=
+Q
+hu  is invariant through the 
+
+discontinuity, while Eq. (5.b) states that the total thrusts to the left and right of the discontinuity are 
+balanced by the force 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+. 
+The relationship between the states u1 and u2 is completely defined if a functional expression 
+for 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+ is given. However, this expression is somehow problematic because very 
+natural assumptions such as stagnant water and hydrostatic pressure distribution for the computation 
+of 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+ (Guinot and Soares-Frazão 2006, Sanders et al. 2008, Mohamed 2014, Guinot 
+et al. 2017) may lead to unphysical conditions where the flow acquires energy through the porosity 
+discontinuity (see the discussion in Chow 1959 and Cozzolino et al. 2018b). 
+To simplify the expression of the relationship between u1 and u2, we conveniently reformulate 
+the generalized Rankine-Hugoniot conditions. Appendix A shows that the force balance of Eqs. (5.a) 
+and (5.b) can be rewritten in the following head-balance form 
+ 
+(6) 
+(
+)
+(
+)
+(
+)
+2
+2
+1 1
+2
+1
+1
+2
+0
+,
+,
+,
+
+
+ 
+−
+=
+−
+= 
+R
+L
+L
+R
+h u
+hu
+H
+H
+H
+u
+u
+u u
+, 
+ 
+ 
+where 
+( )
+(
+)
+2
+2
+=
++
+H
+h
+u
+g
+u
+ is the head corresponding to the generic state u and 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ 
+is the head loss through the porosity discontinuity. The relationship between the head loss 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ and the force 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+ is given by (see Appendix A) 
+ 
+(7) 
+(
+)
+(
+)
+2
+2
+1
+2
+2
+1
+1
+2
+2
+1
+1
+2
+2
+1
+1 2
+1
+1
+,
+,
+,
+3
+3
+,
+,
+,
+4
+2
+
+
+
+
+ 
+ 
+
+
+ 
+
+
+
++
+
+=
++
+−
+−
++
+
+
+
+
+L
+R
+R
+L
+L
+R
+L
+R
+R
+L
+L
+R
+h
+h
+h
+h
+H
+h
+h
+S
+h
+h
+g
+h h
+u u
+u u
+, 
+ 
+implying that the choice of 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+S
+u u
+ in Eq. (5.b) is equivalent to the choice of  
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ in Eq. (6), and vice versa. 
+
+From the mathematical point of view, the head-balance form is equivalent to the force-balance 
+form, but Eq. (6) is more convenient because it allows to easily verify the physical congruence of 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+. In fact, the flow energy cannot increase through the porosity discontinuity, 
+implying that the entropic condition 
+ 
+(8) 
+(
+)
+1
+2
+,
+,
+,
+0
+ 
+
+
+L
+R
+Q H
+u u
+, 
+ 
+where 
+2
+2
+1 1
+
+
+=
+=
+R
+L
+Q
+h u
+hu , must be verified. The head-balance approach allows in principle to easily 
+introduce local effects at geometric discontinuities that do not explicitly appear in Eq. (1), such as 
+non-hydrostatic flow, viscosities, velocity variability along the vertical direction, flow depth and 
+velocity variability along the transverse directions, and the shape of obstacles. Not surprisingly, 
+existing definitions of channel internal boundary conditions such as width discontinuities and 
+junctions from the technical literature are usually given in terms of head loss (Formica 1955, Austin 
+et al. 1970, Cunge et al. 1980, Hager 2010).  
+However, the system of Eq. (2) does not provide additional information to compute the head 
+loss 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+, implying that the hydraulic modeller should use external physical 
+knowledge for its definition. This will be discussed in the following subsection, where the internal 
+structure of the porosity discontinuity between x = 0- and x = 0+ will be examined. 
+ 
+ 
+2.2 Channel analogy and porosity discontinuity definition  
+The 1-d SP model of Eq. (2) coincides with the 1-d SWE model in a rectangular channel with variable 
+width and horizontal bed, where the porosity symbol  takes the place of the width symbol B (Guinot 
+and Soares-Frazão 2006, Sanders et al. 2008, Varra et al. 2021). Like the porosity models written in 
+differential form, where the storage and conveyance porosity must coincide, the width B in 
+
+rectangular channels can be regarded both as i) the channel base-area per unit length (storage) and ii) 
+the transverse space available for the flow (conveyance). In addition, the non-conservative product 
+2
+0.5
+
+
+
+gh
+x  in Eq. (2), which represents the force per unit-width exerted by the obstacles on the 
+flow along x, acts like the corresponding term in the 1-d variable-width SWE model, where it 
+represents the force exerted on the flow by the channel walls. In the following, this channel analogy 
+will be exploited to supply a convenient expression for the head loss 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ through the 
+porosity discontinuity. 
+ 
+2.2.1 Porosity variation through the discontinuity   
+Consider the bottom of Figure 1a, where a strip of unitary width modelling a simplified urban area 
+with obstacles is depicted. The strip is subdivided into two cells, left and right, respectively, with 
+different obstacle densities represented by the porosities L  and R . Obstacles are also present 
+through the cell interface in x = 0, with a density intermediate between those of the two adjacent cells. 
+The corresponding rectangular channel analogue is represented by the top channel of Figure 1a, where 
+the widths at the left and right ends are represented by L  and R , while a monotonic width variation 
+(channel contraction or expansion) connects the left and right reaches. In this case, it is natural to 
+assume that the porosity  through the discontinuity between x = 0- and x = 0+ is described by a 
+monotonic function 
+( )
+ s  in the interval s  [0, 1], with 
+( )
+0
+
+
+
+=
+L  and 
+( )
+1
+
+
+
+=
+R . In Figure 2a, 
+the internal structure of the porosity discontinuity between x = 0- and x = 0+ is exploded to show the 
+relationship between 
+( )
+ s  and the parameter 
+
+
+0,1
+
+s
+. 
+A similar situation is depicted in Figure 1b, but now the obstacles density at the cells interface 
+is greater than those of the two adjacent cells. The corresponding rectangular channel analogue is 
+represented by the top channel of Figure 1b, where a non-monotonic width variation (channel 
+
+constriction) connects the left and right reaches and  the porosity 
+( )
+ s  through the discontinuity 
+varies non-monotonically between L  and R . 
+ 
+Remark 1. In Finite Volume schemes, a homogeneous porosity is assigned to each computational 
+cell and the actual obstacles distribution along the interface between two contiguous cells is canceled. 
+This implies that the simplest application of the channel analogy is to consider a rectangular channel 
+that is normal to the cell interface and symmetrical with respect to its longitudinal axis (as made in 
+Figures 1 and 2). 
+ 
+Thanks to the channel analogy, both the monotonic and non-monotonic choices of 
+( )
+ s are 
+physically viable and lead to numerically stable computations. Examples of models with monotonic 
+porosity variation through the discontinuity are contained in the works by Guinot and Soares-Frazão 
+(2006), Soares-Frazão et al. (2008), Cea and Vázquez-Cendón (2010), Finaud-Guyot et al. (2010), 
+Ferrari et al. (2017), and Cozzolino et al. (2018a,b), while examples with a non-monotonic variation 
+are the models by Sanders et al. (2008), Özgen et al. (2017), Bruwier et al. (2017), and Guinot et al. 
+(2017, 2018, 2022). 
+The preceding discussion suggests that the choice of the porosity discontinuity internal 
+structure can be made considering the underlying urban geometry at each cell interface. This simple 
+observation, which supplies a physically congruent framework, contradicts the unproven statement 
+from the literature (Bruwier et al. 2017, Guinot et al. 2017, 2018, 2022) that only non-monotonic 
+descriptions of the porosity variation through the discontinuity are viable and stable (see the 
+corresponding discussion in Varra et al. 2020). 
+ 
+Remark 2. For the sake of simplicity, only monotonic porosity variations through the discontinuity 
+with 
+
+
+L
+R  will be considered in the rest of the paper (monotonic variations with 
+
+
+L
+R  can be 
+
+discussed by simply mirroring the local reference framework). Having defined the aspect ratio 
+
+
+=
+L
+R
+AR
+, this implies that 
+1
+
+AR
+ will be assumed in the following developments. 
+ 
+ 
+Figure 1. Physical interpretation of the porosity discontinuity between L  and R :  monotonic (a) and 
+non-monotonic porosity variation (b).  
+ 
+2.2.2 Flow depth and velocity variation through the discontinuity   
+To complete the internal description of the porosity discontinuity, it is necessary to specify the 
+variation of flow depth and velocity between x = 0- and x = 0+.  It is assumed that this description is 
+
+a)
+R
+X
+x
+b)
+PRsupplied by the function ( )
+(
+)
+
+
+
+=
+T
+s
+h
+h u
+v
+,  where 
+( )
+h
+s  and 
+( )
+
+u
+s , with s  [0, 1], are the flow 
+depth and velocity through the discontinuity, respectively. The function ( )
+s
+v
+ is characterized by the 
+obvious congruency conditions ( )
+1
+0 =
+v
+u  and ( )
+2
+1 =
+v
+u . In Figures 2b and 2c, the internal structure 
+of the porosity discontinuity between x = 0- and x = 0+ is exploded to show two examples of the 
+relationship between the inner flow depth 
+( )
+h
+s  and the parameter 
+
+
+0,1
+
+s
+. 
+Cozzolino et al. (2017) have proposed that the function ( )
+s
+v
+ is a stationary weak solution of 
+Eq. (2) through the porosity discontinuity, namely a solution of  
+ 
+(9) 
+( )
+( )
+0
+
+
+
+
++
+=
+d
+d
+ds
+ds
+f v
+h v
+ 
+ 
+in the interval 
+
+
+0,1
+
+s
+. If the solution of Eq. (9) exhibits no hydraulic jump (see the example of 
+Figure 2b, where 
+( )
+h
+s  smoothly varies in the interval 
+
+
+0,1
+
+s
+), the relationship between the states 
+1
+u  and 
+2
+u  reduces to the conditions of discharge and total head invariance (see Appendix B) 
+ 
+(10) 
+(
+)
+(
+)
+2
+2
+1 1
+2
+1
+0
+0
+R
+L
+h u
+hu
+H
+H
+
+
+−
+=
+−
+=
+u
+u
+, 
+ 
+which is equivalent to set 
+(
+)
+1
+2
+,
+,
+,
+0
+ 
+
+=
+L
+R
+H
+u u
+ in the head-balance form of Eq. (6). In the case that 
+the porosity varies monotonically between L  and R , the existence of a state u1 connected to u2 by 
+means of Eq. (10) depends on the aspect ratio AR  1 and on 
+(
+)
+2
+F u
+, where 
+( ) =
+F
+u
+gh
+u
+ is the 
+Froude number related to the generic state u. The corresponding discussion is reported in Appendix 
+C. 
+
+ 
+If the solution of Eq. (9) exhibits a hydraulic jump that reverts the incoming supercritical flow 
+into subcritical (see the example of Figure 2c), the total head is not invariant through the porosity 
+discontinuity and the corresponding head loss depends on the position of the hydraulic jump (see 
+Appendices C and D in Varra et al. 2021). The corresponding relationship between the states 
+1
+u  and 
+2
+u , which recovers the head-balance form of Eq. (6) with 
+(
+)
+1
+2
+,
+,
+,
+0
+ 
+
+
+L
+R
+Q H
+u u
+, is not as simple 
+as that of Eq. (10) and it is not reported here for the sake of brevity. 
+ 
+ 
+Figure 2. Internal description of the porosity discontinuity: plan view of the monotonic porosity 
+variation (a); profile view of smooth flow depth variation (b); profile view of flow depth variation 
+with hydraulic jump (c).  
+ 
+2.2.3 Definition of the Generalized Rankine-Hugoniot conditions 
+
+0
+1
+(a)
+Td
+PR
+1
+0'
+0+
+x
+0
+1
+S
+(b)
+hr (s)
+h
+h
+1
+hr (s)
+(c)
+h,
+0'
+0*Having discussed the internal structure of the porosity discontinuity in the preceding sections, we 
+assume the following definition for the generalized Rankine-Hugoniot conditions: 
+ 
+Definition 1. The relationship between u1 and u2, with
+1
+
+
+=
+
+L
+R
+AR
+, is defined by the head-
+balance form of Eq. (6) with the following internal description of the porosity discontinuity: 
+D1) the porosity varies monotonically between L  and R ; 
+D2) the variation of flow depth and velocity through the porosity discontinuity is defined by a weak 
+solution of Eq. (9) in the interval 
+
+
+0,1
+
+s
+, with ( )
+1
+0 =
+v
+u  and ( )
+2
+1 =
+v
+u . 
+ 
+This definition automatically satisfies the entropic condition of Eq. (8), while this is not true 
+for other porosity discontinuity definitions (Guinot and Soares-Frazão 2006, Sanders et al. 2008, 
+Mohamed 2014, Guinot et al. 2017) available in the literature (see the discussion in Cozzolino et al. 
+2018b). In addition, Varra et al. (2021) have demonstrated that the solution to the Riemann problem 
+of Eqs. (2) and (4) always exists if Definition 1 is used to establish the generalized Rankine-Hugoniot 
+conditions. This fundamental result is not granted for alternative porosity discontinuity definitions 
+from the literature. 
+ 
+ 
+2.3 Multiple solutions to the porosity Riemann problem 
+The solution to the 1-d SP Riemann problem of Eqs. (2) and (4), complemented by the generalized 
+Rankine-Hugoniot conditions of Section 2.2.3, always exists but there are cases, depending on the 
+initial conditions 
+L
+u  and 
+R
+u , where the solution is triple (Varra et al. 2021). The field of occurrence 
+of multiple solutions will be explored in the following for the case 
+1
+
+
+=
+
+L
+R
+AR
+ only (a similar 
+discussion for the case 
+1
+
+AR
+ can be drawn by mirroring the reference framework). 
+
+The necessary (but not sufficient) condition for the existence of multiple solutions to the 1-d 
+SP Riemann problem with 
+1
+
+AR
+ is that the right state uR is directed from right to left (uR < 0) and 
+(
+)
+
+R
+sp
+F
+K
+AR , where 
+(
+)
+=
+R
+R
+F
+F u
+ is the Froude number corresponding to uR while the function 
+(
+)
+sp
+K
+AR  is defined in Appendix C. In this condition, the right supercritical flow uR impinging the 
+porosity reduction has energy greater than the minimum required to pass through the geometric 
+discontinuity. For this reason, it is possible to consider not only a solution where the right flow freely 
+passes through the discontinuity, but also solutions where the head of the incoming flow is partially 
+dissipated by means of a standing hydraulic jump through the porosity transition or by a shock that 
+moves backwards (Viero and Defina 2017, Varra et al. 2021) 
+The theoretical limit curve 
+(
+)
+=
+R
+sp
+F
+K
+AR , called lower boundary (LB) of the hysteresis 
+domains (Viero and Defina 2017), is represented in the plane (
+R
+F , AR) of Figure 3 with a black 
+continuous line. The necessary condition 
+(
+)
+
+R
+sp
+F
+K
+AR  for the existence of multiple solutions is 
+satisfied by the points falling in the regions denoted with B and C to the right of the LB curve in 
+Figure 3. The regions B and C are separated by the curve 
+(
+)
+=
+R
+jump
+F
+K
+AR , called upper boundary 
+(UB) of the hysteresis domains (Viero and Defina 2017), which is represented with a dashed line. 
+The dimensionless function 
+(
+)
+jump
+K
+AR , which is characterised by 
+(
+)
+(
+)
+
+jump
+sp
+K
+AR
+K
+AR  for every  
+1
+
+AR
+, is defined in Appendix C. 
+In the regions B and C, one or three different solutions to the 1-d SP Riemann problem of Eqs. 
+(2) and (4) are possible, depending on the initial left state uL (Varra et al. 2021). When uL is such that 
+three alternative solutions (here called T1, T2, and T3) are possible, the solutions differ from each 
+other by the flow condition through the porosity discontinuity, as follows: 
+(T1) the state u2 immediately to the right of the porosity discontinuity coincides with the 
+supercritical flow uR, while the state u1 is supercritical and it is connected to u2 by means of Eq. (10), 
+namely by the conditions of discharge and total head invariance across the discontinuity (Figure 4a); 
+
+(T2) the state u2 coincides with uR but a hydraulic jump is present through the porosity 
+discontinuity, and the state u1 is subcritical or critical, with H(u1) < H(u2) (Figure 4b); 
+(T3) the supercritical flow uR is reverted into the subcritical state u2 by means of a backward 
+moving shock, with head loss; the state u1 is subcritical or critical and it is connected to u2 by means 
+of Eq. (10) (Figure 4c). 
+When u1 is subcritical in the solutions T2 and T3, the flow through the geometric discontinuity 
+is submerged, i.e., it is dominated by the tailwater h1. The main difference between the regions B and 
+C in Figure 3 is the behaviour of the solutions to the Riemann problem when the state uL coincides 
+with the dry bed, i.e., when 
+0
+=
+Lh
+ and there is no tailwater.  In the domain B, three distinct solutions 
+(T1, T2, and T3) are always possible for 
+0
+=
+Lh
+, while a unique solution T1 occurs in the domain C 
+for 
+0
+=
+Lh
+. In other words, a triple solution is possible in the domain B even if there is not a 
+downstream tailwater able to force the establishing of a subcritical flow through the geometric 
+discontinuity (submerged flow), while such a tailwater is required for the existence of a triple solution 
+in the field C. In a sense, the incoming flow falling in region C has always energy sufficient to flush 
+the hydraulic jump of Figure 4b out of the porosity discontinuity when the flow depth downstream is 
+null. 
+The discussion is completed observing that the region A to the left of the curve LB, 
+characterised by 
+(
+)
+1
+
+R
+sp
+F
+K
+AR , refers to flow conditions where the Riemann problem always 
+admits a unique solution. In this case, the supercritical flow uR has not sufficient energy to pass 
+through the porosity reduction and the solution T3 must occur. 
+ 
+
+ 
+Figure 3. Field of occurrence of multiple solutions to the porosity Riemann problem for right 
+supercritical flows uR impinging a porosity reduction with 
+1
+
+
+=
+
+L
+R
+AR
+. Lower (continuous line) 
+and upper (dashed line) boundaries of the hysteresis domains. Hysteresis domains: A (no multiple 
+solutions), B (multiple solutions even in the case hL = 0), C (multiple solutions only for hL > 0). 
+ 
+ 
+Figure 4. Flow conditions through the porosity discontinuity when multiple solutions to the purely 
+1-d SP Riemann problem are possible: profile view of solutions T1 (a), T2 (b) and T3 (c). 
+ 
+ 
+3. Validation of the channel analogy  
+
+Lowerboundary(LB)ofthehysteresisdomain
+0.8-
+Upper boundary (UB) of the hysteresis domain
+0.6-
+R
+c
+0.4 -
+UB
+0.2-
+B
+A
+LB
+0
+1
+FR
+20T1
+T2
+1
+(a)
+1
+1
+(b)
+ui
+ui
+1
+U=UR
+n=
+1
+supercritical
+subcritical/
+supercritical
+critical
+supercritical
+0+
+0'
+0-
+0+
+T3
+u2
+(c)
+1
+uj
+subcritical
+UR
+subcritical/l
+1
+critical
+1
+supercritical
+777777
+0'
+0+The comparison between 1-d SP and 2-d SWE solutions is justified because the 1-d SP Riemann 
+problem is the main ingredient of 2-d SP Finite Volume schemes, which in turn are intended to 
+approximate the solution of 2-d SWE models with obstacles. For this reason, the generalized Rankine-
+Hugoniot conditions of Section 2.2.3 are validated in the present section by comparing several 1-d 
+SP exact Riemann solutions with the corresponding 2-d SWE numerical solutions in a frictionless 
+horizontal rectangular channel with variable width, where a 2-d contraction is used to model the 1-d 
+sudden porosity reduction. All the Riemann problems, whose initial conditions uL and uR with the 
+corresponding Froude numbers FL and FR are reported in Table 1, refer to porosity values 
+0.6
+L
+ =
+ 
+and 
+1
+R
+ = . The exact Riemann solutions are computed with the methods discussed in Varra et al. 
+(2021). 
+ 
+Table 1. Initial flow conditions of the validation Riemann problems. 
+Example 
+hL (m) 
+uL (m/s) 
+FL (-) 
+hR (m) 
+uR (m/s) 
+FR (-) 
+1 
+1.00 
+2.00 
+0.64 
+1.00 
+-0.50 
+-0.16 
+2 
+1.00 
+2.00 
+0.64 
+1.00 
+2.00 
+0.64 
+3 
+1.00 
+5.00 
+1.60 
+1.00 
+2.00 
+0.64 
+4 
+0.30 
+-10.00 
+-5.83 
+1.00 
+2.00 
+0.64 
+5 
+1.00 
+-2.00 
+-0.64 
+1.00 
+-9.40 
+-3.00 
+6 
+1.00 
+7.00 
+2.23 
+1.00 
+-13.00 
+-4.15 
+7 
+1.00 
+-11.00 
+-3.51 
+1.00 
+-13.00 
+-4.15 
+8 
+0.30 
+-4.00 
+-2.33 
+0.30 
+-11.00 
+-6.41 
+ 
+The rectangular channel considered for the 2-d SWE computations has length L = 200 m with 
+a left reach of width BL = 0.60 m and a right reach of width BR = 1.00 m (see Figure 5). The left and 
+right channel reaches are separated by a symmetric linear expansion whose length is Lc = 0.20 m. The 
+2-d SWE computations are accomplished using the Finite Volume scheme described in Cozzolino et 
+al. (2017) on unstructured triangular grid whose average side is s = 0.50 m at the channel ends and 
+s = 0.05 m at the linear expansion. The flow depth h computed at time t = 5 s with the 1-d SP exact 
+
+solution (continuous black line) is compared in Figures 6, 8, 9, and 10, with the corresponding 2-d 
+shallow water numerical results (white dots).  
+ 
+ 
+Figure 5. Plan view of the channel considered for 2-d SWE numerical simulations. Distorted 
+representation (measures in metres). 
+ 
+The results of Riemann problem 1 are represented in Figure 6a. The 1-d exact solution presents 
+two shocks moving to the left and right of the geometric discontinuity, respectively, while subcritical 
+flow conditions that preserves discharge and energy invariance are established through x = 0. Figure 
+6a shows a good correspondence between the 1-d exact and 2-d numerical solutions. In particular, 
+the 1-d model accurately captures the strength of the wave at x = 0, together with the strength and 
+position of the shocks. The intermediate states u1 and u2 immediately to the left and right of the 
+porosity discontinuity computed with the 1-d exact solution nicely correspond to those provided by 
+the 2-d SWE model. 
+A slightly different picture can be drawn for the solution of Riemann problem 2, represented 
+in Figure 6b. The 1-d exact solution exhibits a resonant condition where a rarefaction is attached to 
+the left of the porosity discontinuity, while a shock and a rarefaction are both moving to the right. 
+The state u1 immediately to the left of x = 0 is critical and accelerates through the rapid geometric 
+transition becoming supercritical with preservation of energy and discharge invariance. In turn, the 
+supercritical state u2 issuing from the channel expansion pushes the slowly moving shock to the right 
+
+Lc = 0.20
+1.00
+= 0.60
+II
+-
+R
+B
+B
+-
+x=-100
+x=0
+x= 100of x = 0. With reference to the left rarefaction, Figure 6b shows a good correspondence between the 
+1-d exact and 2-d numerical results. This representation is less satisfactory with reference to the right 
+moving shock, whose shape in the 2-d model is strongly influenced by its vicinity to the channel 
+expansion. Similarly, the 2-d rarefaction moving on the right is quite smoothed with respect to the 1-
+d exact solution. Despite these discrepancies, the intermediate state between the shock and the 
+rarefaction computed with the 1-d exact solution satisfactorily corresponds to the 2-d numerical 
+solution. 
+ 
+ 
+Figure 6. Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions 
+(dots) for the flow depth at time t = 5 s. Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with 
+initial conditions in Table 1.  
+ 
+In the 1-d exact solution of Riemann problem 3, the supercritical states u1 and u2 are connected 
+by the conditions of discharge and head invariance while the state u2 is separated from uR by means 
+
+1.6
+1.2
+(a)
+1
+(b)
+1.2
+0.8
+(u)
+0.8
+h
+h
+0.4
+0.4 -
+I
+-
+-
+I
+0
+0
+-20
+0
+20
+-10
+0
+10
+20
+30
+× (m)
+x (m)
+1.6
+1.2
+(c)
+-
+(d)
+1.2
+0.8
+(w)
+(w)
+0.8
+h
+h
+0.4 
+0.4
+&
+0
+-40
+-20
+0
+20
+40
+-80
+-40
+0
+40
+() x
+() xof an intermediate state and two moving shocks. The comparison with the 2-d SWE results shows 
+that the strength and position of the right moving shock and the left shock position are satisfactorily 
+captured by the 1-d exact solution, while the flow depth of the state between the two shocks supplied 
+by the1-d model is not too far from the 2-d solution. The 2-d free surface profile view in Figure 6c 
+exhibits a complex pattern whose plane view is represented in Figure 7, which shows that the 
+supercritical flow accelerating through the expansion originates a system of transverse oblique 
+shocks. The 1-d exact solution represents this pattern with a single average flow depth, and this 
+explains the discrepancies between 1-d and 2-d models. Nonetheless, the 1-d model captures the 
+general picture of the 2-d SWE solution. 
+ 
+ 
+Figure 7. Plan view of the 2-d SWE numerical solution for Riemann problem 3 with initial conditions 
+in Table 1. Flow depth contours at time t = 5 s.  
+ 
+In Figure 6d, the results of Riemann problem 4 are represented. The 1-d exact solution consists 
+of two rarefactions to the left of x = 0, with the formation of dry bed between the waves, and of an 
+additional rarefaction to the right of the discontinuity. The subcritical flow immediately to the right 
+of x = 0 (state u2) accelerates through the discontinuity becoming critical immediately to the left (state 
+u1) and preserving the invariance of discharge and energy. The comparison between the 1-d exact 
+and 2-d SWE numerical results shows that the former captures the strength of the 2-d waves, together 
+with the flow depth of the states encompassed by the waves. 
+
+0.5
+h (m)
+1.3
+1.1
+(m)
+0
+0.8
+0.55
+0.3
+-0.5
+0
+7
+3
+0.05
+x (m)The example of Figure 8 is particularly interesting because it corresponds to Riemann problem 
+5 with three exact solutions, where AR and 
+R
+F  fall in the hysteresis domain B of Figure 3. In Figure 
+8a, b, c, the three exact solutions T1, T2, and T3, respectively, are represented (see Section 2.3). In 
+Figure 8d, the superposition between the exact solution T3 and the 2-d SWE numerical results shows 
+a good agreement. This demonstrates that the under-determination of the 1-d Riemann problem can 
+be eliminated by resorting to the 2-d SWE model, which takes into account the transverse flow 
+variability. 
+ 
+ 
+Figure 8. Example Riemann problem 5 with initial conditions in Table 1. Profile view for the flow 
+depth solution at time t = 5 s. 1-d SP exact solutions T1 (a), T2 (b) and T3 (c). Comparison between 
+the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d). 
+ 
+The example of Figure 9 corresponds to Riemann problem 6 with three exact solutions, where 
+AR and 
+R
+F  fall into the hysteresis domain C of Figure 3. In Figure 9a, b, c, the three exact solutions 
+T1, T2, and T3, respectively, are represented (see Section 2.3). In Figure 9d, the superposition 
+
+4
+(a)
+3
+h
+2
+4
+1
+(d)
+0
+4
+3
+I
+(b)
+E
+h
+h
+2
+2
+-
+1
+1
+1
+I
+1
+0
+1
+-
+4
+1
+(c)
+1
+3
+h
+1
+2
+0
+-60
+-40
+-20
+0
+20
+1
+x (m)
+1
+0
+09-
+-40
+-20
+0
+20
+x (m)between the T3 exact solution and the 2-d SWE numerical results shows again a good agreement. 
+The comparison between figures 8d and 9d shows that the 2-d SWE model preferably picks up the 
+solution with a shock moving backwards when three exact solutions are possible. 
+ 
+ 
+Figure 9. Example Riemann problem 6 with initial conditions in Table 1. Profile view for the flow 
+depth solution at time t = 5 s. 1-d SP exact solutions T1 (a), T2 (b) and T3 (c). Comparison between 
+the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d). 
+ 
+In the example of Figure 10a, the state uL is such that Riemann problem 7 has one exact 
+solution of type T1, despite AR and 
+R
+F  fall in the hysteresis domain C of Figure 3. Correspondingly, 
+the 2-d SWE solution is characterised by a strong interaction with the geometric discontinuity and by 
+a supercritical flow to the left of x = 0. The comparison between 1-d and 2-d solutions shows that the 
+number of moving waves supplied by the 1-d exact solution is correct, but their strength and position 
+is very different from those of the 2-d numerical solution. In addition, the flow depth of the 
+supercritical states to the left of x = 0 is poorly captured by the 1-d exact solution. The 2-d SWE 
+
+6
+(a)
+E
+4
+h
+6
+(d)
+0
+6
+1
+1
+(b)
+4 
+1
+(w)
+1
+4 
+1
+h
+1
+1
+2
+h
+1
+1
+1
+2 
+0
+1
+6
+(c)
+1
+4
+h
+1
+0
+2
+-60
+-40
+-20
+0
+20
+x (m)
+0
+-60
+-40
+-20
+0
+20
+x (m)numerical solution exhibits a strong interaction with the channel walls in x = 0, with the formation of 
+a system of oblique shocks whose plan view is represented in Figure 11. These shocks, which are 
+typical of supercritical flows in contractions (Ippen and Dawson 1951, Akers and Bokhove 2008, 
+Defina and Viero 2010), introduce intense head loss that explains the discrepancies between 1-d and 
+2-d solutions.  
+ 
+Similar observations can be made for Riemann problem 8, for which AR and 
+R
+F  fall in the 
+hysteresis domain C of Figure 3. The 1-d exact solution is unique and characterized by flow 
+conditions T1 through the porosity reduction (Figure 10b), while the corresponding 2-d SWE solution 
+exhibits a supercritical flow to the left of x = 0 and a strong interaction with the contraction (Figure 
+12) where intense head loss is produced. 
+ 
+From the validation process above, some considerations can be made. The exact solutions to 
+the 1-d SP Riemann problem, computed with the monotonic porosity discontinuity model of Section 
+2.2.3, compare well with the corresponding 2-d SWE numerical solutions in case of subcritical flow 
+through the porosity discontinuity (Figures 6a,d). Overall, the head loss through the discontinuity 
+seems negligible when the flow is subcritical, but a caveat to this observation will be discussed in the 
+next Section. 
+Similarly, the 1-d exact model shows a good behaviour in both the multiplicity domains B and 
+C when three exact solutions are possible. In this case, the solution characterized by subcritical flow 
+through the porosity discontinuity satisfactorily agrees with the 2-d SWE numerical results (Figures 
+8d and 9d).  
+A minor discrepancy is present when the supercritical flow accelerates through a porosity 
+increase. In this case, the 2-d SWE numerical solution is somehow distorted with respect to the 1-d 
+exact solution (Figures 6b,c). 
+A very different picture is evident when the 1-d exact model predicts a single solution 
+characterized by supercritical flow through a porosity reduction (Figures 10a,b). In this case, the 2-d 
+SWE model exhibits a supercritical flow through the contraction, but the head loss introduced by a 
+
+2-d system of oblique shocks makes the 1-d and 2-d solutions very different. This dissipative 
+mechanism has been discussed by Varra et al. (2020) for the first time in the context of Riemann 
+problems on dry bed, but it has been verified here for the general Riemann problem on wet bed. 
+The validation process accomplished in this section suggests that the results of a 2-d SWE 
+numerical model could be systematically used to disambiguate multiple Riemann problem solutions 
+and evaluate the head loss caused by a supercritical flow passing through a porosity reduction, 
+improving the Definition 1 of the generalized Rankine-Hugoniot conditions. This will be made in the 
+next section. 
+ 
+ 
+Figure 10. Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions 
+(dots) for the flow depth at time t = 5 s. Example Riemann problems 7 (a) and 8 (b) with initial 
+conditions in Table 1. 
+
+6
+(a)
+2
+(b)
+7
+1.5 -
+4
+-
+(m)
+一
+:f
+1
+d
+h
+(w)
+2
+h
+0.5
+一
+ul
+一
+1
+0
+0
+-80
+-40
+0
+-80
+-60
+-40
+-20
+0
+20
+x (m)
+x (m) 
+ 
+Figure 11. Plan view of the 2-d SWE numerical solution for Riemann problem 7 with initial 
+conditions in Table 1. Flow depth contours at time t = 5 s. 
+ 
+ 
+Figure 12. Plan view of the 2-d SWE numerical solution for Riemann problem 8 with initial 
+conditions in Table 1. Flow depth contours at time t = 5 s. 
+ 
+ 
+4. Construction of novel generalized Rankine-Hugoniot conditions 
+Consider a horizontal frictionless rectangular channel L = 60 m long with a single width discontinuity 
+at its centre. The channel consists of a right and a left reach of different widths, connected by a linear 
+contraction whose walls are inclined by 45° with respect to the channel axis (see Figure 13). The 
+
+0.5
+h (m)
+4.4
+3.8
+(m)
+3.2
+2.6
+2
+1.4
+-0.5
+0.8
+-2
+-1.5
+-0.5
+0
+0.5
+x (m)0.5
+h (m)
+2.4
+2
+1.6
+1.2
+0.8
+0.4
+-0.5
+0
+-2
+-1.5
+-1
+-0.5
+0
+0.5
+x (m)contraction is short enough to be regarded as a true geometric discontinuity. In order to perform 2-d 
+SWE simulations with different aspect ratio AR = BL/ BR values, the right reach width is fixed to BR 
+= 1 m, while the left reach width BL (with BL < BR) is varied in each test. Free-slip boundary conditions 
+are imposed at the channel walls, while the left and right ends are open. A non-uniform unstructured 
+triangular mesh is used for simulations, with average side s = 0.20 m at the channel ends and s = 
+0.02 m in the vicinity of the geometric transition. 
+ 
+ 
+Figure 13. Plane view of the channel used for 2-d SWE numerical tests with supercritical flows. 
+Distorted representation (measures in metres). 
+ 
+The 2-d Finite Volume SWE numerical model by Cozzolino et al. (2017) is used for 
+approximating the solution of 159 different 2-d Riemann problems in the channel of Figure 13, where 
+AR  [0.1, 0.9]. The tests are characterized by supercritical flow uR (with uR < 0) approaching the 
+contraction and Froude number 
+
+
+1.5,25
+R
+F 
+. In all the tests, the right flow depth is hR = 1 m and 
+the corresponding velocity uR varies accordingly to FR, while the initial left state uL coincides with 
+the dry bed. Simulations are run until steady state conditions are reached through the contraction 
+(generally after t = 20 s). 
+The 1-d SP exact solution is triple (T1, T2, or T3) for the initial conditions falling in region B 
+of Figure 3, while a single T1 solution is predicted for points falling in region C. The examination of 
+numerical results shows that two distinct types of 2-d SWE solutions occur: 
+
+II
+R
+B
+30
+30
+-
+x= -30
+x=0
+x= 30(G1) The supercritical flow entering the geometric discontinuity passes with the formation of 
+a complicate system of oblique shocks through the contraction like in Figures 10a,b; this set of 
+numerical solutions exhibits a behavior that is somehow in between the exact solutions T1 and T2 
+defined in Section 2.3 (Figures 4a,b), because supercritical flow conditions are present at the outlet 
+of the contraction as in T1, but there is also head loss as in T2. 
+(G2) The flow through the contraction is subcritical, while a moving shock propagates 
+upstream; since the channel bed downstream is initially dry, critical flow conditions are established 
+through the contraction outlet; this set of numerical solutions clearly recalls the exact solutions of 
+type T3 in Section 2.3 (Figure 4c), where u1 is critical. 
+ 
+4.1 Modified upper boundary of the hysteresis domain 
+In Figure 14, the 2-d numerical cases corresponding to solution types G1 (black triangles) and G2 
+(white squares) are plotted in the plane (
+R
+F ,  AR), where the upper hysteresis domain limit is also 
+represented. Figure 14 shows that, for a given value of the aspect ratio AR, it exists a limit Froude 
+number 
+(
+)
+*
+K
+AR  such that a G2 solution is obtained for 
+(
+)
+*
+
+R
+F
+K
+AR , while a G1 solution is 
+obtained for 
+(
+)
+*
+
+R
+F
+K
+AR . The locus of the points separating the fields of G1 and G2 solutions is 
+the modified upper boundary curve with equation 
+(
+)
+*
+=
+R
+F
+K
+AR . Ideally, this curve represents the 
+situations for which a standing hydraulic jump is present at the entrance of the contraction. For 
+(
+)
+*
+
+R
+F
+K
+AR , the incoming flow has energy sufficient to push the jump through the contraction, 
+where it is broken into a complicate pattern of transverse standing waves (Figures 11 and 12). 
+Conversely, the incoming flow is not able to sustain the hydraulic jump for 
+(
+)
+*
+
+R
+F
+K
+AR , and a 
+shock moves backwards. 
+The modified upper boundary curve, represented in Figure 14 with a thick black line, is very 
+close to the UB curve defined in Section 2.3 (dashed line curve in Figure 14) for moderate width 
+
+jumps (AR > 0.5), whereas it departs from the UB curve for strong width jumps (AR  0.5). This is 
+not surprising, because the 1-d theory for width and porosity transitions is expected to work better 
+when AR is close to one. The polynomial interpolation of data supplies for the limit 
+(
+)
+*
+=
+R
+F
+K
+AR  
+the expression 
+ 
+(11) 
+(
+)
+(
+)
+6
+*
+1
+=
+=
+
+i
+jump
+i
+i
+K
+AR
+K
+AR
+m AR , 
+ 
+ 
+whose coefficients mi are reported in Table 2. 
+We observe that 
+(
+)
+(
+)
+*
+
+jump
+K
+AR
+K
+AR  for AR > 0.5, meaning that the incoming flow requires 
+greater energy to push the hydraulic jump through the porosity discontinuity with respect to the case 
+without energy loss. This effect, which is due to the modest head loss related to the subcritical flow 
+through the geometric transition in G2 solutions, will be taken into account numerically without a 
+direct evaluation of the energy losses in subcritical conditions. 
+ 
+
+ 
+ 
+Figure 14. 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 1 m impinging 
+a contraction: G1 configuration (black triangles), G2 configuration (white squares); upper hysteresis 
+domain limit (dashed line); modified upper boundary (thick black line). 
+ 
+Table 2. Coefficients for the polynomial interpolation of Eq. (11). 
+m1 
+m2 
+m3 
+m4 
+m5 
+m6 
+0.9448 
+9.8030 
+-24.2944 
+20.1172 
+-3.7583 
+-1.8122 
+ 
+ 
+4.2 Head loss for supercritical flows at contractions 
+The 2-d SWE solutions of type G1 exhibit a head loss H* through the channel contraction. This head 
+loss is evaluated as 
+(
+)
+*
+*
+1
+
+=
+−
+R
+H
+H
+H
+u
+, where 
+*
+1
+H  is an estimate of the head corresponding to the 
+state immediately to the left of the contraction. The quantity 
+*
+1
+H  is indirectly deduced by evaluating 
+the supercritical flow depth to the left of the contraction. 
+
+0.8
+Upper boundary (UB)
+- Modified upper boundary (MUB)
+A^^ 2-d SWE numerical Gl solution
+0.6 -
+ 2-d SWE numerical G2 solution
+R
+0.4 -
+口
+0.2
+口
+口
+口
+口
+口口口
+口
+一
+UB
+口
+口
+口
+口
+MUB
+0
+25The relative head loss 
+(
+)
+*
+*
+ = 
+R
+H
+H u
+ corresponding to the 2-d SWE solutions of type G1 
+is represented with black triangles in the plane (
+*
+2
+,
+
+R
+F ) of Figure 15, where the data corresponding 
+to the same  AR value are connected by a dashed line. Figure 15 shows that the relative head loss 
+*
+  
+moderately varies with 
+2
+R
+F  for a given value of AR. This implies that 
+*
+  mainly depends on the 
+characteristics of the geometric transition, allowing to use a simplified expression in the form 
+(
+)
+*
+*
+ = 
+AR , where a single  
+*
+  value has been attributed to each  AR value by picking the numerical 
+data closer to the modified upper boundary of Figure 14. These points are connected by a continuous 
+grey line in Figure 15. 
+ 
+ 
+ 
+Figure 15. Relative head losses for supercritical flows through a contraction: 2-d SWE numerical 
+results for G1 configuration (black triangles); limit relative head loss (continuous black line); 
+envelope of G1 data closer to the modified upper boundary of Figure 14 (continuous grey line). 
+ 
+
+^  2-d SWE numerical Gl solution
+0.8
+AA
+AR= 0.1
+0.6
+△AR=0.2
+△AR = 0.3
+*
+AR = 0.4
+△AR= 0.5
+0.4 -
+AR = 0.6
+0.2 -
+AR = 0.75
+AR = 0.8
+AR = 0.9
+0
+10
+F,2
+100
+1000
+1
+RIn the same figure, the limit relative head loss 
+(
+)
+#
+#
+ = 
+R
+H
+H u
+ is also represented with a 
+thick black line. The quantity 
+#
+H  is the head loss through the shock in the exact solution of type T3 
+(see Figure 4c) when the celerity of the shock is null (standing hydraulic jump) and the state u1 is 
+critical, i.e., when 
+(
+)
+=
+R
+jump
+F
+K
+AR . In Appendix D, it is shown that the limit relative head loss 
+#
+  
+depends on AR only and the exact expression of 
+(
+)
+#
+#
+ = 
+AR  is given. The ratio 
+#
+*
+
+  is 
+represented in Figure 16 for different values of AR2.  This leads to the following polynomial 
+interpolation 
+ 
+(12) 
+(
+)
+(
+)
+2
+*
+2
+0
+#
+=
+
+= 
+
+i
+i
+i
+AR
+AR
+m AR , 
+ 
+ 
+with interpolation coefficients in Table 3. 
+ 
+Recalling that 
+2 =
+R
+u
+u  in the G1 solutions, from the position 
+(
+)
+*
+1
+2
+,
+,
+,
+ 
+
+= 
+L
+R
+H
+H
+u u
+ it 
+follows that the head loss in Eq. (6) for supercritical flow through a channel contraction, equivalent 
+to a porosity reduction, can be rewritten as 
+ 
+(13) 
+(
+)
+(
+)
+(
+)
+*
+1
+2
+2
+,
+,
+,
+ 
+
+
+
+=
+
+L
+R
+L
+R
+H
+H
+u u
+u
+. 
+ 
+Table 3. Coefficients for the polynomial interpolation of Eq. (12). 
+m0 
+m1 
+m2 
+1.536 
+0.403 
+0.668 
+ 
+
+ 
+Figure 16. Polynomial interpolation of the relative head loss data. Triangles represent the 
+experimental cases enveloped by a thin grey line in Figure 15. 
+ 
+4.3 Novel generalized Rankine-Hugoniot conditions 
+From the preceding discussion, it is possible to give a novel convenient relationship between u1 and 
+u2 at porosity discontinuities. 
+ 
+Definition 2. The relationship between u1 and u2, with
+1
+
+
+=
+
+L
+R
+AR
+, is defined by the head-
+balance form of Eq. (6) with the following internal description of the porosity discontinuity: 
+D1) the porosity varies monotonically between L  and R ; 
+D2) the variation of flow depth and velocity through the porosity discontinuity is defined by a weak 
+solution of Eq. (9) in the interval 
+
+
+0,1
+
+s
+, with ( )
+1
+0 =
+v
+u  and ( )
+2
+1 =
+v
+u ; 
+D3) the state u2 with u2 < 0 is supercritical only if 
+(
+)
+(
+)
+*
+2
+
+F
+K
+AR
+u
+; in this case, the relationship 
+between u1 and u2 is defined by Eq. (6) where 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ is defined by Eq. (13). 
+ 
+
+2.4 -
+2 -
+1.6
+△#
+*
+1.2
+0.8
+0.4 -
+0
+0.2
+0.4
+0.6
+0.8
+1
+AR2To demonstrate the viability of the Definition 2, the exact solution to Riemann problems 7 and 8 of 
+Table 1 is now found using the novel generalized Rankine-Hugoniot conditions. The 1-d exact 
+solutions are represented with a black line in Figure 17, where the corresponding 2-d solution is 
+represented with dots. The comparison with Figure 10, where the exact solutions are obtained with 
+null energy loss, shows that introducing an appropriate definition of 
+(
+)
+1
+2
+,
+,
+,
+ 
+
+L
+R
+H
+u u
+ reduces the 
+discrepancy between the exact 1-d SP exact solution and the corresponding 2-d SWE numerical 
+solution. 
+ 
+ 
+ 
+Figure 17. Profile view of the 1-d SP exact solution with head loss through the geometric 
+discontinuity (continuous black line) and 2-d SWE numerical solution (dots) for the flow depth at 
+time t = 5 s. Example Riemann problems 7 (a) and 8 (b) with initial conditions in Table 1. 
+ 
+ 
+5. Numerical model 
+In the present Section, the solution of the 1-d SP system of Eq. (2), where the initial conditions 
+(
+)
+( )
+x
+x
+0
+0,
+u
+u
+=
+ and the porosity distribution (x) are specified, is approximated by means of the Finite 
+
+6
+(a)
+2
+(b)
+一
+1.5 -
+4 
+一:
+一
+(m)
+一
+l
+1
+d
+h
+ (m)
+()
+2
+h
+0.5
+c:l
+0
+0
+-80
+-40
+0
+-80
+-60
+-40
+-20
+0
+20
+x (m)
+x (m)Volume method. Having partitioned the 1-d physical domain into non-overlapping cells 
+
+
+2
+1
+2
+1 ,
++
+−
+=
+i
+i
+i
+x
+x
+C
+ of uniform length 
+2
+1
+2
+1
+−
++
+−
+=
+
+i
+i
+x
+x
+x
+, we assume that the averaged quantities 
+ 
+(14) 
+( )
+(
+)
+( ) (
+)
+1
+1
+,
+,
+
+
+
+=
+=
+=
+
+
+
+
+i
+i
+T
+n
+n
+n
+n
+n
+i
+i
+i
+i
+i
+C
+C
+i
+x dx
+h
+h u
+I x
+x t
+dx
+x
+x
+u
+u
+ 
+ 
+are approximations in Ci of (x) and (
+)
+, n
+x t
+u
+, respectively, where 
+t
+n
+t n
+
+=
+ is the time level. In 
+Figure 18a, the cell-averaged constant values of (x) and (
+)
+, n
+x t
+u
+ are conceptually depicted, showing 
+the geometric and flow discontinuity at cells interface. 
+If 
+1
+n
+n
+i
+i
+t
+t
+t
++
+ =
+−
+ is the time step length, the solution is advanced in the generic cell by means 
+of the following explicit first-order scheme 
+ 
+(15) 
+(
+)
+(
+)
+1
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+,
+,
+
+
+
+
++
+−
++
+−
++
++
+−
++
++
++
+−
+−
+−
+−
++
+
+
+
+
+
+
+=
+−
+−
++
++
+
+
+
+
+
+
+n
+n
+i
+i
+i
+i
+i
+i
+i
+i
+i
+i
+i
+i
+t
+t
+x
+x
+u
+u
+g u
+u
+g u
+u
+s
+s
+ 
+ 
+where the symbols are defined as follows: 
+1 2
+i
+ +
+ is a numerical approximation of the porosity at the 
+interface i+1/2 between Ci and Ci+1; (
+)
+,
+g u v  is a numerical flux corresponding to the 1-d SWE model 
+in a constant width channel; finally, 
+(
+)
+1 2
+1 2
+1 2
+1 2
+T
+i
+i
+i
+i
+h
+h
+u
+−
+−
+−
+−
++
++
++
++
+=
+u
+and 
+(
+)
+1 2
+1 2
+1 2
+1 2
+T
+i
+i
+i
+i
+h
+h
+u
++
++
++
++
++
++
++
++
+=
+u
+ are 
+flow variables reconstructed to the left and right of the interface i+1/2, respectively, which are 
+involved in the computation of numerical fluxes and non-conservative product approximations. The 
+quantities 
+(
+)
+1 2
+1 2
+0
++
++
+−
+−
+=
+T
+i
+is
+s
+ and  
+(
+)
+1 2
+1 2
+0
+−
+−
++
++
+=
+T
+i
+is
+s
+ in Eq. (15) are the contributions to Ci of the 
+non-conservative products arising from the porosity gradient through the interfaces in xi-1/2 and xi+1/2, 
+respectively, and their computation depends on the variable reconstruction adopted. The 1-d SWE 
+
+numerical flux (
+)
+,
+g u v  is approximated here by means of the HLLE Riemann solver (Cozzolino et 
+al. 2014), although a different exact or approximate SWE Riemann solver could be used as well. 
+In its essence, the numerical scheme above consists of the following procedure. First, the 
+porosity and flow interface variables are reconstructed from the cell-averaged values. Second, the 
+numerical fluxes and porosity gradients contributions at interfaces are computed. Finally, the cell-
+averaged variables are advanced in time by means of Eq. (15). Clearly, the algorithm used to calculate 
+the reconstructed interface variables from the cell-averaged values determines the properties of the 
+scheme. 
+In the following, we assume without loss of generality that 
+1
+
+ +
+
+i
+i , corresponding to 
+1
+1
+  +
+=
+
+i
+i
+AR
+ (the procedure for the case 
+1
+
+ +
+
+i
+i  is easily obtained by mirroring the local 
+reference framework). The basic variable reconstruction by Castro et al. (2007) is first recalled, and 
+then it is modified to introduce head losses through porosity discontinuities and cope with the case of 
+multiple solutions. Finally, the results of the 1-d SP numerical scheme, with both the basic and novel 
+interface variable reconstructions, are compared with the corresponding 2-d SWE numerical results. 
+
+ 
+Figure 18. Side view of two neighbouring cells in the 1-d computational domain: cell-averaged 
+quantities at a generic time level n (a); interface reconstructed variables used in the basic 
+reconstruction by Castro et al. (2007) (b); interface and in-cell reconstructed variables in the novel 
+reconstruction approach (c). 
+ 
+5.1 Basic reconstruction (Castro et al. 2007) 
+
+(a)
+-
+1
+u'
+1
+βi+1
+Pi
+Xi-1/2
+xi
+Xi+1/2
+Xi+1
+Xi+3/2
+x
+1
+u,+1/2l u+1/2
+1
+(b)
+-
+u"
+1
+1
+1Wi+1/2
+Pi+1
+Pi
+X;-1/2
+1x
+Xi+1/2
+Xi+1
+Xi+3/2
+x
+1
+-
+u,+1/2l u+1/2
+1
+(c)
+-
+R +
+-
+1
+1
+ui
+1
+1
+4i+1/2
+Pi+1
+Pi
+X;-1/2
+Xi
+Xi+1/2
+Xi+1
+Xi+3/2
+x
+Ci+1
+C,The well-balanced reconstruction by Castro et al. (2007), originally implemented for the 1-d variable-
+width SWE model, is intended to capture steady state solutions where the discharge and head are 
+uniform through the space domain. Aiming at this, the interface variables 
+1 2
+−
++
+iu
+, 
+1 2
++
++
+iu
+, and 
+1 2
+ +
+i
+, are 
+connected to the cell-averaged variables by means of Eq. (10), keeping the character of the flow 
+(subcritical or supercritical). The reconstruction approach is schematically depicted in Figure 18b, 
+where the porosity discontinuity internal structure is zoomed in to show the variables used for 
+computations. 
+Given the right state 
+1
++
+n
+iu
+, the following inequalities are checked (see Appendix C): 
+ 
+(16) 
+(
+)
+(
+)
+1
++
+
+n
+i
+sb
+F
+K
+AR
+u
+, 
+(
+)
+(
+)
+1
++
+
+n
+i
+sp
+F
+K
+AR
+u
+. 
+ 
+ 
+ 
+With reference to these checks, two options are possible, as follows. 
+ 
+CR.1) If one of the two inequalities in Eq. (16) is satisfied, the right state 
+1
++
+n
+iu
+ can be 
+connected to a state on the interface left side by the conditions of discharge and head 
+invariance (see Section 2.2.2). In this case, the interface porosity 
+1 2
+
+
++
+=
+i
+i  is assumed, 
+and the state 
+(
+)
+1 2
+1 2
+1 2
+1 2
+T
+i
+i
+i
+i
+h
+h
+u
++
++
++
++
++
++
++
++
+=
+u
+ is easily found by solving the system 
+ 
+(17) 
+(
+)
+(
+)
+1
+1
+1
+1 2
+1 2
+1 2
+1
+1 2
+0
+0
+
+
++
++
++
++
++
++
++
++
++
++
++
+−
+=
+−
+=
+n
+n
+i
+i
+i
+i
+i
+i
+n
+i
+i
+h u
+h
+u
+H
+H
+u
+u
+, 
+ 
+ 
+which is obtained by assuming in Eq. (10) the positions 
+1 2
+
+ +
+=
+L
+i
+, 
+1
+
+ +
+=
+R
+i , 
+1
+1 2
++
++
+=
+i
+u
+u
+, and 
+2
+1
++
+=
+n
+i
+u
+u
+. The system of Eq. (17) admits two exact solutions, one corresponding to 
+a subcritical state and the other to a supercritical state (see Valiani and Caleffi 2008 for 
+
+the corresponding exact expressions). The first is chosen if the state 
+1
++
+n
+iu
+ is subcritical, 
+otherwise the supercritical one is kept. Finally, the position 
+1 2
+−
++
+=
+n
+i
+i
+u
+u  is made. 
+ 
+CR.2) If the inequalities of Eq. (16) are not satisfied, the system of Eq. (17) admits no 
+solution with 
+1 2
+
+
++
+=
+i
+i . In this case, 
+1 2
++
++
+iu
+ is found by means of Eq. (17) where the 
+interface porosity 
+1 2
+ +
+i
+ is defined as 
+ 
+(18) 
+(
+)
+(
+)
+3 2
+1 2
+1
+1
+2
+1
+3
+2
+
+
++
++
++
++
+
+
+
+
+=
+
+
++
+
+
+n
+i
+i
+i
+n
+i
+F
+F
+u
+u
+. 
+ 
+ 
+ 
+This choice is equivalent to imposing that 
+1 2
++
++
+iu
+ is critical (see Appendix C). The state 
+1 2
+−
++
+iu
+ is calculated by means of 
+ 
+(19) 
+(
+)
+(
+)
+1 2
+1 2
+1 2
+1 2
+0
+0
+
+
+−
+−
++
++
++
+−
++
+−
+=
+−
+=
+n
+n
+i
+i
+i
+i
+i
+i
+n
+i
+i
+h
+u
+h u
+H
+H
+u
+u
+,  
+ 
+ 
+which is obtained by assuming in Eq. (10) the positions 
+
+=
+L
+i ,  
+1 2
+
+ +
+=
+R
+i
+, 
+1 =
+n
+i
+u
+u , 
+and 
+2
+1 2
+−
++
+=
+i
+u
+u
+. The subcritical solution is kept if the state 
+n
+iu  is subcritical, otherwise the 
+supercritical solution is chosen. The state 
+1 2
+−
++
+iu
+ certainly exists because 
+1 2
+
+ +
+
+i
+i
+ (see 
+Appendix C). 
+ 
+
+From the preceding, it is evident that the reconstruction by Castro et al. (2007) satisfies the 
+inequality 
+
+
+
+
+1
+1 2
+1
+min
+,
+max
+,
+ 
+
+ 
++
++
++
+
+
+i
+i
+i
+i
+i
+, i.e., it ensures the monotonicity of the porosity 
+variation through the discontinuity. 
+The algorithm is completed by using Eqs. (5.a)-(5.b) to express 
+1 2
++
+−
+is
+ and  
+1 2
+−
++
+is
+ as 
+ 
+(20) 
+(
+)
+(
+)
+(
+)
+(
+)
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+
+
+
+
++
++
+−
+−
+−
+−
+−
++
++
++
+=
+−
+=
+−
+n
+i
+i
+i
+i
+i
+n
+i
+i
+i
+i
+i
+s
+f u
+f u
+s
+f u
+f u
+. 
+ 
+ 
+5.2 Novel variable reconstruction  
+The novel variable reconstruction differs from that by Castro et al. (2007) because the case of a 
+supercritical flow impinging on a porosity reduction is treated in a separate way, congruently with 
+the novel definition of Rankine-Hugoniot conditions given in Section 4.3. This is accomplished by 
+computing an in-cell additional reconstructed state 
+(
+)
+,
+,
+,
+,
+1 2
+1 2
+1 2
+1 2
++
++
++
++
++
++
++
++
+=
+T
+R
+R
+R
+R
+i
+i
+i
+i
+h
+h
+u
+u
+ to manage the case of 
+a backwards moving shock between the geometric transition and the state 
+1
++
+n
+iu
+ when a T3 solution 
+(Figure 4c) occurs. In addition, appropriate head loss is introduced to compute the state 
+1 2
++
++
+iu
+ if the 
+flow through the porosity reduction is supercritical. The novel reconstruction approach is 
+schematically depicted in Figure 18c, showing the in-cell additional reconstructed variable 
+,
+1 2
++
++
+R
+iu
+. 
+Given the right state 
+1
++
+n
+iu
+, the cases 
+1
+0
++ 
+n
+iu
+ and 
+1
+0
++ 
+n
+iu
+ are treated separately. 
+ 
+NR.1) If 
+1
+0
++ 
+n
+iu
+, the procedure by Castro et al. (2007) is used to find 
+1 2
+ +
+i
+, 
+1 2
+−
++
+iu
+, and 
+1 2
++
++
+iu
+ (see points CR.1 and CR.2 of Section 5.1). In this case, there is no backwards moving 
+shock and the position 
+,
+1 2
+1
++
++
++
+=
+R
+n
+i
+i
+u
+u
+ is made. 
+
+ 
+NR.2) If 
+1
+0
++ 
+n
+iu
+, the quantity 
+(
+)
+1
++
+n
+i
+F u
+ is compared to 
+(
+)
+sb
+K
+AR  and 
+(
+)
+*
+jump
+K
+AR . Three 
+cases are possible: 
+ 
+NR.2.1) If 
+(
+)
+(
+)
+1
++
+
+n
+i
+sb
+F
+K
+AR
+u
+ occurs, the energy of the subcritical flow is sufficient to 
+pass through the porosity reduction, and the point CR.1 of Section 5.1 supplies 
+1 2
+ +
+i
+, 
+1 2
+−
++
+iu
+, and 
+1 2
++
++
+iu
+. The position 
+,
+1 2
+1
++
++
++
+=
+R
+n
+i
+i
+u
+u
+ is made because there is no backwards moving shock. 
+ 
+NR.2.2) If 
+(
+)
+(
+)
+*
+1
++
+
+n
+i
+jump
+F
+K
+AR
+u
+, the energy of the supercritical flow is sufficient to pass 
+through the porosity reduction with head losses (see Section 4.2). In this case, the interface 
+porosity 
+1 2
+
+
++
+=
+i
+i  is assumed and the state 
+1 2
++
++
+iu
+ is easily found by picking the 
+supercritical solution of the system 
+ 
+(21) 
+(
+)
+(
+)
+(
+)
+1
+1
+1
+1 2
+1 2
+1 2
+1
+1 2
+1 2
+1
+1 2
+1
+0
+,
+,
+,
+
+
+
+
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
+−
+=
+−
+= 
+n
+n
+i
+i
+i
+i
+i
+i
+n
+n
+i
+i
+i
+i
+i
+i
+h u
+h
+u
+H
+H
+H
+u
+u
+u
+u
+, 
+ 
+ 
+which is obtained by assuming in Eq. (10) the positions 
+1 2
+
+ +
+=
+L
+i
+, 
+1
+
+ +
+=
+R
+i , 
+1
+1 2
++
++
+=
+i
+u
+u
+, 
+and 
+2
+1
++
+=
+n
+i
+u
+u
+. The head loss in Eq. (21) is computed using the Eqs. (12) and (13). Finally, 
+the positions 
+1 2
+−
++
+=
+n
+i
+i
+u
+u  and 
+,
+1 2
+1
++
++
++
+=
+R
+n
+i
+i
+u
+u
+ are made. 
+ 
+NR.2.3) If 
+(
+)
+(
+)
+(
+)
+*
+1
++
+
+
+n
+sb
+i
+jump
+K
+AR
+F
+K
+AR
+u
+, the energy of the state 
+1
++
+n
+iu
+ is either 
+insufficient to pass through the porosity reduction or a multiple solution is possible. In both 
+the cases, the Riemann problem solution is characterized by a backwards moving shock 
+
+radiating from the geometric discontinuity. The occurrence of this shock is forced by 
+posing 
+1 2
+
+
++
+=
+i
+i  and assuming that the states 
+1 2
++
++
+iu
+ and 
+,
+1 2
++
++
+R
+iu
+ have the same discharge of 
+the state 
+1
++
+n
+iu
+. The state 
+1 2
++
++
+iu
+ is critical, obtaining 
+ 
+(22) 
+1
+1
+1
+1 2
+1 2
+1 2
+1 2
+1 2
+0
+
+
++
++
++
++
++
++
++
++
++
++
++
++
+−
+=
+= −
+n
+n
+i
+i
+i
+i
+i
+i
+i
+i
+h u
+h
+u
+u
+gh
+, 
+ 
+ 
+while the state 
+,
+1 2
++
++
+R
+iu
+ is subcritical with (
+)
+(
+)
+,
+*
+1 2
++
++
+= −
+R
+i
+jump
+F
+K
+AR
+u
+, obtaining 
+ 
+(23) 
+(
+)
+,
+,
+1
+1
+1 2
+1 2
+,
+,
+*
+1 2
+1 2
+0
++
++
++
++
++
++
++
++
++
++
+−
+=
+= −
+n
+n
+R
+R
+i
+i
+i
+i
+R
+R
+i
+i
+jump
+h u
+h
+u
+u
+gh
+K
+AR
+. 
+ 
+ 
+Finally, the position 
+1 2
+−
++
+=
+n
+i
+i
+u
+u  is made. 
+ 
+The novel reconstruction satisfies the inequality 
+
+
+
+
+1
+1 2
+1
+min
+,
+max
+,
+ 
+
+ 
++
++
++
+
+
+i
+i
+i
+i
+i
+, 
+ensuring the monotonicity of the porosity discontinuity inner description. Having introduced the in-
+cell additional reconstructed state 
+,
+1 2
++
++
+R
+iu
+, the definition of 
+1 2
++
+−
+is
+ in Eq. (20) changes as follows: 
+ 
+(24) 
+(
+)
+(
+)
+,
+1 2
+1 2
+1 2
+1 2
+R
+i
+i
+i
+i
+i
+
+
++
++
++
+−
+−
+−
+−
+=
+−
+s
+f u
+f u
+. 
+ 
+A similar reconstruction approach has been proposed by Varra et al. (2022), where an iterative 
+procedure is used. Nonetheless, the present reconstruction is an improvement because (i) the iterative 
+procedure is avoided and (ii) it is possible to consider the cases where 
+(
+)
+(
+)
+*
+
+jump
+sp
+K
+AR
+K
+AR . In 
+addition, adequate head loss is introduced for supercritical flows through abrupt porosity reductions. 
+
+ 
+5.3 Numerical experiments 
+The 1-d numerical model of Eq. (15), equipped with the variable reconstructions described in 
+Sections 5.1 and 5.2, respectively, is used to approximate the solution of the Riemann problems with 
+initial conditions in Table 1. For the sake of simplicity, 
+x
+  = 0.20 m and 
+t
+  = 0.005 s in all the 
+numerical experiments. 
+ 
+5.3.1 Numerical experiments with the reconstruction by Castro et al. (2007)  
+Figure 19 shows the numerical results (flow depth) to the Riemann problems 1-4 supplied by the 1-d 
+SP model with the reconstruction by Castro et al. (2007) (continuous black line), while the 
+corresponding 2-d SWE results are represented with white dots. For all these problems, the algorithm 
+by Castro et al. (2007) captures the essentials of the 2-d SWE solution, namely the number of waves 
+and their strength, and the flow depth of the intermediate states. The discrepancies between the 1-d 
+and 2-d solutions in Figures 19b and 19c (Riemann problems 2 and 3, respectively) correspond to 
+those discussed with reference to the comparison between Riemann exact solution and 2-d numerical 
+solution (compare with Figures 6c and 6d).  
+ 
+
+ 
+Figure 19. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model 
+with the variable reconstruction by Castro et al. (2007) (continuous black line) and 2-d SWE model 
+(dots). Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with initial conditions in Table 1.  
+ 
+The numerical results to Riemann problems 5-8 are represented in Figure 20. These cases, 
+characterized by a supercritical flow through a porosity reduction, show that the 1-d SP numerical 
+solution with the basic reconstruction by Castro et al. (2007) greatly differs from the corresponding 
+2-d SWE reference solution. 
+The Figures 20a,b refer to Riemann problems (5 and 6, respectively) admitting multiple 
+solutions. While the 2-d SWE model exhibits a backwards moving shock originated from the porosity 
+discontinuity that causes flow energy dissipation, the 1-d SP numerical model with variable 
+
+1.6
+1.2
+(a)
+(b)
+1.2
+3RXIX EOVYX
+0.8
+(w)
+0.8-
+h
+h
+0.4
+0.4 -
+0
+-20
+0
+20
+-10
+0
+10
+20
+30
+(u) x
+x (m)
+1.6
+1.2
+1
+(c)
+(d)
+1.2
+0.8
+h
+h
+0.4 -
+0.4
+1
+1
+0
+0
+-40
+-20 
+0
+20
+40
+-80
+-40
+0
+40
+x (m)
+x (m)reconstruction by Castro et al. (2007) captures the solution with supercritical flow through the 
+discontinuity. This is expected because the reconstruction by Castro et al. (2007) keeps the 
+supercritical character of the flow impinging on the porosity reduction (point CR.1 in Section 5.1). 
+From the preceding, it follows that the numerical scheme by Castro et al. (2007) overestimates the 
+discharge through the geometric discontinuity and the celerity of the advancing shock on the left, 
+while completely neglects the energy dissipation mechanism connected with the backwards moving 
+shock generated by the interaction of the propagating flow with obstacles (Guinot et al. 2017). Varra 
+et al. (2020) have demonstrated that the same defect is shared by other Riemann solvers such as those 
+by Cozzolino et al. (2018b) and Guinot et al. (2017). 
+The Figures 20c,d refer to Riemann problems (7 and 8, respectively) where a supercritical 
+flow impinges on a porosity reduction, but the solution is unique. In these cases, the 1-d SP numerical 
+model misses to capture the 2-d SWE model solution because it lacks an appropriate dissipation 
+mechanism. Interestingly, this is the same discrepancy found when comparing the 1-d exact solution 
+and the 2-d SWE numerical solution (see Figures 10a,b). 
+ 
+
+ 
+Figure 20. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model 
+with the basic variable reconstruction by Castro et al. (2007) (continuous black line) and 2-d SWE 
+model (dots). Example Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table 
+1.  
+ 
+ 
+5.3.2 Numerical experiments with the novel reconstruction  
+The numerical experiments presented in the preceding subsection are repeated using the 1-d SP 
+numerical model with with the novel reconstruction of Section 5.2. For the Riemann problems 1-4, 
+the numerical results supplied by the novel reconstruction, which are not reported here for the sake 
+of brevity, coincide with those supplied by the basic reconstruction by Castro et al. (2007). This is 
+
+4
+6
+(a)
+(b)
+1
+1:
+9
+4 
+三2
+(w)
+h
+h
+2
+1
+1
+1
+0
+0
+-60
+-40
+-20
+0
+20
+-60
+-40
+-20
+0
+20
+(u) x
+x (m)
+6
+1
+(c)
+2
+(d)
+-
+1
+1
+1.5
+1
+4 
+1
+1
+(w)
+(u)
+-
+d
+h
+h
+:3
+0.5
+:1
+0
+0
+-80
+-40
+0
+-80
+-60
+-40
+-20 
+0
+20
+x (m)
+x (m)expected, because these Riemann problems do not refer to cases where a supercritical flow impinges 
+on a porosity reduction. 
+The Figures 21a,b refer to the Riemann problems 5 and 6, respectively, which admit multiple 
+solutions. Contrary to the algorithm by Castro et al. (2007), the novel variable reconstruction captures 
+the solution with the backwards moving shock exhibited by the 2-d SWE model. It is clear that the 
+introduction of the in-cell subcritical state 
+,
+1 2
++
++
+R
+iu
+ between the geometric transition and the right state 
+n
+iu , together with the computation of the interface reaction term by means of Eq. (24), is the ingredient 
+allowing the computation of the physically congruent shock immediately to the discontinuity right-
+side.  
+The comparison between the 1-d SP numerical results obtained with the novel variable 
+reconstruction and the 2-d SWE results for Riemann problems 7 and 8, is represented in Figures 
+21c,d, respectively. The inspection of these figures, referring to cases where the supercritical flow 
+impinging on the porosity reduction remains supercritical with loss of energy, shows that the novel 
+variable reconstruction satisfactorily reproduces the 2-d SWE model results because the required 
+amount of head loss through the geometric discontinuity is introduced. 
+ 
+ 
+
+ 
+Figure 21. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model 
+with the novel variable reconstruction (continuous black line) and 2-d SWE model (dots). Example 
+Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table 1.  
+ 
+ 
+6. Discussion 
+ 
+ 
+In this section, the model presented in the preceding sections is discussed with reference to alternative 
+numerical and conceptual approaches available in the literature, and with reference to the robustness 
+of Eqs. (11) and (12). 
+ 
+
+4
+6
+(a)
+(b)
+3 -
+4 
+三2
+(w)
+h
+h
+2
+1
+1
+一
+0
+0
+-60
+-40
+-20
+0
+20
+-60
+-40
+-20
+0
+20
+(w) x
+x (m)
+6
+1
+(c)
+2
+1
+(d)
+1
+-
+-
+1
+1.5
+1
+4 
+1
+1
+(w) 
+(u)
+h
+h
+:3
+0.5
+:1
+1
+0
+0
+-80
+-40
+0
+-80
+-60
+-40
+-20 
+0
+20
+x (m)
+x (m)6.1 Comparison with the transient momentum dissipation approach  
+The momentum dissipation approach introduced with the DIP numerical model by Guinot et al. 
+(2017) is a numerical device intended to introduce the transient energy dissipation generated by bore 
+reflection at obstacles during transient propagation. In the 1-d case, the DIP model can be written as 
+ 
+(25) 
+(
+)
+(
+)
+1
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+,
+1 2
+,
+1 2
+,
+,
+
+
+
+
++
+−
++
+−
++
++
+−
++
++
++
++
+−
+−
+−
+−
+−
++
+
+
+
+
+
+
+=
+−
+−
++
++
+
+
+
+
+
+
+n
+n
+i
+i
+i
+i
+i
+i
+i
+i
+i
+i
+sta i
+sta i
+i
+i
+t
+t
+x
+x
+u
+u
+M
+g u
+u
+M
+g u
+u
+s
+s
+, 
+ 
+where M is a momentum dissipation matrix defined as (Guinot et al. 2017) 
+ 
+(26) 
+1
+0
+0
+1
+
+
+
+= 
+
+−
+
+
+M
+, 
+ 
+with  > 0 in case 
+1
++ 
+n
+n
+i
+i
+h
+h , while  = 0 otherwise. The momentum dissipation coefficient  
+appearing in the matrix M of Eq. (26) must be calibrated using fine grid 2-d SWE simulations and it 
+is strongly dependent on the urban fabric structure and flow conditions (Guinot et al. 2017). 
+In Eq. (25), the interface porosity 
+1 2
+ +
+i
+ is evaluated from the underlying urban fabric with a 
+non-monotonic approach, which enforces the condition 
+(
+)
+1 2
+1
+min
+,
+
+ 
++
++
+
+i
+i
+i
+ (see Section 2.2.1); the 
+interface contributions 
+,
+1 2
++
+−
+sta i
+s
+ and  
+,
+1 2
+−
++
+sta i
+s
+ of the non-conservative products are computed under the 
+assumption of in-cell stagnant water (Guinot and Soares-Frazão 2006) as 
+ 
+(27) 
+(
+)
+(
+) (
+)
+(
+)
+(
+) (
+)
+2
+,
+1 2
+1 2
+2
+,
+1 2
+1 2
+0
+1
+2
+0
+1
+2
+
+
+
+
++
+−
+−
+−
++
++
+=
+−
+=
+−
+T
+n
+sta i
+i
+i
+i
+T
+n
+sta i
+i
+i
+i
+g h
+g h
+s
+s
+. 
+ 
+
+Finally, the reconstructed variables are defined as: 
+ 
+(28) 
+(
+)
+(
+)
+1 2
+1 2
+1 2
+1
+1
+1
+1
+1 2
+
+
+
+
+−
++
++
++
++
++
++
++
++
++
+=
+=
+T
+n
+n
+n
+i
+i
+i
+i
+i
+i
+T
+n
+n
+n
+i
+i
+i
+i
+i
+i
+h
+h u
+h
+h u
+u
+u
+. 
+ 
+We reinterpret the momentum dissipation approach observing that the 1-d DIP numerical 
+model of Eq. (25) can be rewritten in the form of Eq. (15) by defining the interface contributions of 
+the non-conservative products as 
+ 
+(29) 
+(
+) (
+)
+(
+) (
+)
+1 2
+,
+1 2
+1 2
+1 2
+1 2
+1 2
+1 2
+,
+1 2
+1 2
+1 2
+1 2
+1 2
+,
+,
+
+
++
++
+−
++
+−
+−
+−
+−
+−
+−
+−
+−
+−
++
++
++
++
++
++
++
+=
++
+−
+=
++
+−
+i
+sta i
+i
+i
+i
+i
+i
+sta i
+i
+i
+i
+i
+s
+s
+M
+I g u
+u
+s
+s
+I
+M
+g u
+u
+. 
+ 
+According to this reinterpretation, the momentum dissipation approach is equivalent to 
+evaluating the forces exerted by obstacles at cell interfaces by adding or subtracting a dynamic 
+contribution to the stagnant water hydrostatic thrusts 
+,
+1 2
++
+−
+sta i
+s
+ and  
+,
+1 2
+−
++
+sta i
+s
+. A slightly different 
+definition of the matrix M has been subsequently given in Guinot et al. (2018), but the role of M as 
+a modulator of the forces exerted by obstacles at cell interfaces remains unchanged. 
+The reinterpretation supplied by Eq. (29) allows to recognize the common numerical 
+framework of the DIP model (Guinot et al. 2017) and of the numerical model presented here. 
+Nonetheless, the exact solution of the 1-d SP Riemann problem has been exploited in the present 
+paper to introduce a mechanism of energy dissipation caused by the reflection of advancing waves at 
+porosity discontinuities.  This approach, which has been accomplished by isolating a single local 
+porosity discontinuity and comparing the corresponding 1-d SP and 2-d SWE Riemann solutions (see 
+Section 3), avoids the intricacies caused by the mutual interaction of waves radiating from the 
+obstacles of complex urban fabrics and allows to consider the local geometry. In addition, it avoids 
+
+the introduction of a momentum dissipation mechanism of unclear physical meaning, whose 
+parameters need calibration on a case-by-case basis (Guinot et al. 2017).  
+ 
+6.2 Disambiguation of the porosity Riemann problem 
+The issue of disambiguating multiple solutions to the Riemann problem where a geometric 
+discontinuity is present has been tackled by researchers considering different types of geometric 
+discontinuities or different fluid models (SWE or Euler equations). It is interesting to compare the 
+results obtained in the present paper with those available in the literature. 
+ 
+The exact solution to the Riemann problem for the 1-d SWE model with variable bed elevation 
+exhibits two classes of triple solutions (for convenience, the solutions are called here S1, S2, and S3) 
+when a supercritical flow impinges on a positive bed step (Han and Warnecke 2014, Aleksyuk et al. 
+2022). In the first class of triple solutions, S1 is characterised by a supercritical flow that jumps over 
+the bed step remaining supercritical, while S2 is characterised by a hydraulic jump located through 
+the discontinuity that reverts the incoming supercritical flow into subcritical. Finally, S3 is 
+characterised by a backward shock while subcritical flow conditions are established over the bed step. 
+The second class of triple solutions differs from the first one because the solution S3 is characterised 
+by a backward shock with blockage of the flow at the bed step (step higher than the free surface level). 
+Cozzolino et al. (2014) used a mix of steady state laboratory data (Karki et al. 1972, Hager and 
+Sinniger 1985) and physical reasoning to establish a disambiguation criterion based on discharge 
+minimization. Following this criterion, the physically relevant solution is S3 (backward shock) in 
+both the classes of triple solutions. Aleksyuk and Belikov (2019) found the same result by considering 
+a mathematical argument based on the continuous dependence of solutions on the initial conditions. 
+Han et al. (2013) considered the Riemann problem for the 1-d Euler equations in a 
+compressible duct flow, where triple solutions may occur when a supersonic flow impinges on a pipe 
+diameter reduction. They compared some examples of 1-d exact multiple solutions with the numerical 
+results supplied by a higher dimensions axisymmetric Euler equations model (longitudinal and radial 
+
+direction), founding that the physically relevant solutions were those characterized by a backward 
+shock. 
+A pattern seems to emerge from these results. In all the examples considered, multiple 
+solutions occur when a supercritical flow (SWE model) or a supersonic flow (Euler equations) impact 
+on a cross-section reduction. Despite the variety of mathematical models and means applied for the 
+disambiguation of multiple exact solutions (laboratory and/or numerical experiments, mathematical 
+arguments), the common output to the different procedures is that the physically relevant solution 
+among the alternatives is the one characterised by a backward shock. In SWE models, this is also the 
+solution which minimizes the discharge through the geometric discontinuity. 
+The results presented in Section 4, which confirm this pattern for the 1-d SP model, can be 
+reinterpreted using an argument based on the continuous dependence of the Riemann problem 
+solution on the initial conditions. In fact, when the tailwater is null (hL = 0), the 1-d SP Riemann 
+problem exhibits three exact solutions (T1, T2, and T3) in the region B of Figure 3, which is bounded 
+by the curves LB and UB, and one exact solution in regions A and C (T3 and T1, respectively). A 
+solution with backward shock (T3 of Figure 4c) in region B of Figure 3 can move through LB to a 
+solution T3 of region A by decreasing the initial Froude number 
+R
+F . On the other side, the same T3 
+solution in region B can move through UB to a T1 solution of region C by increasing 
+R
+F  , flushing 
+the hydraulic jump and establishing supercritical flow conditions through the porosity discontinuity. 
+In conclusion, T3 solutions should be considered to the left of UB, and T1 solutions to the right. Of 
+course, the head loss generated by transverse shocks through the porosity discontinuity does not 
+significantly changes this picture. In this case, the limit curve UB is distorted, becoming the modified 
+limit curve MUB of Figure 14. Indeed, the initial conditions to the left of the MUB curve characterize 
+solutions with a backward shock (G2 solutions in Section 4.1), while the initial conditions to the right 
+characterize supercritical flows through the porosity discontinuity (G1 solutions). 
+ 
+
+6.3 Influence of flow depth and channel width on the porosity discontinuity definition 
+The numerical experiments of Section 4 have been conducted considering a fixed depth hR = 1 m of 
+the flow impinging on a channel contraction and fixed width BR = 1 m of the right channel reach. The 
+reader may wonder if these results can be extended to different values of hR and BR. The answer to 
+this question is affirmative but it requires a brief discussion. 
+Recalling that viscosity and density are not modelled by the incompressible SWE model, the 
+most general way to write the expression of the head loss H* suffered by a supercritical flow uR 
+through the contraction is 
+ 
+(30) 
+(
+)
+*
+,
+,
+,
+, ,
+
+=
+L
+R
+R
+R
+c
+H
+f B
+B
+h u
+g L
+. 
+ 
+We observe that n = 7 physical quantities are involved in Eq. (30), and we want to ascertain 
+if the physical equation can be simplified. Aiming at this, we additionally see that only k = 2 
+fundamental mechanics units, namely length and time, are involved because there is not dependency 
+on density. Recalling the Vaschy-Buckingham theorem (Vaschy 1892, Buckingham 1914), it follows 
+that Eq. (30) can be rewritten as an expression involving n – k = 5 dimensionless independent 
+quantities. The dimensionless quantities chosen are 
+ 
+(31) 
+(
+)
+*
+*
+2
+,
+,
+,
+,
+2
+
+
+−
+
+=
+=
+ =
+=
+=
++
+L
+R
+R
+R
+L
+R
+R
+R
+R
+R
+R
+c
+R
+B
+u
+h
+B
+B
+H
+AR
+F
+B
+h
+u
+g
+B
+L
+gh
+, 
+ 
+and it is easy to verify that they are mutually independent, i.e., it is not possible to build one of the 
+dimensionless quantities starting from the others. This justifies why it is possible to simplify Eq. (30) 
+as 
+ 
+(32) 
+(
+)
+*
+2
+,
+,
+,
+
+
+ =
+R
+R
+f
+AR F
+. 
+
+ 
+In a similar manner, a very general way to describe the boundary MUB between the G1 and 
+G2 solutions discussed in Section 4 is to introduce a limit velocity 
+*
+R
+u  discriminating the two types of 
+solution and expressing this velocity as 
+ 
+(33) 
+(
+)
+*
+,
+,
+, ,
+=
+R
+L
+R
+R
+c
+u
+f B
+B
+h
+g L
+. 
+ 
+This physical equation involves n = 6 physical quantities and k = 2 fundamental units, 
+implying that it can be simplified as 
+ 
+(34) 
+(
+)
+*
+,
+,
+
+
+=
+R
+K
+f
+AR
+ 
+ 
+where 
+*
+*
+=
+R
+R
+K
+u
+gh . 
+We observe that the dependence on  does not need to be explicited because this parameter is 
+constant in all the 2-d SWE simulations, since the contraction walls are always inclined by 45° with 
+respect to the channel axis. With reference to the parameter  R , the comparison between Eq. (32) 
+and (12), and between Eq. (34) and (11), respectively, show that the expressions found in Section 4 
+are valid in the case 
+1
+ =
+R
+ because hR = 1 m and BR = 1 m in all the numerical experiments. 
+Nonetheless, we demonstrate that the parameter R  is superfluous because the expressions of Eqs. 
+(32) and (34) can be safely simplified in the form of Eqs. (12) and (11), respectively.  
+ 
+Consider the steady state 1-d SWE in a channel of variable width B = B(x), with solution u = 
+u(x) for given right boundary condition uR: 
+ 
+(35) 
+( )
+( )
+0
++
+=
+dB
+dB
+dx
+dx
+f u
+h u
+. 
+
+ 
+Despite Eq. (35) is a simplification of the 2-d flow through the contraction, it supplies a 
+sufficient insight for the present discussion. We observe that the multiplication of Eq. (35) by the 
+constant k allows to write  
+ 
+(36) 
+( )
+( )
+0
+
+
++
+=
+
+
+
+
+dB
+dB
+k
+dx
+dx
+f u
+h u
+, 
+ 
+which is still satisfied by the solution u = u(x) of Eq. (35). Of course, k can be moved inside the 
+derivative symbols, leading to 
+ 
+(37) 
+( )
+( )
+'
+'
+0
++
+=
+dB
+dB
+dx
+dx
+f u
+h u
+, 
+ 
+where B’(x) = kB(x). In other words, the solution u = u(x) of Eq. (35) for given boundary condition 
+uR does not change if the width is uniformly amplified by a constant k. The uniform amplification of 
+the width affects the parameter  =
+R
+R
+R
+h
+B  but does not affect the parameters
+=
+L
+R
+AR
+B
+B  and 
+=
+R
+R
+R
+F
+u
+gh , implying that Eqs. (32) and (34) depend on AR and 
+2
+R
+F  but not on  R .  
+ 
+To evaluate the robustness of this theoretical approach, we consider twelve additional 2-d 
+SWE simulations with initial and geometrical conditions as follows: AR = 0.3, 0.6; 
+R
+F  = 3.6, 6, 8, 
+11; hR = 0.1, 0.5, 1 m, and BR = 1 m (see Table 4). The values chosen for hR correspond to the three 
+different values ( R  = 0.1, 0.5, and 1) of the dimensionless parameter  R . Six of the chosen flow 
+conditions correspond to points slightly to the left of the MUB curve and six to the right (see Figure 
+22). Table 4 reports, for each simulation, the dimensionless head loss * if the solution is of type G1, 
+while a hyphen indicates a G2-type solution. The inspection of these results shows that the solution 
+
+types (G1 or G2) expected based on the position with respect to the MUB curve are those actually 
+occurring in the 2-d simulations, while the dependence of the relative head loss * on  R is 
+negligible. These observations confirm the theoretical arguments above and justify the application of 
+the formulas in Section 4 to conditions with  R   1.  
+ 
+Table 4. Supercritical 2-d flow impinging on a contraction for BR = 1 m and hR = 0.1, 0.5, and 1 m: 
+relative head losses * for the cases of flow passing through the discontinuity (G1). A hyphen 
+indicates the cases where a backwards moving shock is produced (G2 configurations). 
+ 
+hR (m) 
+AR 
+0.30 
+0.60 
+FR 
+FR 
+8 
+11 
+3.6 
+6 
+0.1 
+- 
+0.57 
+- 
+0.38 
+0.5 
+- 
+0.57 
+- 
+0.38 
+1 
+- 
+0.57 
+- 
+0.38 
+ 
+ 
+
+0.8
+ Modified upper boundary (MUB)
+A^^ 2-d SWE numerical Gl solution
+hr (m) = 0.1 - 0.5 - 1
+ 2-d SWE numerical G2 solution
+0.6 -
+R
+0.4 
+hr (m) = 0.1 - 0.5 - 1
+0.2
+0
+FR
+25Figure 22. 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 0.1, 0.5, and 1 
+m impinging a contraction: G1 configuration (black triangles), G2 configuration (white squares); 
+modified upper boundary (thick black line). 
+ 
+7. Conclusions 
+The solution of the Riemann problem associated to the Single Porosity (SP) Shallow water Equations 
+(SWE) model by Guinot and Soares-Frazão (2006) is the main ingredient for the computation of 
+interface fluxes and obstacle reaction terms in the Binary SP model (Varra et al. 2020) and in the 
+integral approach (Sanders et al. 2008, Guinot et al. 2017). Previous studies (Cozzolino et al. 2018a, 
+Varra et al. 2020, 2021) have shown that the SP Riemann problem presents a fundamental ambiguity 
+consisting in the appearance of multiple exact solutions for certain initial conditions characterized by 
+a supercritical flow impacting on a porosity reduction. This observation prompts the definition of the 
+unique physically congruent Riemann solution among the alternatives and the construction of a 
+numerical scheme able to reproduce this relevant solution. 
+Having recognized that the 1-d SP model is nothing but a crude simplification of the 2-d SWE 
+model in a variable width rectangular channel, the channel analogy (Guinot and Soares-Frazão 2006, 
+Sanders et al. 2008) has been exploited in this paper to disambiguate the multiple 1-d SP Riemann 
+solutions by means of a systematic comparison with the corresponding 2-d SWE numerical solutions 
+at local geometric discontinuities. The conclusion, which corresponds to other similar results obtained 
+in the literature for different mathematical models (Han et al. 2013, Cozzolino et al. 2014, Aleksyuk 
+and Belikov 2019), is that the solution with a backwards moving shock is physically congruent when 
+multiple solutions are possible. Laboratory (Akers and Bokhove 2008, Defina and Viero 2010) and 
+2-d SWE numerical experiments (Varra et al. 2020) show that supercritical flows in channels suffer 
+intense head loss through a width contraction, which corresponds to a porosity reduction. This 
+phenomenon is an additional cause of energy dissipation in porosity models with respect to those 
+already described in the literature (Guinot et al. 2017, 2018). Also in this case, the systematic study 
+
+of 2-d SWE numerical results at isolated geometric discontinuities has supplied the general conditions 
+under which this energy dissipation is present and how it can be evaluated. 
+Based on a modification of the generalized hydrostatic reconstruction by Castro et al. (2007), 
+we have also built an approximate Riemann solver that discriminates the existence of multiple 
+solutions and is able to add adequate head loss in the case of supercritical flow through porosity 
+discontinuities. The comparison between the numerical results supplied by the novel 1-d SP model 
+and the 2-d SWE model shows that the former can reproduce the effects that in 2-d models are caused 
+by the interaction between a supercritical flow and a contraction. This promising numerical approach 
+could be extended to other cases of hyperbolic systems of differential equations where multiple 
+solutions arise such as the SWE and the porous SWE with variable topography.  
+ 
+Aknowledgements 
+Renata Della Morte and Luca Cozzolino want to acknowledge the financial support from the project 
+"Floods in cities: new insights for integrating pluvial flooding into flood risk management plans 
+(INSPIRING)", funded by the Italian Ministry of University and Research under the national 
+programme PRIN2020. 
+ 
+Appendix A. Head-balance form of the porosity discontinuity 
+If 
+1 1
+2 2
+
+
+=
+=
+L
+R
+Q
+hu
+h u  is the unit-width discharge flowing through the porosity discontinuity, the 
+velocities at the two sides of the geometric transition can be rewritten as 
+ 
+(A.1) 
+1
+2
+1
+2
+,
+ 
+
+
+=
+=
+L
+R
+Q
+Q
+u
+u
+h
+h . 
+ 
+The substitution of Eq. (A.1) into Eq. (5.b) leads to  
+ 
+
+(A.2) 
+(
+)
+2
+2
+2
+2
+2
+1
+1
+2
+2
+1
+,
+,
+,
+2 
+
+ 
+
+
+
+
+
+−
++
+−
+=
+
+
+R
+L
+L
+R
+R
+L
+g
+Q
+Q
+h
+h
+S
+h
+h
+u u
+, 
+ 
+while the substitution into the second of Eq. (6) supplies 
+ 
+(A.3) 
+(
+)
+(
+)
+(
+)
+2
+2
+1
+2
+2
+1
+2
+2
+2
+1
+,
+,
+,
+2
+2
+ 
+
+
+
+
+
+=
++
+−
++
+
+
+
+
+
+
+L
+R
+R
+L
+Q
+Q
+H
+h
+h
+g
+h
+g
+h
+u u
+. 
+ 
+The elimination of Q2 between Eqs. (A.2) and (A.3) finally supplies Eq. (7).  
+ 
+Appendix B. Smooth stationary weak solutions of Eq. (2) 
+If the time derivatives are null, Eq. (2) can be rewritten as 
+ 
+(B.1) 
+2
+2
+2
+0
+0
+2
+2
+
+
+
+
+=
+
+ +
+−
+=
+
+
+
+
+d hu
+ds
+d
+gh
+d hu
+gh d
+ds
+ds
+ds
+. 
+ 
+If the porosity and the flow variables are smooth (i.e., continuous with their derivatives), the 
+second of Eq. (B.1) can be rewritten as 
+ 
+(B.2) 
+2
+2
+2
+0
+2
+2
+2
+
+
+
+
+
+
+
++
++
++
+−
+=
+
+
+
+
+dh
+gh d
+d
+u
+d hu
+gh d
+gh
+h
+u
+ds
+ds
+ds
+ds
+ds
+. 
+ 
+Finally, the substitution of the first of Eq. (B.1) into Eq. (B.2) and the cancellation of the terms 
+with opposite sign leads to 
+ 
+ 
+
+(B.3) 
+2
+0
+2
+
+
++
+=
+
+
+
+
+d
+u
+h
+ds
+g
+, 
+ 
+which states that the total head is uniform.  
+ 
+ 
+Appendix C. Discussion of Eq. (10) and of the corresponding Froude limits 
+This Appendix reports in a condensed form the discussion present in classic (Yarnell 1934, Chow 
+1959) and recent (Defina and Susin 2006, Castro et al. 2007, Akers and Bokhove 2008, Varra et al. 
+2021) literature with reference to 1-d flows in channels with variable width. Given the formal analogy 
+between 1-d SWE model with variable width and 1-d porous SWE model, the discussion can be 
+extended to porous models where the porosity symbol replaces the width symbol while the discharge 
+is intended as unit-width discharge. 
+The parametric family of the flow states 
+(
+)
+=
+T
+h
+hu
+u
+ with given width B and discharge 
+=
+Q
+Bhu  is defined by (
+)
+(
+)
+, ,
+=
+T
+Q B h
+h
+Q B
+u
+, where the flow depth h is the parameter. The total 
+head corresponding to the states of this family is a function of the parameter h only, and it is defined 
+by 
+ 
+(C.1) 
+(
+)
+(
+)
+(
+)
+(
+)
+2
+2
+, ,
+, ,
+2
+=
+=
++
+Q
+H Q B h
+H
+Q B h
+h
+g Bh
+u
+. 
+ 
+ 
+The function H(Q, B, h) in Eq. (C.1) is convex with respect to h > 0, it is positive and it has a 
+unique minimum in 
+(
+)
+,
+=
+c
+h
+h Q B . The critical depth 
+(
+)
+,
+ch
+Q B  and the corresponding critical head 
+(
+)
+(
+)
+(
+)
+,
+, ,
+,
+=
+c
+c
+H
+Q B
+H Q B h Q B
+ are defined by (Chow 1959) 
+ 
+
+(C.2) 
+(
+)
+(
+)
+2
+2
+3
+3
+2
+2
+3
+,
+,
+,
+2
+=
+=
+c
+c
+Q
+Q
+h Q B
+H
+Q B
+gB
+gB
+. 
+ 
+The states u with 
+(
+)
+,
+
+c
+h
+h Q B  are characterized by 
+( )
+1
+
+F u
+ and are called supercritical. 
+The states with 
+(
+)
+,
+
+c
+h
+h Q B  are characterized by 
+( )
+0
+1
+
+
+F u
+ and are called subcritical. From the 
+preceding discussion, it follows that a state u with discharge Q has energy sufficient to pass through 
+the cross-section whose width is B only if the corresponding head is not minor than 
+(
+)
+,
+c
+H
+Q B . 
+This observation has consequences for the application of Eq. (10), expressing the invariance 
+of discharge and head. Let BL and BR be the channel widths at the left and right ends, respectively, of 
+a geometric transition, and let 
+(
+)
+2
+2
+2
+2
+=
+T
+h
+h u
+u
+ be the state corresponding to the flow at the right 
+end. It is possible to find the left end state 
+(
+)
+1
+1
+1 1
+=
+T
+h
+hu
+u
+ connected to u2 by means of the discharge 
+and head invariance only if 
+(
+)
+(
+)
+1
+,
+
+c
+L
+H
+H
+Q B
+u
+, namely only if 
+ 
+(C.3) 
+(
+)
+2
+3
+1
+2
+3
+2
+
+L
+Q
+H
+gB
+u
+, 
+ 
+where 
+1 1
+=
+L
+Q
+B hu  is the discharge corresponding to the right state 
+1
+u . The invariance of discharge 
+and head is expressed by 
+2 2
+1 1
+=
+R
+L
+B h u
+B hu  and 
+(
+)
+(
+)
+2
+1
+=
+H
+H
+u
+u
+, implying that the condition of Eq. 
+(C.3) can be rewritten as  
+ 
+(C.4) 
+(
+)
+(
+)
+2
+2
+2
+3
+2
+2
+3
+1
+2
+
+h u
+H
+AR
+g
+u
+, 
+ 
+ 
+
+where 
+=
+L
+R
+AR
+B
+B  is the aspect ratio. If the Eq. (C.4) is satisfied, it exists the state u1 connected to 
+u2 by the invariance of discharge and head with aspect ratio AR. 
+The Eq. (C.4) can be rewritten in dimensionless form as (Yarnell 1934, Chow 1959, Defina 
+and Susin 2006, Akers and Bokhove 2008) 
+ 
+(C.5) 
+(
+)
+(
+)
+2
+
+AR
+f
+F u
+, 
+ 
+ 
+where 
+(
+)
+2
+2
+2
+=
+F
+u
+gh
+u
+ is the Froude number of the state u2 and the function f(x) is defined as 
+ 
+(C.6) 
+( )
+3 2
+2
+3
+,
+0
+2
+
+
+=
+
+
+
++
+
+
+f x
+x
+x
+x
+. 
+ 
+ 
+The function f(x) of Eq. (C.6) is non-negative, strictly increasing for 
+
+
+0,1
+
+x
+, strictly 
+decreasing for 
+1
+
+x
+, and it has a maximum in 
+1
+=
+x
+ with 
+( )
+1
+1
+=
+f
+. The properties of the function f(x) 
+have the following implications. 
+ 
+Case 
+1
+
+AR
+. In case of 
+1
+
+AR
+, Eq. (C.5) is satisfied for every 
+(
+)
+2
+F u
+. In other words, it 
+always exists the state u1 connected to u2 by the invariance of discharge and head when there is a 
+width increase. 
+ 
+Case 
+1
+
+AR
+. In case of 
+1
+
+AR
+, Eq. (C.5) is satisfied by 
+ 
+(C.7)
+(
+)
+(
+)
+2
+
+sb
+F
+K
+AR
+u
+, 
+(
+)
+(
+)
+2
+
+sp
+F
+K
+AR
+u
+, 
+ 
+where the limit Froude numbers 
+(
+)
+sb
+K
+AR  and 
+(
+)
+sp
+K
+AR  are defined as (Varra et al. 2021) 
+ 
+
+(C.8) 
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+(
+)
+3
+2
+1 2
+2
+3
+2
+1 2
+2
+1
+2cos
+arctan
+1
+3
+3
+5
+1
+2cos
+arctan
+1
+3
+3
+
+
+−
+−
+−
+−
+
+
+
+
+=
+−
+−
+
+
+
+
+
+
+
+
+
+
+
+
+=
+−
+−
+
+
+
+
+
+
+
+
+sp
+sb
+K
+AR
+AR
+AR
+K
+AR
+AR
+AR
+. 
+ 
+ 
+The Froude limits defined by Eq. (C.8) are characterised by 
+(
+)
+1
+
+sp
+K
+AR
+ (supercritical) and 
+(
+)
+1
+
+sb
+K
+AR
+ (subcritical) for every 
+1
+
+AR
+. 
+In conclusion, the present discussion shows that two different situations are possible with 
+reference to Eq. (10) where 
+1
+
+
+=
+
+L
+R
+AR
+: 
+a) Given the state u1, it is always possible to find the corresponding state u2. 
+b) Given the state u2, it is possible to find the corresponding state u1 only if one of the two 
+inequalities of Eq. (C.7) is satisfied. In this case, the invariance of the total head through 
+the porosity transition implies that u1 is subcritical [supercritical] if u2 is subcritical 
+[supercritical], and vice versa. When 
+(
+)
+(
+)
+2
+=
+sb
+F
+K
+AR
+u
+ or  
+(
+)
+(
+)
+2
+=
+sp
+F
+K
+AR
+u
+, the 
+state u1 is critical. 
+For 
+1
+
+
+=
+
+L
+R
+AR
+, it is possible to define an additional limit 
+(
+)
+jump
+K
+AR  for the Froude 
+number, as follows. Let the left state u1 be critical and the right state u2 be subcritical with 
+(
+)
+(
+)
+2 = −
+sb
+F
+K
+AR
+u
+ (flow from right to left). The supercritical state 
+#
+2
+u  to the right of the subcritical 
+state u and connected to it by a standing hydraulic jump is characterized by Froude number 
+(
+)
+(
+)
+#
+2 = −
+jump
+F
+K
+AR
+u
+, where 
+ 
+(C.9) 
+(
+)
+(
+)
+(
+)
+(
+)
+3
+2
+2
+8
+1
+1 8
+−
+=
+− +
++
+jump
+sb
+sb
+K
+AR
+K
+AR
+K
+AR
+. 
+ 
+
+Appendix D. Head loss through a standing hydraulic jump 
+The flow depth 
+#
+Rh  corresponding to the state 
+(
+)
+#
+#
+#
+#
+=
+T
+R
+R
+R
+R
+h
+h u
+u
+ connected to uR by means of a 
+hydraulic jump in a rectangular channel is (Chow 1959) 
+ 
+(D.1) 
+(
+)
+2
+#
+1
+1 8
+2
+=
+− +
++
+R
+R
+R
+h
+h
+F
+, 
+ 
+where 
+2
+R
+F  is the squared Froude number corresponding to the state uR. From Eq. (D.1), the ratio 
+#
+R
+R
+h
+h depends on 
+2
+R
+F  only. 
+The discharge is conserved through the hydraulic jump, implying that 
+#
+# =
+R
+R
+R
+R
+h u
+h u . For this 
+reason, the head 
+#
+R
+H  corresponding to the state 
+#
+R
+u  is 
+ 
+(D.2) 
+(
+)
+#
+#
+#
+#
+#
+2
+3
+2
+#
+2
+1
+2
+2
+
+
+
+
+
+
+
+
+=
+=
++
+=
++
+
+
+
+
+
+
+
+
+
+
+
+
+R
+R
+R
+R
+R
+R
+R
+R
+R
+R
+u
+h
+F
+h
+H
+H
+h
+h
+g
+h
+h
+u
+. 
+ 
+Once that Eq. (D.1) is substituted into Eq. (D.2), the relative head loss 
+(
+)
+#
+#
+#
+ = 
+=
+−
+R
+R
+R
+R
+H
+H
+H
+H
+H  can be easily calculated as 
+ 
+(D.3) 
+1
+#
+#
+#
+#
+3
+2
+2
+1
+1
+1
+2
+2
+−
+
+
+
+
+
+
+−
+
+
+ =
+= −
++
++
+
+
+
+
+
+ 
+
+
+
+
+
+R
+R
+R
+R
+R
+R
+R
+R
+R
+H
+H
+h
+F
+h
+F
+H
+h
+h
+, 
+ 
+where 
+(
+)
+=
+R
+R
+H
+H u
+ is the head corresponding to the state uR. From Eqs. (D.1) and (D.3), it is evident 
+that the relative head loss 
+#
+  through the hydraulic jump depends on 
+2
+R
+F  only. 
+
+In the case that the supercritical state uR is connected to the subcritical state u2 by a hydraulic 
+jump (i.e., when 
+#
+2 =
+R
+u
+u ) and u1 is critical (see Figure 4c), 
+2
+R
+F  coincides with 
+(
+)
+2
+jump
+K
+AR  (see 
+Appendix C) and the relative head loss 
+#
+  depends on AR only. 
+ 
+ 
+References 
+Akers B., Bokhove O. (2008) Hydraulic flow through a channel contraction: multiple steady 
+states, Physics of Fluids 20, 056601. Doi: 10.1063/1.2909659. 
+Aleksyuk A.I., Belikov V.V. (2019) The uniqueness of the exact solution of the Riemann 
+problem for the shallow water equations with discontinuous bottom, Journal of Computational 
+Physics 390, 232-248. Doi: 10.1016/j.jcp.2019.04.001. 
+Aleksyuk A.I., Malakhov M.A., Belikov V.V. (2022) The exact Riemann solver for the shallow 
+water equations with a discontinuous bottom, Journal of Computational Physics 450, 110801. Doi: 
+10.1016/j.jcp.2021.110801. 
+Austin L.H., Skogerboe G.V., Bennett R.S. (1970) Subcritical flow at open channel structures. 
+Open 
+channel 
+expansions. 
+Utah 
+State 
+University 
+Reports, 
+638. 
+https://digitalcommons.usu.edu/water_rep/638. 
+Bruwier M., Archambeau P., Erpicum S., Pirotton M., Dewals B. (2017) Shallow-water models 
+with anisotropic porosity and merging for flood modelling on Cartesian grids, Journal of Hydrology 
+554, 693-709. Doi: 10.1016/j.jhydrol.2017.09.051. 
+Buckingham E. (1914) On physically similar systems; illustrations of the use of dimensional 
+equations. Physical Review 4(4), 345-376. 
+Castro M.J., Pardo Milanés A., Parés C. (2007) Well-balanced numerical schemes based on a 
+generalized hydrostatic reconstruction technique, Mathematical Models and Methods in Applied 
+Sciences 17(12), 2055-2113. Doi: 10.1142/S021820250700256X. 
+
+Cea L., Vázquez-Cendón M.E. (2010) Unstructured finite volume discretization of two-
+dimensional depth-averaged shallow water equations with porosity, International Journal for 
+Numerical Methods in Fluids 63(8), 903-930. http://dx.doi.org/10.1002/fld.2107. 
+Chow V.T. (1959) Open channel hydraulics. McGraw-Hill, New York. 
+Cozzolino L., Castaldo R., Cimorelli L., Della Morte R., Pepe V., Varra G., Covelli C., Pianese 
+D. (2018a) Multiple solutions for the Riemann problem in the Porous Shallow water Equations, EPiC 
+Series in Engineering 3, 476-484. Doi: 10.29007/31n4. 
+Cozzolino L., Cimorelli L., Covelli C., Della Morte R., Pianese D. (2014) Boundary conditions 
+in finite volume schemes for the solution of shallow-water equations: the non-submerged broad-
+crested weir, Journal of Hydroinformatics 14(6), 1235-1249. Doi: 10.2166/hydro.2014.100. 
+Cozzolino L., Pepe V., Cimorelli L., D’Aniello A., Della Morte R., Pianese D. (2018b) The 
+solution of the dam-break problem in the Porous Shallow water Equations, Advances in Water 
+Resources 114, 83-101. Doi: 10.1016/j.advwatres.2018.01.026. 
+Cozzolino L., Pepe V., Morlando F., Cimorelli L., D’Aniello A., Della Morte R., Pianese D. 
+(2017) Exact solution of the dam-break problem for constrictions and obstructions in constant width 
+rectangular 
+channels, 
+ASCE 
+Journal 
+of 
+Hydraulic 
+Engineering 
+143(11), 
+04017047. 
+https://doi.org/10.1061/(ASCE)HY.1943-7900.0001368. 
+Cunge J.A., Holly H.M., Verwey A. (1980) Practical aspects of computational river hydraulics, 
+Pitman, Boston. 
+Dal Maso G., LeFloch P.G., Murat F. (1995) Definition and weak stability of nonconservative 
+products, Journal del Mathématiques Pures et Appliqués 74(6), 483-548. 
+Defina A. (2000) Two-dimensional shallow flow equations for partially dry areas, Water 
+Resources Research 36(11), 3251-3264. http://dx.doi.org/10.1029/2000WR900167. 
+Defina A., Susin F.M. (2006) Multiple states in open channel flow, in Vorticity and turbulence 
+effects in fluids structures interactions – Advances in Fluid Mechanics, M. Brocchini and F. Trivellato 
+eds., Wessex Institute of Technology Press, Southampton, 105-130. 
+
+Defina A., Viero D.P. (2010) Open channel flow through a linear contraction, Physics of Fluids 
+22(3), 036602. Doi: 10.1063/1.3370334. 
+Dewals B., Bruwier M., Pirotton M., Erpicum S., Archambeau P. (2021) Porosity Models for 
+Large-Scale 
+Urban 
+Flood 
+Modelling: 
+A 
+Review, 
+Water, 
+13(7), 
+960. 
+https://doi.org/10.3390/w13070960. 
+Ferrari A., Vacondio R., Dazzo S., Mognosa P. (2017) A 1d–2d shallow water equations solver 
+for discontinuous porosity field based on a generalized Riemann problem, Advances in Eater 
+Resources 107, 233-249. Doi: 10.1016/j.advwatres.2017.06.023 
+Finaud-Guyot P., Delenne C., Lhomme J., Guinot V., Llovel C. (2010) An approximate-state 
+Riemann solver for the two-dimensional shallow water equations with porosity, International Journal 
+for Numerical Methods in Fluids 62(12), 1299-1331. Doi: 10.1002/fld.2066. 
+Formica G. (1955) Esperienze preliminari sulle perdite di carico nei canali dovute a 
+cambiamenti di sezione, L’Energia Elettrica 32(7), 554-567 (in Italian). 
+Godlewski E., Raviart P.-A. (1996) Numerical approximation of systems of hyperbolic 
+equations, Springer, New York. 
+Guinot V., Delenne C. (2014) Macroscopic modelling of urban floods, La Houille Blanche 6, 
+19-25. Doi: 10.1051/lhb/2014058. 
+Guinot V., Delenne C., Rousseau A., Boutron O. (2018) Flux closures and source terms models 
+for shallow water models with depth-dependent integral porosity, Advances in Water Resources 122, 
+1-26. Doi: 10.1016/j.advwatres.2018.09.014. 
+Guinot V., Delenne C., Soares-Frazão S. (2022) Self-similar solutions of shallow water 
+equations 
+with 
+porosity, 
+IAHR 
+Journal 
+of 
+Hydraulic 
+Research. 
+Doi: 
+10.1080/00221686.2022.2106598. 
+Guinot V., Sanders B.F., Schubert J.E. (2017) Dual integral porosity shallow water model for 
+urban 
+flood 
+modelling, 
+Advances 
+in 
+Water 
+Resources 
+103, 
+16-31. 
+Doi: 
+10.1016/j.advwatres.2017.02.009. 
+
+Guinot V., Soares-Frazão S. (2006) Flux and source term discretization in two-dimensional 
+shallow water models with porosity on unstructured grids, International Journal for Numerical 
+Methods in Fluids 50(3), 309-345. Doi: 10.1002/fld.1059. 
+Hager W. (2010) Wastewater hydraulics: theory and practice. Springer, Berlin. 
+Hager W. H., Sinniger R. (1985) Flow characteristics of the hydraulic jump in a stilling basin 
+with an abrupt bottom rise. Journal of Hydraulic Research 23(2), 101–113. Doi: 
+10.1080/00221688509499359. 
+Han E., Handke M., Warnecke G. (2013) Criteria for non-uniqueness of Riemann solutions to 
+compressible duct flows, Zeitschrift für Angewandte Mathematik und Mechanik 93(6-7), 465-475. 
+Doi: 10.1002/zamm.201100176. 
+Han E., Warnecke G. (2014) Exact Riemann solutions to Shallow water Equations, Quarterly 
+of Applied Mathematics 72(3), 407-453. Doi: 10.1090/S0033-569X-2014-01353-3. 
+Ion S., Marinescu D., Ion A.V., Cruceanu S.G. (2022) Numerical scheme for solving a porous 
+Saint-Venant type model for water flow on vegetated hillslopes, Applied Numerical Mathematics 
+172, 67-98. Doi: 10.1016/j.apnum.2021.09.019. 
+Ippen A.T., Dawson J.H. (1951) High-velocity flow in open channels: a symposium: Design of 
+Channel 
+Contractions. 
+ASCE 
+Transactions 
+116(1), 
+326–346. 
+https://doi.org/10.1061/TACEAT.0006522. 
+Jung J. (2022) Development of exact solution and finite-volume method of porous shallow 
+water equations for the analysis of urban flooding, Seoul National University, PhD dissertation. 
+https://hdl.handle.net/10371/181177. 
+Karki K. S., Chander S., Malhotra, R. C. (1972) Supercritical flow over sills at incipient jump 
+conditions, 
+ASCE 
+Journal 
+of 
+the 
+Hydraulic 
+Division 
+98(10), 
+1753–1764. 
+Doi: 
+10.1061/JYCEAJ.0003435. 
+
+LeFloch P. (1989) Shock waves for nonlinear hyperbolic systems in nonconservative form, 
+Institute 
+for 
+Mathematics 
+and 
+Its 
+Applications 
+Preprint 
+Series 
+593, 
+Minneapolis. 
+http://hdl.handle.net/11299/5107. 
+Lhomme J. (2006) Modelisation des inondations en milieu urbain: approaches 
+unidimensionelle, bidimensionnelle et macroscopique. Hydrologie. PhD Thesis, Universite 
+Montpellier II – Sciences e Techniques du Languedoc. (in French). 
+Mohamed K. (2014) A finite volume method for numerical simulation of shallow water models 
+with porosity, Computer & Fluids 104, 9-19. Doi: 10.1016/j.compfluid.2014.07.020. 
+Özgen I., Liang D.-f., Hinkelmann R. (2016a) Shallow water equations with depth-dependent 
+anisotropic porosity for subgrid-scale topography, Applied Mathematical Modelling 40(17-18), 
+7447-7473. Doi: 10.1016/j.apm.2015.12.012. 
+Özgen I., Zhao J.-h., Liang D.-f., Hinkelmann R. (2016b) Urban flood modeling using shallow 
+water equations with depth-dependent anisotropic porosity, Journal of Hydrology 541(Part B), 1165-
+1184. Doi: 10.1016/j.jhydrol.2016.08.025. 
+Özgen I., Zhao J.-h., Liang D.-f., Hinkelmann R. (2017) Wave propagation speeds and source 
+term influences in single and integral porosity shallow water equations, Water Science and 
+Engineering 10(4), 275-286. Doi: 10.1016/j.wse.2017.12.003. 
+Sanders B.F., Schubert J.E., Gallegos H.A. (2008) Integral formulation of shallow-water 
+equations with anisotropic porosity for urban flood modelling, Journal of Hydrology 362(1-2), 19-
+38. Doi: 10.1016/j.jhydrol.2008.08.009. 
+Soares-Frazão S., Lhomme J., Guinot V., Zech Y. (2008) Two-dimensional shallow water 
+model with porosity for urban flood modelling, Journal of Hydraulic Research 46(1), 45-64. 
+Valiani A., Caleffi V. (2008) Depth-energy and depth-force relationships in open channel flows: 
+Analytical 
+findings. 
+Advances 
+in 
+Water 
+Resources 
+31(3), 
+447-454. 
+Doi: 
+10.1016/j.advwatres.2007.09.007.  
+
+Varra G., Pepe V., Cimorelli L., Della Morte R., Cozzolino L. (2020) On integral and 
+differential porosity models for urban flooding simulation. Advances in Water Resources 136. Doi: 
+10.1016/j.advwatres.2019.103455. 
+Varra G., Pepe V., Cimorelli L., Della Morte R., Cozzolino L. (2021) The exact solution to the 
+Shallow water Equations Riemann problem at width jumps in rectangular channels. Advances in 
+Water Resources 155. Doi: 10.1016/j.advwatres.2021.103993. 
+Varra G., Della Morte R., Gargano R., Cozzolino L. (2022) Porous Shallow water Equations 
+model with disambiguation of multiple solutions, Environmental Sciences Proceedings 21(1), 55. 
+Doi: 10.3390/environsciproc2022021055. 
+Vaschy A. (1892) Sur les lois de similitude en physique, Annales Tèlègraphiques 19, 25-28. 
+Viero D.P., Defina A. (2017) Extended theory of hydraulic hysteresis in open-channel flow, 
+Journal of Hydraulic Engineering 143(9), 06017014. Doi: 10.1061/(ASCE)HY.1943-7900.0001342. 
+Whitaker S. (1969) Advances in the theory of fluid motion in porous media, Industrial and 
+Engineering Chemistry 61(12), 14-28. Doi: 10.1021/ie50720a004. 
+Yarnell D.L. (1934) Bridge piers as channel obstructions, Technical Bulletin 444, US 
+Department of Agriculture, Washington. https://naldc.nal.usda.gov/download/CAT86200436/PDF. 
+
+Tables List 
+Table 1. Initial flow conditions of the validation Riemann problems. 
+Table 2. Coefficients for the polynomial interpolation of Eq. (11). 
+Table 3. Coefficients for the polynomial interpolation of Eq. (12). 
+Table 4. Supercritical 2-d flow impinging on a contraction for BR = 1 m and hR = 0.1, 0.5, and 1 m: 
+relative head losses * for the cases of flow passing through the discontinuity (G1). A hyphen 
+indicates the cases where a backwards moving shock is produced (G2 configurations). 
+ 
+Figures List 
+Figure 1. Physical interpretation of the porosity discontinuity between L  and R :  monotonic (a) 
+and non-monotonic porosity variation (b). 
+Figure 2. Internal description of the porosity discontinuity: plan view of the monotonic porosity 
+variation (a); profile view of smooth flow depth variation (b); profile view of flow depth 
+variation with hydraulic jump (c). 
+Figure 3. Field of occurrence of multiple solutions to the porosity Riemann problem for right 
+supercritical flows uR impinging a porosity reduction with 
+1
+
+
+=
+
+L
+R
+AR
+. Lower (continuous 
+line) and upper (dashed line) boundaries of the hysteresis domains. Hysteresis domains: A (no 
+multiple solutions), B (multiple solutions even in the case hL = 0), C (multiple solutions only 
+for hL > 0). 
+Figure 4. Flow conditions through the porosity discontinuity when multiple solutions to the purely 1-
+d SP Riemann problem are possible: profile view of solutions T1 (a), T2 (b) and T3 (c). 
+Figure 5. Plan view of the channel considered for 2-d SWE numerical simulations. Distorted 
+representation (measures in metres). 
+
+Figure 6. Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions 
+(dots) for the flow depth at time t = 5 s. Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) 
+with initial conditions in Table 1.  
+Figure 7. Plan view of the 2-d SWE numerical solution for Riemann problem 3 with initial conditions 
+in Table 1. Flow depth contours at time t = 5 s.  
+Figure 8. Example Riemann problem 5 with initial conditions in Table 1. Profile view for the flow 
+depth solution at time t = 5 s. 1-d SP exact solutions T1 (a), T2 (b) and T3 (c). Comparison 
+between the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d). 
+Figure 9. Example Riemann problem 6 with initial conditions in Table 1. Profile view for the flow 
+depth solution at time t = 5 s. 1-d SP exact solutions T1 (a), T2 (b) and T3 (c). Comparison 
+between the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d). 
+Figure 10. Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions 
+(dots) for the flow depth at time t = 5 s. Example Riemann problems 7 (a) and 8 (b) with initial 
+conditions in Table 1. 
+Figure 11. Plan view of the 2-d SWE numerical solution for Riemann problem 7 with initial 
+conditions in Table 1. Flow depth contours at time t = 5 s. 
+Figure 12. Plan view of the 2-d SWE numerical solution for Riemann problem 8 with initial 
+conditions in Table 1. Flow depth contours at time t = 5 s. 
+Figure 13. Plane view of the channel used for 2-d SWE numerical tests with supercritical flows. 
+Distorted representation (measures in metres). 
+Figure 14. 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 1 m impinging 
+a contraction: G1 configuration (black triangles), G2 configuration (white squares); upper 
+hysteresis domain limit (dashed line); modified upper boundary (thick black line). 
+Figure 15. Relative head losses for supercritical flows through a contraction: 2-d SWE numerical 
+results for G1 configuration (black triangles); limit relative head loss (continuous black line); 
+envelope of G1 data closer to the modified upper boundary of Figure 14 (continuous grey line). 
+
+Figure 16. Polynomial interpolation of the relative head loss data. Triangles represent the 
+experimental cases enveloped by a thin grey line in Figure 15. 
+Figure 17. Profile view of the 1-d SP exact solution with head loss through the geometric discontinuity 
+(continuous black line) and 2-d SWE numerical solution (dots) for the flow depth at time t = 5 
+s. Example Riemann problems 7 (a) and 8 (b) with initial conditions in Table 1. 
+Figure 18. Side view of two neighbouring cells in the 1-d computational domain: cell-averaged 
+quantities at a generic time level n (a); interface reconstructed variables used in the basic 
+reconstruction by Castro et al. (2007) (b); interface and in-cell reconstructed variables in the 
+novel reconstruction approach (c). 
+Figure 19. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with 
+the variable reconstruction by Castro et al. (2007) (continuous black line) and 2-d SWE model 
+(dots). Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with initial conditions in Table 
+1.  
+Figure 20. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with 
+the basic variable reconstruction by Castro et al. (2007) (continuous black line) and 2-d SWE 
+model (dots). Example Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in 
+Table 1.  
+Figure 21. Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with 
+the novel variable reconstruction (continuous black line) and 2-d SWE model (dots). Example 
+Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table 1. 
+Figure 22.  2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 0.1, 0.5, and 1 
+m impinging a contraction: G1 configuration (black triangles), G2 configuration (white 
+squares); modified upper boundary (thick black line). 
+ 
+ 
+ 
+
diff --git a/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/load_file.txt b/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d562c177480c10129fbc39e8996b709b839a8294
--- /dev/null
+++ b/wdE0T4oBgHgl3EQf-ALu/content/tmp_files/load_file.txt
@@ -0,0 +1,1696 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf,len=1695
+page_content='This work is licensed under the Creative Commons Attribution-NonCommercial-  NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' To view a copy of this license, visit  http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/licenses/by-nc-nd/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0/ or send a letter to Creative  Commons, PO Box 1866, Mountain View, CA 94042, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Coping with geometric discontinuities in porous shallow water models  Giada Varra1, Renata Della Morte2, Luigi Cimorelli3, Luca Cozzolino4  Abstract  Porosity-based models are a viable alternative to classical two-dimensional (2-d) Shallow water  Equations (SWE) when the interaction of shallow flows with obstacles is modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The exact solution  of the Single Porosity (SP) Riemann problem, which is the building block of numerous porosity  models solved with the Finite Volume method, exhibits an interesting feature, namely the multiplicity  of solutions when a supercritical flow impinges on a sudden porosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the present paper,  this ambiguity is overcome by systematically comparing the solution of the one-dimensional (1-d) SP  Riemann problem with the corresponding 2-d SWE numerical solutions at local porosity  discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' An additional result of this comparison is that the SP Riemann problem should  incorporate an adequate amount of head loss through porosity discontinuities when strongly  supercritical flows are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' An approximate Riemann solver, able to pick the physically  congruent solution among the alternatives and equipped with the required head loss amount, shows  promising results when implemented in a 1-d Single Porosity Finite Volume scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  1 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' of Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Parthenope Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Centro Direzionale di Napoli – Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' C4, 80143 Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' E-mail:  giada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='varra@uniparthenope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='it  2 Full Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' of Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Parthenope Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Centro Direzionale di Napoli – Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' C4, 80143 Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' E-  mail: renata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='dellamorte@uniparthenope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='it  3 Ass Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', DICEA, Federico II Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Via Claudio 21, 80125 Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' E-mail: luigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='cimorelli@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='it  4 Ass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' of Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Parthenope Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Centro Direzionale di Napoli – Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' C4, 80143 Napoli, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' E-  mail: luca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='cozzolino@uniparthenope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='it  CC  S  BY  NC  ND  Subject headings: Shallow water Equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Porous Shallow water Equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Single Porosity  Shallow water model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' urban flooding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Differential equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Riemann problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Corresponding author: Giada Varra  E-mail address: giada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='varra@uniparthenope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='it  Address: Dipartimento di Ingegneria, Università degli Studi di Napoli Parthenope, Isola C4, 80143 Napoli (Italy)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Introduction  Porosity-based shallow water models have been developed over the last two decades to provide a  viable alternative to classical two-dimensional (2-d) Shallow water Equations (SWE) for partially dry  areas and transitional environments (Defina 2000), large‐scale urban flood modelling (Guinot and  Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008) and runoff simulation on vegetated hillslopes (Ion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The available porosity shallow water models differ from each other by the conceptual  formulation and the underlying physical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' However, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2020) have  demonstrated that the Single Porosity (SP) model (Guinot and Soares-Frazão 2006), the Binary Single  Porosity model (BSP, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020), and the integral formulation of the SWE with obstacles by  Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008), constitute a family of models which share the same mathematical structure and  features such as hyperbolicity, presence of non-conservative products, and small disturbance  celerities coinciding with those exhibited by the 2-d SWE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Not surprisingly, the observation  that a common Finite Volume numerical framework can be used for their approximate solution  confirms their common mathematical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The Integral Porosity model (IP, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008),  the Dual Integral Porosity model (DIP, Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017), and the numerical schemes by Cozzolino  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018b) and Cea and Vázquez-Cendón (2010), are all examples of numerical schemes falling in  this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Due to the rapid variations of urban fabric density and flow characteristics through the urban  environment, the numerical fluxes over 2-d Finite Volume cell edges are usually calculated by solving  a local plane SP Riemann problem (Guinot and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Finaud-  Guyot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2010, Cea and Vázquez-Cendón 2010, Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018b, Jung 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The presence  of porosity discontinuities requires the definition of appropriate generalized Rankine-Hugoniot  conditions (LeFloch 1989, Dal Maso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1995), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', relationships between the flow variables at the  two sides of the porosity discontinuity that have a strong influence on the Riemann solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Numerical methods should incorporate the Rankine-Hugoniot conditions to reproduce the  corresponding Riemann exact solutions at porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Regarding these solutions, previous studies (Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018a, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020, 2021)  have shown that they present a fundamental ambiguity consisting in the appearance of multiple exact  solutions for certain initial conditions characterized by a supercritical flow impacting on a porosity  reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' At this point, two problems arise, namely the need i) to solve this ambiguity by finding the  unique physically congruent solution among the alternatives and ii) to construct a numerical scheme  able to reproduce the corresponding relevant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The one-dimensional (1-d) SP model formally  coincides with the 1-d SWE in rectangular channels with variable width, where the porosity symbol  substitutes the width symbol (Guinot and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This physical  analogy is exploited in the present paper to disambiguate the multiple 1-d SP Riemann exact solutions  by means of a systematic comparison with the corresponding 2-d SWE numerical solutions at local  geometric discontinuities, because the 1-d variable-width SWE model is nothing but a crude  simplification of the 2-d SWE model in a rectangular channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Besides the effects of friction (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018), shallow flows in urban environments may  dissipate energy by means of different mechanisms, such as the drag induced by obstacles (Sanders  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008), the propagation of bores reflected by buildings (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, 2018), and local  effects at geometric discontinuities (Guinot and Soares-Frazão 2006, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In porosity  models, the adoption of computational cells of greater size than that usually adopted in shallow water  models causes the loss of geometrical and hydraulic information, which in turn causes the  underestimation of the energy dissipated by the flow propagating through the urban fabric (Guinot et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, 2018, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' To reproduce missing dissipative effects, structural changes have  often been introduced in the original porosity shallow water models, for example altering the physical  momentum fluxes via reduction coefficients (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Laboratory (Akers and Bokhove 2008, Defina and Viero 2010) and 2-d SWE numerical  experiments (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020) show that supercritical flows suffer intense head loss across channel  contractions, implying that a corresponding energy dissipation must be experienced through rapid  porosity reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the present work, this energy dissipation is considered by appropriately  reformulating the generalized Rankine-Hugoniot conditions in a head-balance form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The introduction  of an interface head loss through the definition itself of porosity discontinuity has the advantage of  leaving the structure of the mathematical model unchanged and it can be very naturally used to take  into account, at least partly, the drag forces through urban fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' However, the amount of head loss  to be introduced across the discontinuity needs to be evaluated in a proper way, depending on the  flow characteristics across the geometric transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Also in this case, a systematic study of 2-d SWE  numerical results at isolated geometric discontinuities is conducted to supply the general conditions  under which this energy dissipation is present and how it can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  With the aim of reproducing the effects that in 2-d shallow water models are caused by the  flow interaction with isolated geometric discontinuities, the present work proposes a novel  approximate Riemann solver that discriminates the existence of multiple solutions and considers  adequate head loss in case of supercritical flows at porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This solver is  implemented in a 1-d Finite Volume scheme adopting the Single Porosity formulation of SWE  (Guinot and Soares-Frazão 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The capability of the 1-d numerical model with porosity of  reproducing the effects that in 2-d models are caused by the interaction between the flow and a  geometric transition is assessed against several Riemann problems by comparing the corresponding  results with the ones provided by a reference 2-d SWE numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The present paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The structure of the SP Riemann problem solution,  where the definition by Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018b) is used for generalized Rankine-Hugoniot conditions,  is discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This solution is validated in Section 3 using 2-d SWE numerical  experiments, and a novel definition of porosity discontinuity is given in Section 4 to better reproduce  the 2-d SWE numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Section 5, it is shown how it is possible to construct a numerical  model able to discriminate multiple solutions and introduce the requested head loss amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' These  findings are discussed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally, the paper is closed by a Conclusions section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Mathematical model  In the present Section, the plane Riemann problem for the SP model is reviewed, showing that it  reduces to a 1-d SP Riemann problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The corresponding solution requires the definition of  generalized Rankine-Hugoniot conditions to be used through porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Exploiting the  analogy between the 1-d SP model and the 1-d SWE in rectangular channels with variable width  (Guinot and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008), we introduce and discuss a head-balance form  defining this relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 The 1-d SP model  The plane SP model considered here is an augmented 1-d system obtained from the 2-d SP model  (Guinot and Soares-Frazão 2006) by setting to zero the derivatives with respect to the y-axis and  neglecting the flow resistance components (Ferrari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020):  (1)  2  2  2  0  0  2  2  0  h  hu  t  x  hu  gh  gh  hu  t  x  x  hv  huv  t  x  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a  \uf0b6  \uf0b6  \uf0ec  +  =  \uf0ef \uf0b6  \uf0b6  \uf0ef  \uf0e6  \uf0f6  \uf0b6  \uf0b6  \uf0b6  \uf0ef  +  +  −  =  \uf0ed  \uf0e7  \uf0f7  \uf0b6  \uf0b6  \uf0b6  \uf0e8  \uf0f8  \uf0ef  \uf0ef\uf0b6  \uf0b6  +  =  \uf0ef \uf0b6  \uf0b6  \uf0ee  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The solution of the corresponding Riemann problem is the building block for the computation  of interface numerical fluxes in shock capturing Finite Volume schemes (Godlewski and Raviart  1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1), the symbols have the following meaning: x and y are the space independent variables  of the inertial reference frame Oxy, while t is the time variable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' h(x, y, t) is the flow depth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' u(x, y, t)  and v(x, y, t) are the vertically averaged components of the flow velocity along x and y, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  g is the gravity acceleration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' and the porosity \uf06a(x, y) \uf0ce [0, 1] represents the fraction of urban area not  occupied by buildings and obstacles (storage porosity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the following, the dependence of the  variables on y will be omitted because all the quantities should be considered constant along y (plane  problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The effects of variable bed elevation are neglected here because the focus of the present  work is on obstacle modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Despite a distinction is often made in the urban hydrology literature between storage porosity  \uf06a and conveyance porosity \uf079 (Lhomme 2006, Guinot and Delenne 2014, Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017), the last  being related to the mass and momentum transport (Dewals et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021), the two definitions are  genuinely different only in porosity models written in integral form while they coincide in differential  models (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This result, which derives from a classical proof developed in the theory  of fluid motion in porous media (Whitaker 1969), states that the geometric parameter \uf06a in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1)  should always be interpreted not only as a storage but also as a conveyance porosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In Finite Volume schemes for the approximate solution of the 2-d SP model, the system of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1), where x is a local reference normal to the cell interface, is solved using initial discontinuous  conditions (Guinot and Soares-Frazão 2006, Soares-Frazão et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Cea and  Vázquez-Cendón 2010, Finaud-Guyot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2010, Özgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2016b, Özgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, Guinot et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The presence of the non-conservative product  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  \uf06a  \uf0b6  \uf0b6  gh  x , which models the force per  unit-width exerted by the obstacles on the flow through the cell interface, requires careful  mathematical and numerical treatment because it cannot be recast in divergence form (Cozzolino et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This point is central to the present discussion, and it will be further clarified in the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The first two relations of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1) do not contain the conserved variable hv and can be decoupled  from the third (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021), leading to the 1-d SP model (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2018b)  (2)  ( )  ( )  0  \uf06a  \uf06a  \uf06a  \uf0b6  \uf0b6  \uf0b6  +  +  =  \uf0b6  \uf0b6  \uf0b6  t  x  x  f u  u  h u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2), the meaning of the symbols is as follows:  (  )  =  T  h  hu  u  and  ( ) (  )  2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  =  +  T  hu  gh  hu  f u  are the vectors of the conserved variables and fluxes, respectively, of  the 1-d SWE model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' T is the matrix transpose symbols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ( ) (  )  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  =  −  T  gh  h u  is a vector  representing the hydrostatic thrust per unit-width exerted by obstacles on the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The 1-d system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) is at the core of the solution to the plane Riemann problem for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In fact, once that h and hu are known from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2), hv in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1) is readily computed with the  passive tracer equation (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021):  (3)  0  \uf0b6  \uf0b6  +  =  \uf0b6  \uf0b6  v  v  u  t  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Porosity Riemann problem  In the Riemann problem of the 1-d SP model, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) is solved under the following discontinuous  flow initial conditions and porosity  (4) (  )  ,  0  ,0  ,  0  \uf03c  \uf0ec  = \uf0ed  \uf03e  \uf0ee  L  R  x  x  x  u  u  u  ,  ( )  ,  0  ,  0  \uf06a  \uf06a  \uf06a  \uf03c  \uf0ec  = \uf0ed  \uf03e  \uf0ee  L  R  x  x  x  ,  where  (  )  =  T  L  L  L  L  h  h u  u  and  (  )  =  T  R  R  R  R  h  h u  u  are the states initially to the left and right of the  geometric discontinuity in x = 0, respectively, while \uf06aL  and \uf06aR  are the corresponding porosities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The  solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) with the initial conditions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (4) is self-similar and consists of a sequence of  constant states, the leftmost and rightmost of which are  L  u  and  R  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' All these states are in turn  connected by standing or moving waves (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Being the solution self-similar, it exists  a vector function  ( )  \uf078  w  of the scalar parameter \uf078 such that the Riemann problem solution can be  expressed as  (  )  (  )  ,  =  x t  x t  u  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This implies that the states  (  )  1  1  1  1  =  T  h  h u  u  and  (  )  2  2  2  2  =  T  h  h u  u  immediately to the left and right of the geometric discontinuity, respectively,  are constant in time  because they can be expressed as  (  )  ( )  1  0 ,  0  −  −  =  =  t  u  u  w  and  (  )  ( )  2  0 ,  0  +  +  =  =  t  u  u  w  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Based on the initial conditions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (4), the porosity is uniform to the left and right of the  geometric discontinuity in x = 0, implying that the non-conservative product  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  \uf06a  \uf0b6  \uf0b6  gh  x  is null  and the system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) is conservative for x < 0 and x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' It follows that the moving waves (shock  or rarefactions) coincide with those of the classic 1-d SWE model and that the shocks are defined by  the classic Rankine-Hugoniot conditions (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Vice versa, the non-conservative product  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  \uf06a  \uf0b6  \uf0b6  gh  x  is active through x = 0, implying that the classic Rankine-Hugoniot conditions cannot  be used at porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For this reason, the generalized Rankine-Hugoniot conditions  introduced by LeFloch (1989) and Dal Maso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1995) must be used to define an appropriate  relationship between u1 and u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the SP Riemann problem, the self-similarity of the solution implies  that this relationship is constant in time because u1 and u2 are constant in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Generalized Rankine-Hugoniot conditions at porosity discontinuities  Following the definition introduced by Dal Maso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1995) for hyperbolic systems of differential  equations with non-conservative products, the generalized Rankine-Hugoniot conditions across the  porosity discontinuity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) reduce to (Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018b, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='a)  2  2  1 1  0  \uf06a  \uf06a  −  =  R  L  h u  hu  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b)  (  )  2  2  2  2  2  2  2  1  1 1  1  2  ,  ,  ,  2  2  \uf06a  \uf06a  \uf06a \uf06a  \uf047  \uf0e9  \uf0f9  \uf0e9  \uf0f9  +  −  +  =  \uf0ea  \uf0fa  \uf0ea  \uf0fa  \uf0eb  \uf0fb  \uf0eb  \uf0fb  R  L  L  R  g  g  h  h u  h  hu  S  u u  ,  where  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf047  L  R  S  u u  is the force exerted on the flow by the obstacles across the unit-width porosity  discontinuity in x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='a) states that the unit-width discharge  \uf06a  =  Q  hu  is invariant through the  discontinuity, while Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b) states that the total thrusts to the left and right of the discontinuity are  balanced by the force  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf047  L  R  S  u u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The relationship between the states u1 and u2 is completely defined if a functional expression  for  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf047  L  R  S  u u  is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' However, this expression is somehow problematic because very  natural assumptions such as stagnant water and hydrostatic pressure distribution for the computation  of  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf047  L  R  S  u u  (Guinot and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Mohamed 2014, Guinot  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017) may lead to unphysical conditions where the flow acquires energy through the porosity  discontinuity (see the discussion in Chow 1959 and Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  To simplify the expression of the relationship between u1 and u2, we conveniently reformulate  the generalized Rankine-Hugoniot conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Appendix A shows that the force balance of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='a)  and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b) can be rewritten in the following head-balance form  (6)  (  )  (  )  (  )  2  2  1 1  2  1  1  2  0  ,  ,  ,  \uf06a  \uf06a  \uf06a \uf06a  −  =  −  = \uf044  R  L  L  R  h u  hu  H  H  H  u  u  u u  ,  where  ( )  (  )  2  2  =  +  H  h  u  g  u  is the head corresponding to the generic state u and  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  is the head loss through the porosity discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The relationship between the head loss  (  )  1  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  \uf06a \uf06a  \uf044  L  R  H  u u  and the force  (  )  1  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  \uf06a \uf06a  \uf047  L  R  S  u u  is given by (see Appendix A)  (7)  (  )  (  )  2  2  1  2  2  1  1  2  2  1  1  2  2  1  1 2  1  1  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  3  3  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  4  2  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a \uf06a  \uf06a \uf06a  \uf06a  \uf06a  \uf06a \uf06a  \uf047  \uf0e9  \uf0f9  +  \uf044  =  +  −  −  +  \uf0ea  \uf0fa  \uf0eb  \uf0fb  L  R  R  L  L  R  L  R  R  L  L  R  h  h  h  h  H  h  h  S  h  h  g  h h  u u  u u  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  implying that the choice of  (  )  1  2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  \uf06a \uf06a  \uf047  L  R  S  u u  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b) is equivalent to the choice of  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6), and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  From the mathematical point of view, the head-balance form is equivalent to the force-balance  form, but Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) is more convenient because it allows to easily verify the physical congruence of  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In fact, the flow energy cannot increase through the porosity discontinuity,  implying that the entropic condition  (8)  (  )  1  2  ,  ,  ,  0  \uf06a \uf06a  \uf044  \uf0a3  L  R  Q H  u u  ,  where  2  2  1 1  \uf06a  \uf06a  =  =  R  L  Q  h u  hu , must be verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The head-balance approach allows in principle to easily  introduce local effects at geometric discontinuities that do not explicitly appear in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1), such as  non-hydrostatic flow, viscosities, velocity variability along the vertical direction, flow depth and  velocity variability along the transverse directions, and the shape of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Not surprisingly,  existing definitions of channel internal boundary conditions such as width discontinuities and  junctions from the technical literature are usually given in terms of head loss (Formica 1955, Austin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1970, Cunge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1980, Hager 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  However, the system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) does not provide additional information to compute the head  loss  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  , implying that the hydraulic modeller should use external physical  knowledge for its definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This will be discussed in the following subsection, where the internal  structure of the porosity discontinuity between x = 0- and x = 0+ will be examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Channel analogy and porosity discontinuity definition  The 1-d SP model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) coincides with the 1-d SWE model in a rectangular channel with variable  width and horizontal bed, where the porosity symbol \uf06a takes the place of the width symbol B (Guinot  and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Like the porosity models written in  differential form, where the storage and conveyance porosity must coincide, the width B in  rectangular channels can be regarded both as i) the channel base-area per unit length (storage) and ii)  the transverse space available for the flow (conveyance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In addition, the non-conservative product  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  \uf06a  \uf0b6  \uf0b6  gh  x  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2), which represents the force per unit-width exerted by the obstacles on the  flow along x, acts like the corresponding term in the 1-d variable-width SWE model, where it  represents the force exerted on the flow by the channel walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the following, this channel analogy  will be exploited to supply a convenient expression for the head loss  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  through the  porosity discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Porosity variation through the discontinuity  Consider the bottom of Figure 1a, where a strip of unitary width modelling a simplified urban area  with obstacles is depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The strip is subdivided into two cells, left and right, respectively, with  different obstacle densities represented by the porosities \uf06aL  and \uf06aR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Obstacles are also present  through the cell interface in x = 0, with a density intermediate between those of the two adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The corresponding rectangular channel analogue is represented by the top channel of Figure 1a, where  the widths at the left and right ends are represented by \uf06aL  and \uf06aR , while a monotonic width variation  (channel contraction or expansion) connects the left and right reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, it is natural to  assume that the porosity \uf06a through the discontinuity between x = 0- and x = 0+ is described by a  monotonic function  ( )  \uf06a\uf047 s  in the interval s \uf0ce [0, 1], with  ( )  0  \uf06a  \uf06a  \uf047  =  L  and  ( )  1  \uf06a  \uf06a  \uf047  =  R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Figure 2a,  the internal structure of the porosity discontinuity between x = 0- and x = 0+ is exploded to show the  relationship between  ( )  \uf06a\uf047 s  and the parameter  \uf05b  \uf05d  0,1  \uf0ce  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A similar situation is depicted in Figure 1b, but now the obstacles density at the cells interface  is greater than those of the two adjacent cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The corresponding rectangular channel analogue is  represented by the top channel of Figure 1b, where a non-monotonic width variation (channel  constriction) connects the left and right reaches and  the porosity  ( )  \uf06a\uf047 s  through the discontinuity  varies non-monotonically between \uf06aL  and \uf06aR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Finite Volume schemes, a homogeneous porosity is assigned to each computational  cell and the actual obstacles distribution along the interface between two contiguous cells is canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  This implies that the simplest application of the channel analogy is to consider a rectangular channel  that is normal to the cell interface and symmetrical with respect to its longitudinal axis (as made in  Figures 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Thanks to the channel analogy, both the monotonic and non-monotonic choices of  ( )  \uf06a\uf047 s are  physically viable and lead to numerically stable computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Examples of models with monotonic  porosity variation through the discontinuity are contained in the works by Guinot and Soares-Frazão  (2006), Soares-Frazão et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008), Cea and Vázquez-Cendón (2010), Finaud-Guyot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2010),  Ferrari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017), and Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018a,b), while examples with a non-monotonic variation  are the models by Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008), Özgen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017), Bruwier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017), and Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (2017, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The preceding discussion suggests that the choice of the porosity discontinuity internal  structure can be made considering the underlying urban geometry at each cell interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This simple  observation, which supplies a physically congruent framework, contradicts the unproven statement  from the literature (Bruwier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, 2018, 2022) that only non-monotonic  descriptions of the porosity variation through the discontinuity are viable and stable (see the  corresponding discussion in Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For the sake of simplicity, only monotonic porosity variations through the discontinuity  with \uf06a  \uf06a  \uf0a3  L  R  will be considered in the rest of the paper (monotonic variations with \uf06a  \uf06a  \uf03e  L  R  can be  discussed by simply mirroring the local reference framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Having defined the aspect ratio  \uf06a  \uf06a  =  L  R  AR  , this implies that  1  \uf0a3  AR  will be assumed in the following developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Physical interpretation of the porosity discontinuity between \uf06aL  and \uf06aR :  monotonic (a) and  non-monotonic porosity variation (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Flow depth and velocity variation through the discontinuity  To complete the internal description of the porosity discontinuity, it is necessary to specify the  variation of flow depth and velocity between x = 0- and x = 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' It is assumed that this description is  a)  R  X  x  b)  PRsupplied by the function ( )  (  )  \uf047  \uf047  \uf047  =  T  s  h  h u  v  ,  where  ( )  \uf047h  s  and  ( )  \uf047  u  s , with s \uf0ce [0, 1], are the flow  depth and velocity through the discontinuity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The function ( )  s  v  is characterized by the  obvious congruency conditions ( )  1  0 =  v  u  and ( )  2  1 =  v  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Figures 2b and 2c, the internal structure  of the porosity discontinuity between x = 0- and x = 0+ is exploded to show two examples of the  relationship between the inner flow depth  ( )  \uf047h  s  and the parameter  \uf05b  \uf05d  0,1  \uf0ce  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) have proposed that the function ( )  s  v  is a stationary weak solution of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) through the porosity discontinuity, namely a solution of  (9)  ( )  ( )  0  \uf06a  \uf06a  \uf047  \uf047  +  =  d  d  ds  ds  f v  h v  in the interval  \uf05b  \uf05d  0,1  \uf0ce  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' If the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (9) exhibits no hydraulic jump (see the example of  Figure 2b, where  ( )  \uf047h  s  smoothly varies in the interval  \uf05b  \uf05d  0,1  \uf0ce  s  ), the relationship between the states  1  u  and  2  u  reduces to the conditions of discharge and total head invariance (see Appendix B)  (10)  (  )  (  )  2  2  1 1  2  1  0  0  R  L  h u  hu  H  H  \uf06a  \uf06a  −  =  −  =  u  u  ,  which is equivalent to set  (  )  1  2  ,  ,  ,  0  \uf06a \uf06a  \uf044  =  L  R  H  u u  in the head-balance form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the case that  the porosity varies monotonically between \uf06aL  and \uf06aR , the existence of a state u1 connected to u2 by  means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) depends on the aspect ratio AR \uf0a3 1 and on  (  )  2  F u  , where  ( ) =  F  u  gh  u  is the  Froude number related to the generic state u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The corresponding discussion is reported in Appendix  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  If the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (9) exhibits a hydraulic jump that reverts the incoming supercritical flow  into subcritical (see the example of Figure 2c), the total head is not invariant through the porosity  discontinuity and the corresponding head loss depends on the position of the hydraulic jump (see  Appendices C and D in Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The corresponding relationship between the states  1  u  and  2  u , which recovers the head-balance form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) with  (  )  1  2  ,  ,  ,  0  \uf06a \uf06a  \uf044  \uf03c  L  R  Q H  u u  , is not as simple  as that of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) and it is not reported here for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Internal description of the porosity discontinuity: plan view of the monotonic porosity  variation (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' profile view of smooth flow depth variation (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' profile view of flow depth variation  with hydraulic jump (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content="3 Definition of the Generalized Rankine-Hugoniot conditions  0  1  (a)  Td  PR  1  0'  0+  x  0  1  S  (b)  hr (s)  h  h  1  hr (s)  (c)  h,  0'  0*Having discussed the internal structure of the porosity discontinuity in the preceding sections, we  assume the following definition for the generalized Rankine-Hugoniot conditions:  Definition 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The relationship between u1 and u2, with  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  , is defined by the head-  balance form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) with the following internal description of the porosity discontinuity:  D1) the porosity varies monotonically between \uf06aL  and \uf06aR ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  D2) the variation of flow depth and velocity through the porosity discontinuity is defined by a weak  solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (9) in the interval  \uf05b  \uf05d  0,1  \uf0ce  s  , with ( )  1  0 =  v  u  and ( )  2  1 =  v  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  This definition automatically satisfies the entropic condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (8), while this is not true  for other porosity discontinuity definitions (Guinot and Soares-Frazão 2006, Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008,  Mohamed 2014, Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017) available in the literature (see the discussion in Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In addition, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2021) have demonstrated that the solution to the Riemann problem  of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) and (4) always exists if Definition 1 is used to establish the generalized Rankine-Hugoniot  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This fundamental result is not granted for alternative porosity discontinuity definitions  from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 Multiple solutions to the porosity Riemann problem  The solution to the 1-d SP Riemann problem of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) and (4), complemented by the generalized  Rankine-Hugoniot conditions of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3, always exists but there are cases, depending on the  initial conditions  L  u  and  R  u , where the solution is triple (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The field of occurrence  of multiple solutions will be explored in the following for the case  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  only (a similar  discussion for the case  1  \uf03e  AR  can be drawn by mirroring the reference framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The necessary (but not sufficient) condition for the existence of multiple solutions to the 1-d  SP Riemann problem with  1  \uf0a3  AR  is that the right state uR is directed from right to left (uR < 0) and  (  )  \uf03e  R  sp  F  K  AR , where  (  )  =  R  R  F  F u  is the Froude number corresponding to uR while the function  (  )  sp  K  AR  is defined in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this condition, the right supercritical flow uR impinging the  porosity reduction has energy greater than the minimum required to pass through the geometric  discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For this reason, it is possible to consider not only a solution where the right flow freely  passes through the discontinuity, but also solutions where the head of the incoming flow is partially  dissipated by means of a standing hydraulic jump through the porosity transition or by a shock that  moves backwards (Viero and Defina 2017, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021)  The theoretical limit curve  (  )  =  R  sp  F  K  AR , called lower boundary (LB) of the hysteresis  domains (Viero and Defina 2017), is represented in the plane (  R  F , AR) of Figure 3 with a black  continuous line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The necessary condition  (  )  \uf03e  R  sp  F  K  AR  for the existence of multiple solutions is  satisfied by the points falling in the regions denoted with B and C to the right of the LB curve in  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The regions B and C are separated by the curve  (  )  =  R  jump  F  K  AR , called upper boundary  (UB) of the hysteresis domains (Viero and Defina 2017), which is represented with a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The dimensionless function  (  )  jump  K  AR , which is characterised by  (  )  (  )  \uf0b3  jump  sp  K  AR  K  AR  for every  1  \uf0a3  AR  , is defined in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In the regions B and C, one or three different solutions to the 1-d SP Riemann problem of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (2) and (4) are possible, depending on the initial left state uL (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' When uL is such that  three alternative solutions (here called T1, T2, and T3) are possible, the solutions differ from each  other by the flow condition through the porosity discontinuity, as follows:  (T1) the state u2 immediately to the right of the porosity discontinuity coincides with the  supercritical flow uR, while the state u1 is supercritical and it is connected to u2 by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10),  namely by the conditions of discharge and total head invariance across the discontinuity (Figure 4a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (T2) the state u2 coincides with uR but a hydraulic jump is present through the porosity  discontinuity, and the state u1 is subcritical or critical, with H(u1) < H(u2) (Figure 4b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (T3) the supercritical flow uR is reverted into the subcritical state u2 by means of a backward  moving shock, with head loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' the state u1 is subcritical or critical and it is connected to u2 by means  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  When u1 is subcritical in the solutions T2 and T3, the flow through the geometric discontinuity  is submerged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', it is dominated by the tailwater h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The main difference between the regions B and  C in Figure 3 is the behaviour of the solutions to the Riemann problem when the state uL coincides  with the dry bed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', when  0  =  Lh  and there is no tailwater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the domain B, three distinct solutions  (T1, T2, and T3) are always possible for  0  =  Lh  , while a unique solution T1 occurs in the domain C  for  0  =  Lh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In other words, a triple solution is possible in the domain B even if there is not a  downstream tailwater able to force the establishing of a subcritical flow through the geometric  discontinuity (submerged flow), while such a tailwater is required for the existence of a triple solution  in the field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In a sense, the incoming flow falling in region C has always energy sufficient to flush  the hydraulic jump of Figure 4b out of the porosity discontinuity when the flow depth downstream is  null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The discussion is completed observing that the region A to the left of the curve LB,  characterised by  (  )  1\uf0a3  \uf0a3  R  sp  F  K  AR , refers to flow conditions where the Riemann problem always  admits a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the supercritical flow uR has not sufficient energy to pass  through the porosity reduction and the solution T3 must occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Field of occurrence of multiple solutions to the porosity Riemann problem for right  supercritical flows uR impinging a porosity reduction with  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Lower (continuous line)  and upper (dashed line) boundaries of the hysteresis domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Hysteresis domains: A (no multiple  solutions), B (multiple solutions even in the case hL = 0), C (multiple solutions only for hL > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow conditions through the porosity discontinuity when multiple solutions to the purely  1-d SP Riemann problem are possible: profile view of solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Validation of the channel analogy  Lowerboundary(LB)ofthehysteresisdomain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8-  Upper boundary (UB) of the hysteresis domain  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6-  R  c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  UB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content="2-  B  A  LB  0  1  FR  20T1  T2  1  (a)  1  1  (b)  ui  ui  1  U=UR  n=  1  supercritical  subcritical/  supercritical  critical  supercritical  0+  0'  0-  0+  T3  u2  (c)  1  uj  subcritical  UR  subcritical/l  1  critical  1  supercritical  777777  0'  0+The comparison between 1-d SP and 2-d SWE solutions is justified because the 1-d SP Riemann  problem is the main ingredient of 2-d SP Finite Volume schemes," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' which in turn are intended to  approximate the solution of 2-d SWE models with obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For this reason, the generalized Rankine-  Hugoniot conditions of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 are validated in the present section by comparing several 1-d  SP exact Riemann solutions with the corresponding 2-d SWE numerical solutions in a frictionless  horizontal rectangular channel with variable width, where a 2-d contraction is used to model the 1-d  sudden porosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' All the Riemann problems, whose initial conditions uL and uR with the  corresponding Froude numbers FL and FR are reported in Table 1, refer to porosity values  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  L  \uf06a =  and  1  R  \uf06a = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The exact Riemann solutions are computed with the methods discussed in Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Initial flow conditions of the validation Riemann problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Example  hL (m)  uL (m/s)  FL (-)  hR (m)  uR (m/s)  FR (-)  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='16  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='30  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='64  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='23  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='15  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='51  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='15  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='30  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='41  The rectangular channel considered for the 2-d SWE computations has length L = 200 m with  a left reach of width BL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='60 m and a right reach of width BR = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00 m (see Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The left and  right channel reaches are separated by a symmetric linear expansion whose length is Lc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The  2-d SWE computations are accomplished using the Finite Volume scheme described in Cozzolino et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) on unstructured triangular grid whose average side is \uf044s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='50 m at the channel ends and  \uf044s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='05 m at the linear expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The flow depth h computed at time t = 5 s with the 1-d SP exact  solution (continuous black line) is compared in Figures 6, 8, 9, and 10, with the corresponding 2-d  shallow water numerical results (white dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the channel considered for 2-d SWE numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Distorted  representation (measures in metres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The results of Riemann problem 1 are represented in Figure 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact solution presents  two shocks moving to the left and right of the geometric discontinuity, respectively, while subcritical  flow conditions that preserves discharge and energy invariance are established through x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Figure  6a shows a good correspondence between the 1-d exact and 2-d numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In particular,  the 1-d model accurately captures the strength of the wave at x = 0, together with the strength and  position of the shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The intermediate states u1 and u2 immediately to the left and right of the  porosity discontinuity computed with the 1-d exact solution nicely correspond to those provided by  the 2-d SWE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A slightly different picture can be drawn for the solution of Riemann problem 2, represented  in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact solution exhibits a resonant condition where a rarefaction is attached to  the left of the porosity discontinuity, while a shock and a rarefaction are both moving to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The state u1 immediately to the left of x = 0 is critical and accelerates through the rapid geometric  transition becoming supercritical with preservation of energy and discharge invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In turn, the  supercritical state u2 issuing from the channel expansion pushes the slowly moving shock to the right  Lc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='00  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='60  II R  B  B x=-100  x=0  x= 100of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' With reference to the left rarefaction, Figure 6b shows a good correspondence between the  1-d exact and 2-d numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This representation is less satisfactory with reference to the right  moving shock, whose shape in the 2-d model is strongly influenced by its vicinity to the channel  expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Similarly, the 2-d rarefaction moving on the right is quite smoothed with respect to the 1-  d exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Despite these discrepancies, the intermediate state between the shock and the  rarefaction computed with the 1-d exact solution satisfactorily corresponds to the 2-d numerical  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions  (dots) for the flow depth at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with  initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In the 1-d exact solution of Riemann problem 3, the supercritical states u1 and u2 are connected  by the conditions of discharge and head invariance while the state u2 is separated from uR by means  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  (a)  1  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  (u)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  h  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  I I  0  0  20  0  20  10  0  10  20  30  × (m)  x (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  (c) (d)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  (w)  (w)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  h  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  &  0  40  20  0  20  40  80  40  0  40  () x  () xof an intermediate state and two moving shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The comparison with the 2-d SWE results shows  that the strength and position of the right moving shock and the left shock position are satisfactorily  captured by the 1-d exact solution, while the flow depth of the state between the two shocks supplied  by the1-d model is not too far from the 2-d solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 2-d free surface profile view in Figure 6c  exhibits a complex pattern whose plane view is represented in Figure 7, which shows that the  supercritical flow accelerating through the expansion originates a system of transverse oblique  shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact solution represents this pattern with a single average flow depth, and this  explains the discrepancies between 1-d and 2-d models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Nonetheless, the 1-d model captures the  general picture of the 2-d SWE solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 3 with initial conditions  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In Figure 6d, the results of Riemann problem 4 are represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact solution consists  of two rarefactions to the left of x = 0, with the formation of dry bed between the waves, and of an  additional rarefaction to the right of the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The subcritical flow immediately to the right  of x = 0 (state u2) accelerates through the discontinuity becoming critical immediately to the left (state  u1) and preserving the invariance of discharge and energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The comparison between the 1-d exact  and 2-d SWE numerical results shows that the former captures the strength of the 2-d waves, together  with the flow depth of the states encompassed by the waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  h (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1  (m)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0  7  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='05  x (m)The example of Figure 8 is particularly interesting because it corresponds to Riemann problem  5 with three exact solutions, where AR and  R  F  fall in the hysteresis domain B of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Figure  8a, b, c, the three exact solutions T1, T2, and T3, respectively, are represented (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In  Figure 8d, the superposition between the exact solution T3 and the 2-d SWE numerical results shows  a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This demonstrates that the under-determination of the 1-d Riemann problem can  be eliminated by resorting to the 2-d SWE model, which takes into account the transverse flow  variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problem 5 with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view for the flow  depth solution at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1-d SP exact solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Comparison between  the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The example of Figure 9 corresponds to Riemann problem 6 with three exact solutions, where  AR and  R  F  fall into the hysteresis domain C of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Figure 9a, b, c, the three exact solutions  T1, T2, and T3, respectively, are represented (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Figure 9d, the superposition  4  (a)  3  h  2  4  1  (d)  0  4  3  I  (b)  E  h  h  2  2 1  1  1  I  1  0  1 4  1  (c)  1  3  h  1  2  0  60  40  20  0  20  1  x (m)  1  0  09-  40  20  0  20  x (m)between the T3 exact solution and the 2-d SWE numerical results shows again a good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The comparison between figures 8d and 9d shows that the 2-d SWE model preferably picks up the  solution with a shock moving backwards when three exact solutions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problem 6 with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view for the flow  depth solution at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1-d SP exact solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Comparison between  the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In the example of Figure 10a, the state uL is such that Riemann problem 7 has one exact  solution of type T1, despite AR and  R  F  fall in the hysteresis domain C of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Correspondingly,  the 2-d SWE solution is characterised by a strong interaction with the geometric discontinuity and by  a supercritical flow to the left of x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The comparison between 1-d and 2-d solutions shows that the  number of moving waves supplied by the 1-d exact solution is correct, but their strength and position  is very different from those of the 2-d numerical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In addition, the flow depth of the  supercritical states to the left of x = 0 is poorly captured by the 1-d exact solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 2-d SWE  6  (a)  E  4  h  6  (d)  0  6  1  1  (b)  4  1  (w)  1  4  1  h  1  1  2  h  1  1  1  2  0  1  6  (c)  1  4  h  1  0  2  60  40  20  0  20  x (m)  0  60  40  20  0  20  x (m)numerical solution exhibits a strong interaction with the channel walls in x = 0, with the formation of  a system of oblique shocks whose plan view is represented in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' These shocks, which are  typical of supercritical flows in contractions (Ippen and Dawson 1951, Akers and Bokhove 2008,  Defina and Viero 2010), introduce intense head loss that explains the discrepancies between 1-d and  2-d solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Similar observations can be made for Riemann problem 8, for which AR and  R  F  fall in the  hysteresis domain C of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact solution is unique and characterized by flow  conditions T1 through the porosity reduction (Figure 10b), while the corresponding 2-d SWE solution  exhibits a supercritical flow to the left of x = 0 and a strong interaction with the contraction (Figure  12) where intense head loss is produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  From the validation process above, some considerations can be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The exact solutions to  the 1-d SP Riemann problem, computed with the monotonic porosity discontinuity model of Section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3, compare well with the corresponding 2-d SWE numerical solutions in case of subcritical flow  through the porosity discontinuity (Figures 6a,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Overall, the head loss through the discontinuity  seems negligible when the flow is subcritical, but a caveat to this observation will be discussed in the  next Section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Similarly, the 1-d exact model shows a good behaviour in both the multiplicity domains B and  C when three exact solutions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the solution characterized by subcritical flow  through the porosity discontinuity satisfactorily agrees with the 2-d SWE numerical results (Figures  8d and 9d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A minor discrepancy is present when the supercritical flow accelerates through a porosity  increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the 2-d SWE numerical solution is somehow distorted with respect to the 1-d  exact solution (Figures 6b,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A very different picture is evident when the 1-d exact model predicts a single solution  characterized by supercritical flow through a porosity reduction (Figures 10a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the 2-d  SWE model exhibits a supercritical flow through the contraction, but the head loss introduced by a  2-d system of oblique shocks makes the 1-d and 2-d solutions very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This dissipative  mechanism has been discussed by Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2020) for the first time in the context of Riemann  problems on dry bed, but it has been verified here for the general Riemann problem on wet bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The validation process accomplished in this section suggests that the results of a 2-d SWE  numerical model could be systematically used to disambiguate multiple Riemann problem solutions  and evaluate the head loss caused by a supercritical flow passing through a porosity reduction,  improving the Definition 1 of the generalized Rankine-Hugoniot conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This will be made in the  next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions  (dots) for the flow depth at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 7 (a) and 8 (b) with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  6  (a)  2  (b)  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5 -  4 (m)  一  :f  1  d  h  (w)  2  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  一  ul  一  1  0  0  80  40  0  80  60  40  20  0  20  x (m)  x (m)  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 7 with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 8 with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Construction of novel generalized Rankine-Hugoniot conditions  Consider a horizontal frictionless rectangular channel L = 60 m long with a single width discontinuity  at its centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The channel consists of a right and a left reach of different widths, connected by a linear  contraction whose walls are inclined by 45° with respect to the channel axis (see Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  h (m)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  (m)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  x (m)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  h (m)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  x (m)contraction is short enough to be regarded as a true geometric discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In order to perform 2-d  SWE simulations with different aspect ratio AR = BL/ BR values, the right reach width is fixed to BR  = 1 m, while the left reach width BL (with BL < BR) is varied in each test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Free-slip boundary conditions  are imposed at the channel walls, while the left and right ends are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' A non-uniform unstructured  triangular mesh is used for simulations, with average side \uf044s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='20 m at the channel ends and \uf044s =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='02 m in the vicinity of the geometric transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plane view of the channel used for 2-d SWE numerical tests with supercritical flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Distorted representation (measures in metres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The 2-d Finite Volume SWE numerical model by Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) is used for  approximating the solution of 159 different 2-d Riemann problems in the channel of Figure 13, where  AR \uf0ce [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The tests are characterized by supercritical flow uR (with uR < 0) approaching the  contraction and Froude number  \uf05b  \uf05d  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5,25  R  F \uf0ce  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In all the tests, the right flow depth is hR = 1 m and  the corresponding velocity uR varies accordingly to FR, while the initial left state uL coincides with  the dry bed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Simulations are run until steady state conditions are reached through the contraction  (generally after t = 20 s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The 1-d SP exact solution is triple (T1, T2, or T3) for the initial conditions falling in region B  of Figure 3, while a single T1 solution is predicted for points falling in region C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The examination of  numerical results shows that two distinct types of 2-d SWE solutions occur:  II  R  B  30  30 x= -30  x=0  x= 30(G1) The supercritical flow entering the geometric discontinuity passes with the formation of  a complicate system of oblique shocks through the contraction like in Figures 10a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' this set of  numerical solutions exhibits a behavior that is somehow in between the exact solutions T1 and T2  defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 (Figures 4a,b), because supercritical flow conditions are present at the outlet  of the contraction as in T1, but there is also head loss as in T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (G2) The flow through the contraction is subcritical, while a moving shock propagates  upstream;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' since the channel bed downstream is initially dry, critical flow conditions are established  through the contraction outlet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' this set of numerical solutions clearly recalls the exact solutions of  type T3 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 (Figure 4c), where u1 is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Modified upper boundary of the hysteresis domain  In Figure 14, the 2-d numerical cases corresponding to solution types G1 (black triangles) and G2  (white squares) are plotted in the plane (  R  F ,  AR), where the upper hysteresis domain limit is also  represented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Figure 14 shows that, for a given value of the aspect ratio AR, it exists a limit Froude  number  (  ) K  AR  such that a G2 solution is obtained for  (  ) \uf0a3  R  F  K  AR , while a G1 solution is  obtained for  (  ) \uf03e  R  F  K  AR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The locus of the points separating the fields of G1 and G2 solutions is  the modified upper boundary curve with equation  (  ) =  R  F  K  AR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Ideally, this curve represents the  situations for which a standing hydraulic jump is present at the entrance of the contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For  (  ) \uf03e  R  F  K  AR , the incoming flow has energy sufficient to push the jump through the contraction,  where it is broken into a complicate pattern of transverse standing waves (Figures 11 and 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Conversely, the incoming flow is not able to sustain the hydraulic jump for  (  ) \uf03c  R  F  K  AR , and a  shock moves backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The modified upper boundary curve, represented in Figure 14 with a thick black line, is very  close to the UB curve defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 (dashed line curve in Figure 14) for moderate width  jumps (AR > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5), whereas it departs from the UB curve for strong width jumps (AR \uf0a3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This is  not surprising, because the 1-d theory for width and porosity transitions is expected to work better  when AR is close to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The polynomial interpolation of data supplies for the limit  (  ) =  R  F  K  AR  the expression  (11)  (  )  (  )  6 1  =  =  \uf0e5  i  jump  i  i  K  AR  K  AR  m AR ,  whose coefficients mi are reported in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  We observe that  (  )  (  ) \uf03e  jump  K  AR  K  AR  for AR > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, meaning that the incoming flow requires  greater energy to push the hydraulic jump through the porosity discontinuity with respect to the case  without energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This effect, which is due to the modest head loss related to the subcritical flow  through the geometric transition in G2 solutions, will be taken into account numerically without a  direct evaluation of the energy losses in subcritical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 1 m impinging  a contraction: G1 configuration (black triangles), G2 configuration (white squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' upper hysteresis  domain limit (dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' modified upper boundary (thick black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Coefficients for the polynomial interpolation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  m1  m2  m3  m4  m5  m6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='9448  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8030  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2944  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1172  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='7583  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8122  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Head loss for supercritical flows at contractions  The 2-d SWE solutions of type G1 exhibit a head loss \uf044H* through the channel contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This head  loss is evaluated as  (  ) 1  \uf044  =  −  R  H  H  H  u  , where 1  H  is an estimate of the head corresponding to the  state immediately to the left of the contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The quantity 1  H  is indirectly deduced by evaluating  the supercritical flow depth to the left of the contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  Upper boundary (UB)  Modified upper boundary (MUB)  A^^ 2-d SWE numerical Gl solution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6 -  2-d SWE numerical G2 solution  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  口  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  口  口  口  口  口口口  口  一  UB  口  口  口  口  MUB  0  25The relative head loss  (  ) \uf044 = \uf044  R  H  H u  corresponding to the 2-d SWE solutions of type G1  is represented with black triangles in the plane ( 2  ,  \uf044  R  F ) of Figure 15, where the data corresponding  to the same  AR value are connected by a dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Figure 15 shows that the relative head loss \uf044  moderately varies with  2  R  F  for a given value of AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This implies that \uf044  mainly depends on the  characteristics of the geometric transition, allowing to use a simplified expression in the form  (  ) \uf044 = \uf044  AR , where a single \uf044  value has been attributed to each  AR value by picking the numerical  data closer to the modified upper boundary of Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' These points are connected by a continuous  grey line in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Relative head losses for supercritical flows through a contraction: 2-d SWE numerical  results for G1 configuration (black triangles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' limit relative head loss (continuous black line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  envelope of G1 data closer to the modified upper boundary of Figure 14 (continuous grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ^  2-d SWE numerical Gl solution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  AA  AR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  △AR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  △AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  △AR= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 -  AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='75  AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='9  0  10  F,2  100  1000  1  RIn the same figure, the limit relative head loss  (  )  #  #  \uf044 = \uf044  R  H  H u  is also represented with a  thick black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The quantity  #  \uf044H  is the head loss through the shock in the exact solution of type T3  (see Figure 4c) when the celerity of the shock is null (standing hydraulic jump) and the state u1 is  critical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', when  (  )  =  R  jump  F  K  AR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In Appendix D, it is shown that the limit relative head loss  #  \uf044  depends on AR only and the exact expression of  (  )  #  #  \uf044 = \uf044  AR  is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The ratio  # \uf044  \uf044  is  represented in Figure 16 for different values of AR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This leads to the following polynomial  interpolation  (12)  (  )  (  )  2 2  0  #  =  \uf044  = \uf044  \uf0e5  i  i  i  AR  AR  m AR ,  with interpolation coefficients in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Recalling that  2 =  R  u  u  in the G1 solutions, from the position  (  ) 1  2  ,  ,  ,  \uf06a \uf06a  \uf044  = \uf044  L  R  H  H  u u  it  follows that the head loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) for supercritical flow through a channel contraction, equivalent  to a porosity reduction, can be rewritten as  (13)  (  )  (  )  (  ) 1  2  2  ,  ,  ,  \uf06a \uf06a  \uf06a  \uf06a  \uf044  =  \uf044  L  R  L  R  H  H  u u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Coefficients for the polynomial interpolation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  m0  m1  m2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='536  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='403  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='668  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Polynomial interpolation of the relative head loss data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Triangles represent the  experimental cases enveloped by a thin grey line in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 Novel generalized Rankine-Hugoniot conditions  From the preceding discussion, it is possible to give a novel convenient relationship between u1 and  u2 at porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The relationship between u1 and u2, with  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  , is defined by the head-  balance form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) with the following internal description of the porosity discontinuity:  D1) the porosity varies monotonically between \uf06aL  and \uf06aR ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  D2) the variation of flow depth and velocity through the porosity discontinuity is defined by a weak  solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (9) in the interval  \uf05b  \uf05d  0,1  \uf0ce  s  , with ( )  1  0 =  v  u  and ( )  2  1 =  v  u ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  D3) the state u2 with u2 < 0 is supercritical only if  (  )  (  ) 2  \uf03e  F  K  AR  u  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' in this case, the relationship  between u1 and u2 is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) where  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  2 -  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  △# 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  1  AR2To demonstrate the viability of the Definition 2, the exact solution to Riemann problems 7 and 8 of  Table 1 is now found using the novel generalized Rankine-Hugoniot conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d exact  solutions are represented with a black line in Figure 17, where the corresponding 2-d solution is  represented with dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The comparison with Figure 10, where the exact solutions are obtained with  null energy loss, shows that introducing an appropriate definition of  (  )  1  2  ,  ,  ,  \uf06a \uf06a  \uf044  L  R  H  u u  reduces the  discrepancy between the exact 1-d SP exact solution and the corresponding 2-d SWE numerical  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact solution with head loss through the geometric  discontinuity (continuous black line) and 2-d SWE numerical solution (dots) for the flow depth at  time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 7 (a) and 8 (b) with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Numerical model  In the present Section, the solution of the 1-d SP system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2), where the initial conditions  (  )  ( )  x  x  0  0,  u  u  =  and the porosity distribution \uf06a(x) are specified, is approximated by means of the Finite  6  (a)  2  (b)  一  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5 -  4  一:  一  (m)  一  l  1  d  h  (m)  ()  2  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  c:l  0  0  80  40  0  80  60  40  20  0  20  x (m)  x (m)Volume method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Having partitioned the 1-d physical domain into non-overlapping cells  \uf05b  \uf05d  2  1  2  1 ,  +  −  =  i  i  i  x  x  C  of uniform length  2  1  2  1  −  +  −  =  \uf044  i  i  x  x  x  , we assume that the averaged quantities  (14)  ( )  (  )  ( ) (  )  1  1  ,  ,  \uf06a  \uf06a  \uf06a  =  =  =  \uf044  \uf044  \uf0f2  \uf0f2  i  i  T  n  n  n  n  n  i  i  i  i  i  C  C  i  x dx  h  h u  I x  x t  dx  x  x  u  u  are approximations in Ci of \uf06a(x) and (  )  , n  x t  u  , respectively, where  t  n  t n  \uf044  =  is the time level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In  Figure 18a, the cell-averaged constant values of \uf06a(x) and (  )  , n  x t  u  are conceptually depicted, showing  the geometric and flow discontinuity at cells interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  If  1  n  n  i  i  t  t  t  +  \uf044 =  −  is the time step length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' the solution is advanced in the generic cell by means  of the following explicit first-order scheme  (15)  (  )  (  )  1  1 2  1 2  1 2  1 2  1 2  1 2  1 2  1 2  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  \uf079  \uf079  \uf06a  \uf06a  +  −  +  −  +  +  −  +  +  +  −  −  −  −  +  \uf044  \uf044  \uf0e9  \uf0f9  \uf0e9  \uf0f9  =  −  −  +  +  \uf0eb  \uf0fb  \uf0eb  \uf0fb  \uf044  \uf044  n  n  i  i  i  i  i  i  i  i  i  i  i  i  t  t  x  x  u  u  g u  u  g u  u  s  s  where the symbols are defined as follows:  1 2  i  \uf079 +  is a numerical approximation of the porosity at the  interface i+1/2 between Ci and Ci+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (  )  ,  g u v  is a numerical flux corresponding to the 1-d SWE model  in a constant width channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' finally,  (  )  1 2  1 2  1 2  1 2  T  i  i  i  i  h  h  u  −  −  −  −  +  +  +  +  =  u  and  (  )  1 2  1 2  1 2  1 2  T  i  i  i  i  h  h  u  +  +  +  +  +  +  +  +  =  u  are  flow variables reconstructed to the left and right of the interface i+1/2, respectively, which are  involved in the computation of numerical fluxes and non-conservative product approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The  quantities  (  )  1 2  1 2  0  +  +  −  −  =  T  i  is  s  and  (  )  1 2  1 2  0  −  −  +  +  =  T  i  is  s  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (15) are the contributions to Ci of the  non-conservative products arising from the porosity gradient through the interfaces in xi-1/2 and xi+1/2,  respectively, and their computation depends on the variable reconstruction adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The 1-d SWE  numerical flux (  )  ,  g u v  is approximated here by means of the HLLE Riemann solver (Cozzolino et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2014), although a different exact or approximate SWE Riemann solver could be used as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In its essence, the numerical scheme above consists of the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' First, the  porosity and flow interface variables are reconstructed from the cell-averaged values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Second, the  numerical fluxes and porosity gradients contributions at interfaces are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally, the cell-  averaged variables are advanced in time by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Clearly, the algorithm used to calculate  the reconstructed interface variables from the cell-averaged values determines the properties of the  scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In the following, we assume without loss of generality that  1  \uf06a  \uf06a +  \uf0a3  i  i , corresponding to  1  1  \uf06a \uf06a +  =  \uf0a3  i  i  AR  (the procedure for the case  1  \uf06a  \uf06a +  \uf03e  i  i  is easily obtained by mirroring the local  reference framework).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The basic variable reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) is first recalled, and  then it is modified to introduce head losses through porosity discontinuities and cope with the case of  multiple solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally, the results of the 1-d SP numerical scheme, with both the basic and novel  interface variable reconstructions, are compared with the corresponding 2-d SWE numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Side view of two neighbouring cells in the 1-d computational domain: cell-averaged  quantities at a generic time level n (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' interface reconstructed variables used in the basic  reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' interface and in-cell reconstructed variables in the novel  reconstruction approach (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Basic reconstruction (Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2007)  (a) 1  u\'  1  βi+1  Pi  Xi-1/2  xi  Xi+1/2  Xi+1  Xi+3/2  x  1  u,+1/2l u+1/2  1  (b) u"  1  1  1Wi+1/2  Pi+1  Pi  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-1/2  1x  Xi+1/2  Xi+1  Xi+3/2  x  1 u,+1/2l u+1/2  1  (c) R + 1  1  ui  1  1  4i+1/2  Pi+1  Pi  X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-1/2  Xi  Xi+1/2  Xi+1  Xi+3/2  x  Ci+1  C,The well-balanced reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007), originally implemented for the 1-d variable-  width SWE model, is intended to capture steady state solutions where the discharge and head are  uniform through the space domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Aiming at this, the interface variables  1 2  −  +  iu  ,  1 2  +  +  iu  , and  1 2  \uf079 +  i  , are  connected to the cell-averaged variables by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10), keeping the character of the flow  (subcritical or supercritical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The reconstruction approach is schematically depicted in Figure 18b,  where the porosity discontinuity internal structure is zoomed in to show the variables used for  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Given the right state  1  +  n  iu  , the following inequalities are checked (see Appendix C):  (16)  (  )  (  )  1  +  \uf03c  n  i  sb  F  K  AR  u  ,  (  )  (  )  1  +  \uf03e  n  i  sp  F  K  AR  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  With reference to these checks, two options are possible, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) If one of the two inequalities in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (16) is satisfied, the right state  1  +  n  iu  can be  connected to a state on the interface left side by the conditions of discharge and head  invariance (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the interface porosity  1 2  \uf079  \uf06a  +  =  i  i  is assumed,  and the state  (  )  1 2  1 2  1 2  1 2  T  i  i  i  i  h  h  u  +  +  +  +  +  +  +  +  =  u  is easily found by solving the system  (17)  (  )  (  )  1  1  1  1 2  1 2  1 2  1  1 2  0  0  \uf06a  \uf079  +  +  +  +  +  +  +  +  +  +  +  −  =  −  =  n  n  i  i  i  i  i  i  n  i  i  h u  h  u  H  H  u  u  ,  which is obtained by assuming in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) the positions  1 2  \uf06a  \uf079 +  =  L  i  ,  1  \uf06a  \uf06a +  =  R  i ,  1  1 2  +  +  =  i  u  u  , and  2  1  +  =  n  i  u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (17) admits two exact solutions, one corresponding to  a subcritical state and the other to a supercritical state (see Valiani and Caleffi 2008 for  the corresponding exact expressions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The first is chosen if the state  1  +  n  iu  is subcritical,  otherwise the supercritical one is kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally, the position  1 2  −  +  =  n  i  i  u  u  is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2) If the inequalities of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (16) are not satisfied, the system of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (17) admits no  solution with  1 2  \uf079  \uf06a  +  =  i  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case,  1 2  +  +  iu  is found by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (17) where the  interface porosity  1 2  \uf079 +  i  is defined as  (18)  (  )  (  )  3 2  1 2  1  1  2  1  3  2  \uf079  \uf06a  +  +  +  +  \uf0e6  \uf0f6  \uf0e7  \uf0f7  =  \uf0e7  \uf0f7  +  \uf0e8  \uf0f8  n  i  i  i  n  i  F  F  u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  This choice is equivalent to imposing that  1 2  +  +  iu  is critical (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The state  1 2  −  +  iu  is calculated by means of  (19)  (  )  (  )  1 2  1 2  1 2  1 2  0  0  \uf079  \uf06a  −  −  +  +  +  −  +  −  =  −  =  n  n  i  i  i  i  i  i  n  i  i  h  u  h u  H  H  u  u  ,  which is obtained by assuming in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) the positions \uf06a  \uf06a  =  L  i ,  1 2  \uf06a  \uf079 +  =  R  i  ,  1 =  n  i  u  u ,  and  2  1 2  −  +  =  i  u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The subcritical solution is kept if the state  n  iu  is subcritical, otherwise the  supercritical solution is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The state  1 2  −  +  iu  certainly exists because  1 2  \uf06a  \uf079 +  \uf03c  i  i  (see  Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  From the preceding, it is evident that the reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) satisfies the  inequality  \uf07b  \uf07d  \uf07b  \uf07d  1  1 2  1  min  ,  max  ,  \uf06a \uf06a  \uf079  \uf06a \uf06a  +  +  +  \uf0a3  \uf0a3  i  i  i  i  i  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', it ensures the monotonicity of the porosity  variation through the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The algorithm is completed by using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='a)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b) to express  1 2  +  −  is  and  1 2  −  +  is  as  (20)  (  )  (  )  (  )  (  )  1 2  1 2  1 2  1 2  1 2  1 2  \uf06a  \uf079  \uf079  \uf06a  +  +  −  −  −  −  −  +  +  +  =  −  =  −  n  i  i  i  i  i  n  i  i  i  i  i  s  f u  f u  s  f u  f u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Novel variable reconstruction  The novel variable reconstruction differs from that by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) because the case of a  supercritical flow impinging on a porosity reduction is treated in a separate way, congruently with  the novel definition of Rankine-Hugoniot conditions given in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This is accomplished by  computing an in-cell additional reconstructed state  (  )  ,  ,  ,  ,  1 2  1 2  1 2  1 2  +  +  +  +  +  +  +  +  =  T  R  R  R  R  i  i  i  i  h  h  u  u  to manage the case of  a backwards moving shock between the geometric transition and the state  1  +  n  iu  when a T3 solution  (Figure 4c) occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In addition, appropriate head loss is introduced to compute the state  1 2  +  +  iu  if the  flow through the porosity reduction is supercritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The novel reconstruction approach is  schematically depicted in Figure 18c, showing the in-cell additional reconstructed variable  ,  1 2  +  +  R  iu  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Given the right state  1  +  n  iu  , the cases  1  0  + \uf03c  n  iu  and  1  0  + \uf0b3  n  iu  are treated separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) If  1  0  + \uf0b3  n  iu  , the procedure by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) is used to find  1 2  \uf079 +  i  ,  1 2  −  +  iu  , and  1 2  +  +  iu  (see points CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 and CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, there is no backwards moving  shock and the position  ,  1 2  1  +  +  +  =  R  n  i  i  u  u  is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2) If  1  0  + \uf03c  n  iu  , the quantity  (  )  1  +  n  i  F u  is compared to  (  )  sb  K  AR  and  (  ) jump  K  AR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Three  cases are possible:  NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) If  (  )  (  )  1  +  \uf03c  n  i  sb  F  K  AR  u  occurs, the energy of the subcritical flow is sufficient to  pass through the porosity reduction, and the point CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 supplies  1 2  \uf079 +  i  ,  1 2  −  +  iu  , and  1 2  +  +  iu  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The position  ,  1 2  1  +  +  +  =  R  n  i  i  u  u  is made because there is no backwards moving shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2) If  (  )  (  ) 1  +  \uf03e  n  i  jump  F  K  AR  u  , the energy of the supercritical flow is sufficient to pass  through the porosity reduction with head losses (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the interface  porosity  1 2  \uf079  \uf06a  +  =  i  i  is assumed and the state  1 2  +  +  iu  is easily found by picking the  supercritical solution of the system  (21)  (  )  (  )  (  )  1  1  1  1 2  1 2  1 2  1  1 2  1 2  1  1 2  1  0  ,  ,  ,  \uf06a  \uf079  \uf079  \uf06a  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  −  =  −  = \uf044  n  n  i  i  i  i  i  i  n  n  i  i  i  i  i  i  h u  h  u  H  H  H  u  u  u  u  ,  which is obtained by assuming in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) the positions  1 2  \uf06a  \uf079 +  =  L  i  ,  1  \uf06a  \uf06a +  =  R  i ,  1  1 2  +  +  =  i  u  u  ,  and  2  1  +  =  n  i  u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The head loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (21) is computed using the Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (12) and (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally,  the positions  1 2  −  +  =  n  i  i  u  u  and  ,  1 2  1  +  +  +  =  R  n  i  i  u  u  are made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  NR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3) If  (  )  (  )  (  ) 1  +  \uf0a3  \uf0a3  n  sb  i  jump  K  AR  F  K  AR  u  , the energy of the state  1  +  n  iu  is either  insufficient to pass through the porosity reduction or a multiple solution is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In both  the cases, the Riemann problem solution is characterized by a backwards moving shock  radiating from the geometric discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The occurrence of this shock is forced by  posing  1 2  \uf079  \uf06a  +  =  i  i  and assuming that the states  1 2  +  +  iu  and  ,  1 2  +  +  R  iu  have the same discharge of  the state  1  +  n  iu  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The state  1 2  +  +  iu  is critical, obtaining  (22)  1  1  1  1 2  1 2  1 2  1 2  1 2  0  \uf06a  \uf079  +  +  +  +  +  +  +  +  +  +  +  +  −  =  = −  n  n  i  i  i  i  i  i  i  i  h u  h  u  u  gh  ,  while the state  ,  1 2  +  +  R  iu  is subcritical with (  )  (  )  , 1 2  +  +  = −  R  i  jump  F  K  AR  u  , obtaining  (23)  (  )  ,  ,  1  1  1 2  1 2  ,  , 1 2  1 2  0  +  +  +  +  +  +  +  +  +  +  −  =  = −  n  n  R  R  i  i  i  i  R  R  i  i  jump  h u  h  u  u  gh  K  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Finally, the position  1 2  −  +  =  n  i  i  u  u  is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The novel reconstruction satisfies the inequality  \uf07b  \uf07d  \uf07b  \uf07d  1  1 2  1  min  ,  max  ,  \uf06a \uf06a  \uf079  \uf06a \uf06a  +  +  +  \uf0a3  \uf0a3  i  i  i  i  i  ,  ensuring the monotonicity of the porosity discontinuity inner description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Having introduced the in-  cell additional reconstructed state  ,  1 2  +  +  R  iu  , the definition of  1 2  +  −  is  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (20) changes as follows:  (24)  (  )  (  )  ,  1 2  1 2  1 2  1 2  R  i  i  i  i  i  \uf06a  \uf079  +  +  +  −  −  −  −  =  −  s  f u  f u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A similar reconstruction approach has been proposed by Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022), where an iterative  procedure is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Nonetheless, the present reconstruction is an improvement because (i) the iterative  procedure is avoided and (ii) it is possible to consider the cases where  (  )  (  ) \uf03e  jump  sp  K  AR  K  AR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In  addition, adequate head loss is introduced for supercritical flows through abrupt porosity reductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 Numerical experiments  The 1-d numerical model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (15), equipped with the variable reconstructions described in  Sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2, respectively, is used to approximate the solution of the Riemann problems with  initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For the sake of simplicity,  x  \uf044  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='20 m and  t  \uf044  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='005 s in all the  numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Numerical experiments with the reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007)  Figure 19 shows the numerical results (flow depth) to the Riemann problems 1-4 supplied by the 1-d  SP model with the reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (continuous black line), while the  corresponding 2-d SWE results are represented with white dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For all these problems, the algorithm  by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) captures the essentials of the 2-d SWE solution, namely the number of waves  and their strength, and the flow depth of the intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The discrepancies between the 1-d  and 2-d solutions in Figures 19b and 19c (Riemann problems 2 and 3, respectively) correspond to  those discussed with reference to the comparison between Riemann exact solution and 2-d numerical  solution (compare with Figures 6c and 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model  with the variable reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (continuous black line) and 2-d SWE model  (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The numerical results to Riemann problems 5-8 are represented in Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' These cases,  characterized by a supercritical flow through a porosity reduction, show that the 1-d SP numerical  solution with the basic reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) greatly differs from the corresponding  2-d SWE reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The Figures 20a,b refer to Riemann problems (5 and 6, respectively) admitting multiple  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' While the 2-d SWE model exhibits a backwards moving shock originated from the porosity  discontinuity that causes flow energy dissipation, the 1-d SP numerical model with variable  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  (a)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  3RXIX EOVYX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  (w)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8-  h  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  0  20  0  20  10  0  10  20  30  (u) x  x (m)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  1  (c)  (d)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  h  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  1  1  0  0  40  20  0  20  40  80  40  0  40  x (m)  x (m)reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) captures the solution with supercritical flow through the  discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This is expected because the reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) keeps the  supercritical character of the flow impinging on the porosity reduction (point CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  From the preceding, it follows that the numerical scheme by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) overestimates the  discharge through the geometric discontinuity and the celerity of the advancing shock on the left,  while completely neglects the energy dissipation mechanism connected with the backwards moving  shock generated by the interaction of the propagating flow with obstacles (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Varra  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2020) have demonstrated that the same defect is shared by other Riemann solvers such as those  by Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018b) and Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The Figures 20c,d refer to Riemann problems (7 and 8, respectively) where a supercritical  flow impinges on a porosity reduction, but the solution is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In these cases, the 1-d SP numerical  model misses to capture the 2-d SWE model solution because it lacks an appropriate dissipation  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Interestingly, this is the same discrepancy found when comparing the 1-d exact solution  and the 2-d SWE numerical solution (see Figures 10a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model  with the basic variable reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (continuous black line) and 2-d SWE  model (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Numerical experiments with the novel reconstruction  The numerical experiments presented in the preceding subsection are repeated using the 1-d SP  numerical model with with the novel reconstruction of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For the Riemann problems 1-4,  the numerical results supplied by the novel reconstruction, which are not reported here for the sake  of brevity, coincide with those supplied by the basic reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This is  4  6  (a)  (b)  1  1:  9  4  三2  (w)  h  h  2  1  1  1  0  0  60  40  20  0  20  60  40  20  0  20  (u) x  x (m)  6  1  (c)  2  (d) 1  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  1  4  1  1  (w)  (u) d  h  h  :3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  :1  0  0  80  40  0  80  60  40  20  0  20  x (m)  x (m)expected, because these Riemann problems do not refer to cases where a supercritical flow impinges  on a porosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The Figures 21a,b refer to the Riemann problems 5 and 6, respectively, which admit multiple  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Contrary to the algorithm by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007), the novel variable reconstruction captures  the solution with the backwards moving shock exhibited by the 2-d SWE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' It is clear that the  introduction of the in-cell subcritical state  ,  1 2  +  +  R  iu  between the geometric transition and the right state  n  iu , together with the computation of the interface reaction term by means of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (24), is the ingredient  allowing the computation of the physically congruent shock immediately to the discontinuity right-  side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The comparison between the 1-d SP numerical results obtained with the novel variable  reconstruction and the 2-d SWE results for Riemann problems 7 and 8, is represented in Figures  21c,d, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The inspection of these figures, referring to cases where the supercritical flow  impinging on the porosity reduction remains supercritical with loss of energy, shows that the novel  variable reconstruction satisfactorily reproduces the 2-d SWE model results because the required  amount of head loss through the geometric discontinuity is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model  with the novel variable reconstruction (continuous black line) and 2-d SWE model (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example  Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Discussion  In this section, the model presented in the preceding sections is discussed with reference to alternative  numerical and conceptual approaches available in the literature, and with reference to the robustness  of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  4  6  (a)  (b)  3 -  4  三2  (w)  h  h  2  1  1  一  0  0  60  40  20  0  20  60  40  20  0  20  (w) x  x (m)  6  1  (c)  2  1  (d)  1 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  1  4  1  1  (w)  (u)  h  h  :3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5  :1  1  0  0  80  40  0  80  60  40  20  0  20  x (m)  x (m)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 Comparison with the transient momentum dissipation approach  The momentum dissipation approach introduced with the DIP numerical model by Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (2017) is a numerical device intended to introduce the transient energy dissipation generated by bore  reflection at obstacles during transient propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the 1-d case, the DIP model can be written as  (25)  (  )  (  )  1  1 2  1 2  1 2  1 2  1 2  1 2  1 2  1 2  ,  1 2  ,  1 2  ,  ,  \uf079  \uf079  \uf06a  \uf06a  +  −  +  −  +  +  −  +  +  +  +  −  −  −  −  −  +  \uf044  \uf044  \uf0e9  \uf0f9  \uf0e9  \uf0f9  =  −  −  +  +  \uf0eb  \uf0fb  \uf0eb  \uf0fb  \uf044  \uf044  n  n  i  i  i  i  i  i  i  i  i  i  sta i  sta i  i  i  t  t  x  x  u  u  M  g u  u  M  g u  u  s  s  ,  where M is a momentum dissipation matrix defined as (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017)  (26)  1  0  0  1  \uf06d  \uf0e6  \uf0f6  = \uf0e7  \uf0f7  −  \uf0e8  \uf0f8  M  ,  with \uf06d > 0 in case  1  + \uf03e  n  n  i  i  h  h , while \uf06d = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The momentum dissipation coefficient \uf06d  appearing in the matrix M of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (26) must be calibrated using fine grid 2-d SWE simulations and it  is strongly dependent on the urban fabric structure and flow conditions (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (25), the interface porosity  1 2  \uf079 +  i  is evaluated from the underlying urban fabric with a  non-monotonic approach, which enforces the condition  (  )  1 2  1  min  ,  \uf079  \uf06a \uf06a  +  +  \uf0a3  i  i  i  (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' the  interface contributions  ,  1 2  +  −  sta i  s  and  ,  1 2  −  +  sta i  s  of the non-conservative products are computed under the  assumption of in-cell stagnant water (Guinot and Soares-Frazão 2006) as  (27)  (  )  (  ) (  )  (  )  (  ) (  )  2  ,  1 2  1 2  2  ,  1 2  1 2  0  1  2  0  1  2  \uf06a  \uf079  \uf079  \uf06a  +  −  −  −  +  +  =  −  =  −  T  n  sta i  i  i  i  T  n  sta i  i  i  i  g h  g h  s  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Finally, the reconstructed variables are defined as:  (28)  (  )  (  )  1 2  1 2  1 2  1  1  1  1  1 2  \uf06a  \uf079  \uf06a  \uf079  −  +  +  +  +  +  +  +  +  +  =  =  T  n  n  n  i  i  i  i  i  i  T  n  n  n  i  i  i  i  i  i  h  h u  h  h u  u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  We reinterpret the momentum dissipation approach observing that the 1-d DIP numerical  model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (25) can be rewritten in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (15) by defining the interface contributions of  the non-conservative products as  (29)  (  ) (  )  (  ) (  )  1 2  ,  1 2  1 2  1 2  1 2  1 2  1 2  ,  1 2  1 2  1 2  1 2  1 2  ,  ,  \uf079  \uf079  +  +  −  +  −  −  −  −  −  −  −  −  −  +  +  +  +  +  +  +  =  +  −  =  +  −  i  sta i  i  i  i  i  i  sta i  i  i  i  i  s  s  M  I g u  u  s  s  I  M  g u  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  According to this reinterpretation, the momentum dissipation approach is equivalent to  evaluating the forces exerted by obstacles at cell interfaces by adding or subtracting a dynamic  contribution to the stagnant water hydrostatic thrusts  ,  1 2  +  −  sta i  s  and  ,  1 2  −  +  sta i  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' A slightly different  definition of the matrix M has been subsequently given in Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018), but the role of M as  a modulator of the forces exerted by obstacles at cell interfaces remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The reinterpretation supplied by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (29) allows to recognize the common numerical  framework of the DIP model (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017) and of the numerical model presented here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Nonetheless, the exact solution of the 1-d SP Riemann problem has been exploited in the present  paper to introduce a mechanism of energy dissipation caused by the reflection of advancing waves at  porosity discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This approach, which has been accomplished by isolating a single local  porosity discontinuity and comparing the corresponding 1-d SP and 2-d SWE Riemann solutions (see  Section 3), avoids the intricacies caused by the mutual interaction of waves radiating from the  obstacles of complex urban fabrics and allows to consider the local geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In addition, it avoids  the introduction of a momentum dissipation mechanism of unclear physical meaning, whose  parameters need calibration on a case-by-case basis (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2 Disambiguation of the porosity Riemann problem  The issue of disambiguating multiple solutions to the Riemann problem where a geometric  discontinuity is present has been tackled by researchers considering different types of geometric  discontinuities or different fluid models (SWE or Euler equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' It is interesting to compare the  results obtained in the present paper with those available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The exact solution to the Riemann problem for the 1-d SWE model with variable bed elevation  exhibits two classes of triple solutions (for convenience, the solutions are called here S1, S2, and S3)  when a supercritical flow impinges on a positive bed step (Han and Warnecke 2014, Aleksyuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In the first class of triple solutions, S1 is characterised by a supercritical flow that jumps over  the bed step remaining supercritical, while S2 is characterised by a hydraulic jump located through  the discontinuity that reverts the incoming supercritical flow into subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Finally, S3 is  characterised by a backward shock while subcritical flow conditions are established over the bed step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The second class of triple solutions differs from the first one because the solution S3 is characterised  by a backward shock with blockage of the flow at the bed step (step higher than the free surface level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2014) used a mix of steady state laboratory data (Karki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1972, Hager and  Sinniger 1985) and physical reasoning to establish a disambiguation criterion based on discharge  minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Following this criterion, the physically relevant solution is S3 (backward shock) in  both the classes of triple solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Aleksyuk and Belikov (2019) found the same result by considering  a mathematical argument based on the continuous dependence of solutions on the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2013) considered the Riemann problem for the 1-d Euler equations in a  compressible duct flow, where triple solutions may occur when a supersonic flow impinges on a pipe  diameter reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' They compared some examples of 1-d exact multiple solutions with the numerical  results supplied by a higher dimensions axisymmetric Euler equations model (longitudinal and radial  direction), founding that the physically relevant solutions were those characterized by a backward  shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  A pattern seems to emerge from these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In all the examples considered, multiple  solutions occur when a supercritical flow (SWE model) or a supersonic flow (Euler equations) impact  on a cross-section reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Despite the variety of mathematical models and means applied for the  disambiguation of multiple exact solutions (laboratory and/or numerical experiments, mathematical  arguments), the common output to the different procedures is that the physically relevant solution  among the alternatives is the one characterised by a backward shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In SWE models, this is also the  solution which minimizes the discharge through the geometric discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The results presented in Section 4, which confirm this pattern for the 1-d SP model, can be  reinterpreted using an argument based on the continuous dependence of the Riemann problem  solution on the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In fact, when the tailwater is null (hL = 0), the 1-d SP Riemann  problem exhibits three exact solutions (T1, T2, and T3) in the region B of Figure 3, which is bounded  by the curves LB and UB, and one exact solution in regions A and C (T3 and T1, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' A  solution with backward shock (T3 of Figure 4c) in region B of Figure 3 can move through LB to a  solution T3 of region A by decreasing the initial Froude number  R  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' On the other side, the same T3  solution in region B can move through UB to a T1 solution of region C by increasing  R  F  , flushing  the hydraulic jump and establishing supercritical flow conditions through the porosity discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In conclusion, T3 solutions should be considered to the left of UB, and T1 solutions to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Of  course, the head loss generated by transverse shocks through the porosity discontinuity does not  significantly changes this picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the limit curve UB is distorted, becoming the modified  limit curve MUB of Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Indeed, the initial conditions to the left of the MUB curve characterize  solutions with a backward shock (G2 solutions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1), while the initial conditions to the right  characterize supercritical flows through the porosity discontinuity (G1 solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3 Influence of flow depth and channel width on the porosity discontinuity definition  The numerical experiments of Section 4 have been conducted considering a fixed depth hR = 1 m of  the flow impinging on a channel contraction and fixed width BR = 1 m of the right channel reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The  reader may wonder if these results can be extended to different values of hR and BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The answer to  this question is affirmative but it requires a brief discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Recalling that viscosity and density are not modelled by the incompressible SWE model, the  most general way to write the expression of the head loss \uf044H* suffered by a supercritical flow uR  through the contraction is  (30)  (  ) ,  ,  ,  , ,  \uf044  =  L  R  R  R  c  H  f B  B  h u  g L  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  We observe that n = 7 physical quantities are involved in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (30), and we want to ascertain  if the physical equation can be simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Aiming at this, we additionally see that only k = 2  fundamental mechanics units, namely length and time, are involved because there is not dependency  on density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Recalling the Vaschy-Buckingham theorem (Vaschy 1892, Buckingham 1914), it follows  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (30) can be rewritten as an expression involving n – k = 5 dimensionless independent  quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The dimensionless quantities chosen are  (31)  (  ) 2  ,  ,  ,  ,  2  \uf073  \uf06c  −  \uf044  =  =  \uf044 =  =  =  +  L  R  R  R  L  R  R  R  R  R  R  c  R  B  u  h  B  B  H  AR  F  B  h  u  g  B  L  gh  ,  and it is easy to verify that they are mutually independent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', it is not possible to build one of the  dimensionless quantities starting from the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This justifies why it is possible to simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (30)  as  (32)  (  ) 2  ,  ,  ,  \uf073  \uf06c  \uf044 =  R  R  f  AR F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In a similar manner, a very general way to describe the boundary MUB between the G1 and  G2 solutions discussed in Section 4 is to introduce a limit velocity R  u  discriminating the two types of  solution and expressing this velocity as  (33)  (  ) ,  ,  , ,  =  R  L  R  R  c  u  f B  B  h  g L  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  This physical equation involves n = 6 physical quantities and k = 2 fundamental units,  implying that it can be simplified as  (34)  (  ) ,  ,  \uf073  \uf06c  =  R  K  f  AR  where =  R  R  K  u  gh .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  We observe that the dependence on \uf06c does not need to be explicited because this parameter is  constant in all the 2-d SWE simulations, since the contraction walls are always inclined by 45° with  respect to the channel axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' With reference to the parameter \uf073 R , the comparison between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (32)  and (12), and between Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (34) and (11), respectively, show that the expressions found in Section 4  are valid in the case  1  \uf073 =  R  because hR = 1 m and BR = 1 m in all the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Nonetheless, we demonstrate that the parameter\uf073 R  is superfluous because the expressions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (32) and (34) can be safely simplified in the form of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (12) and (11), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Consider the steady state 1-d SWE in a channel of variable width B = B(x), with solution u =  u(x) for given right boundary condition uR:  (35)  ( )  ( )  0  +  =  dB  dB  dx  dx  f u  h u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Despite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (35) is a simplification of the 2-d flow through the contraction, it supplies a  sufficient insight for the present discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' We observe that the multiplication of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (35) by the  constant k allows to write  (36)  ( )  ( )  0  \uf0e6  \uf0f6  +  =  \uf0e7  \uf0f7  \uf0e8  \uf0f8  dB  dB  k  dx  dx  f u  h u  ,  which is still satisfied by the solution u = u(x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=" Of course, k can be moved inside the  derivative symbols, leading to  (37)  ( )  ( )  '  '  0  +  =  dB  dB  dx  dx  f u  h u  ,  where B’(x) = kB(x)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In other words, the solution u = u(x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (35) for given boundary condition  uR does not change if the width is uniformly amplified by a constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The uniform amplification of  the width affects the parameter \uf073 =  R  R  R  h  B  but does not affect the parameters  =  L  R  AR  B  B  and  =  R  R  R  F  u  gh , implying that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (32) and (34) depend on AR and  2  R  F  but not on \uf073 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  To evaluate the robustness of this theoretical approach, we consider twelve additional 2-d  SWE simulations with initial and geometrical conditions as follows: AR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  R  F  = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6, 6, 8,  11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' hR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, 1 m, and BR = 1 m (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The values chosen for hR correspond to the three  different values (\uf073 R  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, and 1) of the dimensionless parameter \uf073 R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Six of the chosen flow  conditions correspond to points slightly to the left of the MUB curve and six to the right (see Figure  22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Table 4 reports, for each simulation, the dimensionless head loss \uf044* if the solution is of type G1,  while a hyphen indicates a G2-type solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The inspection of these results shows that the solution  types (G1 or G2) expected based on the position with respect to the MUB curve are those actually  occurring in the 2-d simulations, while the dependence of the relative head loss \uf044* on \uf073 R is  negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' These observations confirm the theoretical arguments above and justify the application of  the formulas in Section 4 to conditions with \uf073 R  \uf0b9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Supercritical 2-d flow impinging on a contraction for BR = 1 m and hR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, and 1 m:  relative head losses \uf044* for the cases of flow passing through the discontinuity (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' A hyphen  indicates the cases where a backwards moving shock is produced (G2 configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  hR (m)  AR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='60  \uf07cFR\uf07c  \uf07cFR\uf07c  8  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='38  1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='38  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8  Modified upper boundary (MUB)  A^^ 2-d SWE numerical Gl solution  hr (m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5 - 1  2-d SWE numerical G2 solution  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6 -  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4  hr (m) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5 - 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2  0  FR  25Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, and 1  m impinging a contraction: G1 configuration (black triangles), G2 configuration (white squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  modified upper boundary (thick black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Conclusions  The solution of the Riemann problem associated to the Single Porosity (SP) Shallow water Equations  (SWE) model by Guinot and Soares-Frazão (2006) is the main ingredient for the computation of  interface fluxes and obstacle reaction terms in the Binary SP model (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020) and in the  integral approach (Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008, Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Previous studies (Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2018a,  Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020, 2021) have shown that the SP Riemann problem presents a fundamental ambiguity  consisting in the appearance of multiple exact solutions for certain initial conditions characterized by  a supercritical flow impacting on a porosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This observation prompts the definition of the  unique physically congruent Riemann solution among the alternatives and the construction of a  numerical scheme able to reproduce this relevant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Having recognized that the 1-d SP model is nothing but a crude simplification of the 2-d SWE  model in a variable width rectangular channel, the channel analogy (Guinot and Soares-Frazão 2006,  Sanders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2008) has been exploited in this paper to disambiguate the multiple 1-d SP Riemann  solutions by means of a systematic comparison with the corresponding 2-d SWE numerical solutions  at local geometric discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The conclusion, which corresponds to other similar results obtained  in the literature for different mathematical models (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2013, Cozzolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2014, Aleksyuk  and Belikov 2019), is that the solution with a backwards moving shock is physically congruent when  multiple solutions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Laboratory (Akers and Bokhove 2008, Defina and Viero 2010) and  2-d SWE numerical experiments (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2020) show that supercritical flows in channels suffer  intense head loss through a width contraction, which corresponds to a porosity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This  phenomenon is an additional cause of energy dissipation in porosity models with respect to those  already described in the literature (Guinot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Also in this case, the systematic study  of 2-d SWE numerical results at isolated geometric discontinuities has supplied the general conditions  under which this energy dissipation is present and how it can be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Based on a modification of the generalized hydrostatic reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007),  we have also built an approximate Riemann solver that discriminates the existence of multiple  solutions and is able to add adequate head loss in the case of supercritical flow through porosity  discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The comparison between the numerical results supplied by the novel 1-d SP model  and the 2-d SWE model shows that the former can reproduce the effects that in 2-d models are caused  by the interaction between a supercritical flow and a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' This promising numerical approach  could be extended to other cases of hyperbolic systems of differential equations where multiple  solutions arise such as the SWE and the porous SWE with variable topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Aknowledgements  Renata Della Morte and Luca Cozzolino want to acknowledge the financial support from the project  "Floods in cities: new insights for integrating pluvial flooding into flood risk management plans  (INSPIRING)", funded by the Italian Ministry of University and Research under the national  programme PRIN2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Head-balance form of the porosity discontinuity  If  1 1  2 2  \uf06a  \uf06a  =  =  L  R  Q  hu  h u  is the unit-width discharge flowing through the porosity discontinuity, the  velocities at the two sides of the geometric transition can be rewritten as  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1)  1  2  1  2  ,  \uf06a  \uf06a  =  =  L  R  Q  Q  u  u  h  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The substitution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='b) leads to  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2)  (  )  2  2  2  2  2  1  1  2  2  1  ,  ,  ,  2 \uf06a  \uf06a  \uf06a \uf06a  \uf06a  \uf06a  \uf047  \uf0e9  \uf0f9  −  +  −  =  \uf0eb  \uf0fb  R  L  L  R  R  L  g  Q  Q  h  h  S  h  h  u u  ,  while the substitution into the second of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (6) supplies  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3)  (  )  (  )  (  )  2  2  1  2  2  1  2  2  2  1  ,  ,  ,  2  2  \uf06a \uf06a  \uf06a  \uf06a  \uf0e9  \uf0f9  \uf044  =  +  −  +  \uf0ea  \uf0fa  \uf0ea  \uf0fa  \uf0eb  \uf0fb  L  R  R  L  Q  Q  H  h  h  g  h  g  h  u u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The elimination of Q2 between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3) finally supplies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Smooth stationary weak solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2)  If the time derivatives are null, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2) can be rewritten as  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1)  2  2  2  0  0  2  2  \uf06a  \uf06a  \uf06a  \uf06a  =  \uf0e6  \uf0f6 +  −  =  \uf0e7  \uf0f7  \uf0e8  \uf0f8  d hu  ds  d  gh  d hu  gh d  ds  ds  ds  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  If the porosity and the flow variables are smooth (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', continuous with their derivatives), the  second of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) can be rewritten as  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2)  2  2  2  0  2  2  2  \uf06a  \uf06a  \uf06a  \uf06a  \uf06a  \uf0e6  \uf0f6  +  +  +  −  =  \uf0e7  \uf0f7  \uf0e8  \uf0f8  dh  gh d  d  u  d hu  gh d  gh  h  u  ds  ds  ds  ds  ds  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Finally, the substitution of the first of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2) and the cancellation of the terms  with opposite sign leads to  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3)  2  0  2  \uf0e6  \uf0f6  +  =  \uf0e7  \uf0f7  \uf0e8  \uf0f8  d  u  h  ds  g  ,  which states that the total head is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Discussion of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) and of the corresponding Froude limits  This Appendix reports in a condensed form the discussion present in classic (Yarnell 1934, Chow  1959) and recent (Defina and Susin 2006, Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2007, Akers and Bokhove 2008, Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  2021) literature with reference to 1-d flows in channels with variable width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Given the formal analogy  between 1-d SWE model with variable width and 1-d porous SWE model, the discussion can be  extended to porous models where the porosity symbol replaces the width symbol while the discharge  is intended as unit-width discharge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The parametric family of the flow states  (  )  =  T  h  hu  u  with given width B and discharge  =  Q  Bhu  is defined by (  )  (  )  , ,  =  T  Q B h  h  Q B  u  , where the flow depth h is the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The total  head corresponding to the states of this family is a function of the parameter h only, and it is defined  by  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1)  (  )  (  )  (  )  (  )  2  2  , ,  , ,  2  =  =  +  Q  H Q B h  H  Q B h  h  g Bh  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The function H(Q, B, h) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) is convex with respect to h > 0, it is positive and it has a  unique minimum in  (  )  ,  =  c  h  h Q B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The critical depth  (  )  ,  ch  Q B  and the corresponding critical head  (  )  (  )  (  )  ,  , ,  ,  =  c  c  H  Q B  H Q B h Q B  are defined by (Chow 1959)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2)  (  )  (  )  2  2  3  3  2  2  3  ,  ,  ,  2  =  =  c  c  Q  Q  h Q B  H  Q B  gB  gB  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The states u with  (  )  ,  \uf03c  c  h  h Q B  are characterized by  ( )  1  \uf03e  F u  and are called supercritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The states with  (  )  ,  \uf03e  c  h  h Q B  are characterized by  ( )  0  1  \uf03c  \uf03c  F u  and are called subcritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' From the  preceding discussion, it follows that a state u with discharge Q has energy sufficient to pass through  the cross-section whose width is B only if the corresponding head is not minor than  (  )  ,  c  H  Q B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  This observation has consequences for the application of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10), expressing the invariance  of discharge and head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Let BL and BR be the channel widths at the left and right ends, respectively, of  a geometric transition, and let  (  )  2  2  2  2  =  T  h  h u  u  be the state corresponding to the flow at the right  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' It is possible to find the left end state  (  )  1  1  1 1  =  T  h  hu  u  connected to u2 by means of the discharge  and head invariance only if  (  )  (  )  1  ,  \uf0b3  c  L  H  H  Q B  u  , namely only if  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3)  (  )  2  3  1  2  3  2  \uf0b3  L  Q  H  gB  u  ,  where  1 1  =  L  Q  B hu  is the discharge corresponding to the right state  1  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The invariance of discharge  and head is expressed by  2 2  1 1  =  R  L  B h u  B hu  and  (  )  (  )  2  1  =  H  H  u  u  , implying that the condition of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3) can be rewritten as  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4)  (  )  (  )  2  2  2  3  2  2  3  1  2  \uf0b3  h u  H  AR  g  u  ,  where  =  L  R  AR  B  B  is the aspect ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' If the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4) is satisfied, it exists the state u1 connected to  u2 by the invariance of discharge and head with aspect ratio AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='4) can be rewritten in dimensionless form as (Yarnell 1934, Chow 1959, Defina  and Susin 2006, Akers and Bokhove 2008)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5)  (  )  (  )  2  \uf0b3  AR  f  F u  ,  where  (  )  2  2  2  =  F  u  gh  u  is the Froude number of the state u2 and the function f(x) is defined as  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6)  ( )  3 2  2  3  ,  0  2  \uf0e6  \uf0f6  =  \uf0b3  \uf0e7  \uf0f7  +  \uf0e8  \uf0f8  f x  x  x  x  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The function f(x) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='6) is non-negative, strictly increasing for  \uf05b  \uf05b  0,1  \uf0ce  x  , strictly  decreasing for  1  \uf03e  x  , and it has a maximum in  1  =  x  with  ( )  1  1  =  f  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The properties of the function f(x)  have the following implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Case  1  \uf03e  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In case of  1  \uf03e  AR  , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5) is satisfied for every  (  )  2  F u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In other words, it  always exists the state u1 connected to u2 by the invariance of discharge and head when there is a  width increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Case  1  \uf0a3  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In case of  1  \uf0a3  AR  , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5) is satisfied by  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='7)  (  )  (  )  2  \uf0a3  sb  F  K  AR  u  ,  (  )  (  )  2  \uf0b3  sp  F  K  AR  u  ,  where the limit Froude numbers  (  )  sb  K  AR  and  (  )  sp  K  AR  are defined as (Varra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2021)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8)  (  )  (  )  (  )  (  )  (  )  (  )  3  2  1 2  2  3  2  1 2  2  1  2cos  arctan  1  3  3  5  1  2cos  arctan  1  3  3  \uf070  \uf070  −  −  −  −  \uf0e9  \uf0f9  \uf0e6  \uf0f6  =  −  −  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0eb  \uf0fb  \uf0e9  \uf0f9  \uf0e6  \uf0f6  =  −  −  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0eb  \uf0fb  sp  sb  K  AR  AR  AR  K  AR  AR  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The Froude limits defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='8) are characterised by  (  )  1  \uf0b3  sp  K  AR  (supercritical) and  (  )  1  \uf0a3  sb  K  AR  (subcritical) for every  1  \uf0a3  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In conclusion, the present discussion shows that two different situations are possible with  reference to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (10) where  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  :  a) Given the state u1, it is always possible to find the corresponding state u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  b) Given the state u2, it is possible to find the corresponding state u1 only if one of the two  inequalities of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='7) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' In this case, the invariance of the total head through  the porosity transition implies that u1 is subcritical [supercritical] if u2 is subcritical  [supercritical], and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' When  (  )  (  )  2  =  sb  F  K  AR  u  or  (  )  (  )  2  =  sp  F  K  AR  u  , the  state u1 is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  For  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  , it is possible to define an additional limit  (  )  jump  K  AR  for the Froude  number, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Let the left state u1 be critical and the right state u2 be subcritical with  (  )  (  )  2 = −  sb  F  K  AR  u  (flow from right to left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' The supercritical state  #  2  u  to the right of the subcritical  state u and connected to it by a standing hydraulic jump is characterized by Froude number  (  )  (  )  #  2 = −  jump  F  K  AR  u  , where  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='9)  (  )  (  )  (  )  (  )  3  2  2  8  1  1 8  −  =  − +  +  jump  sb  sb  K  AR  K  AR  K  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Head loss through a standing hydraulic jump  The flow depth  #  Rh  corresponding to the state  (  )  #  #  #  #  =  T  R  R  R  R  h  h u  u  connected to uR by means of a  hydraulic jump in a rectangular channel is (Chow 1959)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1)  (  )  2  #  1  1 8  2  =  − +  +  R  R  R  h  h  F  ,  where  2  R  F  is the squared Froude number corresponding to the state uR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1), the ratio  #  R  R  h  h depends on  2  R  F  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  The discharge is conserved through the hydraulic jump, implying that  #  # =  R  R  R  R  h u  h u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' For this  reason, the head  #  R  H  corresponding to the state  #  R  u  is  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2)  (  )  #  #  #  #  #  2  3  2  #  2  1  2  2  \uf0e9  \uf0f9  \uf0e6  \uf0f6  \uf0e6  \uf0f6  \uf0ea  \uf0fa  =  =  +  =  +  \uf0e7  \uf0f7  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0e8  \uf0f8  \uf0e8  \uf0f8  \uf0eb  \uf0fb  R  R  R  R  R  R  R  R  R  R  u  h  F  h  H  H  h  h  g  h  h  u  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Once that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) is substituted into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2), the relative head loss  (  )  #  #  #  \uf044 = \uf044  =  −  R  R  R  R  H  H  H  H  H  can be easily calculated as  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3)  1  #  #  #  #  3  2  2  1  1  1  2  2  −  \uf0e9  \uf0f9  \uf0e6  \uf0f6  \uf0e9  \uf0f9  −  \uf0ea  \uf0fa  \uf044 =  = −  +  +  \uf0e7  \uf0f7  \uf0ea  \uf0fa  \uf0ea  \uf0fa \uf0eb  \uf0fb  \uf0e8  \uf0f8  \uf0eb  \uf0fb  R  R  R  R  R  R  R  R  R  H  H  h  F  h  F  H  h  h  ,  where  (  )  =  R  R  H  H u  is the head corresponding to the state uR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3), it is evident  that the relative head loss  #  \uf044  through the hydraulic jump depends on  2  R  F  only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  In the case that the supercritical state uR is connected to the subcritical state u2 by a hydraulic  jump (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', when  #  2 =  R  u  u ) and u1 is critical (see Figure 4c),  2  R  F  coincides with  (  )  2  jump  K  AR  (see  Appendix C) and the relative head loss  #  \uf044  depends on AR only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  References  Akers B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Bokhove O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008) Hydraulic flow through a channel contraction: multiple steady  states, Physics of Fluids 20, 056601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2909659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Aleksyuk A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Belikov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2019) The uniqueness of the exact solution of the Riemann  problem for the shallow water equations with discontinuous bottom, Journal of Computational  Physics 390, 232-248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Aleksyuk A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Malakhov M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Belikov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022) The exact Riemann solver for the shallow  water equations with a discontinuous bottom, Journal of Computational Physics 450, 110801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='jcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='110801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Austin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Skogerboe G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Bennett R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1970) Subcritical flow at open channel structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Open  channel  expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Utah  State  University  Reports,  638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  https://digitalcommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='usu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='edu/water_rep/638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Bruwier M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Archambeau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Erpicum S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pirotton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dewals B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) Shallow-water models  with anisotropic porosity and merging for flood modelling on Cartesian grids, Journal of Hydrology  554, 693-709.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='jhydrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='051.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Buckingham E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1914) On physically similar systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' illustrations of the use of dimensional  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Physical Review 4(4), 345-376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Castro M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pardo Milanés A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Parés C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) Well-balanced numerical schemes based on a  generalized hydrostatic reconstruction technique, Mathematical Models and Methods in Applied  Sciences 17(12), 2055-2113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1142/S021820250700256X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cea L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Vázquez-Cendón M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2010) Unstructured finite volume discretization of two-  dimensional depth-averaged shallow water equations with porosity, International Journal for  Numerical Methods in Fluids 63(8), 903-930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1002/fld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Chow V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1959) Open channel hydraulics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' McGraw-Hill, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Castaldo R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pepe V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Varra G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Covelli C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pianese  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018a) Multiple solutions for the Riemann problem in the Porous Shallow water Equations, EPiC  Series in Engineering 3, 476-484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='29007/31n4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Covelli C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pianese D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2014) Boundary conditions  in finite volume schemes for the solution of shallow-water equations: the non-submerged broad-  crested weir, Journal of Hydroinformatics 14(6), 1235-1249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2166/hydro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pepe V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', D’Aniello A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pianese D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018b) The  solution of the dam-break problem in the Porous Shallow water Equations, Advances in Water  Resources 114, 83-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pepe V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Morlando F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', D’Aniello A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pianese D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  (2017) Exact solution of the dam-break problem for constrictions and obstructions in constant width  rectangular  channels,  ASCE  Journal  of  Hydraulic  Engineering  143(11),  04017047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1061/(ASCE)HY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1943-7900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0001368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Cunge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Holly H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Verwey A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1980) Practical aspects of computational river hydraulics,  Pitman, Boston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Dal Maso G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', LeFloch P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Murat F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1995) Definition and weak stability of nonconservative  products, Journal del Mathématiques Pures et Appliqués 74(6), 483-548.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Defina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2000) Two-dimensional shallow flow equations for partially dry areas, Water  Resources Research 36(11), 3251-3264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1029/2000WR900167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Defina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Susin F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2006) Multiple states in open channel flow, in Vorticity and turbulence  effects in fluids structures interactions – Advances in Fluid Mechanics, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Brocchini and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Trivellato  eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Wessex Institute of Technology Press, Southampton, 105-130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Defina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Viero D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2010) Open channel flow through a linear contraction, Physics of Fluids  22(3), 036602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3370334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Dewals B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Bruwier M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pirotton M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Erpicum S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Archambeau P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2021) Porosity Models for  Large-Scale  Urban  Flood  Modelling:  A  Review,  Water,  13(7),  960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3390/w13070960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Ferrari A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Vacondio R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dazzo S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Mognosa P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) A 1d–2d shallow water equations solver  for discontinuous porosity field based on a generalized Riemann problem, Advances in Eater  Resources 107, 233-249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='023  Finaud-Guyot P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Delenne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Lhomme J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Llovel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2010) An approximate-state  Riemann solver for the two-dimensional shallow water equations with porosity, International Journal  for Numerical Methods in Fluids 62(12), 1299-1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1002/fld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Formica G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1955) Esperienze preliminari sulle perdite di carico nei canali dovute a  cambiamenti di sezione, L’Energia Elettrica 32(7), 554-567 (in Italian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Godlewski E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Raviart P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1996) Numerical approximation of systems of hyperbolic  equations, Springer, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Delenne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2014) Macroscopic modelling of urban floods, La Houille Blanche 6,  19-25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1051/lhb/2014058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Delenne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Rousseau A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Boutron O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2018) Flux closures and source terms models  for shallow water models with depth-dependent integral porosity, Advances in Water Resources 122,  1-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Delenne C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Soares-Frazão S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022) Self-similar solutions of shallow water  equations  with  porosity,  IAHR  Journal  of  Hydraulic  Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1080/00221686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2106598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Sanders B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Schubert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) Dual integral porosity shallow water model for  urban  flood  modelling,  Advances  in  Water  Resources  103,  16-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Soares-Frazão S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2006) Flux and source term discretization in two-dimensional  shallow water models with porosity on unstructured grids, International Journal for Numerical  Methods in Fluids 50(3), 309-345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1002/fld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Hager W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2010) Wastewater hydraulics: theory and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Springer, Berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Hager W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Sinniger R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1985) Flow characteristics of the hydraulic jump in a stilling basin  with an abrupt bottom rise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Journal of Hydraulic Research 23(2), 101–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1080/00221688509499359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Han E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Handke M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Warnecke G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2013) Criteria for non-uniqueness of Riemann solutions to  compressible duct flows, Zeitschrift für Angewandte Mathematik und Mechanik 93(6-7), 465-475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1002/zamm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='201100176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Han E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Warnecke G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2014) Exact Riemann solutions to Shallow water Equations, Quarterly  of Applied Mathematics 72(3), 407-453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1090/S0033-569X-2014-01353-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Ion S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Marinescu D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Ion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cruceanu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022) Numerical scheme for solving a porous  Saint-Venant type model for water flow on vegetated hillslopes, Applied Numerical Mathematics  172, 67-98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='apnum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Ippen A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Dawson J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1951) High-velocity flow in open channels: a symposium: Design of  Channel  Contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  ASCE  Transactions  116(1),  326–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1061/TACEAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0006522.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Jung J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022) Development of exact solution and finite-volume method of porous shallow  water equations for the analysis of urban flooding, Seoul National University, PhD dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  https://hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='net/10371/181177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Karki K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Chander S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Malhotra, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1972) Supercritical flow over sills at incipient jump  conditions,  ASCE  Journal  of  the  Hydraulic  Division  98(10),  1753–1764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1061/JYCEAJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0003435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  LeFloch P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1989) Shock waves for nonlinear hyperbolic systems in nonconservative form,  Institute  for  Mathematics  and  Its  Applications  Preprint  Series  593,  Minneapolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  http://hdl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='net/11299/5107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Lhomme J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2006) Modelisation des inondations en milieu urbain: approaches  unidimensionelle, bidimensionnelle et macroscopique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Hydrologie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' PhD Thesis, Universite  Montpellier II – Sciences e Techniques du Languedoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (in French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Mohamed K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2014) A finite volume method for numerical simulation of shallow water models  with porosity, Computer & Fluids 104, 9-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='compfluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Özgen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Liang D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Hinkelmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2016a) Shallow water equations with depth-dependent  anisotropic porosity for subgrid-scale topography, Applied Mathematical Modelling 40(17-18),  7447-7473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='apm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Özgen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Zhao J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Liang D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Hinkelmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2016b) Urban flood modeling using shallow  water equations with depth-dependent anisotropic porosity, Journal of Hydrology 541(Part B), 1165-  1184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='jhydrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='025.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Özgen I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Zhao J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Liang D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Hinkelmann R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) Wave propagation speeds and source  term influences in single and integral porosity shallow water equations, Water Science and  Engineering 10(4), 275-286.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='wse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Sanders B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Schubert J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Gallegos H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008) Integral formulation of shallow-water  equations with anisotropic porosity for urban flood modelling, Journal of Hydrology 362(1-2), 19-  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='jhydrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Soares-Frazão S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Lhomme J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Guinot V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Zech Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008) Two-dimensional shallow water  model with porosity for urban flood modelling, Journal of Hydraulic Research 46(1), 45-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Valiani A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Caleffi V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2008) Depth-energy and depth-force relationships in open channel flows:  Analytical  findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Advances  in  Water  Resources  31(3),  447-454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Varra G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pepe V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2020) On integral and  differential porosity models for urban flooding simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Advances in Water Resources 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='103455.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Varra G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Pepe V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cimorelli L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2021) The exact solution to the  Shallow water Equations Riemann problem at width jumps in rectangular channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Advances in  Water Resources 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='advwatres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='103993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Varra G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Della Morte R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Gargano R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Cozzolino L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2022) Porous Shallow water Equations  model with disambiguation of multiple solutions, Environmental Sciences Proceedings 21(1), 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='3390/environsciproc2022021055.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Vaschy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1892) Sur les lois de similitude en physique, Annales Tèlègraphiques 19, 25-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Viero D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=', Defina A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2017) Extended theory of hydraulic hysteresis in open-channel flow,  Journal of Hydraulic Engineering 143(9), 06017014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1061/(ASCE)HY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1943-7900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='0001342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Whitaker S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1969) Advances in the theory of fluid motion in porous media, Industrial and  Engineering Chemistry 61(12), 14-28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1021/ie50720a004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Yarnell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (1934) Bridge piers as channel obstructions, Technical Bulletin 444, US  Department of Agriculture, Washington.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' https://naldc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='nal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='usda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='gov/download/CAT86200436/PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Tables List  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Initial flow conditions of the validation Riemann problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Coefficients for the polynomial interpolation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Coefficients for the polynomial interpolation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Supercritical 2-d flow impinging on a contraction for BR = 1 m and hR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, and 1 m:  relative head losses \uf044* for the cases of flow passing through the discontinuity (G1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' A hyphen  indicates the cases where a backwards moving shock is produced (G2 configurations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figures List  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Physical interpretation of the porosity discontinuity between \uf06aL  and \uf06aR :  monotonic (a)  and non-monotonic porosity variation (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Internal description of the porosity discontinuity: plan view of the monotonic porosity  variation (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' profile view of smooth flow depth variation (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' profile view of flow depth  variation with hydraulic jump (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Field of occurrence of multiple solutions to the porosity Riemann problem for right  supercritical flows uR impinging a porosity reduction with  1  \uf06a  \uf06a  =  \uf0a3  L  R  AR  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Lower (continuous  line) and upper (dashed line) boundaries of the hysteresis domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Hysteresis domains: A (no  multiple solutions), B (multiple solutions even in the case hL = 0), C (multiple solutions only  for hL > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow conditions through the porosity discontinuity when multiple solutions to the purely 1-  d SP Riemann problem are possible: profile view of solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the channel considered for 2-d SWE numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Distorted  representation (measures in metres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions  (dots) for the flow depth at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d)  with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 3 with initial conditions  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problem 5 with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view for the flow  depth solution at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1-d SP exact solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Comparison  between the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problem 6 with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view for the flow  depth solution at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 1-d SP exact solutions T1 (a), T2 (b) and T3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Comparison  between the T3 exact solution (continuous line) and the 2-d SWE numerical solution (dots) (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact (continuous black line) and 2-d SWE numerical solutions  (dots) for the flow depth at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 7 (a) and 8 (b) with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 7 with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plan view of the 2-d SWE numerical solution for Riemann problem 8 with initial  conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Flow depth contours at time t = 5 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Plane view of the channel used for 2-d SWE numerical tests with supercritical flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Distorted representation (measures in metres).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 1 m impinging  a contraction: G1 configuration (black triangles), G2 configuration (white squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' upper  hysteresis domain limit (dashed line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' modified upper boundary (thick black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Relative head losses for supercritical flows through a contraction: 2-d SWE numerical  results for G1 configuration (black triangles);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' limit relative head loss (continuous black line);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  envelope of G1 data closer to the modified upper boundary of Figure 14 (continuous grey line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Polynomial interpolation of the relative head loss data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Triangles represent the  experimental cases enveloped by a thin grey line in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the 1-d SP exact solution with head loss through the geometric discontinuity  (continuous black line) and 2-d SWE numerical solution (dots) for the flow depth at time t = 5  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 7 (a) and 8 (b) with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Side view of two neighbouring cells in the 1-d computational domain: cell-averaged  quantities at a generic time level n (a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' interface reconstructed variables used in the basic  reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' interface and in-cell reconstructed variables in the  novel reconstruction approach (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with  the variable reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (continuous black line) and 2-d SWE model  (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 1 (a), 2 (b), 3 (c) and 4 (d) with initial conditions in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with  the basic variable reconstruction by Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' (2007) (continuous black line) and 2-d SWE  model (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Profile view of the numerical solution for the flow depth at time t = 5 s: 1-d SP model with  the novel variable reconstruction (continuous black line) and 2-d SWE model (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' Example  Riemann problems 5 (a), 6 (b), 7 (c) and 8 (d) with initial conditions in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='  Figure 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' 2-d SWE numerical results for supercritical flows with BR = 1 m and hR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content='5, and 1  m impinging a contraction: G1 configuration (black triangles), G2 configuration (white  squares);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
+page_content=' modified upper boundary (thick black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQf-ALu/content/2301.02810v1.pdf'}
diff --git a/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/2301.01203v1.pdf.txt b/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/2301.01203v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a510d5c4f99afb65dd8c6b2f1db0f88d2cf731c6
--- /dev/null
+++ b/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/2301.01203v1.pdf.txt
@@ -0,0 +1,3370 @@
+Quantum simulation of exact electron dynamics can be
+more efficient than classical mean-field methods
+Ryan Babbush,1, ∗ William J. Huggins,1 Dominic W. Berry,2 Shu Fay Ung,3 Andrew Zhao,1, 4
+David R. Reichman,3 Hartmut Neven,1 Andrew D. Baczewski,5 and Joonho Lee1, 3, †
+1Google Quantum AI, Venice, CA, United States
+2Department of Physics and Astronomy, Macquarie University, Sydney, NSW, Australia
+3Department of Chemistry, Columbia University, New York, NY, United States
+4Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM, United States
+5Quantum Algorithms and Applications Collaboratory, Sandia National Laboratories, Albuquerque NM, United States
+(Dated: January 4, 2023)
+Quantum algorithms for simulating electronic ground states are slower than popular classical
+mean-field algorithms such as Hartree-Fock and density functional theory, but offer higher accuracy.
+Accordingly, quantum computers have been predominantly regarded as competitors to only the most
+accurate and costly classical methods for treating electron correlation. However, here we tighten
+bounds showing that certain first quantized quantum algorithms enable exact time evolution of
+electronic systems with exponentially less space and polynomially fewer operations in basis set size
+than conventional real-time time-dependent Hartree-Fock and density functional theory. Although
+the need to sample observables in the quantum algorithm reduces the speedup, we show that one can
+estimate all elements of the k-particle reduced density matrix with a number of samples scaling only
+polylogarithmically in basis set size. We also introduce a more efficient quantum algorithm for first
+quantized mean-field state preparation that is likely cheaper than the cost of time evolution. We
+conclude that quantum speedup is most pronounced for finite temperature simulations and suggest
+several practically important electron dynamics problems with potential quantum advantage.
+Introduction
+Quantum computers were first proposed as tools for
+dynamics by Feynman [1] and later shown to be univer-
+sal for that purpose by Lloyd et al. [2]. Like those early
+papers, most work on this topic assumes that the ad-
+vantage of quantum computers for dynamics is that they
+provide an approach to simulation with systematically
+improvable precision but without scaling exponentially.
+Here, we advance and analyze a different idea: certain
+(exact) quantum algorithms for dynamics may be more
+efficient than even classical methods that make uncon-
+trolled approximations. We examine this in the context
+of simulating interacting fermions – systems of relevance
+in fields such as chemistry, physics, and materials science.
+It is often the case that practically relevant ground
+state problems in chemistry and materials science do not
+exhibit strong correlation.
+For those problems, many
+classical heuristic methods work well [3–5].
+Even for
+some strongly correlated systems, there are successful
+polynomial-scaling classical methods [6].
+Here, we ar-
+gue that even if electronic systems are well described
+by mean-field theory, quantum algorithms can achieve
+speedup over classical algorithms for simulating the time
+evolution of such systems.
+We focus on comparing to
+mean-field methods such as real-time time-dependent
+Hartree-Fock and density functional theory due to their
+popularity and well-defined scaling. Nonetheless, many
+∗ corresponding author: ryanbabbush@gmail.com
+† corresponding author: linusjoonho@gmail.com
+of our arguments translate to advantages over other
+known classical approaches to dynamics that are more
+expensive but more accurate than mean-field methods.
+This is a sharp contrast to prior studies of quantum
+algorithms, which have focused on strongly correlated
+ground state problems such as FeMoCo [7–11], P450
+[12], chromium dimers [13] and jellium [14–17], assess-
+ing quantum advantage over only the most accurate and
+costly classical algorithms.
+Quantum algorithms competitive with efficient clas-
+sical algorithms for dynamics have been analyzed in
+contexts outside of fermionic simulation. For example,
+work by Somma [18] showed that certain one-dimensional
+quantum systems, such as harmonic oscillators, could be
+simulated with sublinear complexity in system size. Ex-
+perimentally motivated work by Geller et al. [19] also pro-
+posed simulating quantum systems in a single-excitation
+subspace, a task for which they suggested a constant fac-
+tor speedup was plausible. However, neither work is con-
+nected to the context studied here.
+We begin by analyzing the cost of classical mean-field
+dynamics and recent exact quantum algorithms in first
+quantization, focusing on explaining why there is often a
+quantum speedup in the number of basis functions over
+classical mean-field methods. Next, we analyze the over-
+heads associated with measuring quantities of interest on
+a quantum computer and introduce more efficient meth-
+ods for measuring the one-particle reduced density ma-
+trix in first quantization (which characterizes all mean-
+field observables). Then, we discuss the costs of prepar-
+ing mean-field states on the quantum computer and de-
+scribe new methods that make this cost likely negligible
+compared to the cost of time evolution. Finally, we con-
+arXiv:2301.01203v1  [quant-ph]  3 Jan 2023
+
+2
+clude with a discussion of systems where these techniques
+might lead to practical quantum advantage over classical
+mean-field simulations.
+Classical mean-field dynamics
+Here we will discuss mean-field classical algorithms for
+simulating the dynamics of interacting systems of elec-
+trons and nuclei. Thus, we will focus on the ab initio
+Hamiltonian with η particles discretized using N basis
+functions, which can be expressed as
+H =
+N
+�
+µν
+hµνa†
+µaν + 1
+2
+N
+�
+µνλσ
+(µν|λσ) a†
+µa†
+λaσaν
+(1)
+where a(†)
+µ
+is the fermionic annihilation (creation) opera-
+tor for the µ-th orbital and integral values are given by
+hµν =
+�
+dr φ∗
+µ (r)
+�
+−∇2
+2 + V (r)
+�
+φν (r) ,
+(2)
+(µν|λσ) =
+�
+dr1dr2
+φ∗
+µ (r1) φν (r1) φ∗
+λ (r2) φσ (r2)
+|r1 − r2|
+. (3)
+Here, V (r) is the external potential (perhaps arising from
+the nuclei) and φµ(r) represents a spatial orbital.
+Exact quantum dynamics is encoded by the time-
+dependent Schr¨odinger equation given by
+i ∂
+∂t |ψ (t)⟩ = H |ψ (t)⟩ .
+(4)
+Mean-field dynamics, such as real-time time-dependent
+Hartree-Fock
+(RT-TDHF)
+[20],
+employs
+a
+time-
+dependent variational principle within the space of sin-
+gle Slater determinants (i.e., anti-symmetrized product
+states) to approximate Eq. (4).
+Other methods with
+similar cost such as real-time time-dependent density
+functional theory (RT-TDDFT) rely on a relationship
+between the interacting system and an auxiliary non-
+interacting system to define dynamics within a space
+of single Slater determinants [20–22]. In both methods,
+there are η occupied orbitals, each expressed as a linear
+combination of N basis functions using the coefficient
+matrix, Cocc. The dimension of Cocc is N × η. These
+orbitals then constitute a Slater determinant (i.e., anti-
+symmetric product states), det(Cocc). Storing Cocc on a
+classical computer has space complexity O(Nη log(1/ϵ)).
+As a result of this approximation, we solve the follow-
+ing effective time-dependent equation for the occupied
+orbital coefficients that specify the Slater determinant
+Cocc(t) at a given moment in time:
+i∂Cocc (t)
+∂t
+= F (t) Cocc (t)
+(5)
+where the effective one-body mean-field operator F(t),
+also known as the time-dependent Fock matrix, is
+Fµν(t) = hµν +
+N
+�
+λσ
+�
+(µν|λσ) − (µσ|λν)
+2
+�
+Pσλ(t)
+(6)
+with P(t) = Cocc(t)(Cocc(t))†. While F(t) is an N × N
+dimensional matrix, we can apply it to Cocc(t) without
+explicitly constructing it, thus avoiding a space complex-
+ity of O(N 2 log(1/ϵ)). Using the most common methods
+of applying this matrix to update each of η occupied or-
+bitals in Cocc(t) requires �
+O(N 2η) total operations1.
+However, a recent technique referred to as occ-RI-K by
+Head-Gordon and co-workers [23], and similarly “Adap-
+tively Compressed Exchange” (ACE) [24, 25] by Lin and
+co-workers, further reduces this cost.
+These methods
+leverage the observation that, when restricted to the sub-
+space of the η occupied orbitals, the effective rank of the
+Fock operator scales as O(η). This gives an approach to
+updating the Fock operator that requires only
+�
+O(N η2)
+(7)
+operations. Below we will use gate complexity and the
+number of operations interchangeably when discussing
+the scaling of classical algorithms. Although these tech-
+niques are not implemented in every quantum chemistry
+code, we regard them as the main point of comparison
+to quantum algorithms. We also note that RT-TDDFT
+with hybrid functionals [26] has the same scaling as RT-
+TDHF. Simpler RT-TDDFT methods (i.e., those without
+exact exchange) can achieve better scaling, �
+O(Nη) in a
+plane wave basis, but are often less accurate.
+For finite-temperature simulation, one often needs to
+track M > η orbitals with appreciable occupations, in-
+creasing the space complexity to O(NM log(1/ϵ)). This
+increases the cost of occ-RI-K or ACE mean-field up-
+dates to �
+O(NM 2). At temperatures well above the Fermi
+energy, most orbitals have appreciable occupations so
+M ≃ N. More expensive methods for dynamics that in-
+clude electron correlation in the dynamics tend to scale
+at least linearly in the cost of ground state simulation
+at that level of theory. Thus, speedup over mean-field
+methods implies speedup over more expensive methods.
+In recent years, by leveraging the “nearsightedness”
+of electronic systems [27], “linear-scaling” methods have
+been developed that achieve updates scaling as O(N)
+[28].
+For RT-TDHF and RT-TDDFT, linear-scaling
+comes from the fact that the off-diagonal elements of P
+fall off quickly with distance for the ground state [29]
+and some low-lying excited states [30] in a localized basis.
+One can show that for gapped ground states, the decay
+rate is exponential, whereas for metallic ground states,
+1 Throughout the paper we will use the convention that �
+O(·) im-
+plies big-O notation suppressing polylogarithmic factors.
+
+3
+it is algebraic [27]. However, often such asymptotic be-
+havior only onsets for very large systems, and the onset
+can be highly system-dependent.
+This should be con-
+trasted with the scaling analyzed above and the scaling
+of quantum algorithms (vide infra) that onsets already
+at modest system sizes. Furthermore, the nearsighted-
+ness of electrons does not necessarily hold for dynamics
+of highly excited states and at high temperatures. Due to
+these limitations, we do not focus on comparing quantum
+algorithms and classical linear scaling methods.
+It has also been suggested that one can exploit a low-
+rank structure of occupied orbitals using the quantized
+tensor train format [31]. Assuming the compression of
+orbitals in real space is efficient such that the rank does
+not grow with system size or the number of grid points,
+the storage cost is reduced to ˜O(η), and the update cost is
+˜O(η2). It is unclear how well compression can be realized
+for dynamics problems and finite-temperature problems,
+and to our knowledge, it has been never been deployed
+for those purposes. Accordingly, we do not consider this
+approach as the point of comparison.
+We now discuss how many time steps are required
+to perform time evolution using classical mean-field ap-
+proaches. The number of time steps will depend on the
+target precision as well as the total unitless time
+T = max
+Cocc ∥F∥ t,
+(8)
+where t is duration of time-evolution and ∥ · ∥ denotes
+the spectral norm. This dependence on the norm of F is
+similar to what would be obtained in the case of linear
+differential equations despite the dependence on Cocc;
+see Appendix A for a derivation. We can upper bound T
+by considering its scaling in a local basis, and with open
+boundary conditions. We find
+max
+Cocc∥F∥= O
+�η2/3
+δ
++ 1
+δ2
+�
+= O
+�
+N 1/3η1/3+ N 2/3
+η2/3
+�
+,
+(9)
+where δ = O((η/N)1/3) is the minimum grid spacing.
+The first term comes from the Coulomb operator, and
+the second comes from the kinetic energy operator. This
+scaling for δ comes from taking the computational cell
+volume proportional to η.
+We briefly describe how this scaling for the norm is
+obtained and refer the reader to Appendix A for more
+details. The 1/δ2 term is obtained from the kinetic en-
+ergy term in hµν. When diagonalized, that term will be
+non-zero only when µ = ν with entries scaling as O(1/δ2)
+due to the ∇2 in the expression for hµν.
+That upper
+bounds the spectral norm for this diagonal matrix, and
+the spectral norm is unchanged under change of basis.
+The η2/3/δ comes from the sum in the expression for
+Fµν. To bound the tensor norm of (µν|λσ) − (µσ|λν)/2
+we can bound the norms of the two terms separately. For
+each, the tensor norm can be upper bounded by noting
+that the summing over µν, λσ with normalized vectors
+corresponds to transformations of the individual orbitals
+in the integral defining (µν|λσ). Since orbitals cannot be
+any more compact than width δ, the 1/|r1 − r2| in the
+integral averages to give O(1/δ). There is a further fac-
+tor of η2/3 when accounting for η electrons that cannot
+be any closer than η1/3δ on average.
+The number of time steps required to effect evolution
+to within error ϵ depends on the choice of time integra-
+tor. Many options are available [32–34], and the optimal
+choice depends on implementation details like the basis
+set and pseudization scheme, as well as the desired accu-
+racy [35]. In Appendix A, we argue that the minimum
+number of time steps t/∆t one could hope for by using
+an arbitrarily high order integration scheme of this sort
+is T 1+o(1)/ϵo(1).
+In particular, for an order k integra-
+tor, the error can be bounded as O((∥F∥∆t)k+1), with
+a possibly k-dependent constant factor that is ignored
+in this expression. That means the error for t/∆t time
+steps is O(t∥F∥k+1∆tk). To obtain error no more than
+ϵ, take (t/∆t)k = O((t∥F∥k+1/ϵ), so the number of time
+steps is t/∆t = O(T 1+1/k/ϵ1/k). Plugging Eq. (9) into
+Eq. (8) and multiplying the update cost in Eq. (7) by
+T 1+o(1)/ϵo(1) time steps, we find the number of opera-
+tions required for classical mean-field time-evolution is
+�
+N 4/3η7/3t + N 5/3η4/3t
+� �Nt
+ϵ
+�o(1)
+.
+(10)
+Finally, when performing mean-field dynamics, the
+central quantity of interest is often the one-particle re-
+duced density matrix (1-RDM). The 1-RDM is an N ×N
+matrix defined as a function of time with matrix elements
+ρµν (t) = ⟨ψ (t)| a†
+µaν |ψ (t)⟩ .
+(11)
+The 1-RDM is the central quantity of interest because
+it can be used to reconstruct any observable associated
+with a Slater determinant efficiently. For more general
+states, one would also need higher order RDMs; however,
+all higher order RDMs can be exactly computed from the
+1-RDM via Wick’s theorem when the wavefunction is a
+single Slater determinant [36].
+Thus, when mean-field
+approximations work well, the time-dependent 1-RDM
+can also be used to compute multi-time correlators such
+as Green’s functions and spectral functions.
+Exact quantum dynamics in first quantization
+One of the key advantages of some quantum algorithms
+over mean-field classical methods is the ability to per-
+form dynamics using the compressed representation of
+first quantization. First quantized quantum simulations
+date back to [37–40]. They were first applied to fermionic
+systems in [38] and developed for molecular systems in
+[41, 42].
+In first quantization, one encodes the wave-
+function using η different registers (one for each occupied
+orbital), each of size log N (to index the basis functions
+comprising each occupied orbital). The space complexity
+of first quantized quantum algorithms is O(η log N).
+
+4
+As described previously, mean-field classical methods
+require space complexity of O(Nη log(1/ϵ)) where ϵ is
+the target precision.
+Thus, these quantum algorithms
+require exponentially less space in N.
+Usually, when
+one thinks of quantum computers more efficiently en-
+coding representations of quantum systems, the advan-
+tage comes from the fact that the wavefunction might
+be specified by a Hilbert space vector of dimension
+�N
+η
+�
+and could require as much space to represent explicitly
+on a classical computer. However, this alone cannot give
+exponential quantum advantage in storage in N over clas-
+sical mean-field methods since mean-field methods only
+resolve entanglement arising from anti-symmetry and do
+not attempt to represent wavefunction in the full Hilbert
+space. Instead, the scaling advantage these quantum al-
+gorithms have over mean-field methods is related to the
+ability to store the distribution of each occupied orbital
+over N basis functions, using only log N qubits.
+But
+quantum algorithms require more than the compressed
+representations of first quantization in order to realize a
+scaling advantage over classical mean-field methods; they
+must also have sufficiently low gate complexity in the ba-
+sis size and other parameters.
+Here we will review and tighten bounds for the most
+efficient known quantum algorithms for simulating the
+dynamics of interacting electrons.
+Early first quan-
+tized algorithms for simulating chemistry dynamics such
+as [41, 42] were based on Trotterization of the time-
+evolution operator in a real space basis and utilized the
+quantum Fourier transform to switch between a repre-
+sentation where the potential operator was diagonal and
+the kinetic operator was diagonal.
+This enabled Trot-
+ter steps with gate complexity �
+O(η2) but the number of
+Trotter steps required for the approach of those papers
+scaled worse than linearly in N, η, the simulation time t
+and the desired inverse error in the evolution, 1/ϵ.
+Leveraging recent techniques for bounding Trotter er-
+ror [43–45], in Appendix B we show that using sufficiently
+high order Trotter formulas, the overall gate complexity
+of these algorithms can be reduced to
+�
+N 1/3η7/3t + N 2/3η4/3t
+� �Nt
+ϵ
+�o(1)
+.
+(12)
+This is the lowest reported scaling of any Trotter based
+first quantized quantum chemistry simulation.
+We re-
+mark that the N 1/3η7/3t scaling is dominant whenever
+N < Θ(η3).
+In that regime, it represents a quartic
+speedup in basis size for propagation over the classical
+mean-field scaling given in Eq. (10).
+The first algorithms to achieve sublinear scaling in N
+were those introduced by Babbush et al. [46]. That work
+focused on first quantized simulation in a plane wave
+basis and leveraged the interaction picture simulation
+scheme of [47] to give gate complexity scaling as
+�
+O
+�
+N 1/3η8/3t
+�
+.
+(13)
+When N > Θ(η4), this algorithm is more efficient than
+the Trotter based approach. Since that is also the regime
+where the second term in Eq. (10) dominates that scal-
+ing, this represents a quintic speedup in N, coupled with
+a quadratic slowdown in η, over mean-field classical algo-
+rithms. The work of Su et al. [48] analyzed the constant
+factors in the scaling of this algorithm for use in ground
+state preparation via quantum phase estimation [49]. In
+Appendix C of this work we analyze the constant factors
+in the scaling of this algorithm when deployed for time-
+evolution. Su et al. [48] also introduced algorithms with
+the same scaling as Eq. (13) but in a grid representation
+(see Appendix K therein).
+A key component of the algorithms of [46, 48] is the
+realization of block encodings [50] with just �
+O(η) gates.
+The difficult part of the block encoding is the preparation
+of a superposition state with amplitudes proportional to
+the square root of the Hamiltonian term coefficients. A
+novel quantum algorithm is devised for this purpose in
+[46] which scales only polylogarithmically in basis size.
+The N 1/3 dependence of Eq. (13) enters via the number
+of times the block encoding must be repeated to perform
+time evolution, related to the norm of the potential oper-
+ator. Under certain assumptions, the norm of the poten-
+tial term can be reduced to a polylogarithmic dependence
+on N (see Appendix D for more details). In that case,
+exponential quantum advantage in N is possible.
+We note that second quantized algorithms outperform
+first quantized quantum algorithms in gate complexity
+when N < Θ(η2). This is because while the best scal-
+ing Trotter steps in first quantization require �
+O(η2) gates
+[42], the best scaling Trotter steps in second quantization
+require �
+O(N) gates. As recently shown in [45], such ap-
+proaches lead to a total gate complexity for Trotter based
+second quantized algorithms scaling as
+�
+N 4/3η1/3t + N 5/3
+η2/3 t
+� �Nt
+ϵ
+�o(1)
+.
+(14)
+In the limit that η = Θ(N), this approach has O(N 5/3)
+gate complexity, which is significantly less than the
+O(N 8/3) gate complexity of Trotter based first quantized
+quantum algorithms mentioned here, or the O(N 11/3)
+gate complexity of classical mean-field algorithms. (See
+Appendix E for discussion on the overall quantum
+speedup in different regimes of how N scales in η.) How-
+ever, these second quantized approaches generally require
+at least O(N) qubits. The approach used in [45] to im-
+plement Trotter steps involves the fast multipole method
+[51], which requires O(N log N) qubits as well as the re-
+striction to a grid-like basis. When using such basis sets,
+we expect N ≫ η, and so this space complexity would
+be prohibitive for quantum computers.
+Methods such as fast multipole [51], Barnes-Hut [52],
+or particle-mesh Ewald [53] compute the Coulomb poten-
+tial in time �
+O(η) when implemented within the classical
+random access memory model. If the Coulomb potential
+could be computed with that complexity on a quantum
+computer it would speed up the first quantized Trotter
+
+5
+algorithms discussed here by a factor of O(η). However,
+it is unclear whether such algorithms extend to the quan-
+tum circuit model with the same complexity without un-
+favorable assumptions such as QRAM [54, 55], or with-
+out restricting the maximum number of electrons within
+a region of space (see Appendix E for details). Thus, we
+exclude such approaches from our comparisons here.
+Quantum measurement costs
+In contrast to classical mean-field simulations, on a
+quantum computer, all observables must be sampled
+from the quantum simulation.
+There are a variety of
+techniques for doing this, with the optimal choice de-
+pending on the target precision in the estimated observ-
+able as well as the number and type of observables one
+wishes to measure. For example, when measuring W unit
+norm observables to precision ϵ one could use algorithms
+introduced in [56] which require �
+O(
+√
+W/ϵ) state prepa-
+rations and O(W log(1/ϵ)) ancillae.
+Thus, to measure
+all W = O(N 2) elements of the 1-RDM to a fixed addi-
+tive error in each element, this approach would require
+�
+O(N/ϵ) circuit repetitions. While scaling optimally in ϵ
+for quantum algorithms, this linear scaling in N would
+decrease the speedup over classical mean-field algorithms.
+Instead, here we will focus on measuring the 1-RDM
+with a new variation of the classical shadows method.
+Classical shadows were introduced in [57] and adapted
+for second quantized fermionic systems in [58–61]. Our
+approach is to apply a separate random Clifford channel
+to each of the η different log N sized registers represent-
+ing an occupied orbital. Applying a random Clifford on
+log N qubits requires O(log2N) gates; thus, O(η log2N)
+gates comprise the full channel (a negligible cost relative
+to time-evolution). In Appendix F we prove that repeat-
+ing this procedure �
+O(η/ϵ2) times enables estimation of all
+1-RDM elements to within additive error ϵ. More gen-
+erally, we prove that this same procedure allows for es-
+timating all higher order k-particle RDMs elements with
+�
+O(kkηk/ϵ2) circuit repetitions. In the next section and in
+Appendix G, we describe a way to map second quantized
+representations to first quantization, effectively extend-
+ing the applicability of these classical shadows techniques
+to second quantization as well.
+To give some intuition for how this works, we consider
+the 1-RDM elements in first quantization:
+ρµν (t) = ⟨ψ (t)|
+�
+�
+η
+�
+j=1
+|µ⟩⟨ν|j
+�
+� |ψ (t)⟩ ,
+(15)
+where the subscript j indicates which of the η registers
+the orbital-ν to orbital-µ transition operator acts upon.
+Due to the antisymmetry of the occupied orbital regis-
+ters in first quantization, we could also obtain the 1-RDM
+by measuring the expectation value of an operator such
+as η |p⟩⟨q|1, which acts on just one of the η registers.
+Because η |p⟩⟨q|1 has the Hilbert-Schmidt norm of O(η),
+the standard classical shadows procedure applied to this
+log N sized register would require �
+O(η2/ϵ2) repetitions.
+But we can parallelize the procedure by also collecting
+classical shadows on the other η − 1 registers simultane-
+ously. One way of interpreting the results we prove in
+Appendix F is that, due to antisymmetry, these regis-
+ters are anticorrelated. As a result, collecting shadows
+on all η registers simultaneously reduces the overall cost
+by at least a factor of η. To obtain W elements of the
+1-RDM one will need to perform an offline classical inver-
+sion of the Clifford channel that will scale as �
+O(Wη2/ϵ2);
+of course, any quantum or classical algorithm for estimat-
+ing W quantities must have gate complexity of at least
+W. However, this only needs to be done once and does
+not scale in t. As a comparison, the cost of computing
+1-RDM classically without exploiting sparsity is O(Wη).
+When simulating systems that are well described by
+mean-field theory, all observables can be efficiently ob-
+tained from the time-dependent 1-RDM. However, for
+observables such as the energy that have a norm growing
+in system size or basis size, targeting fixed additive er-
+ror in the 1-RDM elements will not be sufficient for fixed
+additive error in the observable. In such situations, it
+could be preferable to estimate the observable of inter-
+est directly using a combination of block encodings [50]
+and amplitude amplification [62] (see e.g., [63]). Assum-
+ing the cost of block encoding the observable is negligible
+to the cost of time-evolution (true for many observables,
+including energy), this results in needing O(λ/ϵ) circuit
+repetitions, where λ is the 1-norm associated with the
+block encoding of the observable. For example, whereas
+there are many correlation functions with λ = O(1), for
+the energy λ = O(N 1/3η5/3 + N 2/3η1/3) [46]. Multiply-
+ing that to the cost of quantum time-evolution further
+reduces the quantum speedup.
+The final measurement cost to consider is that of re-
+solving observables in time. In some cases, e.g., when
+computing scattering cross sections or reaction rates, one
+might be satisfied measuring the state of the simulation
+at a single point in time t. However, in other situations,
+one might wish to simulate time-evolution up to a maxi-
+mum duration of t, but sample quantities at L different
+points in time. Most quantum simulation methods that
+accomplish this goal scale as O(L) (O(Lt) in the case
+where the points are evenly spaced in time). However,
+the work of [56] shows that this cost can be reduced to
+O(
+√
+Lt), but with an additional additive space complex-
+ity of �
+O(L). Either way, this is another cost that plagues
+quantum but not classical algorithms.
+Quantum state preparation costs
+Initial state preparation can be as simple or as complex
+as the state that one desires to begin the simulation in.
+Since the focus of this paper is outperforming mean-field
+calculations, we will discuss the cost of preparing Slater
+
+6
+determinants within first quantization. For example, one
+may wish to start in the Hartree-Fock state (the lowest
+energy Slater determinant). Classical approaches to com-
+puting the Hartree-Fock state scale as roughly �
+O(Nη2) in
+practice [23, 24]. This is a one-time additive classical cost
+that is not multiplied by the duration of time-evolution
+so it is likely subdominant to other costs.
+Quantum algorithms for preparing Slater determinants
+have focused on the “Givens rotation” approach intro-
+duced in [64] for second quantization. That algorithm
+requires O(Nη) “Givens rotation” unitaries. Such uni-
+taries can be implemented with O(η log N) gates in first
+quantization [48, 65], hence combining that with the se-
+quence of rotations called for in [64] gives an approach to
+preparing Slater determinants in first quantization with
+�
+O(Nη2) gates in total, a relatively high cost. Unlike the
+offline cost to compute the occupied orbital coefficients,
+this state preparation cost would be multiplied by the
+number of measurement repetitions.
+Here, we develop a new algorithm to prepare arbi-
+trary Slater determinants in first quantization with only
+�
+O(Nη) gates.
+The approach is to first generate a su-
+perposition of all of the configurations of occupied or-
+bitals in the Slater determinant while making sure that
+electron registers holding the label of the occupied or-
+bitals are always sorted within each configuration so
+that they are in ascending order. This is necessary be-
+cause without such structure (or guarantees of something
+similar), the next step (anti-symmetrization) could not
+be reversible.
+For this next step, we apply the anti-
+symmetrization procedure introduced in [66], which re-
+quires only O(η log η log N) gates (a negligible additive
+cost). Note that if one did not need the property that
+the configurations were ordered by the electron register,
+then it would be relatively trivial to prepare an arbitrary
+Slater determinant as a product state of η different reg-
+isters, each in an arbitrary superposition over log N bits
+(e.g., using the brute-force state preparation of [67]).
+A high level description of how the superposition of
+“ordered” configurations comprising the Slater determi-
+nant is prepared now follows, with details given in Ap-
+pendix G. The idea is to generate the Slater determi-
+nant in second quantization in an ancilla register us-
+ing the Givens rotation approach of [64], while mapping
+the second quantized representation to a first quantized
+representation one second quantized qubit (orbital) at a
+time.
+One can get away with storing only η non-zero
+qubits (orbitals) at a time in the second quantized rep-
+resentation because the Givens rotation algorithm grad-
+ually produces qubits that do not require further rota-
+tions. Whenever one produces a new qubit in the second
+quantized representation that does not require further
+rotations, one can convert it to the first quantized repre-
+sentation, which zeros that qubit. Thus, the procedure
+only requires O(η) ancilla qubits – a negligible additive
+space overhead.
+A total of O(Nη log N) gates are re-
+quired because for each of O(N) steps one accesses all
+O(η log N) qubits of the first quantized representation.
+In Appendix G, we show the Toffoli complexity can be
+further reduced to O(Nη) with some additional tricks.
+Finally, we note that quantum algorithms can also per-
+form finite temperature simulation by sampling initial
+states from a thermal density matrix in each realization
+of the circuit. For example, if the system is in a regime
+that is well treated by mean-field theory, one can initial-
+ize the system in a Slater determinant that is sampled
+from the thermal Hartree-Fock state [68]. Since the out-
+put of quantum simulations already needs to be sampled
+this does not meaningfully increase the number of quan-
+tum repetitions required. Such an approach would also
+be viable classically (and would allow one to perform
+simulations that only ever treat η occupied orbitals de-
+spite having finite temperature), but would introduce a
+multiplicative O(1/ϵ2) sampling cost. For either proces-
+sor there is the cost of classically computing the thermal
+Hartree-Fock state, but this is a one-time cost not mul-
+tiplied by the duration of time-evolution or O(1/ϵ2).
+Discussion
+We have reviewed and provided new analysis of the
+costs associated with both classical mean-field methods
+and state-of-the-art exact quantum algorithms for dy-
+namics. We introduced new and more efficient strategies
+for initializing Slater determinants in first quantization,
+and for measuring RDMs via classical shadows. We com-
+pare these costs in Table I. Relative to classical mean-
+field methods, we see that when the goal is to sample
+the output of quantum dynamics at zero temperature,
+the best quantum algorithms deliver a seventh power
+speedup in particle number when N < Θ(η2), quartic in
+basis size when Θ(η2) < N < Θ(η3), super-quadratic in
+basis size when Θ(η3) < N < Θ(η4) and quintic in basis
+size but with a quadratic slowdown in η when N > Θ(η4).
+In the extremal regimes of N < Θ(η5/4) and N > Θ(η4),
+the overall speedup in system size is super-quadratic
+(see Appendix E for details).
+These are large enough
+speedups that quantum advantage may persist even de-
+spite quantum error-correction overhead [69]. Note that
+our analyses are based on derivable upper bounds for
+both classical and quantum algorithms over all possible
+input states. Tighter bounds derived over restricted in-
+puts would give asymptotically fewer time steps required
+for both classical and quantum Trotter algorithms [70].
+The story becomes more nuanced when we wish to es-
+timate ϵ-accurate quantities via sampling the quantum
+simulation output at L different time points. For observ-
+ables with norm scaling as O(1) (e.g., simple correlation
+functions or single RDM elements), or those pertaining
+to amplitudes of the state (e.g.
+scattering amplitudes
+or reaction rates) the scaling advantages in system and
+basis size are maintained but at the cost of the quan-
+tum algorithm slowing down by a multiplicative factor
+of at least O(
+√
+L/ϵ). When targeting the 1-RDM (which
+characterizes all observables within mean-field theory) we
+
+7
+Processor
+Algorithm
+Observable
+Space
+Gate complexity
+classical
+T = 0 mean-field with occ-RI-K/ACE [23, 24]
+anything
+�
+O(Nη)
+(N 4/3η7/3t + N 5/3η4/3t)( Nt
+ϵ )o(1)
+classical
+T > 0 mean-field (density matrix) with [23, 24]
+anything
+�
+O(NM)
+(N 4/3M 2η1/3t+ N5/3M2t
+η2/3
+)( Nt
+ϵ )o(1)
+classical T > 0 mean-field (sampled trajectories) with [23, 24]
+anything
+�
+O(Nη)
+( N4/3η7/3t
+ϵ2
++ N5/3η4/3t
+ϵ2
+)( Nt
+ϵ )o(1)
+quantum
+second quantized Trotter grid algorithm [45]
+sample |ψ(t)⟩ O(N log N)
+(N 4/3η1/3t + N5/3t
+η2/3 )( Nt
+ϵ )o(1)
+quantum
+first quantized Trotter grid algorithm here
+sample |ψ(t)⟩
+O(η log N)
+(N 1/3η7/3t +N 2/3η4/3t)( Nt
+ϵ )o(1)
+quantum
+interaction picture plane wave algorithm [46]
+sample |ψ(t)⟩
+O(η log N)
+�
+O(N 1/3η8/3t)
+quantum
+grid basis algorithm from Appendix K of [48]
+sample |ψ(t)⟩
+O(η log N)
+�
+O(N 1/3η8/3t)
+quantum
+new shadows procedure here
+k-RDM(t)
+O(η log N)
+�
+O(kkηkL Csamp/ϵ2)
+quantum
+gradient measurement [56]
+⟨ψ(t)| O |ψ(t)⟩
+�
+O(η + L)
+�
+O(
+√
+L Csamp λ/ϵ)
+quantum
+gradient measurement [56]
+⟨ψ(t)| H |ψ(t)⟩
+�
+O(η + L)
+�
+O(
+√
+LCsampt(N1/3η5/3+N2/3η1/3)
+ϵ
+)
+TABLE I. Costs of exact quantum algorithms and mean-field classical algorithms for simulating fermionic dynamics. N is the
+number of basis functions, η is the number of particles, ϵ is target precision, M is the number of appreciably occupied orbitals
+in a finite temperature (T) simulation (M ≃ N for high T), O is any observable having norm λ that can be block encoded
+with cost less than time-evolution, t is the duration of evolution, L is the number of time points at which we wish to resolve
+quantities and Csamp is the cost of sampling |ψ(t)⟩ with a quantum algorithm. For classical algorithms, gate complexity means
+the number of floating point operations. We are not accounting for the additive time-independent costs of state preparation
+( �
+O(ηN) gates using the procedure of Appendix G), of classically computing initial occupied orbital coefficients, or of classically
+reconstructing the k-RDM given measurement outcomes. Thus, this table reports gate complexities for long-time t simulations.
+In Appendix E we provide a table clarifying which algorithm has optimal gate complexity as a function of N/η.
+maintain speedup in N but at the cost of an additional
+linear slowdown in η. When measuring the total energy,
+the overall speedup becomes tenuous. Thus, the viability
+of quantum advantage with respect to zero temperature
+classical mean-field methods depends sensitively on the
+target precision and particular observables of interest.
+In terms of applications, we expect RT-TDHF to pro-
+vide qualitatively correct dynamics whenever electron
+correlation effects are not pronounced. RT-TDDFT in-
+cludes some aspects of electron correlation but the adi-
+abatic approximation often creates issues [71] and the
+method suffers from self-interaction error [72]. When the
+adiabatic approximation is accurate, self-interaction er-
+ror is not pronounced, and the system does not exhibit
+strong correlation, we expect RT-TDDFT to generate
+qualitatively correct dynamics.
+When there are many
+excited states to consider for spectral properties, it is of-
+ten beneficial to resort to real-time dynamics methods
+instead of linear-response methods. Furthermore, we are
+often interested in real-time non-equilibrium electronic
+dynamics. This is the case for photo-excited molecules
+near metal surfaces [73].
+The time evolution of elec-
+tron density (i.e., the diagonal of the 1-RDM) near the
+molecule is of particular interest due to its implications
+for chemical reactivity and kinetics in the context of het-
+erogeneous catalysis [74]. In this application, the simu-
+lation of nuclear degrees of freedom may be equally im-
+portant, which we will leave for future analysis.
+We see from Table I that prospects for quantum ad-
+vantage are considerably increased at finite tempera-
+tures. Thus, a promising class of problems to consider
+for speedup over mean-field methods is the electronic
+dynamics of either warm dense matter (WDM) [75–78]
+or hot dense matter (HDM) [79].
+The WDM regime
+(where thermal energy is comparable to the Fermi en-
+ergy) is typified by temperatures and densities that re-
+quire the accurate treatment of both quantum and ther-
+mal effects [80, 81]. These conditions occur in planetary
+interiors, experiments involving high-intensity lasers, and
+in inertial confinement fusion experiments as the ablator
+and fuel are compressed into the conditions necessary for
+thermonuclear ignition. Ignition occurs in the hot dense
+matter (HDM) regime (where thermal energy far exceeds
+the Fermi energy). While certain aspects of these systems
+are conspicuously classical, they still present spectra that
+can be challenging to model [82, 83].
+Simulations in either the WDM or HDM regime typ-
+ically rely on large plane wave basis sets and the inclu-
+sion of ten to one-hundred times more partially occupied
+orbitals per atom than would be required at lower tem-
+peratures. Often, the attendant costs are so great that
+it is impractical to implement RT-TDDFT with hybrid
+functionals. Therefore, many calculations necessarily use
+adiabatic semi-local approximations, even on large classi-
+cal high-performance computing systems [75]. Thus, the
+level of practically achievable accuracy can be quite low,
+and the prospect of exactly simulating the dynamics on
+a quantum computer is particularly compelling.
+Although we have focused on assessing quantum
+speedup over mean-field theory, we view the main con-
+tribution of this work as more general. In particular, if
+exact quantum simulations are sometimes more efficient
+than classical mean-field methods, then all levels of the-
+ory in between mean-field and exact diagonalization are
+in scope for possible quantum advantage. Targeting sys-
+tems that require more correlated calculations narrows
+the application space but improves prospects for quan-
+tum advantage due to the unfavorable scaling of the req-
+
+8
+uisite classical algorithms. Thus, it may turn out that the
+domain of systems requiring, say, coupled cluster dynam-
+ics [84–87], might be an even more ideal regime for prac-
+tical quantum advantage, striking a balance in the trade-
+off between the breadth of possible applications and the
+cost of the classical competition.
+Acknowledgments
+The authors thank Alina Kononov, Garnet Kin-Lic
+Chan, Robin Kothari, Alicia Magann, Fionn Malone,
+Jarrod McClean,
+Thomas O’Brien,
+Nicholas Rubin,
+Henry Schurkus, Rolando Somma, and Yuan Su for help-
+ful discussions and feedback.
+We thank Lin Lin for
+bringing our attention to the quantized tensor train for-
+mat in [31] and thank Yuehaw Khoo for a discussion
+related to this.
+DWB worked on this project under a
+sponsored research agreement with Google Quantum AI.
+DWB is supported by Australian Research Council Dis-
+covery Projects DP190102633 and DP210101367. ADB
+acknowledges support from the Advanced Simulation and
+Computing Program and the Sandia LDRD Program.
+Some work on this project occurred while in residence at
+The Kavli Institute for Theoretical Physics, supported
+in part by the National Science Foundation under Grant
+No. NSF PHY-1748958. Sandia National Laboratories
+is a multi-mission laboratory managed and operated by
+National Technology and Engineering Solutions of San-
+dia, LLC, a wholly owned subsidiary of Honeywell In-
+ternational, Inc., for DOE’s National Nuclear Security
+Administration under contract DE-NA0003525.
+[1] Richard P Feynman, “Simulating physics with comput-
+ers,” International Journal of Theoretical Physics 21,
+467–488 (1982).
+[2] Seth Lloyd, “Universal Quantum Simulators,” Science
+273, 1073–1078 (1996).
+[3] Rodney J. Bartlett and Monika Musial, “Coupled-cluster
+theory in quantum chemistry,” Rev. Mod. Phys. 79, 291–
+352 (2007).
+[4] Narbe Mardirossian and Martin Head-Gordon, “Thirty
+years of density functional theory in computational chem-
+istry: an overview and extensive assessment of 200 den-
+sity functionals,” Mol. Phys. 115, 2315–2372 (2017).
+[5] Joonho Lee, Hung Q. Pham,
+and David R. Reichman,
+“Twenty Years of Auxiliary-Field Quantum Monte Carlo
+in Quantum Chemistry: An Overview and Assessment on
+Main Group Chemistry and Bond-Breaking,” J. Chem.
+Theory Comput. 2022 (2022), 10.1021/acs.jctc.2c00802.
+[6] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong,
+Alexander M. Dalzell, Ashutosh Kumar, Phillip Helms,
+Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kas-
+toryano, Ryan Babbush, John Preskill, David R. Re-
+ichman, Earl T. Campbell, Edward F. Valeev, Lin Lin,
+and Garnet Kin-Lic Chan, “Is There Evidence for Expo-
+nential Quantum Advantage in Quantum Chemistry?”
+arXiv:2208.02199 (2022), 10.48550/arxiv.2208.02199.
+[7] Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave
+Wecker,
+and Matthias Troyer, “Elucidating Reac-
+tion Mechanisms on Quantum Computers,” Proceedings
+of the National Academy of Sciences 114, 7555–7560
+(2017).
+[8] Zhendong Li, Junhao Li, Nikesh S. Dattani, C. J. Um-
+rigar,
+and Garnet Kin-Lic Chan, “The electronic com-
+plexity of the ground-state of the FeMo cofactor of nitro-
+genase as relevant to quantum simulations,” The Journal
+of Chemical Physics 150, 024302 (2019).
+[9] Dominic Berry, Craig Gidney, Mario Motta, Jarrod Mc-
+Clean,
+and Ryan Babbush, “Qubitization of Arbitrary
+Basis Quantum Chemistry Leveraging Sparsity and Low
+Rank Factorization,” Quantum 3, 208 (2019).
+[10] Vera von Burg,
+Guang Hao Low,
+Thomas H¨aner,
+Damian S. Steiger, Markus Reiher, Martin Roetteler,
+and Matthias Troyer, “Quantum computing enhanced
+computational catalysis,” Physical Review Research 3,
+033055–033071 (2021).
+[11] Joonho Lee, Dominic W. Berry, Craig Gidney, William J.
+Huggins, Jarrod R. McClean, Nathan Wiebe, and Ryan
+Babbush, “Even More Efficient Quantum Computations
+of Chemistry Through Tensor Hypercontraction,” PRX
+Quantum 2, 030305 (2021).
+[12] Joshua J. Goings, Alec White, Joonho Lee, Christofer S.
+Tautermann, Matthias Degroote, Craig Gidney, Toru Sh-
+iozaki, Ryan Babbush, and Nicholas C. Rubin, “Reliably
+Assessing the Electronic Structure of Cytochrome P450
+on Today’s Classical Computers and Tomorrow’s Quan-
+tum Computers,” Proceedings of the National Academy
+of Sciences 119 (2022), 10.1073/pnas.2203533119.
+[13] V. E. Elfving, B. W. Broer, M. Webber, J. Gavartin,
+M. D. Halls, K. P. Lorton, and A. Bochevarov, “How will
+quantum computers provide an industrially relevant com-
+putational advantage in quantum chemistry?” (2020).
+[14] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James
+McClain, Hartmut Neven,
+and Garnet Kin-Lic Chan,
+“Low-Depth Quantum Simulation of Materials,” Physi-
+cal Review X 8, 011044 (2018).
+[15] Ryan Babbush, Craig Gidney, Dominic Berry, Nathan
+Wiebe,
+Jarrod
+McClean,
+Alexandru
+Paler,
+Austin
+Fowler, and Hartmut Neven, “Encoding Electronic Spec-
+tra in Quantum Circuits with Linear T Complexity,”
+Physical Review X 8, 041015 (2018).
+[16] Ian D. Kivlichan, Craig Gidney, Dominic W. Berry,
+Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang,
+Nicholas Rubin, Austin Fowler, Al´an Aspuru-Guzik,
+Hartmut Neven, and Ryan Babbush, “Improved Fault-
+Tolerant Quantum Simulation of Condensed-Phase Cor-
+related Electrons via Trotterization,” Quantum 4, 296
+(2020).
+[17] Sam McArdle, Earl Campbell, and Yuan Su, “Exploiting
+fermion number in factorized decompositions of the elec-
+tronic structure Hamiltonian,” Physical Review A 105,
+012403 (2022).
+[18] Rolando D. Somma, “Quantum Simulations of One Di-
+mensional Quantum Systems,” arXiv:2203.17006 (2015).
+[19] Michael R. Geller, John M. Martinis, Andrew T. Sorn-
+borger, Phillip C. Stancil, Emily J. Pritchett, Hao You,
+
+9
+and Andrei Galiautdinov, “Universal Quantum Simula-
+tion with Prethreshold Superconducting Qubits: Single-
+Excitation Subspace Method,” arXiv:1505.04990 (2015).
+[20] Andreas
+Dreuw
+and
+Martin
+Head-Gordon,
+“Single-
+Reference ab Initio Methods for the Calculation of Ex-
+cited States of Large Molecules,” Chem. Rev. 105, 4009–
+4037 (2005).
+[21] Erich
+Runge
+and
+Eberhard
+KU
+Gross,
+“Density-
+functional theory for time-dependent systems,” Physical
+review letters 52, 997 (1984).
+[22] Robert Van Leeuwen, “Mapping from densities to poten-
+tials in time-dependent density-functional theory,” Phys-
+ical review letters 82, 3863 (1999).
+[23] Samuel Manzer, Paul R. Horn, Narbe Mardirossian, and
+Martin Head-Gordon, “Fast, accurate evaluation of ex-
+act exchange: The occ-RI-K algorithm,” J. Chem. Phys.
+143, 024113 (2015).
+[24] Lin Lin, “Adaptively Compressed Exchange Operator,”
+J. Chem. Theory Comput. 12, 2242–2249 (2016).
+[25] Weile Jia and Lin Lin, “Fast real-time time-dependent
+hybrid functional calculations with the parallel transport
+gauge and the adaptively compressed exchange formu-
+lation,” Computer Physics Communications 240, 21–29
+(2019).
+[26] Weile Jia and Lin Lin, “Fast real-time time-dependent
+hybrid functional calculations with the parallel transport
+gauge and the adaptively compressed exchange formula-
+tion,” Comput. Phys. Commun. 240, 21–29 (2019).
+[27] E. Prodan and W. Kohn, “Nearsightedness of electronic
+matter,” Proc. Natl. Acad. Sci. U.S.A. 102, 11635–11638
+(2005).
+[28] J¨org Kussmann, Matthias Beer, and Christian Ochsen-
+feld, “Linear-scaling self-consistent field methods for
+large molecules,” WIREs Comput. Mol. Sci. 3, 614–636
+(2013).
+[29] Conn O’Rourke and David R. Bowler, “Linear scaling
+density matrix real time TDDFT: Propagator unitar-
+ity and matrix truncation,” J. Chem. Phys. 143, 102801
+(2015).
+[30] T. J. Zuehlsdorff, N. D. M. Hine, J. S. Spencer, N. M.
+Harrison, D. J. Riley, and P. D. Haynes, “Linear-scaling
+time-dependent density-functional theory in the linear re-
+sponse formalism,” J. Chem. Phys. 139, 064104 (2013).
+[31] Venera Khoromskaia, Boris Khoromskij,
+and Reinhold
+Schneider, “QTT Representation of the Hartree and Ex-
+change Operators in Electronic Structure Calculations,”
+Comput. Methods Appl. Math. 11, 327–341 (2011).
+[32] Alberto Castro, Miguel AL Marques, and Angel Rubio,
+“Propagators for the time-dependent Kohn–Sham equa-
+tions,” The Journal of chemical physics 121, 3425–3433
+(2004).
+[33] Weile Jia, Dong An, Lin-Wang Wang,
+and Lin Lin,
+“Fast real-time time-dependent density functional the-
+ory calculations with the parallel transport gauge,” Jour-
+nal of Chemical Theory and Computation 14, 5645–5652
+(2018).
+[34] Alina Kononov, Cheng-Wei Lee, Tatiane Pereira dos San-
+tos, Brian Robinson, Yifan Yao, Yi Yao, Xavier An-
+drade, Andrew David Baczewski, Emil Constantinescu,
+Alfredo A Correa, Yosuke Kanai, Normand Modine, and
+Andr´e Schleife, “Electron dynamics in extended systems
+within real-time time-dependent density-functional the-
+ory,” MRS Communications , 1–13 (2022).
+[35] Christopher Shepard, Ruiyi Zhou, Dillon C. Yost, Yi Yao,
+and Yosuke Kanai, “Simulating electronic excitation
+and dynamics with real-time propagation approach to
+TDDFT within plane-wave pseudopotential formula-
+tion,” J. Chem. Phys. 155, 100901 (2021).
+[36] Isaiah Shavitt and Rodney J. Bartlett, Many-Body Meth-
+ods in Chemistry and Physics:
+MBPT and Coupled-
+Cluster Theory
+(Cambridge University Press, Cam-
+bridge, England, UK, 2009).
+[37] Stephen Wiesner, “Simulations of Many-Body Quan-
+tum Systems by a Quantum Computer,” arXiv:quant-
+ph/9603028 (1996).
+[38] Daniel S Abrams and Seth Lloyd, “Simulation of Many-
+Body Fermi Systems on a Universal Quantum Com-
+puter,” Physical Review Letters 79, 4 (1997).
+[39] Christof Zalka, “Efficient Simulation of Quantum Sys-
+tems by Quantum Computers,” Fortschritte der Physik
+46, 877–879 (1998).
+[40] Bruce M Boghosian and Washington Taylor, “Simulating
+quantum mechanics on a quantum computer,” Physica
+D-Nonlinear Phenomena 120, 30–42 (1998).
+[41] Daniel A Lidar and Haobin Wang, “Calculating the
+thermal rate constant with exponential speedup on a
+quantum computer,” Physical Review E 59, 2429–2438
+(1999).
+[42] Ivan Kassal, Stephen P Jordan, Peter J Love, Masoud
+Mohseni,
+and Alan Aspuru-Guzik, “Polynomial-time
+quantum algorithm for the simulation of chemical dy-
+namics,” Proceedings of the National Academy of Sci-
+ences 105, 18681–18686 (2008).
+[43] Andrew Childs and Yuan Su, “Nearly optimal lattice sim-
+ulation by product formulas,” Physical Review Letters
+123, 050503 (2019).
+[44] Yuan Su, Hsin-Yuan Huang,
+and Earl T. Campbell,
+“Nearly tight Trotterization of interacting electrons,”
+Quantum 5, 495 (2021).
+[45] Guang Hao Low, Yuan Su, Yu Tong, and Minh C. Tran,
+“On the complexity of implementing Trotter steps,”
+(2022), 10.48550/arxiv.2211.09133.
+[46] Ryan Babbush, Dominic W. Berry, Jarrod R. McClean,
+and Hartmut Neven, “Quantum Simulation of Chemistry
+with Sublinear Scaling in Basis Size,” npj Quantum In-
+formation 5, 92 (2019).
+[47] Guang Hao Low and Nathan Wiebe, “Hamiltonian Sim-
+ulation in the Interaction Picture,” arXiv:1805.00675
+(2018).
+[48] Yuan Su, Dominic Berry, Nathan Wiebe, Nicholas Ru-
+bin, and Ryan Babbush, “Fault-tolerant quantum simu-
+lations of chemistry in first quantization,” PRX Quantum
+4, 040332 (2021).
+[49] Alan Aspuru-Guzik, Anthony D Dutoi, Peter J Love,
+and Martin Head-Gordon, “Simulated Quantum Compu-
+tation of Molecular Energies,” Science 309, 1704 (2005).
+[50] Guang Hao Low and Isaac L Chuang, “Hamiltonian Sim-
+ulation by Qubitization,” Quantum 3, 163 (2019).
+[51] V Rokhlin, “Rapid solution of integral equations of clas-
+sical potential theory,” Journal of Computational Physics
+60, 187–207 (1985).
+[52] Josh Barnes and Piet Hut, “A hierarchical O(N log
+N) force-calculation algorithm,” Nature 324, 446–449
+(1986).
+[53] Tom Darden, Darrin York, and Lee Pedersen, “Particle
+mesh Ewald: An N log N method for Ewald sums in large
+systems,” The Journal of Chemical Physics 98, 10089–
+
+10
+10092 (1993).
+[54] Andrew M. Childs, Jiaqi Leng, Tongyang Li, Jin-Peng
+Liu, and Chenyi Zhang, “Quantum Simulation of Real-
+Space Dynamics,” arXiv:2203.17006 (2022).
+[55] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone,
+“Quantum Random Access Memory,” Physical Review
+Letters 100, 160501 (2008).
+[56] William J. Huggins, Kianna Wan, Jarrod McClean,
+Thomas E. O’Brien, Nathan Wiebe,
+and Ryan Bab-
+bush, “Nearly Optimal Quantum Algorithm for Esti-
+mating Multiple Expectation Values,” arXiv:2111.09283
+(2021), 10.48550/arxiv.2111.09283.
+[57] Hsin-Yuan Huang, Richard Kueng,
+and John Preskill,
+“Predicting many properties of a quantum system from
+very few measurements,” Nature Physics 16, 1050–1057
+(2020).
+[58] Andrew Zhao, Nicholas C. Rubin, and Akimasa Miyake,
+“Fermionic Partial Tomography via Classical Shadows,”
+Physical Review Letters 127, 110504 (2021).
+[59] Kianna Wan, William J. Huggins, Joonho Lee,
+and
+Ryan Babbush, “Matchgate Shadows for Fermionic
+Quantum
+Simulation,”
+arXiv:2207.13723
+(2022),
+10.48550/arxiv.2207.13723.
+[60] B O’Gorman, “Fermionic tomography and learning,”
+arXiv:2207.14787 (2022).
+[61] Guang Hao Low, “Classical shadows of fermions with
+particle number symmetry,” arXiv:2208.08964
+(2022),
+10.48550/arxiv.2208.08964.
+[62] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain
+Tapp, “Quantum amplitude amplification and estima-
+tion,” in Quantum Computation and Information, edited
+by Vitaly I Voloshin, Samuel J. Lomonaco, and Howard
+E. Brandt (American Mathematical Society, Washington
+D.C., 2002) Chap. 3, pp. 53–74.
+[63] Patrick Rall, “Quantum algorithms for estimating phys-
+ical quantities using block encodings,” Physical Review
+A 102, 022408 (2020).
+[64] I.D. Kivlichan,
+J. McClean,
+N. Wiebe,
+C. Gidney,
+A. Aspuru-Guzik, G.K.-L. Chan,
+and R. Babbush,
+“Quantum Simulation of Electronic Structure with Lin-
+ear Depth and Connectivity,” Physical Review Letters
+120 (2018), 10.1103/PhysRevLett.120.110501.
+[65] Alain Delgado, Pablo A. M. Casares, Roberto dos Reis,
+Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-
+Hern´andez, Arne-Christian Voigt, Angus Lowe, Soran
+Jahangiri, M. A. Martin-Delgado, Jonathan E. Mueller,
+and Juan Miguel Arrazola, “Simulating key properties of
+lithium-ion batteries with a fault-tolerant quantum com-
+puter,” Physical Review A 106, 032428 (2022).
+[66] Dominic W Berry, Maria Kieferov´a, Artur Scherer, Yu-
+val R Sanders, Guang Hao Low, Nathan Wiebe, Craig
+Gidney, and Ryan Babbush, “Improved Techniques for
+Preparing Eigenstates of Fermionic Hamiltonians,” npj
+Quantum Information 4, 22 (2018).
+[67] V V Shende, S S Bullock,
+and I L Markov, “Syn-
+thesis of quantum-logic circuits,” IEEE Transactions on
+Computer-Aided Design of Integrated Circuits and Sys-
+tems 25, 1000–1010 (2006).
+[68] N. David Mermin, “Stability of the thermal Hartree-Fock
+approximation,” Annals of Physics 21, 99–121 (1963).
+[69] Ryan Babbush, Jarrod R. McClean, Michael Newman,
+Craig Gidney, Sergio Boixo, and Hartmut Neven, “Focus
+beyond Quadratic Speedups for Error-Corrected Quan-
+tum Advantage,” PRX Quantum 2, 010103 (2021).
+[70] Dong An, Di Fang, and Lin Lin, “Time-dependent un-
+bounded Hamiltonian simulation with vector norm scal-
+ing,” Quantum 5, 459 (2021).
+[71] Makenzie R. Provorse and Christine M. Isborn, “Electron
+dynamics with real-time time-dependent density func-
+tional theory,” Int. J. Quantum Chem. 116, 739–749
+(2016).
+[72] Aron J. Cohen, Paula Mori-Sa’nchez, and Weitao Yang,
+“Insights into Current Limitations of Density Functional
+Theory,” Science 321, 792–794 (2008).
+[73] John C. Tully, “Chemical Dynamics at Metal Surfaces,”
+Annu. Rev. Phys. Chem. 51, 153–178 (2000).
+[74] Rulin Wang, Dong Hou,
+and Xiao Zheng, “Time-
+dependent density-functional theory for real-time elec-
+tronic dynamics on material surfaces,” Phys. Rev. B 88,
+205126 (2013).
+[75] Andrew David Baczewski, L Shulenburger, MP Desjar-
+lais, SB Hansen, and RJ Magyar, “X-ray thomson scat-
+tering in warm dense matter without the chihara decom-
+position,” Physical review letters 116, 115004 (2016).
+[76] Rudolph J Magyar, L Shulenburger, and AD Baczewski,
+“Stopping of deuterium in warm dense deuterium from
+ehrenfest time-dependent density functional theory,”
+(2016).
+[77] Xavier Andrade, S´ebastien Hamel, and Alfredo A Cor-
+rea, “Negative differential conductivity in liquid alu-
+minum from real-time quantum simulations,” The Eu-
+ropean Physical Journal B 91, 1–7 (2018).
+[78] YH Ding, Alexander James White, SX Hu, Ondrej Cer-
+tik,
+and Lee A Collins, “Ab initio studies on the stop-
+ping power of warm dense matter with time-dependent
+orbital-free density functional theory,” Physical Review
+Letters 121, 145001 (2018).
+[79] Stefano Atzeni and J¨urgen Meyer-ter Vehn, The physics
+of inertial fusion: beam plasma interaction, hydrodynam-
+ics, hot dense matter, Vol. 125 (OUP Oxford, 2004).
+[80] Frank Graziani, Michael P Desjarlais, Ronald Redmer,
+and Samuel B Trickey, Frontiers and challenges in warm
+dense matter, Vol. 96 (Springer Science & Business,
+2014).
+[81] Tobias Dornheim, Simon Groth,
+and Michael Bonitz,
+“The uniform electron gas at warm dense matter condi-
+tions,” Physics Reports 744, 1–86 (2018).
+[82] James E Bailey, Taisuke Nagayama, Guillaume Pascal
+Loisel, Gregory Alan Rochau, C Blancard, James Colgan,
+Ph Cosse, G Faussurier, CJ Fontes, F Gilleron, et al.,
+“A higher-than-predicted measurement of iron opacity at
+solar interior temperatures,” Nature 517, 56–59 (2015).
+[83] Taisuke Nagayama, JE Bailey, GP Loisel, GS Dunham,
+GA Rochau, C Blancard, J Colgan, Ph Coss´e, G Faus-
+surier, Christopher John Fontes, et al., “Systematic study
+of l-shell opacity at stellar interior temperatures,” Phys-
+ical review letters 122, 235001 (2019).
+[84] Christian Huber and Tillmann Klamroth, “Explicitly
+time-dependent coupled cluster singles doubles calcula-
+tions of laser-driven many-electron dynamics,” J. Chem.
+Phys. 134, 054113 (2011).
+[85] Takeshi Sato,
+Himadri Pathak,
+Yuki Orimo,
+and
+Kenichi L. Ishikawa, “Communication: Time-dependent
+optimized coupled-cluster method for multielectron dy-
+namics,” J. Chem. Phys. 148, 051101 (2018).
+[86] Philip Shushkov and Thomas F. Miller, “Real-time
+density-matrix coupled-cluster approach for closed and
+open systems at finite temperature,” J. Chem. Phys. 151,
+
+11
+134107 (2019).
+[87] Alec F. White and Garnet Kin-Lic Chan, “Time-
+Dependent Coupled Cluster Theory on the Keldysh Con-
+tour for Nonequilibrium Systems,” J. Chem. Theory
+Comput. 15, 6137–6153 (2019).
+[88] Ian D Kivlichan, Nathan Wiebe, Ryan Babbush,
+and
+Alan Aspuru-Guzik, “Bounding the costs of quantum
+simulation of many-body physics in real space,” Journal
+of Physics A: Mathematical and Theoretical 50, 305301
+(2017).
+[89] Yng-Gwei Chen and John D. Weeks, “Local molecular
+field theory for effective attractions between like charged
+objects in systems with strong Coulomb interactions,”
+Proceedings of the National Academy of Sciences 103,
+7560–7565 (2006).
+[90] Cristina E. Gonz´alez-Espinoza, Paul W. Ayers, Jacek
+Karwowski,
+and Andreas Savin, “Smooth models for
+the Coulomb potential,” Theoretical Chemistry Accounts
+135, 256 (2016).
+[91] J. Carrier, L. Greengard,
+and V. Rokhlin, “A Fast
+Adaptive Multipole Algorithm for Particle Simulations,”
+SIAM Journal on Scientific and Statistical Computing 9,
+669–686 (1988).
+[92] Sergey Bravyi and Dmitri Maslov, “Hadamard-free cir-
+cuits expose the structure of the clifford group,” IEEE
+Trans. Inf. Theory 67, 4546–4563 (2021).
+[93] William J Huggins, Bryan A O’Gorman, Nicholas C Ru-
+bin, David R Reichman, Ryan Babbush,
+and Joonho
+Lee, “Unbiasing fermionic quantum monte carlo with
+a quantum computer,” Nature 603, 416–420 (2022),
+arXiv:2106.16235 [quant-ph].
+[94] Matthieu Lerasle, “Lecture notes:
+Selected topics on
+robust statistical learning theory,” arXiv:1908.10761
+(2019).
+[95] H´ector J Garc´ıa, Igor L Markov, and Andrew W Cross,
+“On the geometry of stabilizer states,” arXiv:1711.07848
+(2017).
+[96] Scott Aaronson and Daniel Gottesman, “Improved sim-
+ulation of stabilizer circuits,” Phys. Rev. A 70 (2004),
+10.1103/physreva.70.052328.
+[97] D Gottesman, The Heisenberg representation of quantum
+computers, Tech. Rep. LA-UR-98-2848; CONF-980788-
+(Los Alamos National Lab., NM (United States), 1998).
+[98] D Gross, F Krahmer,
+and R Kueng, “A partial de-
+randomization of PhaseLift using spherical designs,” J.
+Fourier Anal. Appl. 21, 229–266 (2015).
+Appendix A: Norms and scaling for the nonlinear differential equation governing mean-field evolution
+We have the differential equation
+i∂Cocc (t)
+∂t
+= F (t) Cocc (t)
+(A1)
+where
+Fµν(t) = hµν +
+N
+�
+λσ
+�
+(µν|λσ) − (µσ|λν)
+2
+�
+Pσλ(t)
+(A2)
+with P(t) = Cocc(t)Cocc(t)†. If F were independent of Cocc, then it would imply that taking the nth derivative gives
+(i)n ∂nCocc (t)
+∂tn
+= FnCocc (t) .
+(A3)
+That means the norm of the nth derivative would scale as ∥F∥n (with Cocc normalized).
+Then higher-order methods will typically have an error that scales as the norm of the higher-order derivatives. For
+example, if one were to use a Taylor series up to order k to approximate a time step, then the error for a time step
+of length δt would scale as
+1
+(k + 1)!∥F∥k+1δtk+1.
+(A4)
+This means that if the size of the time step is taken as proportional to 1/∥F∥, then the error may be made exponentially
+small in k. As a result, the total number of time steps used scales as O(∥F∥t). Similar considerations hold for other
+higher-order methods for integration. The dependence of the complexity on ∥F∥ can also be expected from principles
+of scaling, where if F is divided by ∥F∥ but t is also multiplied by ∥F∥, then the same differential equation is obtained.
+In our case where F is dependent on Cocc, the situation is more complicated. This is because taking higher-order
+derivatives of Cocc yields more terms due to the derivatives of Cocc in F. To describe this, let us write, omitting hµν
+for simplicity,
+Fµν(t) = VµνσλCσaC∗
+λa,
+(A5)
+
+12
+with Cσa the matrix entries of Cocc. We are taking a convention that Greek indices are over all orbitals, English
+letters are over electrons, and repeated indices are summed over. Then we would give the derivative as
+i∂Cµb
+∂t
+= VµνσλCσaC∗
+λaCνb.
+(A6)
+We can define an η-norm of V as
+∥V∥η = max
+x,y,z Vµνσλxµy∗
+νzσλ,
+(A7)
+with ∥x∥ = ∥y∥ = ∥z∥ = 1 (i.e., spectral norms are normalized), and z of rank η. To bound this norm, we can
+consider the first term for Vµνσλ, which is
+(µν|λσ) =
+�
+dr1 dr2
+φ∗
+µ (r1) φν (r1) φ∗
+λ (r2) φσ (r2)
+|r1 − r2|
+.
+(A8)
+The multiplication by xµ and sum over µ corresponds to a transformation of φµ to a new orbital, and similarly,
+the sum over ν transforms φν to another new orbital. Since z is of rank η, the sum over λ and σ corresponds to
+transforming the orbital basis for both φλ and φσ, and summing over η of these basis states.
+That is, we can write
+η
+�
+a=1
+�
+dr1 dr2
+φ∗ (r1) χ (r1) ψ∗
+a (r2) θa (r2)
+|r1 − r2|
+,
+(A9)
+for some transformed orbitals φ, χ, ψa, θa. We can then use the fact that |φ∗χ| ≤ |φ|2 + |χ|2, and similarly for ψa and
+θa to upper bound this expression by
+4
+η
+�
+a=1
+�
+dr1 dr2
+|φ (r1) |2|ψa (r2) |2
+|r1 − r2|
+,
+(A10)
+for some choice of φ and ψa. This integral can be maximized when the ψa are orbitals that are clustered as close as
+possible to φ. With neighboring grid points separated by δ, the smallest the average separation can be is O(η1/3δ).
+Then the factor of 1/|r1 − r2| in the integrals will give O(1/[η1/3δ]). Multiplying the sum by η gives O(η2/3/δ).
+The second term for Vµνσλ is (µσ|λν). This is similar to (µν|λσ), but with ν and σ swapped. Then the transfor-
+mation of orbitals gives
+η
+�
+a=1
+�
+dr1 dr2
+φ∗ (r1) χa (r1) ψ∗
+a (r2) θ (r2)
+|r1 − r2|
+,
+(A11)
+for some choice of φ, χa, ψa, θ. The same argument holds, where the sum is maximized with orbitals over a region of
+volume ηδ3 so there are contributions from all η terms in the sum, but 1/|r1 − r2| averages to give O(1/[η1/3δ]). This
+gives the same scaling for the second term for Vµνσλ, and so
+∥V∥η = O(η2/3/δ).
+(A12)
+What this means is that, whenever we have a contraction of the σ, λ indices in Vµνσλ with a normalized matrix of
+rank η, the remaining matrix has norm O(η2/3/δ). That immediately implies that ∥F∥ has this norm. Then applying
+F to the normalized matrix Cocc gives an upper bound on the first derivative O(η2/3/δ).
+Taking the second derivative then yields an expression with 3 terms, where each has V appearing twice and Cocc
+appearing five times. In particular,
+−∂2Cµb
+∂t2
+= Vµνσλ[(VσϵζηCζcC∗
+ηc)CϵaC∗
+λa]Cνb
++ Vµνσλ[Cσa(VλϵζηC∗
+ζcCηc)C∗
+ϵa]Cνb
++ (VµνσλCσaC∗
+λa)(VνϵζηCζcC∗
+ηc)Cϵb
+(A13)
+Only the third line has a simple interpretation as F squared times Cocc (indicated by the brackets).
+The first line has V contracted with Cocc using ζ, η, so the expression in round brackets is a matrix with norm
+O(η2/3/δ). Then in matrix terms, it is multiplied by CϵaC∗
+λa (summed over a), which is a matrix of norm 1 and rank
+
+13
+η. As a result, the expression in square brackets is of norm O(η2/3/δ) and rank η. We can then see that the first V is
+contracted over σ, λ with a matrix of rank η and norm O(η2/3/δ). That implies that the norm of the resulting matrix
+is upper bounded by the square of O(η2/3/δ). That is then multiplied by Cνb which is of norm 1, resulting in the
+overall norm of this line being upper bounded by the square of O(η2/3/δ). Similar considerations hold for the second
+line, so we can upper bound the entire second derivative by an order scaling that is the square of that for ∥F∥.
+In this, the general principle is that wherever we have something of the form CσaC∗
+λa, it is a matrix of norm 1 and
+rank η, and taking the derivative of it yields something that is still of rank η, but with a norm upper bounded by
+O(η2/3/δ). Because we have bounded the norm when contracting V with a general matrix of rank η, that yields a
+factor of O(η2/3/δ) on whatever result we had for the lower-order derivative. The other scenario is where we take the
+derivative of Cνb, which is effectively like multiplying it by F which increases the norm (but not the rank).
+This reasoning holds in general whenever we take the derivative of an expression for the derivative of some order
+to give the derivative of higher order. The norm is multiplied by O(η2/3/δ) for each of the terms. The number of
+terms will increase exponentially with the order. The third derivative has 3×5 terms, where each of the three original
+terms yields five due to the derivatives of Cocc at each location. Then the fourth-order derivative has 3 × 5 × 7 terms
+and so on. In describing the scaling we can ignore this exponential number of terms, and give the upper bound on
+the nth order derivative as O(η2/3/δ) to the power of n. This implies that the appropriate scaling of the time should
+again be T = ∥F∥t.
+Finally we bound the norm of ∥h∥. When using a plane wave basis, hµν will be non-zero only when µ = ν with
+entries scaling as O(1/δ2) due to the ∇2 in the expression for hµν. That gives the scaling of the spectral norm for this
+component, which would be unchanged under a unitary transformation, such as the Fourier transform which maps
+plane waves to an approximately local basis.
+For the dependence of hµν on V (r), the potential will come from nuclei, and for charge-neutral systems the total
+nuclear charge will be the same as the number of electrons. If the nuclear charge were entirely at one location and we
+have a charge-neutral system, then the largest contribution to hµν would be for an approximately local basis, where
+the contribution would scale as η/δ, with the factor of η from the nuclear charge and 1/δ from the inverse distance.
+In most cases that we would be interested in, there would be a more even distribution of nuclear charges through
+the volume. In that case, if the volume scales as η, there would be an average distance O(η1/3). That would result in
+a contribution to hµν of O(η2/3). An orbital localized near one nucleus would give a contribution of O(1/δ) just from
+that nucleus, which may be larger than O(η2/3) if N > η3 but may be ignored in comparison to 1/δ2.
+As a result of these considerations, we can give the upper bound on F in the case without V (r) as
+∥F∥ = O
+�η2/3
+δ
++ 1
+δ2
+�
+.
+(A14)
+In the case with nuclei we obtain the same result, provided the nuclear charges are not clustered any closer than the grid
+spacing. Here δ = O((η/N)1/3) is the minimum grid spacing. This scaling for δ comes from taking the computational
+cell volume proportional to η (a reasonable assumption for both condensed-phase and molecular systems). Thus, the
+scaling becomes
+∥F∥ = O
+�
+N 1/3η1/3 + N 2/3
+η2/3
+�
+.
+(A15)
+In this case we can see that the first term is dominant unless N > η3.
+Appendix B: Proving sublinear gate complexity in basis size for Trotter based methods
+Here we derive the complexity for quantum simulation of the electronic structure problem given in Eq. (12). We
+consider the simulation of the electronic structure problem defined on a spatial grid in first quantization. Such a
+
+14
+Hamiltonian can be expressed as
+H = T + U + V +
+L
+�
+ℓ̸=κ=1
+ζℓζκ
+2 ∥Rℓ − Rκ∥
+(B1)
+T ≈
+η
+�
+i=1
+QFTj
+�
+��
+p∈G
+∥kp∥2
+2
+|p⟩⟨p|j
+�
+� QFT†
+j
+(B2)
+U = −
+η
+�
+j=1
+L
+�
+ℓ=1
+�
+p∈G
+ζℓ
+∥Rℓ − rp∥ |p⟩⟨p|j
+(B3)
+V =
+η
+�
+j̸=k=1
+�
+p,q∈G
+1
+2 ∥rp − rq∥ |p⟩⟨p|j |q⟩⟨q|k
+(B4)
+where QFTj is the usual quantum Fourier transform applied to register j. We emphasize that T is only approximately
+given by the expression involving the QFT. This relation is exact in the continuum limit where N → ∞. For finite-
+sized grids N, it cannot be the case that the QFT completely diagonalizes the momentum operator. Instead, writing
+T this way represents something similar to the approximations made by so-called “discrete value representation”
+methods. Using the QFT means that the evolution can be broken into a product of the evolution under T and the
+one under U + V .
+In the above expression, ℓ and κ index nuclear degrees of freedom; thus, Rℓ represents the positions of nuclei and ζℓ
+the atomic numbers of nuclei. In this appendix, we use L to denote the number of nuclei in our simulation (elsewhere,
+L is the number of time points). Furthermore, we have the following definition of grid points and their frequencies in
+the dual space defined by the QFT:
+rp = p Ω1/3
+N 1/3
+kp = 2πp
+Ω1/3
+p ∈ G
+G =
+�
+−N 1/3 − 1
+2
+, N 1/3 − 1
+2
+�3
+⊂ Z3 ,
+(B5)
+where Ω is the volume of the simulation cell and N is the number of grid points in the cell. Although it is defined
+here in more precise terms, this is essentially the same representation used in the first work on quantum simulating
+chemistry in first quantization, by Kassal et al. [42], well over a decade ago.
+We consider simulation performed using high-order product formulas with a split-operator Trotter step. What we
+mean by the latter is that we will alternate evolution under T (using the QFT) and evolution under U + V . In fact,
+the implementation of each Trotter step that we will pursue is essentially identical to the Trotter steps proposed by
+Kassal et al. [42]. The Trotter step requires �
+O(η2) gates, with the complexity being dominated by computing the
+O(η2) different interactions in the two-electron term. Recently, Low et al. [45] have shown that the number of Trotter
+steps required in second quantization using arbitrarily high order formulas can be as low as
+�
+N 1/3η1/3 + N 2/3
+η2/3
+� t1+o(1)N o(1)
+ϵo(1)
+.
+(B6)
+We note that, curiously, this also closely matches our bound for the norm of the Fock operator (see Eq. (A15)) proved
+in Appendix A. The first term in brackets similarly corresponds to a contribution to the potential from electrons
+grouped as closely as possible in real space, but the reason why this quantity is relevant is very different between the
+two calculations.
+The results for the Trotter error in second quantization also hold for first quantization. As a general principle, we
+can consider the effect of �
+j |p⟩ ⟨q|j on a computational basis state consisting of an anti-symmetric combination of
+lists of electron positions. This removes an electron from orbital q and places it in p. This is performed for every part
+of the anti-symmetric state, preserving its sign. However, for the starting anti-symmetric state the sign is based on
+whether the permutation is even or odd (as compared to ascending order). If moving an electron from q to p passes
+over an odd number of electrons, then the parity of each permutation flips. That means that there is an overall sign
+flip in the basis state.
+Similarly, if we consider the action of a†
+paq on a state a†
+q1 · · · a†
+qη |0⟩, then the aq can be anti-commuted to the
+right to give several sign flips corresponding to the number of a†
+qj operators that are anti-commuted through. This
+corresponds to the number of occupied orbitals before q. Then aqa†
+q gives the identity. Next, anti-commute a†
+p to the
+appropriate location in the list of operators. The sign that is obtained corresponds to the number of a†
+qj operators
+
+15
+that are anti-commuted through, which is the number of electrons before p. There is an overall sign flip if there is an
+odd number of electrons between p and q.
+This can then be extended to products such as
+�
+j
+|p⟩ ⟨q|j
+�
+k
+|r⟩ ⟨s|k .
+(B7)
+The first sum corresponds to a†
+paq in second quantization, and the second sum corresponds to a†
+ras. This means that
+we have the equivalence
+�
+pqrs
+Vpqrs
+�
+j
+|p⟩ ⟨q|j
+�
+k
+|r⟩ ⟨s|k ≡
+�
+pqrs
+Vpqrsa†
+paqa†
+ras.
+(B8)
+The action on an anti-symmetric computational basis state in first quantization has exactly the same effects as that on
+the corresponding second-quantization state with η electrons. Moreover, the action of the operators always preserves
+the electron number in second quantization, so there is a corresponding state in first quantization. Similarly, because
+we are using anti-symmetric states in first quantization, it is impossible to obtain a state with multiple electrons on
+the same orbital. That is because two registers with the same orbital number will give cancellation of terms.
+As a result all operators and states in second-quantization map directly to first quantization, preserving the norms,
+and in particular the error bounds derived in second-quantization hold for first quantization. Therefore, multiplying
+the number of steps in Eq. (B6) by the �
+O(η2) gate complexity required of the first quantized Trotter step from [42]
+gives the following gate complexity for the product formula based time evolution in first quantization:
+�
+N 1/3η7/3 + N 2/3η4/3� t1+o(1)N o(1)
+ϵo(1)
+.
+(B9)
+This is the complexity given in Eq. (12).
+Appendix C: Constant factors for time-evolution in the interaction-picture plane-wave algorithm
+Here we analyze the constant factors in the scaling of the interaction picture based plane wave algorithm from
+Babbush at al. [46] which was analyzed in detail for use in phase estimation by Su et al. [48]. As explained on page 30
+of [48], the number of steps to give total time T using the time evolution approach is λBT/ ln 2, but with a factor of
+3 overhead for amplitude amplification. Using the qubitization approach the number of steps is eλBT. That means
+simulating the time evolution gives an overhead of 3/(e ln 2) ≈ 1.59 over the qubitization. Then in Eq. (154) of
+[48], the total time of evolution is approximately π/(2ϵpha) to give precision ϵpha of the phase estimation. There is
+moreover a (small) term O((λU + λV )2∆E2) in the expression for the number of steps N in [48] that originates from
+the nonlinearity of the sine function in phase estimation, which is not used here.
+As a result, the complexity given in Theorem 5 of [48] can be modified to be appropriate for time evolution simply
+by replacing the formula for the number of steps in Eq. (174) of [48] with
+N = 3T(λ1
+U + λ1
+V /(1 − 1/η))
+Peq ln 2
++ O(1) .
+(C1)
+Here we have replaced π/(2ϵpha) with T, replaced e with e/ ln 2, and removed O((λU + λV )2∆E2). Note that in this
+expression
+λU = η �
+ℓ ζℓ
+πΩ1/3 λν,
+(C2)
+λV = η(η − 1)
+2πΩ1/3 λν,
+(C3)
+λν =
+�
+ν∈G0
+1
+∥ν∥2 ≤ 4πN 1/3,
+(C4)
+λ1
+U ≈ λU, λ1
+V ≈ λV , and Peq is close to 1. This expression together with an appropriate choice of constant factor
+in Ω ∝ η gives the constant factor for the number of steps to use for time evolution. It needs to be multiplied by
+a further complicated expression in Theorem 5 of [48] for the gate complexity of a single step to provide the full
+constant factor for the gate complexity in Eq. (13).
+
+16
+Appendix D: Smoothing the Coulomb operator to exponentially suppresses quantum scaling in basis size
+Here we discuss the fact that if one is willing to introduce a slight systematic bias into the Coulomb operator, it is
+possible to further improve the speedup in N of the quantum algorithm. The N 1/3 dependence enters into the cost
+from the 1-norm of the two-body Coulomb operator, which scales as λ = O(η2Vmax) where Vmax is the maximum
+value of the electron-electron interaction for a single pair of electrons.
+For typical plane wave or grid discretizations we have that Vmax = O(N 1/3/Ω1/3) where Ω is the size of the
+computational cell (for the purpose of the analysis in this paper we assume that Ω = O(η), since that is explicitly the
+case in condensed phase simulations). But we could also take steps to smooth out the cusp in the Coulomb operator
+and thus, lower the energy scale of Vmax. For example, this could be accomplished by taking Vmax to be a constant
+and modifying the real-space form of the two-body Coulomb operator as
+1
+|r1 − r2| →
+1
+|r1 − r2| + Vmax
+.
+(D1)
+Such a strategy has been explored in the context of first quantized quantum algorithms in real space in papers by
+Kivlichan et al. [88] and Childs et al. [54].
+In principle, one could choose Vmax = O(log N) and this would lead to the quantum algorithm scaling exponentially
+better than classical algorithms in N. This would also slightly reduce the cost of classical mean-field algorithms from
+scaling as N 4/3 to scaling as N.
+Of course, using such a drastic cutoff will introduce a significant bias into the
+overall dynamics. In order to avoid this, papers such as [89, 90] have sought to develop Richardson extrapolation type
+schemes where simulations are run with a series of smoothing or cutoff parameters in order to extrapolate the value
+of the observable with zero cutoff. However, questions remain about the convergence of such procedures and it seems
+likely to re-introduce some polynomial dependence on N in order to reach convergence with the continuum limit.
+Nevertheless, the context of this paper is that one might be interested in getting a speedup over low accuracy
+classical algorithms. In that spirit, one could probably make the case that if merely trying to improve in speed over
+mean-field algorithms, the error introduced in imposing a cutoff in the Coulomb operator might be less significant
+than the error due to making the mean-field approximation. Thus, this is perhaps a valid approach when competing
+with such classical methods, and thus might provide an exponential speedup.
+Appendix E: Gate complexity and speedup in various regimes
+Processor
+Algorithm for sampling |ψ(t)⟩
+Regime of optimality
+Space
+Effective gate complexity
+classical zero temp mean-field with occ-RI-K/ACE [23, 24]
+N ≤ Θ(η3)
+�
+O(Nη)
+N 4/3η7/3t(Nt/ϵ)o(1)
+classical zero temp mean-field with occ-RI-K/ACE [23, 24]
+N ≥ Θ(η3)
+�
+O(Nη)
+N 5/3η4/3t(Nt/ϵ)o(1)
+quantum
+second quantized Trotter grid algorithm [45]
+N ≤ Θ(η2)
+O(N log N)
+N 4/3η1/3t(Nt/ϵ)o(1)
+quantum
+first quantized Trotter grid algorithm here
+Θ(η2) ≤ N ≤ Θ(η3)
+O(η log N)
+N 1/3η7/3t(Nt/ϵ)o(1)
+quantum
+first quantized Trotter grid algorithm here
+Θ(η3) ≤ N < Θ(η4)
+O(η log N)
+N 2/3η4/3t(Nt/ϵ)o(1)
+quantum
+qubitization algorithms from [46] or [48]
+N = Θ(η4)
+O(η log N)
+�
+O(N 2/3η4/3t)
+quantum
+interaction picture algorithms from [46] or [48]
+N > Θ(η4)
+O(η log N)
+�
+O(N 1/3η8/3t)
+TABLE II. Best known gate complexities of exact quantum algorithms and classical mean-field algorithms for sampling the
+output of time-evolution, by ratio of basis size to particle number. Here we use the asymptotic Θ(·) notation, which implies the
+union of both an asymptotic upper-bound and an asymptotic lower-bound on the scaling. N is number of basis functions, η is
+number of particles, ϵ is target precision, and t is duration of evolution. “Effective gate complexity” is the leading order scaling
+in the stated regime. All quantum algorithms discussed here require either a plane wave or grid basis. For those basis sets, the
+large space overhead of second quantization likely makes second quantized approaches infeasible in practice. When N = η4,
+the quantum algorithms with the best asymptotic scaling are the plane wave or grid basis qubitization algorithms from [46] or
+[48], respectively, as opposed to the interaction picture algorithms of those same works. This is due to lower polylogarithmic
+factors in the scaling that are suppressed by the �
+O(·) notation.
+Another way to express the results of Table II is as a formula for the leading order scaling if assume that N = Θ(ηα).
+Then, for the classical algorithm we have that the leading gate complexity of the best approach is
+�
+ηβt
+� �Nt
+ϵ
+�o(1)
+where
+N = Θ (ηα)
+and
+β =
+�
+4α+7
+3
+α ≤ 3
+5α+4
+3
+α ≥ 3 .
+(E1)
+
+17
+FIG. 1.
+Plot showing the numerical values of the speedup exponent ratio given in Eq. (E3). We see that a super-quadratic
+speedup of exact quantum algorithms over mean-field classical algorithms is realized when α < 5/4 and when α > 4.
+By contrast, for the quantum algorithm we have that the leading order gate complexity of the best approach is
+�
+ηβt
+� �Nt
+ϵ
+�o(1)
+where
+N = Θ (ηα)
+and
+β =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+4α+1
+3
+α ≤ 2
+α+7
+3
+2 ≤ α ≤ 3
+2α+4
+3
+3 ≤ α ≤ 4
+α+8
+3
+α ≥ 4
+.
+(E2)
+For both classical and quantum expressions, these complexities are sometimes loose by sub-polynomial factors. Finally,
+we compare the speedup that exact quantum algorithms offer over classical mean-field algorithms. We report this as
+exponent of η scaling of classical complexity
+exponent of η scaling of quantum complexity =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(4α + 7) / (4α + 1)
+α ≤ 2
+(4α + 7) / (α + 7)
+2 ≤ α ≤ 3
+(5α + 4) / (2α + 4)
+3 ≤ α ≤ 4
+(5α + 4) / (α + 8)
+α ≥ 4
+if
+N = Θ (ηα) .
+(E3)
+We plot numerical values of this speedup in Figure 1.
+Finally, we discuss the hope that Trotter based first quantized algorithms might be sped up by a factor of �
+O(η) by
+developing more efficient Trotter steps. The bottleneck for Trotter steps is the computation of the Coulomb operator
+since the simulation of the kinetic operator scales as �
+O(η). Thus, it seems promising that fast-multipole [51] Barnes-
+Hut [52], or particle-mesh Ewald [53] type algorithms for computing the Coulomb potential require �
+O(η) operations
+in the classical random access memory (RAM) model. By contrast, the standard way of computing the Coulomb
+potential (involving summing up all
+�η
+2
+�
+pairs of electrons) scales as �
+O(η2). Thus, if one can figure out how to extend
+these better scaling methods to first quantization with �
+O(η) operations in the reversible circuit model (the cost model
+of relevance for this subroutine if executed on a quantum computer), the quantum algorithm would scale as
+�
+N 1/3η4/3t + N 2/3η1/3t
+� �Nt
+ϵ
+�o(1)
+.
+(E4)
+We note that it is straightforward to adapt these algorithms to second quantization with �
+O(N) gate complexity
+[45, 47]. However, translating such algorithms to first quantization with �
+O(N) gate complexity in the quantum circuit
+model is highly non-trivial. This is due to nuances of how adaptive tree-like data structures are constructed and used
+in these algorithms, and it is why the work of [54] decided to invoke the impractical assumption of QRAM in order to
+leverage the fast multipole algorithms. Note further that some of these algorithms such as the original fast multipole
+[51] and particle-mesh Ewald [53] make further assumptions on the state. In particular, if space is partitioned into
+O(η) boxes, then these methods require that no more than k electrons are present in any box, in any configuration
+
+2.4 
+2.3 
+2.2
+ speedup ratio
+2.1
+quadratic speedup threshold
+2.0
+1.9
+1.8
+1.7
+2
+3
+4
+5
+1
+6
+value of α in N= (n~)18
+on which the wavefunction has support. Since electrons tend to repel one another this is often a good assumption
+at low energies, but it is not true for general states. It seems possible to implement a first quantized algorithm with
+�
+O(η k) space complexity and �
+O(η poly(k)) gate complexity by keeping k electron registers for each of these O(η)
+boxes of space. But there also exist versions of these algorithms, e.g. described in [91], which use RAM and an
+adaptive tree structure to give �
+O(η) complexity without any assumptions on the state. Such approaches appear quite
+challenging to port to the quantum circuit model with the same complexity. However, if possible, the first quantized
+fast multipole-based Trotter would scale better than all other known approaches as long as N < η7. When N > η7,
+the first quantized interaction picture algorithm has better scaling.
+Appendix F: Efficient reduced density matrix estimation using classical shadows in first quantization
+1.
+Problem statement
+We consider a system of η identical fermions occupying N ≫ η orbitals. In first-quantization, we represent the
+state of such a system as a wavefunction on η registers of n = ⌈log(N)⌉ qubits. We demand that this wavefunction is
+antisymmetric under the exchange of any two registers in order for it to represent a valid physical state.
+Most physically interesting observables of such a system are captured by the few-body marginals, the reduced
+density matrices. In this section, we concern ourselves with efficiently estimating elements of the k-body reduced
+density matrix (k-RDM) of the first-quantized state |ψ⟩ defined on η identical fermion particles,
+kDj1,...,jk
+i1,...,ik =
+η!
+(η − k)! tr
+�
+|ψ⟩⟨ψ|
+k
+�
+ℓ=1
+|iℓ⟩⟨jℓ|ℓ
+�
+,
+(F1)
+where |i⟩⟨j|ℓ indicates the tensor product of |i⟩⟨j| on the ℓth register with the identity on the other η − 1 registers. We
+can write an equivalent definition (equivalent due to the antisymmetry of the wavefunction),
+kDj1,...,jk
+i1,...,ik =
+�
+x∈Sη
+k
+tr
+�
+|ψ⟩⟨ψ|
+k
+�
+ℓ=1
+|iℓ⟩⟨jℓ|xℓ
+�
+,
+(F2)
+where Sη
+k is the set composed of all possible sequences of length k generating by drawing without replacement from
+[η] := {1, . . . , η}.
+Our goal is to use measurements of the state |ψ⟩ to obtain a classical description of the state with enough information
+to approximate all N 2k elements of the k-RDM. We would like all of these estimates to accurate up to some additive
+error ϵ with probability at least 1 − δ.
+Ideally, our protocol will be efficient not only in terms of the number
+of measurements, but also in terms of the (gate) complexity of implementing each measurement and the classical
+complexity of the required post-processing.
+We will accomplish our goal by applying the classical shadows formalism of Ref. 57. We propose and analyze a
+protocol that requires at most
+m = 64e3 log (N/δ) k (2k + 2e)k ηkϵ−2
+(F3)
+measurements to estimate the k-RDM. Performing these measurements requires acting on each of the particle registers
+with a randomly sampled Clifford circuit and performing a measurement in the computational basis. These circuits
+can be implemented using O(ηn2) one- and two-qubit Clifford gates on a linearly connected array of qubits in depth
+O(n). Each element of the k-RDM requires performing a number of classical operations that scales as
+m′ = O
+��
+n4 + log (1/δ)
+�
+η2k2kϵ−2�
+.
+(F4)
+2.
+The measurement protocol
+The classical shadows formalism of Huang et al. works by choosing an ensemble of random unitaries U on n qubits
+and defining a measurement channel
+M(σ) := EU∼U
+�
+b∈{0,1}n
+U † |b⟩⟨b| U ⟨b|UσU †|b⟩ .
+(F5)
+
+19
+For specific choices of U, the channel M is analytically invertible. Operationally, we obtain the classical shadow of σ by
+repeatedly sampling a unitary U from U, applying the sampled U to a copy of σ, and measuring in the computational
+basis (obtaining the bitstring b). If we collect m such samples, then we call the (potentially unphysical) state
+ˆσ := 1
+m
+m
+�
+i=1
+M−1 �
+U †
+i |bi⟩⟨bi| Ui
+�
+(F6)
+a classical shadow of σ. For an arbitrary observable O, we can define an estimator ˆo of the quantity tr [Oρ] using the
+classical shadow of ρ,
+ˆo := tr [Oˆρ] .
+(F7)
+In expectation, we have that
+⟨ˆσ⟩ = EU∼U
+�
+b∈{0,1}n
+M−1 �
+U † |b⟩⟨b| U
+�
+⟨b|UσU †|b⟩ = M−1 (M (σ)) = σ.
+(F8)
+When we take U to be the uniform distribution over the Clifford group on n qubits, the classical shadows measurement
+channel and its inverse have particularly simple forms [57],2
+M(A) =
+1
+2n + 1A + tr [A]
+2n + 1I,
+(F9)
+M−1(A) = (2n + 1)A − tr [A] I.
+(F10)
+Here, and throughout our analysis of the measurement protocol, we use the symbol I to denote the identity operator
+on a Hilbert space whose dimension is appropriate for the context.
+In this work, we propose and analyze the impact of using an ensemble U that consists of a tensor product of η
+copies of the uniform distribution over n qubit Clifford circuits,
+U =
+η
+�
+j=1
+Cl(2n).
+(F11)
+That is to say, we perform our measurements by independently sampling η n-qubit Clifford unitaries, applying one
+to each particle register, and measuring in the computational basis. We can consider the action of the corresponding
+classical shadow measurement channel and its inverse on an operator X1⊗· · ·⊗Xη that factorizes across the η registers.
+The channel is defined on the whole Hilbert space by linear extension. For the classical shadow measurement channel,
+we have
+M(X1 ⊗ · · · ⊗ Xη) =
+η
+�
+j=1
+�
+�EUj∼Cl(2n)
+�
+bj∈{0,1}n
+U †
+j |bj⟩⟨bj| Uj ⟨bj|UjXjU †
+j |bj⟩
+�
+�
+(F12)
+=
+η
+�
+j=1
+�Xj + tr [Xj] I
+2n + 1
+�
+.
+(F13)
+The inverse, similarly, is given by
+M−1(X1 ⊗ · · · ⊗ Xη) =
+η
+�
+j=1
+((2n + 1) Xj − tr [Xj] I) .
+(F14)
+Due to the antisymmetry of the wavefunction, we have the freedom to choose between a number of different
+observables when estimating the elements of the k-RDM. Consider an arbitrary operator O, and the operator POP †,
+where P is an operator that permutes the particle registers. The expectation values of O and POP † with respect
+to a first-quantized wavefunction are the same (to see this, observe that any sign picked up by acting P † on the ket
+2 Actually, a substantial constant factor savings in the number of
+gates can be obtained by using the canonical form of Ref. 92 and
+simply dropping the permutation at the end of the circuit. See,
+e.g., Ref. 93.
+
+20
+is cancelled out by a corresponding sign obtained from acting P on the bra). We can use this degree of freedom
+to minimize the variance of our measurement protocol. Using the observable from Eq. (F1) to construct a classical
+shadow estimator of a k-RDM element would lead to an unnecessarily large variance, essentially because the observable
+doesn’t take advantage of all of the information present in the state. In contrast, Eq. (F2) defines the k-RDM element
+in terms of a sum over many different permutations of the registers. We conjecture that a measurement protocol
+based on the observable in Eq. (F2) would perform well, but the analysis could be tedious due to the many different
+cases that would arise.
+Rather than using the observables implied by either Eq. (F1) or Eq. (F2) in our classical shadow measurement
+procedure, we instead choose to estimate the k-RDM elements using an observable that involves a sum over a simpler
+set of permutations. Essentially, we break the η registers up into k groups of size η/k and measure the k-RDM element
+using registers from each group. For ease of notation, let us assume that η is divisible by k.3 Formally, we can define
+a set of sequences
+Rk = {1, . . . , η/k} × {η/k + 1, . . . , 2η/k} × · · · × {(k − 1) η/k + 1, . . . , η} .
+(F15)
+Due to the antisymmetry of the wavefunction, we have that
+kDj1,...,jk
+i1,...,ik =
+kk (η!)
+ηk (η − k)!
+�
+x∈Rk
+tr
+�
+|ψ⟩⟨ψ|
+k
+�
+ℓ=1
+|iℓ⟩⟨jℓ|xℓ
+�
+.
+(F16)
+We define an estimator ˆd for the k-RDM element kDj1,...,jk
+i1,...,ik using the classical shadow ˆρ of |ψ⟩,
+ˆd =
+kk (η!)
+ηk (η − k)!
+�
+x∈Rk
+tr
+�
+ˆρ
+k
+�
+ℓ=1
+|iℓ⟩⟨jℓ|xℓ
+�
+.
+(F17)
+In Appendix F 4, we prove that the single-shot variance of this estimator is bounded by
+Var( ˆd) ≤ e3ηk (2k + 2e)k .
+(F18)
+In order to guarantee that our estimates are close to the true value of the k-RDM elements with high probability,
+we need to proceed along the same lines as Ref. 57 and construct a median-of-means estimator to obtain the desired
+rigorous guarantees [94]. To be precise, using Proposition 12 from Ref. 94, we can consider an estimator that divides
+the m total classical shadow samples into K groups of size b, and takes the median of the sample mean obtained by
+averaging the estimates within each group. The probability that this median of means estimator has an error larger
+than 2
+�
+Var( ˆd)/b is at most e−K/8. To bound the error in our estimate by ϵ with a success probability of at least
+1 − δ, this implies that we need
+b = 4 Var( ˆd)/ϵ2,
+(F19)
+K = 8 log (1/δ) .
+(F20)
+The overall number of measurements claimed in Eq. (F3) follows directly from applying a union bound over the failure
+probabilities for estimating all N 2k k-RDM elements.
+The measurement protocol can be summarized as follows. We take a classical shadow of |ψ⟩ with the U defined in
+Eq. (F11) using a number of samples m chosen according to Eq. (F3). For each sample, we evaluate the expectation
+values of the (η/k)k different terms in the sum over Rk (see Eq. (F17)) using generalizations of Gottesman-Knill
+theorem that account for the phase of the quantities involved [95–97]. Breaking the samples into K groups of size b,
+averaging within the groups, and then taking the median of these means then yields the final estimate. The classical
+post-processing costs quoted in Eq. (F4) come from counting the number of n-qubit sized Clifford circuits that need
+to be simulated classically to carry out this procedure.
+3 In the event that η is not exactly divisible by k, one could modify
+the protocol to either use groups of slightly different sizes or to
+only perform the measurements using η′ = k⌊η/k⌋ registers.
+
+21
+3.
+Notation and preliminaries
+Before we proceed to bound the variance of the estimator ˆd for an arbitrary k-RDM element, it is helpful to recall
+a few useful expressions and prove some identities that we will use later.
+We will make use of a formula for the two-fold twirl over the Clifford group and partial trace obtained from Ref. 57,
+EU∼Cl(2n)U † |x⟩⟨x| U ⟨x|UAU †|x⟩ = A + tr(A)I
+2n(2n + 1) .
+(F21)
+For the three-fold twirl and partial trace, we find it convenient to use the identity
+EU∼Cl(2n)U † |x⟩⟨x| U ⟨x|UBU †|x⟩ ⟨x|UCU †|x⟩ =
+1
+2n (2n + 1) (2n + 2) (I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB) .
+(F22)
+This equation is different from the corresponding one considered in previous work (Eq. (S36) of Ref. 57), in that it
+allows for B and C to have non-zero trace. It can be obtained directly from the analysis of Ref. 98.4
+Another small departure we make from some prior work is that we directly consider the variance of estimators for
+the expectation values of non-Hermitian observables. For a classical shadow ˆρ of a state ρ and an estimator ˆo = tr [ˆρO]
+of the expectation value of a (not necessarily Hermitian) operator O, we have
+Var(ˆo) = tr
+�
+ρ
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(O)U †|b⟩ ⟨b|UM−1(O†)U †|b⟩
+�
+− |tr [Oρ]|2
+(F23)
+≤ tr
+�
+ρ
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(O)U †|b⟩ ⟨b|UM−1(O†)U †|b⟩
+�
+.
+(F24)
+This expression can be arrived at from the definition of the variance of a complex-valued random variable applied to
+the classical shadow formalism. We refer the reader to Ref. 59 for a thorough discussion.
+In the course of calculating the variance for the higher-order RDMs, we will find that we repeatedly need to simplify
+certain expressions. Before describing those expressions and showing how they may be simplified, let us define some
+notation used for convenience throughout the rest of our analysis:
+Px = |x⟩⟨x| ,
+(F25)
+Pxy = |x⟩⟨y| ,
+(F26)
+EU = EU∼Cl(2n),
+(F27)
+�
+b
+=
+�
+b∈{0,1}n
+.
+(F28)
+One class of expressions that we will need to simplify are of the form
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pij)U †|b⟩ .
+(F29)
+We can use Eq. (F10) and Eq. (F21) to simplify Eq. (F29),
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pij)U †|b⟩
+(F30)
+= EU
+�
+b
+U †PbU ⟨b|U ((2n + 1) Pij − δi,jI) U †|b⟩
+(F31)
+= Pij.
+(F32)
+4 Note that while the proof of Lemma 7 in Ref. 98 is technically
+for Hermitian matrices, the same proof holds exactly in the non-
+Hermitian case.
+
+22
+Another kind of expression that we will need to simplify is of the form
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pkl)U †|b⟩ .
+(F33)
+Let us consider the first case, and simplify A as defined below,
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pi)U †|b⟩ ⟨b|UM−1(Pi)U †|b⟩ .
+(F34)
+We have
+M−1 (Pi) = (2n + 1) Pi − I
+(F35)
+by an application of Eq. (F10). Now we can apply Eq. (F22) with B = C = (2n + 1) Pi − I.
+Let us simplify the pieces of Eq. (F22) separately before combining them. We have
+BC = CB = ((2n + 1) Pi − I)2
+(F36)
+= (2n + 1) (2n − 1) Pi + I,
+(F37)
+tr [BC] = tr [CB] = 2n(2n + 1) − 1,
+(F38)
+tr [B] = tr [C] = 1.
+(F39)
+As a result,
+I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB
+(F40)
+= 2n (2n + 1) I + 2 (2n + 1) Pi − 2I + 2 (2n + 1) (2n − 1) Pi + 2I
+(F41)
+= 2n (2n + 1) (I + 2Pi) .
+(F42)
+Putting everything together, we have
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pi)U †|b⟩ ⟨b|UM−1(Pi)U †|b⟩
+(F43)
+=
+2n
+2n + 2 (I + 2Pi) .
+(F44)
+Now we consider simplifying the expression
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pji)U †|b⟩ .
+(F45)
+In this case, we can again use Eq. (F22) with
+B = M−1(Pij) = (2n + 1) Pij,
+(F46)
+C = M−1(Pji) = (2n + 1) Pji.
+(F47)
+Working out some of the pieces, we have
+BC = (2n + 1)2 Pi,
+(F48)
+CB = (2n + 1)2 Pj,
+(F49)
+tr [BC] = (2n + 1)2 ,
+(F50)
+tr [B] = tr [C] = 0.
+(F51)
+Therefore,
+I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB
+(F52)
+= (2n + 1)2 (I + Pi + Pj) .
+(F53)
+Finally, we have
+A = EU
+�
+b
+U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pji)U †|b⟩
+(F54)
+= 2n + 1
+2n + 2 (I + Pi + Pj) .
+(F55)
+
+23
+4.
+Variance of the k-RDM with a restricted sum
+Now we are ready to turn to the task of bounding the variance ˆd as defined in Eq. (F17). For now, we neglect the
+coefficient in order to simplify the presentation. Let
+O =
+�
+x∈Rk
+Ox,
+(F56)
+Ox =
+k
+�
+ℓ=1
+|iℓ⟩⟨jℓ|xℓ .
+(F57)
+The variance of the classical shadow estimator ˆo of ⟨O⟩ is bounded by
+Var(ˆo) ≤
+�
+x∈Rk
+�
+y∈Rk
+tr [|ψ⟩⟨ψ| Axy] ,
+(F58)
+Axy =
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(Ox)U †|b⟩ ⟨b|UM−1(O†
+y)U †|b⟩ .
+(F59)
+Because the inverse channel, the random unitaries, and the Ox all factorize across the registers, we can rewrite Axy
+as a tensor product,
+Axy =
+η
+�
+z=1
+Az
+xy,
+(F60)
+where Az
+xy takes one of three forms depending on whether neither, one of, or both of Ox and Oy act non-trivially on
+the zth register. If z /∈ x and z /∈ y, then
+Az
+xy = I.
+(F61)
+If exactly one of z ∈ x or z ∈ y is true, then we can use Eq. (F32) to simplify our expression for Az
+xy. The cases
+are symmetric between z ∈ x and z ∈ y, so we can treat only the first case without loss of generality. Let ℓ denote
+the index of z in x (i.e., xℓ = z). We have
+Az
+xy =
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩
+(F62)
+= |iℓ⟩⟨jℓ|z .
+(F63)
+If z ∈ y we instead have Az
+xy = |jℓ⟩⟨iℓ|z.
+The third case we must consider is where z ∈ x and z ∈ y. Let ℓ denote the index of z in x and y (they must be
+the same because of the way we construct x and y). In this case,
+Az
+xy =
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩ U ⟨b|UM−1(|jℓ⟩⟨iℓ|)U †|b⟩ .
+(F64)
+If iℓ = jℓ we can simplify this expression using Eq. (F44), otherwise we can use Eq. (F55). The combination of these
+two formulas lets us write
+Az
+xy =
+�
+b
+EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩ U ⟨b|UM−1(|jℓ⟩⟨iℓ|)U †|b⟩
+(F65)
+= 2n + 1 − δiℓ,jℓ
+2n + 2
+(I + |iℓ⟩⟨iℓ| + |jℓ⟩⟨jℓ|) .
+(F66)
+Now we will use the antisymmetry of |ψ⟩ to bound the quantity |tr [|ψ⟩⟨ψ| Axy]|. Let
+a = |x ∩ y|,
+b = 2k − 2a.
+(F67)
+The operator Axy acts non-trivially on a + b registers. On a registers, it acts with an operator of the form given in
+Eq. (F66). On the other b registers, it acts as |c⟩⟨d| for some c, d (that can vary per register). Due to the antisymmetry
+of |ψ⟩, we can freely permute the registers without affecting the expectation value.
+
+24
+We can therefore rewrite the expectation value of interest as
+|tr [|ψ⟩⟨ψ| Axy]| =
+����� ⟨ψ|
+� a
+�
+ℓ=1
+2n + 1 − δcℓ,dℓ
+2n + 2
+(I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)
+a+b
+�
+ℓ=a+1
+|cℓ⟩⟨dℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩
+�����
+(F68)
+≤
+����� ⟨ψ|
+� a
+�
+ℓ=1
+(I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)
+a+b
+�
+ℓ=a+1
+|cℓ⟩⟨dℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩
+����� .
+(F69)
+a.
+Removing the off-diagonal terms
+We can simplify the bound in Eq. (F69) by replacing the off-diagonal matrix elements with projectors. To do so,
+we will need the following lemma.
+Lemma 1. Let |ψ⟩ be an arbitrary normalized pure quantum state on n qubits.
+Let O be an arbitrary positive
+semidefinite operator on a qubits, and let |α⟩ and |β⟩ be arbitrary orthonormal quantum states on n − a qubits. Then,
+| ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| ≤ | ⟨ψ|(O ⊗ |φ⟩⟨φ|)|ψ⟩|
+(F70)
+for |φ⟩ = |α⟩ or |φ⟩ = |β⟩.
+Proof. To begin the proof, expand |ψ⟩ as
+|ψ⟩ =
+�
+ij
+cij |i⟩ |j⟩ ,
+(F71)
+where the states {|i⟩} form an eigenbasis for O and the states {|j⟩} are an orthonormal basis such that |α⟩ , |β⟩ ∈ {|j⟩}.
+Then
+| ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| =
+�����
+�
+i
+c∗
+iαciβOii
+�����
+(F72)
+=
+�
+i
+k∗
+iαkiβ,
+(F73)
+where Oii denotes the eigenvalue of O corresponding to the eigenvector |i⟩ and kij is defined implicitly as kij = cij
+√Oii.
+We can consider the quantity in Eq. (F73) as the inner product of two vectors ⃗kα and ⃗kβ. The Cauchy-Schwarz
+inequality tells us that
+�����
+�
+i
+k∗
+iαkiβ
+����� ≤
+�
+�
+�
+�
+��
+i
+k∗
+iαkiα
+� ��
+i
+k∗
+iβkiβ
+�
+.
+(F74)
+We can choose γ ∈ {α, β} such that
+�����
+�
+i
+k∗
+iγkiγ
+����� ≥
+�����
+�
+i
+k∗
+iαkiα
+����� and
+�����
+�
+i
+k∗
+iγkiγ
+����� ≥
+�����
+�
+i
+k∗
+iβkiβ
+����� .
+(F75)
+Therefore, we have that
+| ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| ≤
+�����
+�
+i
+k∗
+iγkiγ
+�����
+(F76)
+=
+�����
+�
+i
+c∗
+iγciγOii
+�����
+(F77)
+= ⟨ψ|(O ⊗ |γ⟩⟨γ|)|ψ⟩
+(F78)
+for either |γ⟩ = |α⟩ or |γ⟩ = |β⟩ We can remove the absolute value bars in the final line because O ⊗|γ⟩⟨γ| is a positive
+semidefinite operator.
+
+25
+Now we can return to our bound from Eq. (F69),
+|tr [|ψ⟩⟨ψ| Axy]| ≤
+����� ⟨ψ|
+� a
+�
+ℓ=1
+(I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)
+a+b
+�
+ℓ=a+1
+|cℓ⟩⟨dℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩
+����� .
+(F79)
+By rearranging the registers, we can apply Lemma 1. Taking |α⟩ to be �a+b
+ℓ=a+1 |cℓ⟩ and ⟨β| to be �a+b
+ℓ=a+1 ⟨dℓ|, we
+can show that either
+|tr [|ψ⟩⟨ψ| Axy]| ≤ ⟨ψ|
+� a
+�
+ℓ=1
+(I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)
+a+b
+�
+ℓ=a+1
+|cℓ⟩⟨cℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩
+(F80)
+holds, or an equivalent expression with |dℓ⟩⟨dℓ| instead of |cℓ⟩⟨cℓ| in the second set of registers. Both cases are identical,
+so we will proceed using the label gℓ for whichever choice is valid in each register.
+We can also simplify the expression in the first registers. We claim that, for each register, we can replace the term
+|cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ| with either 2 |cℓ⟩⟨cℓ| or 2 |dℓ⟩⟨dℓ| without making the expectation value any smaller. This can be seen
+by proceeding register by register, using the linearity of the expectation value. Here again, the choice of cℓ or dℓ in
+each register is immaterial, so we use the label gℓ to denote whichever one is appropriate for each register. Making
+this simplification, we have that
+|tr [|ψ⟩⟨ψ| Axy]| ≤ ⟨ψ|
+� a
+�
+ℓ=1
+(I + 2 |gℓ⟩⟨gℓ|)
+a+b
+�
+ℓ=a+1
+|gℓ⟩⟨gℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩ .
+(F81)
+b.
+Taking advantage of antisymmetry
+Now we will take advantage of the antisymmetry of |ψ⟩ to bound the expectation values in Eq. (F81). It is helpful
+to rewrite the expression in the first set of registers in a different form:
+a
+�
+ℓ=1
+(I + 2 |gℓ⟩⟨gℓ|) =
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+a
+�
+ℓ=1
+W S
+ℓ ,
+(F82)
+where W S
+ℓ = |gℓ⟩⟨gℓ| if ℓ ∈ S and Wℓ = I otherwise. This then leads us to the bound
+|tr [|ψ⟩⟨ψ| Axy]| ≤
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+⟨ψ|
+� a
+�
+ℓ=1
+W S
+ℓ
+a+b
+�
+ℓ=a+1
+|gℓ⟩⟨gℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩ .
+(F83)
+Now that we have obtained this bound, we will proceed to use the antisymmetry of |ψ⟩ to show that
+⟨ψ|
+� a
+�
+ℓ=1
+W S
+ℓ ,
+a+b
+�
+ℓ=a+1
+|gℓ⟩⟨gℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩ ≤
+1
+P(η, |S| + b) = (η − |S| − b)!
+η!
+.
+(F84)
+To do so, let us prove the following lemma,
+Lemma 2. Let |ψ⟩ be a normalized pure state on η registers of n qubits each. Furthermore, let S |ψ⟩ = − |ψ⟩ for any
+operator S that swaps the states of two of the registers. Let {Pi}i∈[k] be a set of projectors onto orthonormal n qubit
+states. Then
+0 ≤ ⟨ψ|
+� k
+�
+i=1
+Pi
+η
+�
+i=k+1
+I
+�
+|ψ⟩ ≤
+1
+P(η, k) = (η − k)!
+η!
+,
+(F85)
+where P(η, k) denotes the number of ways to choose a sequence of k items from a set of size η.
+Proof. Let Sk denote the set of all sequences obtained by choosing k items from the set [η]. Note that two sequences
+with the same elements in different orders are treated as distinct elements of Sk. For a sequence s ∈ Sk we define the
+operator As as the operator that acts on register si with the projector Pi for all i ∈ [k] and acts on the other η − k
+
+26
+registers with the identity operation. Note that all of the operators As are defined using the same set of k projectors
+acting on (potentially) different registers.
+We will prove the claim by showing that
+�
+s∈Sk
+⟨ψ|As|ψ⟩ ≤ 1.
+(F86)
+Clearly the operators {As}s∈Sk are all projectors onto different subspaces.
+In general, these projectors are not
+orthogonal (under the Hilbert-Schmidt inner product). Equivalently, we could say that the +1 eigenspaces of these
+operators are not orthogonal in general.
+However, we can show that |ψ⟩ has no support on states that are in the +1 eigenspace of more than one of these
+projectors. Consider Ax and Ay for x ̸= y. There must be some register ℓ on which they act differently. If Ax and
+Ay both act on register ℓ with distinct projectors Pi and Pj then AxAy = 0 and their eigenspaces have no overlap,
+so we are done. Assume that only one of Ax and Ay acts on register ℓ. Without loss of generality we consider the
+case where Ax acts on register ℓ with the projector Pi. Then, by definition, Ay acts on a different register ℓ′ with
+Pi (since Ay acts with exactly the same projectors as Ax, just on a potentially different set of registers). Due to the
+antisymmetry of |ψ⟩, we therefore have ⟨ψ|AxAy|ψ⟩ = 0.
+Therefore, we can assert that
+�
+s∈Sk
+⟨ψ|As|ψ⟩ ≤ 1.
+(F87)
+This could be seen in more detail by expanding |ψ⟩ in the basis that diagonalizes all of the {As} and applying the
+fact that if Ax |φ⟩ = 1 then Ay |φ⟩ = 0 for all x ̸= y. The antisymmetry of |ψ⟩ also implies that ⟨ψ|Ax|ψ⟩ = ⟨ψ|Ay|ψ⟩
+for all x, y. Therefore, we have that
+|Sk| ⟨ψ|As|ψ⟩ ≤ 1
+(F88)
+for any As. The {As} are all positive semidefinite, so we can bound the expectation value of the particular one from
+Eq. (F85) below by zero and divide by |Sk| = P(η, k) to yield
+0 ≤ ⟨ψ|
+� k
+�
+i=1
+Pi
+η
+�
+i=k+1
+I
+�
+|ψ⟩ ≤
+1
+P(η, k) = (η − k)!
+η!
+,
+(F89)
+completing the proof.
+Eq. (F84) follows directly from this lemma and the fact that we can freely permute the observables between registers
+without changing the expectation value. Now we can return to Eq. (F83) and apply Eq. (F84) to show that
+|tr [|ψ⟩⟨ψ| Axy]| ≤
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+⟨ψ|
+� a
+�
+ℓ=1
+W S
+ℓ
+a+b
+�
+ℓ=a+1
+|gℓ⟩⟨gℓ|
+η
+�
+ℓ=a+b+1
+I
+�
+|ψ⟩
+(F90)
+≤
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+(η − w − b)!
+η!
+(F91)
+=
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+(η − w)!
+η!
+(η − w − b)!
+(η − w)!
+(F92)
+≤
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+(η − w)!
+η!
+(η − a − b)!
+(η − a)!
+,
+(F93)
+
+27
+with the last inequality following from the fact that η − a ≤ η − w. Then we have that
+|tr [|ψ⟩⟨ψ| Axy]| ≤
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+(η − w)!
+η!
+(η − a − b)!
+(η − a)!
+,
+(F94)
+= (η − a − b)!
+(η − a)!
+a
+�
+w=0
+2w
+�
+S⊆[a]:|S|=w
+(η − w)!
+η!
+(F95)
+= (η − a − b)!
+(η − a)!
+a
+�
+w=0
+2w
+�a
+w
+�(η − w)!
+η!
+(F96)
+≤ (η − a − b)!
+(η − a)!
+a
+�
+w=0
+2w
+�a
+w
+�(a − w)!
+a!
+(F97)
+= (η − a − b)!
+(η − a)!
+a
+�
+w=0
+2w
+w!
+(F98)
+≤ (η − a − b)!
+(η − a)!
+∞
+�
+w=0
+2w
+w!
+(F99)
+= (η − a − b)!
+(η − a)!
+e2,
+(F100)
+where the last step is obtained by the application of a well-known formula for the infinite sum of the sequence in
+Eq. (F99).
+c.
+Putting the pieces together
+Having shown that
+|tr [|ψ⟩⟨ψ| Axy]| ≤ e2 (η − a − b)!
+(η − a)!
+,
+(F101)
+we are ready to return to the bound in Eq. (F58), which we recall below:
+Var(ˆo) ≤
+�
+x∈Rk
+�
+y∈Rk
+tr [|ψ⟩⟨ψ| Axy] .
+(F102)
+We then have that
+Var(ˆo) ≤
+�
+x∈Rk
+�
+y∈Rk
+|tr [|ψ⟩⟨ψ| Axy]|
+(F103)
+≤
+�
+x∈Rk
+�
+y∈Rk
+e2 (η − a − b)!
+(η − a)!
+.
+(F104)
+Recall that we defined a and b in Eq. (F67) in the following way,
+a = |x ∩ y|,
+b = 2k − 2a.
+(F105)
+Recall also the definition of the set of sequences Rk from Eq. (F15),
+Rk = {1, . . . , η/k} × {η/k + 1, . . . , 2η/k} × · · · × {(k − 1) η/k + 1, . . . , η} .
+(F106)
+Colloquially, a sequence in Rk indexes a set of k registers, one from the first group of η/k, one from the second group
+of η/k, and so on.
+Let us consider a fixed sequence x ∈ Rk and determine how many sequences y ∈ Rk exist for a specific value of a.
+For a fixed value of a, x and y share a elements. By construction, there are
+�k
+a
+�
+different choices for these a elements
+(because there are k groups and x and y can either match or fail to match in each group). In each of the k −a groups
+
+28
+of registers where x and y don’t match, there are exactly η/k − 1 ways to choose the corresponding element of y.
+Therefore, for a given a and x, we have that
+|{y ∈ Rk : |x ∩ y| = a}| =
+�k
+a
+�
+(η/k − 1)k−a .
+(F107)
+The only way that a particular x or y enters into Eq. (F104) is through a and b, so we can use this fact to take the
+sums over x and y, yielding
+Var(ˆo) ≤
+�
+x∈Rk
+�
+y∈Rk
+e2 (η − a − b)!
+(η − a)!
+(F108)
+≤ e2 �
+x∈Rk
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)k−a (η − 2k + a)!
+(η − a)!
+(F109)
+≤ e2 (η/k)k
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)k−a (η − 2k + a)!
+(η − a)!
+,
+(F110)
+under the assumption that η > 2k so that we don’t have to restrict the sum over a.
+Simplifying the inequality further, we find that
+Var(ˆo) ≤ e2 (η/k)k (η/k − 1)k
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)−a (η − 2k + a)!
+(η − a)!
+(F111)
+≤ e2 (η/k)k (η/k − 1)k
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)−a (η − 2k + a)!
+(η − k)!
+.
+(F112)
+Now we employ the upper and lower bounds from Stirling’s formula (that hold for any integer n > 0),
+√
+2πn
+�n
+e
+�n
+< n! < e
+√
+2πn
+�n
+e
+�n
+.
+(F113)
+We can use these bounds to simplify the ratio of factorials in Eq. (F112),
+(η − 2k + a)!
+(η − k)!
+≤ e
+�
+2π (η − 2k + a)
+�η − 2k + a
+e
+�η−2k+a
+1
+(η − k)!
+(F114)
+≤ e
+�
+2π (η − k)
+�η − k
+e
+�η−2k+a
+1
+(η − k)!
+(F115)
+≤ e
+�η − k
+e
+�η−2k+a �
+e
+η − k
+�η−k
+(F116)
+= e
+�
+e
+η − k
+�k−a
+.
+(F117)
+Using the assumption that η > 2k we can proceed further, yielding
+(η − 2k + a)!
+(η − k)!
+≤ e
+�
+e
+η − k
+�k−a
+(F118)
+≤ e
+�2e
+η
+�k−a
+(F119)
+= e
+�2e
+η
+�k �2e
+η
+�−a
+,
+(F120)
+where we have used the fact that η > 2k implies that η − k > η/2.
+
+29
+We can use Eq. (F120) to further simplify Eq. (F112), finding that,
+Var(ˆo) ≤ e2 (η/k)k (η/k − 1)k
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)−a (η − 2k + a)!
+(η − k)!
+(F121)
+≤ e3 (η/k)k (η/k − 1)k
+k
+�
+a=0
+�k
+a
+�
+(η/k − 1)−a
+�2e
+η
+�k �2e
+η
+�−a
+(F122)
+≤ e3 �
+η2/k2�k
+k
+�
+a=0
+�k
+a
+� � η
+2k
+�−a �2e
+η
+�k �2e
+η
+�−a
+(F123)
+= e3
+�2eη
+k2
+�k
+k
+�
+a=0
+�k
+a
+� � e
+k
+�−a
+.
+(F124)
+Note that we again used the fact that η > 2k implies that η − k > η/2 to simplify the part of the bound involving
+(η/k − 1). Applying the binomial theorem to the sum yields the bound
+Var(ˆo) ≤ e3
+�2eη
+k2
+�k
+e−k (k + e)k
+(F125)
+= e3
+�2η (k + e)
+k2
+�k
+.
+(F126)
+Recall that we defined the estimator ˆo by neglecting the coefficient
+kk(η!)
+ηk(η−k)! in Eq. (F16)’s expression for the k-RDM
+element kDj1,...,jk
+i1,...,ik . If we let ˆd be the estimator for this k-RDM element with the coefficient included, we have that
+Var( ˆd) =
+�
+kk (η!)
+ηk (η − k)!
+�2
+Var(ˆo).
+(F127)
+Therefore, we can bound the desired variance by
+Var( ˆd) ≤
+�
+kk (η!)
+ηk (η − k)!
+�2
+e3
+�2η (k + e)
+k2
+�k
+.
+(F128)
+Simplifying this expression, we obtain
+Var( ˆd) ≤
+�
+kk (η!)
+ηk (η − k)!
+�2
+e3
+�2η (k + e)
+k2
+�k
+(F129)
+= e3
+�
+(η!)
+(η − k)!
+�2 �2 (k + e)
+η
+�k
+(F130)
+≤ e3ηk (2k + 2e)k ,
+(F131)
+which is the bound advertised in Eq. (F18).
+Appendix G: More efficient Slater determinant state preparation in first quantization
+The general principle is to prepare the state in second quantization, then convert it to first quantization. To avoid
+needing to store all N qubits for the second quantized state as it is produced, we convert its qubits to the first
+quantized representation.
+To explain this, we will first explain how a state in the second quantized representation can be converted to the
+first quantized representation. A computational basis state in second quantization consists of a string of N bits with
+η ones and N − η zeros. The procedure is to run through these qubits in sequence and store the locations in η
+registers of size ⌈log N⌉. Let us call the qubit number we consider from the second quantized representation q and
+also record the number of electrons (ones) found so far as ξ. The value of ξ will be stored in an ancilla register of size
+nη = ⌈log(η + 1)⌉.
+We initialize all η registers for the first quantized representation and the ξ register as zero. Then, for q = 1 to N
+we perform the following.
+
+30
+1. Add the value in qubit q to the ξ register, with Toffoli cost nη − 1. If the qubit is in the state |1⟩ then ξ is
+incremented.
+2. Now use qubit q to control unary iteration [15] on the register ξ, which has cost η − 1.
+3. Use this unary iteration to write the value q into register ξ using CNOTs. Because q is iterated classically, only
+CNOTs are needed, with no further Toffolis beyond that needed for the unary iteration. Because the unary
+iteration is controlled by qubit q, in the case where qubit q is in state |0⟩, the unary iteration does not proceed
+and the value of q is not written out.
+4. Now perform unary iteration on ξ again that is not controlled; the cost is η − 2.
+5. We use the unary iteration on ξ to check if the value in register number ξ is q; if it is then we perform a NOT on
+qubit q. This multiply-controlled Toffoli is controlled by ⌈log N⌉ + 1 qubits (including the qubit from the unary
+iteration), so it has a cost of ⌈log N⌉. But, this is done for each of the η registers, for a total cost η⌈log N⌉.
+The last operation ensures that qubit q is set to |0⟩. +That is because, if it is initially |0⟩, then value q is not written
+in register ξ, and the value is not flipped. If it is initially |1⟩, then q is written in register ξ, and the multiply-controlled
+Toffoli flips this qubit to |0⟩.
+So far this procedure gives an ordered list of the electron positions, but we need an antisymmetrized state. To
+obtain that, we apply the procedure in [66] to antisymmetrize with cost O(η log η log N). The total Toffoli cost is
+N (2η + nη − 3 + η⌈log N⌉) + O(η log η log N).
+(G1)
+The dominant cost here is ηN log N from erasing the qubits in the second quantized representation, with the factor
+of log N coming from the need to check all qubits of each register to check if it is q. However, recall that in unary
+iteration it is possible to check if a register is equal to a consecutive sequence of values without this logarithmic
+overhead, and we are considering consecutive values of q.
+To eliminate that overhead, we, therefore, consider simultaneous unary iteration on all of the η registers. That is,
+for each register for the first quantized representation, we also store the qubits needed for unary iteration, as well as
+a control register to ensure we do not iterate on registers that do not have value written into them yet. The control
+qubits will correspond to the value of ξ in unary. Our modified procedure is as follows (with the iteration of q from 1
+to N).
+1. Perform a single step of unary iteration on all η registers with cost η Toffolis.
+2. Add the value in qubit q to the ξ register, with Toffoli cost nη − 1.
+3. Use qubit q to control unary iteration on the register ξ, which has cost η − 1.
+4. Use this unary iteration to write the value q into register ξ, as well as the ⌈log N⌉ ancilla qubits for the unary
+iteration and the control qubit. Again this is performed with CNOTs.
+5. Convert the control qubits to one-hot unary using a sequence of CNOTs.
+6. For each of the η registers, use the control qubit and the unary iteration output to control a NOT on qubit q.
+This has a cost of a single Toffoli for each register, for a toal of η.
+7. Convert the control qubits to from one-hot unary with CNOTs.
+As a result, we have eliminated the log N factor and also eliminated the cost of η − 2 for the unary iteration on ξ
+(because the control qubits are a unary representation of ξ). One might ask if the binary representation of ξ is still
+needed; however, it would be more costly to add increment ξ in unary (about η cost instead of log η). The total Toffoli
+cost of this procedure is now
+N (3η + nη − 2) + O(η log η log N),
+(G2)
+where the order term is the cost for antisymmetrizing. Note that this reduces the Toffoli cost, but there is still a
+Clifford cost of Nη log N from the CNOTs to place the value of q in the first quantized registers.
+Now to efficiently prepare the Slater determinant, we can perform the sequence of Givens rotations on the qubits
+for the second quantized representation. The Givens rotations are performed in a sequence where Givens rotations
+are performed in a layer on qubits 1 to η + 1, then on qubits 2 to η + 2, then 3 to η + 3, and so on. One can find the
+details of the Givens rotations that must be applied in [64]. Generally, layer q of Givens rotations is performed on
+
+31
+qubits q to η + q. After the first layer there are only η + 1 qubits being used, and the first qubit is not accessed again
+in the preparation. Therefore we can convert this qubit to the first quantized representation and erase it. Then there
+are only η qubits actively being used in the second quantized representation, and the next layer will be performed on
+qubits 2 to η + 2, bringing on one more qubit.
+In this way, each time we perform a layer of Givens rotations to prepare the state, we can convert one qubit to
+the first quantized representation, and only η + 1 qubits of the second quantized representation need be used at once,
+which is trivial compared to the number of qubits used for the first quantized representation.
+
diff --git a/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/load_file.txt b/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..23e666ef7eb93cd99f361680cfa24399d40ed589
--- /dev/null
+++ b/x9AzT4oBgHgl3EQfQvs-/content/tmp_files/load_file.txt
@@ -0,0 +1,1354 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf,len=1353
+page_content='Quantum simulation of exact electron dynamics can be  more efficient than classical mean-field methods  Ryan Babbush,1, ∗ William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Huggins,1 Dominic W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Berry,2 Shu Fay Ung,3 Andrew Zhao,1, 4  David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Reichman,3 Hartmut Neven,1 Andrew D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Baczewski,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='5 and Joonho Lee1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' †  1Google Quantum AI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Venice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' United States  2Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Macquarie University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' NSW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Australia  3Department of Chemistry,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Columbia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' United States  4Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' University of New Mexico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Albuquerque,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' United States  5Quantum Algorithms and Applications Collaboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sandia National Laboratories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Albuquerque NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' United States  (Dated: January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 2023)  Quantum algorithms for simulating electronic ground states are slower than popular classical  mean-field algorithms such as Hartree-Fock and density functional theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' but offer higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Accordingly, quantum computers have been predominantly regarded as competitors to only the most  accurate and costly classical methods for treating electron correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, here we tighten  bounds showing that certain first quantized quantum algorithms enable exact time evolution of  electronic systems with exponentially less space and polynomially fewer operations in basis set size  than conventional real-time time-dependent Hartree-Fock and density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Although  the need to sample observables in the quantum algorithm reduces the speedup, we show that one can  estimate all elements of the k-particle reduced density matrix with a number of samples scaling only  polylogarithmically in basis set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We also introduce a more efficient quantum algorithm for first  quantized mean-field state preparation that is likely cheaper than the cost of time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We  conclude that quantum speedup is most pronounced for finite temperature simulations and suggest  several practically important electron dynamics problems with potential quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Introduction  Quantum computers were first proposed as tools for  dynamics by Feynman [1] and later shown to be univer-  sal for that purpose by Lloyd et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Like those early  papers, most work on this topic assumes that the ad-  vantage of quantum computers for dynamics is that they  provide an approach to simulation with systematically  improvable precision but without scaling exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Here, we advance and analyze a different idea: certain  (exact) quantum algorithms for dynamics may be more  efficient than even classical methods that make uncon-  trolled approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We examine this in the context  of simulating interacting fermions – systems of relevance  in fields such as chemistry, physics, and materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  It is often the case that practically relevant ground  state problems in chemistry and materials science do not  exhibit strong correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For those problems, many  classical heuristic methods work well [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Even for  some strongly correlated systems, there are successful  polynomial-scaling classical methods [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Here, we ar-  gue that even if electronic systems are well described  by mean-field theory, quantum algorithms can achieve  speedup over classical algorithms for simulating the time  evolution of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We focus on comparing to  mean-field methods such as real-time time-dependent  Hartree-Fock and density functional theory due to their  popularity and well-defined scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Nonetheless, many  ∗ corresponding author: ryanbabbush@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='com  † corresponding author: linusjoonho@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='com  of our arguments translate to advantages over other  known classical approaches to dynamics that are more  expensive but more accurate than mean-field methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This is a sharp contrast to prior studies of quantum  algorithms, which have focused on strongly correlated  ground state problems such as FeMoCo [7–11], P450  [12], chromium dimers [13] and jellium [14–17], assess-  ing quantum advantage over only the most accurate and  costly classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Quantum algorithms competitive with efficient clas-  sical algorithms for dynamics have been analyzed in  contexts outside of fermionic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example,  work by Somma [18] showed that certain one-dimensional  quantum systems, such as harmonic oscillators, could be  simulated with sublinear complexity in system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Ex-  perimentally motivated work by Geller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [19] also pro-  posed simulating quantum systems in a single-excitation  subspace, a task for which they suggested a constant fac-  tor speedup was plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, neither work is con-  nected to the context studied here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We begin by analyzing the cost of classical mean-field  dynamics and recent exact quantum algorithms in first  quantization, focusing on explaining why there is often a  quantum speedup in the number of basis functions over  classical mean-field methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Next, we analyze the over-  heads associated with measuring quantities of interest on  a quantum computer and introduce more efficient meth-  ods for measuring the one-particle reduced density ma-  trix in first quantization (which characterizes all mean-  field observables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then, we discuss the costs of prepar-  ing mean-field states on the quantum computer and de-  scribe new methods that make this cost likely negligible  compared to the cost of time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Finally, we con-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='01203v1  [quant-ph]  3 Jan 2023  2  clude with a discussion of systems where these techniques  might lead to practical quantum advantage over classical  mean-field simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Classical mean-field dynamics  Here we will discuss mean-field classical algorithms for  simulating the dynamics of interacting systems of elec-  trons and nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, we will focus on the ab initio  Hamiltonian with η particles discretized using N basis  functions, which can be expressed as  H =  N  �  µν  hµνa†  µaν + 1  2  N  �  µνλσ  (µν|λσ) a†  µa†  λaσaν  (1)  where a(†)  µ  is the fermionic annihilation (creation) opera-  tor for the µ-th orbital and integral values are given by  hµν =  �  dr φ∗  µ (r)  �  −∇2  2 + V (r)  �  φν (r) ,  (2)  (µν|λσ) =  �  dr1dr2  φ∗  µ (r1) φν (r1) φ∗  λ (r2) φσ (r2)  |r1 − r2|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (3)  Here, V (r) is the external potential (perhaps arising from  the nuclei) and φµ(r) represents a spatial orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Exact quantum dynamics is encoded by the time-  dependent Schr¨odinger equation given by  i ∂  ∂t |ψ (t)⟩ = H |ψ (t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (4)  Mean-field dynamics, such as real-time time-dependent  Hartree-Fock  (RT-TDHF)  [20],  employs  a  time-  dependent variational principle within the space of sin-  gle Slater determinants (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', anti-symmetrized product  states) to approximate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Other methods with  similar cost such as real-time time-dependent density  functional theory (RT-TDDFT) rely on a relationship  between the interacting system and an auxiliary non-  interacting system to define dynamics within a space  of single Slater determinants [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In both methods,  there are η occupied orbitals, each expressed as a linear  combination of N basis functions using the coefficient  matrix, Cocc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The dimension of Cocc is N × η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' These  orbitals then constitute a Slater determinant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', anti-  symmetric product states), det(Cocc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Storing Cocc on a  classical computer has space complexity O(Nη log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  As a result of this approximation, we solve the follow-  ing effective time-dependent equation for the occupied  orbital coefficients that specify the Slater determinant  Cocc(t) at a given moment in time:  i∂Cocc (t)  ∂t  = F (t) Cocc (t)  (5)  where the effective one-body mean-field operator F(t),  also known as the time-dependent Fock matrix, is  Fµν(t) = hµν +  N  �  λσ  �  (µν|λσ) − (µσ|λν)  2  �  Pσλ(t)  (6)  with P(t) = Cocc(t)(Cocc(t))†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' While F(t) is an N × N  dimensional matrix, we can apply it to Cocc(t) without  explicitly constructing it, thus avoiding a space complex-  ity of O(N 2 log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Using the most common methods  of applying this matrix to update each of η occupied or-  bitals in Cocc(t) requires �  O(N 2η) total operations1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  However, a recent technique referred to as occ-RI-K by  Head-Gordon and co-workers [23], and similarly “Adap-  tively Compressed Exchange” (ACE) [24, 25] by Lin and  co-workers, further reduces this cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  These methods  leverage the observation that, when restricted to the sub-  space of the η occupied orbitals, the effective rank of the  Fock operator scales as O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This gives an approach to  updating the Fock operator that requires only  �  O(N η2)  (7)  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Below we will use gate complexity and the  number of operations interchangeably when discussing  the scaling of classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Although these tech-  niques are not implemented in every quantum chemistry  code, we regard them as the main point of comparison  to quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We also note that RT-TDDFT  with hybrid functionals [26] has the same scaling as RT-  TDHF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Simpler RT-TDDFT methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', those without  exact exchange) can achieve better scaling, �  O(Nη) in a  plane wave basis, but are often less accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For finite-temperature simulation, one often needs to  track M > η orbitals with appreciable occupations, in-  creasing the space complexity to O(NM log(1/ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This  increases the cost of occ-RI-K or ACE mean-field up-  dates to �  O(NM 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' At temperatures well above the Fermi  energy, most orbitals have appreciable occupations so  M ≃ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' More expensive methods for dynamics that in-  clude electron correlation in the dynamics tend to scale  at least linearly in the cost of ground state simulation  at that level of theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, speedup over mean-field  methods implies speedup over more expensive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In recent years, by leveraging the “nearsightedness”  of electronic systems [27], “linear-scaling” methods have  been developed that achieve updates scaling as O(N)  [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For RT-TDHF and RT-TDDFT, linear-scaling  comes from the fact that the off-diagonal elements of P  fall off quickly with distance for the ground state [29]  and some low-lying excited states [30] in a localized basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  One can show that for gapped ground states, the decay  rate is exponential, whereas for metallic ground states,  1 Throughout the paper we will use the convention that �  O(·) im-  plies big-O notation suppressing polylogarithmic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  3  it is algebraic [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, often such asymptotic be-  havior only onsets for very large systems, and the onset  can be highly system-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This should be con-  trasted with the scaling analyzed above and the scaling  of quantum algorithms (vide infra) that onsets already  at modest system sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Furthermore, the nearsighted-  ness of electrons does not necessarily hold for dynamics  of highly excited states and at high temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Due to  these limitations, we do not focus on comparing quantum  algorithms and classical linear scaling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  It has also been suggested that one can exploit a low-  rank structure of occupied orbitals using the quantized  tensor train format [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Assuming the compression of  orbitals in real space is efficient such that the rank does  not grow with system size or the number of grid points,  the storage cost is reduced to ˜O(η), and the update cost is  ˜O(η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' It is unclear how well compression can be realized  for dynamics problems and finite-temperature problems,  and to our knowledge, it has been never been deployed  for those purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Accordingly, we do not consider this  approach as the point of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We now discuss how many time steps are required  to perform time evolution using classical mean-field ap-  proaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The number of time steps will depend on the  target precision as well as the total unitless time  T = max  Cocc ∥F∥ t,  (8)  where t is duration of time-evolution and ∥ · ∥ denotes  the spectral norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This dependence on the norm of F is  similar to what would be obtained in the case of linear  differential equations despite the dependence on Cocc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  see Appendix A for a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We can upper bound T  by considering its scaling in a local basis, and with open  boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We find  max  Cocc∥F∥= O  �η2/3  δ  + 1  δ2  �  = O  �  N 1/3η1/3+ N 2/3  η2/3  �  ,  (9)  where δ = O((η/N)1/3) is the minimum grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The first term comes from the Coulomb operator, and  the second comes from the kinetic energy operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This  scaling for δ comes from taking the computational cell  volume proportional to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We briefly describe how this scaling for the norm is  obtained and refer the reader to Appendix A for more  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The 1/δ2 term is obtained from the kinetic en-  ergy term in hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When diagonalized, that term will be  non-zero only when µ = ν with entries scaling as O(1/δ2)  due to the ∇2 in the expression for hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  That upper  bounds the spectral norm for this diagonal matrix, and  the spectral norm is unchanged under change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The η2/3/δ comes from the sum in the expression for  Fµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To bound the tensor norm of (µν|λσ) − (µσ|λν)/2  we can bound the norms of the two terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For  each, the tensor norm can be upper bounded by noting  that the summing over µν, λσ with normalized vectors  corresponds to transformations of the individual orbitals  in the integral defining (µν|λσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Since orbitals cannot be  any more compact than width δ, the 1/|r1 − r2| in the  integral averages to give O(1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' There is a further fac-  tor of η2/3 when accounting for η electrons that cannot  be any closer than η1/3δ on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The number of time steps required to effect evolution  to within error ϵ depends on the choice of time integra-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Many options are available [32–34], and the optimal  choice depends on implementation details like the basis  set and pseudization scheme, as well as the desired accu-  racy [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In Appendix A, we argue that the minimum  number of time steps t/∆t one could hope for by using  an arbitrarily high order integration scheme of this sort  is T 1+o(1)/ϵo(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In particular, for an order k integra-  tor, the error can be bounded as O((∥F∥∆t)k+1), with  a possibly k-dependent constant factor that is ignored  in this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That means the error for t/∆t time  steps is O(t∥F∥k+1∆tk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To obtain error no more than  ϵ, take (t/∆t)k = O((t∥F∥k+1/ϵ), so the number of time  steps is t/∆t = O(T 1+1/k/ϵ1/k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Plugging Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (9) into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (8) and multiplying the update cost in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (7) by  T 1+o(1)/ϵo(1) time steps, we find the number of opera-  tions required for classical mean-field time-evolution is  �  N 4/3η7/3t + N 5/3η4/3t  � �Nt  ϵ  �o(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (10)  Finally, when performing mean-field dynamics, the  central quantity of interest is often the one-particle re-  duced density matrix (1-RDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The 1-RDM is an N ×N  matrix defined as a function of time with matrix elements  ρµν (t) = ⟨ψ (t)| a†  µaν |ψ (t)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (11)  The 1-RDM is the central quantity of interest because  it can be used to reconstruct any observable associated  with a Slater determinant efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For more general  states, one would also need higher order RDMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' however,  all higher order RDMs can be exactly computed from the  1-RDM via Wick’s theorem when the wavefunction is a  single Slater determinant [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Thus, when mean-field  approximations work well, the time-dependent 1-RDM  can also be used to compute multi-time correlators such  as Green’s functions and spectral functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Exact quantum dynamics in first quantization  One of the key advantages of some quantum algorithms  over mean-field classical methods is the ability to per-  form dynamics using the compressed representation of  first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' First quantized quantum simulations  date back to [37–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' They were first applied to fermionic  systems in [38] and developed for molecular systems in  [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In first quantization, one encodes the wave-  function using η different registers (one for each occupied  orbital), each of size log N (to index the basis functions  comprising each occupied orbital).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The space complexity  of first quantized quantum algorithms is O(η log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  4  As described previously, mean-field classical methods  require space complexity of O(Nη log(1/ϵ)) where ϵ is  the target precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Thus, these quantum algorithms  require exponentially less space in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Usually, when  one thinks of quantum computers more efficiently en-  coding representations of quantum systems, the advan-  tage comes from the fact that the wavefunction might  be specified by a Hilbert space vector of dimension  �N  η  �  and could require as much space to represent explicitly  on a classical computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, this alone cannot give  exponential quantum advantage in storage in N over clas-  sical mean-field methods since mean-field methods only  resolve entanglement arising from anti-symmetry and do  not attempt to represent wavefunction in the full Hilbert  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Instead, the scaling advantage these quantum al-  gorithms have over mean-field methods is related to the  ability to store the distribution of each occupied orbital  over N basis functions, using only log N qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  But  quantum algorithms require more than the compressed  representations of first quantization in order to realize a  scaling advantage over classical mean-field methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' they  must also have sufficiently low gate complexity in the ba-  sis size and other parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Here we will review and tighten bounds for the most  efficient known quantum algorithms for simulating the  dynamics of interacting electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Early first quan-  tized algorithms for simulating chemistry dynamics such  as [41, 42] were based on Trotterization of the time-  evolution operator in a real space basis and utilized the  quantum Fourier transform to switch between a repre-  sentation where the potential operator was diagonal and  the kinetic operator was diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This enabled Trot-  ter steps with gate complexity �  O(η2) but the number of  Trotter steps required for the approach of those papers  scaled worse than linearly in N, η, the simulation time t  and the desired inverse error in the evolution, 1/ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Leveraging recent techniques for bounding Trotter er-  ror [43–45], in Appendix B we show that using sufficiently  high order Trotter formulas, the overall gate complexity  of these algorithms can be reduced to  �  N 1/3η7/3t + N 2/3η4/3t  � �Nt  ϵ  �o(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (12)  This is the lowest reported scaling of any Trotter based  first quantized quantum chemistry simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We re-  mark that the N 1/3η7/3t scaling is dominant whenever  N < Θ(η3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In that regime, it represents a quartic  speedup in basis size for propagation over the classical  mean-field scaling given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The first algorithms to achieve sublinear scaling in N  were those introduced by Babbush et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That work  focused on first quantized simulation in a plane wave  basis and leveraged the interaction picture simulation  scheme of [47] to give gate complexity scaling as  �  O  �  N 1/3η8/3t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (13)  When N > Θ(η4), this algorithm is more efficient than  the Trotter based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Since that is also the regime  where the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (10) dominates that scal-  ing, this represents a quintic speedup in N, coupled with  a quadratic slowdown in η, over mean-field classical algo-  rithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The work of Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [48] analyzed the constant  factors in the scaling of this algorithm for use in ground  state preparation via quantum phase estimation [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In  Appendix C of this work we analyze the constant factors  in the scaling of this algorithm when deployed for time-  evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [48] also introduced algorithms with  the same scaling as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (13) but in a grid representation  (see Appendix K therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  A key component of the algorithms of [46, 48] is the  realization of block encodings [50] with just �  O(η) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The difficult part of the block encoding is the preparation  of a superposition state with amplitudes proportional to  the square root of the Hamiltonian term coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' A  novel quantum algorithm is devised for this purpose in  [46] which scales only polylogarithmically in basis size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The N 1/3 dependence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (13) enters via the number  of times the block encoding must be repeated to perform  time evolution, related to the norm of the potential oper-  ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Under certain assumptions, the norm of the poten-  tial term can be reduced to a polylogarithmic dependence  on N (see Appendix D for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In that case,  exponential quantum advantage in N is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We note that second quantized algorithms outperform  first quantized quantum algorithms in gate complexity  when N < Θ(η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is because while the best scal-  ing Trotter steps in first quantization require �  O(η2) gates  [42], the best scaling Trotter steps in second quantization  require �  O(N) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As recently shown in [45], such ap-  proaches lead to a total gate complexity for Trotter based  second quantized algorithms scaling as  �  N 4/3η1/3t + N 5/3  η2/3 t  � �Nt  ϵ  �o(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (14)  In the limit that η = Θ(N), this approach has O(N 5/3)  gate complexity, which is significantly less than the  O(N 8/3) gate complexity of Trotter based first quantized  quantum algorithms mentioned here, or the O(N 11/3)  gate complexity of classical mean-field algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (See  Appendix E for discussion on the overall quantum  speedup in different regimes of how N scales in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=') How-  ever, these second quantized approaches generally require  at least O(N) qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The approach used in [45] to im-  plement Trotter steps involves the fast multipole method  [51], which requires O(N log N) qubits as well as the re-  striction to a grid-like basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When using such basis sets,  we expect N ≫ η, and so this space complexity would  be prohibitive for quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Methods such as fast multipole [51], Barnes-Hut [52],  or particle-mesh Ewald [53] compute the Coulomb poten-  tial in time �  O(η) when implemented within the classical  random access memory model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If the Coulomb potential  could be computed with that complexity on a quantum  computer it would speed up the first quantized Trotter  5  algorithms discussed here by a factor of O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However,  it is unclear whether such algorithms extend to the quan-  tum circuit model with the same complexity without un-  favorable assumptions such as QRAM [54, 55], or with-  out restricting the maximum number of electrons within  a region of space (see Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, we  exclude such approaches from our comparisons here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Quantum measurement costs  In contrast to classical mean-field simulations, on a  quantum computer, all observables must be sampled  from the quantum simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  There are a variety of  techniques for doing this, with the optimal choice de-  pending on the target precision in the estimated observ-  able as well as the number and type of observables one  wishes to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example, when measuring W unit  norm observables to precision ϵ one could use algorithms  introduced in [56] which require �  O(  √  W/ϵ) state prepa-  rations and O(W log(1/ϵ)) ancillae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Thus, to measure  all W = O(N 2) elements of the 1-RDM to a fixed addi-  tive error in each element, this approach would require  �  O(N/ϵ) circuit repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' While scaling optimally in ϵ  for quantum algorithms, this linear scaling in N would  decrease the speedup over classical mean-field algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Instead, here we will focus on measuring the 1-RDM  with a new variation of the classical shadows method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Classical shadows were introduced in [57] and adapted  for second quantized fermionic systems in [58–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Our  approach is to apply a separate random Clifford channel  to each of the η different log N sized registers represent-  ing an occupied orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Applying a random Clifford on  log N qubits requires O(log2N) gates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' thus, O(η log2N)  gates comprise the full channel (a negligible cost relative  to time-evolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In Appendix F we prove that repeat-  ing this procedure �  O(η/ϵ2) times enables estimation of all  1-RDM elements to within additive error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' More gen-  erally, we prove that this same procedure allows for es-  timating all higher order k-particle RDMs elements with  �  O(kkηk/ϵ2) circuit repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In the next section and in  Appendix G, we describe a way to map second quantized  representations to first quantization, effectively extend-  ing the applicability of these classical shadows techniques  to second quantization as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  To give some intuition for how this works, we consider  the 1-RDM elements in first quantization:  ρµν (t) = ⟨ψ (t)|  �  �  η  �  j=1  |µ⟩⟨ν|j  �  � |ψ (t)⟩ ,  (15)  where the subscript j indicates which of the η registers  the orbital-ν to orbital-µ transition operator acts upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Due to the antisymmetry of the occupied orbital regis-  ters in first quantization, we could also obtain the 1-RDM  by measuring the expectation value of an operator such  as η |p⟩⟨q|1, which acts on just one of the η registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Because η |p⟩⟨q|1 has the Hilbert-Schmidt norm of O(η),  the standard classical shadows procedure applied to this  log N sized register would require �  O(η2/ϵ2) repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  But we can parallelize the procedure by also collecting  classical shadows on the other η − 1 registers simultane-  ously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' One way of interpreting the results we prove in  Appendix F is that, due to antisymmetry, these regis-  ters are anticorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As a result, collecting shadows  on all η registers simultaneously reduces the overall cost  by at least a factor of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To obtain W elements of the  1-RDM one will need to perform an offline classical inver-  sion of the Clifford channel that will scale as �  O(Wη2/ϵ2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  of course, any quantum or classical algorithm for estimat-  ing W quantities must have gate complexity of at least  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, this only needs to be done once and does  not scale in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As a comparison, the cost of computing  1-RDM classically without exploiting sparsity is O(Wη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  When simulating systems that are well described by  mean-field theory, all observables can be efficiently ob-  tained from the time-dependent 1-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, for  observables such as the energy that have a norm growing  in system size or basis size, targeting fixed additive er-  ror in the 1-RDM elements will not be sufficient for fixed  additive error in the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In such situations, it  could be preferable to estimate the observable of inter-  est directly using a combination of block encodings [50]  and amplitude amplification [62] (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Assum-  ing the cost of block encoding the observable is negligible  to the cost of time-evolution (true for many observables,  including energy), this results in needing O(λ/ϵ) circuit  repetitions, where λ is the 1-norm associated with the  block encoding of the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example, whereas  there are many correlation functions with λ = O(1), for  the energy λ = O(N 1/3η5/3 + N 2/3η1/3) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Multiply-  ing that to the cost of quantum time-evolution further  reduces the quantum speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The final measurement cost to consider is that of re-  solving observables in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In some cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', when  computing scattering cross sections or reaction rates, one  might be satisfied measuring the state of the simulation  at a single point in time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, in other situations,  one might wish to simulate time-evolution up to a maxi-  mum duration of t, but sample quantities at L different  points in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Most quantum simulation methods that  accomplish this goal scale as O(L) (O(Lt) in the case  where the points are evenly spaced in time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However,  the work of [56] shows that this cost can be reduced to  O(  √  Lt), but with an additional additive space complex-  ity of �  O(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Either way, this is another cost that plagues  quantum but not classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Quantum state preparation costs  Initial state preparation can be as simple or as complex  as the state that one desires to begin the simulation in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Since the focus of this paper is outperforming mean-field  calculations, we will discuss the cost of preparing Slater  6  determinants within first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example, one  may wish to start in the Hartree-Fock state (the lowest  energy Slater determinant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Classical approaches to com-  puting the Hartree-Fock state scale as roughly �  O(Nη2) in  practice [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is a one-time additive classical cost  that is not multiplied by the duration of time-evolution  so it is likely subdominant to other costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Quantum algorithms for preparing Slater determinants  have focused on the “Givens rotation” approach intro-  duced in [64] for second quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That algorithm  requires O(Nη) “Givens rotation” unitaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Such uni-  taries can be implemented with O(η log N) gates in first  quantization [48, 65], hence combining that with the se-  quence of rotations called for in [64] gives an approach to  preparing Slater determinants in first quantization with  �  O(Nη2) gates in total, a relatively high cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Unlike the  offline cost to compute the occupied orbital coefficients,  this state preparation cost would be multiplied by the  number of measurement repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Here, we develop a new algorithm to prepare arbi-  trary Slater determinants in first quantization with only  �  O(Nη) gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The approach is to first generate a su-  perposition of all of the configurations of occupied or-  bitals in the Slater determinant while making sure that  electron registers holding the label of the occupied or-  bitals are always sorted within each configuration so  that they are in ascending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is necessary be-  cause without such structure (or guarantees of something  similar), the next step (anti-symmetrization) could not  be reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For this next step, we apply the anti-  symmetrization procedure introduced in [66], which re-  quires only O(η log η log N) gates (a negligible additive  cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that if one did not need the property that  the configurations were ordered by the electron register,  then it would be relatively trivial to prepare an arbitrary  Slater determinant as a product state of η different reg-  isters, each in an arbitrary superposition over log N bits  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', using the brute-force state preparation of [67]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  A high level description of how the superposition of  “ordered” configurations comprising the Slater determi-  nant is prepared now follows, with details given in Ap-  pendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The idea is to generate the Slater determi-  nant in second quantization in an ancilla register us-  ing the Givens rotation approach of [64], while mapping  the second quantized representation to a first quantized  representation one second quantized qubit (orbital) at a  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  One can get away with storing only η non-zero  qubits (orbitals) at a time in the second quantized rep-  resentation because the Givens rotation algorithm grad-  ually produces qubits that do not require further rota-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Whenever one produces a new qubit in the second  quantized representation that does not require further  rotations, one can convert it to the first quantized repre-  sentation, which zeros that qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, the procedure  only requires O(η) ancilla qubits – a negligible additive  space overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  A total of O(Nη log N) gates are re-  quired because for each of O(N) steps one accesses all  O(η log N) qubits of the first quantized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In Appendix G, we show the Toffoli complexity can be  further reduced to O(Nη) with some additional tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Finally, we note that quantum algorithms can also per-  form finite temperature simulation by sampling initial  states from a thermal density matrix in each realization  of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example, if the system is in a regime  that is well treated by mean-field theory, one can initial-  ize the system in a Slater determinant that is sampled  from the thermal Hartree-Fock state [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Since the out-  put of quantum simulations already needs to be sampled  this does not meaningfully increase the number of quan-  tum repetitions required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Such an approach would also  be viable classically (and would allow one to perform  simulations that only ever treat η occupied orbitals de-  spite having finite temperature), but would introduce a  multiplicative O(1/ϵ2) sampling cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For either proces-  sor there is the cost of classically computing the thermal  Hartree-Fock state, but this is a one-time cost not mul-  tiplied by the duration of time-evolution or O(1/ϵ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Discussion  We have reviewed and provided new analysis of the  costs associated with both classical mean-field methods  and state-of-the-art exact quantum algorithms for dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We introduced new and more efficient strategies  for initializing Slater determinants in first quantization,  and for measuring RDMs via classical shadows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We com-  pare these costs in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Relative to classical mean-  field methods, we see that when the goal is to sample  the output of quantum dynamics at zero temperature,  the best quantum algorithms deliver a seventh power  speedup in particle number when N < Θ(η2), quartic in  basis size when Θ(η2) < N < Θ(η3), super-quadratic in  basis size when Θ(η3) < N < Θ(η4) and quintic in basis  size but with a quadratic slowdown in η when N > Θ(η4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In the extremal regimes of N < Θ(η5/4) and N > Θ(η4),  the overall speedup in system size is super-quadratic  (see Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  These are large enough  speedups that quantum advantage may persist even de-  spite quantum error-correction overhead [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that  our analyses are based on derivable upper bounds for  both classical and quantum algorithms over all possible  input states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Tighter bounds derived over restricted in-  puts would give asymptotically fewer time steps required  for both classical and quantum Trotter algorithms [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The story becomes more nuanced when we wish to es-  timate ϵ-accurate quantities via sampling the quantum  simulation output at L different time points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For observ-  ables with norm scaling as O(1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', simple correlation  functions or single RDM elements), or those pertaining  to amplitudes of the state (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  scattering amplitudes  or reaction rates) the scaling advantages in system and  basis size are maintained but at the cost of the quan-  tum algorithm slowing down by a multiplicative factor  of at least O(  √  L/ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When targeting the 1-RDM (which  characterizes all observables within mean-field theory) we  7  Processor  Algorithm  Observable  Space  Gate complexity  classical  T = 0 mean-field with occ-RI-K/ACE [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 24]  anything  �  O(Nη)  (N 4/3η7/3t + N 5/3η4/3t)( Nt  ϵ )o(1)  classical  T > 0 mean-field (density matrix) with [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 24]  anything  �  O(NM)  (N 4/3M 2η1/3t+ N5/3M2t  η2/3  )( Nt  ϵ )o(1)  classical T > 0 mean-field (sampled trajectories) with [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 24]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='anything  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(Nη)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='( N4/3η7/3t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='+ N5/3η4/3t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')( Nt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ )o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='second quantized Trotter grid algorithm [45]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='sample |ψ(t)⟩ O(N log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='(N 4/3η1/3t + N5/3t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='η2/3 )( Nt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ )o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='first quantized Trotter grid algorithm here  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='sample |ψ(t)⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='(N 1/3η7/3t +N 2/3η4/3t)( Nt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ )o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='interaction picture plane wave algorithm [46]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='sample |ψ(t)⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(N 1/3η8/3t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='grid basis algorithm from Appendix K of [48]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='sample |ψ(t)⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(N 1/3η8/3t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='new shadows procedure here  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='k-RDM(t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(kkηkL Csamp/ϵ2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='gradient measurement [56]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='⟨ψ(t)| O |ψ(t)⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η + L)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='L Csamp λ/ϵ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='gradient measurement [56]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='⟨ψ(t)| H |ψ(t)⟩  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η + L)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='√  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='LCsampt(N1/3η5/3+N2/3η1/3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='ϵ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Costs of exact quantum algorithms and mean-field classical algorithms for simulating fermionic dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' N is the  number of basis functions, η is the number of particles, ϵ is target precision, M is the number of appreciably occupied orbitals  in a finite temperature (T) simulation (M ≃ N for high T), O is any observable having norm λ that can be block encoded  with cost less than time-evolution, t is the duration of evolution, L is the number of time points at which we wish to resolve  quantities and Csamp is the cost of sampling |ψ(t)⟩ with a quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For classical algorithms, gate complexity means  the number of floating point operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We are not accounting for the additive time-independent costs of state preparation  ( �  O(ηN) gates using the procedure of Appendix G), of classically computing initial occupied orbital coefficients, or of classically  reconstructing the k-RDM given measurement outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, this table reports gate complexities for long-time t simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In Appendix E we provide a table clarifying which algorithm has optimal gate complexity as a function of N/η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  maintain speedup in N but at the cost of an additional  linear slowdown in η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When measuring the total energy,  the overall speedup becomes tenuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, the viability  of quantum advantage with respect to zero temperature  classical mean-field methods depends sensitively on the  target precision and particular observables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In terms of applications, we expect RT-TDHF to pro-  vide qualitatively correct dynamics whenever electron  correlation effects are not pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' RT-TDDFT in-  cludes some aspects of electron correlation but the adi-  abatic approximation often creates issues [71] and the  method suffers from self-interaction error [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When the  adiabatic approximation is accurate, self-interaction er-  ror is not pronounced, and the system does not exhibit  strong correlation, we expect RT-TDDFT to generate  qualitatively correct dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  When there are many  excited states to consider for spectral properties, it is of-  ten beneficial to resort to real-time dynamics methods  instead of linear-response methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Furthermore, we are  often interested in real-time non-equilibrium electronic  dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is the case for photo-excited molecules  near metal surfaces [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The time evolution of elec-  tron density (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', the diagonal of the 1-RDM) near the  molecule is of particular interest due to its implications  for chemical reactivity and kinetics in the context of het-  erogeneous catalysis [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In this application, the simu-  lation of nuclear degrees of freedom may be equally im-  portant, which we will leave for future analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We see from Table I that prospects for quantum ad-  vantage are considerably increased at finite tempera-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, a promising class of problems to consider  for speedup over mean-field methods is the electronic  dynamics of either warm dense matter (WDM) [75–78]  or hot dense matter (HDM) [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The WDM regime  (where thermal energy is comparable to the Fermi en-  ergy) is typified by temperatures and densities that re-  quire the accurate treatment of both quantum and ther-  mal effects [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' These conditions occur in planetary  interiors, experiments involving high-intensity lasers, and  in inertial confinement fusion experiments as the ablator  and fuel are compressed into the conditions necessary for  thermonuclear ignition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Ignition occurs in the hot dense  matter (HDM) regime (where thermal energy far exceeds  the Fermi energy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' While certain aspects of these systems  are conspicuously classical, they still present spectra that  can be challenging to model [82, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Simulations in either the WDM or HDM regime typ-  ically rely on large plane wave basis sets and the inclu-  sion of ten to one-hundred times more partially occupied  orbitals per atom than would be required at lower tem-  peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Often, the attendant costs are so great that  it is impractical to implement RT-TDDFT with hybrid  functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Therefore, many calculations necessarily use  adiabatic semi-local approximations, even on large classi-  cal high-performance computing systems [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, the  level of practically achievable accuracy can be quite low,  and the prospect of exactly simulating the dynamics on  a quantum computer is particularly compelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Although we have focused on assessing quantum  speedup over mean-field theory, we view the main con-  tribution of this work as more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In particular, if  exact quantum simulations are sometimes more efficient  than classical mean-field methods, then all levels of the-  ory in between mean-field and exact diagonalization are  in scope for possible quantum advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Targeting sys-  tems that require more correlated calculations narrows  the application space but improves prospects for quan-  tum advantage due to the unfavorable scaling of the req-  8  uisite classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, it may turn out that the  domain of systems requiring, say, coupled cluster dynam-  ics [84–87], might be an even more ideal regime for prac-  tical quantum advantage, striking a balance in the trade-  off between the breadth of possible applications and the  cost of the classical competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Acknowledgments  The authors thank Alina Kononov, Garnet Kin-Lic  Chan, Robin Kothari, Alicia Magann, Fionn Malone,  Jarrod McClean,  Thomas O’Brien,  Nicholas Rubin,  Henry Schurkus, Rolando Somma, and Yuan Su for help-  ful discussions and feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We thank Lin Lin for  bringing our attention to the quantized tensor train for-  mat in [31] and thank Yuehaw Khoo for a discussion  related to this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  DWB worked on this project under a  sponsored research agreement with Google Quantum AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  DWB is supported by Australian Research Council Dis-  covery Projects DP190102633 and DP210101367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' ADB  acknowledges support from the Advanced Simulation and  Computing Program and the Sandia LDRD Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Some work on this project occurred while in residence at  The Kavli Institute for Theoretical Physics, supported  in part by the National Science Foundation under Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' NSF PHY-1748958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sandia National Laboratories  is a multi-mission laboratory managed and operated by  National Technology and Engineering Solutions of San-  dia, LLC, a wholly owned subsidiary of Honeywell In-  ternational, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', for DOE’s National Nuclear Security  Administration under contract DE-NA0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [1] Richard P Feynman, “Simulating physics with comput-  ers,” International Journal of Theoretical Physics 21,  467–488 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [2] Seth Lloyd, “Universal Quantum Simulators,” Science  273, 1073–1078 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [3] Rodney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Bartlett and Monika Musial, “Coupled-cluster  theory in quantum chemistry,” Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 79, 291–  352 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [4] Narbe Mardirossian and Martin Head-Gordon, “Thirty  years of density functional theory in computational chem-  istry: an overview and extensive assessment of 200 den-  sity functionals,” Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 115, 2315–2372 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [5] Joonho Lee, Hung Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Pham,  and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Reichman,  “Twenty Years of Auxiliary-Field Quantum Monte Carlo  in Quantum Chemistry: An Overview and Assessment on  Main Group Chemistry and Bond-Breaking,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 2022 (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='jctc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2c00802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [6] Seunghoon Lee, Joonho Lee, Huanchen Zhai, Yu Tong,  Alexander M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Dalzell, Ashutosh Kumar, Phillip Helms,  Johnnie Gray, Zhi-Hao Cui, Wenyuan Liu, Michael Kas-  toryano, Ryan Babbush, John Preskill, David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Re-  ichman, Earl T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Campbell, Edward F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Valeev, Lin Lin,  and Garnet Kin-Lic Chan, “Is There Evidence for Expo-  nential Quantum Advantage in Quantum Chemistry?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='02199 (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='02199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [7] Markus Reiher, Nathan Wiebe, Krysta M Svore, Dave  Wecker,  and Matthias Troyer, “Elucidating Reac-  tion Mechanisms on Quantum Computers,” Proceedings  of the National Academy of Sciences 114, 7555–7560  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [8] Zhendong Li, Junhao Li, Nikesh S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Dattani, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Um-  rigar,  and Garnet Kin-Lic Chan, “The electronic com-  plexity of the ground-state of the FeMo cofactor of nitro-  genase as relevant to quantum simulations,” The Journal  of Chemical Physics 150, 024302 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [9] Dominic Berry, Craig Gidney, Mario Motta, Jarrod Mc-  Clean,  and Ryan Babbush, “Qubitization of Arbitrary  Basis Quantum Chemistry Leveraging Sparsity and Low  Rank Factorization,” Quantum 3, 208 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [10] Vera von Burg,  Guang Hao Low,  Thomas H¨aner,  Damian S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Steiger, Markus Reiher, Martin Roetteler,  and Matthias Troyer, “Quantum computing enhanced  computational catalysis,” Physical Review Research 3,  033055–033071 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [11] Joonho Lee, Dominic W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Berry, Craig Gidney, William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Huggins, Jarrod R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' McClean, Nathan Wiebe, and Ryan  Babbush, “Even More Efficient Quantum Computations  of Chemistry Through Tensor Hypercontraction,” PRX  Quantum 2, 030305 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [12] Joshua J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Goings, Alec White, Joonho Lee, Christofer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Tautermann, Matthias Degroote, Craig Gidney, Toru Sh-  iozaki, Ryan Babbush, and Nicholas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rubin, “Reliably  Assessing the Electronic Structure of Cytochrome P450  on Today’s Classical Computers and Tomorrow’s Quan-  tum Computers,” Proceedings of the National Academy  of Sciences 119 (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2203533119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [13] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Elfving, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Broer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Webber, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Gavartin,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Halls, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Lorton, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Bochevarov, “How will  quantum computers provide an industrially relevant com-  putational advantage in quantum chemistry?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [14] Ryan Babbush, Nathan Wiebe, Jarrod McClean, James  McClain, Hartmut Neven,  and Garnet Kin-Lic Chan,  “Low-Depth Quantum Simulation of Materials,” Physi-  cal Review X 8, 011044 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [15] Ryan Babbush, Craig Gidney, Dominic Berry, Nathan  Wiebe,  Jarrod  McClean,  Alexandru  Paler,  Austin  Fowler, and Hartmut Neven, “Encoding Electronic Spec-  tra in Quantum Circuits with Linear T Complexity,”  Physical Review X 8, 041015 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [16] Ian D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Kivlichan, Craig Gidney, Dominic W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Berry,  Nathan Wiebe, Jarrod McClean, Wei Sun, Zhang Jiang,  Nicholas Rubin, Austin Fowler, Al´an Aspuru-Guzik,  Hartmut Neven, and Ryan Babbush, “Improved Fault-  Tolerant Quantum Simulation of Condensed-Phase Cor-  related Electrons via Trotterization,” Quantum 4, 296  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [17] Sam McArdle, Earl Campbell, and Yuan Su, “Exploiting  fermion number in factorized decompositions of the elec-  tronic structure Hamiltonian,” Physical Review A 105,  012403 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [18] Rolando D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Somma, “Quantum Simulations of One Di-  mensional Quantum Systems,” arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='17006 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [19] Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Geller, John M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Martinis, Andrew T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sorn-  borger, Phillip C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Stancil, Emily J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Pritchett, Hao You,  9  and Andrei Galiautdinov, “Universal Quantum Simula-  tion with Prethreshold Superconducting Qubits: Single-  Excitation Subspace Method,” arXiv:1505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='04990 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [20] Andreas  Dreuw  and  Martin  Head-Gordon,  “Single-  Reference ab Initio Methods for the Calculation of Ex-  cited States of Large Molecules,” Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 105, 4009–  4037 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [21] Erich  Runge  and  Eberhard  KU  Gross,  “Density-  functional theory for time-dependent systems,” Physical  review letters 52, 997 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [22] Robert Van Leeuwen, “Mapping from densities to poten-  tials in time-dependent density-functional theory,” Phys-  ical review letters 82, 3863 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [23] Samuel Manzer, Paul R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Horn, Narbe Mardirossian, and  Martin Head-Gordon, “Fast, accurate evaluation of ex-  act exchange: The occ-RI-K algorithm,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  143, 024113 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [24] Lin Lin, “Adaptively Compressed Exchange Operator,”  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 12, 2242–2249 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [25] Weile Jia and Lin Lin, “Fast real-time time-dependent  hybrid functional calculations with the parallel transport  gauge and the adaptively compressed exchange formu-  lation,” Computer Physics Communications 240, 21–29  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [26] Weile Jia and Lin Lin, “Fast real-time time-dependent  hybrid functional calculations with the parallel transport  gauge and the adaptively compressed exchange formula-  tion,” Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 240, 21–29 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [27] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Prodan and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Kohn, “Nearsightedness of electronic  matter,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 102, 11635–11638  (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [28] J¨org Kussmann, Matthias Beer, and Christian Ochsen-  feld, “Linear-scaling self-consistent field methods for  large molecules,” WIREs Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 3, 614–636  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [29] Conn O’Rourke and David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Bowler, “Linear scaling  density matrix real time TDDFT: Propagator unitar-  ity and matrix truncation,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 143, 102801  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Zuehlsdorff, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Hine, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Spencer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Harrison, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Riley, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Haynes, “Linear-scaling  time-dependent density-functional theory in the linear re-  sponse formalism,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 139, 064104 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [31] Venera Khoromskaia, Boris Khoromskij,  and Reinhold  Schneider, “QTT Representation of the Hartree and Ex-  change Operators in Electronic Structure Calculations,”  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 11, 327–341 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [32] Alberto Castro, Miguel AL Marques, and Angel Rubio,  “Propagators for the time-dependent Kohn–Sham equa-  tions,” The Journal of chemical physics 121, 3425–3433  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [33] Weile Jia, Dong An, Lin-Wang Wang,  and Lin Lin,  “Fast real-time time-dependent density functional the-  ory calculations with the parallel transport gauge,” Jour-  nal of Chemical Theory and Computation 14, 5645–5652  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [34] Alina Kononov, Cheng-Wei Lee, Tatiane Pereira dos San-  tos, Brian Robinson, Yifan Yao, Yi Yao, Xavier An-  drade, Andrew David Baczewski, Emil Constantinescu,  Alfredo A Correa, Yosuke Kanai, Normand Modine, and  Andr´e Schleife, “Electron dynamics in extended systems  within real-time time-dependent density-functional the-  ory,” MRS Communications , 1–13 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [35] Christopher Shepard, Ruiyi Zhou, Dillon C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Yost, Yi Yao,  and Yosuke Kanai, “Simulating electronic excitation  and dynamics with real-time propagation approach to  TDDFT within plane-wave pseudopotential formula-  tion,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 155, 100901 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [36] Isaiah Shavitt and Rodney J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Bartlett, Many-Body Meth-  ods in Chemistry and Physics:  MBPT and Coupled-  Cluster Theory  (Cambridge University Press, Cam-  bridge, England, UK, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [37] Stephen Wiesner, “Simulations of Many-Body Quan-  tum Systems by a Quantum Computer,” arXiv:quant-  ph/9603028 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [38] Daniel S Abrams and Seth Lloyd, “Simulation of Many-  Body Fermi Systems on a Universal Quantum Com-  puter,” Physical Review Letters 79, 4 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [39] Christof Zalka, “Efficient Simulation of Quantum Sys-  tems by Quantum Computers,” Fortschritte der Physik  46, 877–879 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [40] Bruce M Boghosian and Washington Taylor, “Simulating  quantum mechanics on a quantum computer,” Physica  D-Nonlinear Phenomena 120, 30–42 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [41] Daniel A Lidar and Haobin Wang, “Calculating the  thermal rate constant with exponential speedup on a  quantum computer,” Physical Review E 59, 2429–2438  (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [42] Ivan Kassal, Stephen P Jordan, Peter J Love, Masoud  Mohseni,  and Alan Aspuru-Guzik, “Polynomial-time  quantum algorithm for the simulation of chemical dy-  namics,” Proceedings of the National Academy of Sci-  ences 105, 18681–18686 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [43] Andrew Childs and Yuan Su, “Nearly optimal lattice sim-  ulation by product formulas,” Physical Review Letters  123, 050503 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [44] Yuan Su, Hsin-Yuan Huang,  and Earl T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Campbell,  “Nearly tight Trotterization of interacting electrons,”  Quantum 5, 495 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [45] Guang Hao Low, Yuan Su, Yu Tong, and Minh C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Tran,  “On the complexity of implementing Trotter steps,”  (2022), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='09133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [46] Ryan Babbush, Dominic W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Berry, Jarrod R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' McClean,  and Hartmut Neven, “Quantum Simulation of Chemistry  with Sublinear Scaling in Basis Size,” npj Quantum In-  formation 5, 92 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [47] Guang Hao Low and Nathan Wiebe, “Hamiltonian Sim-  ulation in the Interaction Picture,” arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='00675  (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [48] Yuan Su, Dominic Berry, Nathan Wiebe, Nicholas Ru-  bin, and Ryan Babbush, “Fault-tolerant quantum simu-  lations of chemistry in first quantization,” PRX Quantum  4, 040332 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [49] Alan Aspuru-Guzik, Anthony D Dutoi, Peter J Love,  and Martin Head-Gordon, “Simulated Quantum Compu-  tation of Molecular Energies,” Science 309, 1704 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [50] Guang Hao Low and Isaac L Chuang, “Hamiltonian Sim-  ulation by Qubitization,” Quantum 3, 163 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [51] V Rokhlin, “Rapid solution of integral equations of clas-  sical potential theory,” Journal of Computational Physics  60, 187–207 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [52] Josh Barnes and Piet Hut, “A hierarchical O(N log  N) force-calculation algorithm,” Nature 324, 446–449  (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [53] Tom Darden, Darrin York, and Lee Pedersen, “Particle  mesh Ewald: An N log N method for Ewald sums in large  systems,” The Journal of Chemical Physics 98, 10089–  10  10092 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [54] Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Childs, Jiaqi Leng, Tongyang Li, Jin-Peng  Liu, and Chenyi Zhang, “Quantum Simulation of Real-  Space Dynamics,” arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='17006 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [55] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone,  “Quantum Random Access Memory,” Physical Review  Letters 100, 160501 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [56] William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Huggins, Kianna Wan, Jarrod McClean,  Thomas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' O’Brien, Nathan Wiebe,  and Ryan Bab-  bush, “Nearly Optimal Quantum Algorithm for Esti-  mating Multiple Expectation Values,” arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='09283  (2021), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='09283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [57] Hsin-Yuan Huang, Richard Kueng,  and John Preskill,  “Predicting many properties of a quantum system from  very few measurements,” Nature Physics 16, 1050–1057  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [58] Andrew Zhao, Nicholas C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rubin, and Akimasa Miyake,  “Fermionic Partial Tomography via Classical Shadows,”  Physical Review Letters 127, 110504 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [59] Kianna Wan, William J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Huggins, Joonho Lee,  and  Ryan Babbush, “Matchgate Shadows for Fermionic  Quantum  Simulation,”  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='13723  (2022),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='13723.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [60] B O’Gorman, “Fermionic tomography and learning,”  arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='14787 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [61] Guang Hao Low, “Classical shadows of fermions with  particle number symmetry,” arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='08964  (2022),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='48550/arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='08964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [62] Gilles Brassard, Peter Høyer, Michele Mosca, and Alain  Tapp, “Quantum amplitude amplification and estima-  tion,” in Quantum Computation and Information, edited  by Vitaly I Voloshin, Samuel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Lomonaco, and Howard  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Brandt (American Mathematical Society, Washington  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', 2002) Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 53–74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [63] Patrick Rall, “Quantum algorithms for estimating phys-  ical quantities using block encodings,” Physical Review  A 102, 022408 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [64] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Kivlichan,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' McClean,  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Wiebe,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Gidney,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Aspuru-Guzik, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chan,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Babbush,  “Quantum Simulation of Electronic Structure with Lin-  ear Depth and Connectivity,” Physical Review Letters  120 (2018), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='110501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [65] Alain Delgado, Pablo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Casares, Roberto dos Reis,  Modjtaba Shokrian Zini, Roberto Campos, Norge Cruz-  Hern´andez, Arne-Christian Voigt, Angus Lowe, Soran  Jahangiri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Martin-Delgado, Jonathan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Mueller,  and Juan Miguel Arrazola, “Simulating key properties of  lithium-ion batteries with a fault-tolerant quantum com-  puter,” Physical Review A 106, 032428 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [66] Dominic W Berry, Maria Kieferov´a, Artur Scherer, Yu-  val R Sanders, Guang Hao Low, Nathan Wiebe, Craig  Gidney, and Ryan Babbush, “Improved Techniques for  Preparing Eigenstates of Fermionic Hamiltonians,” npj  Quantum Information 4, 22 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [67] V V Shende, S S Bullock,  and I L Markov, “Syn-  thesis of quantum-logic circuits,” IEEE Transactions on  Computer-Aided Design of Integrated Circuits and Sys-  tems 25, 1000–1010 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [68] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' David Mermin, “Stability of the thermal Hartree-Fock  approximation,” Annals of Physics 21, 99–121 (1963).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [69] Ryan Babbush, Jarrod R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' McClean, Michael Newman,  Craig Gidney, Sergio Boixo, and Hartmut Neven, “Focus  beyond Quadratic Speedups for Error-Corrected Quan-  tum Advantage,” PRX Quantum 2, 010103 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [70] Dong An, Di Fang, and Lin Lin, “Time-dependent un-  bounded Hamiltonian simulation with vector norm scal-  ing,” Quantum 5, 459 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [71] Makenzie R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Provorse and Christine M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Isborn, “Electron  dynamics with real-time time-dependent density func-  tional theory,” Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Quantum Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 116, 739–749  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [72] Aron J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Cohen, Paula Mori-Sa’nchez, and Weitao Yang,  “Insights into Current Limitations of Density Functional  Theory,” Science 321, 792–794 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [73] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Tully, “Chemical Dynamics at Metal Surfaces,”  Annu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 51, 153–178 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [74] Rulin Wang, Dong Hou,  and Xiao Zheng, “Time-  dependent density-functional theory for real-time elec-  tronic dynamics on material surfaces,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' B 88,  205126 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [75] Andrew David Baczewski, L Shulenburger, MP Desjar-  lais, SB Hansen, and RJ Magyar, “X-ray thomson scat-  tering in warm dense matter without the chihara decom-  position,” Physical review letters 116, 115004 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [76] Rudolph J Magyar, L Shulenburger, and AD Baczewski,  “Stopping of deuterium in warm dense deuterium from  ehrenfest time-dependent density functional theory,”  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [77] Xavier Andrade, S´ebastien Hamel, and Alfredo A Cor-  rea, “Negative differential conductivity in liquid alu-  minum from real-time quantum simulations,” The Eu-  ropean Physical Journal B 91, 1–7 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [78] YH Ding, Alexander James White, SX Hu, Ondrej Cer-  tik,  and Lee A Collins, “Ab initio studies on the stop-  ping power of warm dense matter with time-dependent  orbital-free density functional theory,” Physical Review  Letters 121, 145001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [79] Stefano Atzeni and J¨urgen Meyer-ter Vehn, The physics  of inertial fusion: beam plasma interaction, hydrodynam-  ics, hot dense matter, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 125 (OUP Oxford, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [80] Frank Graziani, Michael P Desjarlais, Ronald Redmer,  and Samuel B Trickey, Frontiers and challenges in warm  dense matter, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 96 (Springer Science & Business,  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [81] Tobias Dornheim, Simon Groth,  and Michael Bonitz,  “The uniform electron gas at warm dense matter condi-  tions,” Physics Reports 744, 1–86 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [82] James E Bailey, Taisuke Nagayama, Guillaume Pascal  Loisel, Gregory Alan Rochau, C Blancard, James Colgan,  Ph Cosse, G Faussurier, CJ Fontes, F Gilleron, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',  “A higher-than-predicted measurement of iron opacity at  solar interior temperatures,” Nature 517, 56–59 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [83] Taisuke Nagayama, JE Bailey, GP Loisel, GS Dunham,  GA Rochau, C Blancard, J Colgan, Ph Coss´e, G Faus-  surier, Christopher John Fontes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', “Systematic study  of l-shell opacity at stellar interior temperatures,” Phys-  ical review letters 122, 235001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [84] Christian Huber and Tillmann Klamroth, “Explicitly  time-dependent coupled cluster singles doubles calcula-  tions of laser-driven many-electron dynamics,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 134, 054113 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [85] Takeshi Sato,  Himadri Pathak,  Yuki Orimo,  and  Kenichi L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Ishikawa, “Communication: Time-dependent  optimized coupled-cluster method for multielectron dy-  namics,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 148, 051101 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [86] Philip Shushkov and Thomas F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Miller, “Real-time  density-matrix coupled-cluster approach for closed and  open systems at finite temperature,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 151,  11  134107 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [87] Alec F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' White and Garnet Kin-Lic Chan, “Time-  Dependent Coupled Cluster Theory on the Keldysh Con-  tour for Nonequilibrium Systems,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Theory  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 15, 6137–6153 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [88] Ian D Kivlichan, Nathan Wiebe, Ryan Babbush,  and  Alan Aspuru-Guzik, “Bounding the costs of quantum  simulation of many-body physics in real space,” Journal  of Physics A: Mathematical and Theoretical 50, 305301  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [89] Yng-Gwei Chen and John D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Weeks, “Local molecular  field theory for effective attractions between like charged  objects in systems with strong Coulomb interactions,”  Proceedings of the National Academy of Sciences 103,  7560–7565 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [90] Cristina E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Gonz´alez-Espinoza, Paul W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Ayers, Jacek  Karwowski,  and Andreas Savin, “Smooth models for  the Coulomb potential,” Theoretical Chemistry Accounts  135, 256 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [91] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Carrier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Greengard,  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rokhlin, “A Fast  Adaptive Multipole Algorithm for Particle Simulations,”  SIAM Journal on Scientific and Statistical Computing 9,  669–686 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [92] Sergey Bravyi and Dmitri Maslov, “Hadamard-free cir-  cuits expose the structure of the clifford group,” IEEE  Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Theory 67, 4546–4563 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [93] William J Huggins, Bryan A O’Gorman, Nicholas C Ru-  bin, David R Reichman, Ryan Babbush,  and Joonho  Lee, “Unbiasing fermionic quantum monte carlo with  a quantum computer,” Nature 603, 416–420 (2022),  arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='16235 [quant-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [94] Matthieu Lerasle, “Lecture notes:  Selected topics on  robust statistical learning theory,” arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='10761  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [95] H´ector J Garc´ıa, Igor L Markov, and Andrew W Cross,  “On the geometry of stabilizer states,” arXiv:1711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='07848  (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [96] Scott Aaronson and Daniel Gottesman, “Improved sim-  ulation of stabilizer circuits,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' A 70 (2004),  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='1103/physreva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='052328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [97] D Gottesman, The Heisenberg representation of quantum  computers, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' LA-UR-98-2848;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' CONF-980788-  (Los Alamos National Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', NM (United States), 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  [98] D Gross, F Krahmer,  and R Kueng, “A partial de-  randomization of PhaseLift using spherical designs,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Fourier Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 21, 229–266 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix A: Norms and scaling for the nonlinear differential equation governing mean-field evolution  We have the differential equation  i∂Cocc (t)  ∂t  = F (t) Cocc (t)  (A1)  where  Fµν(t) = hµν +  N  �  λσ  �  (µν|λσ) − (µσ|λν)  2  �  Pσλ(t)  (A2)  with P(t) = Cocc(t)Cocc(t)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If F were independent of Cocc, then it would imply that taking the nth derivative gives  (i)n ∂nCocc (t)  ∂tn  = FnCocc (t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A3)  That means the norm of the nth derivative would scale as ∥F∥n (with Cocc normalized).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Then higher-order methods will typically have an error that scales as the norm of the higher-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For  example, if one were to use a Taylor series up to order k to approximate a time step, then the error for a time step  of length δt would scale as  1  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='∥F∥k+1δtk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A4)  This means that if the size of the time step is taken as proportional to 1/∥F∥, then the error may be made exponentially  small in k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As a result, the total number of time steps used scales as O(∥F∥t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Similar considerations hold for other  higher-order methods for integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The dependence of the complexity on ∥F∥ can also be expected from principles  of scaling, where if F is divided by ∥F∥ but t is also multiplied by ∥F∥, then the same differential equation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In our case where F is dependent on Cocc, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is because taking higher-order  derivatives of Cocc yields more terms due to the derivatives of Cocc in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To describe this, let us write, omitting hµν  for simplicity,  Fµν(t) = VµνσλCσaC∗  λa,  (A5)  12  with Cσa the matrix entries of Cocc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We are taking a convention that Greek indices are over all orbitals, English  letters are over electrons, and repeated indices are summed over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then we would give the derivative as  i∂Cµb  ∂t  = VµνσλCσaC∗  λaCνb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A6)  We can define an η-norm of V as  ∥V∥η = max  x,y,z Vµνσλxµy∗  νzσλ,  (A7)  with ∥x∥ = ∥y∥ = ∥z∥ = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', spectral norms are normalized), and z of rank η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To bound this norm, we can  consider the first term for Vµνσλ, which is  (µν|λσ) =  �  dr1 dr2  φ∗  µ (r1) φν (r1) φ∗  λ (r2) φσ (r2)  |r1 − r2|  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A8)  The multiplication by xµ and sum over µ corresponds to a transformation of φµ to a new orbital, and similarly,  the sum over ν transforms φν to another new orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Since z is of rank η, the sum over λ and σ corresponds to  transforming the orbital basis for both φλ and φσ, and summing over η of these basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  That is, we can write  η  �  a=1  �  dr1 dr2  φ∗ (r1) χ (r1) ψ∗  a (r2) θa (r2)  |r1 − r2|  ,  (A9)  for some transformed orbitals φ, χ, ψa, θa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We can then use the fact that |φ∗χ| ≤ |φ|2 + |χ|2, and similarly for ψa and  θa to upper bound this expression by  4  η  �  a=1  �  dr1 dr2  |φ (r1) |2|ψa (r2) |2  |r1 − r2|  ,  (A10)  for some choice of φ and ψa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This integral can be maximized when the ψa are orbitals that are clustered as close as  possible to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' With neighboring grid points separated by δ, the smallest the average separation can be is O(η1/3δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Then the factor of 1/|r1 − r2| in the integrals will give O(1/[η1/3δ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Multiplying the sum by η gives O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The second term for Vµνσλ is (µσ|λν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is similar to (µν|λσ), but with ν and σ swapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then the transfor-  mation of orbitals gives  η  �  a=1  �  dr1 dr2  φ∗ (r1) χa (r1) ψ∗  a (r2) θ (r2)  |r1 − r2|  ,  (A11)  for some choice of φ, χa, ψa, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The same argument holds, where the sum is maximized with orbitals over a region of  volume ηδ3 so there are contributions from all η terms in the sum, but 1/|r1 − r2| averages to give O(1/[η1/3δ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This  gives the same scaling for the second term for Vµνσλ, and so  ∥V∥η = O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A12)  What this means is that, whenever we have a contraction of the σ, λ indices in Vµνσλ with a normalized matrix of  rank η, the remaining matrix has norm O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That immediately implies that ∥F∥ has this norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then applying  F to the normalized matrix Cocc gives an upper bound on the first derivative O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Taking the second derivative then yields an expression with 3 terms, where each has V appearing twice and Cocc  appearing five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In particular,  −∂2Cµb  ∂t2  = Vµνσλ[(VσϵζηCζcC∗  ηc)CϵaC∗  λa]Cνb  + Vµνσλ[Cσa(VλϵζηC∗  ζcCηc)C∗  ϵa]Cνb  + (VµνσλCσaC∗  λa)(VνϵζηCζcC∗  ηc)Cϵb  (A13)  Only the third line has a simple interpretation as F squared times Cocc (indicated by the brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The first line has V contracted with Cocc using ζ, η, so the expression in round brackets is a matrix with norm  O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then in matrix terms, it is multiplied by CϵaC∗  λa (summed over a), which is a matrix of norm 1 and rank  13  η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As a result, the expression in square brackets is of norm O(η2/3/δ) and rank η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We can then see that the first V is  contracted over σ, λ with a matrix of rank η and norm O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That implies that the norm of the resulting matrix  is upper bounded by the square of O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That is then multiplied by Cνb which is of norm 1, resulting in the  overall norm of this line being upper bounded by the square of O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Similar considerations hold for the second  line, so we can upper bound the entire second derivative by an order scaling that is the square of that for ∥F∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In this, the general principle is that wherever we have something of the form CσaC∗  λa, it is a matrix of norm 1 and  rank η, and taking the derivative of it yields something that is still of rank η, but with a norm upper bounded by  O(η2/3/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Because we have bounded the norm when contracting V with a general matrix of rank η, that yields a  factor of O(η2/3/δ) on whatever result we had for the lower-order derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The other scenario is where we take the  derivative of Cνb, which is effectively like multiplying it by F which increases the norm (but not the rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This reasoning holds in general whenever we take the derivative of an expression for the derivative of some order  to give the derivative of higher order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The norm is multiplied by O(η2/3/δ) for each of the terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The number of  terms will increase exponentially with the order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The third derivative has 3×5 terms, where each of the three original  terms yields five due to the derivatives of Cocc at each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then the fourth-order derivative has 3 × 5 × 7 terms  and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In describing the scaling we can ignore this exponential number of terms, and give the upper bound on  the nth order derivative as O(η2/3/δ) to the power of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This implies that the appropriate scaling of the time should  again be T = ∥F∥t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Finally we bound the norm of ∥h∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When using a plane wave basis, hµν will be non-zero only when µ = ν with  entries scaling as O(1/δ2) due to the ∇2 in the expression for hµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That gives the scaling of the spectral norm for this  component, which would be unchanged under a unitary transformation, such as the Fourier transform which maps  plane waves to an approximately local basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For the dependence of hµν on V (r), the potential will come from nuclei, and for charge-neutral systems the total  nuclear charge will be the same as the number of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If the nuclear charge were entirely at one location and we  have a charge-neutral system, then the largest contribution to hµν would be for an approximately local basis, where  the contribution would scale as η/δ, with the factor of η from the nuclear charge and 1/δ from the inverse distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In most cases that we would be interested in, there would be a more even distribution of nuclear charges through  the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In that case, if the volume scales as η, there would be an average distance O(η1/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That would result in  a contribution to hµν of O(η2/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' An orbital localized near one nucleus would give a contribution of O(1/δ) just from  that nucleus, which may be larger than O(η2/3) if N > η3 but may be ignored in comparison to 1/δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  As a result of these considerations, we can give the upper bound on F in the case without V (r) as  ∥F∥ = O  �η2/3  δ  + 1  δ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A14)  In the case with nuclei we obtain the same result, provided the nuclear charges are not clustered any closer than the grid  spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Here δ = O((η/N)1/3) is the minimum grid spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This scaling for δ comes from taking the computational  cell volume proportional to η (a reasonable assumption for both condensed-phase and molecular systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, the  scaling becomes  ∥F∥ = O  �  N 1/3η1/3 + N 2/3  η2/3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (A15)  In this case we can see that the first term is dominant unless N > η3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix B: Proving sublinear gate complexity in basis size for Trotter based methods  Here we derive the complexity for quantum simulation of the electronic structure problem given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We  consider the simulation of the electronic structure problem defined on a spatial grid in first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Such a  14  Hamiltonian can be expressed as  H = T + U + V +  L  �  ℓ̸=κ=1  ζℓζκ  2 ∥Rℓ − Rκ∥  (B1)  T ≈  η  �  i=1  QFTj  �  ��  p∈G  ∥kp∥2  2  |p⟩⟨p|j  �  � QFT†  j  (B2)  U = −  η  �  j=1  L  �  ℓ=1  �  p∈G  ζℓ  ∥Rℓ − rp∥ |p⟩⟨p|j  (B3)  V =  η  �  j̸=k=1  �  p,q∈G  1  2 ∥rp − rq∥ |p⟩⟨p|j |q⟩⟨q|k  (B4)  where QFTj is the usual quantum Fourier transform applied to register j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We emphasize that T is only approximately  given by the expression involving the QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This relation is exact in the continuum limit where N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For finite-  sized grids N, it cannot be the case that the QFT completely diagonalizes the momentum operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Instead, writing  T this way represents something similar to the approximations made by so-called “discrete value representation”  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Using the QFT means that the evolution can be broken into a product of the evolution under T and the  one under U + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In the above expression, ℓ and κ index nuclear degrees of freedom;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' thus, Rℓ represents the positions of nuclei and ζℓ  the atomic numbers of nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In this appendix, we use L to denote the number of nuclei in our simulation (elsewhere,  L is the number of time points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Furthermore, we have the following definition of grid points and their frequencies in  the dual space defined by the QFT:  rp = p Ω1/3  N 1/3  kp = 2πp  Ω1/3  p ∈ G  G =  �  −N 1/3 − 1  2  , N 1/3 − 1  2  �3  ⊂ Z3 ,  (B5)  where Ω is the volume of the simulation cell and N is the number of grid points in the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Although it is defined  here in more precise terms, this is essentially the same representation used in the first work on quantum simulating  chemistry in first quantization, by Kassal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [42], well over a decade ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We consider simulation performed using high-order product formulas with a split-operator Trotter step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' What we  mean by the latter is that we will alternate evolution under T (using the QFT) and evolution under U + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In fact,  the implementation of each Trotter step that we will pursue is essentially identical to the Trotter steps proposed by  Kassal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The Trotter step requires �  O(η2) gates, with the complexity being dominated by computing the  O(η2) different interactions in the two-electron term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Recently, Low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [45] have shown that the number of Trotter  steps required in second quantization using arbitrarily high order formulas can be as low as  �  N 1/3η1/3 + N 2/3  η2/3  � t1+o(1)N o(1)  ϵo(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (B6)  We note that, curiously, this also closely matches our bound for the norm of the Fock operator (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (A15)) proved  in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The first term in brackets similarly corresponds to a contribution to the potential from electrons  grouped as closely as possible in real space, but the reason why this quantity is relevant is very different between the  two calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The results for the Trotter error in second quantization also hold for first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As a general principle, we  can consider the effect of �  j |p⟩ ⟨q|j on a computational basis state consisting of an anti-symmetric combination of  lists of electron positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This removes an electron from orbital q and places it in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is performed for every part  of the anti-symmetric state, preserving its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, for the starting anti-symmetric state the sign is based on  whether the permutation is even or odd (as compared to ascending order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If moving an electron from q to p passes  over an odd number of electrons, then the parity of each permutation flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That means that there is an overall sign  flip in the basis state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Similarly, if we consider the action of a†  paq on a state a†  q1 · · · a†  qη |0⟩, then the aq can be anti-commuted to the  right to give several sign flips corresponding to the number of a†  qj operators that are anti-commuted through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This  corresponds to the number of occupied orbitals before q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then aqa†  q gives the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Next, anti-commute a†  p to the  appropriate location in the list of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The sign that is obtained corresponds to the number of a†  qj operators  15  that are anti-commuted through, which is the number of electrons before p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' There is an overall sign flip if there is an  odd number of electrons between p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This can then be extended to products such as  �  j  |p⟩ ⟨q|j  �  k  |r⟩ ⟨s|k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (B7)  The first sum corresponds to a†  paq in second quantization, and the second sum corresponds to a†  ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This means that  we have the equivalence  �  pqrs  Vpqrs  �  j  |p⟩ ⟨q|j  �  k  |r⟩ ⟨s|k ≡  �  pqrs  Vpqrsa†  paqa†  ras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (B8)  The action on an anti-symmetric computational basis state in first quantization has exactly the same effects as that on  the corresponding second-quantization state with η electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Moreover, the action of the operators always preserves  the electron number in second quantization, so there is a corresponding state in first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Similarly, because  we are using anti-symmetric states in first quantization, it is impossible to obtain a state with multiple electrons on  the same orbital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That is because two registers with the same orbital number will give cancellation of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  As a result all operators and states in second-quantization map directly to first quantization, preserving the norms,  and in particular the error bounds derived in second-quantization hold for first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Therefore, multiplying  the number of steps in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (B6) by the �  O(η2) gate complexity required of the first quantized Trotter step from [42]  gives the following gate complexity for the product formula based time evolution in first quantization:  �  N 1/3η7/3 + N 2/3η4/3� t1+o(1)N o(1)  ϵo(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (B9)  This is the complexity given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix C: Constant factors for time-evolution in the interaction-picture plane-wave algorithm  Here we analyze the constant factors in the scaling of the interaction picture based plane wave algorithm from  Babbush at al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [46] which was analyzed in detail for use in phase estimation by Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' As explained on page 30  of [48], the number of steps to give total time T using the time evolution approach is λBT/ ln 2, but with a factor of  3 overhead for amplitude amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Using the qubitization approach the number of steps is eλBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That means  simulating the time evolution gives an overhead of 3/(e ln 2) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='59 over the qubitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (154) of  [48], the total time of evolution is approximately π/(2ϵpha) to give precision ϵpha of the phase estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' There is  moreover a (small) term O((λU + λV )2∆E2) in the expression for the number of steps N in [48] that originates from  the nonlinearity of the sine function in phase estimation, which is not used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  As a result, the complexity given in Theorem 5 of [48] can be modified to be appropriate for time evolution simply  by replacing the formula for the number of steps in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (174) of [48] with  N = 3T(λ1  U + λ1  V /(1 − 1/η))  Peq ln 2  + O(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (C1)  Here we have replaced π/(2ϵpha) with T, replaced e with e/ ln 2, and removed O((λU + λV )2∆E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that in this  expression  λU = η �  ℓ ζℓ  πΩ1/3 λν,  (C2)  λV = η(η − 1)  2πΩ1/3 λν,  (C3)  λν =  �  ν∈G0  1  ∥ν∥2 ≤ 4πN 1/3,  (C4)  λ1  U ≈ λU, λ1  V ≈ λV , and Peq is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This expression together with an appropriate choice of constant factor  in Ω ∝ η gives the constant factor for the number of steps to use for time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' It needs to be multiplied by  a further complicated expression in Theorem 5 of [48] for the gate complexity of a single step to provide the full  constant factor for the gate complexity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  16  Appendix D: Smoothing the Coulomb operator to exponentially suppresses quantum scaling in basis size  Here we discuss the fact that if one is willing to introduce a slight systematic bias into the Coulomb operator, it is  possible to further improve the speedup in N of the quantum algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The N 1/3 dependence enters into the cost  from the 1-norm of the two-body Coulomb operator, which scales as λ = O(η2Vmax) where Vmax is the maximum  value of the electron-electron interaction for a single pair of electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For typical plane wave or grid discretizations we have that Vmax = O(N 1/3/Ω1/3) where Ω is the size of the  computational cell (for the purpose of the analysis in this paper we assume that Ω = O(η), since that is explicitly the  case in condensed phase simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' But we could also take steps to smooth out the cusp in the Coulomb operator  and thus, lower the energy scale of Vmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For example, this could be accomplished by taking Vmax to be a constant  and modifying the real-space form of the two-body Coulomb operator as  1  |r1 − r2| →  1  |r1 − r2| + Vmax  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (D1)  Such a strategy has been explored in the context of first quantized quantum algorithms in real space in papers by  Kivlichan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [88] and Childs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In principle, one could choose Vmax = O(log N) and this would lead to the quantum algorithm scaling exponentially  better than classical algorithms in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This would also slightly reduce the cost of classical mean-field algorithms from  scaling as N 4/3 to scaling as N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Of course, using such a drastic cutoff will introduce a significant bias into the  overall dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In order to avoid this, papers such as [89, 90] have sought to develop Richardson extrapolation type  schemes where simulations are run with a series of smoothing or cutoff parameters in order to extrapolate the value  of the observable with zero cutoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, questions remain about the convergence of such procedures and it seems  likely to re-introduce some polynomial dependence on N in order to reach convergence with the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Nevertheless, the context of this paper is that one might be interested in getting a speedup over low accuracy  classical algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In that spirit, one could probably make the case that if merely trying to improve in speed over  mean-field algorithms, the error introduced in imposing a cutoff in the Coulomb operator might be less significant  than the error due to making the mean-field approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, this is perhaps a valid approach when competing  with such classical methods, and thus might provide an exponential speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix E: Gate complexity and speedup in various regimes  Processor  Algorithm for sampling |ψ(t)⟩  Regime of optimality  Space  Effective gate complexity  classical zero temp mean-field with occ-RI-K/ACE [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 24]  N ≤ Θ(η3)  �  O(Nη)  N 4/3η7/3t(Nt/ϵ)o(1)  classical zero temp mean-field with occ-RI-K/ACE [23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 24]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N ≥ Θ(η3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(Nη)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N 5/3η4/3t(Nt/ϵ)o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='second quantized Trotter grid algorithm [45]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N ≤ Θ(η2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(N log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N 4/3η1/3t(Nt/ϵ)o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='first quantized Trotter grid algorithm here  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='Θ(η2) ≤ N ≤ Θ(η3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N 1/3η7/3t(Nt/ϵ)o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='first quantized Trotter grid algorithm here  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='Θ(η3) ≤ N < Θ(η4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N 2/3η4/3t(Nt/ϵ)o(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='qubitization algorithms from [46] or [48]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N = Θ(η4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(N 2/3η4/3t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='quantum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='interaction picture algorithms from [46] or [48]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='N > Θ(η4)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(η log N)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='O(N 1/3η8/3t)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Best known gate complexities of exact quantum algorithms and classical mean-field algorithms for sampling the  output of time-evolution, by ratio of basis size to particle number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Here we use the asymptotic Θ(·) notation, which implies the  union of both an asymptotic upper-bound and an asymptotic lower-bound on the scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' N is number of basis functions, η is  number of particles, ϵ is target precision, and t is duration of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' “Effective gate complexity” is the leading order scaling  in the stated regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' All quantum algorithms discussed here require either a plane wave or grid basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For those basis sets, the  large space overhead of second quantization likely makes second quantized approaches infeasible in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When N = η4,  the quantum algorithms with the best asymptotic scaling are the plane wave or grid basis qubitization algorithms from [46] or  [48], respectively, as opposed to the interaction picture algorithms of those same works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is due to lower polylogarithmic  factors in the scaling that are suppressed by the �  O(·) notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Another way to express the results of Table II is as a formula for the leading order scaling if assume that N = Θ(ηα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Then, for the classical algorithm we have that the leading gate complexity of the best approach is  �  ηβt  � �Nt  ϵ  �o(1)  where  N = Θ (ηα)  and  β =  �  4α+7  3  α ≤ 3  5α+4  3  α ≥ 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (E1)  17  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Plot showing the numerical values of the speedup exponent ratio given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We see that a super-quadratic  speedup of exact quantum algorithms over mean-field classical algorithms is realized when α < 5/4 and when α > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  By contrast, for the quantum algorithm we have that the leading order gate complexity of the best approach is  �  ηβt  � �Nt  ϵ  �o(1)  where  N = Θ (ηα)  and  β =  �  �  �  �  �  �  �  �  �  4α+1  3  α ≤ 2  α+7  3  2 ≤ α ≤ 3  2α+4  3  3 ≤ α ≤ 4  α+8  3  α ≥ 4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (E2)  For both classical and quantum expressions, these complexities are sometimes loose by sub-polynomial factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Finally,  we compare the speedup that exact quantum algorithms offer over classical mean-field algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We report this as  exponent of η scaling of classical complexity  exponent of η scaling of quantum complexity =  �  �  �  �  �  �  �  �  �  (4α + 7) / (4α + 1)  α ≤ 2  (4α + 7) / (α + 7)  2 ≤ α ≤ 3  (5α + 4) / (2α + 4)  3 ≤ α ≤ 4  (5α + 4) / (α + 8)  α ≥ 4  if  N = Θ (ηα) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (E3)  We plot numerical values of this speedup in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Finally, we discuss the hope that Trotter based first quantized algorithms might be sped up by a factor of �  O(η) by  developing more efficient Trotter steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The bottleneck for Trotter steps is the computation of the Coulomb operator  since the simulation of the kinetic operator scales as �  O(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, it seems promising that fast-multipole [51] Barnes-  Hut [52], or particle-mesh Ewald [53] type algorithms for computing the Coulomb potential require �  O(η) operations  in the classical random access memory (RAM) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' By contrast, the standard way of computing the Coulomb  potential (involving summing up all  �η  2  �  pairs of electrons) scales as �  O(η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Thus, if one can figure out how to extend  these better scaling methods to first quantization with �  O(η) operations in the reversible circuit model (the cost model  of relevance for this subroutine if executed on a quantum computer), the quantum algorithm would scale as  �  N 1/3η4/3t + N 2/3η1/3t  � �Nt  ϵ  �o(1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (E4)  We note that it is straightforward to adapt these algorithms to second quantization with �  O(N) gate complexity  [45, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, translating such algorithms to first quantization with �  O(N) gate complexity in the quantum circuit  model is highly non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This is due to nuances of how adaptive tree-like data structures are constructed and used  in these algorithms, and it is why the work of [54] decided to invoke the impractical assumption of QRAM in order to  leverage the fast multipole algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note further that some of these algorithms such as the original fast multipole  [51] and particle-mesh Ewald [53] make further assumptions on the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In particular, if space is partitioned into  O(η) boxes, then these methods require that no more than k electrons are present in any box, in any configuration  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='2  speedup ratio  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='1  quadratic speedup threshold  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='7  2  3  4  5  1  6  value of α in N= (n~)18  on which the wavefunction has support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Since electrons tend to repel one another this is often a good assumption  at low energies, but it is not true for general states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' It seems possible to implement a first quantized algorithm with  �  O(η k) space complexity and �  O(η poly(k)) gate complexity by keeping k electron registers for each of these O(η)  boxes of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' But there also exist versions of these algorithms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' described in [91], which use RAM and an  adaptive tree structure to give �  O(η) complexity without any assumptions on the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Such approaches appear quite  challenging to port to the quantum circuit model with the same complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, if possible, the first quantized  fast multipole-based Trotter would scale better than all other known approaches as long as N < η7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' When N > η7,  the first quantized interaction picture algorithm has better scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix F: Efficient reduced density matrix estimation using classical shadows in first quantization  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Problem statement  We consider a system of η identical fermions occupying N ≫ η orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In first-quantization, we represent the  state of such a system as a wavefunction on η registers of n = ⌈log(N)⌉ qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We demand that this wavefunction is  antisymmetric under the exchange of any two registers in order for it to represent a valid physical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Most physically interesting observables of such a system are captured by the few-body marginals, the reduced  density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In this section, we concern ourselves with efficiently estimating elements of the k-body reduced  density matrix (k-RDM) of the first-quantized state |ψ⟩ defined on η identical fermion particles,  kDj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',jk  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',ik =  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' tr  �  |ψ⟩⟨ψ|  k  �  ℓ=1  |iℓ⟩⟨jℓ|ℓ  �  ,  (F1)  where |i⟩⟨j|ℓ indicates the tensor product of |i⟩⟨j| on the ℓth register with the identity on the other η − 1 registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We  can write an equivalent definition (equivalent due to the antisymmetry of the wavefunction),  kDj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',jk  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',ik =  �  x∈Sη  k  tr  �  |ψ⟩⟨ψ|  k  �  ℓ=1  |iℓ⟩⟨jℓ|xℓ  �  ,  (F2)  where Sη  k is the set composed of all possible sequences of length k generating by drawing without replacement from  [η] := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , η}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Our goal is to use measurements of the state |ψ⟩ to obtain a classical description of the state with enough information  to approximate all N 2k elements of the k-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We would like all of these estimates to accurate up to some additive  error ϵ with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Ideally, our protocol will be efficient not only in terms of the number  of measurements, but also in terms of the (gate) complexity of implementing each measurement and the classical  complexity of the required post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We will accomplish our goal by applying the classical shadows formalism of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We propose and analyze a  protocol that requires at most  m = 64e3 log (N/δ) k (2k + 2e)k ηkϵ−2  (F3)  measurements to estimate the k-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Performing these measurements requires acting on each of the particle registers  with a randomly sampled Clifford circuit and performing a measurement in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' These circuits  can be implemented using O(ηn2) one- and two-qubit Clifford gates on a linearly connected array of qubits in depth  O(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Each element of the k-RDM requires performing a number of classical operations that scales as  m′ = O  ��  n4 + log (1/δ)  �  η2k2kϵ−2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The measurement protocol  The classical shadows formalism of Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' works by choosing an ensemble of random unitaries U on n qubits  and defining a measurement channel  M(σ) := EU∼U  �  b∈{0,1}n  U † |b⟩⟨b| U ⟨b|UσU †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F5)  19  For specific choices of U, the channel M is analytically invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Operationally, we obtain the classical shadow of σ by  repeatedly sampling a unitary U from U, applying the sampled U to a copy of σ, and measuring in the computational  basis (obtaining the bitstring b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If we collect m such samples, then we call the (potentially unphysical) state  ˆσ := 1  m  m  �  i=1  M−1 �  U †  i |bi⟩⟨bi| Ui  �  (F6)  a classical shadow of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For an arbitrary observable O, we can define an estimator ˆo of the quantity tr [Oρ] using the  classical shadow of ρ,  ˆo := tr [Oˆρ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F7)  In expectation, we have that  ⟨ˆσ⟩ = EU∼U  �  b∈{0,1}n  M−1 �  U † |b⟩⟨b| U  �  ⟨b|UσU †|b⟩ = M−1 (M (σ)) = σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F8)  When we take U to be the uniform distribution over the Clifford group on n qubits, the classical shadows measurement  channel and its inverse have particularly simple forms [57],2  M(A) =  1  2n + 1A + tr [A]  2n + 1I,  (F9)  M−1(A) = (2n + 1)A − tr [A] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F10)  Here, and throughout our analysis of the measurement protocol, we use the symbol I to denote the identity operator  on a Hilbert space whose dimension is appropriate for the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In this work, we propose and analyze the impact of using an ensemble U that consists of a tensor product of η  copies of the uniform distribution over n qubit Clifford circuits,  U =  η  �  j=1  Cl(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F11)  That is to say, we perform our measurements by independently sampling η n-qubit Clifford unitaries, applying one  to each particle register, and measuring in the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We can consider the action of the corresponding  classical shadow measurement channel and its inverse on an operator X1⊗· · ·⊗Xη that factorizes across the η registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The channel is defined on the whole Hilbert space by linear extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For the classical shadow measurement channel,  we have  M(X1 ⊗ · · · ⊗ Xη) =  η  �  j=1  �  �EUj∼Cl(2n)  �  bj∈{0,1}n  U †  j |bj⟩⟨bj| Uj ⟨bj|UjXjU †  j |bj⟩  �  �  (F12)  =  η  �  j=1  �Xj + tr [Xj] I  2n + 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F13)  The inverse, similarly, is given by  M−1(X1 ⊗ · · · ⊗ Xη) =  η  �  j=1  ((2n + 1) Xj − tr [Xj] I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F14)  Due to the antisymmetry of the wavefunction, we have the freedom to choose between a number of different  observables when estimating the elements of the k-RDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Consider an arbitrary operator O, and the operator POP †,  where P is an operator that permutes the particle registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The expectation values of O and POP † with respect  to a first-quantized wavefunction are the same (to see this, observe that any sign picked up by acting P † on the ket  2 Actually, a substantial constant factor savings in the number of  gates can be obtained by using the canonical form of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 92 and  simply dropping the permutation at the end of the circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' See,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  20  is cancelled out by a corresponding sign obtained from acting P on the bra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We can use this degree of freedom  to minimize the variance of our measurement protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Using the observable from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F1) to construct a classical  shadow estimator of a k-RDM element would lead to an unnecessarily large variance, essentially because the observable  doesn’t take advantage of all of the information present in the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In contrast, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F2) defines the k-RDM element  in terms of a sum over many different permutations of the registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We conjecture that a measurement protocol  based on the observable in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F2) would perform well, but the analysis could be tedious due to the many different  cases that would arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Rather than using the observables implied by either Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F1) or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F2) in our classical shadow measurement  procedure, we instead choose to estimate the k-RDM elements using an observable that involves a sum over a simpler  set of permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Essentially, we break the η registers up into k groups of size η/k and measure the k-RDM element  using registers from each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For ease of notation, let us assume that η is divisible by k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='3 Formally, we can define  a set of sequences  Rk = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , η/k} × {η/k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , 2η/k} × · · · × {(k − 1) η/k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , η} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F15)  Due to the antisymmetry of the wavefunction, we have that  kDj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',jk  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',ik =  kk (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �  x∈Rk  tr  �  |ψ⟩⟨ψ|  k  �  ℓ=1  |iℓ⟩⟨jℓ|xℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F16)  We define an estimator ˆd for the k-RDM element kDj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',jk  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',ik using the classical shadow ˆρ of |ψ⟩,  ˆd =  kk (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �  x∈Rk  tr  �  ˆρ  k  �  ℓ=1  |iℓ⟩⟨jℓ|xℓ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F17)  In Appendix F 4, we prove that the single-shot variance of this estimator is bounded by  Var( ˆd) ≤ e3ηk (2k + 2e)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F18)  In order to guarantee that our estimates are close to the true value of the k-RDM elements with high probability,  we need to proceed along the same lines as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 57 and construct a median-of-means estimator to obtain the desired  rigorous guarantees [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To be precise, using Proposition 12 from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 94, we can consider an estimator that divides  the m total classical shadow samples into K groups of size b, and takes the median of the sample mean obtained by  averaging the estimates within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The probability that this median of means estimator has an error larger  than 2  �  Var( ˆd)/b is at most e−K/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To bound the error in our estimate by ϵ with a success probability of at least  1 − δ, this implies that we need  b = 4 Var( ˆd)/ϵ2,  (F19)  K = 8 log (1/δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F20)  The overall number of measurements claimed in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F3) follows directly from applying a union bound over the failure  probabilities for estimating all N 2k k-RDM elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The measurement protocol can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We take a classical shadow of |ψ⟩ with the U defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F11) using a number of samples m chosen according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For each sample, we evaluate the expectation  values of the (η/k)k different terms in the sum over Rk (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F17)) using generalizations of Gottesman-Knill  theorem that account for the phase of the quantities involved [95–97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Breaking the samples into K groups of size b,  averaging within the groups, and then taking the median of these means then yields the final estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The classical  post-processing costs quoted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F4) come from counting the number of n-qubit sized Clifford circuits that need  to be simulated classically to carry out this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  3 In the event that η is not exactly divisible by k, one could modify  the protocol to either use groups of slightly different sizes or to  only perform the measurements using η′ = k⌊η/k⌋ registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Notation and preliminaries  Before we proceed to bound the variance of the estimator ˆd for an arbitrary k-RDM element, it is helpful to recall  a few useful expressions and prove some identities that we will use later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We will make use of a formula for the two-fold twirl over the Clifford group and partial trace obtained from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 57,  EU∼Cl(2n)U † |x⟩⟨x| U ⟨x|UAU †|x⟩ = A + tr(A)I  2n(2n + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F21)  For the three-fold twirl and partial trace, we find it convenient to use the identity  EU∼Cl(2n)U † |x⟩⟨x| U ⟨x|UBU †|x⟩ ⟨x|UCU †|x⟩ =  1  2n (2n + 1) (2n + 2) (I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F22)  This equation is different from the corresponding one considered in previous work (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (S36) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 57), in that it  allows for B and C to have non-zero trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' It can be obtained directly from the analysis of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='4  Another small departure we make from some prior work is that we directly consider the variance of estimators for  the expectation values of non-Hermitian observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For a classical shadow ˆρ of a state ρ and an estimator ˆo = tr [ˆρO]  of the expectation value of a (not necessarily Hermitian) operator O, we have  Var(ˆo) = tr  �  ρ  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(O)U †|b⟩ ⟨b|UM−1(O†)U †|b⟩  �  − |tr [Oρ]|2  (F23)  ≤ tr  �  ρ  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(O)U †|b⟩ ⟨b|UM−1(O†)U †|b⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F24)  This expression can be arrived at from the definition of the variance of a complex-valued random variable applied to  the classical shadow formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We refer the reader to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 59 for a thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In the course of calculating the variance for the higher-order RDMs, we will find that we repeatedly need to simplify  certain expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Before describing those expressions and showing how they may be simplified, let us define some  notation used for convenience throughout the rest of our analysis:  Px = |x⟩⟨x| ,  (F25)  Pxy = |x⟩⟨y| ,  (F26)  EU = EU∼Cl(2n),  (F27)  �  b  =  �  b∈{0,1}n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F28)  One class of expressions that we will need to simplify are of the form  A = EU  �  b  U †PbU ⟨b|UM−1(Pij)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F29)  We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F10) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F21) to simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F29),  A = EU  �  b  U †PbU ⟨b|UM−1(Pij)U †|b⟩  (F30)  = EU  �  b  U †PbU ⟨b|U ((2n + 1) Pij − δi,jI) U †|b⟩  (F31)  = Pij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F32)  4 Note that while the proof of Lemma 7 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' 98 is technically  for Hermitian matrices, the same proof holds exactly in the non-  Hermitian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  22  Another kind of expression that we will need to simplify is of the form  A = EU  �  b  U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pkl)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F33)  Let us consider the first case, and simplify A as defined below,  A = EU  �  b  U †PbU ⟨b|UM−1(Pi)U †|b⟩ ⟨b|UM−1(Pi)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F34)  We have  M−1 (Pi) = (2n + 1) Pi − I  (F35)  by an application of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Now we can apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F22) with B = C = (2n + 1) Pi − I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Let us simplify the pieces of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F22) separately before combining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We have  BC = CB = ((2n + 1) Pi − I)2  (F36)  = (2n + 1) (2n − 1) Pi + I,  (F37)  tr [BC] = tr [CB] = 2n(2n + 1) − 1,  (F38)  tr [B] = tr [C] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F39)  As a result,  I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB  (F40)  = 2n (2n + 1) I + 2 (2n + 1) Pi − 2I + 2 (2n + 1) (2n − 1) Pi + 2I  (F41)  = 2n (2n + 1) (I + 2Pi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F42)  Putting everything together, we have  A = EU  �  b  U †PbU ⟨b|UM−1(Pi)U †|b⟩ ⟨b|UM−1(Pi)U †|b⟩  (F43)  =  2n  2n + 2 (I + 2Pi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F44)  Now we consider simplifying the expression  A = EU  �  b  U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pji)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F45)  In this case, we can again use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F22) with  B = M−1(Pij) = (2n + 1) Pij,  (F46)  C = M−1(Pji) = (2n + 1) Pji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F47)  Working out some of the pieces, we have  BC = (2n + 1)2 Pi,  (F48)  CB = (2n + 1)2 Pj,  (F49)  tr [BC] = (2n + 1)2 ,  (F50)  tr [B] = tr [C] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F51)  Therefore,  I (tr [BC] + tr [B] tr [C]) + B tr [C] + C tr [B] + BC + CB  (F52)  = (2n + 1)2 (I + Pi + Pj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F53)  Finally, we have  A = EU  �  b  U †PbU ⟨b|UM−1(Pij)U †|b⟩ ⟨b|UM−1(Pji)U †|b⟩  (F54)  = 2n + 1  2n + 2 (I + Pi + Pj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F55)  23  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Variance of the k-RDM with a restricted sum  Now we are ready to turn to the task of bounding the variance ˆd as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For now, we neglect the  coefficient in order to simplify the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let  O =  �  x∈Rk  Ox,  (F56)  Ox =  k  �  ℓ=1  |iℓ⟩⟨jℓ|xℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F57)  The variance of the classical shadow estimator ˆo of ⟨O⟩ is bounded by  Var(ˆo) ≤  �  x∈Rk  �  y∈Rk  tr [|ψ⟩⟨ψ| Axy] ,  (F58)  Axy =  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(Ox)U †|b⟩ ⟨b|UM−1(O†  y)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F59)  Because the inverse channel, the random unitaries, and the Ox all factorize across the registers, we can rewrite Axy  as a tensor product,  Axy =  η  �  z=1  Az  xy,  (F60)  where Az  xy takes one of three forms depending on whether neither, one of, or both of Ox and Oy act non-trivially on  the zth register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If z /∈ x and z /∈ y, then  Az  xy = I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F61)  If exactly one of z ∈ x or z ∈ y is true, then we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F32) to simplify our expression for Az  xy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The cases  are symmetric between z ∈ x and z ∈ y, so we can treat only the first case without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let ℓ denote  the index of z in x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=', xℓ = z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We have  Az  xy =  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩  (F62)  = |iℓ⟩⟨jℓ|z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F63)  If z ∈ y we instead have Az  xy = |jℓ⟩⟨iℓ|z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The third case we must consider is where z ∈ x and z ∈ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let ℓ denote the index of z in x and y (they must be  the same because of the way we construct x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In this case,  Az  xy =  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩ U ⟨b|UM−1(|jℓ⟩⟨iℓ|)U †|b⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F64)  If iℓ = jℓ we can simplify this expression using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F44), otherwise we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The combination of these  two formulas lets us write  Az  xy =  �  b  EU∼UU † |b⟩⟨b| U ⟨b|UM−1(|iℓ⟩⟨jℓ|)U †|b⟩ U ⟨b|UM−1(|jℓ⟩⟨iℓ|)U †|b⟩  (F65)  = 2n + 1 − δiℓ,jℓ  2n + 2  (I + |iℓ⟩⟨iℓ| + |jℓ⟩⟨jℓ|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F66)  Now we will use the antisymmetry of |ψ⟩ to bound the quantity |tr [|ψ⟩⟨ψ| Axy]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let  a = |x ∩ y|,  b = 2k − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F67)  The operator Axy acts non-trivially on a + b registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' On a registers, it acts with an operator of the form given in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' On the other b registers, it acts as |c⟩⟨d| for some c, d (that can vary per register).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Due to the antisymmetry  of |ψ⟩, we can freely permute the registers without affecting the expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  24  We can therefore rewrite the expectation value of interest as  |tr [|ψ⟩⟨ψ| Axy]| =  ����� ⟨ψ|  � a  �  ℓ=1  2n + 1 − δcℓ,dℓ  2n + 2  (I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)  a+b  �  ℓ=a+1  |cℓ⟩⟨dℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩  �����  (F68)  ≤  ����� ⟨ψ|  � a  �  ℓ=1  (I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)  a+b  �  ℓ=a+1  |cℓ⟩⟨dℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F69)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Removing the off-diagonal terms  We can simplify the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F69) by replacing the off-diagonal matrix elements with projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To do so,  we will need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let |ψ⟩ be an arbitrary normalized pure quantum state on n qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Let O be an arbitrary positive  semidefinite operator on a qubits, and let |α⟩ and |β⟩ be arbitrary orthonormal quantum states on n − a qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then,  | ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| ≤ | ⟨ψ|(O ⊗ |φ⟩⟨φ|)|ψ⟩|  (F70)  for |φ⟩ = |α⟩ or |φ⟩ = |β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To begin the proof, expand |ψ⟩ as  |ψ⟩ =  �  ij  cij |i⟩ |j⟩ ,  (F71)  where the states {|i⟩} form an eigenbasis for O and the states {|j⟩} are an orthonormal basis such that |α⟩ , |β⟩ ∈ {|j⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Then  | ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| =  �����  �  i  c∗  iαciβOii  �����  (F72)  =  �  i  k∗  iαkiβ,  (F73)  where Oii denotes the eigenvalue of O corresponding to the eigenvector |i⟩ and kij is defined implicitly as kij = cij  √Oii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We can consider the quantity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F73) as the inner product of two vectors ⃗kα and ⃗kβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The Cauchy-Schwarz  inequality tells us that  �����  �  i  k∗  iαkiβ  ����� ≤  �  �  �  �  ��  i  k∗  iαkiα  � ��  i  k∗  iβkiβ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F74)  We can choose γ ∈ {α, β} such that  �����  �  i  k∗  iγkiγ  ����� ≥  �����  �  i  k∗  iαkiα  ����� and  �����  �  i  k∗  iγkiγ  ����� ≥  �����  �  i  k∗  iβkiβ  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F75)  Therefore, we have that  | ⟨ψ|(O ⊗ |α⟩⟨β|)|ψ⟩| ≤  �����  �  i  k∗  iγkiγ  �����  (F76)  =  �����  �  i  c∗  iγciγOii  �����  (F77)  = ⟨ψ|(O ⊗ |γ⟩⟨γ|)|ψ⟩  (F78)  for either |γ⟩ = |α⟩ or |γ⟩ = |β⟩ We can remove the absolute value bars in the final line because O ⊗|γ⟩⟨γ| is a positive  semidefinite operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  25  Now we can return to our bound from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F69),  |tr [|ψ⟩⟨ψ| Axy]| ≤  ����� ⟨ψ|  � a  �  ℓ=1  (I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)  a+b  �  ℓ=a+1  |cℓ⟩⟨dℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F79)  By rearranging the registers, we can apply Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Taking |α⟩ to be �a+b  ℓ=a+1 |cℓ⟩ and ⟨β| to be �a+b  ℓ=a+1 ⟨dℓ|, we  can show that either  |tr [|ψ⟩⟨ψ| Axy]| ≤ ⟨ψ|  � a  �  ℓ=1  (I + |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ|)  a+b  �  ℓ=a+1  |cℓ⟩⟨cℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩  (F80)  holds, or an equivalent expression with |dℓ⟩⟨dℓ| instead of |cℓ⟩⟨cℓ| in the second set of registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Both cases are identical,  so we will proceed using the label gℓ for whichever choice is valid in each register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We can also simplify the expression in the first registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We claim that, for each register, we can replace the term  |cℓ⟩⟨cℓ| + |dℓ⟩⟨dℓ| with either 2 |cℓ⟩⟨cℓ| or 2 |dℓ⟩⟨dℓ| without making the expectation value any smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This can be seen  by proceeding register by register, using the linearity of the expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Here again, the choice of cℓ or dℓ in  each register is immaterial, so we use the label gℓ to denote whichever one is appropriate for each register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Making  this simplification, we have that  |tr [|ψ⟩⟨ψ| Axy]| ≤ ⟨ψ|  � a  �  ℓ=1  (I + 2 |gℓ⟩⟨gℓ|)  a+b  �  ℓ=a+1  |gℓ⟩⟨gℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F81)  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Taking advantage of antisymmetry  Now we will take advantage of the antisymmetry of |ψ⟩ to bound the expectation values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' It is helpful  to rewrite the expression in the first set of registers in a different form:  a  �  ℓ=1  (I + 2 |gℓ⟩⟨gℓ|) =  a  �  w=0  2w  �  S⊆[a]:|S|=w  a  �  ℓ=1  W S  ℓ ,  (F82)  where W S  ℓ = |gℓ⟩⟨gℓ| if ℓ ∈ S and Wℓ = I otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This then leads us to the bound  |tr [|ψ⟩⟨ψ| Axy]| ≤  a  �  w=0  2w  �  S⊆[a]:|S|=w  ⟨ψ|  � a  �  ℓ=1  W S  ℓ  a+b  �  ℓ=a+1  |gℓ⟩⟨gℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F83)  Now that we have obtained this bound, we will proceed to use the antisymmetry of |ψ⟩ to show that  ⟨ψ|  � a  �  ℓ=1  W S  ℓ ,  a+b  �  ℓ=a+1  |gℓ⟩⟨gℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩ ≤  1  P(η, |S| + b) = (η − |S| − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F84)  To do so, let us prove the following lemma,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let |ψ⟩ be a normalized pure state on η registers of n qubits each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Furthermore, let S |ψ⟩ = − |ψ⟩ for any  operator S that swaps the states of two of the registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let {Pi}i∈[k] be a set of projectors onto orthonormal n qubit  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then  0 ≤ ⟨ψ|  � k  �  i=1  Pi  η  �  i=k+1  I  �  |ψ⟩ ≤  1  P(η, k) = (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F85)  where P(η, k) denotes the number of ways to choose a sequence of k items from a set of size η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let Sk denote the set of all sequences obtained by choosing k items from the set [η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that two sequences  with the same elements in different orders are treated as distinct elements of Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For a sequence s ∈ Sk we define the  operator As as the operator that acts on register si with the projector Pi for all i ∈ [k] and acts on the other η − k  26  registers with the identity operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that all of the operators As are defined using the same set of k projectors  acting on (potentially) different registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We will prove the claim by showing that  �  s∈Sk  ⟨ψ|As|ψ⟩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F86)  Clearly the operators {As}s∈Sk are all projectors onto different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In general, these projectors are not  orthogonal (under the Hilbert-Schmidt inner product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Equivalently, we could say that the +1 eigenspaces of these  operators are not orthogonal in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  However, we can show that |ψ⟩ has no support on states that are in the +1 eigenspace of more than one of these  projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Consider Ax and Ay for x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' There must be some register ℓ on which they act differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If Ax and  Ay both act on register ℓ with distinct projectors Pi and Pj then AxAy = 0 and their eigenspaces have no overlap,  so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Assume that only one of Ax and Ay acts on register ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Without loss of generality we consider the  case where Ax acts on register ℓ with the projector Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then, by definition, Ay acts on a different register ℓ′ with  Pi (since Ay acts with exactly the same projectors as Ax, just on a potentially different set of registers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Due to the  antisymmetry of |ψ⟩, we therefore have ⟨ψ|AxAy|ψ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Therefore, we can assert that  �  s∈Sk  ⟨ψ|As|ψ⟩ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F87)  This could be seen in more detail by expanding |ψ⟩ in the basis that diagonalizes all of the {As} and applying the  fact that if Ax |φ⟩ = 1 then Ay |φ⟩ = 0 for all x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The antisymmetry of |ψ⟩ also implies that ⟨ψ|Ax|ψ⟩ = ⟨ψ|Ay|ψ⟩  for all x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Therefore, we have that  |Sk| ⟨ψ|As|ψ⟩ ≤ 1  (F88)  for any As.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The {As} are all positive semidefinite, so we can bound the expectation value of the particular one from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F85) below by zero and divide by |Sk| = P(η, k) to yield  0 ≤ ⟨ψ|  � k  �  i=1  Pi  η  �  i=k+1  I  �  |ψ⟩ ≤  1  P(η, k) = (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F89)  completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F84) follows directly from this lemma and the fact that we can freely permute the observables between registers  without changing the expectation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Now we can return to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F83) and apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F84) to show that  |tr [|ψ⟩⟨ψ| Axy]| ≤  a  �  w=0  2w  �  S⊆[a]:|S|=w  ⟨ψ|  � a  �  ℓ=1  W S  ℓ  a+b  �  ℓ=a+1  |gℓ⟩⟨gℓ|  η  �  ℓ=a+b+1  I  �  |ψ⟩  (F90)  ≤  a  �  w=0  2w  �  S⊆[a]:|S|=w  (η − w − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F91)  =  a  �  w=0  2w  �  S⊆[a]:|S|=w  (η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − w − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F92)  ≤  a  �  w=0  2w  �  S⊆[a]:|S|=w  (η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F93)  27  with the last inequality following from the fact that η − a ≤ η − w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then we have that  |tr [|ψ⟩⟨ψ| Axy]| ≤  a  �  w=0  2w  �  S⊆[a]:|S|=w  (η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F94)  = (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  a  �  w=0  2w  �  S⊆[a]:|S|=w  (η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F95)  = (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  a  �  w=0  2w  �a  w  �(η − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F96)  ≤ (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  a  �  w=0  2w  �a  w  �(a − w)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F97)  = (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  a  �  w=0  2w  w!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F98)  ≤ (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ∞  �  w=0  2w  w!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F99)  = (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  e2,  (F100)  where the last step is obtained by the application of a well-known formula for the infinite sum of the sequence in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Putting the pieces together  Having shown that  |tr [|ψ⟩⟨ψ| Axy]| ≤ e2 (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F101)  we are ready to return to the bound in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F58), which we recall below:  Var(ˆo) ≤  �  x∈Rk  �  y∈Rk  tr [|ψ⟩⟨ψ| Axy] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F102)  We then have that  Var(ˆo) ≤  �  x∈Rk  �  y∈Rk  |tr [|ψ⟩⟨ψ| Axy]|  (F103)  ≤  �  x∈Rk  �  y∈Rk  e2 (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F104)  Recall that we defined a and b in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F67) in the following way,  a = |x ∩ y|,  b = 2k − 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F105)  Recall also the definition of the set of sequences Rk from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F15),  Rk = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , η/k} × {η/k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , 2η/k} × · · · × {(k − 1) η/k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' , η} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F106)  Colloquially, a sequence in Rk indexes a set of k registers, one from the first group of η/k, one from the second group  of η/k, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Let us consider a fixed sequence x ∈ Rk and determine how many sequences y ∈ Rk exist for a specific value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  For a fixed value of a, x and y share a elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' By construction, there are  �k  a  �  different choices for these a elements  (because there are k groups and x and y can either match or fail to match in each group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' In each of the k −a groups  28  of registers where x and y don’t match, there are exactly η/k − 1 ways to choose the corresponding element of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Therefore, for a given a and x, we have that  |{y ∈ Rk : |x ∩ y| = a}| =  �k  a  �  (η/k − 1)k−a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F107)  The only way that a particular x or y enters into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F104) is through a and b, so we can use this fact to take the  sums over x and y, yielding  Var(ˆo) ≤  �  x∈Rk  �  y∈Rk  e2 (η − a − b)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F108)  ≤ e2 �  x∈Rk  k  �  a=0  �k  a  �  (η/k − 1)k−a (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F109)  ≤ e2 (η/k)k  k  �  a=0  �k  a  �  (η/k − 1)k−a (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ,  (F110)  under the assumption that η > 2k so that we don’t have to restrict the sum over a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Simplifying the inequality further, we find that  Var(ˆo) ≤ e2 (η/k)k (η/k − 1)k  k  �  a=0  �k  a  �  (η/k − 1)−a (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F111)  ≤ e2 (η/k)k (η/k − 1)k  k  �  a=0  �k  a  �  (η/k − 1)−a (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F112)  Now we employ the upper and lower bounds from Stirling’s formula (that hold for any integer n > 0),  √  2πn  �n  e  �n  < n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' < e  √  2πn  �n  e  �n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F113)  We can use these bounds to simplify the ratio of factorials in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F112),  (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ≤ e  �  2π (η − 2k + a)  �η − 2k + a  e  �η−2k+a  1  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F114)  ≤ e  �  2π (η − k)  �η − k  e  �η−2k+a  1  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F115)  ≤ e  �η − k  e  �η−2k+a �  e  η − k  �η−k  (F116)  = e  �  e  η − k  �k−a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F117)  Using the assumption that η > 2k we can proceed further, yielding  (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  ≤ e  �  e  η − k  �k−a  (F118)  ≤ e  �2e  η  �k−a  (F119)  = e  �2e  η  �k �2e  η  �−a  ,  (F120)  where we have used the fact that η > 2k implies that η − k > η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  29  We can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F120) to further simplify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F112), finding that,  Var(ˆo) ≤ e2 (η/k)k (η/k − 1)k  k  �  a=0  �k  a  �  (η/k − 1)−a (η − 2k + a)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F121)  ≤ e3 (η/k)k (η/k − 1)k  k  �  a=0  �k  a  �  (η/k − 1)−a  �2e  η  �k �2e  η  �−a  (F122)  ≤ e3 �  η2/k2�k  k  �  a=0  �k  a  � � η  2k  �−a �2e  η  �k �2e  η  �−a  (F123)  = e3  �2eη  k2  �k  k  �  a=0  �k  a  � � e  k  �−a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F124)  Note that we again used the fact that η > 2k implies that η − k > η/2 to simplify the part of the bound involving  (η/k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Applying the binomial theorem to the sum yields the bound  Var(ˆo) ≤ e3  �2eη  k2  �k  e−k (k + e)k  (F125)  = e3  �2η (k + e)  k2  �k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F126)  Recall that we defined the estimator ˆo by neglecting the coefficient  kk(η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk(η−k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F16)’s expression for the k-RDM  element kDj1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',jk  i1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=',ik .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If we let ˆd be the estimator for this k-RDM element with the coefficient included, we have that  Var( ˆd) =  �  kk (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �2  Var(ˆo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F127)  Therefore, we can bound the desired variance by  Var( ˆd) ≤  �  kk (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �2  e3  �2η (k + e)  k2  �k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (F128)  Simplifying this expression, we obtain  Var( ˆd) ≤  �  kk (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  ηk (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �2  e3  �2η (k + e)  k2  �k  (F129)  = e3  �  (η!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=')  (η − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  �2 �2 (k + e)  η  �k  (F130)  ≤ e3ηk (2k + 2e)k ,  (F131)  which is the bound advertised in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' (F18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Appendix G: More efficient Slater determinant state preparation in first quantization  The general principle is to prepare the state in second quantization, then convert it to first quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To avoid  needing to store all N qubits for the second quantized state as it is produced, we convert its qubits to the first  quantized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  To explain this, we will first explain how a state in the second quantized representation can be converted to the  first quantized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' A computational basis state in second quantization consists of a string of N bits with  η ones and N − η zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The procedure is to run through these qubits in sequence and store the locations in η  registers of size ⌈log N⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Let us call the qubit number we consider from the second quantized representation q and  also record the number of electrons (ones) found so far as ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The value of ξ will be stored in an ancilla register of size  nη = ⌈log(η + 1)⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  We initialize all η registers for the first quantized representation and the ξ register as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then, for q = 1 to N  we perform the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Add the value in qubit q to the ξ register, with Toffoli cost nη − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If the qubit is in the state |1⟩ then ξ is  incremented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Now use qubit q to control unary iteration [15] on the register ξ, which has cost η − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Use this unary iteration to write the value q into register ξ using CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Because q is iterated classically, only  CNOTs are needed, with no further Toffolis beyond that needed for the unary iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Because the unary  iteration is controlled by qubit q, in the case where qubit q is in state |0⟩, the unary iteration does not proceed  and the value of q is not written out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Now perform unary iteration on ξ again that is not controlled;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' the cost is η − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' We use the unary iteration on ξ to check if the value in register number ξ is q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' if it is then we perform a NOT on  qubit q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' This multiply-controlled Toffoli is controlled by ⌈log N⌉ + 1 qubits (including the qubit from the unary  iteration), so it has a cost of ⌈log N⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' But, this is done for each of the η registers, for a total cost η⌈log N⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  The last operation ensures that qubit q is set to |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' +That is because, if it is initially |0⟩, then value q is not written  in register ξ, and the value is not flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' If it is initially |1⟩, then q is written in register ξ, and the multiply-controlled  Toffoli flips this qubit to |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  So far this procedure gives an ordered list of the electron positions, but we need an antisymmetrized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' To  obtain that, we apply the procedure in [66] to antisymmetrize with cost O(η log η log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The total Toffoli cost is  N (2η + nη − 3 + η⌈log N⌉) + O(η log η log N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  (G1)  The dominant cost here is ηN log N from erasing the qubits in the second quantized representation, with the factor  of log N coming from the need to check all qubits of each register to check if it is q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' However, recall that in unary  iteration it is possible to check if a register is equal to a consecutive sequence of values without this logarithmic  overhead, and we are considering consecutive values of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  To eliminate that overhead, we, therefore, consider simultaneous unary iteration on all of the η registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' That is,  for each register for the first quantized representation, we also store the qubits needed for unary iteration, as well as  a control register to ensure we do not iterate on registers that do not have value written into them yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The control  qubits will correspond to the value of ξ in unary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Our modified procedure is as follows (with the iteration of q from 1  to N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Perform a single step of unary iteration on all η registers with cost η Toffolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Add the value in qubit q to the ξ register, with Toffoli cost nη − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Use qubit q to control unary iteration on the register ξ, which has cost η − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Use this unary iteration to write the value q into register ξ, as well as the ⌈log N⌉ ancilla qubits for the unary  iteration and the control qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Again this is performed with CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Convert the control qubits to one-hot unary using a sequence of CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' For each of the η registers, use the control qubit and the unary iteration output to control a NOT on qubit q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  This has a cost of a single Toffoli for each register, for a toal of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Convert the control qubits to from one-hot unary with CNOTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  As a result, we have eliminated the log N factor and also eliminated the cost of η − 2 for the unary iteration on ξ  (because the control qubits are a unary representation of ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' One might ask if the binary representation of ξ is still  needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' however, it would be more costly to add increment ξ in unary (about η cost instead of log η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The total Toffoli  cost of this procedure is now  N (3η + nη − 2) + O(η log η log N),  (G2)  where the order term is the cost for antisymmetrizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Note that this reduces the Toffoli cost, but there is still a  Clifford cost of Nη log N from the CNOTs to place the value of q in the first quantized registers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  Now to efficiently prepare the Slater determinant, we can perform the sequence of Givens rotations on the qubits  for the second quantized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' The Givens rotations are performed in a sequence where Givens rotations  are performed in a layer on qubits 1 to η + 1, then on qubits 2 to η + 2, then 3 to η + 3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' One can find the  details of the Givens rotations that must be applied in [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Generally, layer q of Givens rotations is performed on  31  qubits q to η + q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' After the first layer there are only η + 1 qubits being used, and the first qubit is not accessed again  in the preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Therefore we can convert this qubit to the first quantized representation and erase it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content=' Then there  are only η qubits actively being used in the second quantized representation, and the next layer will be performed on  qubits 2 to η + 2, bringing on one more qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
+page_content='  In this way, each time we perform a layer of Givens rotations to prepare the state, we can convert one qubit to  the first quantized representation, and only η + 1 qubits of the second quantized representation need be used at once,  which is trivial compared to the number of qubits used for the first quantized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/x9AzT4oBgHgl3EQfQvs-/content/2301.01203v1.pdf'}
diff --git a/xdE3T4oBgHgl3EQflgqp/content/tmp_files/2301.04608v1.pdf.txt b/xdE3T4oBgHgl3EQflgqp/content/tmp_files/2301.04608v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..26141f4e2e292e76eaf35f028aa8e6e2e86b6084
--- /dev/null
+++ b/xdE3T4oBgHgl3EQflgqp/content/tmp_files/2301.04608v1.pdf.txt
@@ -0,0 +1,1390 @@
+Padding Module: Learning the Padding in Deep
+Neural Networks
+Fahad Alrasheedi
+Department of Computer Science
+University of Nebraska Omaha
+Omaha, USA
+falrasheedi@unomaha.edu
+Xin Zhong
+Department of Computer Science
+University of Nebraska Omaha
+Omaha, USA
+xzhong@unomaha.edu
+Pei-Chi Huang
+Department of Computer Science
+University of Nebraska Omaha
+Omaha, USA
+phuang@unomaha.edu
+Abstract—During the last decades, many studies have been
+dedicated to improving the performance of neural networks, for
+example, the network architectures, initialization, and activation.
+However, investigating the importance and effects of learnable
+padding methods in deep learning remains relatively open. To
+mitigate the gap, this paper proposes a novel trainable Padding
+Module that can be placed in a deep learning model. The Padding
+Module can optimize itself without requiring or influencing
+the model’s entire loss function. To train itself, the Padding
+Module constructs a ground truth and a predictor from the
+inputs by leveraging the underlying structure in the input data
+for supervision. As a result, the Padding Module can learn
+automatically to pad pixels to the border of its input images or
+feature maps. The padding contents are realistic extensions to its
+input data and simultaneously facilitate the deep learning model’s
+downstream task. Experiments have shown that the proposed
+Padding Module outperforms the state-of-the-art competitors and
+the baseline methods. For example, the Padding Module has
+1.23% and 0.44% more classification accuracy than the zero
+padding when tested on the VGG16 and ResNet50.
+Index Terms—Padding Module, Deep Learning, Neural Net-
+works, Trainable Padding
+I. INTRODUCTION
+Deep Neural Networks (DNNs) have significantly improved
+the performance of a wide range of computer vision tasks
+to the extent of being comparable to or exceeding human-
+level in many domains [1], such as image classification [2],
+object recognition [3], and image segmentation [4]. DNNs for
+computer vision have been iteratively improving in different
+aspects such as network architecture [5]–[8], network initial-
+ization [9], [10], optimization [11], [12], and activation [13],
+[14]. While it is intuitive that the salient foreground of an input
+image can control the results of a deep learning model [15],
+[16], researchers have also discovered that the input’s bor-
+ders and corners can dominate the model’s performance re-
+cently [17]–[19]. The study on the importance and effects of
+image borders remains relatively open, and this paper focuses
+on a trainable padding method that process image borders for
+deep learning models.
+Padding refers to the technique of adding extra data to the
+input’s borders so that the input’s width, height, or depth can
+be manipulated. Padding is widely used in Convolutional Neu-
+ral Networks (CNNs) to alter the output size of a convolutional
+This paper has been accepted for publication by the IEEE Access.
+Fig. 1: Five-pixel padding applied to a CIFAR-10 sampled
+image using three different padding methods: A) the zero
+padding, B) the local mean interpolation, and C) the proposed
+Padding Module.
+layer. Without padding, convolutional filters will not process
+the input’s borders and the output size will be reduced. The
+input size can be maintained with padding; we add an extra
+border before the convolution so that the original border can
+be processed [20].
+Traditional padding techniques include zero padding, repli-
+cation padding, and reflection padding. The reflection padding
+reflects the valid data over the borders; the replication padding
+uses the borders themselves as padding values; the zero
+padding specifies the use of zeroes as padding values. The
+replication and reflection padding methods extend the input
+with duplicate contents that may not be realistic; hence, they
+may destroy the original distribution [19]. The zero padding
+may outweigh the replication and reflection padding methods
+in terms of speed due to its computational simplicity. The
+major drawback of the traditional methods is that they are not
+dynamic. Thus, the padding values are always static and not
+optimized during the model training in a way that how they
+could be optimally predicted rationally to the input’s borders.
+More recently, padding methods have been studied aiming
+at a more related and realistic extension of the original
+input [19], [21]. For example, Liu et al. [21] proposed a
+padding method using partial convolution. Nguyen et al. [19]
+used a local mean of a sliding window over the input’s borders
+so the local distributions at the borders before and after the
+padding are consistent. These state-of-the-art padding methods
+outperformed the traditional padding in several tasks such
+as image classification, image segmentation, and image style
+transfer. However, the major disadvantage of the state-of-the-
+art padding methods is that they are not trainable: the padding
+contents are still not optimized.
+arXiv:2301.04608v1  [cs.CV]  11 Jan 2023
+
+(A)
+(B)
+(C)In this paper, we propose a trainable Padding Module that
+can be inserted into selected positions of a deep learning
+model to learn how to pad its inputs. The Padding Module
+can be trained simultaneously with a deep learning model,
+but, it is a self-learner in a way that will not require or
+influence the model’s entire loss function. During the training,
+the Padding Module internally constructs a ground truth from
+the input’s actual borders and trains a predictor considering the
+neighboring areas. The trained Padding Module can produce
+plausible results, as shown in Figure 1. The advantages of our
+work can be summarized as three-fold:
+• The proposed Padding Module introduces a trainable
+method that automatically pads its inputs.
+• The Padding Module extends its input with realistic new
+data that are related to the original data.
+• The Padding Module improves the performance of a
+downstream task of a deep learning model and outper-
+forms the state-of-the-art competitors, e.g., classification.
+The remainder of this paper is organized as follows. In Sec-
+tion II, we review the related work that addressed the padding
+effects on the neural networks performance and discuss how
+the current study fills the gap in the related work. Section III
+discusses our approach for the Padding Module followed by
+evaluation results in Section IV. Finally, Section V concludes
+with the discussion on evaluation and highlights some of the
+future work in this sector.
+II. LITERATURE REVIEW AND RELATED WORK
+Many studies have tried to improve the performance of
+CNNs models from network architecture [22]–[25], differ-
+ent variants of optimization [26]–[28], activations [29]–[32],
+regularization methods [33], [34] and so no. However, little
+attention has been paid to investigating the padding schemes
+during the convolution operation. To assist the kernel, i.e.,
+features extractor, in extracting important features during
+image processing in CNNs, padding layers can be added to
+visit pixels of the images around the corners more times, and
+then increase accuracy. The previous padding methods are
+presented as follows: Section II-A presents the performance
+improvement of neural networks; Section II-B introduces the
+improvement of space design; and Section II-C describes our
+contributions.
+A. Performance Improvement of Neural Networks
+Several studies have proposed padding methods to improve
+the performance of the neural networks [17], [19], [21].
+Innamorati et al. [17] addressed the importance of the
+data at the borders of the input by proposing a convolution
+layer that dealt with corners and borders separately from the
+middle part of the image. They specifically designed filters for
+each corner and border to handle the data at the boundaries,
+including upper, lower, left, and right borders. The boundary
+filters used in the convolution were jointly learned with the
+filter used for the middle part of the image. However, the
+main issue of this study is that the number of filters used to
+deal with the boundaries increases linearly with the size of the
+receptive field.
+Also, Nguyen et al. [19] proposed a padding method that
+could keep the local spatial distribution of the padded area
+consistent with the original input. The proposed method used
+the local means of the input at the borders to produce the
+padding values; they proposed two different variants of the
+padding method: mean-interpolation and mean-reflection. Both
+variants used filters with static values, based on the receptive
+field, in the convolution operation that is supposed to yield
+the padding values maintaining the same distributions as the
+original borders. However, the main issue with this method is
+that they are not learnable.
+Liu et al. [21] proposed a padding layer that uses a partial
+convolution that mainly re-weighted the convolution operation
+based on the ratio of the number of parameters in the filter
+to the number of valid data in the sliding window. In other
+words, they dealt with the padded area as hole areas that need
+to be in-painted, while the data coming from the original image
+were seen as non-hole areas. The main issue of this study is
+that the padding process is not learnable.
+B. Improvement of Spaces Design
+Also, some studies addressed the importance of the padding
+and data at the boundaries in the semantic representation
+learning and converting 360-degree space to 2-dimensional (2-
+D) space respectively [35]–[37].
+Cheng et al. [37] showed the importance of the padding
+method when they converted the 360-degree video to 2-
+dimensional space. They converted the video to six faces.
+Then, they used the reflection padding to connect them to form
+the 2-D space. The reflection padding naturally connected the
+faces compared to the zero-padding, which caused discontinu-
+ity.
+Interesting works were provided by Islam et al. [35], [36]
+in which they showed the importance of zero padding along
+with the data at the borders in encoding the absolute position
+information in the semantic representation learning. They
+showed that the zero padding and the boundaries drove the
+CNN models to encode the spatial information that helped
+the filters where to look for a specific feature; the spatial
+information was eventually propagated over the whole image.
+C. Our Contributions
+The padding methods and their effects on a CNN model’s
+performance are still open areas for researchers to investigate;
+hence, it is worth proposing new padding methods that could
+improve the performance of the CNN models. We propose a
+novel padding method, Padding Module, that could realisti-
+cally extend the input with related data. It learns how to pad
+the input by using the inputs’ borders as a ground truth and
+the neighboring areas of the borders as a predictor. Then, it
+uses a local loss function such as Mean Squared Error (MSE)
+and updates the filters using the local differentiation of the
+loss function with respect to the Padding Module’s filters. The
+following section explains the implementation of the Padding
+Module.
+
+III. THE PROPOSED Padding Module
+This paper presents the Padding Module, a learnable
+padding method that can pad the input with related and
+realistic padding, as shown in Figure 1. The Padding Module
+can be used as a substitute for other padding methods in the
+convolution layer, such as the zero padding, the replication
+padding, and the reflection padding. This section shows how
+the padding procedure (Section III-A) and the backpropagation
+(Section III-B) of the Padding Module work.
+A. Padding Procedure
+Algorithms 1 and 2, respectively, give an overview of the
+forward pass and the back-propagation of the Padding Module.
+The Padding Module first constructs a ground truth and a
+predictor from the input ( shown in step 1 to step 3 in
+Algorithm 1 and explained in Sections III-A1 and III-A2).
+Then, the Padding Module uses the filters being learned to
+produce the actual padding values using the input’s borders
+as a predictor ( shown in steps 4 to 13 in Algorithm 1 and
+explained in Section III-A4). Finally, the Padding Module
+uses the MSE as a loss function to compute the loss value
+and updates the filters during the model’s back-propagation
+( shown in steps 1 to 2 in Algorithm 2 and explained in
+Sections III-A3 and III-B).
+The Padding Module can pad the original input with any
+padding size, (e.g., one-pixel, two-pixels, etc). Indeed, the
+padding process in the Padding Module is iterative ( shown in
+steps 4 to 13 in Algorithm 1). Assume the required padding
+size is three pixels, the padding process will iterate three times
+as follows: (1) padding the original input with one-pixel along
+all the four borders; (2) padding the output of the 1st iteration
+with one-pixel along all the four borders; and (3) padding the
+output of the 2nd iteration with one-pixel along all the four
+borders. Here, to easily explain our method, a simple case of
+padding process was presented here, e.g., one-pixel padding.
+Also, the Padding Module is assigned filters as many as the
+number of channels in the input as explained in Section III-A3.
+Then, we explain the padding process considering a single
+channel. Here, the same procedure is separately applied to
+each channel in case of multiple channels.
+1) Ground Truth T: The Padding Module structures the
+ground truth T by extracting the input’s borders and stacking
+them upon each other vertically to form a four-row matrix.
+However, to stack the left and right borders vertically in
+T, they are transposed from column vectors to row vectors.
+Formally, given M r
+c as an original input with r and c as
+the number of rows and columns respectively; henceforth,
+superscripts and subscripts represent the indexes in the row-
+wise traversal and the column-wise traversal of the input
+respectively. The following is T’s extracting function target
+of the input M r
+c :
+Algorithm 1 Forward Pass
+Input: M r
+c , size, where r and c are the dimensions of a
+matrix, and size is the padding size.
+Output: M r′
+c′ , where r′ = r+2×size, and c′ = c+2×size.
+1: T ← target(M r
+c )
+/* as in Eq.1 */
+2: N ← neighbors(M r
+c )
+/* as in Eq.2 */
+3: P ← padz(padr(N))
+/* as in Eq.3 */
+4: M r′
+c′ ← M r
+c
+/* initial state for M r′
+c′ */
+5: while size ̸= 0 do
+6:
+Nout ← borders(M r′
+c′ )
+/* as in Eq.6 */
+7:
+Pout ← padz(padr(Nout))
+/* as in Eq.3 */
+8:
+O ← fθ(Pout)
+/* as in Eq. 7 */
+9:
+M r′+2
+c′+2 ← O0//padz(M r′
+c′ )//O1
+10:
+M r′+2
+c′+2 ← sides((O2)T , M r′+2
+c′+2 , (O3)T ) /* as in Eq.8
+*/
+11:
+M r′
+c′ ← corners(M r′+2
+c′+2 )
+/* as in Eq.9 */
+12:
+size ← size − 1
+13: end while
+14: return M r′
+c′
+Algorithm 2 Back Propagation
+Input: Gr′
+c′, where c′ and r′ are the same dimensions of the
+output of the forward pass in Algorithm 1.
+Output: Gr
+c, where c and r are the same dimensions of the
+input of the forward pass in Algorithm 1.
+1: Compute the local gradients.
+/* as in Eq.5 */
+2: Update the filter weights
+3: Gr
+c ← strip(Gr′
+c′)
+/* as in Eq.10 */
+4: return Gr
+c
+T = target(M r
+c ) =
+�
+�����
+M 0
+[:]
+M r−1
+[:]
+(M [:]
+0 )
+T
+(M [:]
+c−1)
+T
+�
+�����
+,
+(1)
+where M 0
+[:] is the entire row vector in M r
+c at index 0, M r−1
+[:]
+is the entire row vector in M r
+c at index r − 1, (M [:]
+0 )
+T is the
+transpose of the entire column vector in M r
+c at index 0, and
+(M [:]
+c−1)
+T is the transpose of the entire column vector in M r
+c
+at index c−1. Figure 2. (A) is an example to visually illustrate
+how T is constructed where the first row represents the upper
+border in M r
+c , the second row represents the lower border in
+M r
+c , the third row represents the left border in M r
+c , and the
+last row represents the right border in M r
+c .
+2) Predictor (P): To structure the predictor from the orig-
+inal input M r
+c , the Padding Module extracts the row vectors
+that neighbor the upper border and lower border in M r
+c and the
+transpose of the column vectors that neighbor the left border
+and right border in M r
+c . Then, the Padding Module stacks all
+the extracted neighbors vertically to form a four-row matrix.
+Formally, the predictor’s (denoted as P) extracting function of
+
+Fig. 2: An example to illustrate the steps 1-3 in Algorithm 1. On the left: the input M r
+c with size of (6, 6) pixels; the superscripts
+are the indexes in the row-wise traversal while the subscripts are the indexes in the column-wise traversal of the input. On the
+right: (A) the ground truth T: a result of applying step 1 in Algorithm 1 which is a stack of the borders where the first row,
+second row, third row, and last row are the upper, lower, left, and right borders in the input respectively; and (B) the predictor
+P: a result of applying steps 2 and 3 in Algorithm 1 which is a stack of the neighbors where the first row, second row, third
+row, and last row are neighbors to the upper, lower, left, and right borders in the input respectively, and the stack is padded
+at the left and right sides with reflection padding (pr) and with zero padding (pz).
+M r
+c can be expressed in the following way:
+First, the neighbors in M r
+c are selected and denoted as N
+as follows:
+N = neighbors(M r
+c ) =
+�
+�����
+M 1
+[1:c−1]
+M r−2
+[1:c−1]
+(M [1:r−1]
+1
+)
+T
+(M [1:r−1]
+c−2
+)
+T
+�
+�����
+.
+(2)
+The slice [1 : c − 1] excludes the data in the row vectors at
+the borders due to overlapping with the T, whereas the slice
+[1 : r − 1] excludes the data in the column vectors at the
+borders due to overlapping with T.
+Then, the Padding Module pads the structure as follows:
+P = padz(padr(N)).
+(3)
+First, the padr(.) function pads the structure with one pixel of
+the reflection padding horizontally (the left and right sides);
+then, with one pixel of the zero padding horizontally using the
+padz(.) function can get the final structure for P.
+Each row in P will be used to predict the corresponding
+row in T. For example, the first row in P will be used to
+predict the first row in T representing the upper border in
+the input M r
+c . Figure 2 (B) is an example to visually illustrate
+how the Padding Module constructs the stack of the neighbors
+(as a predictor) where the right and left sides of the stack are
+padded with the reflection padding (named as pr), and the zero
+padding (named as pz).
+3) Filters and the Loss Function: The Padding Module uses
+as many filters as the channels in the input (i.e., filter per
+channel). Also, each filter will be a row vector with a size of
+(1, 3) and a stride of (1, 1); that is because of having each row
+in P as a predictor for the corresponding row in T. Therefore,
+to predict T, the Padding Module convolutes the filters over P;
+it uses its own loss function to optimize the prediction through
+the local differentiation of the loss function with respect to the
+filters.
+The loss function used by the Padding Module is the MSE
+which computes the squared difference between the ground
+truth and the predicted value. The following equation is the
+MSE’s mathematical expression for a single data point:
+MSE(fθ(P), T) =
+4
+�
+a=1
+n
+�
+j=1
+(θT · P a
+j − T a
+j )2,
+(4)
+where f is the convolutional operation parameterized by θ, P
+and T are the predictor and the ground truth extracted from
+the original input M r
+c , a represents the indexes for rows in the
+four-row matrices P and T, and j represents the indexes for
+both the slide windows and columns in P and T respectively.
+Hence, P a
+j is the jth slide window in the row indexed at a in
+P, and T a
+j is the corresponding value in T indexed at the ath
+row and jth column.
+The local differentiation of the Padding Module’s loss func-
+tion and the filters’ updates are achieved during the model’s
+back-propagation; these local gradients are not propagated to
+the previous layer. Besides that, the Padding Module facili-
+ties the back-propagation of the model’s loss function going
+through it to the previous layer as explained in Section III-B.
+The following is the mathematical expression for the local
+gradients (Padding Module’s loss function gradients with
+respect to a single filter for a single data point):
+ϕ
+ϕθm
+MSE(fθ(P), T) = 2
+4
+�
+a=1
+n
+�
+j=1
+(θT · P a
+j − T a
+j )xm,
+(5)
+where xm is a single feature in the P a
+j slide window which
+was multiplied by the corresponding weight, namely θm, in
+
+A) Ground Truth T
+MST
+[Mo
+MI
+M
+Ma
+M
+Me
+MS
+M5
+M2
+M5
+MS
+Input Mr=
+M
+M3
+M3
+M
+Mo
+M
+[M
+MI
+M2
+M?
+M
+M
+LMg
+M2
+M3
+Ms
+MS!
+M
+M2
+M1
+Ms
+M
+M1
+M3
+M
+M2
+M3
+M?
+M
+M
+target(M)
+M3
+M3
+M2
+M3
+M
+M3
+Pad(padr(neighbors(M))
+B) Predictor P
+M2
+M4
+Mo
+Mi
+M2
+M9
+Ma
+M?
+Ma
+M4
+M
+Pz
+pr
+pr
+pz
+LM§
+M5
+M5
+M5
+M5
+M5
+M5
+M5
+M5
+M5
+ME
+Pz
+pr
+pr
+Pz
+M
+M3
+Me
+Mo
+M1
+M5
+Pz
+pr
+Pr
+Pz
+M?
+M
+M2
+Lpz
+M?
+Ms
+M5
+pr
+pr
+pz-the θ during the convolution.
+4) Padding
+Process:
+The
+procedures
+in
+Sec-
+tions III-A1, III-A2, and III-A3 are used to guide the
+Padding Module on learning how to predict the borders of
+the original input, M r
+c , based on the neighboring areas to the
+borders, and then the Padding Module can optimize its filters.
+However, the padding process is shown in steps 4 to 13
+in Algorithm 1; it uses the borders of the input, M r′
+c′ , as
+the predictor. In detail, the padding process iterates until the
+original input is padded with the required padding size. Hence,
+the original input M r
+c is assigned to M r′
+c′ as an initial state
+in step 4 before the padding loop starts. Then, each iteration
+pads the input, M r′
+c′ , with one-pixel, and outputs a new M r′
+c′
+which will be used for the next iteration and so forth. The
+dimensions of an iteration’s output, M r′
+c′ in step 11, are two-
+pixel larger than the dimensions of that iteration’s input, M r′
+c′
+in step 6.
+Minutely, constructing the predictor in the padding process
+is similar to the way that constructs P in Section III-A2 with
+small modifications. To distinguish the notions of neighbors,
+N, and P, in Section III-A2, borders, Nout, and Pout are
+denoted for the extracting function, the function’s output, and
+the predictor, respectively. The following is the mathematical
+expression for the extracting function borders:
+Nout = borders(M r′
+c′ ) =
+�
+�����
+M 0
+[:]
+M r−1
+[:]
+(M [:]
+0 )
+T
+(M [:]
+c−1)
+T
+�
+�����
+,
+(6)
+where M 0
+[:] and M r−1
+[:]
+mean extracting the entire upper and
+lower borders respectively. Whereas, (M [:]
+0 )T and (M [:]
+c−1)T
+mean extracting the transpose of the entire left and right
+borders respectively. Then, the Padding Module pads the
+output Nout using Equation 3 to get the final structure for
+Pout.
+Consequently, convoluting the filters over the Pout will
+produce the padding values for the iteration’s input. The output
+can be expressed as follows:
+O = fθ(Pout),
+(7)
+where f is the convolutional operation parameterized by θ,
+Pout is the predictor, and the O is the output and comes as a
+matrix of four rows. Each row represents the padding values
+for the corresponding area in the iteration’s input, M r′
+c′ , as
+follows: the first row (O0), the second row (O1), the third row
+(O2), and the fourth row (O3) represent the padding values for
+the upper, the lower, the left, and the right areas in the input
+respectively.
+Then, the steps from 9 to 11 are how the produced padding
+values stick around the input M r′
+c′ . First, in step 9, the vertical
+concatenation operator // is used to concatenate the first row
+(O0) with M r′
+c′ , and then concatenates the resulted matrix
+with the second row (O1). However, the rows from O are
+two-pixel wider than the rows of M r′
+c′ ; therefore, to match
+the dimensions of these operands, the Padding Module uses
+padz(.) to pad the M r′
+c′ horizontally with one pixel of the zero
+padding before the concatenation process. Hence, the output’s
+dimensions in step 9, denoted as M r′+2
+c′+2 , are two-pixel larger
+than the input M r′
+c′ . Finally, the algorithm uses sides function
+which can be formally expressed as the following:
+sides((O2)T , M r′+2
+c′+2 , (O3)T ).
+(8)
+This function does not change the dimensions; however, it adds
+respectively the transpose of the third row (O2) and fourth row
+(O3) to the left and right columns of M r′+2
+c′+2 , the concatenated
+matrix with zero values at the left and right columns unless
+the corners already assigned values from the concatenation
+process. To resolve the double-count problem at the corners,
+the Padding Module takes the average of added values in the
+corners by dividing each corner by 2; this averaging function
+is step 12 in Algorithm 1:
+M r′
+c′ = corners(M r′+2
+c′+2 ).
+(9)
+Lastly, as mentioned early in this section that the dimensions
+of the iteration’s output are two-pixel larger than the iteration’s
+input. Hence, the output M r′
+c′ , in Equation 9, has dimensions r′
+and c′ that are updated with the dimensions of M r′+2
+c′+2 , namely
+r′ + 2 and c′ + 2 respectively.
+B. Back-propagation
+As seen in Section III-A3, the Padding Module is not opti-
+mized based on the model’s main loss function; therefore, the
+model does not compute the gradients of its loss function with
+respect to the filters of the Padding Module. However, during
+the mode’s backpropagation, the Padding Module achieves two
+key points as follows:
+1) As shown in step 1 in Algorithm 2, the Padding Module
+optimizes its filters through computing the local gradi-
+ents for its loss function with respect to the filters as
+explained in Section III-A3.
+2) The process also receives Gr′
+c′ which are the gradi-
+ents of the model’s loss function with respect to the
+Padding Module’s output, the original input M r
+c after
+being padded. Therefore, the Padding Module strips
+out the gradients from Gr′
+c′ that represent the gradients
+for the padded areas in the Padding Module’s output;
+the stripping-out process is step 3 in Algorithm 2, and
+formally expressed as follows:
+Gr
+c = strip(Gr′
+c′).
+(10)
+Then, the Padding Module back-propagates to the pre-
+vious layer the Gr
+c, representing the gradients for the
+previous layer’s output. Figure 3 is an example to
+visually illustrate how the back-propagation process in
+the Padding Module is achieved.
+
+Fig. 3: An example to illustrate the back propagation in Algorithm 2. On the top: the input to the Padding Module is of size
+(6, 6), the Padding Module uses one-pixel padding and produces an output of size (8, 8) where the borders p is the computed
+padding values. On the bottom: the back-propagation of the received gradients which is of size (8, 8) where the borders are
+gradients for the padding values gp; the Padding Module strips out gp from the received gradients, and sends the remaining to
+the previous layer. The g stands for gradient; for example gM 0
+0 is the gradient for the pixel at index [0, 0] in the input of the
+Forward Pass.
+IV. EXPERIMENTAL RESULTS AND ANALYSIS
+This section shows the design of the training and testing
+experiments on our Padding Module applied to a downstream
+task, i.e., image classification. The experimental setup is
+presented in Sections IV-A. The quantitative and qualitative
+results are described in Section IV-B and IV-C.
+A. Experiment Setup
+The study used the premium service from Google Colabora-
+tory where a GPU of Tesla T4 was assigned. The experiments
+and comparisons were conducted on the CIFAR-10 dataset
+for a classification task [38]. The CIFAR-10 dataset includes
+a training dataset of 50,000 images and a test dataset of
+10,000 images. The images are of shape (32, 32, 3), dis-
+tributed equally to ten classes of airplane, automobile, bird,
+cat, deer, dog, frog, horse, ship, and truck. The Padding
+Module was applied to different networks namely: VGG16 [8]
+and ResNet50V2 [39]; to make the deeper layers in these
+networks carry out a valid convolution, the images were
+resized to (64, 64, 3) and (224, 224, 3) for the VGG16 and
+the ResNet50V2, respectively.
+The VGG16 is a vanilla-based architecture where the net-
+work shape is wider at the beginning of the network and
+narrowed down as going deep in the network. The pre-trained
+VGG16 was obtained from the keras1 without the top layers
+(the last three dense layers including the original softmax
+layer). Then, we added two fully-connected layers each with
+512 neurons and followed by a dropout layer. On the other
+hand, the ResNet50V2 is made up of blocks where each block
+sends the block’s input through the block itself, and also uses a
+skip connection to directly add the block’s input to the output
+of the input’s flow coming through the block. The process
+is known as the identity function that could help deep layers
+to improve the model’s accuracy. ResNet50V2 is a modified
+version of the ResNet50 [13]. The modification mainly is in
+the arrangement of the block layers; batch normalization [11]
+and ReLU activation [40] are applied to the data flow before
+the convolutional layer in the block. These changes enabled
+the ResNet50V2 to outperform the ResNet50 on the image
+classification task. The ResNet50V2 was downloaded from the
+keras2 without the top layer (the last dense layer which is the
+original softmax layer). Then, two fully-connected layers with
+1024 and 512 neurons were added.
+1VGG16 from the keras: https://keras.io/api/applications/vgg
+2ResNet50V2 from the kera: https://keras.io/api/applications/resnet
+
+The Padding Module
+Output
+Input
+p
+p
+p
+p
+p
+p
+p
+MI
+M2
+M
+MI
+M
+M:
+p
+Mi
+M9
+M
+M?
+p
+Mi
+MS
+MS
+M
+p
+M
+M1
+M2
+M3
+M1
+Ms
+p
+M?
+M?
+M3
+Forward Pass
+Me
+M?
+M?
+M
+M2
+p
+p
+Ma
+Mi
+M2
+M3
+Ms
+M3
+p
+p
+Me
+M1
+M
+MS
+M4
+p
+Ma
+Ms
+p
+LM
+M5
+M5
+MS
+Ms
+MS
+M5
+M5
+M5
+M5
+MS
+p
+MS
+p
+Lp
+p
+p
+p
+p
+p
+p
+pJ
+Input
+Output
+T9p
+9p
+9p
+9p
+9p
+9p
+9p
+9pj
+g mi
+w6
+9 mg
+9 mg
+9p
+9M:
+9Mi
+9 ms
+9 mg
+9g
+9 M2
+Backpropagation
+9p
+9M
+gM
+9m?
+9p
+9p
+9Mi
+9ms
+9p
+9m
+9 ms
+9 ms
+9ms
+9 ms
+M
+9p
+9m
+9m
+9p
+gp
+gp
+9p
+9p
+gr
+gp-Fig. 4: The comparison of three different padding methods on
+the test images: zero padding, mean interpolation padding, and
+the Padding Module when applied to the VGG16 model.
+Fig. 5: The comparison of three different padding methods on
+the test images: zero padding, mean interpolation padding, and
+the Padding Module when applied to the ResNet50V2.
+Moreover, we added a softmax layer with ten outputs for
+both models of VGG16 and ResNet50V2, and then used
+the Adam optimizer [12] for the back-propagation of the
+gradients. Finally, the Padding Module was used before every
+convolutional layer in the VGG16; whereas, we replaced every
+zero padding layer in the ResNet50V2 with the Padding
+Module.
+B. Quantitative Results
+Section IV-B1 compares the proposed Padding Module and
+state-of-art padding solutions by performing the image clas-
+sification task, and then Section IV-B2 discusses an ablation
+study based on our solution.
+1) Image Classification Task:
+We considered the zero
+padding method as a baseline to compare the Padding Module
+with. Moreover, we used the mean interpolation padding
+method [19] as the state-of-art since it outperformed the
+partial convolution padding method [19], [21] in the image
+classification. The main goal of this study, which aligns with
+the literature, is to investigate the padding effect on the
+accuracy of DNN models. Therefore, the accuracy is used as
+a comparison metric between the performance of the Padding
+Module and the benchmark. The accuracy is the percentage of
+correctly classified images over the total number of images in
+the dataset.
+Each model was trained with 100 epochs using the training
+dataset, and tested in each epoch using the test dataset. In
+Figure 4, the Padding Module outperforms both the baseline
+method and the mean interpolation padding method when us-
+ing the VGG16; also, we found that the baseline is comparable
+to the mean interpolation method. As for the Resent50, the
+Padding Module also outperforms the other two paddings as
+shown in Figure 5. We also noticed that the baseline method
+is comparable to the mean interpolation method. Moreover,
+Table I summarizes the average of the last five epochs for the
+three different padding methods and the margin between the
+highest and the second-highest accuracies for the two models.
+As mentioned, the study investigated the effects of the
+Padding Module on the accuracy of DNN models. Importantly,
+we showed that the related and realistic padding could improve
+the accuracy of DNN models ; therefore, the Padding Module
+Padding Method
+VGG16
+ResNet50V2
+Padding Module
+92.92
+95.08
+Zero Padding
+91.43
+94.64
+Mean Interoplation
+91.69
+94.44
+margin
+1.23
+0.44
+TABLE I: The average accuracy of the last five epochs
+for three different padding methods used in VGG16 and
+ResNet50V2. The margin is the difference between each
+model’s highest and second-highest padding methods.
+Fig. 6: MSEs for three Padding Modules placed at different
+positions in the VGG16: 1) module 1: at the beginning, 2)
+module 2: at the middle, and 3) module 3: at the end.
+was able to produce such padding by minimizing the MSE
+in Algorithms 1 and 2. Figure 6 illustrates MSEs for three
+cases of the Padding Module applied in different places (at
+the beginning, in the middle, and at the end) in the VGG16;
+it is evident that the MSEs significantly decreased after only
+two epochs and then stayed flat till the end of the experiment
+for the three cases.
+One natural drawback of the current Padding Module was
+the extra running time caused by constructing the data struc-
+tures and optimizing the filters. Table 2 shows that VGG16 and
+ResNet50V2, on average, doubled the epoch’s time when ap-
+plying the Padding Module (i.e., placing the Padding Module
+before every convolutional layer in the VGG16 and replacing
+every zero padding layer in the ResNet50V2 with the Padding
+Module). On the other side, the accuracy for the VGG16
+and ResNet50V2, respectively, gained margins of 1.49% and
+0.44% when applying the Padding Module compared to the
+
+100
+90
+Accuracy
+80
+Padding Types
+Mean Interpolation
+70
+Ours
+Zero padding
+60
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Epochs100
+95
+Accuracy
+Padding Types
+Mean Interpolation
+90
+Ours
+Zero padding
+85
+0
+10
+20
+30
+40
+50
+60
+70
+80
+0.6
+100
+Epochs30
+MES
+Module
+20
+module 1
+module 2
+10
+module 3
+0
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Epochszero padding (no Padding Module). One remedy to lessen the
+running time problem may be to stop training the Padding
+Module when it significantly decreases the MSE after the first
+two epochs. However, improving the current Padding Module
+including the time complexity can be a further direction for
+further research.
+2) Ablation Study: The experiments in this section were
+conducted as an ablation study where the Padding Module
+was empirically placed at different positions in the VGG16
+model, as shown in Figure 7: at the beginning of the model,
+in the middle, at the end, and the combination of all the three
+places together. We also compared the four scenarios with two
+other scenarios: (1) where the Padding Module was placed in
+all positions (before each convolutional layer) in the model;
+and (2) where the Padding Module was not used but the zero
+padding was used instead. We ran each scenario 100 epochs
+using the training dataset for training and the test dataset for
+evaluation, and averaged the test accuracies of the last five
+epochs for each scenario; Table III illustrates the summary of
+the comparison of the models. We noticed that using a single
+Padding Module with the shallow layers outperformed the case
+of using it with the deep layers. Also, the combination scenario
+showed a superiority over the scenario of a single Padding
+Module. However, the best performance was when the Padding
+Module applied in the scenario of all positions. Finally, all the
+scenarios of applying the Padding Module outperformed the
+scenario of the model with no Padding Module.
+Model
+Zero Padding
+Padding Module
+Margin
+Accuracy
+Time
+Accuracy
+Time
+VGG16
+91.43
+2
+92.92
+4
+1.49
+ResNet50
+94.64
+5
+95.08
+9
+0.44
+TABLE II: On average, the running time doubles for one
+epoch when applying the Padding Module to the VGG16 and
+ResNet50V2 compared to the case of the zero padding (no
+Padding Module applied). Times are shown in a minute-scale.
+The margin is the accuracy difference between the case of
+applying the Padding Module and the zero padding.
+No
+Different places in the VGG16
+Accuracy
+1
+At the beginning
+91.98
+2
+At the middle
+91.8
+3
+At the end
+91.97
+4
+Combination of 1, 2, and 3 togather
+92.18
+5
+All positions
+92.8
+6
+VGG16 with no Padding Module
+91.4
+TABLE III: Placing the Padding Module at different positions
+in the VGG16: 1) at the beginning 2) at the middle 3) at the
+end 4) combination of beginning, middle, end 5) before every
+convolutional layer 6) VGG16 with no Padding Module (zero
+padding instead).
+C. Qualitative Results
+Different padding sizes, such as one-pixel, three-pixel, and
+five-pixel, were used to illustrate how the Padding Module
+can extend the input with related and realistic extensions.
+Also, we compared these different padding sizes with the
+other two methods, namely the zero padding and the mean
+interpolation padding. As shown in Figure 8, the Padding
+Module can learn how to pad the input with related data and
+natural extension; this finding becomes more evident as the
+padding size increases.
+V. FUTURE RESEARCH DIRECTIONS AND CONCLUSION
+This paper proposed a novel padding method: Padding
+Module; that can learn how to pad an input from the input’s
+borders; hence, the input can be realistically extended with
+related data. The Padding Module is a self-learning of its
+weights. To train itself, the Padding Module constructs a
+ground truth and a predictor from the inputs by leveraging
+the underlying structure in the input data for supervision.
+The Padding Module uses convolutional operation over the
+predictor to produce a predicted value that is, in turn, com-
+pared with the ground truth. The Padding Module uses a
+local loss function, independent from the model’s main loss
+function, to minimize the difference between the predicted
+value and the ground truth. Therefore, the Padding Module
+updates its convolutional filters locally during the model’s
+back-propagation. Besides that, the Padding Module back-
+propagates the model’s gradients with respect to the Padding
+Module’s output after stripping out the gradients for the padded
+areas to the previous layer.
+The experimental results showed that the Padding Module
+outperformed the zero-padding and the state-of-art padding in
+the image classification task. In the ablation study, we also
+observed that using a single Padding Module with the shallow
+layers improved the performance slightly better than using it
+with the deep layers in the VGG16 network. On the other
+hand, using three of the Padding Module placed in different
+positions (at the beginning, at the middle, and at the end)
+in the VGG16 outperformed the scenario of a single Padding
+Module. Moreover, placing the Padding Module in all positions
+(before every convolutional layer) in the VGG16 outperformed
+all other scenarios as shown in Table III.
+Our experiments applied the Padding Module to the two
+well-known networks: VGG16 and ResNet50, for the image
+classification task. The VGG16 and ResNet50 networks were
+chosen to represent small and large networks, respectively.
+They, also, were used by the literature; hence, we used them
+to compare the Padding Module with the previous work.
+Although two different networks are only used in one task, we
+shall extend the Padding Module to improve such networks in
+different tasks, including object detection, style transfer, and
+image inpainting. We leave investigating the Padding Module
+in a wide range of tasks for future research.
+Also, the Padding Module learned how to pad the input
+independently of the model’s loss function. However, it is
+possible to optimize the Padding Module’s filters based on
+optimizing the model’s main loss function; this approach will
+be entirely different. Hence, one research direction may be to
+implement a padding method that can optimize its padding
+filters based on the model’s main loss function.
+
+Fig. 7: The selected positions in the VGG16 for the Padding Module: at the beginning, middle, and the end.
+Fig. 8: Three images sampled from CIFAR-10 and padded by different padding methods: zero-padding, mean interpolation,
+and the Padding Module. Each padding method uses three different padding sizes: A) one-pixel, B) three-pixel, C) five-pixel.
+REFERENCES
+[1] K. Eykholt, I. Evtimov, E. Fernandes, B. Li, A. Rahmati, C. Xiao,
+A. Prakash, T. Kohno, and D. Song, “Robust physical-world attacks
+on deep learning visual classification,” in Proceedings of the IEEE
+conference on computer vision and pattern recognition, 2018, pp. 1625–
+1634.
+[2] T. He, Z. Zhang, H. Zhang, Z. Zhang, J. Xie, and M. Li, “Bag of
+tricks for image classification with convolutional neural networks,” in
+Proceedings of the IEEE/CVF Conference on Computer Vision and
+Pattern Recognition, 2019, pp. 558–567.
+[3] J. Huang, V. Rathod, C. Sun, M. Zhu, A. Korattikara, A. Fathi, I. Fischer,
+Z. Wojna, Y. Song, S. Guadarrama et al., “Speed/accuracy trade-offs
+for modern convolutional object detectors,” in Proceedings of the IEEE
+conference on computer vision and pattern recognition, 2017, pp. 7310–
+7311.
+[4] K. He, G. Gkioxari, P. Doll´ar, and R. Girshick, “Mask r-cnn,” in
+Proceedings of the IEEE international conference on computer vision,
+2017, pp. 2961–2969.
+[5] J. Kim, H. Zeng, D. Ghadiyaram, S. Lee, L. Zhang, and A. C.
+Bovik, “Deep convolutional neural models for picture-quality prediction:
+Challenges and solutions to data-driven image quality assessment,” IEEE
+Signal processing magazine, vol. 34, no. 6, pp. 130–141, 2017.
+[6] Y. Li, C. Fang, J. Yang, Z. Wang, X. Lu, and M.-H. Yang, “Universal
+style transfer via feature transforms,” Advances in neural information
+processing systems, vol. 30, 2017.
+[7] A.-D. Nguyen, S. Choi, W. Kim, and S. Lee, “A simple way of
+multimodal and arbitrary style transfer,” in ICASSP 2019-2019 IEEE
+International Conference on Acoustics, Speech and Signal Processing
+(ICASSP).
+IEEE, 2019, pp. 1752–1756.
+[8] K. Simonyan and A. Zisserman, “Very deep convolutional networks for
+large-scale image recognition,” arXiv preprint arXiv:1409.1556, 2014.
+[9] D. Arpit, V. Campos, and Y. Bengio, “How to initialize your network?
+robust initialization for weightnorm & resnets,” Advances in Neural
+Information Processing Systems, vol. 32, 2019.
+[10] K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers:
+Surpassing human-level performance on imagenet classification,” in
+Proceedings of the IEEE international conference on computer vision,
+2015, pp. 1026–1034.
+[11] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep
+network training by reducing internal covariate shift,” in International
+conference on machine learning.
+PMLR, 2015, pp. 448–456.
+[12] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+CoRR, vol. abs/1412.6980, 2014.
+[13] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image
+recognition,” in Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2016, pp. 770–778.
+[14] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely
+connected convolutional networks,” in Proceedings of the IEEE confer-
+ence on computer vision and pattern recognition, 2017, pp. 4700–4708.
+[15] K. Simonyan, A. Vedaldi, and A. Zisserman, “Deep inside convolutional
+networks: Visualising image classification models and saliency maps,”
+arXiv preprint arXiv:1312.6034, 2013.
+[16] B. Zhou, A. Khosla, A. Lapedriza, A. Oliva, and A. Torralba, “Learning
+deep features for discriminative localization,” in Proceedings of the IEEE
+
+Beginning
+Pooling
+Input
+Conv1
+Conv2
+Conv3
+Conv4
+Pooling
+Resize
+Conv5
+Conv6
+Middle
+Conv12
+Pooling
+Pooling
+Conv11
+Conv10
+Conv9
+Conv8
+Conv7
+End
+Pooling
+Flatten
+Dense1
+Dense2
+Conv13
+Softmax
+OutputZero padding
+Mean Interpolation
+Padding Module
+C) Five-pixel padding
+A) One-pixel padding
+A) One-pixel padding
+B) Three-pixel padding
+C) Five-pixel padding
+ A) One-pixel padding
+B) Three-pixel padding
+C) Five-pixel padding
+B) Three-pixel paddingconference on computer vision and pattern recognition, 2016, pp. 2921–
+2929.
+[17] C. Innamorati, T. Ritschel, T. Weyrich, and N. J. Mitra, “Learning on
+the edge: Explicit boundary handling in cnns,” 2018.
+[18] G. Liu, K. J. Shih, T.-C. Wang, F. A. Reda, K. Sapra, Z. Yu, A. Tao,
+and B. Catanzaro, “Partial convolution based padding,” arXiv preprint
+arXiv:1811.11718, 2018.
+[19] A.-D. Nguyen, S. Choi, W. Kim, S. Ahn, J. Kim, and S. Lee, “Dis-
+tribution padding in convolutional neural networks,” in 2019 IEEE
+International Conference on Image Processing (ICIP), 2019, pp. 4275–
+4279.
+[20] S. Albawi, T. A. Mohammed, and S. Al-Zawi, “Understanding of a
+convolutional neural network,” in 2017 International Conference on
+Engineering and Technology (ICET), 2017, pp. 1–6.
+[21] G. Liu, K. J. Shih, T.-C. Wang, F. A. Reda, K. Sapra, Z. Yu, A. Tao,
+and B. Catanzaro, “Partial convolution based padding,” 2018.
+[22] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image
+recognition,” in Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2016, pp. 770–778.
+[23] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger, “Densely
+connected convolutional networks,” in Proceedings of the IEEE confer-
+ence on computer vision and pattern recognition, 2017, pp. 4700–4708.
+[24] T. Takikawa, D. Acuna, V. Jampani, and S. Fidler, “Gated-scnn: Gated
+shape cnns for semantic segmentation,” in Proceedings of the IEEE/CVF
+international conference on computer vision, 2019, pp. 5229–5238.
+[25] I. Radosavovic, R. P. Kosaraju, R. Girshick, K. He, and P. Doll´ar,
+“Designing network design spaces,” in Proceedings of the IEEE/CVF
+conference on computer vision and pattern recognition, 2020, pp.
+10 428–10 436.
+[26] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+arXiv preprint arXiv:1412.6980, 2014.
+[27] S. Ioffe and C. Szegedy, “Batch normalization: Accelerating deep
+network training by reducing internal covariate shift,” in International
+conference on machine learning.
+PMLR, 2015, pp. 448–456.
+[28] L. Liu, H. Jiang, P. He, W. Chen, X. Liu, J. Gao, and J. Han, “On
+the variance of the adaptive learning rate and beyond,” arXiv preprint
+arXiv:1908.03265, 2019.
+[29] A. L. Maas, A. Y. Hannun, A. Y. Ng et al., “Rectifier nonlinearities
+improve neural network acoustic models,” in Proc. icml, vol. 30, no. 1.
+Atlanta, Georgia, USA, 2013, p. 3.
+[30] K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers:
+Surpassing human-level performance on imagenet classification,” in
+Proceedings of the IEEE international conference on computer vision,
+2015, pp. 1026–1034.
+[31] P. Ramachandran, B. Zoph, and Q. V. Le, “Searching for activation
+functions,” arXiv preprint arXiv:1710.05941, 2017.
+[32] M. A. Mercioni and S. Holban, “P-swish: Activation function with learn-
+able parameters based on swish activation function in deep learning,” in
+2020 International Symposium on Electronics and Telecommunications
+(ISETC).
+IEEE, 2020, pp. 1–4.
+[33] J. Tompson, R. Goroshin, A. Jain, Y. LeCun, and C. Bregler, “Efficient
+object localization using convolutional networks,” in Proceedings of the
+IEEE conference on computer vision and pattern recognition, 2015, pp.
+648–656.
+[34] G. Larsson, M. Maire, and G. Shakhnarovich, “Fractalnet: Ultra-deep
+neural networks without residuals,” arXiv preprint arXiv:1605.07648,
+2016.
+[35] M. A. Islam, S. Jia, and N. D. B. Bruce, “How much position informa-
+tion do convolutional neural networks encode?” 2020.
+[36] M. A. Islam, M. Kowal, S. Jia, K. G. Derpanis, and N. D. B. Bruce, “Po-
+sition, padding and predictions: A deeper look at position information
+in cnns,” 2021.
+[37] H.-T. Cheng, C.-H. Chao, J.-D. Dong, H.-K. Wen, T.-L. Liu, and M. Sun,
+“Cube padding for weakly-supervised saliency prediction in 360 videos,”
+2018.
+[38] A. Krizhevsky, G. Hinton et al., “Learning multiple layers of features
+from tiny images,” 2009.
+[39] K. He, X. Zhang, S. Ren, and J. Sun, “Identity mappings in deep residual
+networks,” in European conference on computer vision. Springer, 2016,
+pp. 630–645.
+[40] A. F. Agarap, “Deep learning using rectified linear units (relu),” arXiv
+preprint arXiv:1803.08375, 2018.
+
diff --git a/xdE3T4oBgHgl3EQflgqp/content/tmp_files/load_file.txt b/xdE3T4oBgHgl3EQflgqp/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1e4a494de78ae0b582575e2e90990625c48f631d
--- /dev/null
+++ b/xdE3T4oBgHgl3EQflgqp/content/tmp_files/load_file.txt
@@ -0,0 +1,708 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf,len=707
+page_content='Padding Module: Learning the Padding in Deep  Neural Networks  Fahad Alrasheedi  Department of Computer Science  University of Nebraska Omaha  Omaha, USA  falrasheedi@unomaha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='edu  Xin Zhong  Department of Computer Science  University of Nebraska Omaha  Omaha, USA  xzhong@unomaha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='edu  Pei-Chi Huang  Department of Computer Science  University of Nebraska Omaha  Omaha, USA  phuang@unomaha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='edu  Abstract—During the last decades, many studies have been  dedicated to improving the performance of neural networks, for  example, the network architectures, initialization, and activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  However, investigating the importance and effects of learnable  padding methods in deep learning remains relatively open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To  mitigate the gap, this paper proposes a novel trainable Padding  Module that can be placed in a deep learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding  Module can optimize itself without requiring or influencing  the model’s entire loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To train itself, the Padding  Module constructs a ground truth and a predictor from the  inputs by leveraging the underlying structure in the input data  for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' As a result, the Padding Module can learn  automatically to pad pixels to the border of its input images or  feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The padding contents are realistic extensions to its  input data and simultaneously facilitate the deep learning model’s  downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Experiments have shown that the proposed  Padding Module outperforms the state-of-the-art competitors and  the baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' For example, the Padding Module has  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='23% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='44% more classification accuracy than the zero  padding when tested on the VGG16 and ResNet50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Index Terms—Padding Module, Deep Learning, Neural Net-  works, Trainable Padding  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' INTRODUCTION  Deep Neural Networks (DNNs) have significantly improved  the performance of a wide range of computer vision tasks  to the extent of being comparable to or exceeding human-  level in many domains [1], such as image classification [2],  object recognition [3], and image segmentation [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' DNNs for  computer vision have been iteratively improving in different  aspects such as network architecture [5]–[8], network initial-  ization [9], [10], optimization [11], [12], and activation [13],  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' While it is intuitive that the salient foreground of an input  image can control the results of a deep learning model [15],  [16], researchers have also discovered that the input’s bor-  ders and corners can dominate the model’s performance re-  cently [17]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The study on the importance and effects of  image borders remains relatively open, and this paper focuses  on a trainable padding method that process image borders for  deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Padding refers to the technique of adding extra data to the  input’s borders so that the input’s width, height, or depth can  be manipulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Padding is widely used in Convolutional Neu-  ral Networks (CNNs) to alter the output size of a convolutional  This paper has been accepted for publication by the IEEE Access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1: Five-pixel padding applied to a CIFAR-10 sampled  image using three different padding methods: A) the zero  padding, B) the local mean interpolation, and C) the proposed  Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Without padding, convolutional filters will not process  the input’s borders and the output size will be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The  input size can be maintained with padding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' we add an extra  border before the convolution so that the original border can  be processed [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Traditional padding techniques include zero padding, repli-  cation padding, and reflection padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The reflection padding  reflects the valid data over the borders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' the replication padding  uses the borders themselves as padding values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' the zero  padding specifies the use of zeroes as padding values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The  replication and reflection padding methods extend the input  with duplicate contents that may not be realistic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' hence, they  may destroy the original distribution [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The zero padding  may outweigh the replication and reflection padding methods  in terms of speed due to its computational simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The  major drawback of the traditional methods is that they are not  dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Thus, the padding values are always static and not  optimized during the model training in a way that how they  could be optimally predicted rationally to the input’s borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  More recently, padding methods have been studied aiming  at a more related and realistic extension of the original  input [19], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' For example, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [21] proposed a  padding method using partial convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [19]  used a local mean of a sliding window over the input’s borders  so the local distributions at the borders before and after the  padding are consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' These state-of-the-art padding methods  outperformed the traditional padding in several tasks such  as image classification, image segmentation, and image style  transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, the major disadvantage of the state-of-the-  art padding methods is that they are not trainable: the padding  contents are still not optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='04608v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='CV]  11 Jan 2023  (A)  (B)  (C)In this paper, we propose a trainable Padding Module that  can be inserted into selected positions of a deep learning  model to learn how to pad its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding Module  can be trained simultaneously with a deep learning model,  but, it is a self-learner in a way that will not require or  influence the model’s entire loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' During the training,  the Padding Module internally constructs a ground truth from  the input’s actual borders and trains a predictor considering the  neighboring areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The trained Padding Module can produce  plausible results, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The advantages of our  work can be summarized as three-fold:  The proposed Padding Module introduces a trainable  method that automatically pads its inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The Padding Module extends its input with realistic new  data that are related to the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The Padding Module improves the performance of a  downstream task of a deep learning model and outper-  forms the state-of-the-art competitors, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' In Sec-  tion II, we review the related work that addressed the padding  effects on the neural networks performance and discuss how  the current study fills the gap in the related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Section III  discusses our approach for the Padding Module followed by  evaluation results in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Finally, Section V concludes  with the discussion on evaluation and highlights some of the  future work in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' LITERATURE REVIEW AND RELATED WORK  Many studies have tried to improve the performance of  CNNs models from network architecture [22]–[25], differ-  ent variants of optimization [26]–[28], activations [29]–[32],  regularization methods [33], [34] and so no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, little  attention has been paid to investigating the padding schemes  during the convolution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To assist the kernel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=',  features extractor, in extracting important features during  image processing in CNNs, padding layers can be added to  visit pixels of the images around the corners more times, and  then increase accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The previous padding methods are  presented as follows: Section II-A presents the performance  improvement of neural networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Section II-B introduces the  improvement of space design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' and Section II-C describes our  contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Performance Improvement of Neural Networks  Several studies have proposed padding methods to improve  the performance of the neural networks [17], [19], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Innamorati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [17] addressed the importance of the  data at the borders of the input by proposing a convolution  layer that dealt with corners and borders separately from the  middle part of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' They specifically designed filters for  each corner and border to handle the data at the boundaries,  including upper, lower, left, and right borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The boundary  filters used in the convolution were jointly learned with the  filter used for the middle part of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, the  main issue of this study is that the number of filters used to  deal with the boundaries increases linearly with the size of the  receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Also, Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [19] proposed a padding method that  could keep the local spatial distribution of the padded area  consistent with the original input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The proposed method used  the local means of the input at the borders to produce the  padding values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' they proposed two different variants of the  padding method: mean-interpolation and mean-reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Both  variants used filters with static values, based on the receptive  field, in the convolution operation that is supposed to yield  the padding values maintaining the same distributions as the  original borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, the main issue with this method is  that they are not learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [21] proposed a padding layer that uses a partial  convolution that mainly re-weighted the convolution operation  based on the ratio of the number of parameters in the filter  to the number of valid data in the sliding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' In other  words, they dealt with the padded area as hole areas that need  to be in-painted, while the data coming from the original image  were seen as non-hole areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The main issue of this study is  that the padding process is not learnable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Improvement of Spaces Design  Also, some studies addressed the importance of the padding  and data at the boundaries in the semantic representation  learning and converting 360-degree space to 2-dimensional (2-  D) space respectively [35]–[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [37] showed the importance of the padding  method when they converted the 360-degree video to 2-  dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' They converted the video to six faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Then, they used the reflection padding to connect them to form  the 2-D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The reflection padding naturally connected the  faces compared to the zero-padding, which caused discontinu-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Interesting works were provided by Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' [35], [36]  in which they showed the importance of zero padding along  with the data at the borders in encoding the absolute position  information in the semantic representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' They  showed that the zero padding and the boundaries drove the  CNN models to encode the spatial information that helped  the filters where to look for a specific feature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' the spatial  information was eventually propagated over the whole image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Our Contributions  The padding methods and their effects on a CNN model’s  performance are still open areas for researchers to investigate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  hence, it is worth proposing new padding methods that could  improve the performance of the CNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We propose a  novel padding method, Padding Module, that could realisti-  cally extend the input with related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' It learns how to pad  the input by using the inputs’ borders as a ground truth and  the neighboring areas of the borders as a predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, it  uses a local loss function such as Mean Squared Error (MSE)  and updates the filters using the local differentiation of the  loss function with respect to the Padding Module’s filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The  following section explains the implementation of the Padding  Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' THE PROPOSED Padding Module  This paper presents the Padding Module, a learnable  padding method that can pad the input with related and  realistic padding, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding Module  can be used as a substitute for other padding methods in the  convolution layer, such as the zero padding, the replication  padding, and the reflection padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' This section shows how  the padding procedure (Section III-A) and the backpropagation  (Section III-B) of the Padding Module work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Padding Procedure  Algorithms 1 and 2, respectively, give an overview of the  forward pass and the back-propagation of the Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The Padding Module first constructs a ground truth and a  predictor from the input ( shown in step 1 to step 3 in  Algorithm 1 and explained in Sections III-A1 and III-A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Then, the Padding Module uses the filters being learned to  produce the actual padding values using the input’s borders  as a predictor ( shown in steps 4 to 13 in Algorithm 1 and  explained in Section III-A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Finally, the Padding Module  uses the MSE as a loss function to compute the loss value  and updates the filters during the model’s back-propagation  ( shown in steps 1 to 2 in Algorithm 2 and explained in  Sections III-A3 and III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The Padding Module can pad the original input with any  padding size, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', one-pixel, two-pixels, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Indeed, the  padding process in the Padding Module is iterative ( shown in  steps 4 to 13 in Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Assume the required padding  size is three pixels, the padding process will iterate three times  as follows: (1) padding the original input with one-pixel along  all the four borders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' (2) padding the output of the 1st iteration  with one-pixel along all the four borders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' and (3) padding the  output of the 2nd iteration with one-pixel along all the four  borders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Here, to easily explain our method, a simple case of  padding process was presented here, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', one-pixel padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Also, the Padding Module is assigned filters as many as the  number of channels in the input as explained in Section III-A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Then, we explain the padding process considering a single  channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Here, the same procedure is separately applied to  each channel in case of multiple channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  1) Ground Truth T: The Padding Module structures the  ground truth T by extracting the input’s borders and stacking  them upon each other vertically to form a four-row matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  However, to stack the left and right borders vertically in  T, they are transposed from column vectors to row vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Formally, given M r  c as an original input with r and c as  the number of rows and columns respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' henceforth,  superscripts and subscripts represent the indexes in the row-  wise traversal and the column-wise traversal of the input  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The following is T’s extracting function target  of the input M r  c :  Algorithm 1 Forward Pass  Input: M r  c , size, where r and c are the dimensions of a  matrix, and size is the padding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Output: M r′  c′ , where r′ = r+2×size, and c′ = c+2×size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  1: T ← target(M r  c )  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='1 */  2: N ← neighbors(M r  c )  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='2 */  3: P ← padz(padr(N))  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='3 */  4: M r′  c′ ← M r  c  /* initial state for M r′  c′ */  5: while size ̸= 0 do  6:  Nout ← borders(M r′  c′ )  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='6 */  7:  Pout ← padz(padr(Nout))  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='3 */  8:  O ← fθ(Pout)  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 7 */  9:  M r′+2  c′+2 ← O0//padz(M r′  c′ )//O1  10:  M r′+2  c′+2 ← sides((O2)T , M r′+2  c′+2 , (O3)T ) /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='8  /  11:  M r′  c′ ← corners(M r′+2  c′+2 )  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='9 */  12:  size ← size − 1  13: end while  14: return M r′  c′  Algorithm 2 Back Propagation  Input: Gr′  c′, where c′ and r′ are the same dimensions of the  output of the forward pass in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Output: Gr  c, where c and r are the same dimensions of the  input of the forward pass in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  1: Compute the local gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='5 */  2: Update the filter weights  3: Gr  c ← strip(Gr′  c′)  /* as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='10 */  4: return Gr  c  T = target(M r  c ) =  �  �����  M 0  [:]  M r−1  [:]  (M [:]  0 )  T  (M [:]  c−1)  T  �  �����  ,  (1)  where M 0  [:] is the entire row vector in M r  c at index 0, M r−1  [:]  is the entire row vector in M r  c at index r − 1, (M [:]  0 )  T is the  transpose of the entire column vector in M r  c at index 0, and  (M [:]  c−1)  T is the transpose of the entire column vector in M r  c  at index c−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' (A) is an example to visually illustrate  how T is constructed where the first row represents the upper  border in M r  c , the second row represents the lower border in  M r  c , the third row represents the left border in M r  c , and the  last row represents the right border in M r  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  2) Predictor (P): To structure the predictor from the orig-  inal input M r  c , the Padding Module extracts the row vectors  that neighbor the upper border and lower border in M r  c and the  transpose of the column vectors that neighbor the left border  and right border in M r  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, the Padding Module stacks all  the extracted neighbors vertically to form a four-row matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Formally, the predictor’s (denoted as P) extracting function of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 2: An example to illustrate the steps 1-3 in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the left: the input M r  c with size of (6, 6) pixels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' the superscripts  are the indexes in the row-wise traversal while the subscripts are the indexes in the column-wise traversal of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the  right: (A) the ground truth T: a result of applying step 1 in Algorithm 1 which is a stack of the borders where the first row,  second row, third row, and last row are the upper, lower, left, and right borders in the input respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' and (B) the predictor  P: a result of applying steps 2 and 3 in Algorithm 1 which is a stack of the neighbors where the first row, second row, third  row, and last row are neighbors to the upper, lower, left, and right borders in the input respectively, and the stack is padded  at the left and right sides with reflection padding (pr) and with zero padding (pz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M r  c can be expressed in the following way:  First, the neighbors in M r  c are selected and denoted as N  as follows:  N = neighbors(M r  c ) =  �  �����  M 1  [1:c−1]  M r−2  [1:c−1]  (M [1:r−1]  1  )  T  (M [1:r−1]  c−2  )  T  �  �����  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  (2)  The slice [1 : c − 1] excludes the data in the row vectors at  the borders due to overlapping with the T, whereas the slice  [1 : r − 1] excludes the data in the column vectors at the  borders due to overlapping with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Then, the Padding Module pads the structure as follows:  P = padz(padr(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  (3)  First, the padr(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=') function pads the structure with one pixel of  the reflection padding horizontally (the left and right sides);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  then, with one pixel of the zero padding horizontally using the  padz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=') function can get the final structure for P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Each row in P will be used to predict the corresponding  row in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' For example, the first row in P will be used to  predict the first row in T representing the upper border in  the input M r  c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Figure 2 (B) is an example to visually illustrate  how the Padding Module constructs the stack of the neighbors  (as a predictor) where the right and left sides of the stack are  padded with the reflection padding (named as pr), and the zero  padding (named as pz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  3) Filters and the Loss Function: The Padding Module uses  as many filters as the channels in the input (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', filter per  channel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Also, each filter will be a row vector with a size of  (1, 3) and a stride of (1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' that is because of having each row  in P as a predictor for the corresponding row in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Therefore,  to predict T, the Padding Module convolutes the filters over P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  it uses its own loss function to optimize the prediction through  the local differentiation of the loss function with respect to the  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The loss function used by the Padding Module is the MSE  which computes the squared difference between the ground  truth and the predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The following equation is the  MSE’s mathematical expression for a single data point:  MSE(fθ(P), T) =  4  �  a=1  n  �  j=1  (θT · P a  j − T a  j )2,  (4)  where f is the convolutional operation parameterized by θ, P  and T are the predictor and the ground truth extracted from  the original input M r  c , a represents the indexes for rows in the  four-row matrices P and T, and j represents the indexes for  both the slide windows and columns in P and T respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Hence, P a  j is the jth slide window in the row indexed at a in  P, and T a  j is the corresponding value in T indexed at the ath  row and jth column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The local differentiation of the Padding Module’s loss func-  tion and the filters’ updates are achieved during the model’s  back-propagation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' these local gradients are not propagated to  the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Besides that, the Padding Module facili-  ties the back-propagation of the model’s loss function going  through it to the previous layer as explained in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The following is the mathematical expression for the local  gradients (Padding Module’s loss function gradients with  respect to a single filter for a single data point):  ϕ  ϕθm  MSE(fθ(P), T) = 2  4  �  a=1  n  �  j=1  (θT · P a  j − T a  j )xm,  (5)  where xm is a single feature in the P a  j slide window which  was multiplied by the corresponding weight, namely θm, in  A) Ground Truth T  MST  [Mo  MI  M  Ma  M  Me  MS  M5  M2  M5  MS  Input Mr=  M  M3  M3  M  Mo  M  [M  MI  M2  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M  M  LMg  M2  M3  Ms  MS!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M  M2  M1  Ms  M  M1  M3  M  M2  M3  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M  M  target(M)  M3  M3  M2  M3  M  M3  Pad(padr(neighbors(M))  B) Predictor P  M2  M4  Mo  Mi  M2  M9  Ma  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Ma  M4  M  Pz  pr  pr  pz  LM§  M5  M5  M5  M5  M5  M5  M5  M5  M5  ME  Pz  pr  pr  Pz  M  M3  Me  Mo  M1  M5  Pz  pr  Pr  Pz  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M  M2  Lpz  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Ms  M5  pr  pr  pz-the θ during the convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  4) Padding  Process:  The  procedures  in  Sec-  tions III-A1, III-A2, and III-A3 are used to guide the  Padding Module on learning how to predict the borders of  the original input, M r  c , based on the neighboring areas to the  borders, and then the Padding Module can optimize its filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  However, the padding process is shown in steps 4 to 13  in Algorithm 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' it uses the borders of the input, M r′  c′ , as  the predictor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' In detail, the padding process iterates until the  original input is padded with the required padding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hence,  the original input M r  c is assigned to M r′  c′ as an initial state  in step 4 before the padding loop starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, each iteration  pads the input, M r′  c′ , with one-pixel, and outputs a new M r′  c′  which will be used for the next iteration and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The  dimensions of an iteration’s output, M r′  c′ in step 11, are two-  pixel larger than the dimensions of that iteration’s input, M r′  c′  in step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Minutely, constructing the predictor in the padding process  is similar to the way that constructs P in Section III-A2 with  small modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To distinguish the notions of neighbors,  N, and P, in Section III-A2, borders, Nout, and Pout are  denoted for the extracting function, the function’s output, and  the predictor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The following is the mathematical  expression for the extracting function borders:  Nout = borders(M r′  c′ ) =  �  �����  M 0  [:]  M r−1  [:]  (M [:]  0 )  T  (M [:]  c−1)  T  �  �����  ,  (6)  where M 0  [:] and M r−1  [:]  mean extracting the entire upper and  lower borders respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Whereas, (M [:]  0 )T and (M [:]  c−1)T  mean extracting the transpose of the entire left and right  borders respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, the Padding Module pads the  output Nout using Equation 3 to get the final structure for  Pout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Consequently, convoluting the filters over the Pout will  produce the padding values for the iteration’s input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The output  can be expressed as follows:  O = fθ(Pout),  (7)  where f is the convolutional operation parameterized by θ,  Pout is the predictor, and the O is the output and comes as a  matrix of four rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Each row represents the padding values  for the corresponding area in the iteration’s input, M r′  c′ , as  follows: the first row (O0), the second row (O1), the third row  (O2), and the fourth row (O3) represent the padding values for  the upper, the lower, the left, and the right areas in the input  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Then, the steps from 9 to 11 are how the produced padding  values stick around the input M r′  c′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' First, in step 9, the vertical  concatenation operator // is used to concatenate the first row  (O0) with M r′  c′ , and then concatenates the resulted matrix  with the second row (O1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, the rows from O are  two-pixel wider than the rows of M r′  c′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' therefore, to match  the dimensions of these operands, the Padding Module uses  padz(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=') to pad the M r′  c′ horizontally with one pixel of the zero  padding before the concatenation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hence, the output’s  dimensions in step 9, denoted as M r′+2  c′+2 , are two-pixel larger  than the input M r′  c′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Finally, the algorithm uses sides function  which can be formally expressed as the following:  sides((O2)T , M r′+2  c′+2 , (O3)T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  (8)  This function does not change the dimensions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' however, it adds  respectively the transpose of the third row (O2) and fourth row  (O3) to the left and right columns of M r′+2  c′+2 , the concatenated  matrix with zero values at the left and right columns unless  the corners already assigned values from the concatenation  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To resolve the double-count problem at the corners,  the Padding Module takes the average of added values in the  corners by dividing each corner by 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' this averaging function  is step 12 in Algorithm 1:  M r′  c′ = corners(M r′+2  c′+2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  (9)  Lastly, as mentioned early in this section that the dimensions  of the iteration’s output are two-pixel larger than the iteration’s  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hence, the output M r′  c′ , in Equation 9, has dimensions r′  and c′ that are updated with the dimensions of M r′+2  c′+2 , namely  r′ + 2 and c′ + 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Back-propagation  As seen in Section III-A3, the Padding Module is not opti-  mized based on the model’s main loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' therefore, the  model does not compute the gradients of its loss function with  respect to the filters of the Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, during  the mode’s backpropagation, the Padding Module achieves two  key points as follows:  1) As shown in step 1 in Algorithm 2, the Padding Module  optimizes its filters through computing the local gradi-  ents for its loss function with respect to the filters as  explained in Section III-A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  2) The process also receives Gr′  c′ which are the gradi-  ents of the model’s loss function with respect to the  Padding Module’s output, the original input M r  c after  being padded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Therefore, the Padding Module strips  out the gradients from Gr′  c′ that represent the gradients  for the padded areas in the Padding Module’s output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  the stripping-out process is step 3 in Algorithm 2, and  formally expressed as follows:  Gr  c = strip(Gr′  c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  (10)  Then, the Padding Module back-propagates to the pre-  vious layer the Gr  c, representing the gradients for the  previous layer’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Figure 3 is an example to  visually illustrate how the back-propagation process in  the Padding Module is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 3: An example to illustrate the back propagation in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the top: the input to the Padding Module is of size  (6, 6), the Padding Module uses one-pixel padding and produces an output of size (8, 8) where the borders p is the computed  padding values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the bottom: the back-propagation of the received gradients which is of size (8, 8) where the borders are  gradients for the padding values gp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' the Padding Module strips out gp from the received gradients, and sends the remaining to  the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The g stands for gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' for example gM 0  0 is the gradient for the pixel at index [0, 0] in the input of the  Forward Pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' EXPERIMENTAL RESULTS AND ANALYSIS  This section shows the design of the training and testing  experiments on our Padding Module applied to a downstream  task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The experimental setup is  presented in Sections IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The quantitative and qualitative  results are described in Section IV-B and IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Experiment Setup  The study used the premium service from Google Colabora-  tory where a GPU of Tesla T4 was assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The experiments  and comparisons were conducted on the CIFAR-10 dataset  for a classification task [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The CIFAR-10 dataset includes  a training dataset of 50,000 images and a test dataset of  10,000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The images are of shape (32, 32, 3), dis-  tributed equally to ten classes of airplane, automobile, bird,  cat, deer, dog, frog, horse, ship, and truck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding  Module was applied to different networks namely: VGG16 [8]  and ResNet50V2 [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' to make the deeper layers in these  networks carry out a valid convolution, the images were  resized to (64, 64, 3) and (224, 224, 3) for the VGG16 and  the ResNet50V2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The VGG16 is a vanilla-based architecture where the net-  work shape is wider at the beginning of the network and  narrowed down as going deep in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The pre-trained  VGG16 was obtained from the keras1 without the top layers  (the last three dense layers including the original softmax  layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, we added two fully-connected layers each with  512 neurons and followed by a dropout layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the other  hand, the ResNet50V2 is made up of blocks where each block  sends the block’s input through the block itself, and also uses a  skip connection to directly add the block’s input to the output  of the input’s flow coming through the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The process  is known as the identity function that could help deep layers  to improve the model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' ResNet50V2 is a modified  version of the ResNet50 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The modification mainly is in  the arrangement of the block layers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' batch normalization [11]  and ReLU activation [40] are applied to the data flow before  the convolutional layer in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' These changes enabled  the ResNet50V2 to outperform the ResNet50 on the image  classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The ResNet50V2 was downloaded from the  keras2 without the top layer (the last dense layer which is the  original softmax layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Then, two fully-connected layers with  1024 and 512 neurons were added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  1VGG16 from the keras: https://keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='io/api/applications/vgg  2ResNet50V2 from the kera: https://keras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='io/api/applications/resnet  The Padding Module  Output  Input  p  p  p  p  p  p  p  MI  M2  M  MI  M  M:  p  Mi  M9  M  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  p  Mi  MS  MS  M  p  M  M1  M2  M3  M1  Ms  p  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M3  Forward Pass  Me  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  M  M2  p  p  Ma  Mi  M2  M3  Ms  M3  p  p  Me  M1  M  MS  M4  p  Ma  Ms  p  LM  M5  M5  MS  Ms  MS  M5  M5  M5  M5  MS  p  MS  p  Lp  p  p  p  p  p  p  pJ  Input  Output  T9p  9p  9p  9p  9p  9p  9p  9pj  g mi  w6  9 mg  9 mg  9p  9M:  9Mi  9 ms  9 mg  9g  9 M2  Backpropagation  9p  9M  gM  9m?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  9p  9p  9Mi  9ms  9p  9m  9 ms  9 ms  9ms  9 ms  M  9p  9m  9m  9p  gp  gp  9p  9p  gr  gp-Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 4: The comparison of three different padding methods on  the test images: zero padding, mean interpolation padding, and  the Padding Module when applied to the VGG16 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 5: The comparison of three different padding methods on  the test images: zero padding, mean interpolation padding, and  the Padding Module when applied to the ResNet50V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Moreover, we added a softmax layer with ten outputs for  both models of VGG16 and ResNet50V2, and then used  the Adam optimizer [12] for the back-propagation of the  gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Finally, the Padding Module was used before every  convolutional layer in the VGG16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' whereas, we replaced every  zero padding layer in the ResNet50V2 with the Padding  Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Quantitative Results  Section IV-B1 compares the proposed Padding Module and  state-of-art padding solutions by performing the image clas-  sification task, and then Section IV-B2 discusses an ablation  study based on our solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  1) Image Classification Task:  We considered the zero  padding method as a baseline to compare the Padding Module  with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Moreover, we used the mean interpolation padding  method [19] as the state-of-art since it outperformed the  partial convolution padding method [19], [21] in the image  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The main goal of this study, which aligns with  the literature, is to investigate the padding effect on the  accuracy of DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Therefore, the accuracy is used as  a comparison metric between the performance of the Padding  Module and the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The accuracy is the percentage of  correctly classified images over the total number of images in  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Each model was trained with 100 epochs using the training  dataset, and tested in each epoch using the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' In  Figure 4, the Padding Module outperforms both the baseline  method and the mean interpolation padding method when us-  ing the VGG16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' also, we found that the baseline is comparable  to the mean interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' As for the Resent50, the  Padding Module also outperforms the other two paddings as  shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We also noticed that the baseline method  is comparable to the mean interpolation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Moreover,  Table I summarizes the average of the last five epochs for the  three different padding methods and the margin between the  highest and the second-highest accuracies for the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  As mentioned, the study investigated the effects of the  Padding Module on the accuracy of DNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Importantly,  we showed that the related and realistic padding could improve  the accuracy of DNN models ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' therefore, the Padding Module  Padding Method  VGG16  ResNet50V2  Padding Module  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='92  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='08  Zero Padding  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='43  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='64  Mean Interoplation  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='69  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='44  margin  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='23  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='44  TABLE I: The average accuracy of the last five epochs  for three different padding methods used in VGG16 and  ResNet50V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The margin is the difference between each  model’s highest and second-highest padding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 6: MSEs for three Padding Modules placed at different  positions in the VGG16: 1) module 1: at the beginning, 2)  module 2: at the middle, and 3) module 3: at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  was able to produce such padding by minimizing the MSE  in Algorithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Figure 6 illustrates MSEs for three  cases of the Padding Module applied in different places (at  the beginning, in the middle, and at the end) in the VGG16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  it is evident that the MSEs significantly decreased after only  two epochs and then stayed flat till the end of the experiment  for the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  One natural drawback of the current Padding Module was  the extra running time caused by constructing the data struc-  tures and optimizing the filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Table 2 shows that VGG16 and  ResNet50V2, on average, doubled the epoch’s time when ap-  plying the Padding Module (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', placing the Padding Module  before every convolutional layer in the VGG16 and replacing  every zero padding layer in the ResNet50V2 with the Padding  Module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the other side, the accuracy for the VGG16  and ResNet50V2, respectively, gained margins of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='49% and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='44% when applying the Padding Module compared to the  100  90  Accuracy  80  Padding Types  Mean Interpolation  70  Ours  Zero padding  60  0  10  20  30  40  50  60  70  80  90  100  Epochs100  95  Accuracy  Padding Types  Mean Interpolation  90  Ours  Zero padding  85  0  10  20  30  40  50  60  70  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='6  100  Epochs30  MES  Module  20  module 1  module 2  10  module 3  0  10  20  30  40  50  60  70  80  90  100  Epochszero padding (no Padding Module).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' One remedy to lessen the  running time problem may be to stop training the Padding  Module when it significantly decreases the MSE after the first  two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, improving the current Padding Module  including the time complexity can be a further direction for  further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  2) Ablation Study: The experiments in this section were  conducted as an ablation study where the Padding Module  was empirically placed at different positions in the VGG16  model, as shown in Figure 7: at the beginning of the model,  in the middle, at the end, and the combination of all the three  places together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We also compared the four scenarios with two  other scenarios: (1) where the Padding Module was placed in  all positions (before each convolutional layer) in the model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  and (2) where the Padding Module was not used but the zero  padding was used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We ran each scenario 100 epochs  using the training dataset for training and the test dataset for  evaluation, and averaged the test accuracies of the last five  epochs for each scenario;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Table III illustrates the summary of  the comparison of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We noticed that using a single  Padding Module with the shallow layers outperformed the case  of using it with the deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Also, the combination scenario  showed a superiority over the scenario of a single Padding  Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, the best performance was when the Padding  Module applied in the scenario of all positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Finally, all the  scenarios of applying the Padding Module outperformed the  scenario of the model with no Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Model  Zero Padding  Padding Module  Margin  Accuracy  Time  Accuracy  Time  VGG16  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='43  2  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='92  4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='49  ResNet50  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='64  5  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='08  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='44  TABLE II: On average, the running time doubles for one  epoch when applying the Padding Module to the VGG16 and  ResNet50V2 compared to the case of the zero padding (no  Padding Module applied).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Times are shown in a minute-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The margin is the accuracy difference between the case of  applying the Padding Module and the zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  No  Different places in the VGG16  Accuracy  1  At the beginning  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='98  2  At the middle  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='8  3  At the end  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='97  4  Combination of 1, 2, and 3 togather  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='18  5  All positions  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='8  6  VGG16 with no Padding Module  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='4  TABLE III: Placing the Padding Module at different positions  in the VGG16: 1) at the beginning 2) at the middle 3) at the  end 4) combination of beginning, middle, end 5) before every  convolutional layer 6) VGG16 with no Padding Module (zero  padding instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Qualitative Results  Different padding sizes, such as one-pixel, three-pixel, and  five-pixel, were used to illustrate how the Padding Module  can extend the input with related and realistic extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Also, we compared these different padding sizes with the  other two methods, namely the zero padding and the mean  interpolation padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' As shown in Figure 8, the Padding  Module can learn how to pad the input with related data and  natural extension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' this finding becomes more evident as the  padding size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' FUTURE RESEARCH DIRECTIONS AND CONCLUSION  This paper proposed a novel padding method: Padding  Module;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' that can learn how to pad an input from the input’s  borders;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' hence, the input can be realistically extended with  related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding Module is a self-learning of its  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' To train itself, the Padding Module constructs a  ground truth and a predictor from the inputs by leveraging  the underlying structure in the input data for supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The Padding Module uses convolutional operation over the  predictor to produce a predicted value that is, in turn, com-  pared with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The Padding Module uses a  local loss function, independent from the model’s main loss  function, to minimize the difference between the predicted  value and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Therefore, the Padding Module  updates its convolutional filters locally during the model’s  back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Besides that, the Padding Module back-  propagates the model’s gradients with respect to the Padding  Module’s output after stripping out the gradients for the padded  areas to the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  The experimental results showed that the Padding Module  outperformed the zero-padding and the state-of-art padding in  the image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' In the ablation study, we also  observed that using a single Padding Module with the shallow  layers improved the performance slightly better than using it  with the deep layers in the VGG16 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' On the other  hand, using three of the Padding Module placed in different  positions (at the beginning, at the middle, and at the end)  in the VGG16 outperformed the scenario of a single Padding  Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Moreover, placing the Padding Module in all positions  (before every convolutional layer) in the VGG16 outperformed  all other scenarios as shown in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Our experiments applied the Padding Module to the two  well-known networks: VGG16 and ResNet50, for the image  classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' The VGG16 and ResNet50 networks were  chosen to represent small and large networks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  They, also, were used by the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' hence, we used them  to compare the Padding Module with the previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Although two different networks are only used in one task, we  shall extend the Padding Module to improve such networks in  different tasks, including object detection, style transfer, and  image inpainting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' We leave investigating the Padding Module  in a wide range of tasks for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Also, the Padding Module learned how to pad the input  independently of the model’s loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' However, it is  possible to optimize the Padding Module’s filters based on  optimizing the model’s main loss function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' this approach will  be entirely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hence, one research direction may be to  implement a padding method that can optimize its padding  filters based on the model’s main loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 7: The selected positions in the VGG16 for the Padding Module: at the beginning, middle, and the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 8: Three images sampled from CIFAR-10 and padded by different padding methods: zero-padding, mean interpolation,  and the Padding Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Each padding method uses three different padding sizes: A) one-pixel, B) three-pixel, C) five-pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Eykholt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Evtimov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Fernandes, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Li, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Rahmati, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Xiao,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Prakash, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kohno, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Song, “Robust physical-world attacks  on deep learning visual classification,” in Proceedings of the IEEE  conference on computer vision and pattern recognition, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1625–  1634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [2] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Xie, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Li, “Bag of  tricks for image classification with convolutional neural networks,” in  Proceedings of the IEEE/CVF Conference on Computer Vision and  Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 558–567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Huang, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Rathod, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Korattikara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Fathi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Fischer,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Wojna, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Guadarrama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', “Speed/accuracy trade-offs  for modern convolutional object detectors,” in Proceedings of the IEEE  conference on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 7310–  7311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [4] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Gkioxari, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Doll´ar, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Girshick, “Mask r-cnn,” in  Proceedings of the IEEE international conference on computer vision,  2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 2961–2969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zeng, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ghadiyaram, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Lee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Bovik, “Deep convolutional neural models for picture-quality prediction:  Challenges and solutions to data-driven image quality assessment,” IEEE  Signal processing magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 34, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 130–141, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Lu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Yang, “Universal  style transfer via feature transforms,” Advances in neural information  processing systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Nguyen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Choi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kim, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Lee, “A simple way of  multimodal and arbitrary style transfer,” in ICASSP 2019-2019 IEEE  International Conference on Acoustics, Speech and Signal Processing  (ICASSP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  IEEE, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1752–1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Simonyan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zisserman, “Very deep convolutional networks for  large-scale image recognition,” arXiv preprint arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='1556, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [9] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Arpit, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Campos, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Bengio, “How to initialize your network?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  robust initialization for weightnorm & resnets,” Advances in Neural  Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [10] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, “Delving deep into rectifiers:  Surpassing human-level performance on imagenet classification,” in  Proceedings of the IEEE international conference on computer vision,  2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1026–1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ioffe and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Szegedy, “Batch normalization: Accelerating deep  network training by reducing internal covariate shift,” in International  conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  PMLR, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 448–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [12] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ba, “Adam: A method for stochastic optimization,”  CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' abs/1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [13] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, “Deep residual learning for image  recognition,” in Proceedings of the IEEE conference on computer vision  and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [14] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Van Der Maaten, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Weinberger, “Densely  connected convolutional networks,” in Proceedings of the IEEE confer-  ence on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 4700–4708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Simonyan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Vedaldi, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zisserman, “Deep inside convolutional  networks: Visualising image classification models and saliency maps,”  arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='6034, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [16] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Khosla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Lapedriza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Oliva, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Torralba,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' “Learning  deep features for discriminative localization,”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' in Proceedings of the IEEE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Beginning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Resize  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Middle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='End  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Pooling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Flatten  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Dense1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Dense2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Conv13  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Softmax  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='OutputZero padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Mean Interpolation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='Padding Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='C) Five-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='A) One-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='A) One-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='B) Three-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='C) Five-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='A) One-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='B) Three-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='C) Five-pixel padding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='B) Three-pixel paddingconference on computer vision and pattern recognition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 2016,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 2921–  2929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [17] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Innamorati, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ritschel, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Weyrich, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Mitra, “Learning on  the edge: Explicit boundary handling in cnns,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [18] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Shih, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Reda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sapra, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Yu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Tao,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Catanzaro, “Partial convolution based padding,” arXiv preprint  arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='11718, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Nguyen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Choi, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ahn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kim, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Lee, “Dis-  tribution padding in convolutional neural networks,” in 2019 IEEE  International Conference on Image Processing (ICIP), 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 4275–  4279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Albawi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Mohammed, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Al-Zawi, “Understanding of a  convolutional neural network,” in 2017 International Conference on  Engineering and Technology (ICET), 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [21] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Shih, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Wang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Reda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sapra, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Yu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Tao,  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Catanzaro, “Partial convolution based padding,” 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [22] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, “Deep residual learning for image  recognition,” in Proceedings of the IEEE conference on computer vision  and pattern recognition, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 770–778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [23] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Van Der Maaten, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Weinberger, “Densely  connected convolutional networks,” in Proceedings of the IEEE confer-  ence on computer vision and pattern recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 4700–4708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [24] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Takikawa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Acuna, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Jampani, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Fidler, “Gated-scnn: Gated  shape cnns for semantic segmentation,” in Proceedings of the IEEE/CVF  international conference on computer vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 5229–5238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Radosavovic, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kosaraju, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Girshick, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Doll´ar,  “Designing network design spaces,” in Proceedings of the IEEE/CVF  conference on computer vision and pattern recognition, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  10 428–10 436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [26] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kingma and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ba, “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='6980, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ioffe and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Szegedy, “Batch normalization: Accelerating deep  network training by reducing internal covariate shift,” in International  conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  PMLR, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 448–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [28] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Jiang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Chen, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Gao, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Han, “On  the variance of the adaptive learning rate and beyond,” arXiv preprint  arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='03265, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Maas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hannun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', “Rectifier nonlinearities  improve neural network acoustic models,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' icml, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 30, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  Atlanta, Georgia, USA, 2013, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, “Delving deep into rectifiers:  Surpassing human-level performance on imagenet classification,” in  Proceedings of the IEEE international conference on computer vision,  2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1026–1034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [31] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ramachandran, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zoph, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Le, “Searching for activation  functions,” arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='05941, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [32] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Mercioni and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Holban, “P-swish: Activation function with learn-  able parameters based on swish activation function in deep learning,” in  2020 International Symposium on Electronics and Telecommunications  (ISETC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  IEEE, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 1–4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Tompson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Goroshin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Jain, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' LeCun, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Bregler, “Efficient  object localization using convolutional networks,” in Proceedings of the  IEEE conference on computer vision and pattern recognition, 2015, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  648–656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [34] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Larsson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Maire, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Shakhnarovich, “Fractalnet: Ultra-deep  neural networks without residuals,” arXiv preprint arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='07648,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Islam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Jia, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Bruce, “How much position informa-  tion do convolutional neural networks encode?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [36] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Islam, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Kowal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Jia, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Derpanis, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Bruce, “Po-  sition, padding and predictions: A deeper look at position information  in cnns,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [37] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Cheng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Chao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Dong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Wen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Liu, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun,  “Cube padding for weakly-supervised saliency prediction in 360 videos,”  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Krizhevsky, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=', “Learning multiple layers of features  from tiny images,” 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Ren, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Sun, “Identity mappings in deep residual  networks,” in European conference on computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Springer, 2016,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' 630–645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='  [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content=' Agarap, “Deep learning using rectified linear units (relu),” arXiv  preprint arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
+page_content='08375, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE3T4oBgHgl3EQflgqp/content/2301.04608v1.pdf'}
diff --git a/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/2301.05261v1.pdf.txt b/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/2301.05261v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..304a6e0d38fe8d3de33b3fe8174e707bbce0d3f7
--- /dev/null
+++ b/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/2301.05261v1.pdf.txt
@@ -0,0 +1,7394 @@
+Exact emergent higher symmetries in bosonic lattice models
+Salvatore D. Pace and Xiao-Gang Wen
+Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
+(Dated: January 16, 2023)
+While higher-form symmetries are a powerful tool in studying a quantum many-body system,
+theories with exact higher-form symmetries are rather special and, in a sense, fine-tuned.
+This
+raises an interesting question: can the phases of a microscopic (UV) theory without exact higher-
+form symmetries be exactly characterized by emergent higher-form symmetries? Here we argue the
+answer is yes by constructing effective theories for bosonic lattice Hamiltonian models that only
+capture the system’s dynamics at E < Escale. The emergent symmetries below this energy scale
+are then identified as the exact symmetries of this effective theory.
+We find that the emergent
+higher-form symmetries (excluding 0-form symmetries) are robust against local UV perturbations
+and become exact symmetries of the effective theory in the thermodynamic limit. This result is true
+even for more general higher symmetries, such as non-invertible higher symmetries (i.e., algebraic
+higher symmetries). Therefore, emergent higher symmetries are exact emergent symmetries: they
+are not UV symmetries but constrain the IR as if they were. We apply this framework to three lattice
+models (the quantum clock model and emergent ZN and U(1) p-gauge theory) to identify regions of
+parameter space with energy scales below which higher-form symmetries, and sometimes associated
+’t Hooft anomalies, emerge. Since phases of matter are defined in the thermodynamic limit, this
+implies that a UV theory without exact higher symmetries can have phases exactly characterized
+by emergent higher symmetries. We discuss in detail the physical consequences of this and contrast
+it to emergent 0-symmetries, which are never exact. This emphasizes the importance of identifying
+scale hierarchies and emergent higher symmetries when studying quantum many-body systems.
+CONTENTS
+I. Introduction
+1
+II. Energy scales hierarchies and effective
+Hamiltonians
+3
+A. Low energy exact emergent generalized
+symmetries
+3
+B. A holographic picture for emergent finite
+symmetries
+5
+III. Examples of exact emergent higher-form
+symmetries
+8
+A. Quantum clock model
+10
+1. An exact emergent Z(d)
+N symmetry and
+mixed ’t Hooft anomaly
+11
+B. Emergent ZN p-gauge theory for p ≥ 1
+12
+1. An exact emergent Z(p)
+N symmetry
+13
+2. An exact emergent anomalous
+Z(p)
+N × Z(d−p)
+N
+symmetry
+16
+C. Emergent U(1) p-gauge theory
+19
+1. An exact emergent U(1)(p) symmetry
+19
+2. An exact emergent anomalous
+U(1)(p) × U(1)(d−p−1) symmetry
+21
+IV. Physical consequences of exact emergent
+higher-form symmetries
+23
+V. Conclusion and discussion
+26
+VI. Acknowledgements
+26
+A. Review of discrete differential geometry for
+d-dimensional cubic lattices
+26
+B. A TQFT description of the p-form toric code
+ground states—p-form BF theory
+28
+1. Review of p-form BF theory
+29
+a. ZN p-form gauge theory in the continuum 29
+b. Symmetries
+31
+c. Mixed ’t Hooft anomaly and anomaly
+inflow
+32
+C. Taking the continuum limit—p-form Maxwell
+theory
+33
+1. Review of p-form Maxwell theory
+35
+a. U(1)(p) symmetry
+36
+b. Abelian duality
+37
+c. U(1)(d−p−1) symmetry
+39
+d. Mixed ’t Hooft anomaly and anomaly
+inflow
+39
+References
+41
+I.
+INTRODUCTION
+A longstanding pillar for understanding strongly inter-
+acting quantum many-body systems is to identify and
+understand their symmetries. Indeed, symmetries pro-
+vide powerful constraints and universal characterizations
+of a system’s dynamics and phases. This point of view
+has become increasingly fruitful with modern general-
+izations of symmetry [1–8].
+For instance, topological
+order [9], which provided the first indication that con-
+ventional symmetries [10, 11] are not all-powerful, can
+now be understood in a symmetry framework [6, 12–16].
+These generalizations open up an exciting frontier for
+the discovery of new phases of quantum matter and the
+arXiv:2301.05261v1  [cond-mat.str-el]  12 Jan 2023
+
+2
+conceptual organization/systematic understanding [4] of
+quantum phases. In fact, one may wonder if the gener-
+alization of symmetries can become so general that they
+could capture all possible IR symmetries in gapless liquid
+states, becoming a largely complete universal characteri-
+zation of gapless phases and thus a key to understanding
+gapless liquid states [17–20] and webs of dualities [4, 21–
+23].
+One of the simplest and best-understood generaliza-
+tions of symmetry is so-called higher-form symmetry. For
+ordinary symmetries, charged operators act on a point in
+spacetime and the unitary operator that generates the
+symmetry transformation acts on a codimension-1 hy-
+persurface of spacetime (e.g., all of space at a fixed time
+slice). Higher-form symmetries generalize this by allow-
+ing charged operators to be extended objects [1, 2, 12].
+For a p-form symmetry1, the charged operators act on
+closed p-dimensional subspaces and, and the unitary
+generating the transformation acts on a closed (d − p)-
+dimensional subspace of d-dimensional space.
+Notice
+that an ordinary global symmetry is just a 0-form sym-
+metry.
+Most things 0-form symmetries can do, higher-form
+symmetries can also do.
+For example, higher-form
+symmetries can spontaneously break, giving rise to a
+topological ground state degeneracy (gapless Goldstone
+bosons) when discrete (continuous) [2, 27–33].
+In-
+deed, abelian topological orders reflect discrete 1-form
+symmetries spontaneously breaking, and photons in a
+Coulomb phase arise from U(1) 1-form symmetries spon-
+taneously breaking.
+A higher-form symmetry can also
+have a ’t Hooft anomaly, providing powerful constraints
+on the IR through generalized Lieb-Schultz-Mattis-
+Oshikawa-Hastings theorems and introducing higher-
+form symmetry-protected topological phases [13, 14, 34–
+43].
+These applications of higher-form symmetries make
+them a powerful tool in studying quantum many-body
+systems.
+However, unfortunately, models with exact
+higher-form symmetries are rather special and, in a sense,
+fine-tuned. So, it is natural to wonder if they play a role
+in more typical, physically relevant models. One possi-
+bility is that while they may not be exact microscopic
+symmetries, they could still arise as emergent symme-
+tries. However, experience with emergent ordinary (0-
+form) symmetries causes apprehension since their conse-
+quences are never exact and always approximate [44]. In
+other words, explicitly breaking emergent 0-form sym-
+metries at energy scale E creates O(Eγ) errors, even in
+the thermodynamic limit. Amazingly, common folklore
+suggests that this 0-form symmetry-based intuition does
+not carry over to higher-form symmetries and that they
+can constrain a system exactly even as emergent symme-
+tries [13, 33, 45–49].
+1 Here we will consider only pure p-form symmetries, and not the
+more general p-group symmetries [24–26].
+In this manuscript, we investigate this robustness of
+higher-form symmetries from a UV perspective, consid-
+ering bosonic lattice Hamiltonian models without higher-
+form symmetries. These UV-complete theories are sim-
+ple to work with, well-defined, and relevant to condensed
+matter physics. For this class of models, we demonstrate
+how higher-form symmetries can emerge and, when they
+do, why they nevertheless constrain the IR exactly. This
+result is not rigorously demonstrated and relies on phys-
+ically reasonable conjectures made in order to identify
+low-energy sub-Hilbert spaces.
+We find that at energies with the emergent higher-form
+symmetry, the dynamics of states are affected by the
+emergent higher-form symmetry as if it were an exact
+UV symmetry. More precisely, any coming errors from
+the higher-form symmetries being emergent below a finite
+energy scale E are of order O(e−Lγ), where L is the sys-
+tem size measured by lattice constant, and thus vanish in
+the thermodynamic limit. Our arguments apply to sym-
+metries more general then pure higher-form symmetries,
+such as higher-group symmetries and beyond higher-
+group symmetries (non-invertible higher-form symme-
+tries/algebraic higher symmetries [4, 50, 51]), but do not
+apply to 0-symmetries.
+This suggests the phases of microscopic models with-
+out exact higher symmetries can be exactly character-
+ized by emergent higher symmetries2, except the 0-
+symmetries. To emphasize this, we will refer to emer-
+gent higher symmetries as exact emergent symmetries.
+The rest of this paper is goes as follows:
+In section II, we review the importance of the sepa-
+ration of scales in many-body systems and show how to
+identify low-energy sub-Hilbert spaces for a general class
+of models. We then discuss how to identify the emergent
+symmetries of a low-energy sub-Hilbert space, providing
+motivation from the point of view that symmetries are
+described by algebras of local symmetric operators [52].
+We identify these emergent symmetries by developing an
+effective Hamiltonian description for the low-energy sub-
+Hilbert space using the intuition that the microscopic
+dynamics generate low-energy dynamics.
+In section III, we apply the framework introduced in
+section II to three bosonic lattice models. We start by
+warming up with the ZN quantum clock model in the ZN
+spontaneous symmetry broken phase in subsection III A.
+We then apply the framework to more sophisticated mod-
+els for emergent ZN p-gauge theory and emergent U(1) p-
+gauge theory in subsections III B and III C, respectively.
+These subsections include many technical details, ex-
+plicitly demonstrating how higher-form symmetries can
+emerge in these models and how they become exact sym-
+2 Here the term “p-symmetry” and “higher symmetry” includes
+both higher-form symmetry described by higher-group and al-
+gebraic higher symmetry which are beyond higher-group.
+For
+example, 0-symmetry includes both group-like 0-form symmetry
+and beyond-group algebraic symmetry.
+
+3
+metries of the effective mid-IR Hamiltonians in the ther-
+modynamic limit.
+In section IV, we summarize the general lessons learned
+and the physical consequences from exact emergent
+higher-form symmetries. In particular, we first discuss
+why, from the point of view of effective Hamiltonians,
+breaking a higher-form symmetry in the UV does not au-
+tomatically break it at lower energies. Then we discuss
+how exact emergent higher-form symmetries can charac-
+terize phases of theories that do not have UV higher-form
+symmetries. In particular, how exact emergent higher-
+form symmetries can spontaneously break and how they
+can protect symmetry-protected topological phases. An
+important takeaway is to understand how exact emergent
+higher-form symmetries characterize phases, one should
+first partition the parameter space by both the theory’s
+exact emergent IR symmetries and not just its exact UV
+symmetries. These partitions do not necessarily resemble
+the system’s phase diagram and instead lay a foundation
+from which its phases can be labeled and characterized
+using generalized symmetries.
+In section V, we conclude and discuss some open ques-
+tions arising from this work.
+The appendix includes three additional sections which
+supplement the main text. Firstly, in appendix section A,
+we review the discrete differential geometry notation (in a
+non-rigorous fashion) used throughout section III. Then,
+appendix sections B and C show how the deconfined
+phases of the models for emergent ZN p-gauge theory and
+emergent U(1) p-gauge theory, respectively, are related
+to the corresponding quantum field theory descriptions:
+p-form BF theory and p-form Maxwell theory. The re-
+lated subsections review the higher-form symmetries and
+’t Hooft anomalies in these quantum field theories.
+II.
+ENERGY SCALES HIERARCHIES AND
+EFFECTIVE HAMILTONIANS
+A.
+Low energy exact emergent generalized
+symmetries
+Consider a lattice bosonic quantum system described
+by the local Hamiltonian H and whose total Hilbert space
+V is tensor product decomposable: V = �
+i Vi. Since H
+includes the exact interactions at the microscopic scale
+and describes the system at all energies throughout the
+entire parameter space, we refer to it as the UV Hamilto-
+nian, adopting the language used in field theory. While,
+in theory, the system’s physical properties can be ex-
+tracted from H, this proves much too difficult in prac-
+tice [53].
+A guiding principle to overcome this daunting prob-
+lem is the separation of energy scales (assuming there is
+no UV/IR mixing [54–57]). We will always denote the
+lowest energy scale as EIR and refer to the sub-Hilbert
+space VIR = span{|En⟩ | En < EIR}, where |En⟩ is an
+energy eigenstate, as the IR. Furthermore, we will al-
+E(III)
+UV
+E(IV)
+UV
+E(IV)
+mid−IR
+E(IV)
+IR
+E(I)
+UV
+E(I)
+mid−IR I
+E(I)
+IR
+E(I)
+mid−IR II
+E(II)
+UV
+E(II)
+IR
+I
+IV
+II
+III
+Parameter Space
+FIG. 1. The parameter space of a many-body Hamiltonian
+can be partitioned by its differing hierarchies of energy scales.
+A schematic depiction of this is shown here. The parameter
+space is partitioned into four regions, labeled I, II, III, and
+IV, with their differing energy scale hierarchies shown.
+ways denote the largest possible energy value as EUV.
+However, there can be other interesting energy scales
+between the IR and the UV scales, which we will call
+mid-IR energies Emid-IR and refer to the sub-Hilbert
+space Vmid-IR = span{|En⟩ | En < Emid-IR} as the mid-
+IR. Generally, there can be multiple of these mid-IR
+scales, and as demonstrated in Fig. 1, different regions
+of parameter space will have a different hierarchy of en-
+ergy scales. One reason to introduce the notion of the
+mid-IR is to make the connection between lattice models
+and quantum field theories clearer. Indeed, in condensed
+matter physics, the UV theory is a lattice model and
+quantum field theories are generally mid-IR theories.
+It is helpful to organize a quantum many-body system
+around its scale hierarchies because the low-energy eigen-
+states will often have additional structures absent from
+high-energy eigenstates. For example, these additional
+structures could reflect the presence of new, emergent
+symmetries. Indeed, as depicted in Fig. 2, states with
+energy EIR ≤ E < Emid-IR have equal or more symmetry
+than states with E ≥ Emid-IR but equal or less symmetry
+than states with E < EIR.
+It is nontrivial to systematically identify the scale hi-
+erarchies of a general UV Hamiltonian. Here we will spe-
+cialize to a typical situation where the UV Hamiltonian
+can be written as
+H = H0 + H1,
+(1)
+where a mid-IR scale Emid-IR of H0 is known (e.g., an en-
+ergy gap of a quasiparticle), and V(H0)
+mid-IR is spanned by en-
+ergy eigenstates of H0 satisfying ⟨Pi⟩ = 0,3 where {Pi} is
+a collection of mutually commuting local projection oper-
+ators. In other words, {Pi} act on |ψmid-IR⟩ ∈ V(H0)
+mid-IR as
+3 Here, we will use the notation that an operator with the subscript
+i acts only on degrees of freedom near site i.
+
+4
+Pi |ψmid-IR⟩ = 0, while Pi |ψ⟩ ̸= 0 for |ψ⟩ ̸∈ V(H0)
+mid-IR. Fol-
+lowing the previous discussion, the low energy physics of
+H0 below Emid-IR can have additional structures which
+are determined by Pi = 0. In fact, as we will soon ar-
+gue, these special structures commonly include emergent
+symmetries.
+Now, let us include the perturbation H1 in Eq. (1).
+We will assume H1 includes terms that mix states with
+⟨Pi⟩ = 0 and states with ⟨Pi⟩ ̸= 0. Because of H1, en-
+ergy eigenstates of H are a superposition of states with
+⟨Pi⟩ = 0 and states with ⟨Pi⟩ ̸= 0 and, therefore, the sub-
+Hilbert space spanned by states satisfying ⟨Pi⟩ = 0 is not
+a mid-IR of H. Consequently, it appears that any spe-
+cial structures arising due to Pi = 0 (such as emergent
+symmetry) are destroyed by the H1 term.
+On the other hand, if all parameters in H1 are much
+smaller than those in H0, it is tempting to think that
+the mid-IR of H is closely related to the ⟨Pi⟩ = 0 states.
+This intuition motivates one to introduce the parameter
+s ∈ [0, 1] and family of Hamiltonians
+H(s) = H0 + s H1,
+(2)
+from which the mid-IR of H can be constructed from
+the mid-IR of H0 by slowly tuning s = 0 to s = 1 [46].
+Indeed, let us denote the nth many-body energy eigen-
+state of H(s) as |ψ(s)
+n ⟩ and define the unitary operator
+Vs = �
+n |ψ(s)
+n ⟩⟨ψ(0)
+n | which satisfies Vs|ψ(0)
+n ⟩ = |ψ(s)
+n ⟩ and
+⟨ψ(0)
+n |A|ψ(0)
+n ⟩ = ⟨ψ(s)
+n |VsAV †
+s |ψ(s)
+n ⟩
+(3)
+for any operator A. Therefore, the ⟨P⟩ = 0 sector of H0
+is related to the ⟨V1PV †
+1 ⟩ = 0 sector of H. However, this
+definition of Vs is unphysical since VsPV †
+s is likely to
+be nonlocal even if P is local.
+Ref. 46 found a local
+unitary operator ULU that approximates Vs very well,
+while ensuring local operators remain local when dressed.
+An explicit form of ULU is [58]
+ULU = S′
+�
+exp
+�
+i
+� 1
+0
+ds′ Ds′
+��
+,
+Ds ≡ i
+�
+dt F(t)eiH(s)t∂sH(s)e−iH(s)t,
+(4)
+where S′ denotes s′-ordering and F(t) is a function sat-
+isfying a particular set of requirements, such as F(t) =
+−F(−t) so that Ds is anti-Hermitian.
+Motivated by those results, here we assume that there
+exists a proper local unitary operator ULU with the fol-
+lowing properties: (1) it maps a local operator Oi to
+a local operator �Oi ≡ ULUOiU †
+LU (with some fattening);
+(2) it maps the nth eigenstate |ψ(0)
+n ⟩ of H(0) with eigen-
+value E(0)
+n
+to a superposition of some eigenstates |ψ(1)
+n′ ⟩ of
+H(1) with eigenvalue E(1)
+n
+− δ < E(1)
+n′ < E(1)
+n
++ δ, where
+δ ≪ Emin-IR and E(1)
+n
+is the eigenvalue of the nth eigen-
+state |ψ(1)
+n ⟩ of H(1). If such a unitary operator satisfying
+the aforementioned properties does not exist for a partic-
+ular H in parameter space, it means that the mid-IR does
+not exist at that point of parameter space (e.g., due to
+the gapped quasiparticles defining Emid-IR condensing).
+We will obtain our results based on this conjecture.
+Consider an eigenstate |ψ(0)
+n ⟩ of H(0) with eigenvalue
+E(0)
+n
+≪ Emid-IR. Thus |ψ(0)
+n ⟩ satisfies Pi|ψ(0)
+n ⟩ = 0. ULU
+will map it to some eigenstates of H(1) with eigenvalues
+much less then Emid-IR, which satisfy �Pi|ψ(1)
+n ⟩ = 0, where
+�Pi = ULUPiU †
+LU. The above is true for all the low energy
+eigenstates of H(0). We can therefore identify the mid-
+IR of H as the sub-Hilbert space spanned by the mid-IR
+eigenstates of H(0) transformed by ULU. Furthermore,
+since the mid-IR states of H(0) satisfied Pi = 0, the mid-
+IR states of H satisfy �Pi = 0. Notice that { �Pi} is also a
+set of mutually commuting local projectors whose �Pi = 0
+space corresponds to the low energy sub-Hilbert space of
+H, V(H)
+mid-IR. Thus the exact low energy structures of H0
+specified by the projectors Pi become the exact low en-
+ergy structures of H specified by the projectors �Pi. Since
+{Pi} and { �Pi} are related by a local unitary transforma-
+tion ULU, we exact the two exact low energy structures
+are equivalent. This result was obtained in Ref. 46 for
+emergent Z2 and U(1) gauge symmetry (i.e. the low en-
+ergy subspace satisfies that Gauss’s law �ρi = 0 exactly
+and is exactly gauge invariant.) We believe such a result
+remains to be valid for more general situations.
+Having identified a mid-IR of H, we now discuss how
+to determine if, and what kind of, additional structures
+emerge at E < Emid-IR. In this paper, we will investi-
+gate the scenario where these additional structures are
+emergent symmetries. To identify emergent symmetries,
+it is helpful to adopt the perspective that a symmetry is
+described/defined by an algebra of local symmetric oper-
+ators [52]. For instance, if the UV symmetries are gener-
+ated by the unitaries {Ug}, i.e. [Ug, H0] = [Ug, H1] = 0,
+then the associated algebra of local symmetric operators
+is
+AUV = {OUV | OUVUg = UgOUV ∀ g},
+(5)
+where OUV is a local operator acting on the full Hilbert
+space V. Indeed, given A, one can recover the symme-
+try transformation operators by finding all operators that
+commute with the elements of A while also acting non-
+trivially on some physical operators4.
+Using this view, the mid-IR symmetries are described
+by the algebra of local symmetric operators
+Amid-IR = {Omid-IR | Omid-IR �Pi = �PiOmid-IR ∀ i,
+∀ Omid-IR ∈ AUV}.
+(6)
+Indeed, the symmetry transformations of the symmetry
+are given by all the operators that commute with Amid-IR:
+Tmid-IR = {Umid-IR | Umid-IROmid-IR = Omid-IRUmid-IR,
+∀ Omid-IR ∈ Amid-IR}.
+(7)
+4 The latter requirement is required to avoid including gauge re-
+dundancies.
+
+5
+GUV ⊂ Gmid−IR ⊂ GIR
+EUV
+Emid−IR
+EIR
+UV, GUV
+mid-IR, Gmid−IR
+IR, GIR
+Energy
+FIG. 2.
+The symmetries of a quantum many-body system
+generally depend on the energy scale of an observer. While
+microscopic (UV) symmetries GUV are noticeable at all en-
+ergy scales, at lower energies an observer may discover addi-
+tional, emergent symmetries. These emergent symmetries can
+be ordinary symmetries, anomalous symmetries, higher-form
+symmetries, non-invertible symmetries, etc. [2, 6, 7].
+Tmid-IR will include the UV symmetries but could include
+additional emergent symmetries, reflecting the possibil-
+ity depicted in Fig. 2. In fact, it should generally be the
+case that the emergent symmetries of H determined by
+the algebra Amid-IR are exact symmetry of H0. We refer
+to emergent symmetries identified by Amid-IR as exact
+emergent symmetries to emphasize that at E < Emid-IR,
+they are equally impactful as exact UV symmetries. In-
+deed, exact emergent symmetries are not approximate,
+and Amid-IR cannot describe approximate symmetries.
+The above description of symmetry using AUV is very
+general and capable of describing all generalizations of
+symmetries, with or without ’t Hooft anomalies. Further-
+more, for finite symmetries, an algebra of local symmet-
+ric operators determines a non-degenerate braided fusion
+(higher) category (i.e. a topological order in one higher
+dimension), which is referred to as categorical symmetry
+and is a direct description of the symmetry [19, 52]. Iden-
+tifying exact emergent symmetries using Amid-IR, speci-
+fied by local commuting projectors �Pi, includes all types
+of emergent generalized symmetries but does not include
+emergent 0-form symmetries. Indeed, emergent 0-form
+symmetries are usually not exact and instead approxi-
+mate symmetries.
+On the other hand, as long as the
+lattice is free from defects (e.g., no lattice sites are miss-
+ing), emergent higher-form symmetries are always exact.
+We will provide a more detailed discussion on this point
+in the next subsection and next section.
+One can possibly discover all of the emergent symme-
+tries of a system in a Hamiltonian-independent way by
+constructing A at all energy scales for each energy hierar-
+chy in parameter space. However, it is desirable to have
+a Hamiltonian description reflecting the emergent sym-
+metries. The symmetries that emerge at E < Emid-IR are
+hidden from the UV Hamiltonian H since it describes the
+dynamics of both states with �Pi = 0 and �Pi = 1. There-
+fore, to make the emergent symmetries manifest, we will
+develop an effective mid-IR theory Hmid-IR that describes
+only the dynamics of states with �Pi = 0. The symmetries
+of Hmid-IR should include the UV symmetries but could
+also include additional ones, which we will identify as
+emergent symmetries. Since H is a sum of only opera-
+tors in AUV, we expect that Hmid-IR should be a sum of
+only operators in Amid-IR.
+The effective mid-IR Hamiltonian should act only on
+the mid-IR Hilbert space Vmid-IR.
+Here, we develop
+Hmid-IR using the physical intuition that the UV dynam-
+ics generate the mid-IR dynamics. Indeed, decomposing
+the UV theory as a sum over local terms H = �
+i H(i),
+consider the amplitude
+⟨ψ| eiH |ψ⟩ =
+∞
+�
+n=0
+in
+n!
+�
+i0···in
+⟨ψ| H(i0) · · · H(in) |ψ⟩ .
+(8)
+When |ψ⟩ ∈ Vmid-IR,
+we require that Hmid-IR satis-
+fies ⟨ψ| eiHmid-IR |ψ⟩ = ⟨ψ| eiH |ψ⟩ under the constraint
+that all terms in Hmid-IR commute with { �Pi}.
+No-
+tice that since the amplitudes must match, terms in
+Hmid-IR can only be constructed from {H(i0) · · · H(in)}
+for n = 0, 1, · · · , ∞.
+Therefore, the most general form
+of Hmid-IR is a sum of operators constructed from all
+combinations of the terms in H, or equivalently �H, that
+commute with �Pi. Letting A denote the set of all of these
+operators, we define Hmid-IR as
+Hmid-IR =
+�
+O∈A
+COO,
+(9)
+where {CO} are constants chosen such that the ampli-
+tudes match. Therefore, as expected, Hmid-IR is a sum
+over operators in Amid-IR, but A ⊂ Amid-IR since A in-
+cludes only those which can generated from the terms in
+�H.
+The constants {CO} are renormalized versions of the
+UV parameters. On physical grounds, we require the ef-
+fective mid-IR theory to be a local Hamiltonian. There-
+fore, the greater number of terms from �H involved in
+O or the larger the region of the lattice O acts on, the
+smaller CO is. The ability to define a local effective mid-
+IR Hamiltonian is a requirement of the mid-IR to be well
+defined. This definition of the effective mid-IR Hamilto-
+nian is physically reasonable but not rigorous. We will
+not present a rigorous justification or proof of Hmid-IR,
+leaving it for future work. Here, we state Eq. (9) with
+the restrictions on CO as the conjectured form of Hmid-IR,
+and the rest of this paper is dedicated to examining the
+consequences of this conjecture.
+B.
+A holographic picture for emergent finite
+symmetries
+One way to obtain a systematic understanding of
+strongly correlated gapless states is to find all their uni-
+versal characterizations. Low-energy emergent symme-
+tries provide a possible candidate for such universal char-
+acterizations. As mentioned previously, emergent sym-
+
+6
+M
+QFTano
+FIG. 3.
+A symmetric system with symmetry R, after be-
+ing restricted in the symmetric sub-Hilbert space in a closed
+space, can be viewed as a system (denoted as QFTano) with a
+non-invertible gravitational anomaly. Low energy properties
+of QFTano can be exactly simulated by a boundary of a topo-
+logical order M in one higher dimension. QFTano uniquely
+determines the bulk topological order M.
+metries can be very general. They can include group-
+like symmetries, anomalous symmetries, (anomalous)
+higher-form symmetries, (anomalous) algebraic higher-
+form symmetries (also called non-invertible symmetries),
+etc. When these symmetries are finite, Refs. 4 and 51
+proposed a unified description of all types of symmetries
+in terms of topological orders in one higher dimension,
+called categorical symmetry5 [4, 50] or symmetry topo-
+logical field theory (TFT) [59]. This has the following
+physical meaning: given an anomaly-free system6 (de-
+noted as QFTaf) with a symmetry R, its low energy
+properties within the symmetric sub-Hilbert space in a
+closed space are exactly simulated by a boundary of the
+corresponding topological order M in one higher dimen-
+sion. This is because, when restricted to symmetric sub-
+Hilbert space in a closed space, the system can be viewed
+as a system (denoted as QFTano) with a non-invertible
+gravitational anomaly [50, 60], which is the same as topo-
+logical order M in one higher dimension [61, 62] (see
+Fig. 3).
+Certainly, there are many physically distinguishable
+systems with the same finite symmetry R.
+Neverthe-
+less, for each such system, we can find a boundary of M
+to exactly simulate it at low energies, since M can also
+have many physically distinguishable boundaries. Thus
+using bulk topological order M to described a symmetry
+R means that there is an one-to-one correspondence be-
+tween systems with the symmetry R and boundaries of
+bulk topological order M, such that the local low energy
+properties of the corresponding symmetric system and
+boundary of M are identical. For example, the low en-
+ergy spectrum in the symmetric sub-Hilbert space of the
+symmetric system is identical to the low energy spectrum
+of the corresponding boundary of M.
+The boundary of M in Fig. 3 does not simulate all of
+QFTaf’s low energy properties since QFTano only de-
+scribes the symmetric sub-Hilbert space of QFTaf. In-
+deed, truly simulating QFTaf requires the boundary of
+5 This
+is
+not
+to
+be
+confused
+with
+non-invertible
+symme-
+try/algebraic higher-form symmetry, which are sometimes also
+referred to as categorical symmetry.
+6 Throughout section II B, an anomaly-free system means a system
+with a lattice UV completion.
+R
+R
+M
+M
+=
+QFT
+QFTano
+QFTano
+af
+FIG. 4.
+The low energy properties QFTaf with symmetry
+R can be exactly simulated by a slab of topological order M
+with two boundaries QFTano and �
+R. The stacking of the two
+boundaries through the bulk topological order is denoted as
+QFTano ⊠M �
+R.
+M to capture all states in the Hilbert space. This can
+be achieved by adding an additional gapped boundary
+�R of M [4, 8], as shown in Fig. 4. We will denote the
+composition of the topological order M with two bound-
+aries QFTano and �R as QFTano ⊠M �R. Provided that
+the topological order M and the boundary �R have an
+infinite energy gap, the low-energy properties of QFTaf
+are described by the composite system QFTano ⊠M �R,
+and thus
+QFTaf = QFTano ⊠M �R.
+(10)
+If an anomaly-free system QFTaf admits a decomposi-
+tion QFTaf = QFTano ⊠M �R, then we say the anomaly-
+free system QFTaf is described by the categorical sym-
+metry M.
+The boundary �R contains gapped excitations that can
+generally be of various dimensions. If the spatial dimen-
+sion of the boundary is n, these excitations are described
+by a fusion n-category, which we will denote as �R. On
+the other hand, the bulk topological order M in (n + 1)-
+dimensional space will generally also have numerous bulk
+excitations of various dimensions, which are described by
+a braided fusion n-category denoted as M. The bound-
+ary fusion n-category �R uniquely determines the bulk
+braided fusion n-category M. In fact, M and �R are re-
+lated to one another by
+M = Z( �R),
+(11)
+where Z( �R) is the center of �R. When n = 1, Z( �R) is
+the Drinfeld center of �R and, if �R = Rep(G), M is the
+quantum double of �R.
+In the decomposition QFTaf = QFTano ⊠M �R, if �R is
+a local fusion n-category7, then QFTaf has an anomaly-
+free symmetry described by R, the dual of �R. Because R
+7 A fusion n-category �
+R is local if there exists a fusion n-category
+R such that Z(R) = Z( �
+R) and R ⊠M �
+R = nVec, where nVec is
+the braided fusion n-category describing excitations in a trivial
+topological order (i.e. above a trivial product state). The two
+local fusion n-categories R and �
+R are then said dual to each
+other.
+
+7
+and �R are dual, the anomaly-free symmetry R also deter-
+mines the categorical symmetry M = Z(R). However, if
+�R is not local, we cannot say QFTaf has an anomaly-free
+symmetry, although its symmetries are still described by
+the categorical symmetry M.
+We note that in Ref. 8,
+the pair ( �R, M) is regarded as a generalized symmetry
+regardless if �R is local or not.
+Using QFTano ⊠M �R to describe the symmetries of
+QFTaf is very general and provides a unifying formal-
+ism capable of describing all generalizations of symme-
+try. In fact, as we will now argue, QFTano ⊠M �R is also
+able to describe the exact emergent symmetries of QFTaf
+discussed in the previous subsection. Therefore, one may
+view QFTano ⊠M �R as the definition of (exact emergent)
+symmetry.
+Recall from the previous subsection the general Hamil-
+tonian Eq. (1), where a mid-IR of H0 was known and
+spanned by states satisfying Pi |ψ⟩ = 0 for local com-
+muting projectors {Pi}. There, we found that the ex-
+act emergent symmetries in the mid-IR are determined
+by {Pi}. The exact emergent symmetries of H could be
+found by dressing operators with ULU. This is of course
+still true here, but for simplicity we will just consider H0
+instead of the full H0 + H1. The results will hold even af-
+ter we include an arbitrary perturbation H1 to H0 as we
+discussed in the last subsection. Without a loss of gen-
+erality, we will assume that H0 for the finite symmetry
+case has the form
+H0 =
+�
+i
+Oi + Emid-IR
+�
+i
+Pi,
+[Oi, Pj] = 0.
+(12)
+Here Oi are local operators and [Oi, Pj] = 0 is required,
+since Oi is assumed not to mix the Pi = 0 states and
+Pi = 1 states.
+Let us now consider how exact emergent symmetries
+arise from Pi = 0 starting from ( �R, M) which is related
+to the physical theory H0 by QFTano ⊠M �R. In doing so,
+we will also see why 0-form symmetries cannot be exact
+emergent symmetries.
+It is believed that a topological order with a gapped
+boundary can be realized by commuting projector model,
+which also has a commuting projector Hamiltonian real-
+izing the gapped boundary. Therefore, the �R-boundary
+and the M-bulk in Fig. 4 can be realized by a commuting
+projector model. Pi in H0 are those commuting projec-
+tors, where H0 is is viewed as a Hamiltonian that de-
+scribes the slab in Fig. 4 which is a model in one lower
+dimension if the slab has a finite thickness compare to
+lattice spacing. Oi’s in H0 are the boundary Hamilto-
+nian terms describing the QFTano boundary in Fig. 4.
+Since the thickness of the bulk M is finite, the local op-
+erators Oi in H0 can also contain operators that connect
+the QFTano boundary and the �R boundary in Fig. 4. By
+definition, these Oi must commute with Pi in order to be
+allowed in H0. If a symmetry on the boundary QFTano
+is going to be explicitly broken, there must be an al-
+lowed operator which transfers symmetry charge from �R
+R
+QFTano
+FIG. 5. A lattice realization of Fig. 4, where qubits live on
+the links. The bulk Hamitonian contains the star terms (the
+diamonds around a vertex) and the plaquette terms (the di-
+amonds inside a square). The intra-boundary terms act on
+qubits near a boundary, while the inter-boundary terms act
+on qubits that connect two boundaries (the vertical line).
+to QFTano. For a p-form symmetry, its charges are cre-
+ated by operators acting on a p-dimensional subspace,
+and therefore transferring a symmetry charge from �R to
+QFTano requires a (p + 1)-dimensional operator. For a
+0-form symmetry, such an operator would include a fi-
+nite number of local operators acting in a line from �R to
+QFTano. This is allowed in the set of local operators {Oi}
+and therefore allowed in H0.
+Thus, the projectors Pi
+cannot produce exact emergent 0-form symmetries. For
+p-form symmetries with p > 0, any inter-boundary oper-
+ators that transfer symmetry charge are non-contractible
+extended operators, acting on the whole system. These
+operators are not local and are not included in the set of
+local operators {Oi} and therefore the symmetry is pre-
+served. Thus the projectors Pi can give rise to an exact
+emergent higher-form symmetry even when including all
+the local low energy operators Oi (regardless if they are
+inter-boundary or intra-boundary).
+The above discussion is pretty general, so let us give
+an example to construct a model with an exact emergent
+Z(1)
+2
+symmetry in 1+1D. We consider M to be the toric
+code model (i.e. the Z2 lattice gauge theory) and �R is
+the Z2-charge condensed boundary (the so-called rough
+boundary [63]). The slab QFTano ⊠M �R in Fig. 4 be-
+comes the lattice shown in Fig. 5, with qubits residing
+on the links.
+The bulk Hamiltonian that give rise to a toric code
+model contains star terms Pvert
+i
+= (1 − Z1Z2Z3Z4)/2
+acting on the four qubits on the four links con-
+necting
+to
+each
+vertex
+and
+plaquette
+terms
+Pplaq
+i
+= (1 − X1X2X3X4)/2 acting on the four qubits
+on the four links around each square (see the diamond
+around a vertex and inside a square, respectively, in
+Fig. 5). The �R boundary has truncated plaquette terms
+Pbdry
+i
+= (1 − X1X2X3)/2 acting on the three qubits
+on the three links around each open square (see the
+up-side-down triangle in Fig. 5 in the �R boundary).
+The lattice model Fig. 5 can be viewed as a 1+1D
+lattice model H0, and the above three types of terms,
+Pvert
+i
+, Pplaq
+i
+, and Pbdry
+i
+, correspond to the projectors Pi
+in Eq. (12). The Oi operators in H0 can be any operator
+that commutes with Pi. Thus the local energy dynamics
+
+8
+R
+i+1/2
+i
+i+1
+i−1
+i−1/2
+QFTano
+FIG. 6. A thin slab limite of Fig. 5, where qubits live on the
+links.
+controlled by Oi operators are constrained by the local
+projectors Pi’s. Consequently, the local projectors Pi’s
+cam give rise to an emergent symmetry. But what is this
+emergent symmetry?
+We note that the Oi operators can be divided into two
+classes:
+intra-boundary operators and inter-boundary
+operators.
+The intra-boundary operators only act on
+the qubits near the boundary QFTano8.
+The inter-
+boundary operators can act on qubits that connect the
+two boundaries. An example of inter-boundary opera-
+tors is X1X2 · · · Xn acting on the vertical links connect-
+ing the two boundaries (see Fig. 5), which commute with
+the projectors Pvert
+i
+, Pplaq
+i
+, and Pbdry
+i
+.
+To make our explicit discussion as simple as possi-
+ble, let us take a thin slab limit of Fig. 5, which gives
+us Fig. 6 where we label vertical (horizontal) links by
+i (i + 1
+2).
+In this limit, the only remain projector is
+Pbdry
+i
+:
+Pi = 1
+2(1 − XiXi+ 1
+2 Xi+1).
+The allowed local
+boundary operators Oi must commute with Pi’s. From
+the thin slab limit, we find that all such Oi’s are gen-
+erated by taking products of the operators Xi, Xi+ 1
+2 ,
+and Zi− 1
+2 ZiZi+ 1
+2 . Notice that while Xi+ 1
+2 , XiXi+ 1
+2 Xi+1,
+Zi− 1
+2 ZiZi+ 1
+2 are intra-boundary operators, the Xi’s are
+inter-boundary operators.
+Let us first restrict ourselves to the local operators Oi
+that are intra-boundary operators. We will consider the
+full set up, where Oi includes intra and inter-boundary
+operators, after.
+In this case, we will obtain an exact
+emergent Z2 0-form symmetry.
+To see this result, we
+note that the intra-boundary local operators form the
+algebra of the local symmetric operator
+A(intra-only)
+mid-IR
+= {Xi+ 1
+2 , XiXi+ 1
+2 Xi+1, Zi− 1
+2 ZiZi+ 1
+2 }.
+(13)
+The operators (local or non-local) that commute with
+A(intra-only)
+mid-IR
+are generated by
+T (intra-only)
+mid-IR
+= {XiXi+ 1
+2 Xi+1,
+�
+i
+(Zi− 1
+2 ZiZi+ 1
+2 )}. (14)
+8 The commuting projectors Pbdry
+i
+etc already give the �
+R bound-
+ary a large energy gap. The intra-boundary operators near �
+R
+can only be combinations of those commuting projectors. There
+are no new operators.
+and give rise to all symmetry transformations. Using the
+language of operator algebra, the symmetry transforma-
+tions T (intra-only)
+mid-IR
+are the double commutant of the local
+projectors {Pi}. From T (intra-only)
+mid-IR
+, we find that when re-
+stricted to the intra-boundary operators there are two ex-
+act emergent symmetries enforced by the projectors Pi’s.
+XiXi+ 1
+2 Xi+1 generates symmetry transformations which
+act on loops of the slab, and therefore corresponds to a
+Z(1)
+2
+symmetry. �
+i(Zi− 1
+2 ZiZi+ 1
+2 ) acts on the entire lat-
+tice and therefore corresponds to a Z(0)
+2
+symmetry. Note
+that when the (2 + 1)D system QFTano ⊠M �R is mapped
+to the (1 + 1)D system QFTaf, the Z(0)
+2
+symmetry still
+acts on the entire lattice while the Z(1)
+2
+symmetry now
+acts on a single lattice site.
+From the previous general discussion, we expect that
+exact emergent 0-form symmetries will be explicitly bro-
+ken by inter-boundary operators.
+Let’s verify this ex-
+pectation in this example by now including the inter-
+boundary operators. Doing so, the algebra of local sym-
+metric operators becomes
+Amid-IR = {Xi+ 1
+2 , Xi, Zi− 1
+2 ZiZi+ 1
+2 }.
+(15)
+The symmetry transformations and now generated by
+Tmid-IR = {XiXi+ 1
+2 Xi+1}.
+(16)
+So, the Z(1)
+2
+symmetry is still present, but the Z(0)
+2
+sym-
+metry is now explicitly broken. This is because the inter-
+boundary operators Xi’s also commute with the projec-
+tors Pi but transfer the charges of the Z(0)
+2
+symmetry
+between the two boundaries.
+As a result, those inter-
+boundary operators break the Z(0)
+2
+symmetry previously
+enforced by the projectors Pi’s. The operators that could
+transform the Z(1)
+2
+symmetry charges between the two
+boundaries were not included in Amid-IR since they are
+not local operators, and as a result, the Z(1)
+2
+symmetry
+is exact.
+III.
+EXAMPLES OF EXACT EMERGENT
+HIGHER-FORM SYMMETRIES
+In this section, we apply the framework discussed in
+section II A to three lattice models.
+We organize our
+discussion around the energy hierarchies of these models
+and discover emergent higher-form symmetries by deriv-
+ing effective Hamiltonians. These symmetries have been
+noticed previously in the literature in similar models, but
+typically from an IR point of view. Here we take a UV
+point of view, starting from lattice models for which these
+symmetries are not exact, and emphasizing that as emer-
+gent symmetries they are exact emergent symmetries.
+This means that they are symmetries that emerge be-
+low an energy scale yet constrain the IR as if they were
+UV symmetries. Each subsection is dedicated to a single
+
+9
+K
+U
+I
+III
+II
+J
+U
+E(I)
+UV
+UV
+E(II)
+UV
+E(II)
+IR
+UV
+IR
+E(III)
+UV
+E(III)
+mid−IR
+E(III)
+IR
+UV
+mid-IR
+IR
+Model C:
+Model D:
+GUV
+GUV
+GUV
+GIR = GUV × U(1)(p)
+GUV
+GIR = GUV × ℤ(p)
+N
+GUV
+Gmid−IR = GUV × U(1)(p)
+GIR = GUV × U(1)(p) × U(1)(d−p−1)
+GUV
+Gmid−IR = GUV × ℤ(p)
+N
+GIR = GUV × ℤ(p)
+N × ℤ(d−p)
+N
+Model D:
+Model C:
+Model C
+Model D
+ Parameter Space
+HUV
+FIG. 7. (left) We partition the parameter space of models C (Eq. (66)) and D (Eq. (35)) into three regions labeled I, II, and
+III, which we structure our discussions in sections III B and III C around. (right) In these regions, we identify energy scale
+hierarchies and the exact emergent symmetries at each scale. The exact emergent U(1)(d−p−1) symmetry in the IR of region
+III for model C is trivial at the lattice scale, but its action becomes nontrivial in the continuum limit. These regions are not
+necessarily distinct phases of the models. Indeed, region I and II are likely in the same phase while region III is a different
+phase. This is emphasized in the left figure where solid lines indicate a phase transition while dashed lines do not.
+one of these models and is entirely self-contained. So, if
+desired, the reader should feel free to read only the sub-
+section(s) on their favorite model(s). For the reader who
+reads more than one, we apologize for any inconvenience
+due to subsections overlapping.
+All of the models we consider are described by Hamil-
+tonians governing degrees of freedom on a d-dimensional
+cubic spatial lattice with continuous time.
+We exten-
+sively use discrete differential geometry notation, which
+we review (in a non-rigorous fashion) in appendix sec-
+tion A. The remainder of this section is organized as fol-
+lowed.
+In subsection III A, as a warm-up we consider the quan-
+tum clock model Eq. (20) which has a UV ZN symmetry.
+When this ZN symmetry is spontaneously broken, we
+find there is an exact emergent Z(d)
+N symmetry at energies
+below the domain wall gap. The IR symmetry operators
+form a projective representation of ZN × Z(d)
+N , signaling
+the presence of a mixed ’t Hooft anomaly that protects
+the ground state degeneracy in the SSB phase.
+In subsection III B, we consider a model of emergent
+ZN p-gauge theory Eq. (35), which we call model D, and
+in subsection III C, we consider a model of emergent U(1)
+p-gauge theory Eq. (66), which we call model C. The ex-
+act emergent symmetries and energy scale hierarchies of
+these models are summarized in Fig. 7. We want to em-
+phasize that the left panel of Fig. 7 is a schematic depic-
+tion and that the precise shapes of the regions and the
+nature of the boundaries between them are not investi-
+gated. Region II ∪ III exists in parameter space wherever
+there are gapped gauge charge excitations. The creation
+operator for the gauge charges is exactly known in the
+K, J → 0 limit, and the implicitly defined local unitary
+ULU from section II A is used to construct the creation
+operator away from this single point in parameter space.
+We do not construct an explicit form for ULU and thus
+cannot rigorously investigate the shape of region II ∪ III
+in parameter space. Since region III corresponds to the
+deconfined phase of the gauge theory, assuming d is large
+enough that the exact emergent p-form symmetry can
+spontaneously break, we expect it to be a finite-sized re-
+gion in the thermodynamic limit. Region II corresponds
+to the confined phase, where the gauge charges are con-
+fined. When J = 0, it is easy to confirm the exact emer-
+gent p-form symmetry is present, but for finite J, it re-
+lies on the existence of ULU and whether or not the gauge
+charges are genuinely gapped excitations. Fig. 7 portrays
+the possibility that they are and that region II exists
+even when J ̸= 0, and the discussion throughout these
+examples will reflect this possibility. Another possibility
+is that region II exists only for J → 0; outside of the
+deconfined phase, the exact emergent symmetry would
+then only exist along a portion of the vertical axis of the
+parameter space.
+Let us summarize the results of model C and D in
+the familiar case p = 1. We first consider model D with
+p = 1, d = 3, and trivial GUV, which is emergent 3+1D
+Z2 lattice gauge theory. In Eq. (35), the Gauss’s law is
+
+10
+enforced at a large energy scale U which represents the
+energy gap of a Z2-gauge charge excitation. Furthermore,
+the term creating a pair of Z2-charges has a energy scale
+J (which explicitly breaks a UV Z(1)
+2
+symmetry) and the
+term creating Z2-fluxes has an energy scale K (which
+explicitly breaks a UV Z(2)
+2
+symmetry). Thus a large J
+drives a Z2-charge condensation transition (i.e. a Higgs
+transition) and a large K drives a Z2-flux condensation
+transition (i.e. a confinement transition). The region III
+in Fig. 7 corresponds to the deconfined phase of the Z2
+gauge theory, which has an exact emergent IR symmetry
+Z(1)
+2
+×Z(2)
+2
+(below the energy gap of dressed Z2-charges).
+The symmetry operators of the Z(1)
+2
+symmetry and Z(2)
+2
+symmetry anti-commute if their intersect at odd number
+of points.
+Thus model D realizes the exact emergent
+Z(1)
+2
+× Z(2)
+2
+in a projective representation, and therefore
+there is a mixed ’t Hooft anomaly between Z(1)
+2
+and Z(2)
+2
+symmetries. Such a mixed anomaly can be described by
+anomaly in-flow using an invertible topological quantum
+field theory in one higher dimension, described by the
+following action amplitude (see Eq. (B57))
+e−S = eiπ
+�
+M5 A∪ ˆ
+A,
+(17)
+where A is a Z2-valued 2-cocycle field and ˆ
+A is a Z2-
+valued 3-cocycle field. A is the background gauge field
+from gauging the Z(1)
+2
+symmetry, while ˆ
+A is the back-
+ground gauge field from gauging the Z(2)
+2
+symmetry. We
+note that, following the discussion in section (II B), once
+making A and ˆ
+A dynamical, Eq. (17) can also be viewed
+as the action amplitude in the path integral describing
+the categorical symmetry (i.e. the topological order in
+one higher dimension) for the anomalous Z(1)
+2
+× Z(2)
+2
+sym-
+metry.
+Due to the mixed ’t Hooft anomaly, the system must
+have degenerate ground states on a 3-dimensional torus
+as long as Z(1)
+2
+× Z(2)
+2
+is present in the IR. Since the
+emergent symmetry Z(1)
+2
+× Z(2)
+2
+remains to be exact
+against local UV perturbations, the ground states on a
+3-dimensional torus also remain degenerate against any
+perturbations. This is a way to understand the robust-
+ness of topological order using the robustness of exact
+emergent symmetry.
+The exact emergent mid-IR Z(1)
+2
+symmetry (existing
+at energies below ∼ U) is present on both sides of the
+confinement transition, II ↔ III, and is a consequence of
+weak Z2 charge fluctuations. The exact emergent mid-IR
+symmetry Z(1)
+2
+controls this transition and its unphysical
+part corresponds to the exact emergent Z2 gauge redun-
+dancy [46].
+Next, let us consider model C with p = 1, d = 3, and
+trivial GUV, which is emergent 3+1D U(1) lattice gauge
+theory.
+In Eq. (66), the Gauss’s law is enforced at a
+large energy scale U which is the energy gap of a U(1)-
+gauge charge. A term creating a pair of U(1)-charge has
+a energy scale J and a term creating U(1)-flux fluctu-
+ation has an energy scale K. Thus a large J drives a
+U(1)-charge condensation transition (i.e. a Higgs transi-
+tion) and a large K drives a U(1)-monopole condensa-
+tion transition (i.e. a confinement transition). The re-
+gion III corresponds to the deconfined phase of the U(1)
+gauge theory, which has an exact emergent IR symmetry
+U(1)(1) × U(1)(1) (below the energy gap of U(1)-charges
+and U(1)-monopoles) in the continuum limit.
+There is a mixed ’t Hooft anomaly between the two
+U(1)(1) symmetries. It is described by an invertible topo-
+logical quantum field theory in one higher dimension (see
+Eq. (C77))
+e−S = ei2π
+�
+M5 A∧d ˆ
+A,
+(18)
+where A, ˆ
+A are two R/Z-valued 2-form fields. A is the
+background gauge field from gauging the first U(1)(1)
+symmetry, while ˆ
+A is the background gauge field from
+gauging the second U(1)(1) symmetry. Speculating that
+categorical symmetry can be generalized to continuous
+symmetry, once making A and ˆ
+A dynamical, Eq. (18)
+can also be viewed as the action amplitude describing the
+categorical symmetry for the anomalous U(1)(1)×U(1)(1)
+symmetry.
+A result of the mixed ’t Hooft anomaly of the exact
+emergent U(1)(1) × U(1)(1) symmetry is that as long as
+the U(1)-charges and U(1)-monopoles have large energy
+gap the system must be gapless [64, 65] no matter how
+strong of the interaction between the U(1) photons. This
+would mean that as long as these excitations have a
+large energy gap, then due the exact emergent anoma-
+lous U(1)(1) × U(1)(1) symmetry, the system should re-
+main gapless even when a strong interaction between the
+U(1) photons drives a phase transition.
+The exact emergent mid-IR U(1)(1) symmetry (exist-
+ing at energies below ∼ U) is present on both sides of
+confinement transition II ↔ III. Indeed, it controls the
+transition and its unphysical part corresponds to the ex-
+act emergent U(1) gauge redundancy [46].
+A.
+Quantum clock model
+In this section, we will apply the framework discussed
+in section II to the quantum clock model. Consider ZN
+quantum rotors residing on the 0-cells (sites) of the spa-
+tial d-dimensional cubic lattice. A ZN quantum rotor is
+an N-level system described by clock operators Xc0 and
+Zc0. These unitary operators are N-dimensional gener-
+alizations of the Pauli matrices, satisfying
+Z�c0Xc0 = ωδc0,�c0 Xc0Z�c0,
+XN
+c0 = ZN
+c0 = 1,
+(19)
+where ω ≡ ei2π/N. The eigenvalues of the clock opera-
+tors are {1, ω, ω2, · · · , ωN−1} ≃ ZN.
+
+11
+The Hamiltonian of the quantum clock model is
+H = −J
+2
+�
+⟨c0�c0⟩
+�
+X†
+c0X�c0 + X†
+�c0Xc0
+�
+− K
+2
+�
+c0
+�
+Zc0 + Z†
+c0
+�
+.
+(20)
+where the first sum is over pairs of nearest neighbor 0-
+cells c0 and �c0 and the second sum is over all 0-cells. This
+theory has an exact ZN 0-form—Z(0)
+N —symmetry, whose
+symmetry operator is generated by the unitary
+U =
+�
+c0
+Zc0.
+(21)
+The charged operators of this symmetry are Xc0, which
+from the clock operator algebra transform as
+Xc0 → UXc0U † = e2π i/NXc0.
+(22)
+1.
+An exact emergent Z(d)
+N
+symmetry and mixed ’t Hooft
+anomaly
+Assuming that d > 0, when K/J ≪ 1, the quantum
+clock model at zero temperature is in a Z(0)
+N spontaneous
+symmetry broken (SSB) phase. Indeed, in the tractable
+K/J → 0 limit, the quantum clock model is
+H
+����
+K/J→0
+= −J
+2
+�
+⟨c0�c0⟩
+�
+X†
+c0X�c0 + X†
+�c0Xc0
+�
+.
+(23)
+The
+ground
+state
+in
+this
+limit
+satisfies
+X†
+c0X�c0 |vac⟩ = |vac⟩
+for
+all
+neighboring
+0-cells.
+Consequently,
+since
+for
+any
+0-cells
+c0
+and
+c′
+0,
+X†
+c0Xc′
+0 = �
+c1∈O1
+�
+�c0∈∂c1 X�c0 where ∂O1 = {−c0, c′
+0},
+⟨X†
+c0Xc′
+0⟩ = 1. Therefore, in this limit the Z(0)
+N symmetry
+is spontaneously broken.
+In a Z(0)
+N
+SSB phase,
+there are gapped (d − 1)-
+dimensional topological defects—domain walls—carrying
+ZN topological charge which populate the excited states.
+Indeed, in the K/J → 0 limit, the topological defect den-
+sity ˆρ for a state |ψ⟩ is defined by
+�
+c0∈∂c1
+Xc0 |ψ⟩ = exp
+�2πi
+N (∗ ˆρ)c1
+�
+|ψ⟩ ,
+(24)
+where X−c0 ≡ X†
+c0 and (∗ ˆρ)c1 ≡ ˆρ∗ c1.
+From the ZN
+clock algebra Eq. (34), �
+c0∈∂c1 Xc0 and Zc0 satisfy
+Zc0
+�
+�c0∈∂c1
+X�c0 = ω(−1)c0
+�
+�c0∈∂c1
+X�c0Zc0,
+(25)
+for all c0 ∈ ∂c1, where ω = e2π i/N and (−1)c0 is the
+sign in front of c0 in ∂c1.
+Using this,
+let’s act
+�
+�c0∈∂c1 X�c0 on the state (∗ Z)ˆcd |0⟩, with ∗ ˆcd ∈ ∂c1 and
+�
+�c0∈∂c1 X�c0 |0⟩ = |0⟩:
+�
+�c0∈∂c1
+X�c0(∗ Z)ˆcd|0⟩ = ω(−1)∗ ˆcd (∗ Z)ˆcd |0⟩ .
+(26)
+Because this is true for all ∗ ˆcd ∈ ∂c1, acting (∗ Z)ˆcd onto
+|0⟩ causes (∗ ˆρ)c1 ̸= 0 for all c1 ∈ δ ∗ ˆcd. Therefore, the
+operator (∗ �Z)ˆcd excites a topological defect on ∂ˆcd.
+In the K/J → 0 limit of the Z(0)
+N
+SSB phase, the
+fact that the ground state satisfies X†
+c0X�c0 |vac⟩ = |vac⟩
+means that the topological defect density in the ground
+states is zero for all d-cells of the dual lattice. Equiva-
+lently, this can also be see from the fact that the ground
+state satisfies ⟨Zc0⟩ = 0. So, because the topological de-
+fect creation operator does not have a vev, the topological
+defects are not condensed and thus gapped.
+The domain wall gap provides a candidate energy scale
+below which new structures may emerge. Indeed, when
+K/J = 0, there exists a low energy sub-Hilbert space
+spanned by states satisfying ⟨ˆρˆcd−1⟩ = 0.
+This is the
+ground state subspace and defines the IR of the SSB
+phase. However, when K/J ̸= 0, there no longer exists a
+low-energy sub-Hilbert space spanned by states satisfy-
+ing ⟨ˆρˆcd−1⟩ = 0. This is because the K term in H causes
+the ⟨ˆρˆcd−1⟩ = 0 and ⟨ˆρˆcd−1⟩ ̸= 0 states to mix. However, a
+corresponding low-energy sub-Hilbert space can be iden-
+tified using ULU from section II A to continue any oper-
+ator A which we understand at K/J = 0 to a (fattened)
+local operator �A ≡ ULUAU †
+LU with the same expectation
+values of A but at K/J ̸= 0. Therefore, there exists a low-
+energy sub-Hilbert space for both K/J = 0 and K/J ̸= 0
+spanned by states satisfying ⟨�ˆρˆcd−1⟩ = 0.
+Because the
+Z(0)
+N
+SSB phase is gapped, we can use the local unitary
+from the quasi-adiabatic continuation, Eq. (4), to con-
+tinue local operators throughout the entirety of the Z(0)
+N
+SSB phase.
+Because ULU in Eq. (4) is constructed only from terms
+in H, operators that commute with every term in H are
+unaffected by ULU. In particular, because the Z(0)
+N sym-
+metry operator commutes with each term in H, the sym-
+metry operator U in Eq. (21) satisfies
+U = �U =
+�
+c0
+�Z0.
+(27)
+Having identified the IR of the SSB phase, we’d now
+like to find an effective IR theory. As we learned in sec-
+tion II, the effective IR Hamiltonian HIR is a sum of all
+terms allowed in the IR that can be constructed from
+the terms in �H. The terms in �H are ( �
+X†
+c0 �
+X�c0 + h.c.) and
+( �Zc0 + h.c.).
+In
+the
+IR,
+because
+�ˆρˆcd−1 = 0,
+from
+Eq.
+(24),
+�
+c0∈∂c1 �
+Xc0 ≡ X†
+c0 �
+X�c0 = 1 and therefore ( �
+X†
+c0 �
+X�c0 + h.c.)
+does not contribute in HIR. The operators ( �Zc0 + h.c.)
+are not allowed operators in the IR since they excite
+dressed topological defects.
+However, allowed IR op-
+erators can be constructed from �Zc0.
+Indeed, because
+(∗ �Z)ˆcd excites a dressed topological defect on ∂ˆcd, acting
+(∗ �Z′)ˆcd0 on any d-cycle of the dual lattice ˆCd does not ex-
+cite dressed topological defects since ∂2 = 0. Therefore,
+
+12
+the IR allowed operator constructed from �Zc0 is
+�T †[ ˆCd] =
+�
+ˆcd∈ ˆ
+Cd
+(∗ �Z)ˆcd.
+(28)
+This is a product of �Zc0 over all 0-cells in a connected
+part of the spatial lattice. Indeed, denoting a connected
+part of the spatial lattice as M, we can instead consider
+�T †[M] =
+�
+c0∈M
+�Zc0.
+(29)
+Interestingly, this is just the Z(0)
+N
+symmetry operator,
+and therefore does not need to be dressed by ULU:
+�T †[M] = T †[M].
+The effective IR theory, therefore, includes all terms
+constructed from
+�T.
+Denoting the set of all con-
+nected part of the spatial lattice with ZN coefficients as
+H0(Md; ZN), the effective IR Hamiltonian is
+HIR = −J
+�
+M∈H0(Md;ZN)
+κ|M| �T[M],
+(30)
+where κ ∼ K/J and |M| is the number of 0-cells form
+which M is constructed.
+Since K/J ≪ 1 in the SSB
+phase, in the thermodynamic limit where |M| → ∞, the
+effective IR Hamiltonian becomes a constant
+HIR
+����
+L→∞
+= 0.
+(31)
+This simply reflects the fact that the Z(0)
+N
+SSB phase is
+a gapped phase.
+The effective IR Hamiltonian of the Z(0)
+N SSB phase has
+a new symmetry which was not present in the UV theory.
+Indeed, the IR has the symmetry where the IR-allowed
+operator �T transforms as
+�T[ ˆCd] �→ e
+2π i
+N �T[ ˆCd].
+(32)
+This cannot be a transformation by an arbitrary phase
+eiα and must be e
+2π i
+N
+in order for ( �T[ ˆCd])N = 1 to re-
+main satisfied. The transformation can be accomplished
+by transforming �Zc0→ e
+2π i
+N �Zc0 for only a single lattice
+site. Therefore, the operator that causes this transfor-
+mation is
+�ˆU[c0] = �
+Xc0.
+(33)
+At first glance of Eq. (33), one sees a local transforma-
+tion and may think it is a conventional gauge transfor-
+mation. However, it is not because a physical operator
+transforms nontrivially under it: �T[ ˆCd] transforms by a
+nontrivial element of ZN.
+Therefore, since the physi-
+cal operators are supported on d-cycles, Eq. (33) is the
+symmetry operator of a ZN d-form symmetry—a Z(d)
+N
+symmetry. Notice that since the Z(0)
+N SSB phase only ex-
+ists when d > 0, this symmetry is always a higher-form
+symmetry.
+This emergent Z(d)
+N
+symmetry has been noted previ-
+ously throughout the literature [49, 66, 67]. Here we find
+that it is an exact emergent symmetry, meaning that de-
+spite it being an emergent symmetry, it is an exact sym-
+metry of the effective IR Hamiltonian in the thermody-
+namic limit. Therefore, it constrains the IR as if it were
+an exact UV symmetry. Furthermore, this exact emer-
+gent symmetry is present throughout the entire Z(0)
+N SSB
+phase. Indeed, away from the tractable K/J = 0 point,
+local unitary operators dress the K/J = 0 charged and
+symmetry operators.
+So, we have found that in the IR of the Z(0)
+N SSB phase,
+there is an exact emergent Z(0)
+N × Z(d)
+N
+symmetry. How-
+ever, the Z(0)
+N
+and Z(d)
+N
+symmetries are not independent
+of one another, there is a mixed ’t Hooft anomaly. The
+fact that the Z(0)
+N × Z(d)
+N
+symmetry is anomalous can be
+noticed from the fact the symmetry operator of the Z(0)
+N
+symmetry U of Eq. (27) is charged under the Z(d)
+N
+sym-
+metry, reflecting an obstruction to gauging both symme-
+tries9.
+To summarize, when considering only the UV theory,
+it appeared that there was only an exact Z(0)
+N
+symme-
+try in all phases of the model. However, by construct-
+ing the effective IR Hamiltonian of the Z(N)
+N
+SSB phase,
+we learned that there is actually an exact anomalous
+Z(0)
+N × Z(d)
+N symmetry in the IR.
+B.
+Emergent ZN p-gauge theory for p ≥ 1
+In this section, we will apply the framework discussed
+in section II to a model for emergent ZN p-gauge theory
+which we call model D. Consider ZN quantum rotors re-
+siding on each p-cell of the spatial d-dimensional cubic
+lattice with p > 0. A ZN quantum rotor is an N-level
+system described by clock operators Xcp and Zcp. These
+unitary operators are N-dimensional generalizations of
+the Pauli matrices, satisfying
+Z�cpXcp = ωδcp,�cp XcpZ�cp,
+XN
+cp = ZN
+cp = 1,
+(34)
+where ω ≡ ei2π/N. The eigenvalues of the clock opera-
+tors are {1, ω, ω2, · · · , ωN−1} ≃ ZN.
+The microscopic model we consider is described by the
+9 See footnote 14.
+
+13
++
+−
++
+−
++
+−
++
+−
+−
++
+−
+−
++
+model C: 
+    
+ρcp−1 = ∑
+Lz
+±
+model D: 
+  
+τz
+cp−1 = ∏Z
+±
++
++
+−
+FIG. 8.
+Graphical representation of the operator ρcp−1 of
+model C (see Eq. (67)) and the operator τ z
+cp−1 of model D
+(see Eq. (35)) in three spatial dimensions for (first row) p = 1,
+(second row) p = 2, and (third row) p = 3. Depending on the
+operator, the disks on the p-cells labeled by ± either denote
+the sign in front of Lz belonging to that p-cell in the sum for
+ρcp−1, or whether the Z operator belonging to that p-cell is
+Z+ ≡ Z or Z− ≡ Z† in the product for τ z
+cp−1. The (p − 1)-cell
+the operator is associated with is colored purple.
+Hamiltonian
+HUV= −U
+2
+�
+cp−1
+τ z
+cp−1 − K
+2
+�
+cp
+Zcp + J
+2
+�
+cp
+Xcp + h.c.,
+τ z
+cp−1 ≡
+�
+cp∈δcp−1
+Zcp.
+(35)
+The sum �
+cp is over all p-cells of the spatial lattice. The
+product �
+cp∈δcp−1 in the definition of τ z
+cp−1 is over the
+coboundary of cp−1, defined in Eq. (A3) of the appendix,
+with the convention Z−cp = Z†
+cp. Note that we need to
+require p ≥ 1 in order for (p − 1)-cell cp−1 to be well
+defined. τ z is generally a product of 2(d − p + 1) opera-
+tors, examples of which are shown in Fig. 8. Note that
+because the eigenvalues of Zcp are elements of ZN, the
+eigenvalues of τ z
+cp−1 are also ZN. Lastly, since HUV has
+terms linear in Zcp and Xcp, there are no transformation
+of Zcp or Xcp that leave HUV invariant and thus no UV
+symmetries in this theory.
+L+
++
+−
+± ρ = ± 1
++
+−
++
+−
+−
+−
++
++
+−
++
++
+−
+−
+−
++
++
+−
++
+X
+± τz = e±2πi/N
+model C
+model D
+FIG. 9.
+Graphical representation of the excitation created
+by L+
+cp in model C and the excitation created by Xcp in
+model D, shown in three spatial dimensions for (first row)
+p = 1, (second row) p = 2, and (third row) p = 3. The yel-
+low disk represents the L+
+cp or Xcp operator, depending on
+the model, acting on the U(1) or ZN rotor belonging to that
+p-cell. For model C, the purple disk labeled by ± represents
+the ± sign in ρcp−1(L+
+cp |0⟩) = ±(L+
+cp |0⟩) for that (p − 1)-cell.
+For model D, the disk labeled by ± represents the ± sign in
+τ z
+cp−1(Xcp |0⟩) = ω±1(Xcp |0⟩) for that (p − 1)-cell.
+1.
+An exact emergent Z(p)
+N
+symmetry
+When J = K = 0, there exists a low energy sub-
+Hilbert space spanned by states satisfying ⟨τ z
+cp−1⟩ = 1.
+Indeed, the first term in HUV introduces an energetic
+penalty for states satisfying ⟨ψ| (τ z
+cp−1 + h.c.) |ψ⟩ ̸= 1.
+We interpret such states in the J = 0 and U ≫ K
+limit as describing a gapped excitation, a segment of
+which resides on the (p − 1)-cell cp−1.
+Indeed, from
+the clock operators’ algebra, τ z
+cp−1
+and Xcp
+satisfy
+τ z
+cp−1Xcp = ω(−1)cp Xcpτ z
+cp−1 for all cp−1 ∈ ∂cp, where
+(−1)cp denotes the sign in front of cp in the expression
+for δcp−1. Using this, let us act τ z
+cp−1 on the state Xcp |0⟩,
+where cp−1 ∈ ∂cp and τ z
+cp−1 |0⟩ = |0⟩:
+τ z
+cp−1
+�
+Xcp |0⟩
+�
+= ω(−1)cp �
+Xcp |0⟩
+�
+.
+(36)
+Because this is true for all cp−1 ∈ ∂cp, Xcp excites the
+aforementioned excitation on ∂cp, examples of which are
+shown in Fig. 9.
+We’ll refer to these bosonic (p − 1)-
+dimensional (in space) excitations as “charges.”
+How-
+ever, note that since XN
+cp = 1, exciting N charges is the
+same as not exciting any. Thus, the charge number takes
+values in ZN.
+It is tempting to consider the charge gap as a candidate
+energy scale below which new physics emerges. However,
+
+14
+when J ̸= 0, there no longer exists a low-energy sub-
+Hilbert space spanned by states satisfying ⟨τ z
+cp−1⟩ = 1.
+This is because the J term in HUV causes the ⟨τ z
+cp−1⟩ = 1
+and ⟨τ z
+cp−1⟩ ̸= 1 states to mix. However, a correspond-
+ing low-energy sub-Hilbert space can be identified us-
+ing ULU from section II A to continue any operator A
+which we understand at J = 0 to a (fattened) local oper-
+ator �A ≡ U (1)
+LUA(U (1)
+LU)† with the same expectation values
+of A but at J ̸= 0. Therefore, when U ≫ K, there ex-
+ists a low-energy sub-Hilbert space for both J = 0 and
+J ̸= 0 spanned by states satisfying ⟨�τ z
+cp−1⟩ = 1. We view
+states with ⟨�τ z
+cp−1⟩ ̸= 1 as having dressed charges excited.
+Since the undressed charges are created using Xcp, these
+dressed charges are created using �
+Xcp. We will not find
+an explicit form for U (1)
+LU and thus will not precisely know
+throughout how much of parameter space the dressed
+(fattened) operators can be defined without violating the
+assumptions of U (1)
+LU. Instead, we will assume that such
+an operator exists and can access a greater than measure-
+zero part of parameter space, and will investigate the
+consequences of this conjecture.
+At this point in our investigations, we cannot tell if the
+dressed charge gap ∆dressed-charge is an IR scale or a mid-
+IR I scale or a mid-II scale, etc. In section III B 2, we will
+learn that it is a mid-IR scale in region III of parameter
+space but an IR scale in region II of parameter space
+(see Fig. 7). For the rest of this section, however, we will
+adopt the language from the perspective of region III and
+call the dressed charge gap a mid-IR scale.
+Given the mid-IR scale Emid-IR ≡ ∆dressed-charge, we
+would now like to the find an effective mid-IR theory,
+which by definition is a theory only describing states at
+energies E < Emid-IR. As we learned in section II, the ef-
+fective mid-IR Hamiltonian Hmid-IR is a sum of all terms
+allowed in the mid-IR that can be constructed from the
+terms in �HUV. The terms in �HUV are �τ z
+cp−1, �Zcp, and
+�
+Xcp. In the mid-IR, �τ z
+cp−1 = 1, so it is trivial and will not
+contribute in Hmid-IR. Furthermore, since �Zcp commutes
+with �τ z
+cp−1, it does not excite any dressed charges. Thus
+�Zcp is an allowed mid-IR operator from which terms in
+Hmid-IR can be constructed. The operators �
+Xcp are not
+allowed operators in the mid-IR since they excite charges.
+However, allowed mid-IR operators can be constructed
+from �
+Xcp. Indeed, since �
+Xcp excites a dressed charge on
+∂cp, acting �
+Xcp on any p-cycle does not excite dressed
+charges since ∂ = 0. Therefore, an allowed operator in
+†
+†
+†
+˜L+
+model C
+˜
+X
+model D
+†
+†
+†
+†
+†
+†
+†
+†
+†
+FIG. 10.
+Graphical representation of the Wilson operator
+�
+W †(Cp) of model C (Eq. (70)) and model D (Eq. (38)) sup-
+ported on Cp = ∂cp+1 in three spatial dimensions for (first
+row) p = 0, (second row) p = 1, and (third row) p = 2. For
+model C (D), the yellow colored disks denote �L+
+cp ( �
+Xcp) op-
+erators belonging to that p-cell, the product of which yields
+�
+W †(Cp). Discs labeled by † denote the hermitian conjugate
+of the operator. In each row, the (p + 1)-cell cp+1 satisfying
+Cp = ∂cp+1 is colored green.
+the mid-IR is10
+�
+W †[Cp] =
+�
+cp∈Cp
+�
+Xcp.
+(38)
+We call �
+W † the Wilson operator. It has the interpretation
+of exciting a dressed charge, transporting it along a p-
+cycle, and then ultimately annihilates it. Fig. 10 shows
+a few graphical representations of the Wilson operator
+supported on the smallest possible p-cycle: the boundary
+of a (p + 1)-cell. Note that the Wilson operator satisfies
+�
+W †[Cp] = �
+W[−Cp].
+The effective mid-IR theory, therefore, includes all
+terms constructed from �
+W and �Z. Denoting the set of
+all oriented p-cycles with ZN coefficients on the spatial
+10 Describing the extended object �
+W † using local operators intro-
+duces a gauge redundancy. Indeed, �
+W † is invariant under
+�
+Xcp → Ξcp �
+Xcp
+with
+�
+cp∈Cp
+Ξcp = 1 ∀ Cp.
+(37)
+For ( �
+Xcp)N = 1 to remain invariant,
+the gauge parameter
+must satisfy Ξcp ∈ ZN.
+When Ξcp = �
+cp−1∈∂cp λcp−1, with
+λcp−1 ∈ ZN and λ−cp−1 = λ†
+cp−1, this is the canonical ZN gauge
+redundancy.
+Otherwise, Eq. (37) corresponds to large gauge
+transformations.
+
+15
+lattice Md as Zp(Md; ZN), the mid-IR Hamiltonian takes
+the form
+H(III)
+mid-IR =−κU
+2
+�
+cp
+( �Zcp+ h.c.)− U
+2
+�
+Cp∈Zp(Md;ZN)
+ε|Cp|�
+W[Cp] + · · ·,
+(39)
+where κ ∼ K/U, ε ∼ J/U, |Cp| is the number of p-cells
+form which Cp is constructed, and the · · ·’s include all
+other possible terms constructed from both ( �Zcp + h.c.)
+and �
+W[Cp]. Since the dressed charge gap is the mid-IR
+scale of region III but the IR of region II,
+H(III)
+mid-IR ≡ H(II)
+IR .
+(40)
+The effective mid-IR theory is only well defined pro-
+vided ε < 1.
+As a consequence, since the lattice is
+simply connected, the Wilson operators supported on
+nontrivial p-cycles are exponentially suppressed by εLp,
+where L is the linear system size.
+Therefore, denot-
+ing contractible p-cycles with ZN coefficients on the
+spatial lattice Md as Bp(Md; ZN), terms with Wilson
+operators supported on cycles in the homology class
+Hp(Md; ZN) = Zp(Md; ZN)/Bp(Md; ZN) vanish in the
+thermodynamic limit. Therefore, in the thermodynamic
+limit H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+L→∞
+=−κU
+2
+�
+cp
+( �Zcp+ h.c.)− U
+2
+�
+Cp∈Bp(Md;ZN)
+ε|Cp|�
+W[Cp] + · · ·,
+(41)
+where
+the
+· · ·’s
+include
+all
+possible
+terms
+con-
+structed from both ( �Zcp + h.c.) and �
+W[Cp] with only
+Cp ∈ Bp(Md; ZN).
+In the thermodynamic limit, H(III)
+mid-IR is invariant under
+transforming the UV operator as
+�
+Xcp→ e
+2π i
+N Γcp �
+Xcp,
+(dΓ)cp+1= 0,
+Γcp ∈ Z. (42)
+Here (dΓ)cp+1 is the lattice exterior derivative of Γcp,
+defined as
+(dΓ)cp+1 ≡
+�
+cp∈∂cp+1
+Γcp.
+(43)
+Furthermore, Γcp is required to take integer values in
+order for (Xcp)N = 1 to be invariant under the transfor-
+mation. Under this transformation, the Wilson operator
+transforms as
+�
+W[Cp] �→ e
+2π i
+N
+�
+cp∈Cp Γcp �
+W[Cp],
+(44)
+which leaves Eq. (41) unchanged because �
+cp∈Cp Γcp = 0
+for Cp ∈ Bp(Md; ZN) since (dΓ)cp+1 = 0. This transfor-
+mation is importantly different than the gauge trans-
+formations Eq. (37) because the Wilson operator trans-
+forms nontrivially under it. Indeed, the Wilson operators
+supported on p-cycles in Hp(Md; ZN) transform under
+Eq. (42) by a nontrivial element of ZN since Γcp ∈ Z.
+exp[iα˜Lz]
+†
+†
+†
+model C
+model D
+˜Z
+†
+†
+†
+†
+†
+†
+†
+†
+†
+FIG. 11. Graphical representation of the symmetry operator
+for the U(1)(p) symmetry in model C and the Z(p)
+N
+symme-
+try in model D, supported on the boundary of a (d − p + 1)-
+cell of the dual lattice in d = 3 spatial dimensions for (first
+row) p = 1, (second row) p = 2, and (third row) p = 3. For
+model C (D), the blue colored disks denote �L+
+cp ( �
+Xcp) opera-
+tors belonging to that p-cell, the product of which yields the
+symmetry operator. Discs labeled by † denote the hermitian
+conjugate of the operator. In each row, the aforementioned
+(d − p + 1)-cell of the dual lattice is colored in red.
+Therefore, since the physical operators are supported on
+a p-cycle, Eq. (42) corresponds to the transformation of
+a ZN p-form symmetry—a Z(p)
+N
+symmetry. Throughout
+the remainder of this section, we will always assume to
+be working in the thermodynamic limit.
+The symmetry operator of this Z(p)
+N symmetry is
+�U(ˆΣd−p) =
+�
+ˆcd−p∈ˆΣd−p
+(∗ �Z)ˆcd−p,
+(45)
+where ˆΣd−p is a (d − p)-cycle of the dual lattice, and
+(∗ �Z)ˆcd−p ≡ �Z∗ ˆcd−p. We again see why this is a p-form
+symmetry: its symmetry operator acts on a (d − p)-cycle.
+Fig. 11 shows a graphical representation of �U(ˆΣd−p)
+when ˆΣd−p = ∂ˆcd−p+1.
+To confirm that Eq. (45) is the symmetry operator,
+first note that �Zcp �
+Xcp �Z†
+cp = e2π i/N �
+Xcp. Therefore, let-
+ting #(A, B) denote the intersection number between the
+
+16
+chains A and B,
+�U(ˆΣd−p)�
+W[Cp]�U †(ˆΣd−p) = e
+2π i
+N #(Cp,ˆΣd−p)�
+W[Cp]. (46)
+Introducing Γcp, the Poincar´e dual of ˆΣd−p with respect
+to the spatial lattice, given by
+(∗ Γ)ˆcd−p(ˆΣd−p) =
+�
+ˆyd−p∈ˆΣd−p
+δˆyd−p,ˆcd−p,
+(47)
+the
+intersection
+number
+can
+be
+written
+as
+#(Cp, ˆΣd−p) = �
+cp∈Cp Γcp.
+Since
+∂ ˆΣd−p = 0,
+Γcp
+satisfies (δ ∗ Γ)ˆcd−p−1 = 0, which from Eq (A6) is equiva-
+lent to (dΓ)cp+1 = 0. Therefore, since Γcp ∈ Z, Eq. (46)
+correctly reproduces the Z(p)
+N
+symmetry transformation
+Eq. (44).
+The Z(p)
+N
+symmetry is a symmetry of the effec-
+tive mid-IR theory but not the UV theory.
+There-
+fore, it is an emergent symmetry, emerging at energies
+E < ∆dressed-charge. In the thermodynamic limit, since it
+is an exact symmetry of the effective mid-IR theory, we
+say that the Z(p)
+N symmetry is an exact emergent symme-
+try. Because p > 0, this result that an emergent symme-
+try can act like an exact symmetry only applies to p-form
+symmetries with p > 0.
+The exact emergent Z(p)
+N
+symmetry is not present
+throughout the entire parameter space of HUV. Whether
+or not it emerges depends on both the existence of the
+mid-IR and the existence of the effective mid-IR Hamil-
+tonian. As mentioned at the start of this subsection, the
+mid-IR only exists when U ≫ K. The effective mid-IR
+Hamiltonian is only well defined provided that the infi-
+nite series expansions converge, which required that ε < 1
+and so J/U ≪ 1. Thus, the Z(p)
+N symmetry only emerges
+when K/U ≪ 1 and J/U ≪ 1.
+2.
+An exact emergent anomalous Z(p)
+N × Z(d−p)
+N
+symmetry
+We have seen that at energies below the dressed charge
+gap, there is an exact emergent Z(p)
+N symmetry. Here, we
+will search for additional energy scales where new sym-
+metries can emerge. In fact, such a lower-energy scale
+only exists in region III of parameter space (see Fig. 7).
+This makes the dressed charge gap the IR scale of region
+II but the mid-IR scale of region III. Here, we will iden-
+tify the lower energy scale of region III with the gap of
+the topological defects in the Z(p)
+N spontaneous symmetry
+breaking (SSB) phase. Since there are no other energy
+scales in region III, this is an IR scale.
+To show this, let us first discuss when the Z(p)
+N symme-
+try is spontaneously broken11. To gain some intuition,
+11 See footnote 23 for the definition of spontaneous symmetry
+breaking.
+we consider two tractable limits of the effective mid-IR
+theory. The first limit is in region II when J/U = 0 but
+K/U ̸= 0 such that H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+J/U=0
+= −κU
+2
+�
+cp
+�
+�Zcp + h.c.
+�
+.
+(48)
+The ground state in this limit satisfies �Zcp |vac⟩ = |vac⟩
+for all cp. Because acting �
+W †[Cp] onto a state changes the
+value of ⟨ �Zcp⟩ for each cp ∈ Cp, ⟨�
+W †[Cp]⟩ = 0 for all Cp.
+Consequently, this limit lies in a Z(p)
+N
+symmetric phase.
+The other tractable limit is in region III K/U = 0 but
+J/U ̸= 0 such that H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+K/U=0
+= −U
+2
+�
+Cp∈Bp(Md;ZN)
+ε|Cp| �
+W[Cp].
+(49)
+The
+ground
+state
+in
+this
+limit
+clearly
+satisfies
+�
+W[Cp] |vac⟩ = |vac⟩ for all Cp ∈ Bp(Md; ZN), and conse-
+quently ⟨�
+W †[Cp]⟩ = 1 for all trivial p-cycles. Therefore,
+in this limit the Z(p)
+N symmetry is spontaneously broken.
+According to the higher Coleman-Mermin-Wagner the-
+orem, in (d + 1)-dimensional spacetime, a Z(p)
+N symmetry
+at zero temperature can spontaneously break in the ther-
+modynamic limit when d > p [2, 28]. Therefore, when
+d ≤ p, there is no stable Z(p)
+N SSB phase, and any SSB fea-
+tures are unique to K/U = 0. However, when d > p, we
+expect a stable SSB phase even for K/U ̸= 0. For small
+κ ∼ K/U and small ε ∼ J/U, a reasonable expectation is
+that the SSB phase occurs when κ ≲ εmin |Cp| = ε2(p+1),
+which is equivalent to K/U ≪ (J/U)2(p+1). This iden-
+tifies region II of parameter space as the Z(p)
+N
+symmet-
+ric phase while region III of parameter space is the Z(p)
+N
+SSB phase. In fact, the boundary between regions II and
+III in Fig. 7 is a depiction of the boundary between the
+Z(p)
+N
+symmetric and Z(p)
+N
+SSB phases. We leave a more
+detailed investigation of this phase transition to future
+work.
+Since the order parameter for Z(p)
+N
+symmetry break-
+ing is the vacuum expectation value of �
+W †[Cp], the Wil-
+son operator can detect topological defects related to the
+nontrivial mappings ⟨�
+W †[Cp]⟩ : Zp(Md; ZN) �→ ZN. The
+topological defects excited in a state |ψ⟩ are probed by
+acting the Wilson operator on a trivial p-cycle Cp:12
+�
+W †[Cp] |ψ⟩ = exp
+�2πi
+N
+ˆQ(Cp)
+�
+|ψ⟩ .
+(50)
+Here, the eigenvalues of ˆQ(Cp) are the net number of
+topological defects enclosed by Cp.
+12 This is a natural generalization of the p = 0 case, where the topo-
+logical defects are domain walls (see Eq. (24)).
+
+17
+The topological defects can be characterized locally us-
+ing the topological defect density ˆρ which is defined im-
+plicitly as
+ˆQ(Cp = ∂Op+1) ≡
+�
+cp+2∈Op+1
+(∗ ˆρ)cp+1.
+(51)
+We can find an implicit expression for the topological
+defect density by plugging in Cp = ∂Op+1 into Eq. (50)
+and using Stoke’s theorem. Doing so, we find
+�
+cp∈∂cp+1
+�
+Xcp |ψ⟩ = exp
+�2πi
+N (∗ ˆρ)cp+1
+�
+|ψ⟩ .
+(52)
+Since ˆρ is supported on a (d − p − 1)-cell of the dual lat-
+tice, the topological defects are (d − p − 1)-dimensional
+excitations in space.
+Note that because �
+XN
+cp = 1, the
+eigenvalues of (∗ ˆρ)cp+1 take values in ZN 13.
+From the ZN clock algebra Eq. (34), �
+W †[∂cp+1] and
+�Zcp satisfy
+�Zcp�
+W †[∂cp+1] = ω(−1)cp �
+W †[∂cp+1] �Zcp,
+(53)
+for all cp ∈ ∂cp+1, where ω = e2π i/N and (−1)cp is the
+sign in front of cp in ∂cp+1.
+Using this, let’s act
+�
+W †[∂cp+1] on the state (∗ �Z)ˆcd−p |0⟩, with ∗ ˆcd−p ∈ ∂cp+1
+and �
+W †[∂cp+1] |0⟩ = |0⟩:
+�
+W †[∂cp+1](∗ �Z)ˆcd−p|0⟩ = ω(−1)∗ ˆcd−p (∗ �Z)ˆcd−p |0⟩ .
+(54)
+Because
+this
+is
+true
+for
+all
+∗ ˆcd−p ∈ ∂cp+1,
+act-
+ing
+(∗ �Z)ˆcd−p
+onto
+|0⟩
+causes
+(∗ ˆρ)cp+1 ̸= 0
+for
+all
+cp+1 ∈ δ ∗ ˆcd−p (see Fig. 12).
+Therefore, the operator
+(∗ �Z)ˆcd−p excites a topological defect on ∂ˆcd−p
+In the J/U = 0 limit of the Z(p)
+N
+symmetric phase,
+as
+previously
+discussed
+the
+ground
+state
+satisfies
+�Zcp |vac⟩ = |vac⟩ and thus ⟨�
+W †[Cp]⟩ = 0. From Eqs. (50)
+and (52), ⟨�
+W †[Cp]⟩ = 0 implies the topological defect
+density in the ground state fluctuates wildly.
+There-
+fore,
+the topological defects are condensed.
+This
+can also be seen from �Zcp |vac⟩ = |vac⟩ implying that
+⟨(∗ �Z)ˆcd−p⟩ = 1. The topological defect creation opera-
+tor having a nonzero vev implies that the topological de-
+fects are condensed. On the other hand, in the K/U = 0
+limit of the Z(p)
+N
+SSB phase, the ground state satisfies
+�
+W[Cp] |vac⟩ = |vac⟩ and, therefore, (∗ ˆρ)cp+1 |vac⟩ = 0.
+Thus, topological defects do not populate the ground
+state, so they are gapped excitations.
+This can also
+13 Note that for p = 1, the Z(1)
+N
+SSB phase has ZN topological
+order and is the deconfined phase of ZN gauge theory. In this
+case, we find (d − 2)-dimensional topological defects carrying ZN
+topological charge. These are the m excitations of ZN topological
+order in d-dimensional space.
+˜Z
+± ( * ̂ρ) = ± 1 mod N
+−
++
+−
++
+−
++
++
+−
++
++
+−
+−
++
++
+−
+−
+−
+−
++
++
+FIG. 12.
+Graphical representation of the topological de-
+fects created by (∗ �Z)ˆcd−p in model D, shown in three spa-
+tial dimensions for (first row) p = 0, (second row) p = 1, and
+(third row) p = 2.
+The blue disk represents the (∗ �Z)ˆcd−p
+operator acting on the ZN
+rotor belonging to that p-
+cell.
+The disks labeled by ± represents the ± sign in
+�
+W †[∂cp+1](∗ �Z)ˆcd−p |0⟩ = exp
+�
+± 2πi
+N
+�
+(∗ �Z)ˆcd−p |0⟩, and thus
+the value of (∗ ˆρ)cp+1 for that cp+1 (see Eq. (52)).
+be seen from the fact that �
+W[Cp] |vac⟩ = |vac⟩ implies
+⟨(∗ �Z)ˆcd−p⟩ = 0 and so the topological defects are not con-
+densed.
+Other regions besides the extreme limits of these
+phases can be explored using ULU from section II A. Since
+we are interested in identifying energy scales below which
+new structures can emerge, we restrict ourselves to the
+phase where the topological defects have a gap— the
+Z(p)
+N SSB phase—which is region III of parameter space.
+Indeed, region III will have an IR scale defined by the
+gapped dressed topological defects. Since the defects are
+condensed in region II, they do not give rise to additional
+energy scales in region II.
+When K/U = 0, a low energy sub-Hilbert space exists
+in the mid-IR spanned by states satisfying ⟨ˆρˆcd−p−1⟩ = 0.
+So when K/U = 0, the IR energy scale is the topolog-
+ical defect’s gap: EIR = ∆defect ∼ J2p+2/U 2p+1.
+How-
+ever, when K/U ̸= 0, this sub-Hilbert space is no longer a
+low-energy sector because the κU term in H(III)
+mid-IR causes
+the ⟨ˆρˆcd−p−1⟩ = 0 and ⟨ˆρˆcd−p−1⟩ ̸= 0 states to mix. We
+can use the local unitary discussed in section II A, which
+
+18
+we’ll denote as U (2)
+LU, to continue the K/U = 0 states
+within the Z(p)
+N
+SSB phase. In general, U (2)
+LU is different
+than the local unitary U (1)
+LU used in the previous sub-
+section.
+We’ll denote an operator A continued using
+U (2)
+LU as A′ ≡ U (2)
+LUA(U (2)
+LU)†. Therefore, the IR of region
+III for both K/U = 0 and K/U ̸= 0 is the sub-Hilbert
+space spanned by states satisfying ⟨ˆρ′
+ˆcd−p−1⟩ = 0.
+We
+view states with ⟨ˆρ′
+ˆcd−p−1⟩ ̸= 0 as having gapped dressed
+topological defects excited and the IR scale is their energy
+gap EIR = ∆dressed-defect. Because the Z(p)
+N SSB phase is
+gapped, we can use the local unitary from the quasi-
+adiabatic continuation, Eq. (4), to continue local opera-
+tors throughout the entirety of the Z(p)
+N SSB phase.
+Because U (2)
+LU in Eq. (4) is constructed only from terms
+in H(III)
+mid-IR, operators that commute with every term in
+H(III)
+mid-IR are unaffected by U (2)
+LU. In particular, because
+the Z(p)
+N
+symmetry operator commutes with each term
+in H(III)
+mid-IR, the symmetry operator �U(ˆΣd−p) in Eq. (45)
+satisfies
+�U(ˆΣd−p) = �U ′(ˆΣd−p) =
+�
+ˆcd−p∈ˆΣd−p
+(∗ �Z′)ˆcd−p.
+(55)
+Having identified the IR of region III, we’d now like to
+find an effective IR theory. As we learned in section II,
+the effective IR Hamiltonian H(III)
+IR
+is a sum of all terms
+allowed in the IR that can be constructed from the terms
+in H(III)′
+mid-IR. The terms in H(III)′
+mid-IR are all constructed from
+�
+W ′[Cp] and ( �Z′
+cp + h.c.). In the IR, because ˆρ′
+ˆcd−p−1 = 0,
+�
+W ′[Cp] = 1 and does not contribute in H(III)
+IR . The op-
+erators ( �Z′
+cp + h.c.) are not allowed operators in the IR
+since they excite dressed topological defects. However,
+allowed IR operators can be constructed from �Z′
+cp. In-
+deed, because (∗ �Z′)ˆcd−p excites a dressed topological de-
+fect on ∂ˆcd−p, acting (∗ �Z′)ˆcd−p on any (d − p)-cycle of
+the dual lattice ˆCd−p does not excite dressed topological
+defects since ∂2 = 0. Therefore, the IR allowed operator
+constructed from �Z′
+cp is
+�T ′†[ ˆCd−p] =
+�
+ˆcd−p∈ ˆ
+Cd−p
+(∗ �Z′)ˆcd−p.
+(56)
+We call �T ′† the ’t Hooft operator. Interestingly, this is
+just the Z(p)
+N symmetry operator, and therefore does not
+need to be dressed by U (2)
+LU: �T ′†[ ˆCd−p] = �T †[ ˆCd−p].
+The effective mid-IR theory, therefore, includes all
+terms constructed from �T ′. Denoting the set of all con-
+tractible oriented (d − p)-cycles with ZN coefficients on
+the dual spatial lattice ˆ
+Md as Bd−p( ˆ
+Md; ZN), the effec-
+tive IR Hamiltonian is
+H(III)
+IR
+= −U
+�
+ˆ
+Cd−p∈Bd−p( ˆ
+Md;ZN)
+κ| ˆ
+Cd−p| �T ′[ ˆCd−p],
+(57)
+where κ ∼ K/U has been renormalized and | ˆCd−p| is the
+number of (d − p)-cells form which ˆCd−p is constructed.
+The effective IR Hamiltonian of region III has a new
+symmetry which was not present in the mid-IR Hamil-
+tonian for region III. Indeed, H(III)
+IR
+is invariant under
+transforming the UV operator as
+(∗ �Z′)ˆcd−p→ e
+2π i
+N ˆΓˆcd−p (∗ �Z′)ˆcd−p,
+(dˆΓ)ˆcd−p+1= 0,
+(58)
+where ˆΓˆcd−p ∈ Z such that ( �Z′
+cp)N = 1 is invariant under
+the transformation.
+Under this transformation, the ’t
+Hooft operator transforms as
+�T ′[ ˆCd−p] �→ e
+2π i
+N
+�
+ˆcd−p∈ ˆ
+Cd−p
+ˆΓˆcd−p �T ′[ ˆCd−p],
+(59)
+which
+leaves
+Eq.
+(57)
+unchanged
+because
+�
+ˆcd−p∈ ˆ
+Cd−p ˆΓˆcd−p = 0 for
+ˆCd−p ∈ Bd−p( ˆ
+Md; ZN) since
+(dˆΓ)ˆcd−p+1 = 0.
+This transformation is not a gauge
+transformations since the ’t Hooft operator—a physical
+operator—transforms nontrivially under it.
+Indeed,
+the ’t Hooft operators supported on (d − p)-cycles in
+Hd−p( ˆ
+Md; ZN) transform under Eq. (59) by a nontrivial
+element of ZN since ˆΓˆcd−p ∈ Z.
+Therefore, since the
+physical operators are supported on a (d − p)-cycle,
+Eq. (59) corresponds to the transformation of a ZN
+(d − p)-form symmetry—a Z(d−p)
+N
+symmetry.
+In fact,
+this is an exact emergent symmetry because it is not an
+exact symmetry of the UV but is an exact symmetry
+of the IR. It is straight forward to confirm that the
+symmetry operator of this Z(d−p)
+N
+symmetry is
+�ˆU
+′
+(Σp) =
+�
+cp∈Σp
+�
+X′
+cp,
+(60)
+where Σp is a p-cycle of the lattice. We again see why
+this is a (d − p)-form symmetry: its symmetry operator
+acts on a p-cycle.
+So, we have found that in the IR of region III, there
+is an exact emergent Z(p)
+N × Z(d−p)
+N
+symmetry. However,
+the Z(p)
+N
+and Z(d−p)
+N
+symmetries are not independent of
+one another, there is a mixed ’t Hooft anomaly.
+The
+fact that the Z(p)
+N × Z(d−p)
+N
+symmetry is anomalous can
+be noticed by considering the symmetry operators when
+supported on an open subspace—the disorder operators.
+Indeed, the Z(p)
+N symmetry operator, Eq. (55), on an open
+(d − p)-dimensional subspace of the dual lattice is
+�U ′( ˆOd−p) =
+�
+ˆcd−p∈ ˆ
+Od−p
+(∗ �Z′)ˆcd−p,
+(61)
+while the Z(d−p)
+N
+symmetry operator, Eq. (60), on an open
+p-dimensional subspace of the direct lattice is
+�ˆU
+′
+(Op) =
+�
+cp∈Op
+�
+X′
+cp.
+(62)
+
+19
+When ˆOd−p and Op intersect, these operators generally
+do not commute. This is a manifestation of the mixed ’t
+Hooft anomaly between the Z(p)
+N and Z(d−p)
+N
+symmetries,
+reflecting an obstruction to gauging both symmetries14.
+Before moving onto the next section, we note one fi-
+nal thing about the effective IR Hamiltonian of region
+III. Because the IR is also the dressed charge free sec-
+tor, �τ ′z
+cp−1 ≡ �
+cp∈δcp−1 �Z′
+cp = 1 for all cp−1. However, this
+also implies that �
+ˆcd−p∈∂ ∗ cp−1(∗ �Z′)ˆcd−p = 1 and, there-
+fore, �T ′[ ˆCd−p] = 1 for all ˆCd−p ∈ Bd−p( ˆ
+Md; ZN).
+Be-
+cause of this, H(III)
+IR
+of Eq. (57) is really just a constant:
+H(III)
+IR
+= 0,
+(63)
+This reflects the fact that region III is a gapped phase,
+so the IR is the degenerate ground states of HUV in this
+phase. The ground state is defined as the state with no
+dressed charges or dressed topological defects:
+�
+cp∈δcp−1
+�Z′
+cp |vac⟩ = |vac⟩ ,
+�
+cp∈∂cp+1
+�
+X′
+cp |vac⟩ = |vac⟩ .
+(64)
+This ground state is also the ground state of the p-form
+toric code Hamiltonian
+HpTC = −
+�
+cp−1
+�
+� �
+cp∈δcp−1
+�Z′
+cp
+�
+� −
+�
+cp+1
+�
+� �
+cp∈∂cp+1
+�
+X′
+cp
+�
+� + h.c..
+(65)
+Importantly, this is not the p-form toric code model of
+the UV ZN clock operators: the clock operators in HpTC
+are dressed by the two local unitary operators used when
+identifying the mid-IR and the IR of region III. Further-
+more, this p-form toric code model captures the ground
+state of the entire Z(p)
+N
+SSB phase, and the the UV pa-
+rameters are hidden in the local unitaries dressing Xcp
+and Zcp.
+The emergent anomalous Z(p)
+N × Z(d−p)
+N
+symmetry
+in the ground state is the same as the anomalous
+Z(p)
+N × Z(d−p)
+N
+symmetry of p-form BF theory.
+This is
+no accident, and it is well known that the vacuum of BF
+theory is the same as the ground states of the toric code.
+In appendix section B, we show how this connection can
+be made exact by deriving the topological quantum field
+theory description for the ground states of the lattice
+model.
+14 Gauging a symmetry U is the procedure of adding additional
+degrees of freedom such that the theory becomes invariant un-
+der the gauged symmetry operator Ugauged.
+Ugauged acts on
+both open and closed subspaces and physical states must sat-
+isfy Ugauged |ψ⟩ = |ψ⟩.
+A contradiction arises when different
+Ugauged no longer commute, reflecting an obstruction to gaug-
+ing the symmetry (a ’t Hooft anomaly).
+For example, con-
+sider U(1)
+gaugeU(2)
+gauge = −U(2)
+gaugeU(1)
+gauge.
+Since U(1,2)
+gauge |ψ⟩ = |ψ⟩,
+this leads to the contradiction |ψ⟩ = − |ψ⟩.
+C.
+Emergent U(1) p-gauge theory
+In this section, we will apply the framework discussed
+in section II to a model for emergent U(1) p-gauge the-
+ory which we call model C. Consider U(1) quantum ro-
+tors residing on each p-cell of the spatial d-dimensional
+cubic lattice with p > 0. Each rotor can be viewed as
+a particle on an infinitesimal circle, whose position we
+denote as the angle Θ, carrying angular momentum Lz.
+The operators Lz and Θ are hermitian and satisfy the
+canonical commutation relation [Θcp, Lz
+�cp] = iδcp,�cp, so
+Lz = −i ∂
+∂Θ. Additionally, since the eigenvalue of Θ is
+an angle, the eigenvalues of Lz are integers.
+The microscopic model we consider is described by the
+Hamiltonian
+HUV= U
+2
+�
+cp−1
+ρ2
+cp−1+ K
+2
+�
+cp
+(Lz
+cp)2+J
+2
+�
+cp
+�
+L+
+cp+ h.c.
+�
+,
+ρcp−1 =
+�
+cp∈δcp−1
+Lz
+cp.
+(66)
+The sum �
+cp is over all p-cells of the spatial lattice.
+The sum �
+cp∈δcp−1 in the definition of ρcp−1 is over the
+coboundary of cp−1, defined in Eq. (A3) of the appendix.
+Using the definition of δcp, the expression for ρcp−1 can
+be written as (see Fig. 8)
+ρ(x)µ1···µp−1=
+�
+ν
+Lz(x)νµ1...µp−1 − Lz(x − ˆν)νµ1...µp−1.
+(67)
+Note that because the eigenvalues of Lz are integers, the
+eigenvalues of ρ are also integers. In the third term of
+HUV, L+
+cp = (L−
+cp)† = exp
+�
+iΘcp
+�
+is the raising operator
+for Lz
+cp. Lastly, since HUV is invariant under the trans-
+formation Lz
+cp → −Lz
+cp, this theory has a UV Z(0)
+2
+sym-
+metry.
+1.
+An exact emergent U(1)(p) symmetry
+When J = K = 0, there exists a low energy sub-
+Hilbert space spanned by states satisfying ⟨ρcp−1⟩ = 0.
+Indeed, the first term in HUV introduces an energetic
+penalty for states satisfying ⟨ψ| ρcp−1 |ψ⟩ ̸= 0. We inter-
+pret such states in the J = 0 and U ≫ K limit as describ-
+ing a gapped excitation, a segment of which resides on
+the (p − 1)-cell cp−1. Indeed, from the canonical commu-
+tation relation satisfied by Lz and Θ, the rotor operators
+of the same p-cell satisfy [Lz, L±] = ±L±.
+Therefore,
+ρcp−1 and L±
+cp satisfy [ρcp−1, L±
+cp] = ±(−1)cpL±
+cp for all
+cp−1 ∈ ∂cp, where (−1)cp denotes the sign in front of cp
+in the expression for δcp−1. Using this, let us act ρcp−1
+on the state L±
+cp |0⟩, with cp−1 ∈ ∂cp and ρcp−1 |0⟩ = 0:
+ρcp−1
+�
+L±
+cp |0⟩
+�
+= ±(−1)cp �
+L±
+cp |0⟩
+�
+.
+(68)
+
+20
+Because this is true for all cp−1 ∈ ∂cp, L±
+cp excites the
+aforementioned excitation on ∂cp, examples of which are
+shown in Fig. 9.
+We’ll refer to these bosonic (p − 1)-
+dimensional (in space) excitations as “charges.”
+It is tempting to consider the charge gap as a candidate
+energy scale below which new physics emerges. However,
+when J ̸= 0, there no longer exists a low-energy sub-
+Hilbert space spanned by states satisfying ⟨ρcp−1⟩ = 0.
+This is because the J term in HUV causes the ⟨ρcp−1⟩ = 0
+and ⟨ρcp−1⟩ ̸= 0 states to mix. However, a correspond-
+ing low-energy sub-Hilbert space can be identified us-
+ing ULU from section II A to continue any operator A
+which we understand at J = 0 to a (fattened) local oper-
+ator �A ≡ ULUA(ULU)† with the same expectation values
+of A but at J ̸= 0. Therefore, when U ≫ K, there ex-
+ists a low-energy sub-Hilbert space for both J = 0 and
+J ̸= 0 spanned by states satisfying ⟨�ρcp−1⟩ = 0. We view
+states with ⟨�ρcp−1⟩ ̸= 0 as having dressed charges excited.
+Since the undressed charges are created using L±
+cp, these
+dressed charges are created using �L±
+cp. We will not find
+an explicit form for ULU and thus will not precisely know
+throughout how much of parameter space the dressed
+(fattened) operators can be defined without violating the
+assumptions of ULU. Instead, we will assume that such an
+operator exists and can access a greater than measure-
+zero part of parameter space, and will investigate the
+consequences of this conjecture.
+At this point in our investigations, we cannot tell if the
+dressed charge gap ∆dressed-charge is an IR scale or a mid-
+IR I scale or a mid-II scale, etc. In section III C 2, we will
+learn that it is a mid-IR scale in region III of parameter
+space but an IR scale in region II of parameter space
+(see Fig. 7). For the rest of this section, however, we will
+adopt the language from the perspective of region III and
+call the dressed charge gap a mid-IR scale.
+Given the mid-IR scale Emid-IR ≡ ∆dressed-charge, we
+would now like to the find an effective mid-IR theory,
+which by definition is a theory only describing states at
+energies E < Emid-IR. As we learned in section II, the ef-
+fective mid-IR Hamiltonian Hmid-IR is a sum of all terms
+allowed in the mid-IR that can be constructed from the
+terms in �HUV. The terms in �HUV are �ρ2
+cp−1, (�Lz
+cp)2, and
+�L±
+cp. In the mid-IR, �ρ2
+cp−1 = 0 so it will not appear in
+Hmid-IR. Furthermore, since (�Lz
+cp)2 commutes with �ρ2
+cp−1,
+it does not excite any dressed charges. Thus (�Lz
+cp)2 is an
+allowed mid-IR operator from which terms in Hmid-IR can
+be constructed. The operators �L±
+cp are not allowed oper-
+ators in the mid-IR since they excite charges. However,
+allowed mid-IR operators can be constructed from �L±
+cp.
+Indeed, since �L+
+cp excites a dressed charge on ∂cp, acting
+�L+
+cp on any p-cycle does not excite dressed charges since
+∂ = 0. Therefore, an allowed operator in the mid-IR is15
+�
+W †[Cp] =
+�
+cp∈Cp
+�L+
+cp.
+(70)
+We call �
+W † the Wilson operator. It has the interpretation
+of exciting a dressed charge, transporting it along a p-
+cycle, and then ultimately annihilates it. Fig. 10 shows
+a few graphical representations of the Wilson operator
+supported on the smallest possible p-cycle: the boundary
+of a (p + 1)-cell. Note that the Wilson operator satisfies
+�
+W †[Cp] = �
+W[−Cp].
+The effective mid-IR theory, therefore, includes all
+terms constructed from �
+W and �L2
+z.
+Denoting the set
+of all oriented p-cycles with integer coefficients on the
+spatial lattice Md as Zp(Md; Z), the mid-IR Hamiltonian
+takes the form
+H(III)
+mid-IR = κU
+2
+�
+cp
+(�Lz
+cp)2− U
+2
+�
+Cp∈Zp(Md;Z)
+ε|Cp|�
+W[Cp] + · · ·, (71)
+where κ ∼ K/U, ε ∼ J/U, |Cp| is the number of p-cells
+form which Cp is constructed, and the · · ·’s include all
+other possible terms constructed from both (�Lz
+cp)2 and
+�
+W[Cp]. Since the dressed charge gap is the mid-IR scale
+of region III but the IR of region II,
+H(III)
+mid-IR ≡ H(II)
+IR .
+(72)
+The effective mid-IR theory is only well defined pro-
+vided ε < 1.
+As a consequence, since the lattice is
+simply connected, the Wilson operators supported on
+nontrivial p-cycles are exponentially suppressed by εLp,
+where L is the linear system size16.
+Therefore, de-
+noting contractible p-cycles with integer coefficients on
+the spatial lattice Md as Bp(Md; Z), terms with Wil-
+son operators supported on cycles in the homology class
+Hp(Md; Z) = Zp(Md; Z)/Bp(Md; Z) vanish in the ther-
+modynamic limit. Therefore, in the thermodynamic limit
+H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+L→∞
+= κU
+2
+�
+cp
+(�Lz
+cp)2− U
+2
+�
+Cp∈Bp(Md;Z)
+ε|Cp|�
+W[Cp] + · · ·, (73)
+15 Describing the extended object �
+W † using local operators intro-
+duces a gauge redundancy. Indeed, �
+W † is invariant under
+�L+
+cp → e i Ξcp �L+
+cp
+with
+�
+cp∈Cp
+Ξcp ∈ 2πZ ∀ Cp.
+(69)
+When Ξcp = (dχ)cp (where d is the lattice exterior derivative,
+see Eq. (75)), �Θcp transforms as the canonical U(1) gauge redun-
+dancy �Θcp → �Θcp + (dχ)cp. For Ξcp ̸= (dχ)cp, Eq. (69) corre-
+sponds to large gauge transformations.
+16 When p = 1, the mid-IR theory Eq. (71) can be thought of as a
+lattice regularization of the Landau-Ginzburg string field theory
+developed in Ref. 33. However, whereas the suppression of large
+loops needed to be inserted by hand in the string field theory,
+here it automatically arises from the UV theory Eq. (66).
+
+21
+where the · · ·’s include all possible terms constructed
+from both (�Lz
+cp)2 and �
+W[Cp] with only Cp ∈ Bp(Md; Z).
+In the thermodynamic limit, H(III)
+mid-IR is invariant under
+transforming the UV operator as
+�L+
+cp → eiΓcp �L+
+cp,
+with
+(dΓ)cp+1 = 0.
+(74)
+Here (dΓ)cp+1 is the lattice exterior derivative of Γcp,
+defined as
+(dΓ)cp+1 ≡
+�
+cp∈∂cp+1
+Γcp.
+(75)
+Under this transformation, the Wilson operator trans-
+forms as
+�
+W[Cp] → ei �
+cp∈Cp Γcp �
+W[Cp],
+(76)
+which leaves Eq. (73) unchanged because �
+cp∈Cp Γcp = 0
+for Cp ∈ Bp(Md; Z) since (dΓ)cp+1 = 0.
+This transfor-
+mation is importantly different than the gauge trans-
+formations Eq. (69) because the Wilson operator trans-
+forms nontrivially under it. Indeed, the Wilson opera-
+tors supported on p-cycles in Hp(Md; Z) transform un-
+der Eq. (74) by a nontrivial element of U(1). Therefore,
+since the physical operators are supported on a p-cycle
+and pick up a phase under the symmetry transformation,
+Eq. (74) corresponds to the transformation of a U(1) p-
+form symmetry—a U(1)(p) symmetry. Throughout the
+remainder of this section, we will always assume to be
+working in the thermodynamic limit.
+The symmetry operator of this U(1)(p) symmetry is
+�Uα(ˆΣd−p) =
+�
+ˆcd−p∈ˆΣd−p
+exp
+�
+iα (∗ �Lz)ˆcd−p
+�
+,
+(77)
+where α ∈ [0, 2π), ˆΣd−p is a (d − p)-cycle of the dual lat-
+tice, and (∗ �Lz)ˆcd−p ≡ �Lz
+∗ ˆcd−p.
+We again see why this
+is a p-form symmetry: its symmetry operator acts on a
+(d − p)-cycle. Fig. 11 shows a graphical representation of
+�Uα(ˆΣd−p) when ˆΣd−p = ∂ˆcd−p+1. Furthermore, since Lz
+has integer eigenvalues, the charge operator
+�QˆΣd−p =
+�
+ˆcd−p∈ˆΣd−p
+(∗ �Lz)ˆcd−p
+(78)
+also
+has
+integer
+eigenvalues.
+Note
+that
+when
+ˆΣd−p = ∂ ˆOd−p+1, using Eq. (A6) �QˆΣd−p can be rewritten
+as
+�Q∂ ˆ
+Od−p+1=
+�
+ˆcd−p+1∈Od−p+1
+(−1)p(∗ �ρ )ˆcd−p+1.
+(79)
+To confirm that Eq. (77) is the symmetry operator,
+first note that eiα�Lz
+cp �L+
+cp e−iα�Lz
+cp = eiα�L+
+cp. Therefore,
+letting #(A, B) denote the intersection number between
+the chains A and B,
+�Uα(ˆΣd−p)�
+W[Cp]�U †
+α(ˆΣd−p) = eiα#(Cp,ˆΣd−p)�
+W[Cp]. (80)
+Introducing Γcp, the Poincar´e dual of ˆΣd−p with respect
+to the spatial lattice, given by
+(∗ Γ)ˆcd−p(ˆΣd−p) =
+�
+ˆyd−p∈ˆΣd−p
+δˆyd−p,ˆcd−p,
+(81)
+the
+intersection
+number
+can
+be
+written
+as
+#(Cp, ˆΣd−p) = �
+cp∈Cp Γcp.
+Since
+∂ ˆΣd−p = 0,
+Γcp
+satisfies
+(δ ∗ Γ)ˆcd−p−1 = 0,
+which
+from
+Eq
+(A6)
+is
+equivalent to (dΓ)cp+1 = 0.
+Therefore, Eq. (80) cor-
+rectly reproduces the U(1)(p) symmetry transformation
+Eq. (76).
+The U(1)(p) symmetry is a symmetry of the effec-
+tive mid-IR theory but not the UV theory.
+There-
+fore, it is an emergent symmetry, emerging at energies
+E < ∆dressed-charge. In the thermodynamic limit, since it
+is an exact symmetry of the effective mid-IR theory, we
+say that the U(1)(p) symmetry is an exact emergent sym-
+metry. Since p > 0, this result that an emergent symme-
+try can act like an exact symmetry only applies to p-form
+symmetries with p > 0.
+The exact emergent U(1)(p) symmetry is not present
+throughout the entire parameter space of HUV. Whether
+or not it emerges depends on both the existence of the
+mid-IR and the existence of the effective mid-IR Hamil-
+tonian.
+As mentioned at the start of this subsection,
+the mid-IR only exists when U ≫ K. The effective mid-
+IR Hamiltonian is only well defined provided that the
+infinite series expansions converge, which required that
+ε < 1 and so J/U ≪ 1. Thus, the U(1)(p) symmetry only
+emerges when K/U ≪ 1 and J/U ≪ 1.
+2.
+An exact emergent anomalous U(1)(p) × U(1)(d−p−1)
+symmetry
+We have seen that at energies below the dressed charge
+gap, there is an exact emergent U(1)(p) symmetry. Here,
+we will search for additional energy scales where new
+symmetries can emerge. In fact, such a lower-energy scale
+only exists in region III of parameter space (see Fig. 7).
+This makes the dressed charge gap the IR scale of region
+II but the mid-IR scale of region III. Here, we will iden-
+tify the lower energy scale of region III with the gap of
+the topological defects in the U(1)(p) spontaneous sym-
+metry breaking (SSB) phase. Since there are no other
+energy scales in region III, this is an IR scale.
+To show this, let us first discuss when the U(1)(p) sym-
+metry is spontaneously broken17. To gain some intuition,
+we consider two tractable limits of the effective mid-IR
+theory. The first limit is in region II when J/U = 0 but
+17 See footnote 23 for the definition of spontaneous symmetry
+breaking.
+
+22
+K/U ̸= 0 such that H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+J/U=0
+= κU
+2
+�
+cp
+(�Lz
+cp)2.
+(82)
+The ground state in this limit satisfies �Lz
+cp |vac⟩ = 0 for all
+cp. Because acting �
+W †[Cp] onto a state changes the value
+of ⟨�Lz
+cp⟩ for each cp ∈ Cp, ⟨�
+W †[Cp]⟩ = 0 for all Cp. Con-
+sequently, this limit lies in a U(1)(p) symmetric phase.
+The other tractable limit is in region III K/U = 0 but
+J/U ̸= 0 such that H(III)
+mid-IR becomes
+H(III)
+mid-IR
+����
+K/U=0
+= −U
+2
+�
+Cp∈Bp(Md;Z)
+ε|Cp| �
+W[Cp].
+(83)
+The
+ground
+state
+in
+this
+limit
+clearly
+satisfies
+�
+W[Cp] |vac⟩ = |vac⟩ for all Cp ∈ Bp(Md; ZN), and conse-
+quently ⟨�
+W †[Cp]⟩ = 1 for all trivial p-cycles. Therefore,
+in this limit the U(1)(p) symmetry is spontaneously bro-
+ken.
+According to the higher Coleman-Mermin-Wagner the-
+orem, in (d + 1)-dimensional spacetime, a U(1)(p) sym-
+metry at zero temperature can spontaneously break in
+the thermodynamic limit when d > p + 1 [2, 28]. There-
+fore, when d ≤ p + 1, there is no stable U(1)(p) SSB
+phase, and any SSB features are unique to K/U = 0.
+However, when d > p + 1, we expect a stable SSB phase
+even for K/U ̸= 0.
+For small κ ∼ K/U and small
+ε ∼ J/U, a reasonable expectation is that the SSB phase
+occurs when κ ≲ εmin |Cp| = ε2(p+1), which is equivalent
+to K/U ≪ (J/U)2(p+1). This identifies region II of pa-
+rameter space as the U(1)(p) symmetric phase while re-
+gion III of parameter space is the U(1)(p) SSB phase. In
+fact, the boundary between regions II and III in Fig. 7
+is a depiction of the boundary between the U(1)(p) sym-
+metric and U(1)(p) SSB phases. We leave a more detailed
+investigation of this phase transition to future work.
+Since the order parameter for U(1)(p) symmetry break-
+ing is the vacuum expectation value of �
+W †[Cp], the Wil-
+son operator can detect topological defects related to the
+nontrivial mappings ⟨�
+W †[Cp]⟩ : Zp(Md; Z) �→ U(1). The
+topological defects excited in a state |ψ⟩ are probed by re-
+peatedly acting the Wilson operator over a trivial (p + 1)-
+cycle Cp+1:18
+�
+cp+1∈Cp+1
+�
+W †[∂cp+1] |ψ⟩ = exp
+�
+2πi ˆQ(Cp+1)
+�
+|ψ⟩ . (84)
+Here, ˆQ(Cp+1) is a winding number and yields the net
+number of topological defects enclosed by Cp+1. Plugging
+18 This is a natural generalization of the p = 0 case, where the topo-
+logical defects are vortices.
+in �
+W † and using Stoke’s theorem, it can be expressed as
+ˆQ(Cp+1) = 1
+2π
+�
+cp+1∈Cp+1
+Fcp+1,
+(85)
+where Fcp+1 = (d�Θ)cp+1 mod 2π.
+Using the identity
+x mod n = x − n ⌊x/n⌉, where ⌊·⌉ rounds its input to the
+nearest integer, Fcp+1 can be written as
+Fcp+1 ≡ (d�Θ)cp+1 + ωcp+1,
+(86)
+where ωcp+1 ≡ −2π
+�
+(d�Θ)cp+1/(2π)
+�
+.
+The topological defects can be characterized locally us-
+ing the topological defect density ˆρ which is defined im-
+plicitly as
+ˆQ(Cp+1 = ∂Op+2) ≡
+�
+cp+2∈Op+2
+(∗ ˆρ)cp+2.
+(87)
+Plugging in Cp+1 = ∂Op+2 into Eq. (85) and using
+Stoke’s theorem, we find
+(∗ ˆρ)cp+2 = 1
+2π (dF)cp+2.
+(88)
+Since ˆρ is supported on a (d − p − 2)-cell of the dual lat-
+tice, the topological defects are (d − p − 2)-dimensional
+excitations in space, residing on the boundary of a
+(d − p − 1)-cell of the dual lattice (see Fig. 13).
+Note
+that because the eigenvalues of ωcp+1 take values in 2πZ,
+the eigenvalues of (∗ ˆρ)cp+2 are integers19.
+In the J/U → 0 limit of the U(1)(p) symmetric phase,
+the ground state satisfies �Lz
+cp |vac⟩ = 0. Since �Lz
+cp and
+�Θcp are conjugate variables, in the ground state �Θcp
+fluctuates wildly. Consequently, Fcp+1 fluctuates wildly
+and the ground state satisfies (dF)cp+2 |vac⟩ ̸= 0, so
+the topological defects are condensed.
+On the other
+hand, in the K/U → 0 limit of the U(1)(p) SSB phase,
+the ground state satisfies �
+W[Cp] |vac⟩ = |vac⟩ for all
+Cp ∈ Bp(Md; ZN).
+In terms of the dressed phase vari-
+able �Θcp, this implies that (d�Θ)cp+1 |vac⟩ = 2πn with
+n ∈ Z. Consequently, Fcp+1 |vac⟩ = 0 for all (p + 1)-cells
+and therefore ˆρˆcd−p−2 |vac⟩ = 0. Thus, topological defects
+do not populate the ground state and are gapped excita-
+tions.
+However, these topological defects cannot be observed
+directly in the lattice model20. Indeed since Θcp always
+19 Note that for p = 1, the U(1)(1) SSB phase is a Coulomb phase.
+In this case, we find (d − 3)-dimensional topological defects car-
+rying integer topological charge. These are the Dirac magnetic
+“monopoles” of the Coulomb phase, whose interpretation as
+topological defects was also noted in Ref. 33.
+20 One could instead consider a Villain type Hamiltonian model
+for which these topological defects are observable even in the
+UV/mid-IR [68–70].
+Nevertheless, these different UV lattice
+models should have the same IR effective field theory.
+
+23
++
+−
+−
++
+−
++
+−
++
++
+−
++
+−
+    
+   
+̂ρ ̂cd−p−2 = ∑
+⌊
+d ˜Θ
+2π ⌉
+±
++
+−
+FIG. 13. Graphical representation of ˆρˆcd−p−2 (see Eq. (88))
+in three spatial dimensions for (first row) p = 0 and (second
+row) p = 1. The disks labeled by ± denote the sign in front
+of
+�
+d�Θ/(2π)
+�
+belonging to that (p + 1)-cell in the sum for
+ˆρˆcd−p−2. The direct lattice is colored in black while the dual
+lattice is in red. Furthermore, the (d − p − 2)-cell of the dual
+lattice ˆρˆcd−p−2 is associated with is highlighted in blue.
+appears as L+
+cp = eiΘcp , (∗ ˆρ)cp+2 too always appears as
+ei2π(∗ ˆρ)cp+2 and so ˆρˆcd−p−2 ∼ ˆρˆcd−p−2 + 1 on the lattice.
+An additional sign that they are trivial on the lattice
+comes attempting to construct an operator which does
+not create any topological defects (the equivalent of the
+Wilson operator from the last section).
+Indeed, they
+are created any operator which causes the eigenvalue
+of d�Θcp+1 to jump through the boundary of its range
+[0, 2π) (e.g., going from π to 3π). If an operator can do
+this for only a single (p + 1)-cell cp+1, it would excite
+a single dressed topological defect residing on ∂ ∗ cp+1.
+It is tempting to consider the operator exp[ix(∗ �Lz)ˆcd−p]
+since it shifts d�Θcp+1 by x(−1)∗ ˆcd−p. In fact, if x = 2π, it
+shifts
+�
+d�Θ
+2π
+�
+cp+1 by (−1)∗ ˆcd−p: the sign of ∗ ˆcd−p in ∂cp+1.
+However, steaming from the fact that ∂2 = 0, this shift
+of
+�
+d�Θ
+2π
+�
+cp+1 does not cause d
+�
+d�Θ
+2π
+�
+cp+2 to shift. Thus,
+we find that an operator that does no excite topological
+charges is
+�T †[∂ ˆOd−p] =
+�
+ˆcd−p∈ ˆ
+Od−p
+exp
+�
+2πi(∗ �Lz)ˆcd−p
+�
+(89)
+where ˆOd−p is an open (d − p)-dimensional subspace of
+the dual lattice. We call �T ′† the ’t Hooft operator, which
+is generally supported on (d − p − 1)-cycles of the dual
+lattice. Physically �T ′† excites a dressed topological de-
+fect, transports it along the (d − p − 1)-cycle, and ulti-
+mately annihilates it. However, since the eigenvalues of
+�Lz are integers, the operator �T † is in fact trivial.
+While the topological defects are unobservable on the
+lattice, their effects emergent in the continuum limit.
+The general paradigm for most lattice models is that
+the effective IR theory deep into a phase of matter is a
+continuum quantum field theory which reflects only the
+universal properties of that phase21. Finding the IR ef-
+fective field theory involves going deep into the U(1)(p)
+SSB phase and taking the continuum limit. Deep into the
+SSB phase, the effective IR hamiltonian Eq. (73) includes
+only the leading order in κ and ε terms:
+Hdeep IR ≈ κU
+2
+�
+cp
+�
+�L′z
+cp
+�2
++ ε2p+2U
+2
+�
+cp+1
+�
+F ′
+cp+1
+�2
+. (90)
+In the field theory, these higher-order terms could con-
+tribute as higher-derivative terms, but do not affect the
+deep IR.
+Appendix section C, shows how we take the contin-
+uum limit of Hdeep IR, doing so carefully to capture the
+topologically nontrivial parts of the quantum fields from
+the lattice operators. We find that the IR effective field
+theory is compact p-form Maxwell theory, described by
+the path integral
+Zdeep IR =
+�
+D[a]
+�
+ωa∈2πHp+1(X;Z)
+ei
+�
+X Ldeep IR,
+Ldeep IR = − 1
+2g2 Fa ∧ ∗ Fa,
+(91)
+where
+a
+is
+a
+p-form
+in
+Minkowski
+spacetime
+X,
+Hp+1(X; Z) is the (p + 1)th de Rham cohomology group
+with integral periods, and Fa ≡ da + ωa. This field the-
+ory describes the dynamical fluctuations of the
+(d−1)!
+p!(d−p−1)!
+p-form Goldstone bosons of the U(1)(p) SSB phase [31].
+Furthermore, as reviewed in depth in appendix sec-
+tion C 1, it has an anomalous U(1)(p) ×U(1)(d−p−1) sym-
+metry [2]. Therefore, deep into the U(1)(p) SSB phase
+of the lattice model, a new symmetry emerges.
+How-
+ever, the U(1)(p) and U(1)(d−p−1) symmetries are not
+independent of one another, there is a mixed ’t Hooft
+anomaly. Furthermore, the field theory also an emergent
+Lorentz invariance, and in terms of the UV parameters,
+the “speed of light” is c = Uεp+1√κ ∼ Jp+1
+U p
+�
+K
+U .
+IV.
+PHYSICAL CONSEQUENCES OF EXACT
+EMERGENT HIGHER-FORM SYMMETRIES
+In the previous section, we applied the framework in-
+troduced in section II to three lattice models without
+21 There are fascinating counter-examples where exotic lattice mod-
+els exhibit UV/IR mixing [54–57]. In these cases, the UV lattice
+details sneak into the definitions of the IR (and mid-IRs), causing
+the IR to exhibit a sensitive dependency on UV details. Conse-
+quently, since the thermodynamic limit and continuum limit do
+not commute, the effective IR field theory is no longer a conven-
+tional continuum quantum field theory.
+
+24
+exact higher-form symmetries and found that in particu-
+lar regions of parameter space, there are exact emergent
+higher-form symmetries below particular energy scales.
+In this section, we summarize the general lessons learned.
+Furthermore, we discuss the physical consequences of
+these general lessons and why the phases of a microscopic
+(UV) theory without exact higher-form symmetries be
+exactly characterized by emergent higher-form symme-
+tries.
+It is well known that emergent 0-form symmetries are
+never exact; thus, their consequences are always approx-
+imate. However, the emergent higher-form symmetries
+we identified are exact emergent symmetries; thus, they
+can exactly constrain the IR in the regions of parameter
+space they emerge. This important difference between
+0-form and higher-form symmetries is natural from the
+point of view of an effective mid-IR theory. Firstly, note
+how from Eq. (9), if the UV theory has a symmetry,
+any effective mid-IR theory will also have that symme-
+try. However, if the symmetry is explicitly broken in the
+UV theory, the effective mid-IR theory will generically
+include terms charged under the symmetry. For 0-form
+symmetries, the charged operators are local, so terms
+including them will persist in the thermodynamic limit.
+For higher-form symmetries, however, the charged opera-
+tors are non-local, so terms including them will vanish in
+the thermodynamic limit. A general consequence is that
+while emergent 0-form symmetries are never exact, emer-
+gent higher-form symmetries are always exact emergent
+symmetries whenever spacetime is simply connected.
+The exact emergent higher-form symmetries are robust
+against any local perturbations of the UV theory.
+In
+other words, adding local terms to the UV will not change
+the fact that the higher-form symmetry emerges nor the
+fact that it becomes an exact symmetry of the effective
+theory in the thermodynamic limit. Indeed, adding any
+local term to HUV does not affect the form of Hmid-IR
+because the perturbation will either not survive the pro-
+jection to the mid-IR or simply renormalize the effective
+Hamiltonian’s parameters. In this sense, exact emergent
+higher-form symmetries are topologically robust, and any
+physical properties arising from their existence are also
+robust to local UV perturbations.
+An immediate consequence of an exact emergent
+higher-form symmetry emerging at E < Emid-IR is that
+mid-IR states and mid-IR observables22 are organized
+into its representations. This implies the existence of an
+exact emergent conservation law obeyed at E < Emid-IR.
+Furthermore, this gives rise to selection rules on the cor-
+relation functions of mid-IR allowed operators [2, 71].
+In particular, mid-IR operators charged under the exact
+22 A mid-IR state is a quantum state in the the sub-Hilbert space
+spanned by energy-eigenstates with E < Emid-IR. A mid-IR ob-
+servable is an operator that takes states in the sub-Hilbert space
+spanned by energy eigenstates with E < Emid-IR to only other
+states in that sub-Hilbert space.
+̂x
+̂y
+̂t
+Emergent Symmetry
+No Emergent Symmetry
+FIG. 14. The existence of an exact emergent p-form symmetry
+constrains the way a p-dimensional membrane excitation can
+decay. Shown here is a cartoon of the different decay processes
+in spacetime for p = 1. The world sheet of the 1-dimensional
+membrane excitation is colored blue, while the worldline of
+charge excitations is orange.
+emergent symmetry must have a zero vacuum expecta-
+tion value.
+The presence of an exact emergent higher-form sym-
+metry also affects the dynamics of a quantum-many body
+system in a phase where the exact emergent higher-form
+symmetry is not spontaneously broken. Indeed, if a p-
+form symmetry (with p > 0) emerges at E < Emid-IR, it
+constraints the dynamics of mid-IR states with excited
+p-dimensional membrane excitations created by an oper-
+ator charged under the p-form symmetry. Indeed, con-
+sider starting in the state �
+W †[Cp] |vac⟩, where Cp is con-
+tractible and |vac⟩ is the ground state. The lifetime of
+the p-dimensional membrane excitation is qualitatively
+different depending on whether or not the exact emer-
+gent p-form symmetry is present. When the symmetry
+is present, the only way for the p-dimensional membrane
+excitation to decay is for it to contract to a point, so
+the lifetime of the p-dimensional membrane excitation
+depends on |Cp|.
+This also implies that if there is a
+trap potential for the p-dimensional membrane excita-
+tion (i.e., a term in the Hamiltonian such that there is a
+p-dimensional membrane excitation on Cp in the ground
+state), the exact emergent symmetry ensures that its life-
+time is infinity. In the absence of the symmetry, the p-
+dimensional membrane excitation can decay by breaking
+apart. Thus, without the emergent symmetry the lifetime
+of the p-dimensional membrane excitation is independent
+of its size |Cp|. Fig. 14 shows a cartoon depicting these
+differing behaviors.
+Because emergent higher-form symmetries are exact
+emergent symmetries, they characterize the IR of the
+parameter space region where they emerge as if they
+were UV symmetries. This, combined with the fact that
+emergent higher-form symmetries become exact emer-
+gent symmetries in the thermodynamic, implies that
+emergent higher-form symmetries characterize phases of
+matter with the same power as exact UV symmetries.
+One way that exact emergent higher-form symmetries
+characterize phases, which we encountered in our exam-
+ples, is by their ability to become spontaneously bro-
+
+25
+ken in the thermodynamic limit23. Indeed, since sponta-
+neous symmetry breaking is diagnosed using the ground
+state, for which an emergent higher-form symmetry is
+exact, an emergent higher-form symmetry can be spon-
+taneously broken in the same way a UV symmetry can
+be spontaneously broken. A consequence of this is that a
+phase with an emergent discrete higher-form symmetry
+spontaneously broken has an exact ground state degen-
+eracy which depends on spacetime’s topology. Another
+consequence is that a phase with an emergent contin-
+uous higher-form symmetry spontaneously broken has
+Goldstone bosons.
+If the continuous higher-form sym-
+metry emerges at E < Emid-IR, these Goldstone bosons
+are exactly gapless for mid-IR states. However, for states
+in the sub-Hilbert space spanned by energy eigenstates
+with E ≥ Emid-IR, the Goldstone bosons acquire a gap24.
+This is very different from a 0-form continuous symme-
+try where even weakly breaking the symmetry in the
+UV gaps out the Goldstone boson [44].
+However, be-
+cause exact emergent higher-form symmetries are topo-
+logically robust, a local UV perturbation does not gap out
+the higher-form Goldstone bosons nor lift the topological
+ground state degeneracy. The last physical consequence
+we will note is that the SSB phase of a p-form symmetry
+has gapped topological defect excitations, which formally
+arise due to nontrivial mappings from p-cycles to the or-
+der parameter manifold [33, 73]. From the examples in
+section III, when an anomaly-free U(1)(p) (Z(p)
+N ) symme-
+try spontaneously breaks in d-dimensional space, there
+are d − p − 2 (d − p − 1) dimensional topological defects
+carrying Z (ZN) topological charge.
+Another way exact emergent higher-form symmetries
+can characterize phases, which we did not encounter
+in our examples, is by giving rise to nontrivial emer-
+gent symmetry-protected topological (SPT) orders25.
+For instance, if a higher-form symmetry emerges at
+E < Emid-IR and is realized anomalously on the bound-
+23 To be precise, by spontaneous symmetry breaking, we mean
+a phase where an order parameter constructed from oper-
+ators charged under a symmetry acquires a nonzero vac-
+uum expectation value in the thermodynamic limit.
+For in-
+stance, |vac⟩ has spontaneously broken a U(1)(p) symmetry if
+⟨vac| W[Cp] |vac⟩ ̸= 0 for Cp ∈ Bp(Md) as |Cp| → ∞. Note that
+when p = 0, C0 ∈ B0(Md) is an oriented collection of two lat-
+tice points C0 = {−x, y} and so W[C0] = W †(x)W(y), which
+makes a connection to the historical perspective of long-range
+order. Physically, ⟨vac| W[Cp] |vac⟩ ̸= 0 implies that objects car-
+rying (neutral amounts) of symmetry charge have condensed. So,
+when p = 0 SSB implies the condensation of particles while when
+p = 1 SSB implies the condensation of loops [15, 72].
+24 This is a familiar concept in the p = 1 case where electric screen-
+ing causes the photon to acquire a gap (e.g., plasmas and metals).
+25 If an SPT phase is protected by a G symmetry, the G symme-
+try is realized anomalously on the system’s boundary, endowing
+the boundary with gapless/degenerate modes or topological or-
+der [74].
+The presence of the bulk SPT order is reflected by
+the ability to nevertheless couple a background G gauge field in
+the bulk+boundary theory since the ’t Hooft anomaly on the
+boundary is canceled by an anomaly in-flow mechanism [75].
+ary, a corresponding nontrivial SPT order could also
+emerge at E < Emid-IR and cause the system to be in
+an SPT phase. The boundary could then have sponta-
+neously broken exact emergent higher-form symmetries,
+therefore hosting abelian topological orders or gapless
+(for E < Emid-IR) higher-form Goldstone bosons, or con-
+tain additional gapless degrees of freedom.
+This em-
+phasizes an important distinction between SPT phases
+protected by 0-form symmetries and those protected by
+higher-form symmetries. 0-form SPT phases cannot oc-
+cur in regions of parameter space where the 0-form sym-
+metry is explicitly broken in the UV. Indeed, explicitly
+breaking a 0-form symmetry in the UV breaks the sym-
+metry at all energy scales, hence preventing the boundary
+from realizing the symmetry anomalously, an anomaly
+inflow mechanism from occurring, etc. Amazingly, this
+is not true for higher-form SPT phases. Indeed, as we
+have seen, breaking a higher-form symmetry in the UV
+does not guarantee it remains broken in the IR due to
+the topological robustness of emergent higher-form sym-
+metries. Therefore, while there may not be any higher-
+form symmetries at E > Emid-IR, there could be an exact
+emergent higher-form symmetry at E < Emid-IR which is
+realized anomalously on the boundary. If so, the system
+will have nontrivial exact emergent SPT order if there
+also exists a corresponding anomaly inflow mechanism
+at E < Emid-IR that allows one to turn on background
+gauge fields of the higher-form symmetry. Therefore, to
+understand higher-form SPT phases, instead of partition-
+ing parameter space by the UV higher-form symmetries,
+as done for 0-form SPT phases, one must partition pa-
+rameter space by the exact emergent higher-form sym-
+metries.
+The final physical consequence of exact emergent
+higher-form symmetries that we will emphasize is their
+ability to be anomalous.
+When a ’t Hooft anomaly is
+present, the ground state cannot be a trivial product
+state, and the phase is guaranteed to be gapless, topo-
+logically ordered, or a gapped SSB phase.
+Therefore,
+in the presence of an anomalous symmetry, the ground
+state cannot be made into a trivial product state with-
+out getting rid of the ’t Hooft anomaly. Exact emergent
+higher-form symmetries can also be anomalous. Indeed,
+this was the case in all of the examples we considered
+in section III: in the SSB phases discussed, there were
+exact emergent anomalous higher-form symmetries that
+guaranteed these phases would have degenerate ground
+states or gapless modes in the thermodynamic limit. In
+fact, the ground state degeneracies and the gaplessness of
+Goldstone bosons in these phases were protected by the
+’t Hooft anomaly, and the only way to eliminate them
+would be to destroy the exact emergent anomalous sym-
+metry. In the example, this could be done by condensing
+either the gauge charges (if present) or the topological de-
+fects. Thus, if their gaps were held at infinity, the ground
+state in the SSB phase could never become a trivial prod-
+uct state due to the ’t Hooft anomaly.
+
+26
+V.
+CONCLUSION AND DISCUSSION
+In this paper, we have investigated the robustness of
+emergent higher-form symmetries from a UV perspec-
+tive, considering bosonic lattice Hamiltonian models. By
+identifying low-energy sub-Hilbert spaces using ULU from
+section II A, in section II we discussed how to write down
+effective Hamiltonians to identify emergent symmetries,
+which we applied to three lattice models in section III.
+We found that even when the lattice models did not have
+exact UV higher-form symmetries, there could be emer-
+gent higher-form symmetries whose effects in the IR are
+the same as if they were UV symmetries.
+To empha-
+size this robustness, we referred to emergent higher-form
+symmetries as exact emergent symmetries. Using this,
+we argued that exact emergent higher-form symmetries
+can exactly characterize the phases of microscopic models
+without exact UV higher-form symmetries. The physical
+consequences of this were summarized in detail in sec-
+tion IV.
+There are many exciting follow-up questions. Firstly,
+in section II, we arrived at our expression for the effective
+Hamiltonian in a non-rigorous manner. While its defini-
+tion is physically very reasonable, it would be desirable
+to have a rigorous derivation for its form.
+A possible
+starting point for such a proof could be building off of
+perturbatively defined effective Hamiltonians found using
+Brillouin-Wigner perturbation theory [76] or Schrieffer-
+Wolff transformations [77].
+It would also be interesting to investigate if emergent
+higher-form symmetries are no longer exact emergent
+symmetries, and instead approximate symmetries, in lat-
+tice models with lattice defects. Indeed, the reason we
+found why emergent higher-form symmetries act as ex-
+act symmetries boiled down to the fact that their charged
+operators are supported on nontrivial cycles and there-
+fore did not appear in the effective Hamiltonian in the
+thermodynamic limit. However, imagine removing lat-
+tice sites and introducing nontrivial cycles that did not
+involve a sub-extensive number of cells.
+Then, like 0-
+form symmetries, operators charged under the higher-
+form symmetry would appear in the effective Hamilto-
+nian if the UV theory did not have that higher-form
+symmetry, thus causing the emergent higher-form sym-
+metry to be approximate. This could be confirmed in
+spontaneous higher-form symmetry broken phases by in-
+vestigating if removing lattice sites lifts the topological
+ground state degeneracy/gaps the Goldstone bosons
+Another important follow-up would be an in-depth
+study of region II in Fig. 7. As discussed at the begin-
+ning of section III, the size of region II and the boundary
+between region II and I in models C and D is not rigor-
+ously investigated. It would be interesting to numerically
+investigate these models C and D to better understand
+region II in parameter space. Alternatively, if one could
+explicitly construct the local unitary ULU, it may be pos-
+sible for this question to be addressed analytically.
+Lastly, while we discussed symmetry-protected topo-
+logical (SPT) phases protected by exact emergent higher-
+form symmetries in section IV, the models we consid-
+ered in section III did not have SPT phases. It would
+be interesting to modify the models for emergent ZN p-
+gauge theory and U(1) p-gauge theory so they can have
+SPT phases protected by their exact emergent higher-
+form symmetries. In fact, the confined phase (the Z(p)
+N
+and U(1)(p) symmetric phases) of these gauge theories
+becomes an SPT upon adding a topological θ term to
+the Lagrangian [2, 42, 43]. Therefore, it would be inter-
+esting if one could modify the UV theories of the emer-
+gent gauge theory models to have the 2π quantized topo-
+logical term appear in the mid-IR effective Hamiltonian
+and investigate the resulting emergent SPT order from a
+Hamiltonian perspective.
+VI.
+ACKNOWLEDGEMENTS
+We are grateful for fun and helpful discussions with
+Arkya Chatterjee, Hart Goldman, Ethan Lake, Ho Tat
+Lam, Yu Leon Liu, and Carolyn Zhang. S.D.P. is sup-
+ported by the National Science Foundation Graduate Re-
+search Fellowship under Grant No. 2141064 and by the
+Henry W. Kendall Fellowship. This work is partially sup-
+ported by NSF DMR-2022428 and by the Simons Collab-
+oration on Ultra-Quantum Matter, which is a grant from
+the Simons Foundation (651446, XGW).
+Appendix A: Review of discrete differential
+geometry for d-dimensional cubic lattices
+In this appendix section, we review relevant parts of
+discrete differential geometry (in a non-rigorous fashion)
+used throughout the main text.
+Consider a cubic lat-
+tice in d-dimensional space with periodic boundary con-
+ditions, denoted by Md.
+While a Bravais lattice is a
+collection of lattice sites x ∈ Zd, it is useful to view it
+as also formed by higher-dimensional objects, like links,
+plaquettes, cubes, etc. We call a p-dimensional object
+a p-cell, with 0 ≤ p ≤ d. So, a 0-cell is a lattice site, a
+1-cell is a link, a 2-cell is a plaquette, etc. This does not
+add additional structures to the lattice, but instead is
+just a useful way of organizing the lattice sites. Indeed,
+denoting a p-cell associated with site x as cp(x)µ1µ2···µp,
+where µ1 < µ2 < · · · < µp and µi ∈ {1, 2, · · · , d}, a p-cell
+of the cubic lattice is the set of 2p lattice sites26
+cp(x)µ1µ2···µp= {x} ∪ {x + ˆµi | 1 ≤ i ≤ p}
+∪ {x + ˆµi + ˆµj | 1 ≤ i < j ≤ p}
+∪ · · · ∪ {x + ˆµ1 + . . . + ˆµp},
+(A1)
+26 We adopt the discrete differential geometry and exterior calculus
+notations and conventions used in Ref. 78.
+
+27
+p = 1
+p = 2
+p = 1
+p = 2
+p = 3
+FIG. 15.
+The p-cells of the d-dimensional cubic lattice
+are equivalently the 0-cells—the sites—of some other d-
+dimensional lattice. Shown here are examples of this equiv-
+alent lattice (drawn in pink) embedded in the conventional
+unit cell of the cubic lattice (drawn in black). (First row) In
+2 dimensions, the 1-cells form another square lattice, rotated
+by 45 degrees, whose lattice constant is 1/
+√
+2 times that of the
+original square lattice. The 2-cells also form another square
+lattice, which is the original shifted by the vector (ˆµ1 + ˆµ2)/2.
+(Second row) In 3 dimensions, both the 1-cells and also the 2-
+cells form a lattice of corner-sharing octahedra with a lattice
+constant that is 1/
+√
+2 times the cubic lattice’s. When p = 1,
+the octagons are centered at the cubic lattice’s 0-cells. When
+p = 2, the octagons are centered at the cubic lattices 3-cells.
+Lastly, the 3-cells form another cubic lattice of the same size,
+but shifted by the vector (ˆµ1 + ˆµ2 + ˆµ3)/2.
+where ˆµi is the unit vector in the µi-direction.
+It
+is often convenient to drop the requirement that the
+indices are canonically ordered (i.e., that they satisfy
+µ1 < µ2 < · · · < µp < ν) and instead let cp(x)µ1µ2···µp
+obey the relation cp(x)···µ1µ2··· = −cp(x)···µ2µ1···.
+The
+p-cells of the d-dimensional cubic lattice are equiva-
+lently viewed as the 0-cells of some other lattice in d-
+dimensions, as demonstrated for d = 2 and 3 in Fig. 15.
+Introducing the concept of p-cells is strictly unneces-
+sary but very convenient because “sewing” p-cells to-
+gether gives a natural way to form p-dimensional sub-
+spaces of the lattice. Furthermore these subspaces can
+also be given an orientation by defining an orientation
+structure to the lattice.
+A nice local scheme for the
+lattice orientation is a branching structure, where the
+orientation on each 1-cell is chosen such that a collec-
+tion of 1-cells cannot form an oriented closed loop. A
+canonical orientation on all other p-cells then follows from
+the branching structure. We use the branching structure
+where each 1-cell c1(x)µ has an arrow pointing in the ˆµ
+direction (see Fig. 16). However, it is important to note
+that the choice of lattice orientation is a formal conven-
+tion, and choosing different branching structures does not
+affect the physics27.
+A p-cell can be related to (p − 1) cells using the bound-
+ary operator ∂. The boundary operator acting on a p-
+27 However, according to a conjecture from Ref. 79, observables are
+independent of the branching structure only if the continuum
+effective field theory is free of a framing anomaly [80].
+̂x
+̂y
+̂z
+FIG. 16. Example of the branching structure used for a chunk
+of the cubic lattice in three-dimensional space.
+cell—∂cp—is the oriented sum of (p − 1)-cells on the
+boundary of cp. For the branching structure we use, it is
+given by
+∂cp(x)µ1···µp=
+p
+�
+k=1
+(−1)k+1�
+cp−1(x + ˆµk)µ1···
+oµk···µp
+−cp−1(x)µ1···
+oµk···µp
+�
+,
+(A2)
+where the notation
+oµk indicates that the µk index is omit-
+ted. From its definition, the boundary operator satisfies
+∂2cp = 0 for any p-cell.
+Furthermore, as there are no
+(−1)-cells, the boundary operator acting on a 0-cell is
+defined to be zero.
+On the other hand, a p-cell can be related to (p + 1)-
+cells using the coboundary operator δ.
+The cobound-
+ary operator acting on a p-cell—δcp—is an oriented sum
+of all (p + 1)-cells whose boundary includes cp. For the
+branching structure we use, it is given by
+δcp(x)µ1···µp =
+�
+ν
+cp+1(x)νµ1...µp − cp+1(x − ˆν)νµ1...µp.
+(A3)
+From its definition, the coboundary operator satisfies
+δ2cp = 0 for any p-cell.
+Furthermore, as there are no
+(d + 1)-cells, the coboundary operator acting on a d-cell
+is defined to be zero.
+Lastly, the lattice has an associated dual lattice. The
+dual lattice has its lattice sites centered at the d-cells
+of the direct lattice. For the cubic lattice, one way to
+relate a dual lattice site ˆx to a direct lattice site x is by
+ˆx = x + 1
+2 ˆr with ˆr = �
+i ˆµi.
+Each p-cell cp on the direct lattice is associated with
+a (d − p)-cell ˆcd−p on the dual lattice.
+This is imple-
+mented by the dual operator ∗. A p-cell cp(x)µ1···µp (with
+canonical ordering µ1 < · · · < µp) and a (d − p)-cell of
+the dual lattice ˆcd−p(ˆx)µ1···µd−p (with canonical ordering
+µ1 < · · · < µd−p) are related to one another by
+∗ cp(x)µ1···µp = ϵµ1···µpµp+1···µd
+(A4)
+× ˆcd−p(ˆx − ˆµp+1 − . . . − ˆµd)µp+1···µd,
+∗ ˆcp(ˆx)µ1...µp = ϵµ1···µpµp+1···µd
+(A5)
+× cd−p(x + ˆµ1 + . . . + ˆµp)µp+1···µd,
+where summation is not implied on the right hand side.
+Here ϵ is the Levi-Civita symbol, which takes into ac-
+
+28
+count the lattice’s and dual lattice’s relative orienta-
+tions. From the definition of ∗, acting ∗ twice on a p-cell
+of the direct (dual) lattice yields ∗ ∗ cp = (−1)p(d−p)cp
+(∗ ∗ ˆcp = (−1)p(d−p)ˆcp).
+Furthermore, from the defini-
+tions of the boundary, coboundary, and dual operators,
+they are related to one another by
+δcp = (−1)d(p+1)+1 ∗ ∂ ∗ cp,
+(A6)
+which, equivalently, is ∗ δcp = (−1)p∂ ∗ cp.
+Appendix B: A TQFT description of the p-form
+toric code ground states—p-form BF theory
+In section III B 2 of the main text, we found that the
+ground states of the Z(p)
+N
+SSB phase is equivalent to
+the ground states of the p-form toric code Hamiltonian
+Eq. (65) in terms of the dressed clock operators. In this
+appendix section, we relate the lattice description of the
+ground states to an equivalent topological quantum field
+theory description. Doing so demonstrates the connec-
+tion between exact emergent higher-form symmetries in
+lattice models and exact higher-form symmetries in La-
+grangian quantum field theories, where higher-form sym-
+metries are most commonly studied.
+The ground states of the Z(p)
+N
+SSB phase are defined
+by Eq. (64).
+To develop a field theory description of
+these ground states, we take inspiration from Ref. 81 and
+parameterize the clock operators as
+Xcp = exp
+�
+iΘcp
+�
+,
+Zcp = exp
+�
+i(∗ Φ)cp
+�
+.
+(B1)
+The dressed clock operators in the Z(p)
+N
+SSB phase then
+become
+�
+X′
+cp = exp
+�
+i �Θ′
+cp
+�
+,
+(B2)
+�Z′
+cp = exp
+�
+i(∗ �Φ′)cp
+�
+.
+(B3)
+Note that in order for ( �
+X′
+cp)N = ( �Z′
+cp)N = 1, it must
+be that the eigenvalues of �Θ′
+cp and (∗ �Φ′)cp satisfy
+�Θ′
+cp ∈ 2πZ/N, and (∗ �Φ′)cp ∈ 2πZ/N.
+Furthermore, in
+order for the clock operators algebra Eq. (34) to be sat-
+isfied, �Θ′
+cp and (∗ �Φ′)cp must obey the commutation re-
+lation [�Θ′
+cp, (∗ �Φ′)�cp] = 2π i
+N δcp,�cp.
+In terms of �Θ′
+cp and
+(∗ �Φ′)cp, the constraints Eq. (64) defining the IR are
+N
+2π δ(∗ �Φ′)cp−1 = N
+2π (d�Θ′)cp+1 = 0.
+(B4)
+The the lattice Heisenberg operators
+�Θ′
+cp(t) and
+(∗ �Φ′)cp(t) are related to their continuum counterparts
+�Θ′(t, x) and ∗ �Φ′(t, x) by
+�Θ′
+cp =
+�
+cp
+�Θ′,
+(∗ �Φ′)cp =
+�
+cp
+∗ �Φ′,
+(B5)
+where
+�
+cp denotes spatial integration over the p-cell cp.
+For simplicity, we will work locally and treat the con-
+tinuum quantum fields as differential forms in space (�Θ′
+is a p-form while �Φ′ is a (d − p)-form), mapping from
+spacetime to R/2πZ. One of the many conveniences of
+using the discrete exterior calculus notation is that in
+the continuum limit these lattice operators become their
+continuum versions. So, the constraint Eq. (B4) in the
+continuum limit becomes
+N
+2π d† ∗ �Φ′ = N
+2π d�Θ′ = 0.
+(B6)
+Here, d† is the adjoint of d, which when acting on a
+p-form is given by d† ≡ (−1)d(p+1)+1 ∗ d ∗.
+The lattice Hamiltonian in the IR is just the ground
+state energy, which we’ll set to zero. Thus, the continuum
+limit of H(III)
+IR , Eq. (63), is
+H(III)
+IR
+= 0.
+(B7)
+To write down the path integral, we can find the
+Lorentzian action from H(III)
+IR
+and then perform a func-
+tional integral over all dynamical fields.
+However, the
+path integral only integrates over field configurations sat-
+isfying Eq. (B6). Taking these constraints into account,
+the path integral in Lorentzian signature is
+Z(III)
+IR
+=
+�
+D[�Θ′]D[�Φ′] δ
+� N
+2π d† ∗ �Φ′
+�
+(B8)
+× δ
+� N
+2π d�Θ′
+�
+ei
+�
+dtddxL(III)
+IR ,
+L(III)
+IR
+=
+N
+2πp!(∗ �Φ′)i1···ip∂t �Θ′
+i1···ip
+The first term in L(III)
+IR
+enforces the equal-time commu-
+tation relation
+[�Θ′
+i1···ip(x), (∗ �Φ′)j1···jp(y)]
+p!
+= 2πi
+N δi1
+[j1· · · δip
+jp]δd(x − y).
+(B9)
+Since we work locally, we do not enforce the constraint
+that the holonomies of �Θ′ and ∗ �Φ′ are restricted to values
+in 2πZ/N.
+All that is left is to massage this path integral into a
+more familiar form. We can rewrite both of the func-
+tional delta functions by integrating in new fields acting
+as Lagrange multipliers and modifying the action. For
+instance, the first delta function can be rewritten using
+a (p − 1)-form Lagrange multiplier λ:
+δ
+� N
+2π d†∗ �Φ′
+�
+=
+�
+Dλ e
+i N
+2π
+�
+λi1···ip−1 ( d† ∗ �
+Φ′)i1···ip−1
+(p−1)!
+. (B10)
+Likewise, the second delta function can be rewritten us-
+ing a (p + 1)-form Lagrange multiplier ∗ η:
+δ
+� N
+2π d�Θ′
+�
+=
+�
+Dη e
+i N
+2π
+�
+(∗ η)i1···ip+1 ( d �
+Θ′)i1···ip+1
+(p+1)!
+.
+(B11)
+
+29
+Plugging these expressions into the Eq. (B8), the path
+integral becomes
+Z(III)
+IR
+=
+�
+D[�Θ′]D[�Φ′]D[λ]D[η] ei
+�
+dtddxL(III)
+IR ,
+(B12)
+L(III)
+IR = N
+2π
+�
+(∗ �Φ′)i1···ip∂t �Θ′
+i1···ip
+p!
++λi1···ip−1(d†∗ �Φ′)i1···ip−1
+(p − 1)!
++(∗ η)i1···ip+1(d�Θ′)i1···ip+1
+(p + 1)!
+�
+.
+Plugging in the components of d�Θ′ and d† ∗ �Φ′ and sim-
+plifying, L(III)
+IR
+can be rewritten as
+L(III)
+IR
+=
+N
+2πp!
+�
+(∗ �Φ′)i1···ip
+�
+∂t �Θ′
+i1···ip + p ∂[i1 λ i2···ip]
+�
++ (∗ η)i1i2···ip+1∂[i1 �Θ′
+i2···ip+1]
+�
+.
+(B13)
+Let’s
+now
+introduce
+the
+p-form
+a
+and
+(d − p)-
+form b in spacetime whose spatial components are
+ai1···ip = �Θ′
+i1···ip
+and
+bi1···id−p = (−1)d−p�Φ′
+i1···id−p
+and
+timelike
+components
+are
+a0i2···ip = −λi2···ip
+and
+b0i2···id−p = −ηi2···id−p.
+Note
+that
+with
+this
+identification,
+(∗ η)i1···ip+1 = −(∗ b)i1···ip+1
+and
+(∗ �Φ′)i1···ip = (∗ b)0i1···ip.
+Plugging these in, the path
+integral becomes
+Z(III)
+IR
+=
+�
+D[a]D[b] ei
+�
+dtddxL(III)
+IR ,
+L(III)
+IR
+=
+N
+2πp!
+�
+(∗ b)0i1···ip
+�
+∂tai1···ip + (−1)pp ∂[i1 a i2···ip]0
+�
+− (∗ b)i1i2···ip+1∂[i1 a i2···ip+1]
+�
+.
+(B14)
+The
+term
+in
+brackets
+can
+be
+rewritten
+using
+∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 = (p + 1)∂[0ai1···ip].
+Furthermore, working in flat spacetime, X is equipped
+with Minkowski metric (−, +, · · · +). Using it and sum-
+ming over spacetime indices µ, L(III)
+IR
+can be rewritten
+as
+L(III)
+IR
+= − N
+2π
+�(∗ b)µ1µ2···µp+1∂[µ1 a µ2···ip+1]
+p!
+�
+.
+(B15)
+Lasting, using differential forms notation, we arrive at
+our final expression for the path integral
+Z(III)
+IR
+=
+�
+D[a]D[b] ei
+�
+L(III)
+IR ,
+L(III)
+IR
+= N
+2π b ∧ da.
+(B16)
+As anticipated, the low-energy effective field theory,
+which describes the ground states of the Z(p)
+N
+symme-
+try broken phase, is p-form ZN gauge theory.
+This is
+arguably the simplest field theory with an anomalous
+Z(p)
+N × Z(d−p)
+N
+symmetry [66].
+1.
+Review of p-form BF theory
+In the remainder of this appendix section, we will re-
+view p-form BF theory, focusing on its symmetries and
+anomalies, working in D = d + 1 dimensional spacetime.
+From canonical quantization, the fields a and b satisfy
+the equal-time commutation relations
+[aµ1···µp(x), bµp+1···µd(y)] = 2πi
+N ϵ0µ1···µdδd(x − y).
+(B17)
+a.
+ZN p-form gauge theory in the continuum
+We start by review how p-form BF theory can be
+obtained by condensing charge-N gauge charges in p-
+form Maxwell theory [82–84].
+p-form Maxwell theory
+is reviewed in appendix section C 1. This will be use-
+ful as we’ll encounter dual representations of this theory
+which will make the symmetry analysis of the next sec-
+tion easier. However, for the reader who would like to
+jump straight to p-form BF theory, they should skip to
+Eq. (B28).
+We modify p-form Maxwell theory Eq. (C19) by intro-
+ducing the dynamical (p − 1)-form bosonic field H and
+the gauge redundancy
+a → a + dχ,
+H → H + Nχ,
+(B18)
+where N ∈ Z. A gauge-invariant globally defined quan-
+tity in terms of only H is FH = dH + ωH,
+where
+ωH ∈ 2πHp(X; Z).
+FH
+satisfies
+a
+Bianchi
+iden-
+tity
+1
+2π ∗ dFH = 0 and its periods are quantized as
+�
+FH ∈ 2πZ. We note that the local part of FH, dH,
+is only there when p ≥ 1. When p = 0, FH has no lo-
+cal fluctuations but only purely topological contributions
+from the 0-form ωH.
+With the additional degrees of freedom provided by H,
+we can now introduce the Wilson operator
+Wa,H(O) = exp
+�
+i
+�
+O
+Na − dH
+�
+,
+= exp
+�
+iN
+�
+O
+a
+�
+exp
+�
+−i
+�
+∂O
+H
+�
+,
+= W N
+a (O)W †
+H(∂O).
+(B19)
+where O is an open p-submanifold. Physically Wa,H(O)
+is an operator that creates a charge excitation on ∂O,
+but one carrying N-units of a-charge. H is the contin-
+uum phase operator of created fractionalized charges that
+carries N-units of a-charge. Minimally coupling FH to a
+in light of the gauge redundancy Eq. (B18), we find the
+
+30
+partition function
+Z =
+�
+D[a]D[H]
+�
+ωa
+2π ∈Hp+1(X;Z)
+ωH
+2π ∈Hp(X;Z)
+e−
+�
+X L,
+L =
+1
+2g2 |Fa|2 + v2
+2 |FH − Na|2.
+(B20)
+Expanding out the new term, we can rewrite the La-
+grangian density as
+L =
+1
+2g2 |Fa|2 + v2
+2 |FH|2 + N 2v2
+2
+|a|2 − Nv2a ∧ ∗ FH.
+(B21)
+Thus, we find that L ⊃ −a∧ ∗ J with J = Nv2FH for
+which generally d†J ̸= 0. Furthermore, there is now a
+mass term for a: L ⊃ N 2v2
+2
+|a|2. Indeed, the new term
+added to p-form Maxwell theory is essentially a Higgs
+term with H the phase of the Higgs field and v ∈ R the
+vev of the Higgs field. The gauge redundancy described
+by Eq. (B18) is a ZN gauge redundancy, reflecting how
+the initial U(1) gauge redundancy has been Higgsed down
+to a ZN gauge redundancy.
+As we saw in appendix section C 1 b, these types of
+theories have a generalized “particle-vortex” like du-
+ality called abelian duality.
+For instance, since the
+action’s dependency on H is entirely in the form of
+FH we can dualize H → ˆH using the same method
+shown in section C 1 b.
+Indeed, dualizing H to the
+(D − p − 1)-form ˆH satisfying
+�
+F ˆ
+H ∈ 2πZ and dF ˆ
+H
+(where F ˆ
+H = d ˆH + ω ˆ
+H), the Euclidean Lagrangian be-
+comes
+L = |Fa|2
+2g2 + |F ˆ
+H|2
+8π2v2 − iN
+2π a ∧ F ˆ
+H.
+(B22)
+The
+Euclidean
+path
+integral
+now
+integrates
+over
+the
+dynamical
+fields
+a
+and
+ˆH
+and
+sums
+over
+ωa ∈ 2πHp+1(X; Z) and ω ˆ
+H ∈ 2πHD−p(X; Z). We note
+that without changing the action amplitude, the La-
+grangian density can be rewritten as
+L = |Fa|2
+2g2 + |F ˆ
+H|2
+8π2v2 − iN
+2π
+ˆH ∧ Fa.
+(B23)
+Locally, we have just integrated by parts in the BF term.
+However, keeping track of the globally nontrivial parts of
+F ˆ
+H and Fa makes showing this difficult (it’s most natu-
+rally seen using Deligne-Beilinson cohomology [85]).
+Utilizing abelian duality we have found two representa-
+tions for the theory: the (a, H) representation Eq. (B20)
+and the (a, ˆH) representation Eqs. (B22) and (B23), the
+latter being dual only locally.
+In representation (a, ˆH), the deep IR is governed by
+p-form BF theory. Here, the deep IR refers to energies
+below the gap of a and ˆH, which have a gap through
+topological mass generation.
+Indeed, to find their en-
+ergy gaps, first note that in the (a, ˆH) representation,
+the Lorentzian action is
+S =
+�
+X
+�
+−|Fa|2
+2g2 − |F ˆ
+H|2
+8π2v2 + N
+2π a ∧ F ˆ
+H
+�
+.
+(B24)
+Since this theory is Gaussian, we can show that the a∧F ˆ
+H
+term causes all excitations to be gapped using the equa-
+tions of motion. Varying the action, we find that
+δS =
+�
+X
+�
+− 1
+g2
+�
+d†Fa + (−1)p(D−p) Ng2
+2π ∗ F ˆ
+H
+�
+∧ ∗ δa
+−
+1
+4π2v2
+�
+d†F ˆ
+H − (−1)p2+D2πNv2 ∗ Fa
+�
+∧ ∗ δ ˆH
+�
+,
+and therefore the classical equations of motion are
+d†Fa + (−1)p(D−p) Ng2
+2π ∗ F ˆ
+H = 0,
+d†F ˆ
+H − (−1)p2+D2πNv2 ∗ Fa = 0
+(B25)
+We can now decouple ∗ Fa and ∗ F ˆ
+H to get the equations
+�
+d† d + N 2g2v2�
+∗ F ˆ
+H = 0,
+�
+d† d + N 2g2v2�
+∗ Fa = 0.
+(B26)
+Using that d† ∗ Fb, ˆ
+H = 0, we can then rewrite this in
+terms of the Hodge Laplacian δ = d† d + dd† as
+�
+δ + N 2g2v2�
+∗ F ˆ
+H = 0,
+�
+δ + N 2g2v2�
+∗ Fa = 0.
+.
+(B27)
+Therefore,
+we
+see
+that
+the
+p-form
+∗ F ˆ
+H
+and
+the
+(D − p − 1)-form ∗ Fa both have an energy gap Ngv.
+To go below the energy gap into the deep IR, we take
+the limit g → ∞ and v → ∞. In this limit, the Euclidean
+path integral becomes
+ZBF =
+�
+D[a]D[ ˆH]
+�
+ω ˆ
+H
+2π ∈HD−p(X;Z)
+e−
+�
+X LBF ,
+LBF = − iN
+2π a ∧ F ˆ
+H.
+(B28)
+This is p-form BF theory, and it is in terms of the
+p-form bosonic field a, which is the U(1) gauge field
+we started with and ˆH, which is the abelian dual of
+the Higgs field phase.
+Taking the deep IR limit us-
+ing the Lagrangian density in this representation written
+as Eq. (B23), the Lagrangian in the topological limit is
+equivalent to LBF = − iN
+2π ˆH∧Fa.
+Plugging in F ˆ
+H = d ˆH + ω ˆ
+H into Eq. (B28), the path
+integral becomes
+ZBF =
+�
+D[a]D[ ˆH]
+�
+ω ˆ
+H
+2π ∈HD−p(X;Z)
+e
+i N
+2π
+� (a∧d ˆ
+H+a∧ω ˆ
+H)
+(B29)
+Integrating by parts on the first term and using Poincar´e
+duality on the second term, we can rewrite this as
+ZBF =
+�
+D[a]D[ ˆH]
+�
+ω∈Hp(X;Z)
+exp
+� iN
+2π
+�
+X
+ˆH ∧ da + iN
+�
+ω
+a
+�
+.
+(B30)
+
+31
+Integrating over ˆH and summing over ω, the path integral
+becomes
+ZBF =
+�
+D[a] δ(da) δ
+��
+a ∈ 2πZ
+N
+�
+.
+(B31)
+Notice that if we would have instead started with the
+Lagrangian density written as LBF = − iN
+2π ˆH∧Fa, upon
+integrating out a and ωa we’d get
+ZBF =
+�
+D[ ˆH] δ(d ˆH) δ
+��
+ˆH ∈ 2πZ
+N
+�
+.
+(B32)
+Having massaged p-form BF theory into Eq. (B31), we
+find that the U(1) gauge fields are closed forms and have
+quantized holonomies28. This has an important effect on
+the Wilson operators
+Wa[Cp] = exp
+�
+i
+�
+Cp
+a
+�
+,
+(B34)
+W ˆ
+H[CD−p−1] = exp
+�
+i
+�
+CD−p−1
+ˆH
+�
+.
+(B35)
+Indeed, since the holonomies of a and ˆH are quantized,
+the Wilson operators are constrained as
+Wa ∈ ZN,
+W ˆ
+H ∈ ZN.
+(B36)
+Thus, because the holonomies are quantized the Wilson
+operators satisfy
+(Wa)N = (W ˆ
+H)N = 1.
+(B37)
+When Cp ∈ Bp(X) (i.e., there exists an Op+1 such that
+Cp = ∂Op+1), Wa can be written as
+Wa[Cp] ≡ exp
+�
+i
+�
+Cp
+a
+�
+= exp
+�
+i
+�
+Op+1
+da
+�
+.
+Therefore, since da = 0, Wa supported on a contractible
+cycle is trivial element of ZN:
+Wa [Cp ∈ Bp(X)] = 1.
+(B38)
+This is also true for W ˆ
+H:
+W ˆ
+H [CD−p−1 ∈ BD−p−1(X)] = 1.
+(B39)
+28 This can also be deduced from the equations of motion. Indeed,
+in the (a, H) representation, Eq. (B20), the H equations of mo-
+tion in the deep IR are
+FH = Na.
+(B33)
+Therefore, because of the Bianchi identity dFH = 0, a is a
+closed p-form. Furthermore, since FH satisfies
+�
+FH = 2πZ, the
+holonomies of a are quantized as
+�
+a = 2πZ/N.
+Therefore, the Wilson operators are topological opera-
+tors, meaning that they can be continuously deformed:
+W[C + ∂O] = W[C].
+(B40)
+Therefore, at low-energies, any contractible Wilson oper-
+ators can condense into vacuum, but for non-contractible
+Wilson operators only N of them can condense into vac-
+uum.
+We can evaluate ZBF by considering the Wilson op-
+erators, which is convenient as we do not have to worry
+about performing the path integral in the presence of
+gauge redundancies.
+Indeed, the path integral simply
+counts the number of nontrivial Wilson operators W(C)
+which satisfy W(C)N = 1, and thus the number of con-
+figurations a ∈ Hp(X; ZN):
+ZBF =
+�
+a∈Hp(X;ZN)
+1,
+= |Hp(X, ZN)|.
+The coefficients of Hp(X; ZN) are ZN to preserve the
+condition Eq. (B36).
+The number of ground states is
+given by the partition function evaluated on X = R × M,
+where M is a space. Therefore, there are |Hp(M; ZN)|
+degenerate ground states.
+b.
+Symmetries
+Having reviewed the basics of p-form BF theory in the
+previous section, we now turn to identifying the theory’s
+symmetries, showing that there is a Z(p)
+N × Z(d−p)
+N
+sym-
+metry [66].
+Let’s first consider the symmetries manifest in the
+(a, H) representation, Eq. (B20). The path integral is
+invariant under the transformation
+a → a + Γ,
+FH → FH + NΓ,
+(B41)
+with dΓ = 0.
+Since FH satisfies
+�
+FH ∈ 2πZ, in order
+to shift FH → FH + NΓ we require that
+�
+Γ ∈ 2πZ/N.
+When Γ = dω, the transformation Eq. (B41) becomes
+the gauge transformation Eq. (B18).
+Therefore, the
+Γ that correspond to physical transformations are
+N
+2πΓ ∈ Hp(X; Z).
+The quantization condition of the periods of Γ has a
+significant consequence.
+Indeed, note that the Wilson
+operator Wa (Eq. (B34)) is charged under this symmetry.
+Because of the quantization condition
+�
+Γ ∈ 2πZ/N, it
+transforms as
+Wa[Cp] → ei
+�
+ΓWa[Cp]
+ei
+�
+Γ ∈ ZN.
+(B42)
+Since the charged operators are p-dimensional and trans-
+form by an element of ZN, this is a Z(p)
+N symmetry.
+It’s tempting to think that there may be an additional
+symmetry associated with the H field. Indeed, Eq. (B20)
+
+32
+is invariant under the transformation H → H + ω for
+dω = 0. However, there are no physical observables that
+transform under this. Indeed, while the Wilson operator
+exp
+�
+i
+�
+H
+�
+picks up a phase, it is not a physical opera-
+tor since it is not invariant under the gauge redundancy
+Eq. (B18). Since there are no charged operators under
+H → H + ω, it does not have physical implications and
+is therefore not a symmetry.
+The
+Z(p)
+N
+symmetry
+can
+also
+be
+seen
+in
+the
+(a, ˆH) representation,
+when the Lagrangian is de-
+scribed by Eq. (B22).
+Indeed, under the symme-
+try transformation, the action amplitude transforms as
+exp[−S] → exp[−S] exp[−δS], where
+exp[−δS] = exp
+�
+− iN
+2π
+�
+Γ ∧ F ˆ
+H
+�
+,
+(B43)
+with
+N
+2πΓ ∈ Hp(X; Z). Plugging in F ˆ
+H = d ˆH + ω ˆ
+H, the
+phase factor exp[−δS] becomes
+exp[−δS] = e− i N
+2π
+�
+Γ∧d ˆ
+H e− i N
+2π
+�
+Γ∧ω ˆ
+H
+= e
+i N
+2π (−1)p �
+dΓ∧ ˆ
+H e−2π i
+�
+N Γ
+2π
+∧
+ω ˆ
+H
+2π
+= exp
+�
+−2πi
+� N Γ
+2π
+∧ ω ˆ
+H
+2π
+�
+,
+where we used integration by parts and that dΓ = 0. Re-
+call that N
+2πΓ ∈ Hp(X; Z) and ω ˆ
+H
+2π ∈ HD−p(X; Z). Then,
+since the wedge product preserves integral de Rham
+cohomology classes,
+N Γ
+2π ∧ ω ˆ
+H
+2π ∈ HD(X; Z).
+Therefore,
+exp[−δS] becomes
+exp[−δS] = exp[−2πiZ] = 1,
+(B44)
+and thus the action amplitude is invariant.
+The Lagrangian in the (a, ˆH) representation can also
+be written as Eq. (B23) without changing the partition
+function. In this form, following the same argument used
+to show that there is a Z(p)
+N
+symmetry, we find there is
+also Z(d−p)
+N
+symmetry. Indeed, the action amplitude is
+invariant under ˆH → ˆH + ˆΓ where
+N
+2π ˆΓ ∈ Hd−p(X; Z).
+The charged operator of this Z(d−p)
+N
+symmetry is the Wil-
+son operator W ˆ
+H = exp
+�
+i
+� ˆH
+�
+, which transforms as
+W ˆ
+H(C) → ei
+� ˆΓW ˆ
+H(C)
+ei
+� ˆΓ ∈ ZN.
+(B45)
+To summarize, the symmetry of p-form BF theory is
+Z(p)
+N × Z(d−p)
+N
+. The symmetry transformations associated
+with Z(p)
+N and Z(d−p)
+N
+are
+Z(p)
+N :
+a → a + 2π
+N Γ,
+Γ ∈ Hp(X; Z),
+Z(d−p)
+N
+:
+ˆH → ˆH + 2π
+N
+ˆΓ,
+ˆΓ ∈ Hd−p(X; Z),
+(B46)
+respectively. Note how only in the (a, ˆH) representation
+are both of these symmetries manifest.
+In the (a, H)
+representations, the Z(d−p)
+N
+symmetry is hidden as the
+charged operators are ’t Hooft operators.
+The symmetry operator of the Z(p)
+N symmetry is
+U(Σ) = exp
+�
+i
+�
+Σ
+ˆH
+�
+,
+= exp
+�
+i
+�
+Md
+ˆH ∧ Γ
+�
+,
+(B47)
+where Γ is the Poincar´e dual of the p-cycle Σ with respect
+to space Md. Indeed, using the equal-time commutation
+relation (B17), which form Eq. (B22) is
+[aµ1···µp(x), ˆHµp+1···µd(y)] = 2πi
+N ϵ0µ1···µdδd(x − y),
+(B48)
+we have that
+U(Σ)Wa(C)U †(Σ) = e
+2π i
+N
+�
+C ΓWa(C),
+= e
+2π i
+N #(Σ,C)Wa(C).
+(B49)
+Interesting the symmetry operator of Z(p)
+N
+was the
+charged operator of Z(d−p)
+N
+: U = W ˆ
+H. Similarly, the sym-
+metry operator of the Z(d−p)
+N
+symmetry is
+ˆU(ˆΣ) = exp
+�
+i
+�
+ˆΣ
+a
+�
+,
+(B50)
+which is just Wa.
+c.
+Mixed ’t Hooft anomaly and anomaly inflow
+In the last section, we found that p-form BF theory
+has an Z(p)
+N
+and Z(d−p)
+N
+symmetry. However, these sym-
+metries are not independent of one another: the symme-
+try operator of one symmetry is a charged operator of
+the other symmetry. In other words, these two symme-
+try operators do not commute. This is a manifestation of
+the fact that the Z(p)
+N × Z(d−p)
+N
+symmetry is anomalous.
+In this section, we will turn on a background gauge field
+for these symmetries to learn more about this mixed ’t
+Hooft anomaly.
+Let’s first turn on a background gauge field for the
+Z(p)
+N symmetry. We introduce the background gauge field
+A ∈ 2π
+N Hp+1(X; Z) and the gauge redundancy
+a → a + β,
+A → A + dβ.
+(B51)
+Minimally coupling A, the p-form BF theory path inte-
+gral becomes
+Z[A] =
+�
+D[a]D[ ˆH] e−
+�
+X L[A],
+L = − iN
+2π
+�
+a ∧ F ˆ
+H + (−1)pA ∧ ˆH
+�
+.
+(B52)
+
+33
+We next turn on a background gauge field for the
+Z(d−p)
+N
+symmetry, introducing the background gauge field
+ˆ
+A ∈ 2π
+N Hd−p+1(X; Z) and the gauge redundancy
+ˆH → ˆH + ζ,
+ˆ
+A → ˆ
+A + dζ.
+(B53)
+Minimally coupling ˆ
+A, Eq. (B52) becomes
+Z[A, ˆ
+A] =
+�
+D[a]D[ ˆH] e−
+�
+X L[A, ˆ
+A],
+L = − iN
+2π
+�
+a ∧ (F ˆ
+H − ˆ
+A) + (−1)pA ∧ ˆH
+�
+.
+(B54)
+When the Z(p)
+N
+background gauge field is turned off
+(i.e, A = 0), the path integral is gauge invariant. How-
+ever, when the Z(p)
+N background gauge field is turned on,
+there are no local counter terms which can be added such
+that the path integral is invariant under Eq. (B53). In-
+deed, under this gauge transformation, the path integral
+transforms as
+Z[A, ˆ
+A] → Z[A, ˆ
+A] exp
+�
+(−1)p iN
+2π
+�
+X
+A ∧ ζ
+�
+.
+(B55)
+We thus see that there is an obstruction to coupling a
+background gauge field of both symmetries, and hence a
+mixed ’t Hooft anomaly.
+a ’t Hooft anomaly can be classified by an invertible
+theory in one higher-dimension whose boundary realizes
+the anomalous symmetry.
+Let’s now extend the back-
+ground fields to one higher-dimension and have X be the
+boundary of the new spacetime Y . The phase picked up
+by Z[A, ˆ
+A] in Eq. (B55) can then be written as
+exp
+�
+(−1)p iN
+2π
+�
+X
+A ∧ ζ
+�
+= exp
+�
+− iN
+2π
+�
+Y
+A ∧ dζ
+�
+.
+(B56)
+In order to make the theory gauge invariant and cancel
+out the phase Z picks up, let’s consider the invertible
+theory
+Zinv[A, ˆ
+A] = exp
+� iN
+2π
+�
+Y
+A ∧ ˆ
+A
+�
+.
+(B57)
+Under the gauge transformation Eq. (B53), Zinv trans-
+forms as
+Zinv[A, ˆ
+A] → Zinv[A, ˆ
+A] exp
+� iN
+2π
+�
+Y
+A ∧ dζ
+�
+.
+(B58)
+This is the same as the inverse of the phase picked up by
+Z. Therefore, we replace Z with
+Z[A, ˆ
+A] → Zinv[A, ˆ
+A]Z[A, ˆ
+A],
+(B59)
+which is now gauge invariant. The path integral of the
+new gauge invariant theory is
+Z[A, ˆ
+A] =
+�
+D[a]D[ ˆH] e−
+�
+Y Lbulk+dLboundary,
+Lbulk = − iN
+2π A ∧ ˆ
+A,
+Lboundary = − iN
+2π
+�
+a ∧ F ˆ
+H + (−1)pA ∧ ˆH
+�
+.
+Given that we have found the theory in Y spacetime
+such that the mixed ’t Hooft anomaly is canceled, we can
+now close the one higher-dimensional spacetime.
+Let-
+ting ∂Y = ∅, the topological part of the path integral
+is Zinv[A, ˆ
+A], Eq. (B57). Since A ∈ 2π
+N Hp+1(Y ; Z) and
+ˆ
+A ∈ 2π
+N Hd−p+1(Y ; Z), A∧ ˆ
+A ∈ 4π2
+N 2 Hd+2(Y ; Z).
+There-
+fore, the mixed anomaly is classified by
+Zinv[A, ˆ
+A] = exp
+� i2π
+N Z
+�
+∈ ZN.
+(B60)
+The form of Zinv and the ZN classification agrees with
+the d = 3, p = 1 lattice result found in Ref. 43, which
+started from a bulk twisted U(1) gauge theory in the
+confined phase.
+Appendix C: Taking the continuum limit—p-form
+Maxwell theory
+In this appendix section, we present how we take the
+continuum limit of Eq. (90) in section III C 2 of the main
+text. Doing so also demonstrates the connection between
+exact emergent higher-form symmetries in lattice models
+and exact higher-form symmetries in Lagrangian quan-
+tum field theories, where higher-form symmetries are
+most commonly studied.
+The lattice Heisenberg operators Lz
+cp(t) and Θcp(t)
+in the IR are dressed by two local unitary opera-
+tors U (1)
+LU and U (2)
+LU and denoted as �L′z
+cp(t) and �Θ′
+cp(t).
+Furthermore, in the mid-IR we defined the variable
+ωcp+1 ≡ −2π
+�
+(d�Θ)cp+1/(2π)
+�
+,
+which in the IR was
+dressed by U (2)
+LU and denoted as ω′
+cp+1(t). Therefore, the
+three elementary operators in the IR are �L′z
+cp(t), �Θ′
+cp(t),
+and ω′
+cp+1(t). We relate these lattice operators to their
+continuum counterparts �L′z(t, x), �Θ′(t, x), and ω′(t, x)
+by
+�L′z
+cp =
+�
+cp
+�L′z,
+�Θ′
+cp =
+�
+cp
+�Θ′,
+ω′
+cp+1 =
+�
+cp+1
+ω′, (C1)
+where, for instance,
+�
+cp denotes spatial integral over the
+p-cell cp.
+The continuum quantum fields are globally
+differential forms in space M (�L′z and �Θ′ are p-forms
+while ω′ is a (p + 1)-form), mapping from spacetime X
+to R. One of the many conveniences of using the dis-
+crete exterior calculus notation is that in the contin-
+uum limit, these lattice operators become their contin-
+
+34
+uum versions.
+Indeed, for instance, the lattice opera-
+tor F ′
+cp+1 = (d�Θ′)cp+1 + (ω′)cp+1 is related to its contin-
+uum version by F ′
+cp+1 =
+�
+cp+1 F ′ where F ′ = d�Θ′ + ω′.
+So, taking the continuum limit of Eq. (90), the deep IR
+continuum Hamiltonian is
+Hdeep IR=
+�
+ddx
+�
+κU
+2
+|�L′z
+i1···ip|2
+p!
++ ε2p+2U
+2
+|F ′
+i1···ip+1|2
+(p + 1)!
+�
+,
+(C2)
+where, for instance, |�L′z
+i1···ip|2 ≡ �d
+i1,··· ,ip=1(�L′z
+i1···ip)2.
+To write down the path integral, we can find the
+Lorentzian action from H(III)
+IR
+and then perform a func-
+tional integral over all dynamical fields.
+However, the
+path integral only integrates over field configurations
+obeying particular constraints:
+1. Firstly, since the IR does not include dressed charge
+excitations, we only integrate over �L′z configura-
+tions satisfying �ρ ′ = 0. The expression for �ρ ′
+cp−1
+in Eq. (66) can be rewritten using discrete exterior
+calculus notation as �ρ ′
+cp−1 ∼ (∗ d ∗ �L′z)cp−1.
+So,
+�ρ ′ = 0 in the continuum limit is the Gauss law
+∂j �L′z
+ji1···ip−1 = 0.
+(C3)
+Despite there being no dressed charges in the IR,
+�L′z can still be sourced along nontrivial p-cycles.
+Indeed, because �L′z
+cp ∈ Z on the lattice, in the con-
+tinuum limit, the flux of �L′z is quantized
+�
+Cd−p
+∗ �L′z ∈ Z,
+(C4)
+where Cd−p is any closed (d − p)-submanifold of
+space.
+Notice how using Eq. (C3), when Cd−p
+is contractible, this is automatically satisfied since
+the left-hand side is zero by Stoke’s theorem. So,
+this additional constraint only affects Cd−p in the
+(d − p)th homology group: Cd−p ∈ Hd−p(M; Z).
+2. Additionally, since the IR does not include dressed
+topological defects, we only integrate over �Θ′ and
+ω′ configurations satisfying ˆρ′ = 0.
+The expres-
+sion for (∗ ˆρ′)cp+2 of Eq. (88) in the continuum
+limit becomes ˆρ′ =
+1
+2π ∗ dF ′, and ˆρ′ = 0 becomes
+the Bianchi identity
+1
+2π ∗ dF ′ = 0.
+(C5)
+Despite there being no dressed topological defects
+in the IR, ∗ F ′ can still be sourced along nontrivial
+(d − p)-cycles. Indeed, because ω′
+cp ∈ 2πZ on the
+lattice, in the continuum limit, the flux of ∗ F ′ is
+quantized
+�
+Cp+1
+F ′ ∈ 2πZ,
+(C6)
+where Cp+1 is any closed (p + 1)-submanifold of
+space. Notice how plugging in F ′ = d�Θ′ + ω′, the
+�Θ′ vanishes and Eqs. (C5) and (C6) only put con-
+straints on ω′. In fact, because ω′ is not an exact
+form, these constraints can be summarized as
+ω′
+2π ∈ Hp+1(M; Z),
+(C7)
+where Hp+1(M; Z) denotes the (p + 1)th de Rham
+cohomology group with integral periods.
+From the above, we learn that there are three con-
+straints given by Eqs. (C3),
+(C4), and (C7) that must
+be added by hand into the path integral. Taking into ac-
+count these constraints, the path integral in Lorentzian
+signature is
+Zdeep IR =
+�
+D[�Θ′] D�L′z
+�
+ω′∈2πHp+1(M;Z)
+δ(∂i1 �L′z
+i1···ip)
+(C8)
+× δ
+��
+∗ �L′z ∈ Z
+�
+ei
+�
+X dtddx Ldeep IR
+Ldeep IR =
+�L′z
+i1···ip∂t �Θ′
+i1···ip
+p!
+−
+�
+κU
+2
+|�L′z
+i1···ip|2
+p!
++ ε2p+2U
+2
+|F ′
+i1···ip+1|2
+(p + 1)!
+�
+.
+The first term in Ldeep IR enforces the equal-time com-
+mutation relation
+[�Θ′
+i1···ip(x), �L′z
+j1···jp(y)]
+p!
+= iδi1
+[j1· · · δip
+jp]δd(x − y), (C9)
+and the second term in parenthesis is Hamiltonian den-
+sity from Eq. (C2)
+This expression of the path integral is perfectly correct.
+However, to get it into a more familiar form, we rewrite
+this phase space path integral as a coordinate space path
+integral. To do so, we rewrite both of the functional delta
+functions by integrating in new fields and modifying the
+action. For the delta function enforcing the �L′z quanti-
+zation condition, we can rewrite it by summing over all
+nontrivial closed p-submanifolds in space
+δ
+��
+∗ �L′z ∈ Z
+�
+=
+�
+C∈Hd−p(M)
+exp
+�
+2πi
+�
+C
+∗ �L′z
+�
+.
+(C10)
+Using Poincar´e duality to write 2π
+�
+C ∗ �L′z =
+�
+X ∗ �L′z∧η,
+where η/2π is the poincare dual of C with respect to
+spacetime, this can be rewritten as
+δ
+��
+∗ �L′z ∈ Z
+�
+=
+�
+η∈2πHp(M;Z)
+ei
+�
+X dtddx
+�
+L′z
+i1···ip ηi1···ip
+p!
+.
+(C11)
+
+35
+Since C is closed and has integer coefficients, η is a closed
+form and satisfies
+�
+η ∈ 2πZ. The delta function enforc-
+ing Gauss law can be rewritten using a (p − 1)-form La-
+grange multiplier
+δ(∂i1 �L′z
+i1···ip) =
+�
+Dλei
+�
+X λi2···ip∂i1 �L′z
+i1i2···ip/(p−1)!.
+(C12)
+Plugging in both of these delta functions, the path in-
+tegral takes the cumbersome form
+Zdeep IR =
+�
+D[�Θ′] D�L′z Dλ
+�
+ω′∈2πHp+1(M;Z)
+η∈2πHp(M;Z)
+ei
+�
+X dtddx Ldeep IR,
+Ldeep IR =
+λi2···ip∂i1 �L′z
+i1i2···ip
+(p − 1)!
++
+�L′z
+i1···ipηi1···ip
+p!
+(C13)
++
+�L′z
+i1···ip∂t �Θ′
+i1···ip
+p!
+−
+κU|�L′z
+i1···ip|2
+2p!
+−
+ε2p+2U|F ′
+i1···ip+1|2
+2(p + 1)!
+.
+It is now straight forward to integrate out the �L′z field.
+Assuming spacetime is closed, we integrate by parts on
+the first term in Ldeep IR and use the anti-symmetry of
+�L′z to rewrite Ldeep IR as
+Ldeep IR =
+�L′z
+i1···ip
+�
+∂t �Θ′
+i1···ip − p ∂[i1λi2···ip] + ηi1···ip
+�
+p!
+−
+κU|�L′z
+i1···ip|2
+2p!
+−
+ε2p+2U|F ′
+i1···ip+1|2
+2(p + 1)!
+.
+(C14)
+Integrating
+out
+�L′z,
+rescaling
+t → t/(Uεp+1√κ),
+λ → Uεp+1√κ λ, and η → Uεp+1√κ η, and introducing
+g = 1/(εp+1√
+U), the path integral becomes
+Zdeep IR =
+�
+D[�Θ′] Dλ
+�
+ω′∈2πHp+1(M;Z)
+η∈2πHp(M;Z)
+ei
+�
+X dtddx Ldeep IR,
+(C15)
+Ldeep IR=
+|∂t �Θ′
+i1···ip− p∂[i1λi2···ip] +ηi1···ip|2
+2g2p!
+−
+|F ′
+i1···ip+1|2
+2g2(p + 1)!.
+Having found the coordinate path integral, we now
+massage this to get it into a canonical form.
+Namely,
+let’s introduce the p-form a and (p + 1)-form ωa in
+spacetime whose spatial components are ai1···ip = �Θ′
+i1···ip
+and (ωa)i1···ip+1 = ω′
+i1···ip+1 and timelike components are
+a0i2···ip = λi2···ip and (ωa)0i1···ip = ηi1···ip. Using a and
+letting Fa = da + ωa, we can reexpress the path integral
+as
+Zdeep IR =
+�
+D[a]
+�
+ωa∈2πHp+1(X;Z)
+ei
+�
+X dtddx Ldeep IR,
+(C16)
+Ldeep IR = |∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 + (ωa)0i1···ip|2
+2g2p!
+− |(Fa)i1···ip+1|2
+2g2(p + 1)! .
+Using ∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 = (p + 1)∂[0ai1···ip],
+the first term in Ldeep IR can be rewritten in terms of Fa:
+Ldeep IR =
+1
+2g2
+�|(Fa)0i1···ip|2
+p!
+− |(Fa)i1···ip+1|2
+(p + 1)!
+�
+.
+(C17)
+Furthermore, working in flat spacetime, X is equipped
+with Minkowski metric (−, +, · · · +).
+Thus, summing
+over spacetime indices µ = 0, · · · , d, Ldeep IR can be
+rewritten as
+Ldeep IR = −
+1
+2g2(p + 1)!(Fa)µ1···µp+1(Fa)µ1···µp+1
+(C18)
+Lastly, using differential forms notation, we arrive at our
+final expression for the path integral
+Zdeep IR =
+�
+D[a]
+�
+ωa∈2πHp+1(X;Z)
+ei
+�
+X Ldeep IR,
+Ldeep IR = − 1
+2g2 |Fa|2,
+(C19)
+where |F|2 ≡ F ∧ ∗ F. This is exactly p-form Maxwell the-
+ory, as stated in the main text.
+1.
+Review of p-form Maxwell theory
+In the remainder of this appendix section, we will re-
+view p-form Maxwell theory, focusing on its symmetries
+and anomalies, working in D = d + 1 dimensional space-
+time.
+The
+canonical
+momentum
+field
+Π
+is
+locally
+a
+(D − p − 1)-form associated with a codimension-1 sub-
+manifold of X in a Lorentzian signature. We choose this
+to be the 0-direction. Then, Π’s components are defined
+by varying the action with respect to ∂0aµ1···µp which
+yields
+Π = − 1
+g2 ∗ Fa.
+(C20)
+Therefore, from canonical quantization we have the
+equal-time commutation relations
+�
+aµ1···µp(x), (∗ Fa)µp+1···µd(y)
+g2
+�
+= iϵ0µ1···µdδd(x − y).
+(C21)
+
+36
+a.
+U(1)(p) symmetry
+The action amplitude is only a function of the field
+strength, and so the path integral is invariant under a
+being shifted by a closed p-form:
+a �→ a + Γ
+where
+dΓ = 0.
+(C22)
+However, just because the action amplitude is invariant
+does not mean it is a symmetry transformation.
+In-
+deed, it could be a gauge redundancy.
+If it is a gen-
+uine symmetry, it must transform observables nontriv-
+ially.
+These physical operators are the Wilson opera-
+tors Wa(Cp) = exp
+�
+i
+�
+Cp a
+�
+, and under the transforma-
+tion Eq. (C22) they transform as
+Wa (Cp) �→ exp
+�
+i
+�
+Cp
+Γ
+�
+Wa (Cp) .
+(C23)
+Since there are no restrictions on the holonomies
+of a, Γ satisfies
+�
+Γ ∈ R.
+Thus exp
+�
+i
+�
+Cp Γ
+�
+∈ U(1)
+and Eq. (C23) is the symmetry transformation of a
+U(1)(p) symmetry (i.e., an operator supported on a p-
+submanifold picks up a phase). Note that when Cp is
+a boundary, then by Stokes theorem
+�
+Cp Γ = 0. Thus,
+only Wilson loops defined on non-contractible Cp can be
+charged under the U(1)(p) symmetry.
+Given that Wa (Cp) are the physical quantities of the
+theory instead of a, not all transformations a → a + Γ
+with dΓ = 0 necessarily correspond to symmetry trans-
+formations.
+For Γ that satisfy exp
+�
+i
+�
+Cp Γ
+�
+= 1, the
+transformation a → a + Γ is instead a gauge transfor-
+mations corresponding to formal redundancies29. Let’s
+denote Γ that correspond to gauge redundancies as
+Γgauge, and since exp
+�
+i
+�
+Cp Γgauge
+�
+= 1, their periods
+are
+�
+Cp Γgauge ∈ 2πZ. In the case where Γgauge is exact
+(Γgauge = dω, which is the canonical gauge transforma-
+tion), the periods are zero by Stokes theorem. However,
+there also exist closed but not exact Γgauge whose periods
+are 2πZ. Indeed, these are Γgauge ∈ 2πHp(X; Z), and
+these types of gauge transformations are the so-called
+large gauge transformations. The physical transforma-
+tion on a therefore require that Γ be closed but not exact
+and the periods of Γ are not in 2πZ.
+Given that we’ve identified a U(1)(p) symmetry, let’s
+now find the symmetry transformation operator which
+performs said transformation.
+We can do so using
+Noether’s theorem and the fact that the conserved charge
+operator is the generator of the corresponding symmetry.
+29 This gives rise to an intriguing explanation to why gauge redun-
+dancies appear in theories that describe physical systems: the
+physical system’s constituents are extended objects but the for-
+malism used describes them is in terms of strictly local fields. It
+is physicists’ attachment to locality that infects the formalism
+with redundancies.
+A nice trick to find the conserved current is to introduc-
+ing a closed (p + 1)-form background field A and the
+gauge redundancy
+a �→ a + β,
+A �→ A + dβ,
+(C24)
+where dβ ̸= 0. Minimally coupling the background field
+such that the theory is gauge invariant, the action be-
+comes
+S[A] = − 1
+2g2
+�
+X
+|Fa − A|2.
+(C25)
+The conserved current J corresponding to the symmetry
+will minimally couple to A as the term
+�
+A∧ ∗ J. Ex-
+panding out the terms in S[A], we find that
+J = 1
+g2 Fa.
+(C26)
+The requirement that
+�
+A∧ ∗ J is gauge invariant enforces
+the expected conservation law d ∗ J = 0. Therefore, the
+charge operator Q ≡
+�
+∗ J is supported on a (D − p − 1)-
+dimensional closed surface Σ. The symmetry transforma-
+tion Uα = eiαQ is
+Uα (Σ) = exp
+�
+iα
+�
+Σ
+∗ Fa
+g2
+�
+,
+(C27)
+where α ∈ [0, 2π) parametrizes the U(1) transformation.
+Uα(Σ) is a topological operator. In fact, any operator
+of the form
+U(Σ) = exp
+�
+i
+�
+Σ
+η
+�
+,
+where dη = 0, is.
+Indeed, let’s continuously deform
+Σ → Σ′. Given that σ is homotopically equivalent to Σ′,
+the surfaces differ by a boundary and thus we can write
+Σ′ = Σ + ∂δ. Using this, we find that:
+U(Σ + ∂δ) = exp
+�
+iα
+�
+Σ+∂δ
+η
+�
+,
+= exp
+�
+iα
+��
+Σ
+η +
+�
+∂δ
+η
+��
+,
+= exp
+�
+�iα
+�
+�
+�
+Σ
+η +
+�
+δ
+dη
+����
+=0
+�
+�
+�
+�,
+= exp
+�
+iα
+�
+Σ
+η
+�
+,
+= U(Σ).
+(C28)
+So, number of transformations is given by number of ho-
+motopic equivalence classes for nontrivial contractible p-
+dimensional surfaces.
+Let’s now see how Uα acts on Wa. First, note that
+the closed manifold Σ ⊂ M, where M denotes a closed
+codimension-1 submanifold (e.g., a fixed imaginary-time
+
+37
+slice of Euclidean spacetime). Let the p-form ˆΣ be the
+Poincar´e dual of Σ with respect to M so the symmetry
+operator can be written as Uα (Σ) = exp
+�
+iα
+g2
+�
+M ∗ Fa∧ˆΣ
+�
+.
+Because Σ is closed, the Poincar´e dual satisfies dˆΣ = 0.
+From the Baker–Campbell–Hausdorff formula, the Wil-
+son operator transforms as
+Uα (Σ) Wa (C) U †
+α (Σ) = e
+α
+g2 [
+�
+M ∗ Fa∧ˆΣ,
+�
+C a]Wa (C) .
+From canonical commutation relations Eq. (C21), the
+commutator in the exponential is
+��
+M
+∗ Fa ∧ ˆΣ,
+�
+C
+a
+�
+= ig2
+�
+C
+ˆΣ,
+(C29)
+and thus we find
+Uα (Σ) Wa (C) U †
+α (Σ) = exp
+�
+iα
+�
+C
+ˆΣ
+�
+Wa (C) . (C30)
+Comparing this to Eq. (C23), we identify Γ ≡ αˆΣ. Nev-
+ertheless, we find that q =
+�
+C ˆΣ is the charge of the
+operator Wa (C). We can rewrite the phase noting that
+�
+C ˆΣ =
+�
+Σ∩C 1, which is the intersection number between
+Σ and C in M: # (Σ, C). Therefore q = # (Σ, C) ∈ Z
+and the U(1)(p) symmetry transformation is
+Uα (Σ) Wa (C) U †
+α (Σ) = exp [iα # (Σ, C)] Wa (C) .
+(C31)
+b.
+Abelian duality
+To make the other symmetry manifest we must effec-
+tively change the representation of our degrees of free-
+dom using Abelian duality [86] to dualize the field a to
+the field ˆa.
+This is a particle-vortex duality since, as
+we will see, the sources (topological defects) of a corre-
+spond to the topological defects (sources) of ˆa. Abelian
+duality is further useful since it maps the strong coupling
+limit in the a-representation to a weak coupling limit ˆa-
+representation.
+Let’s now dualize (change representation from a to
+ˆa) the quantum theory. Instead of integrating over the
+equivalence classes of a and summing over ωa, we can in-
+stead integrate over Fa since the action only depends on
+Fa. However, in doing so we have to ensure that we only
+integrate over Fa which satisfy the Bianchi identity30
+1
+2π ∗ dFa = 0,
+(C32)
+30 If there were topological defects described by a current
+ˆ
+J ,
+then the Bianchi identity would be modified to
+1
+2π ∗ dFa = ˆ
+J .
+The presence of topological defects would be captured by the
+field strength globally being give by Fa = db + ωa + d†βa. The
+periods of Fa would be
+�
+Fa =
+�
+(ωa + d†βa) ∈ 2πZ and now
+dFa = d(d†βa) ̸= 0.
+and which obey the quantization condition
+�
+Cp+1
+Fa ∈ 2πZ,
+(C33)
+for all Cp+1 ∈ Hp+1(X). So, with these two constraints
+in mind we can change variables and write the Euclidean
+path integral as
+Z =
+�
+DFa δ
+�∗ dFa
+2π
+�
+δ
+�� Fa
+2π ∈ Z
+�
+e−
+�
+X L,
+L =
+1
+2g2 |Fa|2.
+(C34)
+Let’s now rewrite the functional delta functions by in-
+tegrating in fields. Firstly, introducing the (D − p − 2)-
+form ˆa, we write
+δ
+�∗ dFa
+2π
+�
+=
+�
+Dˆa exp
+�
+i
+�
+X
+ˆa ∧ ∗
+�∗ dFa
+2π
+��
+,
+=
+�
+Dˆa exp
+� i
+2π
+�
+X
+dˆa ∧ Fa
+�
+,
+(C35)
+where we absorbed any minus signs from the ∗ ∗ and inte-
+grating by parts into ˆa. As for the second delta function,
+we can rewrite it as
+δ
+�� Fa
+2π ∈ Z
+�
+=
+�
+ˆωˆa∈Hp+1(X)
+exp
+�
+2πi
+��
+ˆωˆa
+Fa
+2π
+��
+,
+=
+�
+ωˆa∈2πHD−p−1(X;Z)
+exp
+� i
+2π
+�
+X
+ωˆa ∧ Fa
+�
+.
+(C36)
+We start off by summing over all closed (p + 1)-
+submanifold ˆωˆa, and then using Poincar´e duality we
+change
+that
+sum
+to
+a
+sum
+over
+all
+dual
+closed
+(D − p − 1)-forms ωˆa/(2π).
+Because we sum over ˆωˆa
+with integer coefficients, ωˆa satisfies the quantization
+condition
+�
+ωˆa ∈ 2πZ (the factor of 2π is for conve-
+nience).
+Thus the product of the delta functions give
+δ
+�∗ dFa
+2π
+�
+δ
+�� Fa
+2π ∈ Z
+�
+=
+�
+Dˆa
+�
+ωˆa∈2πHD−p−1(X;Z)
+exp
+� i
+2π
+�
+X
+(dˆa + ωˆa) ∧ Fa
+�
+=
+�
+Dˆa
+�
+ωˆa∈2πHD−p−1(X;Z)
+exp
+� i
+2π
+�
+X
+Fˆa ∧ Fa
+�
+,
+(C37)
+where we introduced Fˆa = dˆa + ωˆa. Note that from the
+quantization condition
+�
+ωˆa ∈ 2πZ, we have that
+�
+Fˆa ∈ 2πZ.
+(C38)
+Furthermore, because ωˆa is closed, Fˆa also satisfies a
+Bianchi identity
+1
+2π ∗ dFˆa = 0.
+(C39)
+
+38
+Returning back to the Euclidean path integral (C34),
+using Eq. (C37) it becomes
+Z[X, g] =
+�
+DFaDˆa
+�
+ωˆa∈2πHD−p−1(X;Z)
+e−
+�
+X L,
+L[Fa, Fˆa; g] =
+1
+2g2 |Fa|2 − i
+2π Fˆa ∧ Fa.
+(C40)
+Let’s
+now
+complete
+the
+square,
+and
+introducing
+G = Fa − i g2
+2π ∗ Fˆa, the path integral becomes
+Z =
+�
+DG Dˆa
+�
+ωˆa∈2πHD−p−1(X;Z)
+e−
+�
+X L,
+L =
+1
+2g2 |G|2 + g2
+8π2 |Fˆa|2.
+(C41)
+We can now integrate out G and arrive at the path inte-
+gral only in terms of the dual field ˆa:
+Z[X, g] =
+�
+D[ˆa]
+�
+ωˆa∈2πHD−p−1(X;Z)
+exp
+�
+− g2
+8π2
+�
+X
+|Fˆa|2
+�
+. (C42)
+Remarkably, this theory has the same form as what
+we started with, expect now that initial p-form a is a
+(D − p − 2)-form ˆa and the coupling constant g is now
+2π/g. Thus, strongly coupling (g ≫ 1) in the a represen-
+tation gets mapped to weak coupling in the ˆa represen-
+tation, and vice versa.
+Having gone through the process of dualizing a to ˆa,
+let’s now see how operators in terms of a transform un-
+der dualizing. We introduce the map S which takes an
+operator in the a representation to the ˆa representation.
+Let’s first check to see what the field strength Fa maps
+to by inserting Fa into the path integral. This is quite
+easy to find, noting that when completing the square we
+did a change of variables Fa = G + i g2
+2π ∗ Fˆa, the insertion
+becomes
+⟨Fa⟩a = ⟨G⟩ˆa +
+�
+i g2
+2π ∗ Fˆa
+�
+ˆa
+.
+(C43)
+We use the notation that ⟨·⟩a is the vev evaluated in
+the a-representation and ⟨·⟩ˆa is the vev evaluated in the
+ˆa-representation. Since the G part of the action is Gaus-
+sian, ⟨G⟩ = 0 and so ⟨Fa⟩ = ⟨i g2
+2π ∗ Fˆa⟩. Thus, we find
+that in Euclidean signature
+S : Fa �→ i g2
+2π ∗ Fˆa.
+(C44)
+In the Lorentzian signature, this would of course become
+S : Fa �→ g2
+2π ∗ Fˆa. We can dualizing ˆa back to a by simply
+repeating the same steps as before to find
+S : Fˆa �→ i 2π
+g2 ∗ Fa.
+(C45)
+Therefore, Dualizing a twice gives us S2 : Fa �→ i2 ∗ ∗ Fa
+which using the expression for ∗ ∗ in Euclidean spacetime
+becomes
+S2 : Fa �→ (−1)D(p+1)+pFa.
+(C46)
+When both D and p are even, dualizing twice acts as
+S2 : Fa �→ Fa. However, if D or p or both are odd, then
+S2 : Fa �→ −Fa and thus one must dualize four times to
+get the identity map: S4 : Fa �→ Fa.
+Repeating the above argument, we find that d†Fa and
+∗ dFa get mapped to ∗ dFˆa and d†Fˆa, respectively, and
+vice versa. Thus, the excitations (topological defects) of
+a are the topological defects (excitations) of ˆa. Hence,
+abelian duality is a particle-vortex type duality.
+We emphasize that the above mappings does not im-
+ply that Fa = i g2
+2π ∗ Fˆa. Instead, while operators linear
+in Fa simply have Fa replaced with i g2
+2π ∗ Fˆa, more care
+is required to find the dual representation of operators
+nonlinear in Fa. For instance, (Fa)2 does not become
+(i g2
+2π ∗ Fˆa)2 due to the addition terms pick up when squar-
+ing Fa = G + i g2
+2π ∗ Fˆa.
+Now let’s see what Wilson operator of a gets mapped
+to.
+We will assume that Wa is supported on a con-
+tractible manifold C = ∂M. However, there are infinitely
+many such M whose boundary is C. As such, to avoid
+this ambiguity we simply sum over all such M:
+Wa[C] =
+�
+M:∂M=C
+exp
+�
+i
+�
+∂M
+a
+�
+,
+=
+�
+M:∂M=C
+exp
+�
+i
+�
+M
+Fa
+�
+.
+(C47)
+Let’s now introduce
+ˆ
+M, the Poincar´e dual of M with
+respect to X, which satisfies
+� ˆ
+M ∈ 2πZ. The Poincar´e
+dual ˆC is related to ˆ
+M by ˆC = d ˆ
+M/(2π). Thus, using
+Poincar´e duality, we rewrite Wa as
+Wa[C] =
+�
+D ˆ
+M δ(d ˆ
+M − 2π ˆC) exp
+� i
+2π
+�
+X
+Fa ∧ ˆ
+M
+�
+.
+Inserting this into the path integral after the dual field
+strength ˆa has been integrated in gives
+Z =
+�
+DFaD ˆ
+MD[ˆa]
+�
+ωˆa∈2πHD−p−1(X;Z)
+δ(d ˆ
+M − 2π ˆC) e−
+�
+X L
+L =
+1
+2g2 |Fa|2 + i
+2π (Fˆa − ˆ
+M) ∧ Fa.
+(C48)
+Now integrating out Fa, the resulting theory is
+Z =
+�
+D ˆ
+MD[ˆa]
+�
+ωˆa∈2πHD−p−1(X;Z)
+δ(d ˆ
+M − 2π ˆC) e−
+�
+X L,
+L = g2
+8π2 |Fˆa − ˆ
+M|2
+(C49)
+
+39
+Note that this can be written as
+Z =
+�
+D[ˆa]
+�
+ωˆa∈2πHD−p−1(X;Z)
+T[C] e−
+�
+X L,
+T =
+�
+D ˆ
+M δ(d ˆ
+M − 2π ˆC) e−
+�
+X
+g2
+8π2 [−2Fˆa∧ ∗ ˆ
+M+| ˆ
+M|2],
+L = g2
+8π2 |Fˆa|2
+We thus see that a contractible Wilson loop dualizes to
+T, which is a rather cumbersome expression.
+The Path integral Eq. (C49) is the path integral
+we found without the Wilson loop insertion Eq. (C42)
+but now with the connection ˆ
+M which is restrained to
+d ˆ
+M = 2π ˆC.
+We can get ride of the
+ˆ
+M connection in
+Eq. (C49) and have the path integral look similar to
+Eq. (C42) if we let ˆa be singular field not defined on
+C:
+Z =
+�
+D[ˆa]
+�
+ωˆa∈2πHD−p−1(X;Z)
+e−
+�
+X/C L,
+L = g2
+8π2 |Fˆa|2,
+(C50)
+and with
+�
+σ Fˆa = 2π for any open submanifold σ which
+have a nonzero intersection number with C. Therefore,
+the Wilson loop in the a representation has become a ’t
+Hooft loop in the ˆa representation.
+c.
+U(1)(d−p−1) symmetry
+When the theory is in the a representation, it would ap-
+pear that the model only has a U(1)(p) symmetry. How-
+ever, upon dualizing a to ˆa, the path integral Eq. (C42)
+took a similar form in terms of Fˆa but with g replaced by
+2π/g. Thus, we find that there is a new globally defined
+differential (D − p − 2)-form ˆa which shifting by a closed
+form leaves the path integral invariant.
+Following the
+same process as used in investigating the U(1)(p) sym-
+metry, this transformation has a physical part associated
+with a U(1)(D−p−2) symmetry.
+Everything about this U(1)(D−p−2) symmetry follows
+in a similar fashion from the U(1)(p) case. In particular,
+the symmetry transformation acts on ˆa as
+ˆa �→ ˆa + ˆΓ
+where
+dˆΓ = 0
+(C51)
+and the charged operators are Wilson operators in terms
+of ˆa
+Wˆa (C) = exp
+�
+i
+�
+C
+ˆa
+�
+.
+(C52)
+These correspond to the ’t Hooft operators in the a rep-
+resentation. The Noether’s current associated with this
+U(1)(D−p−2) symmetry is
+ˆJ = g2
+4π2 Fˆa,
+(C53)
+and thus the symmetry operator is
+ˆUˆα (Σ) = exp
+�
+i ˆα g2
+4π2
+�
+Σ
+∗ Fˆa
+�
+,
+(C54)
+where ˆα ∈ [0, 2π) parametrizes the U(1) transformation.
+To see what ˆU is in the a representation, let’s start
+with the operator exp
+�
+iθ
+�
+Σ Fa
+�
+and find its image under
+S. We’ve already done this calculation while finding the
+image of the Wilson operator. This time, we simply do
+not sum over all Σ. We therefore have that (in Lorentzian
+signature)
+S : eiθ
+�
+Σ Fa �→ eiθ
+�
+Σ
+g2
+2π ∗ Fˆa−i θ2g2
+2
+�
+X |ˆΣ|2.
+(C55)
+Setting θ =
+ˆα
+2π, the U(1)(D−p−2) symmetry operator in
+the a-representation is
+ˆUˆα(Σ) = exp
+�
+i ˆα
+�
+Σ
+Fa
+2π + i ˆα2g2
+8π2
+�
+X
+|ˆΣ|2
+�
+(C56)
+since under S it transforms to Eq. (C54). However, note
+that the term
+�
+X |ˆΣ|2 is an overall phase and therefore
+does not affect the symmetry transformation.
+So, we
+can drop this overall phase and treat the U(1)(D−p−2)
+symmetry operator in the a-representation instead as
+ˆUˆα(Σ) = exp
+�
+i ˆα
+�
+Σ
+Fa
+2π
+�
+.
+(C57)
+From this expression, it is easy to see that the Noether
+current of the U(1)(D−p−2) symmetry in the a represen-
+tation ˆJ satisfies
+∗ ˆJ = 1
+2π Fa.
+(C58)
+The fact the ˆJ is conserved is simply a reflection of the
+Bianchi identity.
+The charged objects under the U(1)(D−p−2) symmetry
+are the Wilson loops of ˆa, or equivalently the ’t Hooft
+loops of a. To see so in the ˆa representation is straight
+forward and follows exactly as how we found that the
+Wilson loops of a were charged operators of U(1)(p) in
+the a representation. So, let’s instead see that the Wilson
+loops of ˆa are charged operators of U(1)(D−p−2) in the a
+representation
+Let’s denote the ’t Hooft loop of a supported on the
+closed (D − p − 2) submanifold C as Ta(C). A part of
+the definition of Ta(C) is that in it’s presence
+�
+Σ Fˆb = 2π
+for any open submanifold Σ which have a nonzero inter-
+section number with C. Therefore
+ˆUˆα(σ)Ta(C) ˆU †
+ˆα(σ) = exp [i ˆα # (Σ, C)] Ta (C) .
+(C59)
+d.
+Mixed ’t Hooft anomaly and anomaly inflow
+In the last section, we reviewed that p-form Maxwell
+theory has a U(1)(p) and a U(1)(d−p−1) symmetry. How-
+ever, it turns out that these two symmetries are not fully
+
+40
+independent from one another: there is a mixed ’t Hooft
+anomaly preventing us from simultaneously turning on a
+background gauge field of both symmetries.
+The main idea of gauging a symmetry is to add addi-
+tional degrees of freedom such that the theory is invari-
+ant under the symmetry operator even when it is sup-
+ported on open submanifolds31. For instance, let’s start
+off in the in the a representation and gauge the U(1)(p)
+symmetry. The U(1)(p) symmetry operator Uα(Σ) (see
+Eq. (C27)) is supported on the closed submanifold Σ,
+and acting it on a shifted a by ˆΣ, the Poincar´e dual of
+Σ. Because Σ is a closed submanifold (i.e., ∂Σ = ∅), the
+Poincar´e dual ˆΣ is a closed form (i.e., dˆΣ = 0). Thus,
+gauging the symmetry requires the theory to be invari-
+ant under Uα(σ) for ∂Σ ̸= ∅, and therefore invariant un-
+der shifting a by a ˆΣ such that dˆΣ ̸= 0.
+Let’s first turn on a background field A of the U(1)(p)
+symmetry, which satisfies
+�
+FA ∈ 2πZ and dFA = 0, and
+includes the gauge redundancy
+a �→ a + β,
+A �→ A + dβ,
+(C60)
+where dβ ̸= 0.
+If we were to gauge the symmetry, A
+would be a dynamical field rather than a background
+gauge field.
+Nevertheless, minimally coupling A to a,
+the path integral becomes
+Z[A] =
+�
+D[a] e−
+�
+X L,
+L =
+1
+2g2 |Fa − A|2.
+(C61)
+We now dualize a to ˆa to see how A couples to ˆa. Re-
+peating the first steps of abelian duality is reviewed in
+section C 1 b, we can rewrite the path integral as
+Z[A] =
+�
+DFaD[ˆa] e−
+�
+X L,
+L =
+1
+2g2 |Fa − A|2 − i
+2π Fˆa ∧ Fa.
+(C62)
+In the A = 0 theory, at this step in dualizing we com-
+pleted the square. But instead, let’s first make the change
+of variables Fa = K + A such that the path integral be-
+comes
+Z[A] =
+�
+DKD[ˆa] e−
+�
+X L,
+L =
+1
+2g2 |K|2 − i
+2π Fˆa ∧ K − i
+2π A ∧ Fˆa.
+(C63)
+31 This can be thought of as a generalization of the 0-form sym-
+metry case, where gauging a symmetry is typically thought of
+as requiring the theory to be invariant under local symmetry
+transformations.
+The two first terms in L that depend on K are the same
+two terms we encountered when dualizing the theory pre-
+viously. The A dependency is entirely in the third term.
+Since it does not include K, upon integrating out K we
+will arrive at the same dual theory as before plus this
+third term. Therefore, in the ˆa representation, A mini-
+mally couples as
+Z[A] =
+�
+D[ˆa] e−
+�
+X L,
+L = g2
+8π2 |Fˆa|2 − i
+2π A ∧ Fˆa.
+(C64)
+Note that because dFˆa = 0 and spacetime is closed, the
+path integral in the ˆa-representation is still invariant un-
+der the gauge transformation Eq (C60).
+Eq. (C64) reveals that coupling a U(1)(p) symmetry
+background gauge field is equivalent to adding a topologi-
+cal term in the ˆa representation. This new term has a no-
+ticeable effect. The Noether current for the U(1)(D−p−2)
+symmetry, ˆJ =
+g2
+4π2 Fˆa, is no longer conserved:
+d† ˆJ = 1
+2π ∗ dA.
+(C65)
+This is a manifestation of the mixed ’t Hooft anomaly.
+Let’s not turn off A but turn on a background gauge
+field for the U(1)(D−p−2) symmetry.
+From our above
+discussion, this means that we introduce a background
+field ˆ
+A and the gauge redundancy
+ˆa �→ ˆa + ˆβ,
+ˆ
+A �→ ˆ
+A + dˆβ,
+(C66)
+where dˆβ ̸= 0.
+Inspired by how A coupled to ˆa in
+Eq. (C63), we minimally couple ˆ
+A to a in a similar fash-
+ion and consider
+Z[ ˆ
+A] =
+�
+D[a] e−
+�
+X L,
+L = 1
+2g2 |Fa|2 − i
+2π
+ˆ
+A ∧ Fa,
+(C67)
+Note that because Fa closed, shifting ˆ
+A by an exact form
+does not change the path integral and thus this is cor-
+rectly gauge invariant. To check if this way of coupling
+ˆ
+A to a is correct, we note that we expect in the ˆa repre-
+sentation that ˆ
+A couples to ˆa in the form Fˆa − ˆ
+A. Let’s
+check this by dualizing a to ˆa in Eq. (C67). Repeating
+the first steps of abelian duality, we can rewrite the path
+integral as
+Z[ ˆ
+A] =
+�
+DFaD[ˆa] e−
+�
+X L,
+L =
+1
+2g2 |Fa|2 − i
+2π (Fˆa − ˆ
+A) ∧ Fa.
+(C68)
+which integrating out Fa yields
+Z[ ˆ
+A] =
+�
+D[ˆa]D[ ˆ
+A] e−
+�
+X L,
+L = g2
+8π2 |Fˆa − ˆ
+A|2,
+(C69)
+
+41
+as expected. Returning back to Eq. (C67), due to the new
+topological term, the U(1)(p) symmetry Noether current
+J =
+1
+g2 Fa is no longer conserved:
+d†J = 1
+2π ∗ d ˆ
+A.
+(C70)
+Once again, this is a manifestation of the mixed ’t Hooft
+anomaly.
+Let us now turn on both of the background gauge fields
+A and
+ˆ
+A.
+Working in the a representation and using
+what we just found, the path integral becomes
+Z[A, ˆ
+A] =
+�
+D[a] e−
+�
+X L,
+L =
+1
+2g2 |Fa − A|2 − i
+2π
+ˆ
+A ∧ Fa.
+(C71)
+This path integral is invariant under the gauge transfor-
+mation Eq. (C66). However, due to the second term in L,
+this path integral is no longer invariant under the gauge
+transformation Eq. (C60), and transforms as
+Z[A, ˆ
+A] �→ Z[A, ˆ
+A] exp
+� i
+2π
+�
+X
+ˆ
+A ∧ dβ
+�
+.
+(C72)
+One could add a local-counter term to remedy this, and
+the path integral becomes
+Z[A, ˆ
+A] =
+�
+D[a] e−
+�
+X L,
+L =
+1
+2g2 |Fa − A|2 − i
+2π
+ˆ
+A ∧ (Fa − A).
+(C73)
+This is indeed now invariant under the gauge transforma-
+tion Eq. (C60). However, it is no longer invariant under
+the gauge transformation Eq. (C66), transforming as
+Z[A, ˆ
+A] �→ Z[A, ˆ
+A] exp
+�
+− i
+2π
+�
+X
+dˆβ ∧ A
+�
+.
+(C74)
+In fact, there are no local counter terms which will make
+the path integral gauge invariant when both A and ˆ
+A are
+turned on. Thus, the U(1)(p) × U(1)(d−p−1) symmetry is
+anomalous.
+If the theory is going to be gauge invariant, we must do
+something more drastic. Indeed, we can make the theory
+invariant under the gauge transformations by introducing
+the (D + 1)-dimensional spacetime Y such that X = ∂Y
+and extending the background gauge fields A and ˆ
+A into
+Y . To get a hint as to why, let’s return to Eq. (C72)
+and note that using Stoke’s theorem we can rewrite the
+integral in the exponential as
+�
+X=∂Y
+ˆ
+A ∧ dβ =
+�
+Y
+d( ˆ
+A ∧ dβ) =
+�
+Y
+d ˆ
+A ∧ dβ
+(C75)
+This then motivates the new gauge invariant partition
+function
+Z[A, ˆ
+A] = exp
+�
+− i
+2π
+�
+Y
+d ˆ
+A ∧ A
+� �
+D[a] e−
+�
+∂Y L,
+L =
+1
+2g2 |Fa − A|2 − i
+2π
+ˆ
+A ∧ Fa.
+(C76)
+Indeed, Z[A, ˆ
+A] is invariant under the gauge trans-
+formations Eq. (C72) since the phase picked up from
+�
+D[ˆa] e−
+�
+∂Y L cancels with the phase we added.
+Thus, we see the ’t Hooft anomaly through the mod-
+ern prospective of anomaly inflow. The theory Lboundary
+with A and ˆ
+A was not well defined in D-dimensions. In-
+stead, in order to turn on both A and
+ˆ
+A, it costs us
+having the theory Lboundary resides on the boundary of
+an invertible theory Lbulk in (D + 1)-dimensions.
+The
+invertible theory is [87]
+Zinv[A, ˆ
+A] = exp
+�
+− i
+2π
+�
+Y
+d ˆ
+A ∧ A
+�
+.
+(C77)
+We remark that in the above discussion, we have as-
+sumed that dFa = 0 and dFˆa = 0. In other words, we
+have assumed the absence of the electric and magnetic-
+branes in the path integral.
+So the above results are
+valid only at energies below the energy scales of these
+excitations where we have an exact emergent U(1)(p) ×
+U(1)(d−p−1) symmetry.
+[1] Z. Nussinov and G. Ortiz, Sufficient symmetry con-
+ditions for topological quantum order, Proceedings of
+the National Academy of Sciences 106, 16944 (2009),
+arXiv:cond-mat/0605316.
+[2] D. Gaiotto, A. Kapustin, N. Seiberg, and B. Willett, Gen-
+eralized global symmetries, J. High Energ. Phys. 2015
+(2), 172, arXiv:1412.5148.
+[3] R. Thorngren and Y. Wang, Fusion Category Symme-
+try I: Anomaly In-Flow and Gapped Phases,
+(2019),
+arXiv:1912.02817.
+[4] L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng,
+Algebraic higher symmetry and categorical symmetry: A
+holographic and entanglement view of symmetry, Physi-
+cal Review Research 2, 043086 (2020), arXiv:2005.14178.
+[5] L. Lootens, J. Fuchs, J. Haegeman, C. Schweigert, and
+F. Verstraete, Matrix product operator symmetries and
+intertwiners in string-nets with domain walls, SciPost
+Phys. 10, 053 (2021), arXiv:2008.11187.
+[6] J. McGreevy, Generalized Symmetries in Condensed
+Matter, (2022), arXiv:2204.03045.
+[7] C. C´ordova, T. T. Dumitrescu, K. Intriligator, and S.-
+H. Shao, Snowmass White Paper: Generalized Symme-
+tries in Quantum Field Theory and Beyond,
+(2022),
+arXiv:2205.09545.
+[8] D. S. Freed, G. W. Moore, and C. Teleman, Topo-
+logical symmetry in quantum field theory,
+(2022),
+
+42
+arXiv:2209.07471.
+[9] X.-G. Wen, Topological orders in rigid states, Int. J. Mod.
+Phys. B 04, 239 (1990).
+[10] L. D. Landau, Theory of phase transformations I, Phys.
+Z. Sowjetunion 11, 26 (1937).
+[11] V. L. Ginzburg and L. D. Landau, On the theory of su-
+perconductivity, Zh. Eksp. Teor. Fiz. 20, 1064 (1950).
+[12] Z. Nussinov and G. Ortiz, A symmetry principle for
+topological quantum order, Ann. Phys. 324, 977 (2009),
+arXiv:cond-mat/0702377.
+[13] X.-G. Wen, Emergent (anomalous) higher symmetries
+from topological order and from dynamical electromag-
+netic field in condensed matter systems, Phys. Rev. B
+99, 205139 (2019), arXiv:1812.02517.
+[14] P.-S. Hsin, H. T. Lam, and N. Seiberg, Comments on
+one-form global symmetries and their gauging in 3d and
+4d, SciPost Phys. 6, 039 (2019), arXiv:1812.04716.
+[15] M. Qi, L. Radzihovsky, and M. Hermele, Fracton phases
+via exotic higher-form symmetry-breaking, Annals of
+Physics 424, 168360 (2021), arXiv:2010.02254.
+[16] J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri,
+and S.-H. Shao, Higher central charges and topological
+boundaries in 2+1-dimensional TQFTs, SciPost Phys.
+13, 067 (2022), arXiv:2107.13091.
+[17] L. Kong and H. Zheng, Categories of quantum liquids I,
+(2020), arXiv:2011.02859.
+[18] L. Kong and H. Zheng, Categories of quantum liquids II,
+(2021), arXiv:2107.03858.
+[19] L. Kong and H. Zheng, Categories of quantum liquids
+III, (2022), arXiv:2201.05726.
+[20] A. Chatterjee and X.-G. Wen, Holographic theory for
+the emergence and the symmetry protection of gapless-
+ness and for continuous phase transitions, arXiv e-prints
+(2022), arXiv:2205.06244.
+[21] N. Seiberg, T. Senthil, C. Wang, and E. Witten, A duality
+web in 2 + 1 dimensions and condensed matter physics,
+Annals of Physics 374, 395 (2016), arXiv:1606.01989.
+[22] T. Senthil, D. T. Son, C. Wang, and C. Xu, Duality be-
+tween (2 + 1) d quantum critical points, Phys. Rep. 827,
+1 (2019), arXiv:1810.05174.
+[23] L. Lootens, C. Delcamp, G. Ortiz, and F. Verstraete,
+Category-theoretic recipe for dualities in one-dimensional
+quantum lattice models, (2021), arXiv:2112.09091.
+[24] E. Sharpe, Notes on generalized global symmetries in
+QFT, Fortschr. Phys. 63, 659 (2015), arXiv:1508.04770.
+[25] F. Benini, C. C´ordova, and P.-S. Hsin, On 2-group global
+symmetries and their anomalies, J. High Energ. Phys.
+2019 (3), 118, arXiv:1803.09336.
+[26] C. C´ordova, T. T. Dumitrescu, and K. Intriligator, Ex-
+ploring 2-group global symmetries, J. High Energ. Phys.
+2019 (2), 184, arXiv:1802.04790.
+[27] A. Kovner and B. Rosenstein, New look at QED4: the
+photon as a Goldstone boson and the topological in-
+terpretation of electric charge, Phys. Rev. D 49, 5571
+(1994), arXiv:hep-th/9210154.
+[28] E. Lake, Higher-form symmetries and spontaneous sym-
+metry breaking, (2018), arXiv:1802.07747.
+[29] D. Hofman and N. Iqbal, Goldstone modes and pho-
+tonization for higher form symmetries, SciPost Phys. 6,
+006 (2019), arXiv:1802.09512.
+[30] K.-S. Kim and Y. Hirono, Higher-form symmetries and d-
+wave superconductors from doped Mott insulators, Phys.
+Rev. B 100, 085142 (2019), arXiv:1905.04617.
+[31] Y.
+Hidaka,
+Y.
+Hirono,
+and
+R.
+Yokokura,
+Count-
+ing Nambu-Goldstone Modes of Higher-Form Global
+Symmetries, Phys. Rev. Lett. 126, 071601 (2021),
+arXiv:2007.15901.
+[32] N. Yamamoto and R. Yokokura, Topological mass gen-
+eration in gapless systems, Phys. Rev. D 104, 025010
+(2021), arXiv:2009.07621.
+[33] N. Iqbal and J. McGreevy, Mean string field theory:
+Landau-Ginzburg theory for 1-form symmetries, SciPost
+Phys. 13, 114 (2022), arXiv:2106.12610.
+[34] A. Kapustin and R. Thorngren, Higher symmetry and
+gapped phases of gauge theories, in Algebra, Geome-
+try, and Physics in the 21st Century. Progress in Math-
+ematics. Progress in Mathematics, Vol. 324, edited by
+D. Auroux, L. Katzarkov, T. Pantev, Y. Soibelman, and
+Y. Tschinkel (Birkh¨auser, Cham., 2017) pp. 177–202,
+arXiv:1309.4721.
+[35] R. Thorngren and C. von Keyserlingk, Higher SPT’s
+and
+a
+generalization
+of
+anomaly
+in-flow,
+(2015),
+arXiv:1511.02929.
+[36] B.
+Yoshida,
+Topological
+phases
+with
+generalized
+global symmetries, Phys. Rev. B 93, 155131 (2016),
+arXiv:1508.03468.
+[37] D.
+Gaiotto,
+A.
+Kapustin,
+Z.
+Komargodski,
+and
+N. Seiberg, Theta,
+time reversal and temperature,
+Journal
+of
+High
+Energy
+Physics
+2017,
+1
+(2017),
+arXiv:1703.00501.
+[38] R. Kobayashi, K. Shiozaki, Y. Kikuchi, and S. Ryu, Lieb-
+Schultz-mattis type theorem with higher-form symmetry
+and the quantum dimer models, Phys. Rev. B 99, 014402
+(2019), arXiv:1805.05367.
+[39] Z. Wan and J. Wang, Non-Abelian Gauge Theories,
+Sigma Models,
+Higher Anomalies,
+Symmetries,
+and
+Cobordisms, Annals of Mathematical Sciences and Ap-
+plications 4, 107 (2019), arXiv:1812.11967.
+[40] L. Tsui and X.-G. Wen, Lattice models that realize zn-1-
+symmetry protected topological states for even n, Phys.
+Rev. B 101, 035101 (2020), arXiv:1908.02613.
+[41] C.-M. Jian, X.-C. Wu, Y. Xu, and C. Xu, Physics of
+Symmetry Protected Topological phases involving Higher
+Symmetries and their Applications, Phys. Rev. B 103,
+064426 (2021), arXiv:2009.00023.
+[42] B. Moy,
+H. Goldman,
+R. Sohal, and E. Fradkin,
+Theory
+of
+oblique
+topological
+insulators,
+(2022),
+arXiv:2206.07725.
+[43] S. D. Pace and X.-G. Wen, Emergent Higher-Symmetry
+Protected Topological Orders in the Confined Phase of
+U(1) Gauge Theory, (2022), arXiv:2207.03544.
+[44] S. Weinberg, Approximate Symmetries and Pseudo-
+Goldstone Bosons, Phys. Rev. Lett. 29, 1698 (1972).
+[45] D. Foerster, H. Nielsen, and M. Ninomiya, Dynamical
+stability of local gauge symmetry creation of light from
+chaos, Phys. Lett. B 94, 135 (1980).
+[46] M. B. Hastings and X.-G. Wen, Quasiadiabatic con-
+tinuation of quantum states: The stability of topolog-
+ical ground-state degeneracy and emergent gauge in-
+variance, Phys. Rev. B 72, 045141 (2005), arXiv:cond-
+mat/0503554.
+[47] Z. Komargodski, A. Sharon, R. Thorngren, and X. Zhou,
+Comments on abelian Higgs models and persistent order,
+SciPost Phys. 6, 003 (2019), arXiv:1705.04786.
+[48] C. C´ordova, K. Ohmori, and T. Rudelius, General-
+ized symmetry breaking scales and weak gravity conjec-
+tures, Journal of High Energy Physics 11, 154 (2022),
+arXiv:2202.05866.
+
+43
+[49] R. Verresen,
+U. Borla,
+A. Vishwanath,
+S. Moroz,
+and R. Thorngren, Higgs Condensates are Symmetry-
+Protected Topological Phases:
+I. Discrete Symmetries
+(2022), arXiv:2211.01376.
+[50] W. Ji and X.-G. Wen, Categorical symmetry and nonin-
+vertible anomaly in symmetry-breaking and topological
+phase transitions, Phys. Rev. Research 2, 033417 (2020),
+arXiv:1912.13492.
+[51] L. Kong, T. Lan, X.-G. Wen, Z.-H. Zhang, and H. Zheng,
+Classification of topological phases with finite internal
+symmetries in all dimensions, J. High Energ. Phys. 2020
+(9), 93, arXiv:2003.08898.
+[52] A. Chatterjee and X.-G. Wen, Algebra of local symmetric
+operators and braided fusion n-category – symmetry is a
+shadow of topological order, (2022), arXiv:2203.03596.
+[53] R. B. Laughlin and D. Pines, The Theory of Everything,
+Proceedings of the National Academy of Sciences 97, 28
+(2000).
+[54] N. Seiberg and S.-H. Shao, Exotic symmetries, duality,
+and fractons in 2+1-dimensional quantum field theory,
+SciPost Phys. 10, 027 (2021), arXiv:2003.10466.
+[55] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao,
+Low-energy limit of some exotic lattice theories and
+UV/IR mixing, Phys. Rev. B 104, 235116 (2021),
+arXiv:2108.00020.
+[56] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao,
+Global dipole symmetry, compact Lifshitz theory, tensor
+gauge theory, and fractons, Phys. Rev. B 106, 045112
+(2022), arXiv:2201.10589.
+[57] S. D. Pace and X.-G. Wen, Position-dependent excita-
+tions and UV/IR mixing in the ZN rank-2 toric code
+and its low-energy effective field theory, Phys. Rev. B
+106, 045145 (2022), arXiv:2204.07111.
+[58] S. Bravyi, M. B. Hastings, and S. Michalakis, Topo-
+logical quantum order: stability under local perturba-
+tions, Journal of mathematical physics 51, 093512 (2010),
+arXiv:1001.0344.
+[59] F. Apruzzi, F. Bonetti, I. Garc´ıa Etxebarria, S. S. Hos-
+seini, and S. Schafer-Nameki, Symmetry TFTs from
+String Theory, arXiv e-prints , arXiv:2112.02092 (2021),
+arXiv:2112.02092.
+[60] W. Ji and X.-G. Wen, Non-invertible anomalies and
+mapping-class-group transformation of anomalous par-
+tition functions, Phys. Rev. Research 1, 033054 (2019),
+arXiv:1905.13279.
+[61] X.-G.
+Wen,
+Classifying
+gauge
+anomalies
+through
+symmetry-protected trivial orders and classifying gravi-
+tational anomalies through topological orders, Phys. Rev.
+D 88, 045013 (2013), arXiv:1303.1803.
+[62] L. Kong and X.-G. Wen, Braided fusion categories,
+gravitational anomalies, and the mathematical frame-
+work for topological orders in any dimensions,
+(2014),
+arXiv:1405.5858.
+[63] A. Kitaev and L. Kong, Models for gapped bound-
+aries and domain walls, Commun. Math. Phys. 313, 351
+(2012), arXiv:1104.5047.
+[64] X.-G. Wen, Gapless boundary excitations in the quantum
+hall states and in the chiral spin states, Phys. Rev. B 43,
+11025 (1991).
+[65] L. Delacr´etaz, D. Hofman, and G. Mathys, Superfluids
+as higher-form anomalies, SciPost Physics 8, 047 (2020),
+arXiv:1908.06977.
+[66] A. Kapustin and N. Seiberg, Coupling a QFT to a
+TQFT and duality, J. High Energ. Phys. 2014 (4), 1,
+arXiv:1401.0740.
+[67] N. Iqbal and J. McGreevy, Toward a 3d Ising model with
+a weakly-coupled string theory dual, SciPost Phys. 9, 019
+(2020), arXiv:2003.04349.
+[68] P. Gorantla, H. T. Lam, N. Seiberg, and S.-H. Shao, A
+modified Villain formulation of fractons and other exotic
+theories, Journal of Mathematical Physics 62, 102301
+(2021), arXiv:2103.01257.
+[69] M. Cheng and N. Seiberg, Lieb-Schultz-Mattis, Lut-
+tinger, and ’t Hooft – anomaly matching in lattice sys-
+tems (2022), arXiv:2211.12543.
+[70] L. Fazza and T. Sulejmanpasic, Lattice Quantum Vil-
+lain Hamiltonians: Compact scalars, U(1) gauge theo-
+ries, fracton models and Quantum Ising model dualities
+(2022), arXiv:2211.13047.
+[71] P.-S. Hsin and A. Turzillo, Symmetry-enriched quantum
+spin liquids in (3 + 1)d, Journal of High Energy Physics
+2020, 1 (2020), arXiv:1904.11550.
+[72] M. A. Levin and X.-G. Wen, String-net condensation: A
+physical mechanism for topological phases, Phys. Rev. B
+71, 045110 (2005), arXiv:cond-mat/0404617.
+[73] Y. L. Liu and S. D. Pace, (unpublished).
+[74] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Symmetry
+protected topological orders and the group cohomology of
+their symmetry group, Phys. Rev. B 87, 155114 (2013),
+arXiv:1106.4772.
+[75] C. G. Callan Jr. and J. A. Harvey, Anomalies and fermion
+zero modes on strings and domain walls, Nuclear Physics
+B 250, 427 (1985).
+[76] I. Hubaˇc and S. Wilson, Brillouin-Wigner Perturbation
+Theory, in Brillouin-Wigner Methods for Many-Body
+Systems (Springer, 2010) pp. 37–68.
+[77] S. Bravyi, D. P. DiVincenzo, and D. Loss, Schrieffer-Wolff
+transformation for quantum many-body systems, Annals
+of physics 326, 2793 (2011), arXiv:1105.0675v1.
+[78] T. Sulejmanpasic and C. Gattringer, Abelian gauge the-
+ories on the lattice: θ-Terms and compact gauge the-
+ory with(out) monopoles, Nuclear Physics B 943, 114616
+(2019), arXiv:1901.02637.
+[79] Z. Wan, J. Wang, and X.-G. Wen, (3 + 1)d boundaries
+with gravitational anomaly of (4+1)d invertible topolog-
+ical order for branch-independent bosonic systems, Phys.
+Rev. B 106, 045127 (2022), arXiv:2112.12148.
+[80] E. Witten, Quantum field theory and the Jones polyno-
+mial, Commun.Math. Phys. 121, 351 (1989).
+[81] K. Slagle and Y. B. Kim, Quantum field theory of
+X-cube fracton topological order and robust degener-
+acy from geometry, Phys. Rev. B 96, 195139 (2017),
+arXiv:1708.04619.
+[82] J. Maldacena, N. Seiberg, and G. Moore, D-brane charges
+in five-brane backgrounds, J. High Energy Phys. 2001
+(10), 005, hep-th/0108152.
+[83] T. Hansson, V. Oganesyan, and S. Sondhi, Supercon-
+ductors are topologically ordered, Ann. Phys. 313, 497
+(2004), arXiv:cond-mat/0404327.
+[84] T. Banks and N. Seiberg, Symmetries and strings in field
+theory and gravity, Phys. Rev. D 83, 084019 (2011),
+arXiv:1011.5120.
+[85] P. Mathieu and F. Thuillier, Abelian BF theory and
+Turaev-Viro invariant, Journal of Mathematical Physics
+57, 022306 (2016), arXiv:1509.04236.
+[86] E. Witten, On S-duality in Abelian Gauge Theory, Se-
+lecta Mathematica 1, 383 (1995), arXiv:hep-th/9505186.
+[87] C.-T. Hsieh, Y. Tachikawa, and K. Yonekura, Anomaly
+
+44
+inflow
+and
+p-form
+gauge
+theories,
+Communica-
+tions
+in
+Mathematical
+Physics
+391,
+495
+(2022),
+arXiv:2003.11550.
+
diff --git a/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/load_file.txt b/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..b747cf4338a837e0a5ec127e918e62520df9e863
--- /dev/null
+++ b/ydE4T4oBgHgl3EQfxw3J/content/tmp_files/load_file.txt
@@ -0,0 +1,2323 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf,len=2322
+page_content='Exact emergent higher symmetries in bosonic lattice models  Salvatore D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pace and Xiao-Gang Wen  Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA  (Dated: January 16, 2023)  While higher-form symmetries are a powerful tool in studying a quantum many-body system,  theories with exact higher-form symmetries are rather special and, in a sense, fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This  raises an interesting question: can the phases of a microscopic (UV) theory without exact higher-  form symmetries be exactly characterized by emergent higher-form symmetries?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here we argue the  answer is yes by constructing effective theories for bosonic lattice Hamiltonian models that only  capture the system’s dynamics at E < Escale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The emergent symmetries below this energy scale  are then identified as the exact symmetries of this effective theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We find that the emergent  higher-form symmetries (excluding 0-form symmetries) are robust against local UV perturbations  and become exact symmetries of the effective theory in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This result is true  even for more general higher symmetries, such as non-invertible higher symmetries (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', algebraic  higher symmetries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, emergent higher symmetries are exact emergent symmetries: they  are not UV symmetries but constrain the IR as if they were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We apply this framework to three lattice  models (the quantum clock model and emergent ZN and U(1) p-gauge theory) to identify regions of  parameter space with energy scales below which higher-form symmetries, and sometimes associated  ’t Hooft anomalies, emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since phases of matter are defined in the thermodynamic limit, this  implies that a UV theory without exact higher symmetries can have phases exactly characterized  by emergent higher symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We discuss in detail the physical consequences of this and contrast  it to emergent 0-symmetries, which are never exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This emphasizes the importance of identifying  scale hierarchies and emergent higher symmetries when studying quantum many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  CONTENTS  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Introduction  1  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Energy scales hierarchies and effective  Hamiltonians  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Low energy exact emergent generalized  symmetries  3  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A holographic picture for emergent finite  symmetries  5  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Examples of exact emergent higher-form  symmetries  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Quantum clock model  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An exact emergent Z(d)  N symmetry and  mixed ’t Hooft anomaly  11  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Emergent ZN p-gauge theory for p ≥ 1  12  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An exact emergent Z(p)  N symmetry  13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An exact emergent anomalous  Z(p)  N × Z(d−p)  N  symmetry  16  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Emergent U(1) p-gauge theory  19  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An exact emergent U(1)(p) symmetry  19  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An exact emergent anomalous  U(1)(p) × U(1)(d−p−1) symmetry  21  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Physical consequences of exact emergent  higher-form symmetries  23  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Conclusion and discussion  26  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Acknowledgements  26  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Review of discrete differential geometry for  d-dimensional cubic lattices  26  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A TQFT description of the p-form toric code  ground states—p-form BF theory  28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Review of p-form BF theory  29  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN p-form gauge theory in the continuum 29  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Symmetries  31  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Mixed ’t Hooft anomaly and anomaly  inflow  32  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Taking the continuum limit—p-form Maxwell  theory  33  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Review of p-form Maxwell theory  35  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' U(1)(p) symmetry  36  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Abelian duality  37  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' U(1)(d−p−1) symmetry  39  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Mixed ’t Hooft anomaly and anomaly  inflow  39  References  41  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  INTRODUCTION  A longstanding pillar for understanding strongly inter-  acting quantum many-body systems is to identify and  understand their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, symmetries pro-  vide powerful constraints and universal characterizations  of a system’s dynamics and phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This point of view  has become increasingly fruitful with modern general-  izations of symmetry [1–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For instance, topological  order [9], which provided the first indication that con-  ventional symmetries [10, 11] are not all-powerful, can  now be understood in a symmetry framework [6, 12–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  These generalizations open up an exciting frontier for  the discovery of new phases of quantum matter and the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='05261v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='str-el]  12 Jan 2023  2  conceptual organization/systematic understanding [4] of  quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, one may wonder if the gener-  alization of symmetries can become so general that they  could capture all possible IR symmetries in gapless liquid  states, becoming a largely complete universal characteri-  zation of gapless phases and thus a key to understanding  gapless liquid states [17–20] and webs of dualities [4, 21–  23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  One of the simplest and best-understood generaliza-  tions of symmetry is so-called higher-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For  ordinary symmetries, charged operators act on a point in  spacetime and the unitary operator that generates the  symmetry transformation acts on a codimension-1 hy-  persurface of spacetime (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', all of space at a fixed time  slice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Higher-form symmetries generalize this by allow-  ing charged operators to be extended objects [1, 2, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For a p-form symmetry1, the charged operators act on  closed p-dimensional subspaces and, and the unitary  generating the transformation acts on a closed (d − p)-  dimensional subspace of d-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Notice  that an ordinary global symmetry is just a 0-form sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Most things 0-form symmetries can do, higher-form  symmetries can also do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For example, higher-form  symmetries can spontaneously break, giving rise to a  topological ground state degeneracy (gapless Goldstone  bosons) when discrete (continuous) [2, 27–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In-  deed, abelian topological orders reflect discrete 1-form  symmetries spontaneously breaking, and photons in a  Coulomb phase arise from U(1) 1-form symmetries spon-  taneously breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A higher-form symmetry can also  have a ’t Hooft anomaly, providing powerful constraints  on the IR through generalized Lieb-Schultz-Mattis-  Oshikawa-Hastings theorems and introducing higher-  form symmetry-protected topological phases [13, 14, 34–  43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  These applications of higher-form symmetries make  them a powerful tool in studying quantum many-body  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, unfortunately, models with exact  higher-form symmetries are rather special and, in a sense,  fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, it is natural to wonder if they play a role  in more typical, physically relevant models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' One possi-  bility is that while they may not be exact microscopic  symmetries, they could still arise as emergent symme-  tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, experience with emergent ordinary (0-  form) symmetries causes apprehension since their conse-  quences are never exact and always approximate [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In  other words, explicitly breaking emergent 0-form sym-  metries at energy scale E creates O(Eγ) errors, even in  the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Amazingly, common folklore  suggests that this 0-form symmetry-based intuition does  not carry over to higher-form symmetries and that they  can constrain a system exactly even as emergent symme-  tries [13, 33, 45–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  1 Here we will consider only pure p-form symmetries, and not the  more general p-group symmetries [24–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this manuscript, we investigate this robustness of  higher-form symmetries from a UV perspective, consid-  ering bosonic lattice Hamiltonian models without higher-  form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These UV-complete theories are sim-  ple to work with, well-defined, and relevant to condensed  matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For this class of models, we demonstrate  how higher-form symmetries can emerge and, when they  do, why they nevertheless constrain the IR exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This  result is not rigorously demonstrated and relies on phys-  ically reasonable conjectures made in order to identify  low-energy sub-Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We find that at energies with the emergent higher-form  symmetry, the dynamics of states are affected by the  emergent higher-form symmetry as if it were an exact  UV symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' More precisely, any coming errors from  the higher-form symmetries being emergent below a finite  energy scale E are of order O(e−Lγ), where L is the sys-  tem size measured by lattice constant, and thus vanish in  the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Our arguments apply to sym-  metries more general then pure higher-form symmetries,  such as higher-group symmetries and beyond higher-  group symmetries (non-invertible higher-form symme-  tries/algebraic higher symmetries [4, 50, 51]), but do not  apply to 0-symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This suggests the phases of microscopic models with-  out exact higher symmetries can be exactly character-  ized by emergent higher symmetries2, except the 0-  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To emphasize this, we will refer to emer-  gent higher symmetries as exact emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The rest of this paper is goes as follows:  In section II, we review the importance of the sepa-  ration of scales in many-body systems and show how to  identify low-energy sub-Hilbert spaces for a general class  of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We then discuss how to identify the emergent  symmetries of a low-energy sub-Hilbert space, providing  motivation from the point of view that symmetries are  described by algebras of local symmetric operators [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We identify these emergent symmetries by developing an  effective Hamiltonian description for the low-energy sub-  Hilbert space using the intuition that the microscopic  dynamics generate low-energy dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In section III, we apply the framework introduced in  section II to three bosonic lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We start by  warming up with the ZN quantum clock model in the ZN  spontaneous symmetry broken phase in subsection III A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We then apply the framework to more sophisticated mod-  els for emergent ZN p-gauge theory and emergent U(1) p-  gauge theory in subsections III B and III C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  These subsections include many technical details, ex-  plicitly demonstrating how higher-form symmetries can  emerge in these models and how they become exact sym-  2 Here the term “p-symmetry” and “higher symmetry” includes  both higher-form symmetry described by higher-group and al-  gebraic higher symmetry which are beyond higher-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For  example, 0-symmetry includes both group-like 0-form symmetry  and beyond-group algebraic symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  3  metries of the effective mid-IR Hamiltonians in the ther-  modynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In section IV, we summarize the general lessons learned  and the physical consequences from exact emergent  higher-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In particular, we first discuss  why, from the point of view of effective Hamiltonians,  breaking a higher-form symmetry in the UV does not au-  tomatically break it at lower energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Then we discuss  how exact emergent higher-form symmetries can charac-  terize phases of theories that do not have UV higher-form  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In particular, how exact emergent higher-  form symmetries can spontaneously break and how they  can protect symmetry-protected topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An  important takeaway is to understand how exact emergent  higher-form symmetries characterize phases, one should  first partition the parameter space by both the theory’s  exact emergent IR symmetries and not just its exact UV  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These partitions do not necessarily resemble  the system’s phase diagram and instead lay a foundation  from which its phases can be labeled and characterized  using generalized symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In section V, we conclude and discuss some open ques-  tions arising from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The appendix includes three additional sections which  supplement the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Firstly, in appendix section A,  we review the discrete differential geometry notation (in a  non-rigorous fashion) used throughout section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Then,  appendix sections B and C show how the deconfined  phases of the models for emergent ZN p-gauge theory and  emergent U(1) p-gauge theory, respectively, are related  to the corresponding quantum field theory descriptions:  p-form BF theory and p-form Maxwell theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The re-  lated subsections review the higher-form symmetries and  ’t Hooft anomalies in these quantum field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ENERGY SCALES HIERARCHIES AND  EFFECTIVE HAMILTONIANS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Low energy exact emergent generalized  symmetries  Consider a lattice bosonic quantum system described  by the local Hamiltonian H and whose total Hilbert space  V is tensor product decomposable: V = �  i Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since H  includes the exact interactions at the microscopic scale  and describes the system at all energies throughout the  entire parameter space, we refer to it as the UV Hamilto-  nian, adopting the language used in field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' While,  in theory, the system’s physical properties can be ex-  tracted from H, this proves much too difficult in prac-  tice [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A guiding principle to overcome this daunting prob-  lem is the separation of energy scales (assuming there is  no UV/IR mixing [54–57]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will always denote the  lowest energy scale as EIR and refer to the sub-Hilbert  space VIR = span{|En⟩ | En < EIR}, where |En⟩ is an  energy eigenstate, as the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, we will al-  E(III)  UV  E(IV)  UV  E(IV)  mid−IR  E(IV)  IR  E(I)  UV  E(I)  mid−IR I  E(I)  IR  E(I)  mid−IR II  E(II)  UV  E(II)  IR  I  IV  II  III  Parameter Space  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The parameter space of a many-body Hamiltonian  can be partitioned by its differing hierarchies of energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A schematic depiction of this is shown here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The parameter  space is partitioned into four regions, labeled I, II, III, and  IV, with their differing energy scale hierarchies shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ways denote the largest possible energy value as EUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, there can be other interesting energy scales  between the IR and the UV scales, which we will call  mid-IR energies Emid-IR and refer to the sub-Hilbert  space Vmid-IR = span{|En⟩ | En < Emid-IR} as the mid-  IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Generally, there can be multiple of these mid-IR  scales, and as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 1, different regions  of parameter space will have a different hierarchy of en-  ergy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' One reason to introduce the notion of the  mid-IR is to make the connection between lattice models  and quantum field theories clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, in condensed  matter physics, the UV theory is a lattice model and  quantum field theories are generally mid-IR theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is helpful to organize a quantum many-body system  around its scale hierarchies because the low-energy eigen-  states will often have additional structures absent from  high-energy eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For example, these additional  structures could reflect the presence of new, emergent  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2, states with  energy EIR ≤ E < Emid-IR have equal or more symmetry  than states with E ≥ Emid-IR but equal or less symmetry  than states with E < EIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is nontrivial to systematically identify the scale hi-  erarchies of a general UV Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here we will spe-  cialize to a typical situation where the UV Hamiltonian  can be written as  H = H0 + H1,  (1)  where a mid-IR scale Emid-IR of H0 is known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', an en-  ergy gap of a quasiparticle), and V(H0)  mid-IR is spanned by en-  ergy eigenstates of H0 satisfying ⟨Pi⟩ = 0,3 where {Pi} is  a collection of mutually commuting local projection oper-  ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In other words, {Pi} act on |ψmid-IR⟩ ∈ V(H0)  mid-IR as  3 Here, we will use the notation that an operator with the subscript  i acts only on degrees of freedom near site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  4  Pi |ψmid-IR⟩ = 0, while Pi |ψ⟩ ̸= 0 for |ψ⟩ ̸∈ V(H0)  mid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fol-  lowing the previous discussion, the low energy physics of  H0 below Emid-IR can have additional structures which  are determined by Pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, as we will soon ar-  gue, these special structures commonly include emergent  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Now, let us include the perturbation H1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We will assume H1 includes terms that mix states with  ⟨Pi⟩ = 0 and states with ⟨Pi⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because of H1, en-  ergy eigenstates of H are a superposition of states with  ⟨Pi⟩ = 0 and states with ⟨Pi⟩ ̸= 0 and, therefore, the sub-  Hilbert space spanned by states satisfying ⟨Pi⟩ = 0 is not  a mid-IR of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consequently, it appears that any spe-  cial structures arising due to Pi = 0 (such as emergent  symmetry) are destroyed by the H1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  On the other hand, if all parameters in H1 are much  smaller than those in H0, it is tempting to think that  the mid-IR of H is closely related to the ⟨Pi⟩ = 0 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This intuition motivates one to introduce the parameter  s ∈ [0, 1] and family of Hamiltonians  H(s) = H0 + s H1,  (2)  from which the mid-IR of H can be constructed from  the mid-IR of H0 by slowly tuning s = 0 to s = 1 [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, let us denote the nth many-body energy eigen-  state of H(s) as |ψ(s)  n ⟩ and define the unitary operator  Vs = �  n |ψ(s)  n ⟩⟨ψ(0)  n | which satisfies Vs|ψ(0)  n ⟩ = |ψ(s)  n ⟩ and  ⟨ψ(0)  n |A|ψ(0)  n ⟩ = ⟨ψ(s)  n |VsAV †  s |ψ(s)  n ⟩  (3)  for any operator A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the ⟨P⟩ = 0 sector of H0  is related to the ⟨V1PV †  1 ⟩ = 0 sector of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, this  definition of Vs is unphysical since VsPV †  s is likely to  be nonlocal even if P is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 46 found a local  unitary operator ULU that approximates Vs very well,  while ensuring local operators remain local when dressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An explicit form of ULU is [58]  ULU = S′  �  exp  �  i  � 1  0  ds′ Ds′  ��  ,  Ds ≡ i  �  dt F(t)eiH(s)t∂sH(s)e−iH(s)t,  (4)  where S′ denotes s′-ordering and F(t) is a function sat-  isfying a particular set of requirements, such as F(t) =  −F(−t) so that Ds is anti-Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Motivated by those results, here we assume that there  exists a proper local unitary operator ULU with the fol-  lowing properties: (1) it maps a local operator Oi to  a local operator �Oi ≡ ULUOiU †  LU (with some fattening);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (2) it maps the nth eigenstate |ψ(0)  n ⟩ of H(0) with eigen-  value E(0)  n  to a superposition of some eigenstates |ψ(1)  n′ ⟩ of  H(1) with eigenvalue E(1)  n  − δ < E(1)  n′ < E(1)  n  + δ, where  δ ≪ Emin-IR and E(1)  n  is the eigenvalue of the nth eigen-  state |ψ(1)  n ⟩ of H(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' If such a unitary operator satisfying  the aforementioned properties does not exist for a partic-  ular H in parameter space, it means that the mid-IR does  not exist at that point of parameter space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', due to  the gapped quasiparticles defining Emid-IR condensing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We will obtain our results based on this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Consider an eigenstate |ψ(0)  n ⟩ of H(0) with eigenvalue  E(0)  n  ≪ Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus |ψ(0)  n ⟩ satisfies Pi|ψ(0)  n ⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ULU  will map it to some eigenstates of H(1) with eigenvalues  much less then Emid-IR, which satisfy �Pi|ψ(1)  n ⟩ = 0, where  �Pi = ULUPiU †  LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The above is true for all the low energy  eigenstates of H(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We can therefore identify the mid-  IR of H as the sub-Hilbert space spanned by the mid-IR  eigenstates of H(0) transformed by ULU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore,  since the mid-IR states of H(0) satisfied Pi = 0, the mid-  IR states of H satisfy �Pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Notice that { �Pi} is also a  set of mutually commuting local projectors whose �Pi = 0  space corresponds to the low energy sub-Hilbert space of  H, V(H)  mid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus the exact low energy structures of H0  specified by the projectors Pi become the exact low en-  ergy structures of H specified by the projectors �Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since  {Pi} and { �Pi} are related by a local unitary transforma-  tion ULU, we exact the two exact low energy structures  are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This result was obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 46 for  emergent Z2 and U(1) gauge symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' the low en-  ergy subspace satisfies that Gauss’s law �ρi = 0 exactly  and is exactly gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=') We believe such a result  remains to be valid for more general situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Having identified a mid-IR of H, we now discuss how  to determine if, and what kind of, additional structures  emerge at E < Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this paper, we will investi-  gate the scenario where these additional structures are  emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To identify emergent symmetries,  it is helpful to adopt the perspective that a symmetry is  described/defined by an algebra of local symmetric oper-  ators [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For instance, if the UV symmetries are gener-  ated by the unitaries {Ug}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' [Ug, H0] = [Ug, H1] = 0,  then the associated algebra of local symmetric operators  is  AUV = {OUV | OUVUg = UgOUV ∀ g},  (5)  where OUV is a local operator acting on the full Hilbert  space V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, given A, one can recover the symme-  try transformation operators by finding all operators that  commute with the elements of A while also acting non-  trivially on some physical operators4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using this view, the mid-IR symmetries are described  by the algebra of local symmetric operators  Amid-IR = {Omid-IR | Omid-IR �Pi = �PiOmid-IR ∀ i,  ∀ Omid-IR ∈ AUV}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (6)  Indeed, the symmetry transformations of the symmetry  are given by all the operators that commute with Amid-IR:  Tmid-IR = {Umid-IR | Umid-IROmid-IR = Omid-IRUmid-IR,  ∀ Omid-IR ∈ Amid-IR}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (7)  4 The latter requirement is required to avoid including gauge re-  dundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  5  GUV ⊂ Gmid−IR ⊂ GIR  EUV  Emid−IR  EIR  UV, GUV  mid-IR, Gmid−IR  IR, GIR  Energy  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The symmetries of a quantum many-body system  generally depend on the energy scale of an observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' While  microscopic (UV) symmetries GUV are noticeable at all en-  ergy scales, at lower energies an observer may discover addi-  tional, emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These emergent symmetries can  be ordinary symmetries, anomalous symmetries, higher-form  symmetries, non-invertible symmetries, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' [2, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Tmid-IR will include the UV symmetries but could include  additional emergent symmetries, reflecting the possibil-  ity depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, it should generally be the  case that the emergent symmetries of H determined by  the algebra Amid-IR are exact symmetry of H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We refer  to emergent symmetries identified by Amid-IR as exact  emergent symmetries to emphasize that at E < Emid-IR,  they are equally impactful as exact UV symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In-  deed, exact emergent symmetries are not approximate,  and Amid-IR cannot describe approximate symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The above description of symmetry using AUV is very  general and capable of describing all generalizations of  symmetries, with or without ’t Hooft anomalies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Further-  more, for finite symmetries, an algebra of local symmet-  ric operators determines a non-degenerate braided fusion  (higher) category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' a topological order in one higher  dimension), which is referred to as categorical symmetry  and is a direct description of the symmetry [19, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Iden-  tifying exact emergent symmetries using Amid-IR, speci-  fied by local commuting projectors �Pi, includes all types  of emergent generalized symmetries but does not include  emergent 0-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, emergent 0-form  symmetries are usually not exact and instead approxi-  mate symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  On the other hand, as long as the  lattice is free from defects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', no lattice sites are miss-  ing), emergent higher-form symmetries are always exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We will provide a more detailed discussion on this point  in the next subsection and next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  One can possibly discover all of the emergent symme-  tries of a system in a Hamiltonian-independent way by  constructing A at all energy scales for each energy hierar-  chy in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, it is desirable to have  a Hamiltonian description reflecting the emergent sym-  metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The symmetries that emerge at E < Emid-IR are  hidden from the UV Hamiltonian H since it describes the  dynamics of both states with �Pi = 0 and �Pi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' There-  fore, to make the emergent symmetries manifest, we will  develop an effective mid-IR theory Hmid-IR that describes  only the dynamics of states with �Pi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The symmetries  of Hmid-IR should include the UV symmetries but could  also include additional ones, which we will identify as  emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since H is a sum of only opera-  tors in AUV, we expect that Hmid-IR should be a sum of  only operators in Amid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective mid-IR Hamiltonian should act only on  the mid-IR Hilbert space Vmid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Here, we develop  Hmid-IR using the physical intuition that the UV dynam-  ics generate the mid-IR dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, decomposing  the UV theory as a sum over local terms H = �  i H(i),  consider the amplitude  ⟨ψ| eiH |ψ⟩ =  ∞  �  n=0  in  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  i0···in  ⟨ψ| H(i0) · · · H(in) |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (8)  When |ψ⟩ ∈ Vmid-IR,  we require that Hmid-IR satis-  fies ⟨ψ| eiHmid-IR |ψ⟩ = ⟨ψ| eiH |ψ⟩ under the constraint  that all terms in Hmid-IR commute with { �Pi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  No-  tice that since the amplitudes must match, terms in  Hmid-IR can only be constructed from {H(i0) · · · H(in)}  for n = 0, 1, · · · , ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, the most general form  of Hmid-IR is a sum of operators constructed from all  combinations of the terms in H, or equivalently �H, that  commute with �Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Letting A denote the set of all of these  operators, we define Hmid-IR as  Hmid-IR =  �  O∈A  COO,  (9)  where {CO} are constants chosen such that the ampli-  tudes match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, as expected, Hmid-IR is a sum  over operators in Amid-IR, but A ⊂ Amid-IR since A in-  cludes only those which can generated from the terms in  �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The constants {CO} are renormalized versions of the  UV parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' On physical grounds, we require the ef-  fective mid-IR theory to be a local Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' There-  fore, the greater number of terms from �H involved in  O or the larger the region of the lattice O acts on, the  smaller CO is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The ability to define a local effective mid-  IR Hamiltonian is a requirement of the mid-IR to be well  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This definition of the effective mid-IR Hamilto-  nian is physically reasonable but not rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will  not present a rigorous justification or proof of Hmid-IR,  leaving it for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here, we state Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (9) with  the restrictions on CO as the conjectured form of Hmid-IR,  and the rest of this paper is dedicated to examining the  consequences of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A holographic picture for emergent finite  symmetries  One way to obtain a systematic understanding of  strongly correlated gapless states is to find all their uni-  versal characterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Low-energy emergent symme-  tries provide a possible candidate for such universal char-  acterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As mentioned previously, emergent sym-  6  M  QFTano  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A symmetric system with symmetry R, after be-  ing restricted in the symmetric sub-Hilbert space in a closed  space, can be viewed as a system (denoted as QFTano) with a  non-invertible gravitational anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Low energy properties  of QFTano can be exactly simulated by a boundary of a topo-  logical order M in one higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' QFTano uniquely  determines the bulk topological order M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  metries can be very general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' They can include group-  like symmetries, anomalous symmetries, (anomalous)  higher-form symmetries, (anomalous) algebraic higher-  form symmetries (also called non-invertible symmetries),  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When these symmetries are finite, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4 and 51  proposed a unified description of all types of symmetries  in terms of topological orders in one higher dimension,  called categorical symmetry5 [4, 50] or symmetry topo-  logical field theory (TFT) [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This has the following  physical meaning: given an anomaly-free system6 (de-  noted as QFTaf) with a symmetry R, its low energy  properties within the symmetric sub-Hilbert space in a  closed space are exactly simulated by a boundary of the  corresponding topological order M in one higher dimen-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is because, when restricted to symmetric sub-  Hilbert space in a closed space, the system can be viewed  as a system (denoted as QFTano) with a non-invertible  gravitational anomaly [50, 60], which is the same as topo-  logical order M in one higher dimension [61, 62] (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Certainly, there are many physically distinguishable  systems with the same finite symmetry R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Neverthe-  less, for each such system, we can find a boundary of M  to exactly simulate it at low energies, since M can also  have many physically distinguishable boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus  using bulk topological order M to described a symmetry  R means that there is an one-to-one correspondence be-  tween systems with the symmetry R and boundaries of  bulk topological order M, such that the local low energy  properties of the corresponding symmetric system and  boundary of M are identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For example, the low en-  ergy spectrum in the symmetric sub-Hilbert space of the  symmetric system is identical to the low energy spectrum  of the corresponding boundary of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The boundary of M in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 3 does not simulate all of  QFTaf’s low energy properties since QFTano only de-  scribes the symmetric sub-Hilbert space of QFTaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In-  deed, truly simulating QFTaf requires the boundary of  5 This  is  not  to  be  confused  with  non-invertible  symme-  try/algebraic higher-form symmetry, which are sometimes also  referred to as categorical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  6 Throughout section II B, an anomaly-free system means a system  with a lattice UV completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  R  R  M  M  =  QFT  QFTano  QFTano  af  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The low energy properties QFTaf with symmetry  R can be exactly simulated by a slab of topological order M  with two boundaries QFTano and �  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The stacking of the two  boundaries through the bulk topological order is denoted as  QFTano ⊠M �  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  M to capture all states in the Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This can  be achieved by adding an additional gapped boundary  �R of M [4, 8], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will denote the  composition of the topological order M with two bound-  aries QFTano and �R as QFTano ⊠M �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Provided that  the topological order M and the boundary �R have an  infinite energy gap, the low-energy properties of QFTaf  are described by the composite system QFTano ⊠M �R,  and thus  QFTaf = QFTano ⊠M �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (10)  If an anomaly-free system QFTaf admits a decomposi-  tion QFTaf = QFTano ⊠M �R, then we say the anomaly-  free system QFTaf is described by the categorical sym-  metry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The boundary �R contains gapped excitations that can  generally be of various dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' If the spatial dimen-  sion of the boundary is n, these excitations are described  by a fusion n-category, which we will denote as �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' On  the other hand, the bulk topological order M in (n + 1)-  dimensional space will generally also have numerous bulk  excitations of various dimensions, which are described by  a braided fusion n-category denoted as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The bound-  ary fusion n-category �R uniquely determines the bulk  braided fusion n-category M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, M and �R are re-  lated to one another by  M = Z( �R),  (11)  where Z( �R) is the center of �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When n = 1, Z( �R) is  the Drinfeld center of �R and, if �R = Rep(G), M is the  quantum double of �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the decomposition QFTaf = QFTano ⊠M �R, if �R is  a local fusion n-category7, then QFTaf has an anomaly-  free symmetry described by R, the dual of �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because R  7 A fusion n-category �  R is local if there exists a fusion n-category  R such that Z(R) = Z( �  R) and R ⊠M �  R = nVec, where nVec is  the braided fusion n-category describing excitations in a trivial  topological order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' above a trivial product state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The two  local fusion n-categories R and �  R are then said dual to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  7  and �R are dual, the anomaly-free symmetry R also deter-  mines the categorical symmetry M = Z(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, if  �R is not local, we cannot say QFTaf has an anomaly-free  symmetry, although its symmetries are still described by  the categorical symmetry M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We note that in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 8,  the pair ( �R, M) is regarded as a generalized symmetry  regardless if �R is local or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using QFTano ⊠M �R to describe the symmetries of  QFTaf is very general and provides a unifying formal-  ism capable of describing all generalizations of symme-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, as we will now argue, QFTano ⊠M �R is also  able to describe the exact emergent symmetries of QFTaf  discussed in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, one may  view QFTano ⊠M �R as the definition of (exact emergent)  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Recall from the previous subsection the general Hamil-  tonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (1), where a mid-IR of H0 was known and  spanned by states satisfying Pi |ψ⟩ = 0 for local com-  muting projectors {Pi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' There, we found that the ex-  act emergent symmetries in the mid-IR are determined  by {Pi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The exact emergent symmetries of H could be  found by dressing operators with ULU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is of course  still true here, but for simplicity we will just consider H0  instead of the full H0 + H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The results will hold even af-  ter we include an arbitrary perturbation H1 to H0 as we  discussed in the last subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Without a loss of gen-  erality, we will assume that H0 for the finite symmetry  case has the form  H0 =  �  i  Oi + Emid-IR  �  i  Pi,  [Oi, Pj] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (12)  Here Oi are local operators and [Oi, Pj] = 0 is required,  since Oi is assumed not to mix the Pi = 0 states and  Pi = 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let us now consider how exact emergent symmetries  arise from Pi = 0 starting from ( �R, M) which is related  to the physical theory H0 by QFTano ⊠M �R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In doing so,  we will also see why 0-form symmetries cannot be exact  emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is believed that a topological order with a gapped  boundary can be realized by commuting projector model,  which also has a commuting projector Hamiltonian real-  izing the gapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the �R-boundary  and the M-bulk in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4 can be realized by a commuting  projector model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pi in H0 are those commuting projec-  tors, where H0 is is viewed as a Hamiltonian that de-  scribes the slab in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4 which is a model in one lower  dimension if the slab has a finite thickness compare to  lattice spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Oi’s in H0 are the boundary Hamilto-  nian terms describing the QFTano boundary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since the thickness of the bulk M is finite, the local op-  erators Oi in H0 can also contain operators that connect  the QFTano boundary and the �R boundary in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' By  definition, these Oi must commute with Pi in order to be  allowed in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' If a symmetry on the boundary QFTano  is going to be explicitly broken, there must be an al-  lowed operator which transfers symmetry charge from �R  R  QFTano  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A lattice realization of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4, where qubits live on  the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The bulk Hamitonian contains the star terms (the  diamonds around a vertex) and the plaquette terms (the di-  amonds inside a square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The intra-boundary terms act on  qubits near a boundary, while the inter-boundary terms act  on qubits that connect two boundaries (the vertical line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  to QFTano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For a p-form symmetry, its charges are cre-  ated by operators acting on a p-dimensional subspace,  and therefore transferring a symmetry charge from �R to  QFTano requires a (p + 1)-dimensional operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For a  0-form symmetry, such an operator would include a fi-  nite number of local operators acting in a line from �R to  QFTano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is allowed in the set of local operators {Oi}  and therefore allowed in H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus, the projectors Pi  cannot produce exact emergent 0-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For  p-form symmetries with p > 0, any inter-boundary oper-  ators that transfer symmetry charge are non-contractible  extended operators, acting on the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These  operators are not local and are not included in the set of  local operators {Oi} and therefore the symmetry is pre-  served.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus the projectors Pi can give rise to an exact  emergent higher-form symmetry even when including all  the local low energy operators Oi (regardless if they are  inter-boundary or intra-boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The above discussion is pretty general, so let us give  an example to construct a model with an exact emergent  Z(1)  2  symmetry in 1+1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We consider M to be the toric  code model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' the Z2 lattice gauge theory) and �R is  the Z2-charge condensed boundary (the so-called rough  boundary [63]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The slab QFTano ⊠M �R in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 4 be-  comes the lattice shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5, with qubits residing  on the links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The bulk Hamiltonian that give rise to a toric code  model contains star terms Pvert  i  = (1 − Z1Z2Z3Z4)/2  acting on the four qubits on the four links con-  necting  to  each  vertex  and  plaquette  terms  Pplaq  i  = (1 − X1X2X3X4)/2 acting on the four qubits  on the four links around each square (see the diamond  around a vertex and inside a square, respectively, in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The �R boundary has truncated plaquette terms  Pbdry  i  = (1 − X1X2X3)/2 acting on the three qubits  on the three links around each open square (see the  up-side-down triangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5 in the �R boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The lattice model Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5 can be viewed as a 1+1D  lattice model H0, and the above three types of terms,  Pvert  i  , Pplaq  i  , and Pbdry  i  , correspond to the projectors Pi  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The Oi operators in H0 can be any operator  that commutes with Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus the local energy dynamics  8  R  i+1/2  i  i+1  i−1  i−1/2  QFTano  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A thin slab limite of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5, where qubits live on the  links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  controlled by Oi operators are constrained by the local  projectors Pi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consequently, the local projectors Pi’s  cam give rise to an emergent symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' But what is this  emergent symmetry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We note that the Oi operators can be divided into two  classes:  intra-boundary operators and inter-boundary  operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The intra-boundary operators only act on  the qubits near the boundary QFTano8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The inter-  boundary operators can act on qubits that connect the  two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' An example of inter-boundary opera-  tors is X1X2 · · · Xn acting on the vertical links connect-  ing the two boundaries (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5), which commute with  the projectors Pvert  i  , Pplaq  i  , and Pbdry  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To make our explicit discussion as simple as possi-  ble, let us take a thin slab limit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 5, which gives  us Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 6 where we label vertical (horizontal) links by  i (i + 1  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this limit, the only remain projector is  Pbdry  i  :  Pi = 1  2(1 − XiXi+ 1  2 Xi+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The allowed local  boundary operators Oi must commute with Pi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From  the thin slab limit, we find that all such Oi’s are gen-  erated by taking products of the operators Xi, Xi+ 1  2 ,  and Zi− 1  2 ZiZi+ 1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Notice that while Xi+ 1  2 , XiXi+ 1  2 Xi+1,  Zi− 1  2 ZiZi+ 1  2 are intra-boundary operators, the Xi’s are  inter-boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let us first restrict ourselves to the local operators Oi  that are intra-boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will consider the  full set up, where Oi includes intra and inter-boundary  operators, after.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this case, we will obtain an exact  emergent Z2 0-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To see this result, we  note that the intra-boundary local operators form the  algebra of the local symmetric operator  A(intra-only)  mid-IR  = {Xi+ 1  2 , XiXi+ 1  2 Xi+1, Zi− 1  2 ZiZi+ 1  2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (13)  The operators (local or non-local) that commute with  A(intra-only)  mid-IR  are generated by  T (intra-only)  mid-IR  = {XiXi+ 1  2 Xi+1,  �  i  (Zi− 1  2 ZiZi+ 1  2 )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (14)  8 The commuting projectors Pbdry  i  etc already give the �  R bound-  ary a large energy gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The intra-boundary operators near �  R  can only be combinations of those commuting projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' There  are no new operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  and give rise to all symmetry transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using the  language of operator algebra, the symmetry transforma-  tions T (intra-only)  mid-IR  are the double commutant of the local  projectors {Pi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From T (intra-only)  mid-IR  , we find that when re-  stricted to the intra-boundary operators there are two ex-  act emergent symmetries enforced by the projectors Pi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  XiXi+ 1  2 Xi+1 generates symmetry transformations which  act on loops of the slab, and therefore corresponds to a  Z(1)  2  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' �  i(Zi− 1  2 ZiZi+ 1  2 ) acts on the entire lat-  tice and therefore corresponds to a Z(0)  2  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note  that when the (2 + 1)D system QFTano ⊠M �R is mapped  to the (1 + 1)D system QFTaf, the Z(0)  2  symmetry still  acts on the entire lattice while the Z(1)  2  symmetry now  acts on a single lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From the previous general discussion, we expect that  exact emergent 0-form symmetries will be explicitly bro-  ken by inter-boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s verify this ex-  pectation in this example by now including the inter-  boundary operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Doing so, the algebra of local sym-  metric operators becomes  Amid-IR = {Xi+ 1  2 , Xi, Zi− 1  2 ZiZi+ 1  2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (15)  The symmetry transformations and now generated by  Tmid-IR = {XiXi+ 1  2 Xi+1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (16)  So, the Z(1)  2  symmetry is still present, but the Z(0)  2  sym-  metry is now explicitly broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is because the inter-  boundary operators Xi’s also commute with the projec-  tors Pi but transfer the charges of the Z(0)  2  symmetry  between the two boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  As a result, those inter-  boundary operators break the Z(0)  2  symmetry previously  enforced by the projectors Pi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The operators that could  transform the Z(1)  2  symmetry charges between the two  boundaries were not included in Amid-IR since they are  not local operators, and as a result, the Z(1)  2  symmetry  is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  EXAMPLES OF EXACT EMERGENT  HIGHER-FORM SYMMETRIES  In this section, we apply the framework discussed in  section II A to three lattice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We organize our  discussion around the energy hierarchies of these models  and discover emergent higher-form symmetries by deriv-  ing effective Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These symmetries have been  noticed previously in the literature in similar models, but  typically from an IR point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here we take a UV  point of view, starting from lattice models for which these  symmetries are not exact, and emphasizing that as emer-  gent symmetries they are exact emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This means that they are symmetries that emerge be-  low an energy scale yet constrain the IR as if they were  UV symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Each subsection is dedicated to a single  9  K  U  I  III  II  J  U  E(I)  UV  UV  E(II)  UV  E(II)  IR  UV  IR  E(III)  UV  E(III)  mid−IR  E(III)  IR  UV  mid-IR  IR  Model C:  Model D:  GUV  GUV  GUV  GIR = GUV × U(1)(p)  GUV  GIR = GUV × ℤ(p)  N  GUV  Gmid−IR = GUV × U(1)(p)  GIR = GUV × U(1)(p) × U(1)(d−p−1)  GUV  Gmid−IR = GUV × ℤ(p)  N  GIR = GUV × ℤ(p)  N × ℤ(d−p)  N  Model D:  Model C:  Model C  Model D  Parameter Space  HUV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (left) We partition the parameter space of models C (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (66)) and D (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (35)) into three regions labeled I, II, and  III, which we structure our discussions in sections III B and III C around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (right) In these regions, we identify energy scale  hierarchies and the exact emergent symmetries at each scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The exact emergent U(1)(d−p−1) symmetry in the IR of region  III for model C is trivial at the lattice scale, but its action becomes nontrivial in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These regions are not  necessarily distinct phases of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, region I and II are likely in the same phase while region III is a different  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is emphasized in the left figure where solid lines indicate a phase transition while dashed lines do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  one of these models and is entirely self-contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, if  desired, the reader should feel free to read only the sub-  section(s) on their favorite model(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the reader who  reads more than one, we apologize for any inconvenience  due to subsections overlapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  All of the models we consider are described by Hamil-  tonians governing degrees of freedom on a d-dimensional  cubic spatial lattice with continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We exten-  sively use discrete differential geometry notation, which  we review (in a non-rigorous fashion) in appendix sec-  tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The remainder of this section is organized as fol-  lowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In subsection III A, as a warm-up we consider the quan-  tum clock model Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (20) which has a UV ZN symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  When this ZN symmetry is spontaneously broken, we  find there is an exact emergent Z(d)  N symmetry at energies  below the domain wall gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The IR symmetry operators  form a projective representation of ZN × Z(d)  N , signaling  the presence of a mixed ’t Hooft anomaly that protects  the ground state degeneracy in the SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In subsection III B, we consider a model of emergent  ZN p-gauge theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (35), which we call model D, and  in subsection III C, we consider a model of emergent U(1)  p-gauge theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (66), which we call model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The ex-  act emergent symmetries and energy scale hierarchies of  these models are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We want to em-  phasize that the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7 is a schematic depic-  tion and that the precise shapes of the regions and the  nature of the boundaries between them are not investi-  gated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Region II ∪ III exists in parameter space wherever  there are gapped gauge charge excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The creation  operator for the gauge charges is exactly known in the  K, J → 0 limit, and the implicitly defined local unitary  ULU from section II A is used to construct the creation  operator away from this single point in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We do not construct an explicit form for ULU and thus  cannot rigorously investigate the shape of region II ∪ III  in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since region III corresponds to the  deconfined phase of the gauge theory, assuming d is large  enough that the exact emergent p-form symmetry can  spontaneously break, we expect it to be a finite-sized re-  gion in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Region II corresponds  to the confined phase, where the gauge charges are con-  fined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When J = 0, it is easy to confirm the exact emer-  gent p-form symmetry is present, but for finite J, it re-  lies on the existence of ULU and whether or not the gauge  charges are genuinely gapped excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7 portrays  the possibility that they are and that region II exists  even when J ̸= 0, and the discussion throughout these  examples will reflect this possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Another possibility  is that region II exists only for J → 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' outside of the  deconfined phase, the exact emergent symmetry would  then only exist along a portion of the vertical axis of the  parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let us summarize the results of model C and D in  the familiar case p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We first consider model D with  p = 1, d = 3, and trivial GUV, which is emergent 3+1D  Z2 lattice gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (35), the Gauss’s law is  10  enforced at a large energy scale U which represents the  energy gap of a Z2-gauge charge excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore,  the term creating a pair of Z2-charges has a energy scale  J (which explicitly breaks a UV Z(1)  2  symmetry) and the  term creating Z2-fluxes has an energy scale K (which  explicitly breaks a UV Z(2)  2  symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus a large J  drives a Z2-charge condensation transition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' a Higgs  transition) and a large K drives a Z2-flux condensation  transition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' a confinement transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The region III  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7 corresponds to the deconfined phase of the Z2  gauge theory, which has an exact emergent IR symmetry  Z(1)  2  ×Z(2)  2  (below the energy gap of dressed Z2-charges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The symmetry operators of the Z(1)  2  symmetry and Z(2)  2  symmetry anti-commute if their intersect at odd number  of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus model D realizes the exact emergent  Z(1)  2  × Z(2)  2  in a projective representation, and therefore  there is a mixed ’t Hooft anomaly between Z(1)  2  and Z(2)  2  symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Such a mixed anomaly can be described by  anomaly in-flow using an invertible topological quantum  field theory in one higher dimension, described by the  following action amplitude (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B57))  e−S = eiπ  �  M5 A∪ ˆ  A,  (17)  where A is a Z2-valued 2-cocycle field and ˆ  A is a Z2-  valued 3-cocycle field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A is the background gauge field  from gauging the Z(1)  2  symmetry, while ˆ  A is the back-  ground gauge field from gauging the Z(2)  2  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We  note that, following the discussion in section (II B), once  making A and ˆ  A dynamical, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (17) can also be viewed  as the action amplitude in the path integral describing  the categorical symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' the topological order in  one higher dimension) for the anomalous Z(1)  2  × Z(2)  2  sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Due to the mixed ’t Hooft anomaly, the system must  have degenerate ground states on a 3-dimensional torus  as long as Z(1)  2  × Z(2)  2  is present in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since the  emergent symmetry Z(1)  2  × Z(2)  2  remains to be exact  against local UV perturbations, the ground states on a  3-dimensional torus also remain degenerate against any  perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is a way to understand the robust-  ness of topological order using the robustness of exact  emergent symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The exact emergent mid-IR Z(1)  2  symmetry (existing  at energies below ∼ U) is present on both sides of the  confinement transition, II ↔ III, and is a consequence of  weak Z2 charge fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The exact emergent mid-IR  symmetry Z(1)  2  controls this transition and its unphysical  part corresponds to the exact emergent Z2 gauge redun-  dancy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Next, let us consider model C with p = 1, d = 3, and  trivial GUV, which is emergent 3+1D U(1) lattice gauge  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (66), the Gauss’s law is enforced at a  large energy scale U which is the energy gap of a U(1)-  gauge charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A term creating a pair of U(1)-charge has  a energy scale J and a term creating U(1)-flux fluctu-  ation has an energy scale K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus a large J drives a  U(1)-charge condensation transition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' a Higgs transi-  tion) and a large K drives a U(1)-monopole condensa-  tion transition (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' a confinement transition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The re-  gion III corresponds to the deconfined phase of the U(1)  gauge theory, which has an exact emergent IR symmetry  U(1)(1) × U(1)(1) (below the energy gap of U(1)-charges  and U(1)-monopoles) in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There is a mixed ’t Hooft anomaly between the two  U(1)(1) symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It is described by an invertible topo-  logical quantum field theory in one higher dimension (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C77))  e−S = ei2π  �  M5 A∧d ˆ  A,  (18)  where A, ˆ  A are two R/Z-valued 2-form fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A is the  background gauge field from gauging the first U(1)(1)  symmetry, while ˆ  A is the background gauge field from  gauging the second U(1)(1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Speculating that  categorical symmetry can be generalized to continuous  symmetry, once making A and ˆ  A dynamical, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (18)  can also be viewed as the action amplitude describing the  categorical symmetry for the anomalous U(1)(1)×U(1)(1)  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A result of the mixed ’t Hooft anomaly of the exact  emergent U(1)(1) × U(1)(1) symmetry is that as long as  the U(1)-charges and U(1)-monopoles have large energy  gap the system must be gapless [64, 65] no matter how  strong of the interaction between the U(1) photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This  would mean that as long as these excitations have a  large energy gap, then due the exact emergent anoma-  lous U(1)(1) × U(1)(1) symmetry, the system should re-  main gapless even when a strong interaction between the  U(1) photons drives a phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The exact emergent mid-IR U(1)(1) symmetry (exist-  ing at energies below ∼ U) is present on both sides of  confinement transition II ↔ III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, it controls the  transition and its unphysical part corresponds to the ex-  act emergent U(1) gauge redundancy [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Quantum clock model  In this section, we will apply the framework discussed  in section II to the quantum clock model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consider ZN  quantum rotors residing on the 0-cells (sites) of the spa-  tial d-dimensional cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A ZN quantum rotor is  an N-level system described by clock operators Xc0 and  Zc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These unitary operators are N-dimensional gener-  alizations of the Pauli matrices, satisfying  Z�c0Xc0 = ωδc0,�c0 Xc0Z�c0,  XN  c0 = ZN  c0 = 1,  (19)  where ω ≡ ei2π/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The eigenvalues of the clock opera-  tors are {1, ω, ω2, · · · , ωN−1} ≃ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  11  The Hamiltonian of the quantum clock model is  H = −J  2  �  ⟨c0�c0⟩  �  X†  c0X�c0 + X†  �c0Xc0  �  − K  2  �  c0  �  Zc0 + Z†  c0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (20)  where the first sum is over pairs of nearest neighbor 0-  cells c0 and �c0 and the second sum is over all 0-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This  theory has an exact ZN 0-form—Z(0)  N —symmetry, whose  symmetry operator is generated by the unitary  U =  �  c0  Zc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (21)  The charged operators of this symmetry are Xc0, which  from the clock operator algebra transform as  Xc0 → UXc0U † = e2π i/NXc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (22)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An exact emergent Z(d)  N  symmetry and mixed ’t Hooft  anomaly  Assuming that d > 0, when K/J ≪ 1, the quantum  clock model at zero temperature is in a Z(0)  N spontaneous  symmetry broken (SSB) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, in the tractable  K/J → 0 limit, the quantum clock model is  H  ����  K/J→0  = −J  2  �  ⟨c0�c0⟩  �  X†  c0X�c0 + X†  �c0Xc0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (23)  The  ground  state  in  this  limit  satisfies  X†  c0X�c0 |vac⟩ = |vac⟩  for  all  neighboring  0-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Consequently,  since  for  any  0-cells  c0  and  c′  0,  X†  c0Xc′  0 = �  c1∈O1  �  �c0∈∂c1 X�c0 where ∂O1 = {−c0, c′  0},  ⟨X†  c0Xc′  0⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, in this limit the Z(0)  N symmetry  is spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In a Z(0)  N  SSB phase,  there are gapped (d − 1)-  dimensional topological defects—domain walls—carrying  ZN topological charge which populate the excited states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, in the K/J → 0 limit, the topological defect den-  sity ˆρ for a state |ψ⟩ is defined by  �  c0∈∂c1  Xc0 |ψ⟩ = exp  �2πi  N (∗ ˆρ)c1  �  |ψ⟩ ,  (24)  where X−c0 ≡ X†  c0 and (∗ ˆρ)c1 ≡ ˆρ∗ c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From the ZN  clock algebra Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (34), �  c0∈∂c1 Xc0 and Zc0 satisfy  Zc0  �  �c0∈∂c1  X�c0 = ω(−1)c0  �  �c0∈∂c1  X�c0Zc0,  (25)  for all c0 ∈ ∂c1, where ω = e2π i/N and (−1)c0 is the  sign in front of c0 in ∂c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using this,  let’s act  �  �c0∈∂c1 X�c0 on the state (∗ Z)ˆcd |0⟩, with ∗ ˆcd ∈ ∂c1 and  �  �c0∈∂c1 X�c0 |0⟩ = |0⟩:  �  �c0∈∂c1  X�c0(∗ Z)ˆcd|0⟩ = ω(−1)∗ ˆcd (∗ Z)ˆcd |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (26)  Because this is true for all ∗ ˆcd ∈ ∂c1, acting (∗ Z)ˆcd onto  |0⟩ causes (∗ ˆρ)c1 ̸= 0 for all c1 ∈ δ ∗ ˆcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the  operator (∗ �Z)ˆcd excites a topological defect on ∂ˆcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the K/J → 0 limit of the Z(0)  N  SSB phase, the  fact that the ground state satisfies X†  c0X�c0 |vac⟩ = |vac⟩  means that the topological defect density in the ground  states is zero for all d-cells of the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Equiva-  lently, this can also be see from the fact that the ground  state satisfies ⟨Zc0⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, because the topological de-  fect creation operator does not have a vev, the topological  defects are not condensed and thus gapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The domain wall gap provides a candidate energy scale  below which new structures may emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, when  K/J = 0, there exists a low energy sub-Hilbert space  spanned by states satisfying ⟨ˆρˆcd−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is the  ground state subspace and defines the IR of the SSB  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, when K/J ̸= 0, there no longer exists a  low-energy sub-Hilbert space spanned by states satisfy-  ing ⟨ˆρˆcd−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is because the K term in H causes  the ⟨ˆρˆcd−1⟩ = 0 and ⟨ˆρˆcd−1⟩ ̸= 0 states to mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, a  corresponding low-energy sub-Hilbert space can be iden-  tified using ULU from section II A to continue any oper-  ator A which we understand at K/J = 0 to a (fattened)  local operator �A ≡ ULUAU †  LU with the same expectation  values of A but at K/J ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, there exists a low-  energy sub-Hilbert space for both K/J = 0 and K/J ̸= 0  spanned by states satisfying ⟨�ˆρˆcd−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because the  Z(0)  N  SSB phase is gapped, we can use the local unitary  from the quasi-adiabatic continuation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (4), to con-  tinue local operators throughout the entirety of the Z(0)  N  SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because ULU in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (4) is constructed only from terms  in H, operators that commute with every term in H are  unaffected by ULU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In particular, because the Z(0)  N sym-  metry operator commutes with each term in H, the sym-  metry operator U in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (21) satisfies  U = �U =  �  c0  �Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (27)  Having identified the IR of the SSB phase, we’d now  like to find an effective IR theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As we learned in sec-  tion II, the effective IR Hamiltonian HIR is a sum of all  terms allowed in the IR that can be constructed from  the terms in �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The terms in �H are ( �  X†  c0 �  X�c0 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=') and  ( �Zc0 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In  the  IR,  because  �ˆρˆcd−1 = 0,  from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (24),  �  c0∈∂c1 �  Xc0 ≡ X†  c0 �  X�c0 = 1 and therefore ( �  X†  c0 �  X�c0 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=')  does not contribute in HIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The operators ( �Zc0 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=')  are not allowed operators in the IR since they excite  dressed topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, allowed IR op-  erators can be constructed from �Zc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, because  (∗ �Z)ˆcd excites a dressed topological defect on ∂ˆcd, acting  (∗ �Z′)ˆcd0 on any d-cycle of the dual lattice ˆCd does not ex-  cite dressed topological defects since ∂2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  12  the IR allowed operator constructed from �Zc0 is  �T †[ ˆCd] =  �  ˆcd∈ ˆ  Cd  (∗ �Z)ˆcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (28)  This is a product of �Zc0 over all 0-cells in a connected  part of the spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, denoting a connected  part of the spatial lattice as M, we can instead consider  �T †[M] =  �  c0∈M  �Zc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (29)  Interestingly, this is just the Z(0)  N  symmetry operator,  and therefore does not need to be dressed by ULU:  �T †[M] = T †[M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective IR theory, therefore, includes all terms  constructed from  �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Denoting the set of all con-  nected part of the spatial lattice with ZN coefficients as  H0(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), the effective IR Hamiltonian is  HIR = −J  �  M∈H0(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  κ|M| �T[M],  (30)  where κ ∼ K/J and |M| is the number of 0-cells form  which M is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since K/J ≪ 1 in the SSB  phase, in the thermodynamic limit where |M| → ∞, the  effective IR Hamiltonian becomes a constant  HIR  ����  L→∞  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (31)  This simply reflects the fact that the Z(0)  N  SSB phase is  a gapped phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective IR Hamiltonian of the Z(0)  N SSB phase has  a new symmetry which was not present in the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, the IR has the symmetry where the IR-allowed  operator �T transforms as  �T[ ˆCd] �→ e  2π i  N �T[ ˆCd].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (32)  This cannot be a transformation by an arbitrary phase  eiα and must be e  2π i  N  in order for ( �T[ ˆCd])N = 1 to re-  main satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The transformation can be accomplished  by transforming �Zc0→ e  2π i  N �Zc0 for only a single lattice  site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the operator that causes this transfor-  mation is  �ˆU[c0] = �  Xc0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (33)  At first glance of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (33), one sees a local transforma-  tion and may think it is a conventional gauge transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, it is not because a physical operator  transforms nontrivially under it: �T[ ˆCd] transforms by a  nontrivial element of ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, since the physi-  cal operators are supported on d-cycles, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (33) is the  symmetry operator of a ZN d-form symmetry—a Z(d)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Notice that since the Z(0)  N SSB phase only ex-  ists when d > 0, this symmetry is always a higher-form  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This emergent Z(d)  N  symmetry has been noted previ-  ously throughout the literature [49, 66, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here we find  that it is an exact emergent symmetry, meaning that de-  spite it being an emergent symmetry, it is an exact sym-  metry of the effective IR Hamiltonian in the thermody-  namic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, it constrains the IR as if it were  an exact UV symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, this exact emer-  gent symmetry is present throughout the entire Z(0)  N SSB  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, away from the tractable K/J = 0 point,  local unitary operators dress the K/J = 0 charged and  symmetry operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So, we have found that in the IR of the Z(0)  N SSB phase,  there is an exact emergent Z(0)  N × Z(d)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' How-  ever, the Z(0)  N  and Z(d)  N  symmetries are not independent  of one another, there is a mixed ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  fact that the Z(0)  N × Z(d)  N  symmetry is anomalous can be  noticed from the fact the symmetry operator of the Z(0)  N  symmetry U of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (27) is charged under the Z(d)  N  sym-  metry, reflecting an obstruction to gauging both symme-  tries9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To summarize, when considering only the UV theory,  it appeared that there was only an exact Z(0)  N  symme-  try in all phases of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, by construct-  ing the effective IR Hamiltonian of the Z(N)  N  SSB phase,  we learned that there is actually an exact anomalous  Z(0)  N × Z(d)  N symmetry in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Emergent ZN p-gauge theory for p ≥ 1  In this section, we will apply the framework discussed  in section II to a model for emergent ZN p-gauge theory  which we call model D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consider ZN quantum rotors re-  siding on each p-cell of the spatial d-dimensional cubic  lattice with p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A ZN quantum rotor is an N-level  system described by clock operators Xcp and Zcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These  unitary operators are N-dimensional generalizations of  the Pauli matrices, satisfying  Z�cpXcp = ωδcp,�cp XcpZ�cp,  XN  cp = ZN  cp = 1,  (34)  where ω ≡ ei2π/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The eigenvalues of the clock opera-  tors are {1, ω, ω2, · · · , ωN−1} ≃ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The microscopic model we consider is described by the  9 See footnote 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  13  +  −  +  −  +  −  +  −  −  +  −  −  +  model C:  ρcp−1 = ∑  Lz  ±  model D:  τz  cp−1 = ∏Z  ±  +  +  −  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Graphical representation of the operator ρcp−1 of  model C (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (67)) and the operator τ z  cp−1 of model D  (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (35)) in three spatial dimensions for (first row) p = 1,  (second row) p = 2, and (third row) p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Depending on the  operator, the disks on the p-cells labeled by ± either denote  the sign in front of Lz belonging to that p-cell in the sum for  ρcp−1, or whether the Z operator belonging to that p-cell is  Z+ ≡ Z or Z− ≡ Z† in the product for τ z  cp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The (p − 1)-cell  the operator is associated with is colored purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Hamiltonian  HUV= −U  2  �  cp−1  τ z  cp−1 − K  2  �  cp  Zcp + J  2  �  cp  Xcp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=',  τ z  cp−1 ≡  �  cp∈δcp−1  Zcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (35)  The sum �  cp is over all p-cells of the spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  product �  cp∈δcp−1 in the definition of τ z  cp−1 is over the  coboundary of cp−1, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (A3) of the appendix,  with the convention Z−cp = Z†  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that we need to  require p ≥ 1 in order for (p − 1)-cell cp−1 to be well  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' τ z is generally a product of 2(d − p + 1) opera-  tors, examples of which are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that  because the eigenvalues of Zcp are elements of ZN, the  eigenvalues of τ z  cp−1 are also ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lastly, since HUV has  terms linear in Zcp and Xcp, there are no transformation  of Zcp or Xcp that leave HUV invariant and thus no UV  symmetries in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  L+  +  −  ± ρ = ± 1  +  −  +  −  −  −  +  +  −  +  +  −  −  −  +  +  −  +  X  ± τz = e±2πi/N  model C  model D  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Graphical representation of the excitation created  by L+  cp in model C and the excitation created by Xcp in  model D, shown in three spatial dimensions for (first row)  p = 1, (second row) p = 2, and (third row) p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The yel-  low disk represents the L+  cp or Xcp operator, depending on  the model, acting on the U(1) or ZN rotor belonging to that  p-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For model C, the purple disk labeled by ± represents  the ± sign in ρcp−1(L+  cp |0⟩) = ±(L+  cp |0⟩) for that (p − 1)-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For model D, the disk labeled by ± represents the ± sign in  τ z  cp−1(Xcp |0⟩) = ω±1(Xcp |0⟩) for that (p − 1)-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An exact emergent Z(p)  N  symmetry  When J = K = 0, there exists a low energy sub-  Hilbert space spanned by states satisfying ⟨τ z  cp−1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, the first term in HUV introduces an energetic  penalty for states satisfying ⟨ψ| (τ z  cp−1 + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=') |ψ⟩ ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We interpret such states in the J = 0 and U ≫ K  limit as describing a gapped excitation, a segment of  which resides on the (p − 1)-cell cp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, from  the clock operators’ algebra, τ z  cp−1  and Xcp  satisfy  τ z  cp−1Xcp = ω(−1)cp Xcpτ z  cp−1 for all cp−1 ∈ ∂cp, where  (−1)cp denotes the sign in front of cp in the expression  for δcp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using this, let us act τ z  cp−1 on the state Xcp |0⟩,  where cp−1 ∈ ∂cp and τ z  cp−1 |0⟩ = |0⟩:  τ z  cp−1  �  Xcp |0⟩  �  = ω(−1)cp �  Xcp |0⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (36)  Because this is true for all cp−1 ∈ ∂cp, Xcp excites the  aforementioned excitation on ∂cp, examples of which are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We’ll refer to these bosonic (p − 1)-  dimensional (in space) excitations as “charges.”  How-  ever, note that since XN  cp = 1, exciting N charges is the  same as not exciting any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the charge number takes  values in ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is tempting to consider the charge gap as a candidate  energy scale below which new physics emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  14  when J ̸= 0, there no longer exists a low-energy sub-  Hilbert space spanned by states satisfying ⟨τ z  cp−1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is because the J term in HUV causes the ⟨τ z  cp−1⟩ = 1  and ⟨τ z  cp−1⟩ ̸= 1 states to mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, a correspond-  ing low-energy sub-Hilbert space can be identified us-  ing ULU from section II A to continue any operator A  which we understand at J = 0 to a (fattened) local oper-  ator �A ≡ U (1)  LUA(U (1)  LU)† with the same expectation values  of A but at J ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, when U ≫ K, there ex-  ists a low-energy sub-Hilbert space for both J = 0 and  J ̸= 0 spanned by states satisfying ⟨�τ z  cp−1⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We view  states with ⟨�τ z  cp−1⟩ ̸= 1 as having dressed charges excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since the undressed charges are created using Xcp, these  dressed charges are created using �  Xcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will not find  an explicit form for U (1)  LU and thus will not precisely know  throughout how much of parameter space the dressed  (fattened) operators can be defined without violating the  assumptions of U (1)  LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Instead, we will assume that such  an operator exists and can access a greater than measure-  zero part of parameter space, and will investigate the  consequences of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  At this point in our investigations, we cannot tell if the  dressed charge gap ∆dressed-charge is an IR scale or a mid-  IR I scale or a mid-II scale, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In section III B 2, we will  learn that it is a mid-IR scale in region III of parameter  space but an IR scale in region II of parameter space  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the rest of this section, however, we will  adopt the language from the perspective of region III and  call the dressed charge gap a mid-IR scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Given the mid-IR scale Emid-IR ≡ ∆dressed-charge, we  would now like to the find an effective mid-IR theory,  which by definition is a theory only describing states at  energies E < Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As we learned in section II, the ef-  fective mid-IR Hamiltonian Hmid-IR is a sum of all terms  allowed in the mid-IR that can be constructed from the  terms in �HUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The terms in �HUV are �τ z  cp−1, �Zcp, and  �  Xcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the mid-IR, �τ z  cp−1 = 1, so it is trivial and will not  contribute in Hmid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, since �Zcp commutes  with �τ z  cp−1, it does not excite any dressed charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus  �Zcp is an allowed mid-IR operator from which terms in  Hmid-IR can be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The operators �  Xcp are not  allowed operators in the mid-IR since they excite charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, allowed mid-IR operators can be constructed  from �  Xcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, since �  Xcp excites a dressed charge on  ∂cp, acting �  Xcp on any p-cycle does not excite dressed  charges since ∂ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, an allowed operator in  †  †  †  ˜L+  model C  ˜  X  model D  †  †  †  †  †  †  †  †  †  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Graphical representation of the Wilson operator  �  W †(Cp) of model C (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (70)) and model D (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (38)) sup-  ported on Cp = ∂cp+1 in three spatial dimensions for (first  row) p = 0, (second row) p = 1, and (third row) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For  model C (D), the yellow colored disks denote �L+  cp ( �  Xcp) op-  erators belonging to that p-cell, the product of which yields  �  W †(Cp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Discs labeled by † denote the hermitian conjugate  of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In each row, the (p + 1)-cell cp+1 satisfying  Cp = ∂cp+1 is colored green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  the mid-IR is10  �  W †[Cp] =  �  cp∈Cp  �  Xcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (38)  We call �  W † the Wilson operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It has the interpretation  of exciting a dressed charge, transporting it along a p-  cycle, and then ultimately annihilates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 10 shows  a few graphical representations of the Wilson operator  supported on the smallest possible p-cycle: the boundary  of a (p + 1)-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that the Wilson operator satisfies  �  W †[Cp] = �  W[−Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective mid-IR theory, therefore, includes all  terms constructed from �  W and �Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Denoting the set of  all oriented p-cycles with ZN coefficients on the spatial  10 Describing the extended object �  W † using local operators intro-  duces a gauge redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, �  W † is invariant under  �  Xcp → Ξcp �  Xcp  with  �  cp∈Cp  Ξcp = 1 ∀ Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (37)  For ( �  Xcp)N = 1 to remain invariant,  the gauge parameter  must satisfy Ξcp ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  When Ξcp = �  cp−1∈∂cp λcp−1, with  λcp−1 ∈ ZN and λ−cp−1 = λ†  cp−1, this is the canonical ZN gauge  redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Otherwise, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (37) corresponds to large gauge  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  15  lattice Md as Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), the mid-IR Hamiltonian takes  the form  H(III)  mid-IR =−κU  2  �  cp  ( �Zcp+ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=')− U  2  �  Cp∈Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  ε|Cp|�  W[Cp] + · · ·,  (39)  where κ ∼ K/U, ε ∼ J/U, |Cp| is the number of p-cells  form which Cp is constructed, and the · · ·’s include all  other possible terms constructed from both ( �Zcp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=')  and �  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since the dressed charge gap is the mid-IR  scale of region III but the IR of region II,  H(III)  mid-IR ≡ H(II)  IR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (40)  The effective mid-IR theory is only well defined pro-  vided ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  As a consequence, since the lattice is  simply connected, the Wilson operators supported on  nontrivial p-cycles are exponentially suppressed by εLp,  where L is the linear system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, denot-  ing contractible p-cycles with ZN coefficients on the  spatial lattice Md as Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), terms with Wilson  operators supported on cycles in the homology class  Hp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) = Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN)/Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) vanish in the  thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, in the thermodynamic  limit H(III)  mid-IR becomes  H(III)  mid-IR  ����  L→∞  =−κU  2  �  cp  ( �Zcp+ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=')− U  2  �  Cp∈Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  ε|Cp|�  W[Cp] + · · ·,  (41)  where  the  · ·’s  include  all  possible  terms  con-  structed from both ( �Zcp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=') and �  W[Cp] with only  Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the thermodynamic limit, H(III)  mid-IR is invariant under  transforming the UV operator as  �  Xcp→ e  2π i  N Γcp �  Xcp,  (dΓ)cp+1= 0,  Γcp ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (42)  Here (dΓ)cp+1 is the lattice exterior derivative of Γcp,  defined as  (dΓ)cp+1 ≡  �  cp∈∂cp+1  Γcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (43)  Furthermore, Γcp is required to take integer values in  order for (Xcp)N = 1 to be invariant under the transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Under this transformation, the Wilson operator  transforms as  �  W[Cp] �→ e  2π i  N  �  cp∈Cp Γcp �  W[Cp],  (44)  which leaves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (41) unchanged because �  cp∈Cp Γcp = 0  for Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) since (dΓ)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This transfor-  mation is importantly different than the gauge trans-  formations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (37) because the Wilson operator trans-  forms nontrivially under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, the Wilson operators  supported on p-cycles in Hp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) transform under  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (42) by a nontrivial element of ZN since Γcp ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  exp[iα˜Lz]  †  †  †  model C  model D  ˜Z  †  †  †  †  †  †  †  †  †  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Graphical representation of the symmetry operator  for the U(1)(p) symmetry in model C and the Z(p)  N  symme-  try in model D, supported on the boundary of a (d − p + 1)-  cell of the dual lattice in d = 3 spatial dimensions for (first  row) p = 1, (second row) p = 2, and (third row) p = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For  model C (D), the blue colored disks denote �L+  cp ( �  Xcp) opera-  tors belonging to that p-cell, the product of which yields the  symmetry operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Discs labeled by † denote the hermitian  conjugate of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In each row, the aforementioned  (d − p + 1)-cell of the dual lattice is colored in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, since the physical operators are supported on  a p-cycle, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (42) corresponds to the transformation of  a ZN p-form symmetry—a Z(p)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Throughout  the remainder of this section, we will always assume to  be working in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The symmetry operator of this Z(p)  N symmetry is  �U(ˆΣd−p) =  �  ˆcd−p∈ˆΣd−p  (∗ �Z)ˆcd−p,  (45)  where ˆΣd−p is a (d − p)-cycle of the dual lattice, and  (∗ �Z)ˆcd−p ≡ �Z∗ ˆcd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We again see why this is a p-form  symmetry: its symmetry operator acts on a (d − p)-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 11 shows a graphical representation of �U(ˆΣd−p)  when ˆΣd−p = ∂ˆcd−p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To confirm that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (45) is the symmetry operator,  first note that �Zcp �  Xcp �Z†  cp = e2π i/N �  Xcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, let-  ting #(A, B) denote the intersection number between the  16  chains A and B,  �U(ˆΣd−p)�  W[Cp]�U †(ˆΣd−p) = e  2π i  N #(Cp,ˆΣd−p)�  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (46)  Introducing Γcp, the Poincar´e dual of ˆΣd−p with respect  to the spatial lattice, given by  (∗ Γ)ˆcd−p(ˆΣd−p) =  �  ˆyd−p∈ˆΣd−p  δˆyd−p,ˆcd−p,  (47)  the  intersection  number  can  be  written  as  #(Cp, ˆΣd−p) = �  cp∈Cp Γcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since  ∂ ˆΣd−p = 0,  Γcp  satisfies (δ ∗ Γ)ˆcd−p−1 = 0, which from Eq (A6) is equiva-  lent to (dΓ)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, since Γcp ∈ Z, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (46)  correctly reproduces the Z(p)  N  symmetry transformation  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The Z(p)  N  symmetry is a symmetry of the effec-  tive mid-IR theory but not the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There-  fore, it is an emergent symmetry, emerging at energies  E < ∆dressed-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the thermodynamic limit, since it  is an exact symmetry of the effective mid-IR theory, we  say that the Z(p)  N symmetry is an exact emergent symme-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because p > 0, this result that an emergent symme-  try can act like an exact symmetry only applies to p-form  symmetries with p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The exact emergent Z(p)  N  symmetry is not present  throughout the entire parameter space of HUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Whether  or not it emerges depends on both the existence of the  mid-IR and the existence of the effective mid-IR Hamil-  tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As mentioned at the start of this subsection, the  mid-IR only exists when U ≫ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The effective mid-IR  Hamiltonian is only well defined provided that the infi-  nite series expansions converge, which required that ε < 1  and so J/U ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the Z(p)  N symmetry only emerges  when K/U ≪ 1 and J/U ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An exact emergent anomalous Z(p)  N × Z(d−p)  N  symmetry  We have seen that at energies below the dressed charge  gap, there is an exact emergent Z(p)  N symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here, we  will search for additional energy scales where new sym-  metries can emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, such a lower-energy scale  only exists in region III of parameter space (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This makes the dressed charge gap the IR scale of region  II but the mid-IR scale of region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here, we will iden-  tify the lower energy scale of region III with the gap of  the topological defects in the Z(p)  N spontaneous symmetry  breaking (SSB) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since there are no other energy  scales in region III, this is an IR scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To show this, let us first discuss when the Z(p)  N symme-  try is spontaneously broken11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To gain some intuition,  11 See footnote 23 for the definition of spontaneous symmetry  breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  we consider two tractable limits of the effective mid-IR  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The first limit is in region II when J/U = 0 but  K/U ̸= 0 such that H(III)  mid-IR becomes  H(III)  mid-IR  ����  J/U=0  = −κU  2  �  cp  �  �Zcp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (48)  The ground state in this limit satisfies �Zcp |vac⟩ = |vac⟩  for all cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because acting �  W †[Cp] onto a state changes the  value of ⟨ �Zcp⟩ for each cp ∈ Cp, ⟨�  W †[Cp]⟩ = 0 for all Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Consequently, this limit lies in a Z(p)  N  symmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The other tractable limit is in region III K/U = 0 but  J/U ̸= 0 such that H(III)  mid-IR becomes  H(III)  mid-IR  ����  K/U=0  = −U  2  �  Cp∈Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  ε|Cp| �  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (49)  The  ground  state  in  this  limit  clearly  satisfies  �  W[Cp] |vac⟩ = |vac⟩ for all Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), and conse-  quently ⟨�  W †[Cp]⟩ = 1 for all trivial p-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  in this limit the Z(p)  N symmetry is spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  According to the higher Coleman-Mermin-Wagner the-  orem, in (d + 1)-dimensional spacetime, a Z(p)  N symmetry  at zero temperature can spontaneously break in the ther-  modynamic limit when d > p [2, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, when  d ≤ p, there is no stable Z(p)  N SSB phase, and any SSB fea-  tures are unique to K/U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, when d > p, we  expect a stable SSB phase even for K/U ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For small  κ ∼ K/U and small ε ∼ J/U, a reasonable expectation is  that the SSB phase occurs when κ ≲ εmin |Cp| = ε2(p+1),  which is equivalent to K/U ≪ (J/U)2(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This iden-  tifies region II of parameter space as the Z(p)  N  symmet-  ric phase while region III of parameter space is the Z(p)  N  SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, the boundary between regions II and  III in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7 is a depiction of the boundary between the  Z(p)  N  symmetric and Z(p)  N  SSB phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We leave a more  detailed investigation of this phase transition to future  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since the order parameter for Z(p)  N  symmetry break-  ing is the vacuum expectation value of �  W †[Cp], the Wil-  son operator can detect topological defects related to the  nontrivial mappings ⟨�  W †[Cp]⟩ : Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) �→ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  topological defects excited in a state |ψ⟩ are probed by  acting the Wilson operator on a trivial p-cycle Cp:12  �  W †[Cp] |ψ⟩ = exp  �2πi  N  ˆQ(Cp)  �  |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (50)  Here, the eigenvalues of ˆQ(Cp) are the net number of  topological defects enclosed by Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  12 This is a natural generalization of the p = 0 case, where the topo-  logical defects are domain walls (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (24)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  17  The topological defects can be characterized locally us-  ing the topological defect density ˆρ which is defined im-  plicitly as  ˆQ(Cp = ∂Op+1) ≡  �  cp+2∈Op+1  (∗ ˆρ)cp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (51)  We can find an implicit expression for the topological  defect density by plugging in Cp = ∂Op+1 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (50)  and using Stoke’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Doing so, we find  �  cp∈∂cp+1  �  Xcp |ψ⟩ = exp  �2πi  N (∗ ˆρ)cp+1  �  |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (52)  Since ˆρ is supported on a (d − p − 1)-cell of the dual lat-  tice, the topological defects are (d − p − 1)-dimensional  excitations in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Note that because �  XN  cp = 1, the  eigenvalues of (∗ ˆρ)cp+1 take values in ZN 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From the ZN clock algebra Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (34), �  W †[∂cp+1] and  �Zcp satisfy  �Zcp�  W †[∂cp+1] = ω(−1)cp �  W †[∂cp+1] �Zcp,  (53)  for all cp ∈ ∂cp+1, where ω = e2π i/N and (−1)cp is the  sign in front of cp in ∂cp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using this, let’s act  �  W †[∂cp+1] on the state (∗ �Z)ˆcd−p |0⟩, with ∗ ˆcd−p ∈ ∂cp+1  and �  W †[∂cp+1] |0⟩ = |0⟩:  �  W †[∂cp+1](∗ �Z)ˆcd−p|0⟩ = ω(−1)∗ ˆcd−p (∗ �Z)ˆcd−p |0⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (54)  Because  this  is  true  for  all  ∗ ˆcd−p ∈ ∂cp+1,  act-  ing  (∗ �Z)ˆcd−p  onto  |0⟩  causes  (∗ ˆρ)cp+1 ̸= 0  for  all  cp+1 ∈ δ ∗ ˆcd−p (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, the operator  (∗ �Z)ˆcd−p excites a topological defect on ∂ˆcd−p  In the J/U = 0 limit of the Z(p)  N  symmetric phase,  as  previously  discussed  the  ground  state  satisfies  �Zcp |vac⟩ = |vac⟩ and thus ⟨�  W †[Cp]⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (50)  and (52), ⟨�  W †[Cp]⟩ = 0 implies the topological defect  density in the ground state fluctuates wildly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There-  fore,  the topological defects are condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This  can also be seen from �Zcp |vac⟩ = |vac⟩ implying that  ⟨(∗ �Z)ˆcd−p⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The topological defect creation opera-  tor having a nonzero vev implies that the topological de-  fects are condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' On the other hand, in the K/U = 0  limit of the Z(p)  N  SSB phase, the ground state satisfies  �  W[Cp] |vac⟩ = |vac⟩ and, therefore, (∗ ˆρ)cp+1 |vac⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus, topological defects do not populate the ground  state, so they are gapped excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This can also  13 Note that for p = 1, the Z(1)  N  SSB phase has ZN topological  order and is the deconfined phase of ZN gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this  case, we find (d − 2)-dimensional topological defects carrying ZN  topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These are the m excitations of ZN topological  order in d-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ˜Z  ± ( * ̂ρ) = ± 1 mod N  −  +  −  +  −  +  +  −  +  +  −  −  +  +  −  −  −  −  +  +  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Graphical representation of the topological de-  fects created by (∗ �Z)ˆcd−p in model D, shown in three spa-  tial dimensions for (first row) p = 0, (second row) p = 1, and  (third row) p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The blue disk represents the (∗ �Z)ˆcd−p  operator acting on the ZN  rotor belonging to that p-  cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The disks labeled by ± represents the ± sign in  �  W †[∂cp+1](∗ �Z)ˆcd−p |0⟩ = exp  �  ± 2πi  N  �  (∗ �Z)ˆcd−p |0⟩, and thus  the value of (∗ ˆρ)cp+1 for that cp+1 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (52)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  be seen from the fact that �  W[Cp] |vac⟩ = |vac⟩ implies  ⟨(∗ �Z)ˆcd−p⟩ = 0 and so the topological defects are not con-  densed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Other regions besides the extreme limits of these  phases can be explored using ULU from section II A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since  we are interested in identifying energy scales below which  new structures can emerge, we restrict ourselves to the  phase where the topological defects have a gap— the  Z(p)  N SSB phase—which is region III of parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, region III will have an IR scale defined by the  gapped dressed topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since the defects are  condensed in region II, they do not give rise to additional  energy scales in region II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  When K/U = 0, a low energy sub-Hilbert space exists  in the mid-IR spanned by states satisfying ⟨ˆρˆcd−p−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So when K/U = 0, the IR energy scale is the topolog-  ical defect’s gap: EIR = ∆defect ∼ J2p+2/U 2p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  How-  ever, when K/U ̸= 0, this sub-Hilbert space is no longer a  low-energy sector because the κU term in H(III)  mid-IR causes  the ⟨ˆρˆcd−p−1⟩ = 0 and ⟨ˆρˆcd−p−1⟩ ̸= 0 states to mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We  can use the local unitary discussed in section II A, which  18  we’ll denote as U (2)  LU, to continue the K/U = 0 states  within the Z(p)  N  SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In general, U (2)  LU is different  than the local unitary U (1)  LU used in the previous sub-  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We’ll denote an operator A continued using  U (2)  LU as A′ ≡ U (2)  LUA(U (2)  LU)†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the IR of region  III for both K/U = 0 and K/U ̸= 0 is the sub-Hilbert  space spanned by states satisfying ⟨ˆρ′  ˆcd−p−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We  view states with ⟨ˆρ′  ˆcd−p−1⟩ ̸= 0 as having gapped dressed  topological defects excited and the IR scale is their energy  gap EIR = ∆dressed-defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because the Z(p)  N SSB phase is  gapped, we can use the local unitary from the quasi-  adiabatic continuation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (4), to continue local opera-  tors throughout the entirety of the Z(p)  N SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because U (2)  LU in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (4) is constructed only from terms  in H(III)  mid-IR, operators that commute with every term in  H(III)  mid-IR are unaffected by U (2)  LU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In particular, because  the Z(p)  N  symmetry operator commutes with each term  in H(III)  mid-IR, the symmetry operator �U(ˆΣd−p) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (45)  satisfies  �U(ˆΣd−p) = �U ′(ˆΣd−p) =  �  ˆcd−p∈ˆΣd−p  (∗ �Z′)ˆcd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (55)  Having identified the IR of region III, we’d now like to  find an effective IR theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As we learned in section II,  the effective IR Hamiltonian H(III)  IR  is a sum of all terms  allowed in the IR that can be constructed from the terms  in H(III)′  mid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The terms in H(III)′  mid-IR are all constructed from  �  W ′[Cp] and ( �Z′  cp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the IR, because ˆρ′  ˆcd−p−1 = 0,  �  W ′[Cp] = 1 and does not contribute in H(III)  IR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The op-  erators ( �Z′  cp + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=') are not allowed operators in the IR  since they excite dressed topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  allowed IR operators can be constructed from �Z′  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In-  deed, because (∗ �Z′)ˆcd−p excites a dressed topological de-  fect on ∂ˆcd−p, acting (∗ �Z′)ˆcd−p on any (d − p)-cycle of  the dual lattice ˆCd−p does not excite dressed topological  defects since ∂2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the IR allowed operator  constructed from �Z′  cp is  �T ′†[ ˆCd−p] =  �  ˆcd−p∈ ˆ  Cd−p  (∗ �Z′)ˆcd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (56)  We call �T ′† the ’t Hooft operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Interestingly, this is  just the Z(p)  N symmetry operator, and therefore does not  need to be dressed by U (2)  LU: �T ′†[ ˆCd−p] = �T †[ ˆCd−p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective mid-IR theory, therefore, includes all  terms constructed from �T ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Denoting the set of all con-  tractible oriented (d − p)-cycles with ZN coefficients on  the dual spatial lattice ˆ  Md as Bd−p( ˆ  Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), the effec-  tive IR Hamiltonian is  H(III)  IR  = −U  �  ˆ  Cd−p∈Bd−p( ˆ  Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  κ| ˆ  Cd−p| �T ′[ ˆCd−p],  (57)  where κ ∼ K/U has been renormalized and | ˆCd−p| is the  number of (d − p)-cells form which ˆCd−p is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective IR Hamiltonian of region III has a new  symmetry which was not present in the mid-IR Hamil-  tonian for region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, H(III)  IR  is invariant under  transforming the UV operator as  (∗ �Z′)ˆcd−p→ e  2π i  N ˆΓˆcd−p (∗ �Z′)ˆcd−p,  (dˆΓ)ˆcd−p+1= 0,  (58)  where ˆΓˆcd−p ∈ Z such that ( �Z′  cp)N = 1 is invariant under  the transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Under this transformation, the ’t  Hooft operator transforms as  �T ′[ ˆCd−p] �→ e  2π i  N  �  ˆcd−p∈ ˆ  Cd−p  ˆΓˆcd−p �T ′[ ˆCd−p],  (59)  which  leaves  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (57)  unchanged  because  �  ˆcd−p∈ ˆ  Cd−p ˆΓˆcd−p = 0 for  ˆCd−p ∈ Bd−p( ˆ  Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) since  (dˆΓ)ˆcd−p+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This transformation is not a gauge  transformations since the ’t Hooft operator—a physical  operator—transforms nontrivially under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed,  the ’t Hooft operators supported on (d − p)-cycles in  Hd−p( ˆ  Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) transform under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (59) by a nontrivial  element of ZN since ˆΓˆcd−p ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, since the  physical operators are supported on a (d − p)-cycle,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (59) corresponds to the transformation of a ZN  (d − p)-form symmetry—a Z(d−p)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In fact,  this is an exact emergent symmetry because it is not an  exact symmetry of the UV but is an exact symmetry  of the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It is straight forward to confirm that the  symmetry operator of this Z(d−p)  N  symmetry is  �ˆU  ′  (Σp) =  �  cp∈Σp  �  X′  cp,  (60)  where Σp is a p-cycle of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We again see why  this is a (d − p)-form symmetry: its symmetry operator  acts on a p-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So, we have found that in the IR of region III, there  is an exact emergent Z(p)  N × Z(d−p)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  the Z(p)  N  and Z(d−p)  N  symmetries are not independent of  one another, there is a mixed ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The  fact that the Z(p)  N × Z(d−p)  N  symmetry is anomalous can  be noticed by considering the symmetry operators when  supported on an open subspace—the disorder operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, the Z(p)  N symmetry operator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (55), on an open  (d − p)-dimensional subspace of the dual lattice is  �U ′( ˆOd−p) =  �  ˆcd−p∈ ˆ  Od−p  (∗ �Z′)ˆcd−p,  (61)  while the Z(d−p)  N  symmetry operator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (60), on an open  p-dimensional subspace of the direct lattice is  �ˆU  ′  (Op) =  �  cp∈Op  �  X′  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (62)  19  When ˆOd−p and Op intersect, these operators generally  do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is a manifestation of the mixed ’t  Hooft anomaly between the Z(p)  N and Z(d−p)  N  symmetries,  reflecting an obstruction to gauging both symmetries14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Before moving onto the next section, we note one fi-  nal thing about the effective IR Hamiltonian of region  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because the IR is also the dressed charge free sec-  tor, �τ ′z  cp−1 ≡ �  cp∈δcp−1 �Z′  cp = 1 for all cp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, this  also implies that �  ˆcd−p∈∂ ∗ cp−1(∗ �Z′)ˆcd−p = 1 and, there-  fore, �T ′[ ˆCd−p] = 1 for all ˆCd−p ∈ Bd−p( ˆ  Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Be-  cause of this, H(III)  IR  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (57) is really just a constant:  H(III)  IR  = 0,  (63)  This reflects the fact that region III is a gapped phase,  so the IR is the degenerate ground states of HUV in this  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The ground state is defined as the state with no  dressed charges or dressed topological defects:  �  cp∈δcp−1  �Z′  cp |vac⟩ = |vac⟩ ,  �  cp∈∂cp+1  �  X′  cp |vac⟩ = |vac⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (64)  This ground state is also the ground state of the p-form  toric code Hamiltonian  HpTC = −  �  cp−1  �  � �  cp∈δcp−1  �Z′  cp  �  � −  �  cp+1  �  � �  cp∈∂cp+1  �  X′  cp  �  � + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='.  (65)  Importantly, this is not the p-form toric code model of  the UV ZN clock operators: the clock operators in HpTC  are dressed by the two local unitary operators used when  identifying the mid-IR and the IR of region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Further-  more, this p-form toric code model captures the ground  state of the entire Z(p)  N  SSB phase, and the the UV pa-  rameters are hidden in the local unitaries dressing Xcp  and Zcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The emergent anomalous Z(p)  N × Z(d−p)  N  symmetry  in the ground state is the same as the anomalous  Z(p)  N × Z(d−p)  N  symmetry of p-form BF theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is  no accident, and it is well known that the vacuum of BF  theory is the same as the ground states of the toric code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In appendix section B, we show how this connection can  be made exact by deriving the topological quantum field  theory description for the ground states of the lattice  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  14 Gauging a symmetry U is the procedure of adding additional  degrees of freedom such that the theory becomes invariant un-  der the gauged symmetry operator Ugauged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Ugauged acts on  both open and closed subspaces and physical states must sat-  isfy Ugauged |ψ⟩ = |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A contradiction arises when different  Ugauged no longer commute, reflecting an obstruction to gaug-  ing the symmetry (a ’t Hooft anomaly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For example, con-  sider U(1)  gaugeU(2)  gauge = −U(2)  gaugeU(1)  gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since U(1,2)  gauge |ψ⟩ = |ψ⟩,  this leads to the contradiction |ψ⟩ = − |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Emergent U(1) p-gauge theory  In this section, we will apply the framework discussed  in section II to a model for emergent U(1) p-gauge the-  ory which we call model C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consider U(1) quantum ro-  tors residing on each p-cell of the spatial d-dimensional  cubic lattice with p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Each rotor can be viewed as  a particle on an infinitesimal circle, whose position we  denote as the angle Θ, carrying angular momentum Lz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The operators Lz and Θ are hermitian and satisfy the  canonical commutation relation [Θcp, Lz  �cp] = iδcp,�cp, so  Lz = −i ∂  ∂Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Additionally, since the eigenvalue of Θ is  an angle, the eigenvalues of Lz are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The microscopic model we consider is described by the  Hamiltonian  HUV= U  2  �  cp−1  ρ2  cp−1+ K  2  �  cp  (Lz  cp)2+J  2  �  cp  �  L+  cp+ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  ,  ρcp−1 =  �  cp∈δcp−1  Lz  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (66)  The sum �  cp is over all p-cells of the spatial lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The sum �  cp∈δcp−1 in the definition of ρcp−1 is over the  coboundary of cp−1, defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (A3) of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using the definition of δcp, the expression for ρcp−1 can  be written as (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 8)  ρ(x)µ1···µp−1=  �  ν  Lz(x)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='µp−1 − Lz(x − ˆν)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='µp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (67)  Note that because the eigenvalues of Lz are integers, the  eigenvalues of ρ are also integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the third term of  HUV, L+  cp = (L−  cp)† = exp  �  iΘcp  �  is the raising operator  for Lz  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lastly, since HUV is invariant under the trans-  formation Lz  cp → −Lz  cp, this theory has a UV Z(0)  2  sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An exact emergent U(1)(p) symmetry  When J = K = 0, there exists a low energy sub-  Hilbert space spanned by states satisfying ⟨ρcp−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, the first term in HUV introduces an energetic  penalty for states satisfying ⟨ψ| ρcp−1 |ψ⟩ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We inter-  pret such states in the J = 0 and U ≫ K limit as describ-  ing a gapped excitation, a segment of which resides on  the (p − 1)-cell cp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, from the canonical commu-  tation relation satisfied by Lz and Θ, the rotor operators  of the same p-cell satisfy [Lz, L±] = ±L±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore,  ρcp−1 and L±  cp satisfy [ρcp−1, L±  cp] = ±(−1)cpL±  cp for all  cp−1 ∈ ∂cp, where (−1)cp denotes the sign in front of cp  in the expression for δcp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using this, let us act ρcp−1  on the state L±  cp |0⟩, with cp−1 ∈ ∂cp and ρcp−1 |0⟩ = 0:  ρcp−1  �  L±  cp |0⟩  �  = ±(−1)cp �  L±  cp |0⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (68)  20  Because this is true for all cp−1 ∈ ∂cp, L±  cp excites the  aforementioned excitation on ∂cp, examples of which are  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We’ll refer to these bosonic (p − 1)-  dimensional (in space) excitations as “charges.”  It is tempting to consider the charge gap as a candidate  energy scale below which new physics emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  when J ̸= 0, there no longer exists a low-energy sub-  Hilbert space spanned by states satisfying ⟨ρcp−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is because the J term in HUV causes the ⟨ρcp−1⟩ = 0  and ⟨ρcp−1⟩ ̸= 0 states to mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, a correspond-  ing low-energy sub-Hilbert space can be identified us-  ing ULU from section II A to continue any operator A  which we understand at J = 0 to a (fattened) local oper-  ator �A ≡ ULUA(ULU)† with the same expectation values  of A but at J ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, when U ≫ K, there ex-  ists a low-energy sub-Hilbert space for both J = 0 and  J ̸= 0 spanned by states satisfying ⟨�ρcp−1⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We view  states with ⟨�ρcp−1⟩ ̸= 0 as having dressed charges excited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since the undressed charges are created using L±  cp, these  dressed charges are created using �L±  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We will not find  an explicit form for ULU and thus will not precisely know  throughout how much of parameter space the dressed  (fattened) operators can be defined without violating the  assumptions of ULU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Instead, we will assume that such an  operator exists and can access a greater than measure-  zero part of parameter space, and will investigate the  consequences of this conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  At this point in our investigations, we cannot tell if the  dressed charge gap ∆dressed-charge is an IR scale or a mid-  IR I scale or a mid-II scale, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In section III C 2, we will  learn that it is a mid-IR scale in region III of parameter  space but an IR scale in region II of parameter space  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the rest of this section, however, we will  adopt the language from the perspective of region III and  call the dressed charge gap a mid-IR scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Given the mid-IR scale Emid-IR ≡ ∆dressed-charge, we  would now like to the find an effective mid-IR theory,  which by definition is a theory only describing states at  energies E < Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As we learned in section II, the ef-  fective mid-IR Hamiltonian Hmid-IR is a sum of all terms  allowed in the mid-IR that can be constructed from the  terms in �HUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The terms in �HUV are �ρ2  cp−1, (�Lz  cp)2, and  �L±  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the mid-IR, �ρ2  cp−1 = 0 so it will not appear in  Hmid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, since (�Lz  cp)2 commutes with �ρ2  cp−1,  it does not excite any dressed charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus (�Lz  cp)2 is an  allowed mid-IR operator from which terms in Hmid-IR can  be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The operators �L±  cp are not allowed oper-  ators in the mid-IR since they excite charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  allowed mid-IR operators can be constructed from �L±  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, since �L+  cp excites a dressed charge on ∂cp, acting  �L+  cp on any p-cycle does not excite dressed charges since  ∂ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, an allowed operator in the mid-IR is15  �  W †[Cp] =  �  cp∈Cp  �L+  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (70)  We call �  W † the Wilson operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It has the interpretation  of exciting a dressed charge, transporting it along a p-  cycle, and then ultimately annihilates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 10 shows  a few graphical representations of the Wilson operator  supported on the smallest possible p-cycle: the boundary  of a (p + 1)-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that the Wilson operator satisfies  �  W †[Cp] = �  W[−Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The effective mid-IR theory, therefore, includes all  terms constructed from �  W and �L2  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Denoting the set  of all oriented p-cycles with integer coefficients on the  spatial lattice Md as Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z), the mid-IR Hamiltonian  takes the form  H(III)  mid-IR = κU  2  �  cp  (�Lz  cp)2− U  2  �  Cp∈Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ε|Cp|�  W[Cp] + · · ·, (71)  where κ ∼ K/U, ε ∼ J/U, |Cp| is the number of p-cells  form which Cp is constructed, and the · · ·’s include all  other possible terms constructed from both (�Lz  cp)2 and  �  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since the dressed charge gap is the mid-IR scale  of region III but the IR of region II,  H(III)  mid-IR ≡ H(II)  IR .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (72)  The effective mid-IR theory is only well defined pro-  vided ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  As a consequence, since the lattice is  simply connected, the Wilson operators supported on  nontrivial p-cycles are exponentially suppressed by εLp,  where L is the linear system size16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, de-  noting contractible p-cycles with integer coefficients on  the spatial lattice Md as Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z), terms with Wil-  son operators supported on cycles in the homology class  Hp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) = Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z)/Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) vanish in the ther-  modynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, in the thermodynamic limit  H(III)  mid-IR becomes  H(III)  mid-IR  ����  L→∞  = κU  2  �  cp  (�Lz  cp)2− U  2  �  Cp∈Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ε|Cp|�  W[Cp] + · · ·, (73)  15 Describing the extended object �  W † using local operators intro-  duces a gauge redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, �  W † is invariant under  �L+  cp → e i Ξcp �L+  cp  with  �  cp∈Cp  Ξcp ∈ 2πZ ∀ Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (69)  When Ξcp = (dχ)cp (where d is the lattice exterior derivative,  see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (75)), �Θcp transforms as the canonical U(1) gauge redun-  dancy �Θcp → �Θcp + (dχ)cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For Ξcp ̸= (dχ)cp, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (69) corre-  sponds to large gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  16 When p = 1, the mid-IR theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (71) can be thought of as a  lattice regularization of the Landau-Ginzburg string field theory  developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, whereas the suppression of large  loops needed to be inserted by hand in the string field theory,  here it automatically arises from the UV theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  21  where the · · ·’s include all possible terms constructed  from both (�Lz  cp)2 and �  W[Cp] with only Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the thermodynamic limit, H(III)  mid-IR is invariant under  transforming the UV operator as  �L+  cp → eiΓcp �L+  cp,  with  (dΓ)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (74)  Here (dΓ)cp+1 is the lattice exterior derivative of Γcp,  defined as  (dΓ)cp+1 ≡  �  cp∈∂cp+1  Γcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (75)  Under this transformation, the Wilson operator trans-  forms as  �  W[Cp] → ei �  cp∈Cp Γcp �  W[Cp],  (76)  which leaves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (73) unchanged because �  cp∈Cp Γcp = 0  for Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) since (dΓ)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This transfor-  mation is importantly different than the gauge trans-  formations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (69) because the Wilson operator trans-  forms nontrivially under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, the Wilson opera-  tors supported on p-cycles in Hp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) transform un-  der Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (74) by a nontrivial element of U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  since the physical operators are supported on a p-cycle  and pick up a phase under the symmetry transformation,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (74) corresponds to the transformation of a U(1) p-  form symmetry—a U(1)(p) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Throughout the  remainder of this section, we will always assume to be  working in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The symmetry operator of this U(1)(p) symmetry is  �Uα(ˆΣd−p) =  �  ˆcd−p∈ˆΣd−p  exp  �  iα (∗ �Lz)ˆcd−p  �  ,  (77)  where α ∈ [0, 2π), ˆΣd−p is a (d − p)-cycle of the dual lat-  tice, and (∗ �Lz)ˆcd−p ≡ �Lz  ∗ ˆcd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We again see why this  is a p-form symmetry: its symmetry operator acts on a  (d − p)-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 11 shows a graphical representation of  �Uα(ˆΣd−p) when ˆΣd−p = ∂ˆcd−p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, since Lz  has integer eigenvalues, the charge operator  �QˆΣd−p =  �  ˆcd−p∈ˆΣd−p  (∗ �Lz)ˆcd−p  (78)  also  has  integer  eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Note  that  when  ˆΣd−p = ∂ ˆOd−p+1, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (A6) �QˆΣd−p can be rewritten  as  �Q∂ ˆ  Od−p+1=  �  ˆcd−p+1∈Od−p+1  (−1)p(∗ �ρ )ˆcd−p+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (79)  To confirm that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (77) is the symmetry operator,  first note that eiα�Lz  cp �L+  cp e−iα�Lz  cp = eiα�L+  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  letting #(A, B) denote the intersection number between  the chains A and B,  �Uα(ˆΣd−p)�  W[Cp]�U †  α(ˆΣd−p) = eiα#(Cp,ˆΣd−p)�  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (80)  Introducing Γcp, the Poincar´e dual of ˆΣd−p with respect  to the spatial lattice, given by  (∗ Γ)ˆcd−p(ˆΣd−p) =  �  ˆyd−p∈ˆΣd−p  δˆyd−p,ˆcd−p,  (81)  the  intersection  number  can  be  written  as  #(Cp, ˆΣd−p) = �  cp∈Cp Γcp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since  ∂ ˆΣd−p = 0,  Γcp  satisfies  (δ ∗ Γ)ˆcd−p−1 = 0,  which  from  Eq  (A6)  is  equivalent to (dΓ)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (80) cor-  rectly reproduces the U(1)(p) symmetry transformation  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The U(1)(p) symmetry is a symmetry of the effec-  tive mid-IR theory but not the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There-  fore, it is an emergent symmetry, emerging at energies  E < ∆dressed-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the thermodynamic limit, since it  is an exact symmetry of the effective mid-IR theory, we  say that the U(1)(p) symmetry is an exact emergent sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since p > 0, this result that an emergent symme-  try can act like an exact symmetry only applies to p-form  symmetries with p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The exact emergent U(1)(p) symmetry is not present  throughout the entire parameter space of HUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Whether  or not it emerges depends on both the existence of the  mid-IR and the existence of the effective mid-IR Hamil-  tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  As mentioned at the start of this subsection,  the mid-IR only exists when U ≫ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The effective mid-  IR Hamiltonian is only well defined provided that the  infinite series expansions converge, which required that  ε < 1 and so J/U ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the U(1)(p) symmetry only  emerges when K/U ≪ 1 and J/U ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An exact emergent anomalous U(1)(p) × U(1)(d−p−1)  symmetry  We have seen that at energies below the dressed charge  gap, there is an exact emergent U(1)(p) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here,  we will search for additional energy scales where new  symmetries can emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, such a lower-energy scale  only exists in region III of parameter space (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This makes the dressed charge gap the IR scale of region  II but the mid-IR scale of region III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here, we will iden-  tify the lower energy scale of region III with the gap of  the topological defects in the U(1)(p) spontaneous sym-  metry breaking (SSB) phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since there are no other  energy scales in region III, this is an IR scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To show this, let us first discuss when the U(1)(p) sym-  metry is spontaneously broken17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To gain some intuition,  we consider two tractable limits of the effective mid-IR  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The first limit is in region II when J/U = 0 but  17 See footnote 23 for the definition of spontaneous symmetry  breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  22  K/U ̸= 0 such that H(III)  mid-IR becomes  H(III)  mid-IR  ����  J/U=0  = κU  2  �  cp  (�Lz  cp)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (82)  The ground state in this limit satisfies �Lz  cp |vac⟩ = 0 for all  cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because acting �  W †[Cp] onto a state changes the value  of ⟨�Lz  cp⟩ for each cp ∈ Cp, ⟨�  W †[Cp]⟩ = 0 for all Cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Con-  sequently, this limit lies in a U(1)(p) symmetric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The other tractable limit is in region III K/U = 0 but  J/U ̸= 0 such that H(III)  mid-IR becomes  H(III)  mid-IR  ����  K/U=0  = −U  2  �  Cp∈Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ε|Cp| �  W[Cp].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (83)  The  ground  state  in  this  limit  clearly  satisfies  �  W[Cp] |vac⟩ = |vac⟩ for all Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN), and conse-  quently ⟨�  W †[Cp]⟩ = 1 for all trivial p-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  in this limit the U(1)(p) symmetry is spontaneously bro-  ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  According to the higher Coleman-Mermin-Wagner the-  orem, in (d + 1)-dimensional spacetime, a U(1)(p) sym-  metry at zero temperature can spontaneously break in  the thermodynamic limit when d > p + 1 [2, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' There-  fore, when d ≤ p + 1, there is no stable U(1)(p) SSB  phase, and any SSB features are unique to K/U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, when d > p + 1, we expect a stable SSB phase  even for K/U ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For small κ ∼ K/U and small  ε ∼ J/U, a reasonable expectation is that the SSB phase  occurs when κ ≲ εmin |Cp| = ε2(p+1), which is equivalent  to K/U ≪ (J/U)2(p+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This identifies region II of pa-  rameter space as the U(1)(p) symmetric phase while re-  gion III of parameter space is the U(1)(p) SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In  fact, the boundary between regions II and III in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7  is a depiction of the boundary between the U(1)(p) sym-  metric and U(1)(p) SSB phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We leave a more detailed  investigation of this phase transition to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since the order parameter for U(1)(p) symmetry break-  ing is the vacuum expectation value of �  W †[Cp], the Wil-  son operator can detect topological defects related to the  nontrivial mappings ⟨�  W †[Cp]⟩ : Zp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) �→ U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  topological defects excited in a state |ψ⟩ are probed by re-  peatedly acting the Wilson operator over a trivial (p + 1)-  cycle Cp+1:18  �  cp+1∈Cp+1  �  W †[∂cp+1] |ψ⟩ = exp  �  2πi ˆQ(Cp+1)  �  |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (84)  Here, ˆQ(Cp+1) is a winding number and yields the net  number of topological defects enclosed by Cp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Plugging  18 This is a natural generalization of the p = 0 case, where the topo-  logical defects are vortices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  in �  W † and using Stoke’s theorem, it can be expressed as  ˆQ(Cp+1) = 1  2π  �  cp+1∈Cp+1  Fcp+1,  (85)  where Fcp+1 = (d�Θ)cp+1 mod 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using the identity  x mod n = x − n ⌊x/n⌉, where ⌊·⌉ rounds its input to the  nearest integer, Fcp+1 can be written as  Fcp+1 ≡ (d�Θ)cp+1 + ωcp+1,  (86)  where ωcp+1 ≡ −2π  �  (d�Θ)cp+1/(2π)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The topological defects can be characterized locally us-  ing the topological defect density ˆρ which is defined im-  plicitly as  ˆQ(Cp+1 = ∂Op+2) ≡  �  cp+2∈Op+2  (∗ ˆρ)cp+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (87)  Plugging in Cp+1 = ∂Op+2 into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (85) and using  Stoke’s theorem, we find  (∗ ˆρ)cp+2 = 1  2π (dF)cp+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (88)  Since ˆρ is supported on a (d − p − 2)-cell of the dual lat-  tice, the topological defects are (d − p − 2)-dimensional  excitations in space, residing on the boundary of a  (d − p − 1)-cell of the dual lattice (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Note  that because the eigenvalues of ωcp+1 take values in 2πZ,  the eigenvalues of (∗ ˆρ)cp+2 are integers19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the J/U → 0 limit of the U(1)(p) symmetric phase,  the ground state satisfies �Lz  cp |vac⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since �Lz  cp and  �Θcp are conjugate variables, in the ground state �Θcp  fluctuates wildly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consequently, Fcp+1 fluctuates wildly  and the ground state satisfies (dF)cp+2 |vac⟩ ̸= 0, so  the topological defects are condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  On the other  hand, in the K/U → 0 limit of the U(1)(p) SSB phase,  the ground state satisfies �  W[Cp] |vac⟩ = |vac⟩ for all  Cp ∈ Bp(Md;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In terms of the dressed phase vari-  able �Θcp, this implies that (d�Θ)cp+1 |vac⟩ = 2πn with  n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Consequently, Fcp+1 |vac⟩ = 0 for all (p + 1)-cells  and therefore ˆρˆcd−p−2 |vac⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, topological defects  do not populate the ground state and are gapped excita-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, these topological defects cannot be observed  directly in the lattice model20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed since Θcp always  19 Note that for p = 1, the U(1)(1) SSB phase is a Coulomb phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this case, we find (d − 3)-dimensional topological defects car-  rying integer topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' These are the Dirac magnetic  “monopoles” of the Coulomb phase, whose interpretation as  topological defects was also noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  20 One could instead consider a Villain type Hamiltonian model  for which these topological defects are observable even in the  UV/mid-IR [68–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Nevertheless, these different UV lattice  models should have the same IR effective field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  23  +  −  −  +  −  +  −  +  +  −  +  −  ̂ρ ̂cd−p−2 = ∑  ⌊  d ˜Θ  2π ⌉  ±  +  −  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Graphical representation of ˆρˆcd−p−2 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (88))  in three spatial dimensions for (first row) p = 0 and (second  row) p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The disks labeled by ± denote the sign in front  of  �  d�Θ/(2π)  �  belonging to that (p + 1)-cell in the sum for  ˆρˆcd−p−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The direct lattice is colored in black while the dual  lattice is in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, the (d − p − 2)-cell of the dual  lattice ˆρˆcd−p−2 is associated with is highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  appears as L+  cp = eiΘcp , (∗ ˆρ)cp+2 too always appears as  ei2π(∗ ˆρ)cp+2 and so ˆρˆcd−p−2 ∼ ˆρˆcd−p−2 + 1 on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An additional sign that they are trivial on the lattice  comes attempting to construct an operator which does  not create any topological defects (the equivalent of the  Wilson operator from the last section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, they  are created any operator which causes the eigenvalue  of d�Θcp+1 to jump through the boundary of its range  [0, 2π) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', going from π to 3π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' If an operator can do  this for only a single (p + 1)-cell cp+1, it would excite  a single dressed topological defect residing on ∂ ∗ cp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is tempting to consider the operator exp[ix(∗ �Lz)ˆcd−p]  since it shifts d�Θcp+1 by x(−1)∗ ˆcd−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, if x = 2π, it  shifts  �  d�Θ  2π  �  cp+1 by (−1)∗ ˆcd−p: the sign of ∗ ˆcd−p in ∂cp+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, steaming from the fact that ∂2 = 0, this shift  of  �  d�Θ  2π  �  cp+1 does not cause d  �  d�Θ  2π  �  cp+2 to shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus,  we find that an operator that does no excite topological  charges is  �T †[∂ ˆOd−p] =  �  ˆcd−p∈ ˆ  Od−p  exp  �  2πi(∗ �Lz)ˆcd−p  �  (89)  where ˆOd−p is an open (d − p)-dimensional subspace of  the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We call �T ′† the ’t Hooft operator, which  is generally supported on (d − p − 1)-cycles of the dual  lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Physically �T ′† excites a dressed topological de-  fect, transports it along the (d − p − 1)-cycle, and ulti-  mately annihilates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, since the eigenvalues of  �Lz are integers, the operator �T † is in fact trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  While the topological defects are unobservable on the  lattice, their effects emergent in the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The general paradigm for most lattice models is that  the effective IR theory deep into a phase of matter is a  continuum quantum field theory which reflects only the  universal properties of that phase21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Finding the IR ef-  fective field theory involves going deep into the U(1)(p)  SSB phase and taking the continuum limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Deep into the  SSB phase, the effective IR hamiltonian Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (73) includes  only the leading order in κ and ε terms:  Hdeep IR ≈ κU  2  �  cp  �  �L′z  cp  �2  + ε2p+2U  2  �  cp+1  �  F ′  cp+1  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (90)  In the field theory, these higher-order terms could con-  tribute as higher-derivative terms, but do not affect the  deep IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Appendix section C, shows how we take the contin-  uum limit of Hdeep IR, doing so carefully to capture the  topologically nontrivial parts of the quantum fields from  the lattice operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We find that the IR effective field  theory is compact p-form Maxwell theory, described by  the path integral  Zdeep IR =  �  D[a]  �  ωa∈2πHp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X Ldeep IR,  Ldeep IR = − 1  2g2 Fa ∧ ∗ Fa,  (91)  where  a  is  a  p-form  in  Minkowski  spacetime  X,  Hp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) is the (p + 1)th de Rham cohomology group  with integral periods, and Fa ≡ da + ωa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This field the-  ory describes the dynamical fluctuations of the  (d−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (d−p−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  p-form Goldstone bosons of the U(1)(p) SSB phase [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, as reviewed in depth in appendix sec-  tion C 1, it has an anomalous U(1)(p) ×U(1)(d−p−1) sym-  metry [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, deep into the U(1)(p) SSB phase  of the lattice model, a new symmetry emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  How-  ever, the U(1)(p) and U(1)(d−p−1) symmetries are not  independent of one another, there is a mixed ’t Hooft  anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, the field theory also an emergent  Lorentz invariance, and in terms of the UV parameters,  the “speed of light” is c = Uεp+1√κ ∼ Jp+1  U p  �  K  U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  PHYSICAL CONSEQUENCES OF EXACT  EMERGENT HIGHER-FORM SYMMETRIES  In the previous section, we applied the framework in-  troduced in section II to three lattice models without  21 There are fascinating counter-examples where exotic lattice mod-  els exhibit UV/IR mixing [54–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In these cases, the UV lattice  details sneak into the definitions of the IR (and mid-IRs), causing  the IR to exhibit a sensitive dependency on UV details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Conse-  quently, since the thermodynamic limit and continuum limit do  not commute, the effective IR field theory is no longer a conven-  tional continuum quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  24  exact higher-form symmetries and found that in particu-  lar regions of parameter space, there are exact emergent  higher-form symmetries below particular energy scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this section, we summarize the general lessons learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, we discuss the physical consequences of  these general lessons and why the phases of a microscopic  (UV) theory without exact higher-form symmetries be  exactly characterized by emergent higher-form symme-  tries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is well known that emergent 0-form symmetries are  never exact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' thus, their consequences are always approx-  imate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, the emergent higher-form symmetries  we identified are exact emergent symmetries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' thus, they  can exactly constrain the IR in the regions of parameter  space they emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This important difference between  0-form and higher-form symmetries is natural from the  point of view of an effective mid-IR theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Firstly, note  how from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (9), if the UV theory has a symmetry,  any effective mid-IR theory will also have that symme-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, if the symmetry is explicitly broken in the  UV theory, the effective mid-IR theory will generically  include terms charged under the symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For 0-form  symmetries, the charged operators are local, so terms  including them will persist in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For higher-form symmetries, however, the charged opera-  tors are non-local, so terms including them will vanish in  the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A general consequence is that  while emergent 0-form symmetries are never exact, emer-  gent higher-form symmetries are always exact emergent  symmetries whenever spacetime is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The exact emergent higher-form symmetries are robust  against any local perturbations of the UV theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In  other words, adding local terms to the UV will not change  the fact that the higher-form symmetry emerges nor the  fact that it becomes an exact symmetry of the effective  theory in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, adding any  local term to HUV does not affect the form of Hmid-IR  because the perturbation will either not survive the pro-  jection to the mid-IR or simply renormalize the effective  Hamiltonian’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this sense, exact emergent  higher-form symmetries are topologically robust, and any  physical properties arising from their existence are also  robust to local UV perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  An immediate consequence of an exact emergent  higher-form symmetry emerging at E < Emid-IR is that  mid-IR states and mid-IR observables22 are organized  into its representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This implies the existence of an  exact emergent conservation law obeyed at E < Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, this gives rise to selection rules on the cor-  relation functions of mid-IR allowed operators [2, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In particular, mid-IR operators charged under the exact  22 A mid-IR state is a quantum state in the the sub-Hilbert space  spanned by energy-eigenstates with E < Emid-IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A mid-IR ob-  servable is an operator that takes states in the sub-Hilbert space  spanned by energy eigenstates with E < Emid-IR to only other  states in that sub-Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ̂x  ̂y  ̂t  Emergent Symmetry  No Emergent Symmetry  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The existence of an exact emergent p-form symmetry  constrains the way a p-dimensional membrane excitation can  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shown here is a cartoon of the different decay processes  in spacetime for p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The world sheet of the 1-dimensional  membrane excitation is colored blue, while the worldline of  charge excitations is orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  emergent symmetry must have a zero vacuum expecta-  tion value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The presence of an exact emergent higher-form sym-  metry also affects the dynamics of a quantum-many body  system in a phase where the exact emergent higher-form  symmetry is not spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, if a p-  form symmetry (with p > 0) emerges at E < Emid-IR, it  constraints the dynamics of mid-IR states with excited  p-dimensional membrane excitations created by an oper-  ator charged under the p-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, con-  sider starting in the state �  W †[Cp] |vac⟩, where Cp is con-  tractible and |vac⟩ is the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The lifetime of  the p-dimensional membrane excitation is qualitatively  different depending on whether or not the exact emer-  gent p-form symmetry is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When the symmetry  is present, the only way for the p-dimensional membrane  excitation to decay is for it to contract to a point, so  the lifetime of the p-dimensional membrane excitation  depends on |Cp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This also implies that if there is a  trap potential for the p-dimensional membrane excita-  tion (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', a term in the Hamiltonian such that there is a  p-dimensional membrane excitation on Cp in the ground  state), the exact emergent symmetry ensures that its life-  time is infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the absence of the symmetry, the p-  dimensional membrane excitation can decay by breaking  apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, without the emergent symmetry the lifetime  of the p-dimensional membrane excitation is independent  of its size |Cp|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 14 shows a cartoon depicting these  differing behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because emergent higher-form symmetries are exact  emergent symmetries, they characterize the IR of the  parameter space region where they emerge as if they  were UV symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This, combined with the fact that  emergent higher-form symmetries become exact emer-  gent symmetries in the thermodynamic, implies that  emergent higher-form symmetries characterize phases of  matter with the same power as exact UV symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  One way that exact emergent higher-form symmetries  characterize phases, which we encountered in our exam-  ples, is by their ability to become spontaneously bro-  25  ken in the thermodynamic limit23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, since sponta-  neous symmetry breaking is diagnosed using the ground  state, for which an emergent higher-form symmetry is  exact, an emergent higher-form symmetry can be spon-  taneously broken in the same way a UV symmetry can  be spontaneously broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A consequence of this is that a  phase with an emergent discrete higher-form symmetry  spontaneously broken has an exact ground state degen-  eracy which depends on spacetime’s topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Another  consequence is that a phase with an emergent contin-  uous higher-form symmetry spontaneously broken has  Goldstone bosons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  If the continuous higher-form sym-  metry emerges at E < Emid-IR, these Goldstone bosons  are exactly gapless for mid-IR states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, for states  in the sub-Hilbert space spanned by energy eigenstates  with E ≥ Emid-IR, the Goldstone bosons acquire a gap24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is very different from a 0-form continuous symme-  try where even weakly breaking the symmetry in the  UV gaps out the Goldstone boson [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, be-  cause exact emergent higher-form symmetries are topo-  logically robust, a local UV perturbation does not gap out  the higher-form Goldstone bosons nor lift the topological  ground state degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The last physical consequence  we will note is that the SSB phase of a p-form symmetry  has gapped topological defect excitations, which formally  arise due to nontrivial mappings from p-cycles to the or-  der parameter manifold [33, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From the examples in  section III, when an anomaly-free U(1)(p) (Z(p)  N ) symme-  try spontaneously breaks in d-dimensional space, there  are d − p − 2 (d − p − 1) dimensional topological defects  carrying Z (ZN) topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Another way exact emergent higher-form symmetries  can characterize phases, which we did not encounter  in our examples, is by giving rise to nontrivial emer-  gent symmetry-protected topological (SPT) orders25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For instance, if a higher-form symmetry emerges at  E < Emid-IR and is realized anomalously on the bound-  23 To be precise, by spontaneous symmetry breaking, we mean  a phase where an order parameter constructed from oper-  ators charged under a symmetry acquires a nonzero vac-  uum expectation value in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For in-  stance, |vac⟩ has spontaneously broken a U(1)(p) symmetry if  ⟨vac| W[Cp] |vac⟩ ̸= 0 for Cp ∈ Bp(Md) as |Cp| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that  when p = 0, C0 ∈ B0(Md) is an oriented collection of two lat-  tice points C0 = {−x, y} and so W[C0] = W †(x)W(y), which  makes a connection to the historical perspective of long-range  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Physically, ⟨vac| W[Cp] |vac⟩ ̸= 0 implies that objects car-  rying (neutral amounts) of symmetry charge have condensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So,  when p = 0 SSB implies the condensation of particles while when  p = 1 SSB implies the condensation of loops [15, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  24 This is a familiar concept in the p = 1 case where electric screen-  ing causes the photon to acquire a gap (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', plasmas and metals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  25 If an SPT phase is protected by a G symmetry, the G symme-  try is realized anomalously on the system’s boundary, endowing  the boundary with gapless/degenerate modes or topological or-  der [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The presence of the bulk SPT order is reflected by  the ability to nevertheless couple a background G gauge field in  the bulk+boundary theory since the ’t Hooft anomaly on the  boundary is canceled by an anomaly in-flow mechanism [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ary, a corresponding nontrivial SPT order could also  emerge at E < Emid-IR and cause the system to be in  an SPT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The boundary could then have sponta-  neously broken exact emergent higher-form symmetries,  therefore hosting abelian topological orders or gapless  (for E < Emid-IR) higher-form Goldstone bosons, or con-  tain additional gapless degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This em-  phasizes an important distinction between SPT phases  protected by 0-form symmetries and those protected by  higher-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 0-form SPT phases cannot oc-  cur in regions of parameter space where the 0-form sym-  metry is explicitly broken in the UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, explicitly  breaking a 0-form symmetry in the UV breaks the sym-  metry at all energy scales, hence preventing the boundary  from realizing the symmetry anomalously, an anomaly  inflow mechanism from occurring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Amazingly, this  is not true for higher-form SPT phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, as we  have seen, breaking a higher-form symmetry in the UV  does not guarantee it remains broken in the IR due to  the topological robustness of emergent higher-form sym-  metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, while there may not be any higher-  form symmetries at E > Emid-IR, there could be an exact  emergent higher-form symmetry at E < Emid-IR which is  realized anomalously on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' If so, the system  will have nontrivial exact emergent SPT order if there  also exists a corresponding anomaly inflow mechanism  at E < Emid-IR that allows one to turn on background  gauge fields of the higher-form symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, to  understand higher-form SPT phases, instead of partition-  ing parameter space by the UV higher-form symmetries,  as done for 0-form SPT phases, one must partition pa-  rameter space by the exact emergent higher-form sym-  metries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The final physical consequence of exact emergent  higher-form symmetries that we will emphasize is their  ability to be anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  When a ’t Hooft anomaly is  present, the ground state cannot be a trivial product  state, and the phase is guaranteed to be gapless, topo-  logically ordered, or a gapped SSB phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore,  in the presence of an anomalous symmetry, the ground  state cannot be made into a trivial product state with-  out getting rid of the ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Exact emergent  higher-form symmetries can also be anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed,  this was the case in all of the examples we considered  in section III: in the SSB phases discussed, there were  exact emergent anomalous higher-form symmetries that  guaranteed these phases would have degenerate ground  states or gapless modes in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In  fact, the ground state degeneracies and the gaplessness of  Goldstone bosons in these phases were protected by the  ’t Hooft anomaly, and the only way to eliminate them  would be to destroy the exact emergent anomalous sym-  metry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the example, this could be done by condensing  either the gauge charges (if present) or the topological de-  fects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, if their gaps were held at infinity, the ground  state in the SSB phase could never become a trivial prod-  uct state due to the ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  26  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  CONCLUSION AND DISCUSSION  In this paper, we have investigated the robustness of  emergent higher-form symmetries from a UV perspec-  tive, considering bosonic lattice Hamiltonian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' By  identifying low-energy sub-Hilbert spaces using ULU from  section II A, in section II we discussed how to write down  effective Hamiltonians to identify emergent symmetries,  which we applied to three lattice models in section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We found that even when the lattice models did not have  exact UV higher-form symmetries, there could be emer-  gent higher-form symmetries whose effects in the IR are  the same as if they were UV symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To empha-  size this robustness, we referred to emergent higher-form  symmetries as exact emergent symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using this,  we argued that exact emergent higher-form symmetries  can exactly characterize the phases of microscopic models  without exact UV higher-form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The physical  consequences of this were summarized in detail in sec-  tion IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There are many exciting follow-up questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Firstly,  in section II, we arrived at our expression for the effective  Hamiltonian in a non-rigorous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' While its defini-  tion is physically very reasonable, it would be desirable  to have a rigorous derivation for its form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A possible  starting point for such a proof could be building off of  perturbatively defined effective Hamiltonians found using  Brillouin-Wigner perturbation theory [76] or Schrieffer-  Wolff transformations [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It would also be interesting to investigate if emergent  higher-form symmetries are no longer exact emergent  symmetries, and instead approximate symmetries, in lat-  tice models with lattice defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, the reason we  found why emergent higher-form symmetries act as ex-  act symmetries boiled down to the fact that their charged  operators are supported on nontrivial cycles and there-  fore did not appear in the effective Hamiltonian in the  thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, imagine removing lat-  tice sites and introducing nontrivial cycles that did not  involve a sub-extensive number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Then, like 0-  form symmetries, operators charged under the higher-  form symmetry would appear in the effective Hamilto-  nian if the UV theory did not have that higher-form  symmetry, thus causing the emergent higher-form sym-  metry to be approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This could be confirmed in  spontaneous higher-form symmetry broken phases by in-  vestigating if removing lattice sites lifts the topological  ground state degeneracy/gaps the Goldstone bosons  Another important follow-up would be an in-depth  study of region II in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As discussed at the begin-  ning of section III, the size of region II and the boundary  between region II and I in models C and D is not rigor-  ously investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It would be interesting to numerically  investigate these models C and D to better understand  region II in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Alternatively, if one could  explicitly construct the local unitary ULU, it may be pos-  sible for this question to be addressed analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Lastly, while we discussed symmetry-protected topo-  logical (SPT) phases protected by exact emergent higher-  form symmetries in section IV, the models we consid-  ered in section III did not have SPT phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It would  be interesting to modify the models for emergent ZN p-  gauge theory and U(1) p-gauge theory so they can have  SPT phases protected by their exact emergent higher-  form symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, the confined phase (the Z(p)  N  and U(1)(p) symmetric phases) of these gauge theories  becomes an SPT upon adding a topological θ term to  the Lagrangian [2, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, it would be inter-  esting if one could modify the UV theories of the emer-  gent gauge theory models to have the 2π quantized topo-  logical term appear in the mid-IR effective Hamiltonian  and investigate the resulting emergent SPT order from a  Hamiltonian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We are grateful for fun and helpful discussions with  Arkya Chatterjee, Hart Goldman, Ethan Lake, Ho Tat  Lam, Yu Leon Liu, and Carolyn Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' is sup-  ported by the National Science Foundation Graduate Re-  search Fellowship under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2141064 and by the  Henry W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kendall Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This work is partially sup-  ported by NSF DMR-2022428 and by the Simons Collab-  oration on Ultra-Quantum Matter, which is a grant from  the Simons Foundation (651446, XGW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Appendix A: Review of discrete differential  geometry for d-dimensional cubic lattices  In this appendix section, we review relevant parts of  discrete differential geometry (in a non-rigorous fashion)  used throughout the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Consider a cubic lat-  tice in d-dimensional space with periodic boundary con-  ditions, denoted by Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  While a Bravais lattice is a  collection of lattice sites x ∈ Zd, it is useful to view it  as also formed by higher-dimensional objects, like links,  plaquettes, cubes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We call a p-dimensional object  a p-cell, with 0 ≤ p ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, a 0-cell is a lattice site, a  1-cell is a link, a 2-cell is a plaquette, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This does not  add additional structures to the lattice, but instead is  just a useful way of organizing the lattice sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed,  denoting a p-cell associated with site x as cp(x)µ1µ2···µp,  where µ1 < µ2 < · · · < µp and µi ∈ {1, 2, · · · , d}, a p-cell  of the cubic lattice is the set of 2p lattice sites26  cp(x)µ1µ2···µp= {x} ∪ {x + ˆµi | 1 ≤ i ≤ p}  ∪ {x + ˆµi + ˆµj | 1 ≤ i < j ≤ p}  ∪ · · · ∪ {x + ˆµ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' + ˆµp},  (A1)  26 We adopt the discrete differential geometry and exterior calculus  notations and conventions used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  27  p = 1  p = 2  p = 1  p = 2  p = 3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The p-cells of the d-dimensional cubic lattice  are equivalently the 0-cells—the sites—of some other d-  dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shown here are examples of this equiv-  alent lattice (drawn in pink) embedded in the conventional  unit cell of the cubic lattice (drawn in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (First row) In  2 dimensions, the 1-cells form another square lattice, rotated  by 45 degrees, whose lattice constant is 1/  √  2 times that of the  original square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The 2-cells also form another square  lattice, which is the original shifted by the vector (ˆµ1 + ˆµ2)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (Second row) In 3 dimensions, both the 1-cells and also the 2-  cells form a lattice of corner-sharing octahedra with a lattice  constant that is 1/  √  2 times the cubic lattice’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When p = 1,  the octagons are centered at the cubic lattice’s 0-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When  p = 2, the octagons are centered at the cubic lattices 3-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Lastly, the 3-cells form another cubic lattice of the same size,  but shifted by the vector (ˆµ1 + ˆµ2 + ˆµ3)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  where ˆµi is the unit vector in the µi-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It  is often convenient to drop the requirement that the  indices are canonically ordered (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', that they satisfy  µ1 < µ2 < · · · < µp < ν) and instead let cp(x)µ1µ2···µp  obey the relation cp(x)···µ1µ2··· = −cp(x)···µ2µ1···.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The  p-cells of the d-dimensional cubic lattice are equiva-  lently viewed as the 0-cells of some other lattice in d-  dimensions, as demonstrated for d = 2 and 3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Introducing the concept of p-cells is strictly unneces-  sary but very convenient because “sewing” p-cells to-  gether gives a natural way to form p-dimensional sub-  spaces of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore these subspaces can  also be given an orientation by defining an orientation  structure to the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A nice local scheme for the  lattice orientation is a branching structure, where the  orientation on each 1-cell is chosen such that a collec-  tion of 1-cells cannot form an oriented closed loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A  canonical orientation on all other p-cells then follows from  the branching structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We use the branching structure  where each 1-cell c1(x)µ has an arrow pointing in the ˆµ  direction (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, it is important to note  that the choice of lattice orientation is a formal conven-  tion, and choosing different branching structures does not  affect the physics27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A p-cell can be related to (p − 1) cells using the bound-  ary operator ∂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The boundary operator acting on a p-  27 However, according to a conjecture from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 79, observables are  independent of the branching structure only if the continuum  effective field theory is free of a framing anomaly [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ̂x  ̂y  ̂z  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Example of the branching structure used for a chunk  of the cubic lattice in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  cell—∂cp—is the oriented sum of (p − 1)-cells on the  boundary of cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the branching structure we use, it is  given by  ∂cp(x)µ1···µp=  p  �  k=1  (−1)k+1�  cp−1(x + ˆµk)µ1···  oµk···µp  −cp−1(x)µ1···  oµk···µp  �  ,  (A2)  where the notation  oµk indicates that the µk index is omit-  ted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From its definition, the boundary operator satisfies  ∂2cp = 0 for any p-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, as there are no  (−1)-cells, the boundary operator acting on a 0-cell is  defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  On the other hand, a p-cell can be related to (p + 1)-  cells using the coboundary operator δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The cobound-  ary operator acting on a p-cell—δcp—is an oriented sum  of all (p + 1)-cells whose boundary includes cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the  branching structure we use, it is given by  δcp(x)µ1···µp =  �  ν  cp+1(x)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='µp − cp+1(x − ˆν)νµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='µp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (A3)  From its definition, the coboundary operator satisfies  δ2cp = 0 for any p-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, as there are no  (d + 1)-cells, the coboundary operator acting on a d-cell  is defined to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Lastly, the lattice has an associated dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  dual lattice has its lattice sites centered at the d-cells  of the direct lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the cubic lattice, one way to  relate a dual lattice site ˆx to a direct lattice site x is by  ˆx = x + 1  2 ˆr with ˆr = �  i ˆµi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Each p-cell cp on the direct lattice is associated with  a (d − p)-cell ˆcd−p on the dual lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is imple-  mented by the dual operator ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A p-cell cp(x)µ1···µp (with  canonical ordering µ1 < · · · < µp) and a (d − p)-cell of  the dual lattice ˆcd−p(ˆx)µ1···µd−p (with canonical ordering  µ1 < · · · < µd−p) are related to one another by  ∗ cp(x)µ1···µp = ϵµ1···µpµp+1···µd  (A4)  × ˆcd−p(ˆx − ˆµp+1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' − ˆµd)µp+1···µd,  ∗ ˆcp(ˆx)µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='µp = ϵµ1···µpµp+1···µd  (A5)  × cd−p(x + ˆµ1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' + ˆµp)µp+1···µd,  where summation is not implied on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Here ϵ is the Levi-Civita symbol, which takes into ac-  28  count the lattice’s and dual lattice’s relative orienta-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' From the definition of ∗, acting ∗ twice on a p-cell  of the direct (dual) lattice yields ∗ ∗ cp = (−1)p(d−p)cp  (∗ ∗ ˆcp = (−1)p(d−p)ˆcp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, from the defini-  tions of the boundary, coboundary, and dual operators,  they are related to one another by  δcp = (−1)d(p+1)+1 ∗ ∂ ∗ cp,  (A6)  which, equivalently, is ∗ δcp = (−1)p∂ ∗ cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Appendix B: A TQFT description of the p-form  toric code ground states—p-form BF theory  In section III B 2 of the main text, we found that the  ground states of the Z(p)  N  SSB phase is equivalent to  the ground states of the p-form toric code Hamiltonian  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (65) in terms of the dressed clock operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this  appendix section, we relate the lattice description of the  ground states to an equivalent topological quantum field  theory description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Doing so demonstrates the connec-  tion between exact emergent higher-form symmetries in  lattice models and exact higher-form symmetries in La-  grangian quantum field theories, where higher-form sym-  metries are most commonly studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The ground states of the Z(p)  N  SSB phase are defined  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To develop a field theory description of  these ground states, we take inspiration from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 81 and  parameterize the clock operators as  Xcp = exp  �  iΘcp  �  ,  Zcp = exp  �  i(∗ Φ)cp  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B1)  The dressed clock operators in the Z(p)  N  SSB phase then  become  �  X′  cp = exp  �  i �Θ′  cp  �  ,  (B2)  �Z′  cp = exp  �  i(∗ �Φ′)cp  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B3)  Note that in order for ( �  X′  cp)N = ( �Z′  cp)N = 1, it must  be that the eigenvalues of �Θ′  cp and (∗ �Φ′)cp satisfy  �Θ′  cp ∈ 2πZ/N, and (∗ �Φ′)cp ∈ 2πZ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, in  order for the clock operators algebra Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (34) to be sat-  isfied, �Θ′  cp and (∗ �Φ′)cp must obey the commutation re-  lation [�Θ′  cp, (∗ �Φ′)�cp] = 2π i  N δcp,�cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In terms of �Θ′  cp and  (∗ �Φ′)cp, the constraints Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (64) defining the IR are  N  2π δ(∗ �Φ′)cp−1 = N  2π (d�Θ′)cp+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B4)  The the lattice Heisenberg operators  �Θ′  cp(t) and  (∗ �Φ′)cp(t) are related to their continuum counterparts  �Θ′(t, x) and ∗ �Φ′(t, x) by  �Θ′  cp =  �  cp  �Θ′,  (∗ �Φ′)cp =  �  cp  ∗ �Φ′,  (B5)  where  �  cp denotes spatial integration over the p-cell cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For simplicity, we will work locally and treat the con-  tinuum quantum fields as differential forms in space (�Θ′  is a p-form while �Φ′ is a (d − p)-form), mapping from  spacetime to R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' One of the many conveniences of  using the discrete exterior calculus notation is that in  the continuum limit these lattice operators become their  continuum versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, the constraint Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B4) in the  continuum limit becomes  N  2π d† ∗ �Φ′ = N  2π d�Θ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B6)  Here, d† is the adjoint of d, which when acting on a  p-form is given by d† ≡ (−1)d(p+1)+1 ∗ d ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The lattice Hamiltonian in the IR is just the ground  state energy, which we’ll set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the continuum  limit of H(III)  IR , Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (63), is  H(III)  IR  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B7)  To write down the path integral, we can find the  Lorentzian action from H(III)  IR  and then perform a func-  tional integral over all dynamical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, the  path integral only integrates over field configurations sat-  isfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Taking these constraints into account,  the path integral in Lorentzian signature is  Z(III)  IR  =  �  D[�Θ′]D[�Φ′] δ  � N  2π d† ∗ �Φ′  �  (B8)  × δ  � N  2π d�Θ′  �  ei  �  dtddxL(III)  IR ,  L(III)  IR  =  N  2πp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (∗ �Φ′)i1···ip∂t �Θ′  i1···ip  The first term in L(III)  IR  enforces the equal-time commu-  tation relation  [�Θ′  i1···ip(x), (∗ �Φ′)j1···jp(y)]  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  = 2πi  N δi1  [j1· · · δip  jp]δd(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B9)  Since we work locally, we do not enforce the constraint  that the holonomies of �Θ′ and ∗ �Φ′ are restricted to values  in 2πZ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  All that is left is to massage this path integral into a  more familiar form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We can rewrite both of the func-  tional delta functions by integrating in new fields acting  as Lagrange multipliers and modifying the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For  instance, the first delta function can be rewritten using  a (p − 1)-form Lagrange multiplier λ:  δ  � N  2π d†∗ �Φ′  �  =  �  Dλ e  i N  2π  �  λi1···ip−1 ( d† ∗ �  Φ′)i1···ip−1  (p−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B10)  Likewise, the second delta function can be rewritten us-  ing a (p + 1)-form Lagrange multiplier ∗ η:  δ  � N  2π d�Θ′  �  =  �  Dη e  i N  2π  �  (∗ η)i1···ip+1 ( d �  Θ′)i1···ip+1  (p+1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B11)  29  Plugging these expressions into the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B8), the path  integral becomes  Z(III)  IR  =  �  D[�Θ′]D[�Φ′]D[λ]D[η] ei  �  dtddxL(III)  IR ,  (B12)  L(III)  IR = N  2π  �  (∗ �Φ′)i1···ip∂t �Θ′  i1···ip  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  +λi1···ip−1(d†∗ �Φ′)i1···ip−1  (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  +(∗ η)i1···ip+1(d�Θ′)i1···ip+1  (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Plugging in the components of d�Θ′ and d† ∗ �Φ′ and sim-  plifying, L(III)  IR  can be rewritten as  L(III)  IR  =  N  2πp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  (∗ �Φ′)i1···ip  �  ∂t �Θ′  i1···ip + p ∂[i1 λ i2···ip]  �  + (∗ η)i1i2···ip+1∂[i1 �Θ′  i2···ip+1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B13)  Let’s  now  introduce  the  p-form  a  and  (d − p)-  form b in spacetime whose spatial components are  ai1···ip = �Θ′  i1···ip  and  bi1···id−p = (−1)d−p�Φ′  i1···id−p  and  timelike  components  are  a0i2···ip = −λi2···ip  and  b0i2···id−p = −ηi2···id−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Note  that  with  this  identification,  (∗ η)i1···ip+1 = −(∗ b)i1···ip+1  and  (∗ �Φ′)i1···ip = (∗ b)0i1···ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Plugging these in, the path  integral becomes  Z(III)  IR  =  �  D[a]D[b] ei  �  dtddxL(III)  IR ,  L(III)  IR  =  N  2πp!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  (∗ b)0i1···ip  �  ∂tai1···ip + (−1)pp ∂[i1 a i2···ip]0  �  − (∗ b)i1i2···ip+1∂[i1 a i2···ip+1]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B14)  The  term  in  brackets  can  be  rewritten  using  ∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 = (p + 1)∂[0ai1···ip].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, working in flat spacetime, X is equipped  with Minkowski metric (−, +, · · · +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using it and sum-  ming over spacetime indices µ, L(III)  IR  can be rewritten  as  L(III)  IR  = − N  2π  �(∗ b)µ1µ2···µp+1∂[µ1 a µ2···ip+1]  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B15)  Lasting, using differential forms notation, we arrive at  our final expression for the path integral  Z(III)  IR  =  �  D[a]D[b] ei  �  L(III)  IR ,  L(III)  IR  = N  2π b ∧ da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B16)  As anticipated, the low-energy effective field theory,  which describes the ground states of the Z(p)  N  symme-  try broken phase, is p-form ZN gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is  arguably the simplest field theory with an anomalous  Z(p)  N × Z(d−p)  N  symmetry [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Review of p-form BF theory  In the remainder of this appendix section, we will re-  view p-form BF theory, focusing on its symmetries and  anomalies, working in D = d + 1 dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From canonical quantization, the fields a and b satisfy  the equal-time commutation relations  [aµ1···µp(x), bµp+1···µd(y)] = 2πi  N ϵ0µ1···µdδd(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B17)  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  ZN p-form gauge theory in the continuum  We start by review how p-form BF theory can be  obtained by condensing charge-N gauge charges in p-  form Maxwell theory [82–84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  p-form Maxwell theory  is reviewed in appendix section C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This will be use-  ful as we’ll encounter dual representations of this theory  which will make the symmetry analysis of the next sec-  tion easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, for the reader who would like to  jump straight to p-form BF theory, they should skip to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We modify p-form Maxwell theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C19) by intro-  ducing the dynamical (p − 1)-form bosonic field H and  the gauge redundancy  a → a + dχ,  H → H + Nχ,  (B18)  where N ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A gauge-invariant globally defined quan-  tity in terms of only H is FH = dH + ωH,  where  ωH ∈ 2πHp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  FH  satisfies  a  Bianchi  iden-  tity  1  2π ∗ dFH = 0 and its periods are quantized as  �  FH ∈ 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We note that the local part of FH, dH,  is only there when p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' When p = 0, FH has no lo-  cal fluctuations but only purely topological contributions  from the 0-form ωH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  With the additional degrees of freedom provided by H,  we can now introduce the Wilson operator  Wa,H(O) = exp  �  i  �  O  Na − dH  �  ,  = exp  �  iN  �  O  a  �  exp  �  −i  �  ∂O  H  �  ,  = W N  a (O)W †  H(∂O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B19)  where O is an open p-submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Physically Wa,H(O)  is an operator that creates a charge excitation on ∂O,  but one carrying N-units of a-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' H is the contin-  uum phase operator of created fractionalized charges that  carries N-units of a-charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Minimally coupling FH to a  in light of the gauge redundancy Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B18), we find the  30  partition function  Z =  �  D[a]D[H]  �  ωa  2π ∈Hp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ωH  2π ∈Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e−  �  X L,  L =  1  2g2 |Fa|2 + v2  2 |FH − Na|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B20)  Expanding out the new term, we can rewrite the La-  grangian density as  L =  1  2g2 |Fa|2 + v2  2 |FH|2 + N 2v2  2  |a|2 − Nv2a ∧ ∗ FH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B21)  Thus, we find that L ⊃ −a∧ ∗ J with J = Nv2FH for  which generally d†J ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, there is now a  mass term for a: L ⊃ N 2v2  2  |a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, the new term  added to p-form Maxwell theory is essentially a Higgs  term with H the phase of the Higgs field and v ∈ R the  vev of the Higgs field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The gauge redundancy described  by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B18) is a ZN gauge redundancy, reflecting how  the initial U(1) gauge redundancy has been Higgsed down  to a ZN gauge redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  As we saw in appendix section C 1 b, these types of  theories have a generalized “particle-vortex” like du-  ality called abelian duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For instance, since the  action’s dependency on H is entirely in the form of  FH we can dualize H → ˆH using the same method  shown in section C 1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, dualizing H to the  (D − p − 1)-form ˆH satisfying  �  F ˆ  H ∈ 2πZ and dF ˆ  H  (where F ˆ  H = d ˆH + ω ˆ  H), the Euclidean Lagrangian be-  comes  L = |Fa|2  2g2 + |F ˆ  H|2  8π2v2 − iN  2π a ∧ F ˆ  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B22)  The  Euclidean  path  integral  now  integrates  over  the  dynamical  fields  a  and  ˆH  and  sums  over  ωa ∈ 2πHp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) and ω ˆ  H ∈ 2πHD−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We note  that without changing the action amplitude, the La-  grangian density can be rewritten as  L = |Fa|2  2g2 + |F ˆ  H|2  8π2v2 − iN  2π  ˆH ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B23)  Locally, we have just integrated by parts in the BF term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, keeping track of the globally nontrivial parts of  F ˆ  H and Fa makes showing this difficult (it’s most natu-  rally seen using Deligne-Beilinson cohomology [85]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Utilizing abelian duality we have found two representa-  tions for the theory: the (a, H) representation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B20)  and the (a, ˆH) representation Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B22) and (B23), the  latter being dual only locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In representation (a, ˆH), the deep IR is governed by  p-form BF theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Here, the deep IR refers to energies  below the gap of a and ˆH, which have a gap through  topological mass generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, to find their en-  ergy gaps, first note that in the (a, ˆH) representation,  the Lorentzian action is  S =  �  X  �  −|Fa|2  2g2 − |F ˆ  H|2  8π2v2 + N  2π a ∧ F ˆ  H  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B24)  Since this theory is Gaussian, we can show that the a∧F ˆ  H  term causes all excitations to be gapped using the equa-  tions of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Varying the action, we find that  δS =  �  X  �  − 1  g2  �  d†Fa + (−1)p(D−p) Ng2  2π ∗ F ˆ  H  �  ∧ ∗ δa  −  1  4π2v2  �  d†F ˆ  H − (−1)p2+D2πNv2 ∗ Fa  �  ∧ ∗ δ ˆH  �  ,  and therefore the classical equations of motion are  d†Fa + (−1)p(D−p) Ng2  2π ∗ F ˆ  H = 0,  d†F ˆ  H − (−1)p2+D2πNv2 ∗ Fa = 0  (B25)  We can now decouple ∗ Fa and ∗ F ˆ  H to get the equations  �  d† d + N 2g2v2�  ∗ F ˆ  H = 0,  �  d† d + N 2g2v2�  ∗ Fa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B26)  Using that d† ∗ Fb, ˆ  H = 0, we can then rewrite this in  terms of the Hodge Laplacian δ = d† d + dd† as  �  δ + N 2g2v2�  ∗ F ˆ  H = 0,  �  δ + N 2g2v2�  ∗ Fa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B27)  Therefore,  we  see  that  the  p-form  ∗ F ˆ  H  and  the  (D − p − 1)-form ∗ Fa both have an energy gap Ngv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To go below the energy gap into the deep IR, we take  the limit g → ∞ and v → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this limit, the Euclidean  path integral becomes  ZBF =  �  D[a]D[ ˆH]  �  ω ˆ  H  2π ∈HD−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e−  �  X LBF ,  LBF = − iN  2π a ∧ F ˆ  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B28)  This is p-form BF theory, and it is in terms of the  p-form bosonic field a, which is the U(1) gauge field  we started with and ˆH, which is the abelian dual of  the Higgs field phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Taking the deep IR limit us-  ing the Lagrangian density in this representation written  as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B23), the Lagrangian in the topological limit is  equivalent to LBF = − iN  2π ˆH∧Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Plugging in F ˆ  H = d ˆH + ω ˆ  H into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B28), the path  integral becomes  ZBF =  �  D[a]D[ ˆH]  �  ω ˆ  H  2π ∈HD−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e  i N  2π  � (a∧d ˆ  H+a∧ω ˆ  H)  (B29)  Integrating by parts on the first term and using Poincar´e  duality on the second term, we can rewrite this as  ZBF =  �  D[a]D[ ˆH]  �  ω∈Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  exp  � iN  2π  �  X  ˆH ∧ da + iN  �  ω  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B30)  31  Integrating over ˆH and summing over ω, the path integral  becomes  ZBF =  �  D[a] δ(da) δ  ��  a ∈ 2πZ  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B31)  Notice that if we would have instead started with the  Lagrangian density written as LBF = − iN  2π ˆH∧Fa, upon  integrating out a and ωa we’d get  ZBF =  �  D[ ˆH] δ(d ˆH) δ  ��  ˆH ∈ 2πZ  N  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B32)  Having massaged p-form BF theory into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B31), we  find that the U(1) gauge fields are closed forms and have  quantized holonomies28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This has an important effect on  the Wilson operators  Wa[Cp] = exp  �  i  �  Cp  a  �  ,  (B34)  W ˆ  H[CD−p−1] = exp  �  i  �  CD−p−1  ˆH  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B35)  Indeed, since the holonomies of a and ˆH are quantized,  the Wilson operators are constrained as  Wa ∈ ZN,  W ˆ  H ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B36)  Thus, because the holonomies are quantized the Wilson  operators satisfy  (Wa)N = (W ˆ  H)N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B37)  When Cp ∈ Bp(X) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', there exists an Op+1 such that  Cp = ∂Op+1), Wa can be written as  Wa[Cp] ≡ exp  �  i  �  Cp  a  �  = exp  �  i  �  Op+1  da  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, since da = 0, Wa supported on a contractible  cycle is trivial element of ZN:  Wa [Cp ∈ Bp(X)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B38)  This is also true for W ˆ  H:  W ˆ  H [CD−p−1 ∈ BD−p−1(X)] = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B39)  28 This can also be deduced from the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed,  in the (a, H) representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B20), the H equations of mo-  tion in the deep IR are  FH = Na.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B33)  Therefore, because of the Bianchi identity dFH = 0, a is a  closed p-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Furthermore, since FH satisfies  �  FH = 2πZ, the  holonomies of a are quantized as  �  a = 2πZ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, the Wilson operators are topological opera-  tors, meaning that they can be continuously deformed:  W[C + ∂O] = W[C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B40)  Therefore, at low-energies, any contractible Wilson oper-  ators can condense into vacuum, but for non-contractible  Wilson operators only N of them can condense into vac-  uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We can evaluate ZBF by considering the Wilson op-  erators, which is convenient as we do not have to worry  about performing the path integral in the presence of  gauge redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, the path integral simply  counts the number of nontrivial Wilson operators W(C)  which satisfy W(C)N = 1, and thus the number of con-  figurations a ∈ Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN):  ZBF =  �  a∈Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='ZN)  1,  = |Hp(X, ZN)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The coefficients of Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN) are ZN to preserve the  condition Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The number of ground states is  given by the partition function evaluated on X = R × M,  where M is a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, there are |Hp(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' ZN)|  degenerate ground states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Symmetries  Having reviewed the basics of p-form BF theory in the  previous section, we now turn to identifying the theory’s  symmetries, showing that there is a Z(p)  N × Z(d−p)  N  sym-  metry [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s first consider the symmetries manifest in the  (a, H) representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The path integral is  invariant under the transformation  a → a + Γ,  FH → FH + NΓ,  (B41)  with dΓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since FH satisfies  �  FH ∈ 2πZ, in order  to shift FH → FH + NΓ we require that  �  Γ ∈ 2πZ/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  When Γ = dω, the transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B41) becomes  the gauge transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore, the  Γ that correspond to physical transformations are  N  2πΓ ∈ Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The quantization condition of the periods of Γ has a  significant consequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, note that the Wilson  operator Wa (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B34)) is charged under this symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because of the quantization condition  �  Γ ∈ 2πZ/N, it  transforms as  Wa[Cp] → ei  �  ΓWa[Cp]  ei  �  Γ ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B42)  Since the charged operators are p-dimensional and trans-  form by an element of ZN, this is a Z(p)  N symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It’s tempting to think that there may be an additional  symmetry associated with the H field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B20)  32  is invariant under the transformation H → H + ω for  dω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, there are no physical observables that  transform under this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, while the Wilson operator  exp  �  i  �  H  �  picks up a phase, it is not a physical opera-  tor since it is not invariant under the gauge redundancy  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since there are no charged operators under  H → H + ω, it does not have physical implications and  is therefore not a symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The  Z(p)  N  symmetry  can  also  be  seen  in  the  (a, ˆH) representation,  when the Lagrangian is de-  scribed by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, under the symme-  try transformation, the action amplitude transforms as  exp[−S] → exp[−S] exp[−δS], where  exp[−δS] = exp  �  − iN  2π  �  Γ ∧ F ˆ  H  �  ,  (B43)  with  N  2πΓ ∈ Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Plugging in F ˆ  H = d ˆH + ω ˆ  H, the  phase factor exp[−δS] becomes  exp[−δS] = e− i N  2π  �  Γ∧d ˆ  H e− i N  2π  �  Γ∧ω ˆ  H  = e  i N  2π (−1)p �  dΓ∧ ˆ  H e−2π i  �  N Γ  2π  ∧  ω ˆ  H  2π  = exp  �  −2πi  � N Γ  2π  ∧ ω ˆ  H  2π  �  ,  where we used integration by parts and that dΓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Re-  call that N  2πΓ ∈ Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) and ω ˆ  H  2π ∈ HD−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Then,  since the wedge product preserves integral de Rham  cohomology classes,  N Γ  2π ∧ ω ˆ  H  2π ∈ HD(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Therefore,  exp[−δS] becomes  exp[−δS] = exp[−2πiZ] = 1,  (B44)  and thus the action amplitude is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The Lagrangian in the (a, ˆH) representation can also  be written as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B23) without changing the partition  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In this form, following the same argument used  to show that there is a Z(p)  N  symmetry, we find there is  also Z(d−p)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, the action amplitude is  invariant under ˆH → ˆH + ˆΓ where  N  2π ˆΓ ∈ Hd−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The charged operator of this Z(d−p)  N  symmetry is the Wil-  son operator W ˆ  H = exp  �  i  � ˆH  �  , which transforms as  W ˆ  H(C) → ei  � ˆΓW ˆ  H(C)  ei  � ˆΓ ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B45)  To summarize, the symmetry of p-form BF theory is  Z(p)  N × Z(d−p)  N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The symmetry transformations associated  with Z(p)  N and Z(d−p)  N  are  Z(p)  N :  a → a + 2π  N Γ,  Γ ∈ Hp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z),  Z(d−p)  N  :  ˆH → ˆH + 2π  N  ˆΓ,  ˆΓ ∈ Hd−p(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z),  (B46)  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note how only in the (a, ˆH) representation  are both of these symmetries manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In the (a, H)  representations, the Z(d−p)  N  symmetry is hidden as the  charged operators are ’t Hooft operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The symmetry operator of the Z(p)  N symmetry is  U(Σ) = exp  �  i  �  Σ  ˆH  �  ,  = exp  �  i  �  Md  ˆH ∧ Γ  �  ,  (B47)  where Γ is the Poincar´e dual of the p-cycle Σ with respect  to space Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, using the equal-time commutation  relation (B17), which form Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B22) is  [aµ1···µp(x), ˆHµp+1···µd(y)] = 2πi  N ϵ0µ1···µdδd(x − y),  (B48)  we have that  U(Σ)Wa(C)U †(Σ) = e  2π i  N  �  C ΓWa(C),  = e  2π i  N #(Σ,C)Wa(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B49)  Interesting the symmetry operator of Z(p)  N  was the  charged operator of Z(d−p)  N  : U = W ˆ  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Similarly, the sym-  metry operator of the Z(d−p)  N  symmetry is  ˆU(ˆΣ) = exp  �  i  �  ˆΣ  a  �  ,  (B50)  which is just Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Mixed ’t Hooft anomaly and anomaly inflow  In the last section, we found that p-form BF theory  has an Z(p)  N  and Z(d−p)  N  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, these sym-  metries are not independent of one another: the symme-  try operator of one symmetry is a charged operator of  the other symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In other words, these two symme-  try operators do not commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is a manifestation of  the fact that the Z(p)  N × Z(d−p)  N  symmetry is anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In this section, we will turn on a background gauge field  for these symmetries to learn more about this mixed ’t  Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s first turn on a background gauge field for the  Z(p)  N symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We introduce the background gauge field  A ∈ 2π  N Hp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) and the gauge redundancy  a → a + β,  A → A + dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B51)  Minimally coupling A, the p-form BF theory path inte-  gral becomes  Z[A] =  �  D[a]D[ ˆH] e−  �  X L[A],  L = − iN  2π  �  a ∧ F ˆ  H + (−1)pA ∧ ˆH  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B52)  33  We next turn on a background gauge field for the  Z(d−p)  N  symmetry, introducing the background gauge field  ˆ  A ∈ 2π  N Hd−p+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) and the gauge redundancy  ˆH → ˆH + ζ,  ˆ  A → ˆ  A + dζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B53)  Minimally coupling ˆ  A, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B52) becomes  Z[A, ˆ  A] =  �  D[a]D[ ˆH] e−  �  X L[A, ˆ  A],  L = − iN  2π  �  a ∧ (F ˆ  H − ˆ  A) + (−1)pA ∧ ˆH  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B54)  When the Z(p)  N  background gauge field is turned off  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e, A = 0), the path integral is gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' How-  ever, when the Z(p)  N background gauge field is turned on,  there are no local counter terms which can be added such  that the path integral is invariant under Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In-  deed, under this gauge transformation, the path integral  transforms as  Z[A, ˆ  A] → Z[A, ˆ  A] exp  �  (−1)p iN  2π  �  X  A ∧ ζ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B55)  We thus see that there is an obstruction to coupling a  background gauge field of both symmetries, and hence a  mixed ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  a ’t Hooft anomaly can be classified by an invertible  theory in one higher-dimension whose boundary realizes  the anomalous symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s now extend the back-  ground fields to one higher-dimension and have X be the  boundary of the new spacetime Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The phase picked up  by Z[A, ˆ  A] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B55) can then be written as  exp  �  (−1)p iN  2π  �  X  A ∧ ζ  �  = exp  �  − iN  2π  �  Y  A ∧ dζ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B56)  In order to make the theory gauge invariant and cancel  out the phase Z picks up, let’s consider the invertible  theory  Zinv[A, ˆ  A] = exp  � iN  2π  �  Y  A ∧ ˆ  A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B57)  Under the gauge transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B53), Zinv trans-  forms as  Zinv[A, ˆ  A] → Zinv[A, ˆ  A] exp  � iN  2π  �  Y  A ∧ dζ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B58)  This is the same as the inverse of the phase picked up by  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, we replace Z with  Z[A, ˆ  A] → Zinv[A, ˆ  A]Z[A, ˆ  A],  (B59)  which is now gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The path integral of the  new gauge invariant theory is  Z[A, ˆ  A] =  �  D[a]D[ ˆH] e−  �  Y Lbulk+dLboundary,  Lbulk = − iN  2π A ∧ ˆ  A,  Lboundary = − iN  2π  �  a ∧ F ˆ  H + (−1)pA ∧ ˆH  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Given that we have found the theory in Y spacetime  such that the mixed ’t Hooft anomaly is canceled, we can  now close the one higher-dimensional spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let-  ting ∂Y = ∅, the topological part of the path integral  is Zinv[A, ˆ  A], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (B57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since A ∈ 2π  N Hp+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) and  ˆ  A ∈ 2π  N Hd−p+1(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z), A∧ ˆ  A ∈ 4π2  N 2 Hd+2(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  There-  fore, the mixed anomaly is classified by  Zinv[A, ˆ  A] = exp  � i2π  N Z  �  ∈ ZN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (B60)  The form of Zinv and the ZN classification agrees with  the d = 3, p = 1 lattice result found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 43, which  started from a bulk twisted U(1) gauge theory in the  confined phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Appendix C: Taking the continuum limit—p-form  Maxwell theory  In this appendix section, we present how we take the  continuum limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (90) in section III C 2 of the main  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Doing so also demonstrates the connection between  exact emergent higher-form symmetries in lattice models  and exact higher-form symmetries in Lagrangian quan-  tum field theories, where higher-form symmetries are  most commonly studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The lattice Heisenberg operators Lz  cp(t) and Θcp(t)  in the IR are dressed by two local unitary opera-  tors U (1)  LU and U (2)  LU and denoted as �L′z  cp(t) and �Θ′  cp(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Furthermore, in the mid-IR we defined the variable  ωcp+1 ≡ −2π  �  (d�Θ)cp+1/(2π)  �  ,  which in the IR was  dressed by U (2)  LU and denoted as ω′  cp+1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the  three elementary operators in the IR are �L′z  cp(t), �Θ′  cp(t),  and ω′  cp+1(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We relate these lattice operators to their  continuum counterparts �L′z(t, x), �Θ′(t, x), and ω′(t, x)  by  �L′z  cp =  �  cp  �L′z,  �Θ′  cp =  �  cp  �Θ′,  ω′  cp+1 =  �  cp+1  ω′, (C1)  where, for instance,  �  cp denotes spatial integral over the  p-cell cp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The continuum quantum fields are globally  differential forms in space M (�L′z and �Θ′ are p-forms  while ω′ is a (p + 1)-form), mapping from spacetime X  to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' One of the many conveniences of using the dis-  crete exterior calculus notation is that in the contin-  uum limit, these lattice operators become their contin-  34  uum versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, for instance, the lattice opera-  tor F ′  cp+1 = (d�Θ′)cp+1 + (ω′)cp+1 is related to its contin-  uum version by F ′  cp+1 =  �  cp+1 F ′ where F ′ = d�Θ′ + ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So, taking the continuum limit of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (90), the deep IR  continuum Hamiltonian is  Hdeep IR=  �  ddx  �  κU  2  |�L′z  i1···ip|2  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  + ε2p+2U  2  |F ′  i1···ip+1|2  (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  ,  (C2)  where, for instance, |�L′z  i1···ip|2 ≡ �d  i1,··· ,ip=1(�L′z  i1···ip)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To write down the path integral, we can find the  Lorentzian action from H(III)  IR  and then perform a func-  tional integral over all dynamical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, the  path integral only integrates over field configurations  obeying particular constraints:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Firstly, since the IR does not include dressed charge  excitations, we only integrate over �L′z configura-  tions satisfying �ρ ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The expression for �ρ ′  cp−1  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (66) can be rewritten using discrete exterior  calculus notation as �ρ ′  cp−1 ∼ (∗ d ∗ �L′z)cp−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So,  �ρ ′ = 0 in the continuum limit is the Gauss law  ∂j �L′z  ji1···ip−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C3)  Despite there being no dressed charges in the IR,  �L′z can still be sourced along nontrivial p-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, because �L′z  cp ∈ Z on the lattice, in the con-  tinuum limit, the flux of �L′z is quantized  �  Cd−p  ∗ �L′z ∈ Z,  (C4)  where Cd−p is any closed (d − p)-submanifold of  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Notice how using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C3), when Cd−p  is contractible, this is automatically satisfied since  the left-hand side is zero by Stoke’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So,  this additional constraint only affects Cd−p in the  (d − p)th homology group: Cd−p ∈ Hd−p(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Additionally, since the IR does not include dressed  topological defects, we only integrate over �Θ′ and  ω′ configurations satisfying ˆρ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The expres-  sion for (∗ ˆρ′)cp+2 of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (88) in the continuum  limit becomes ˆρ′ =  1  2π ∗ dF ′, and ˆρ′ = 0 becomes  the Bianchi identity  1  2π ∗ dF ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C5)  Despite there being no dressed topological defects  in the IR, ∗ F ′ can still be sourced along nontrivial  (d − p)-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, because ω′  cp ∈ 2πZ on the  lattice, in the continuum limit, the flux of ∗ F ′ is  quantized  �  Cp+1  F ′ ∈ 2πZ,  (C6)  where Cp+1 is any closed (p + 1)-submanifold of  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Notice how plugging in F ′ = d�Θ′ + ω′, the  �Θ′ vanishes and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C5) and (C6) only put con-  straints on ω′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, because ω′ is not an exact  form, these constraints can be summarized as  ω′  2π ∈ Hp+1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z),  (C7)  where Hp+1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z) denotes the (p + 1)th de Rham  cohomology group with integral periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From the above, we learn that there are three con-  straints given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C3),  (C4), and (C7) that must  be added by hand into the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Taking into ac-  count these constraints, the path integral in Lorentzian  signature is  Zdeep IR =  �  D[�Θ′] D�L′z  �  ω′∈2πHp+1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  δ(∂i1 �L′z  i1···ip)  (C8)  × δ  ��  ∗ �L′z ∈ Z  �  ei  �  X dtddx Ldeep IR  Ldeep IR =  �L′z  i1···ip∂t �Θ′  i1···ip  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  �  κU  2  |�L′z  i1···ip|2  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  + ε2p+2U  2  |F ′  i1···ip+1|2  (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The first term in Ldeep IR enforces the equal-time com-  mutation relation  [�Θ′  i1···ip(x), �L′z  j1···jp(y)]  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  = iδi1  [j1· · · δip  jp]δd(x − y), (C9)  and the second term in parenthesis is Hamiltonian den-  sity from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C2)  This expression of the path integral is perfectly correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  However, to get it into a more familiar form, we rewrite  this phase space path integral as a coordinate space path  integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To do so, we rewrite both of the functional delta  functions by integrating in new fields and modifying the  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For the delta function enforcing the �L′z quanti-  zation condition, we can rewrite it by summing over all  nontrivial closed p-submanifolds in space  δ  ��  ∗ �L′z ∈ Z  �  =  �  C∈Hd−p(M)  exp  �  2πi  �  C  ∗ �L′z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C10)  Using Poincar´e duality to write 2π  �  C ∗ �L′z =  �  X ∗ �L′z∧η,  where η/2π is the poincare dual of C with respect to  spacetime, this can be rewritten as  δ  ��  ∗ �L′z ∈ Z  �  =  �  η∈2πHp(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X dtddx  �  L′z  i1···ip ηi1···ip  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C11)  35  Since C is closed and has integer coefficients, η is a closed  form and satisfies  �  η ∈ 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The delta function enforc-  ing Gauss law can be rewritten using a (p − 1)-form La-  grange multiplier  δ(∂i1 �L′z  i1···ip) =  �  Dλei  �  X λi2···ip∂i1 �L′z  i1i2···ip/(p−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='.  (C12)  Plugging in both of these delta functions, the path in-  tegral takes the cumbersome form  Zdeep IR =  �  D[�Θ′] D�L′z Dλ  �  ω′∈2πHp+1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  η∈2πHp(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X dtddx Ldeep IR,  Ldeep IR =  λi2···ip∂i1 �L′z  i1i2···ip  (p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  +  �L′z  i1···ipηi1···ip  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C13)  +  �L′z  i1···ip∂t �Θ′  i1···ip  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  κU|�L′z  i1···ip|2  2p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  ε2p+2U|F ′  i1···ip+1|2  2(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  It is now straight forward to integrate out the �L′z field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Assuming spacetime is closed, we integrate by parts on  the first term in Ldeep IR and use the anti-symmetry of  �L′z to rewrite Ldeep IR as  Ldeep IR =  �L′z  i1···ip  �  ∂t �Θ′  i1···ip − p ∂[i1λi2···ip] + ηi1···ip  �  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  κU|�L′z  i1···ip|2  2p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  ε2p+2U|F ′  i1···ip+1|2  2(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C14)  Integrating  out  �L′z,  rescaling  t → t/(Uεp+1√κ),  λ → Uεp+1√κ λ, and η → Uεp+1√κ η, and introducing  g = 1/(εp+1√  U), the path integral becomes  Zdeep IR =  �  D[�Θ′] Dλ  �  ω′∈2πHp+1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  η∈2πHp(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X dtddx Ldeep IR,  (C15)  Ldeep IR=  |∂t �Θ′  i1···ip− p∂[i1λi2···ip] +ηi1···ip|2  2g2p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  −  |F ′  i1···ip+1|2  2g2(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='.  Having found the coordinate path integral, we now  massage this to get it into a canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Namely,  let’s introduce the p-form a and (p + 1)-form ωa in  spacetime whose spatial components are ai1···ip = �Θ′  i1···ip  and (ωa)i1···ip+1 = ω′  i1···ip+1 and timelike components are  a0i2···ip = λi2···ip and (ωa)0i1···ip = ηi1···ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using a and  letting Fa = da + ωa, we can reexpress the path integral  as  Zdeep IR =  �  D[a]  �  ωa∈2πHp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X dtddx Ldeep IR,  (C16)  Ldeep IR = |∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 + (ωa)0i1···ip|2  2g2p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  − |(Fa)i1···ip+1|2  2g2(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Using ∂0ai1···ip + (−1)pp ∂[i1ai2···ip]0 = (p + 1)∂[0ai1···ip],  the first term in Ldeep IR can be rewritten in terms of Fa:  Ldeep IR =  1  2g2  �|(Fa)0i1···ip|2  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  − |(Fa)i1···ip+1|2  (p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C17)  Furthermore, working in flat spacetime, X is equipped  with Minkowski metric (−, +, · · · +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus, summing  over spacetime indices µ = 0, · · · , d, Ldeep IR can be  rewritten as  Ldeep IR = −  1  2g2(p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (Fa)µ1···µp+1(Fa)µ1···µp+1  (C18)  Lastly, using differential forms notation, we arrive at our  final expression for the path integral  Zdeep IR =  �  D[a]  �  ωa∈2πHp+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  ei  �  X Ldeep IR,  Ldeep IR = − 1  2g2 |Fa|2,  (C19)  where |F|2 ≡ F ∧ ∗ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is exactly p-form Maxwell the-  ory, as stated in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Review of p-form Maxwell theory  In the remainder of this appendix section, we will re-  view p-form Maxwell theory, focusing on its symmetries  and anomalies, working in D = d + 1 dimensional space-  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The  canonical  momentum  field  Π  is  locally  a  (D − p − 1)-form associated with a codimension-1 sub-  manifold of X in a Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We choose this  to be the 0-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Then, Π’s components are defined  by varying the action with respect to ∂0aµ1···µp which  yields  Π = − 1  g2 ∗ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C20)  Therefore, from canonical quantization we have the  equal-time commutation relations  �  aµ1···µp(x), (∗ Fa)µp+1···µd(y)  g2  �  = iϵ0µ1···µdδd(x − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C21)  36  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  U(1)(p) symmetry  The action amplitude is only a function of the field  strength, and so the path integral is invariant under a  being shifted by a closed p-form:  a �→ a + Γ  where  dΓ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C22)  However, just because the action amplitude is invariant  does not mean it is a symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  In-  deed, it could be a gauge redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  If it is a gen-  uine symmetry, it must transform observables nontriv-  ially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  These physical operators are the Wilson opera-  tors Wa(Cp) = exp  �  i  �  Cp a  �  , and under the transforma-  tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C22) they transform as  Wa (Cp) �→ exp  �  i  �  Cp  Γ  �  Wa (Cp) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C23)  Since there are no restrictions on the holonomies  of a, Γ satisfies  �  Γ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus exp  �  i  �  Cp Γ  �  ∈ U(1)  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C23) is the symmetry transformation of a  U(1)(p) symmetry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', an operator supported on a p-  submanifold picks up a phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that when Cp is  a boundary, then by Stokes theorem  �  Cp Γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus,  only Wilson loops defined on non-contractible Cp can be  charged under the U(1)(p) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Given that Wa (Cp) are the physical quantities of the  theory instead of a, not all transformations a → a + Γ  with dΓ = 0 necessarily correspond to symmetry trans-  formations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  For Γ that satisfy exp  �  i  �  Cp Γ  �  = 1, the  transformation a → a + Γ is instead a gauge transfor-  mations corresponding to formal redundancies29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Let’s  denote Γ that correspond to gauge redundancies as  Γgauge, and since exp  �  i  �  Cp Γgauge  �  = 1, their periods  are  �  Cp Γgauge ∈ 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In the case where Γgauge is exact  (Γgauge = dω, which is the canonical gauge transforma-  tion), the periods are zero by Stokes theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However,  there also exist closed but not exact Γgauge whose periods  are 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, these are Γgauge ∈ 2πHp(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Z), and  these types of gauge transformations are the so-called  large gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The physical transforma-  tion on a therefore require that Γ be closed but not exact  and the periods of Γ are not in 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Given that we’ve identified a U(1)(p) symmetry, let’s  now find the symmetry transformation operator which  performs said transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We can do so using  Noether’s theorem and the fact that the conserved charge  operator is the generator of the corresponding symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  29 This gives rise to an intriguing explanation to why gauge redun-  dancies appear in theories that describe physical systems: the  physical system’s constituents are extended objects but the for-  malism used describes them is in terms of strictly local fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' It  is physicists’ attachment to locality that infects the formalism  with redundancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  A nice trick to find the conserved current is to introduc-  ing a closed (p + 1)-form background field A and the  gauge redundancy  a �→ a + β,  A �→ A + dβ,  (C24)  where dβ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Minimally coupling the background field  such that the theory is gauge invariant, the action be-  comes  S[A] = − 1  2g2  �  X  |Fa − A|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C25)  The conserved current J corresponding to the symmetry  will minimally couple to A as the term  �  A∧ ∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ex-  panding out the terms in S[A], we find that  J = 1  g2 Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C26)  The requirement that  �  A∧ ∗ J is gauge invariant enforces  the expected conservation law d ∗ J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, the  charge operator Q ≡  �  ∗ J is supported on a (D − p − 1)-  dimensional closed surface Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The symmetry transforma-  tion Uα = eiαQ is  Uα (Σ) = exp  �  iα  �  Σ  ∗ Fa  g2  �  ,  (C27)  where α ∈ [0, 2π) parametrizes the U(1) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Uα(Σ) is a topological operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In fact, any operator  of the form  U(Σ) = exp  �  i  �  Σ  η  �  ,  where dη = 0, is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Indeed, let’s continuously deform  Σ → Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Given that σ is homotopically equivalent to Σ′,  the surfaces differ by a boundary and thus we can write  Σ′ = Σ + ∂δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Using this, we find that:  U(Σ + ∂δ) = exp  �  iα  �  Σ+∂δ  η  �  ,  = exp  �  iα  ��  Σ  η +  �  ∂δ  η  ��  ,  = exp  �  �iα  �  �  �  Σ  η +  �  δ  dη  ����  =0  �  �  �  �,  = exp  �  iα  �  Σ  η  �  ,  = U(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C28)  So, number of transformations is given by number of ho-  motopic equivalence classes for nontrivial contractible p-  dimensional surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s now see how Uα acts on Wa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' First, note that  the closed manifold Σ ⊂ M, where M denotes a closed  codimension-1 submanifold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', a fixed imaginary-time  37  slice of Euclidean spacetime).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Let the p-form ˆΣ be the  Poincar´e dual of Σ with respect to M so the symmetry  operator can be written as Uα (Σ) = exp  �  iα  g2  �  M ∗ Fa∧ˆΣ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because Σ is closed, the Poincar´e dual satisfies dˆΣ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From the Baker–Campbell–Hausdorff formula, the Wil-  son operator transforms as  Uα (Σ) Wa (C) U †  α (Σ) = e  α  g2 [  �  M ∗ Fa∧ˆΣ,  �  C a]Wa (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From canonical commutation relations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C21), the  commutator in the exponential is  ��  M  ∗ Fa ∧ ˆΣ,  �  C  a  �  = ig2  �  C  ˆΣ,  (C29)  and thus we find  Uα (Σ) Wa (C) U †  α (Σ) = exp  �  iα  �  C  ˆΣ  �  Wa (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C30)  Comparing this to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C23), we identify Γ ≡ αˆΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Nev-  ertheless, we find that q =  �  C ˆΣ is the charge of the  operator Wa (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We can rewrite the phase noting that  �  C ˆΣ =  �  Σ∩C 1, which is the intersection number between  Σ and C in M: # (Σ, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore q = # (Σ, C) ∈ Z  and the U(1)(p) symmetry transformation is  Uα (Σ) Wa (C) U †  α (Σ) = exp [iα # (Σ, C)] Wa (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C31)  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Abelian duality  To make the other symmetry manifest we must effec-  tively change the representation of our degrees of free-  dom using Abelian duality [86] to dualize the field a to  the field ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  This is a particle-vortex duality since, as  we will see, the sources (topological defects) of a corre-  spond to the topological defects (sources) of ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Abelian  duality is further useful since it maps the strong coupling  limit in the a-representation to a weak coupling limit ˆa-  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s now dualize (change representation from a to  ˆa) the quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Instead of integrating over the  equivalence classes of a and summing over ωa, we can in-  stead integrate over Fa since the action only depends on  Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, in doing so we have to ensure that we only  integrate over Fa which satisfy the Bianchi identity30  1  2π ∗ dFa = 0,  (C32)  30 If there were topological defects described by a current  ˆ  J ,  then the Bianchi identity would be modified to  1  2π ∗ dFa = ˆ  J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The presence of topological defects would be captured by the  field strength globally being give by Fa = db + ωa + d†βa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The  periods of Fa would be  �  Fa =  �  (ωa + d†βa) ∈ 2πZ and now  dFa = d(d†βa) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  and which obey the quantization condition  �  Cp+1  Fa ∈ 2πZ,  (C33)  for all Cp+1 ∈ Hp+1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, with these two constraints  in mind we can change variables and write the Euclidean  path integral as  Z =  �  DFa δ  �∗ dFa  2π  �  δ  �� Fa  2π ∈ Z  �  e−  �  X L,  L =  1  2g2 |Fa|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C34)  Let’s now rewrite the functional delta functions by in-  tegrating in fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Firstly, introducing the (D − p − 2)-  form ˆa, we write  δ  �∗ dFa  2π  �  =  �  Dˆa exp  �  i  �  X  ˆa ∧ ∗  �∗ dFa  2π  ��  ,  =  �  Dˆa exp  � i  2π  �  X  dˆa ∧ Fa  �  ,  (C35)  where we absorbed any minus signs from the ∗ ∗ and inte-  grating by parts into ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As for the second delta function,  we can rewrite it as  δ  �� Fa  2π ∈ Z  �  =  �  ˆωˆa∈Hp+1(X)  exp  �  2πi  ��  ˆωˆa  Fa  2π  ��  ,  =  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  exp  � i  2π  �  X  ωˆa ∧ Fa  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C36)  We start off by summing over all closed (p + 1)-  submanifold ˆωˆa, and then using Poincar´e duality we  change  that  sum  to  a  sum  over  all  dual  closed  (D − p − 1)-forms ωˆa/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Because we sum over ˆωˆa  with integer coefficients, ωˆa satisfies the quantization  condition  �  ωˆa ∈ 2πZ (the factor of 2π is for conve-  nience).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus the product of the delta functions give  δ  �∗ dFa  2π  �  δ  �� Fa  2π ∈ Z  �  =  �  Dˆa  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  exp  � i  2π  �  X  (dˆa + ωˆa) ∧ Fa  �  =  �  Dˆa  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  exp  � i  2π  �  X  Fˆa ∧ Fa  �  ,  (C37)  where we introduced Fˆa = dˆa + ωˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Note that from the  quantization condition  �  ωˆa ∈ 2πZ, we have that  �  Fˆa ∈ 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C38)  Furthermore, because ωˆa is closed, Fˆa also satisfies a  Bianchi identity  1  2π ∗ dFˆa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C39)  38  Returning back to the Euclidean path integral (C34),  using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C37) it becomes  Z[X, g] =  �  DFaDˆa  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e−  �  X L,  L[Fa, Fˆa;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' g] =  1  2g2 |Fa|2 − i  2π Fˆa ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C40)  Let’s  now  complete  the  square,  and  introducing  G = Fa − i g2  2π ∗ Fˆa, the path integral becomes  Z =  �  DG Dˆa  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e−  �  X L,  L =  1  2g2 |G|2 + g2  8π2 |Fˆa|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C41)  We can now integrate out G and arrive at the path inte-  gral only in terms of the dual field ˆa:  Z[X, g] =  �  D[ˆa]  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  exp  �  − g2  8π2  �  X  |Fˆa|2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C42)  Remarkably, this theory has the same form as what  we started with, expect now that initial p-form a is a  (D − p − 2)-form ˆa and the coupling constant g is now  2π/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, strongly coupling (g ≫ 1) in the a represen-  tation gets mapped to weak coupling in the ˆa represen-  tation, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Having gone through the process of dualizing a to ˆa,  let’s now see how operators in terms of a transform un-  der dualizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We introduce the map S which takes an  operator in the a representation to the ˆa representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s first check to see what the field strength Fa maps  to by inserting Fa into the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This is quite  easy to find, noting that when completing the square we  did a change of variables Fa = G + i g2  2π ∗ Fˆa, the insertion  becomes  ⟨Fa⟩a = ⟨G⟩ˆa +  �  i g2  2π ∗ Fˆa  �  ˆa  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C43)  We use the notation that ⟨·⟩a is the vev evaluated in  the a-representation and ⟨·⟩ˆa is the vev evaluated in the  ˆa-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Since the G part of the action is Gaus-  sian, ⟨G⟩ = 0 and so ⟨Fa⟩ = ⟨i g2  2π ∗ Fˆa⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, we find  that in Euclidean signature  S : Fa �→ i g2  2π ∗ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C44)  In the Lorentzian signature, this would of course become  S : Fa �→ g2  2π ∗ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We can dualizing ˆa back to a by simply  repeating the same steps as before to find  S : Fˆa �→ i 2π  g2 ∗ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C45)  Therefore, Dualizing a twice gives us S2 : Fa �→ i2 ∗ ∗ Fa  which using the expression for ∗ ∗ in Euclidean spacetime  becomes  S2 : Fa �→ (−1)D(p+1)+pFa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C46)  When both D and p are even, dualizing twice acts as  S2 : Fa �→ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, if D or p or both are odd, then  S2 : Fa �→ −Fa and thus one must dualize four times to  get the identity map: S4 : Fa �→ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Repeating the above argument, we find that d†Fa and  ∗ dFa get mapped to ∗ dFˆa and d†Fˆa, respectively, and  vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the excitations (topological defects) of  a are the topological defects (excitations) of ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hence,  abelian duality is a particle-vortex type duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We emphasize that the above mappings does not im-  ply that Fa = i g2  2π ∗ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Instead, while operators linear  in Fa simply have Fa replaced with i g2  2π ∗ Fˆa, more care  is required to find the dual representation of operators  nonlinear in Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For instance, (Fa)2 does not become  (i g2  2π ∗ Fˆa)2 due to the addition terms pick up when squar-  ing Fa = G + i g2  2π ∗ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Now let’s see what Wilson operator of a gets mapped  to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We will assume that Wa is supported on a con-  tractible manifold C = ∂M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, there are infinitely  many such M whose boundary is C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' As such, to avoid  this ambiguity we simply sum over all such M:  Wa[C] =  �  M:∂M=C  exp  �  i  �  ∂M  a  �  ,  =  �  M:∂M=C  exp  �  i  �  M  Fa  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C47)  Let’s now introduce  ˆ  M, the Poincar´e dual of M with  respect to X, which satisfies  � ˆ  M ∈ 2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The Poincar´e  dual ˆC is related to ˆ  M by ˆC = d ˆ  M/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, using  Poincar´e duality, we rewrite Wa as  Wa[C] =  �  D ˆ  M δ(d ˆ  M − 2π ˆC) exp  � i  2π  �  X  Fa ∧ ˆ  M  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Inserting this into the path integral after the dual field  strength ˆa has been integrated in gives  Z =  �  DFaD ˆ  MD[ˆa]  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  δ(d ˆ  M − 2π ˆC) e−  �  X L  L =  1  2g2 |Fa|2 + i  2π (Fˆa − ˆ  M) ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C48)  Now integrating out Fa, the resulting theory is  Z =  �  D ˆ  MD[ˆa]  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  δ(d ˆ  M − 2π ˆC) e−  �  X L,  L = g2  8π2 |Fˆa − ˆ  M|2  (C49)  39  Note that this can be written as  Z =  �  D[ˆa]  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  T[C] e−  �  X L,  T =  �  D ˆ  M δ(d ˆ  M − 2π ˆC) e−  �  X  g2  8π2 [−2Fˆa∧ ∗ ˆ  M+| ˆ  M|2],  L = g2  8π2 |Fˆa|2  We thus see that a contractible Wilson loop dualizes to  T, which is a rather cumbersome expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The Path integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C49) is the path integral  we found without the Wilson loop insertion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C42)  but now with the connection ˆ  M which is restrained to  d ˆ  M = 2π ˆC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  We can get ride of the  ˆ  M connection in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C49) and have the path integral look similar to  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C42) if we let ˆa be singular field not defined on  C:  Z =  �  D[ˆa]  �  ωˆa∈2πHD−p−1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Z)  e−  �  X/C L,  L = g2  8π2 |Fˆa|2,  (C50)  and with  �  σ Fˆa = 2π for any open submanifold σ which  have a nonzero intersection number with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore,  the Wilson loop in the a representation has become a ’t  Hooft loop in the ˆa representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  U(1)(d−p−1) symmetry  When the theory is in the a representation, it would ap-  pear that the model only has a U(1)(p) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' How-  ever, upon dualizing a to ˆa, the path integral Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C42)  took a similar form in terms of Fˆa but with g replaced by  2π/g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, we find that there is a new globally defined  differential (D − p − 2)-form ˆa which shifting by a closed  form leaves the path integral invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Following the  same process as used in investigating the U(1)(p) sym-  metry, this transformation has a physical part associated  with a U(1)(D−p−2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Everything about this U(1)(D−p−2) symmetry follows  in a similar fashion from the U(1)(p) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In particular,  the symmetry transformation acts on ˆa as  ˆa �→ ˆa + ˆΓ  where  dˆΓ = 0  (C51)  and the charged operators are Wilson operators in terms  of ˆa  Wˆa (C) = exp  �  i  �  C  ˆa  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C52)  These correspond to the ’t Hooft operators in the a rep-  resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The Noether’s current associated with this  U(1)(D−p−2) symmetry is  ˆJ = g2  4π2 Fˆa,  (C53)  and thus the symmetry operator is  ˆUˆα (Σ) = exp  �  i ˆα g2  4π2  �  Σ  ∗ Fˆa  �  ,  (C54)  where ˆα ∈ [0, 2π) parametrizes the U(1) transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  To see what ˆU is in the a representation, let’s start  with the operator exp  �  iθ  �  Σ Fa  �  and find its image under  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We’ve already done this calculation while finding the  image of the Wilson operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This time, we simply do  not sum over all Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' We therefore have that (in Lorentzian  signature)  S : eiθ  �  Σ Fa �→ eiθ  �  Σ  g2  2π ∗ Fˆa−i θ2g2  2  �  X |ˆΣ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C55)  Setting θ =  ˆα  2π, the U(1)(D−p−2) symmetry operator in  the a-representation is  ˆUˆα(Σ) = exp  �  i ˆα  �  Σ  Fa  2π + i ˆα2g2  8π2  �  X  |ˆΣ|2  �  (C56)  since under S it transforms to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, note  that the term  �  X |ˆΣ|2 is an overall phase and therefore  does not affect the symmetry transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So, we  can drop this overall phase and treat the U(1)(D−p−2)  symmetry operator in the a-representation instead as  ˆUˆα(Σ) = exp  �  i ˆα  �  Σ  Fa  2π  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C57)  From this expression, it is easy to see that the Noether  current of the U(1)(D−p−2) symmetry in the a represen-  tation ˆJ satisfies  ∗ ˆJ = 1  2π Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C58)  The fact the ˆJ is conserved is simply a reflection of the  Bianchi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The charged objects under the U(1)(D−p−2) symmetry  are the Wilson loops of ˆa, or equivalently the ’t Hooft  loops of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To see so in the ˆa representation is straight  forward and follows exactly as how we found that the  Wilson loops of a were charged operators of U(1)(p) in  the a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' So, let’s instead see that the Wilson  loops of ˆa are charged operators of U(1)(D−p−2) in the a  representation  Let’s denote the ’t Hooft loop of a supported on the  closed (D − p − 2) submanifold C as Ta(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A part of  the definition of Ta(C) is that in it’s presence  �  Σ Fˆb = 2π  for any open submanifold Σ which have a nonzero inter-  section number with C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore  ˆUˆα(σ)Ta(C) ˆU †  ˆα(σ) = exp [i ˆα # (Σ, C)] Ta (C) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C59)  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Mixed ’t Hooft anomaly and anomaly inflow  In the last section, we reviewed that p-form Maxwell  theory has a U(1)(p) and a U(1)(d−p−1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' How-  ever, it turns out that these two symmetries are not fully  40  independent from one another: there is a mixed ’t Hooft  anomaly preventing us from simultaneously turning on a  background gauge field of both symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The main idea of gauging a symmetry is to add addi-  tional degrees of freedom such that the theory is invari-  ant under the symmetry operator even when it is sup-  ported on open submanifolds31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' For instance, let’s start  off in the in the a representation and gauge the U(1)(p)  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The U(1)(p) symmetry operator Uα(Σ) (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C27)) is supported on the closed submanifold Σ,  and acting it on a shifted a by ˆΣ, the Poincar´e dual of  Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Because Σ is a closed submanifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', ∂Σ = ∅), the  Poincar´e dual ˆΣ is a closed form (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', dˆΣ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus,  gauging the symmetry requires the theory to be invari-  ant under Uα(σ) for ∂Σ ̸= ∅, and therefore invariant un-  der shifting a by a ˆΣ such that dˆΣ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s first turn on a background field A of the U(1)(p)  symmetry, which satisfies  �  FA ∈ 2πZ and dFA = 0, and  includes the gauge redundancy  a �→ a + β,  A �→ A + dβ,  (C60)  where dβ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  If we were to gauge the symmetry, A  would be a dynamical field rather than a background  gauge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Nevertheless, minimally coupling A to a,  the path integral becomes  Z[A] =  �  D[a] e−  �  X L,  L =  1  2g2 |Fa − A|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C61)  We now dualize a to ˆa to see how A couples to ˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Re-  peating the first steps of abelian duality is reviewed in  section C 1 b, we can rewrite the path integral as  Z[A] =  �  DFaD[ˆa] e−  �  X L,  L =  1  2g2 |Fa − A|2 − i  2π Fˆa ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C62)  In the A = 0 theory, at this step in dualizing we com-  pleted the square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' But instead, let’s first make the change  of variables Fa = K + A such that the path integral be-  comes  Z[A] =  �  DKD[ˆa] e−  �  X L,  L =  1  2g2 |K|2 − i  2π Fˆa ∧ K − i  2π A ∧ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C63)  31 This can be thought of as a generalization of the 0-form sym-  metry case, where gauging a symmetry is typically thought of  as requiring the theory to be invariant under local symmetry  transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The two first terms in L that depend on K are the same  two terms we encountered when dualizing the theory pre-  viously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The A dependency is entirely in the third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Since it does not include K, upon integrating out K we  will arrive at the same dual theory as before plus this  third term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Therefore, in the ˆa representation, A mini-  mally couples as  Z[A] =  �  D[ˆa] e−  �  X L,  L = g2  8π2 |Fˆa|2 − i  2π A ∧ Fˆa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C64)  Note that because dFˆa = 0 and spacetime is closed, the  path integral in the ˆa-representation is still invariant un-  der the gauge transformation Eq (C60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C64) reveals that coupling a U(1)(p) symmetry  background gauge field is equivalent to adding a topologi-  cal term in the ˆa representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' This new term has a no-  ticeable effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The Noether current for the U(1)(D−p−2)  symmetry, ˆJ =  g2  4π2 Fˆa, is no longer conserved:  d† ˆJ = 1  2π ∗ dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C65)  This is a manifestation of the mixed ’t Hooft anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let’s not turn off A but turn on a background gauge  field for the U(1)(D−p−2) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  From our above  discussion, this means that we introduce a background  field ˆ  A and the gauge redundancy  ˆa �→ ˆa + ˆβ,  ˆ  A �→ ˆ  A + dˆβ,  (C66)  where dˆβ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Inspired by how A coupled to ˆa in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C63), we minimally couple ˆ  A to a in a similar fash-  ion and consider  Z[ ˆ  A] =  �  D[a] e−  �  X L,  L = 1  2g2 |Fa|2 − i  2π  ˆ  A ∧ Fa,  (C67)  Note that because Fa closed, shifting ˆ  A by an exact form  does not change the path integral and thus this is cor-  rectly gauge invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To check if this way of coupling  ˆ  A to a is correct, we note that we expect in the ˆa repre-  sentation that ˆ  A couples to ˆa in the form Fˆa − ˆ  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Let’s  check this by dualizing a to ˆa in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Repeating  the first steps of abelian duality, we can rewrite the path  integral as  Z[ ˆ  A] =  �  DFaD[ˆa] e−  �  X L,  L =  1  2g2 |Fa|2 − i  2π (Fˆa − ˆ  A) ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C68)  which integrating out Fa yields  Z[ ˆ  A] =  �  D[ˆa]D[ ˆ  A] e−  �  X L,  L = g2  8π2 |Fˆa − ˆ  A|2,  (C69)  41  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Returning back to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C67), due to the new  topological term, the U(1)(p) symmetry Noether current  J =  1  g2 Fa is no longer conserved:  d†J = 1  2π ∗ d ˆ  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C70)  Once again, this is a manifestation of the mixed ’t Hooft  anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Let us now turn on both of the background gauge fields  A and  ˆ  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Working in the a representation and using  what we just found, the path integral becomes  Z[A, ˆ  A] =  �  D[a] e−  �  X L,  L =  1  2g2 |Fa − A|2 − i  2π  ˆ  A ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C71)  This path integral is invariant under the gauge transfor-  mation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, due to the second term in L,  this path integral is no longer invariant under the gauge  transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C60), and transforms as  Z[A, ˆ  A] �→ Z[A, ˆ  A] exp  � i  2π  �  X  ˆ  A ∧ dβ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C72)  One could add a local-counter term to remedy this, and  the path integral becomes  Z[A, ˆ  A] =  �  D[a] e−  �  X L,  L =  1  2g2 |Fa − A|2 − i  2π  ˆ  A ∧ (Fa − A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C73)  This is indeed now invariant under the gauge transforma-  tion Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' However, it is no longer invariant under  the gauge transformation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C66), transforming as  Z[A, ˆ  A] �→ Z[A, ˆ  A] exp  �  − i  2π  �  X  dˆβ ∧ A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C74)  In fact, there are no local counter terms which will make  the path integral gauge invariant when both A and ˆ  A are  turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thus, the U(1)(p) × U(1)(d−p−1) symmetry is  anomalous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  If the theory is going to be gauge invariant, we must do  something more drastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Indeed, we can make the theory  invariant under the gauge transformations by introducing  the (D + 1)-dimensional spacetime Y such that X = ∂Y  and extending the background gauge fields A and ˆ  A into  Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' To get a hint as to why, let’s return to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C72)  and note that using Stoke’s theorem we can rewrite the  integral in the exponential as  �  X=∂Y  ˆ  A ∧ dβ =  �  Y  d( ˆ  A ∧ dβ) =  �  Y  d ˆ  A ∧ dβ  (C75)  This then motivates the new gauge invariant partition  function  Z[A, ˆ  A] = exp  �  − i  2π  �  Y  d ˆ  A ∧ A  � �  D[a] e−  �  ∂Y L,  L =  1  2g2 |Fa − A|2 − i  2π  ˆ  A ∧ Fa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C76)  Indeed, Z[A, ˆ  A] is invariant under the gauge trans-  formations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' (C72) since the phase picked up from  �  D[ˆa] e−  �  ∂Y L cancels with the phase we added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Thus, we see the ’t Hooft anomaly through the mod-  ern prospective of anomaly inflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' The theory Lboundary  with A and ˆ  A was not well defined in D-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In-  stead, in order to turn on both A and  ˆ  A, it costs us  having the theory Lboundary resides on the boundary of  an invertible theory Lbulk in (D + 1)-dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  The  invertible theory is [87]  Zinv[A, ˆ  A] = exp  �  − i  2π  �  Y  d ˆ  A ∧ A  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  (C77)  We remark that in the above discussion, we have as-  sumed that dFa = 0 and dFˆa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' In other words, we  have assumed the absence of the electric and magnetic-  branes in the path integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  So the above results are  valid only at energies below the energy scales of these  excitations where we have an exact emergent U(1)(p) ×  U(1)(d−p−1) symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [1] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Nussinov and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ortiz, Sufficient symmetry con-  ditions for topological quantum order, Proceedings of  the National Academy of Sciences 106, 16944 (2009),  arXiv:cond-mat/0605316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gaiotto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kapustin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Willett, Gen-  eralized global symmetries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2015  (2), 172, arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='5148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thorngren and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wang, Fusion Category Symme-  try I: Anomaly In-Flow and Gapped Phases,  (2019),  arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [4] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zhang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zheng,  Algebraic higher symmetry and categorical symmetry: A  holographic and entanglement view of symmetry, Physi-  cal Review Research 2, 043086 (2020), arXiv:2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='14178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [5] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lootens, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fuchs, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Haegeman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Schweigert, and  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Verstraete, Matrix product operator symmetries and  intertwiners in string-nets with domain walls, SciPost  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 10, 053 (2021), arXiv:2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='11187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [6] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' McGreevy, Generalized Symmetries in Condensed  Matter, (2022), arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='03045.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' C´ordova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Dumitrescu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Intriligator, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao, Snowmass White Paper: Generalized Symme-  tries in Quantum Field Theory and Beyond,  (2022),  arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='09545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Freed, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Moore, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Teleman, Topo-  logical symmetry in quantum field theory,  (2022),  42  arXiv:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='07471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [9] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Topological orders in rigid states, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 04, 239 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [10] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Landau, Theory of phase transformations I, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sowjetunion 11, 26 (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [11] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ginzburg and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Landau, On the theory of su-  perconductivity, Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 20, 1064 (1950).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [12] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Nussinov and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ortiz, A symmetry principle for  topological quantum order, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 324, 977 (2009),  arXiv:cond-mat/0702377.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Emergent (anomalous) higher symmetries  from topological order and from dynamical electromag-  netic field in condensed matter systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B  99, 205139 (2019), arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [14] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hsin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lam, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, Comments on  one-form global symmetries and their gauging in 3d and  4d, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 6, 039 (2019), arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [15] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Qi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Radzihovsky, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hermele, Fracton phases  via exotic higher-form symmetry-breaking, Annals of  Physics 424, 168360 (2021), arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [16] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kaidi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Komargodski, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ohmori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seifnashri,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao, Higher central charges and topological  boundaries in 2+1-dimensional TQFTs, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  13, 067 (2022), arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='13091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [17] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zheng, Categories of quantum liquids I,  (2020), arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zheng, Categories of quantum liquids II,  (2021), arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='03858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zheng, Categories of quantum liquids  III, (2022), arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='05726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Chatterjee and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Holographic theory for  the emergence and the symmetry protection of gapless-  ness and for continuous phase transitions, arXiv e-prints  (2022), arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='06244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [21] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Senthil, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wang, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Witten, A duality  web in 2 + 1 dimensions and condensed matter physics,  Annals of Physics 374, 395 (2016), arXiv:1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='01989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [22] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Senthil, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Son, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wang, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Xu, Duality be-  tween (2 + 1) d quantum critical points, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 827,  1 (2019), arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='05174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lootens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Delcamp, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ortiz, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Verstraete,  Category-theoretic recipe for dualities in one-dimensional  quantum lattice models, (2021), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='09091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sharpe, Notes on generalized global symmetries in  QFT, Fortschr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 63, 659 (2015), arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04770.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [25] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Benini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' C´ordova, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hsin, On 2-group global  symmetries and their anomalies, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  2019 (3), 118, arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='09336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' C´ordova, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Dumitrescu, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Intriligator, Ex-  ploring 2-group global symmetries, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  2019 (2), 184, arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [27] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kovner and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rosenstein, New look at QED4: the  photon as a Goldstone boson and the topological in-  terpretation of electric charge, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D 49, 5571  (1994), arXiv:hep-th/9210154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [28] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lake, Higher-form symmetries and spontaneous sym-  metry breaking, (2018), arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='07747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [29] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hofman and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Iqbal, Goldstone modes and pho-  tonization for higher form symmetries, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 6,  006 (2019), arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='09512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [30] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kim and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hirono, Higher-form symmetries and d-  wave superconductors from doped Mott insulators, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 100, 085142 (2019), arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Hidaka,  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Hirono,  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Yokokura,  Count-  ing Nambu-Goldstone Modes of Higher-Form Global  Symmetries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 126, 071601 (2021),  arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='15901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [32] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Yamamoto and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Yokokura, Topological mass gen-  eration in gapless systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D 104, 025010  (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='07621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [33] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Iqbal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' McGreevy, Mean string field theory:  Landau-Ginzburg theory for 1-form symmetries, SciPost  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 13, 114 (2022), arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='12610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [34] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kapustin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thorngren, Higher symmetry and  gapped phases of gauge theories, in Algebra, Geome-  try, and Physics in the 21st Century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Progress in Math-  ematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Progress in Mathematics, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 324, edited by  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Auroux, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Katzarkov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pantev, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Soibelman, and  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Tschinkel (Birkh¨auser, Cham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=', 2017) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 177–202,  arXiv:1309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='4721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [35] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thorngren and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' von Keyserlingk, Higher SPT’s  and  a  generalization  of  anomaly  in-flow,  (2015),  arXiv:1511.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02929.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [36] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Yoshida,  Topological  phases  with  generalized  global symmetries, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 93, 155131 (2016),  arXiv:1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='03468.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [37] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Gaiotto,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Kapustin,  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Komargodski,  and  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, Theta,  time reversal and temperature,  Journal  of  High  Energy  Physics  2017,  1  (2017),  arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='00501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [38] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shiozaki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kikuchi, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ryu, Lieb-  Schultz-mattis type theorem with higher-form symmetry  and the quantum dimer models, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 99, 014402  (2019), arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='05367.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [39] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wang, Non-Abelian Gauge Theories,  Sigma Models,  Higher Anomalies,  Symmetries,  and  Cobordisms, Annals of Mathematical Sciences and Ap-  plications 4, 107 (2019), arXiv:1812.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='11967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [40] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Tsui and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Lattice models that realize zn-1-  symmetry protected topological states for even n, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 101, 035101 (2020), arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [41] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Jian, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Xu, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Xu, Physics of  Symmetry Protected Topological phases involving Higher  Symmetries and their Applications, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 103,  064426 (2021), arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='00023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [42] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Moy,  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Goldman,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sohal, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fradkin,  Theory  of  oblique  topological  insulators,  (2022),  arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='07725.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [43] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pace and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Emergent Higher-Symmetry  Protected Topological Orders in the Confined Phase of  U(1) Gauge Theory, (2022), arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='03544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [44] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Weinberg, Approximate Symmetries and Pseudo-  Goldstone Bosons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 29, 1698 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [45] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Foerster, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Nielsen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ninomiya, Dynamical  stability of local gauge symmetry creation of light from  chaos, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 94, 135 (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [46] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hastings and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Quasiadiabatic con-  tinuation of quantum states: The stability of topolog-  ical ground-state degeneracy and emergent gauge in-  variance, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 72, 045141 (2005), arXiv:cond-  mat/0503554.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [47] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Komargodski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sharon, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thorngren, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zhou,  Comments on abelian Higgs models and persistent order,  SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 6, 003 (2019), arXiv:1705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [48] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' C´ordova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ohmori, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rudelius, General-  ized symmetry breaking scales and weak gravity conjec-  tures, Journal of High Energy Physics 11, 154 (2022),  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='05866.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  43  [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Verresen,  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Borla,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Vishwanath,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Moroz,  and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thorngren, Higgs Condensates are Symmetry-  Protected Topological Phases:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Discrete Symmetries  (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='01376.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [50] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ji and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Categorical symmetry and nonin-  vertible anomaly in symmetry-breaking and topological  phase transitions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Research 2, 033417 (2020),  arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='13492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [51] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zhang, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Zheng,  Classification of topological phases with finite internal  symmetries in all dimensions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2020  (9), 93, arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='08898.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [52] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Chatterjee and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Algebra of local symmetric  operators and braided fusion n-category – symmetry is a  shadow of topological order, (2022), arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='03596.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [53] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Laughlin and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pines, The Theory of Everything,  Proceedings of the National Academy of Sciences 97, 28  (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [54] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao, Exotic symmetries, duality,  and fractons in 2+1-dimensional quantum field theory,  SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 10, 027 (2021), arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='10466.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [55] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gorantla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao,  Low-energy limit of some exotic lattice theories and  UV/IR mixing, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 104, 235116 (2021),  arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='00020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [56] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gorantla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao,  Global dipole symmetry, compact Lifshitz theory, tensor  gauge theory, and fractons, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 106, 045112  (2022), arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='10589.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [57] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pace and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Position-dependent excita-  tions and UV/IR mixing in the ZN rank-2 toric code  and its low-energy effective field theory, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B  106, 045145 (2022), arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='07111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [58] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Bravyi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hastings, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Michalakis, Topo-  logical quantum order: stability under local perturba-  tions, Journal of mathematical physics 51, 093512 (2010),  arXiv:1001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='0344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [59] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Apruzzi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Bonetti, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Garc´ıa Etxebarria, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hos-  seini, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Schafer-Nameki, Symmetry TFTs from  String Theory, arXiv e-prints , arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02092 (2021),  arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02092.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [60] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Ji and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Non-invertible anomalies and  mapping-class-group transformation of anomalous par-  tition functions, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Research 1, 033054 (2019),  arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='13279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [61] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Wen,  Classifying  gauge  anomalies  through  symmetry-protected trivial orders and classifying gravi-  tational anomalies through topological orders, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  D 88, 045013 (2013), arXiv:1303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [62] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Braided fusion categories,  gravitational anomalies, and the mathematical frame-  work for topological orders in any dimensions,  (2014),  arXiv:1405.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='5858.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [63] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kitaev and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kong, Models for gapped bound-  aries and domain walls, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 313, 351  (2012), arXiv:1104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='5047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [64] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Gapless boundary excitations in the quantum  hall states and in the chiral spin states, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 43,  11025 (1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [65] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Delacr´etaz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hofman, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Mathys, Superfluids  as higher-form anomalies, SciPost Physics 8, 047 (2020),  arXiv:1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='06977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [66] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kapustin and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, Coupling a QFT to a  TQFT and duality, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2014 (4), 1,  arXiv:1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='0740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [67] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Iqbal and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' McGreevy, Toward a 3d Ising model with  a weakly-coupled string theory dual, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 9, 019  (2020), arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [68] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gorantla, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Lam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Shao, A  modified Villain formulation of fractons and other exotic  theories, Journal of Mathematical Physics 62, 102301  (2021), arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='01257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [69] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Cheng and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, Lieb-Schultz-Mattis, Lut-  tinger, and ’t Hooft – anomaly matching in lattice sys-  tems (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='12543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [70] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Fazza and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sulejmanpasic, Lattice Quantum Vil-  lain Hamiltonians: Compact scalars, U(1) gauge theo-  ries, fracton models and Quantum Ising model dualities  (2022), arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='13047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [71] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hsin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Turzillo, Symmetry-enriched quantum  spin liquids in (3 + 1)d, Journal of High Energy Physics  2020, 1 (2020), arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='11550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [72] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Levin and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, String-net condensation: A  physical mechanism for topological phases, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B  71, 045110 (2005), arXiv:cond-mat/0404617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [73] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Liu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Pace, (unpublished).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [74] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Liu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, Symmetry  protected topological orders and the group cohomology of  their symmetry group, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 87, 155114 (2013),  arXiv:1106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='4772.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [75] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Callan Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Harvey, Anomalies and fermion  zero modes on strings and domain walls, Nuclear Physics  B 250, 427 (1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [76] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hubaˇc and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wilson, Brillouin-Wigner Perturbation  Theory, in Brillouin-Wigner Methods for Many-Body  Systems (Springer, 2010) pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 37–68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [77] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Bravyi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' DiVincenzo, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Loss, Schrieffer-Wolff  transformation for quantum many-body systems, Annals  of physics 326, 2793 (2011), arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='0675v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [78] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sulejmanpasic and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Gattringer, Abelian gauge the-  ories on the lattice: θ-Terms and compact gauge the-  ory with(out) monopoles, Nuclear Physics B 943, 114616  (2019), arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='02637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [79] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wang, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Wen, (3 + 1)d boundaries  with gravitational anomaly of (4+1)d invertible topolog-  ical order for branch-independent bosonic systems, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 106, 045127 (2022), arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='12148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [80] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Witten, Quantum field theory and the Jones polyno-  mial, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 121, 351 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [81] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Slagle and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Kim, Quantum field theory of  X-cube fracton topological order and robust degener-  acy from geometry, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' B 96, 195139 (2017),  arXiv:1708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04619.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [82] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Maldacena, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Moore, D-brane charges  in five-brane backgrounds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 2001  (10), 005, hep-th/0108152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [83] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hansson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Oganesyan, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Sondhi, Supercon-  ductors are topologically ordered, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' 313, 497  (2004), arXiv:cond-mat/0404327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [84] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Banks and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Seiberg, Symmetries and strings in field  theory and gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' D 83, 084019 (2011),  arXiv:1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='5120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [85] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Mathieu and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Thuillier, Abelian BF theory and  Turaev-Viro invariant, Journal of Mathematical Physics  57, 022306 (2016), arXiv:1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='04236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [86] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Witten, On S-duality in Abelian Gauge Theory, Se-  lecta Mathematica 1, 383 (1995), arXiv:hep-th/9505186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='  [87] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Hsieh, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Tachikawa, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content=' Yonekura, Anomaly  44  inflow  and  p-form  gauge  theories,  Communica-  tions  in  Mathematical  Physics  391,  495  (2022),  arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
+page_content='11550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydE4T4oBgHgl3EQfxw3J/content/2301.05261v1.pdf'}
diff --git a/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/2301.00017v1.pdf.txt b/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/2301.00017v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c6c790cecbd7be46de822caee41cf2331a670ac8
--- /dev/null
+++ b/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/2301.00017v1.pdf.txt
@@ -0,0 +1,858 @@
+Draft version January 3, 2023
+Typeset using LATEX twocolumn style in AASTeX61
+BROAD EMISSION LINES IN OPTICAL SPECTRA OF HOT DUST-OBSCURED GALAXIES CAN
+CONTRIBUTE SIGNIFICANTLY TO JWST/NIRCAM PHOTOMETRY
+Jed McKinney,1 Luke Finnerty,2 Caitlin M. Casey,1 Maximilien Franco,1 Arianna S. Long,1, ∗
+Seiji Fujimoto,1, 3, 4, ∗ Jorge A. Zavala,5 Olivia Cooper,1 Hollis Akins,1 Alexandra Pope,6 Lee Armus,7
+B. T. Soifer,8 Kirsten Larson,9 Keith Matthews,8 Jason Melbourne,8 and Michael Cushing10
+1Department of Astronomy, The University of Texas at Austin, 2515 Speedway Blvd Stop C1400, Austin, TX 78712, USA
+2Department of Physics & Astronomy, 430 Portola Plaza, University of California, Los Angeles, CA 90095, USA
+3Cosmic Dawn Center (DAWN), Jagtvej 128, DK2200 Copenhagen N, Denmark
+4Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK2100 Copenhagen Ø, Denmark
+5National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
+6Department of Astronomy, University of Massachusetts, Amherst, MA 01003, USA.
+7IPAC, California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
+8Division of Physics, Math, and Astronomy, California Institute of Technology, 1200 E California Blvd., Pasadena, CA 91125, USA
+9AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+10Department of Physics and Astronomy, University of Toledo, 2801 West Bancroft St., Toledo, OH 43606, USA
+ABSTRACT
+Selecting the first galaxies at z > 7 − 10 from JWST surveys is complicated by z < 6 contaminants with degenerate
+photometry. For example, strong optical nebular emission lines at z < 6 may mimic JWST/NIRCam photometry of
+z > 7 − 10 Lyman Break Galaxies (LBGs). Dust-obscured 3 < z < 6 galaxies in particular are potentially important
+contaminants, and their faint rest-optical spectra have been historically difficult to observe. A lack of optical emission
+line and continuum measures for 3 < z < 6 dusty galaxies now makes it difficult to test their expected JWST/NIRCam
+photometry for degenerate solutions with NIRCam dropouts. Towards this end, we quantify the contribution by strong
+emission lines to NIRCam photometry in a physically motivated manner by stacking 21 Keck II/NIRES spectra of
+hot, dust-obscured, massive (log M∗/M⊙ ≳ 10 − 11) and infrared (IR) luminous galaxies at z ∼ 1 − 4. We derive an
+average spectrum and measure strong narrow (broad) [OIII]5007 and Hα features with equivalent widths of 130 ± 20 ˚A
+(150±50 ˚A) and 220±30 ˚A (540±80 ˚A) respectively. These features can increase broadband NIRCam fluxes by factors
+of 1.2−1.7 (0.2−0.6 mag). Due to significant dust-attenuation (AV ∼ 6), we find Hα+[NII] to be significantly brighter
+than [OIII]+Hβ, and therefore find that emission-line dominated contaminants of high−z galaxy searches can only
+reproduce moderately blue perceived UV continua of Sλ ∝ λβ with β > −1.5 and z > 4. While there are some redshifts
+(z ∼ 3.75) where our stack is more degenerate with the photometry of z > 10 LBGs between λrest ∼ 0.3 − 0.8 µm,
+redder filter coverage beyond λobs > 3.5 µm and far-IR/sub-mm follow-up may be useful for breaking the degeneracy
+and making a crucial separation between two fairly unconstrained populations, dust-obscured galaxies at z∼ 3 − 6 and
+LBGs at z> 10.
+∗ NASA Hubble Fellow
+arXiv:2301.00017v1  [astro-ph.GA]  30 Dec 2022
+
+2
+1. INTRODUCTION
+A major objective baked into the design of JWST is
+detecting the light from the first galaxies residing at
+ultra-high redshifts (z > 10). Delivering on its promise,
+more than 30 galaxy candidates with photometric red-
+shift solutions favoring z > 10 were identified within the
+first months of publicly available data (Bradley et al.
+2022; Donnan et al. 2022; Naidu et al. 2022a; Harikane
+et al. 2022; Finkelstein et al. 2022a; Yu-Yang Hsiao et al.
+2022). Assessing the fidelity of these samples is criti-
+cal, particularly because the statistics assuming current
+z > 10 − 15 candidates are real may or may not violate
+ΛCDM predictions (Naidu et al. 2022a; Boylan-Kolchin
+2022; Labbe et al. 2022; Maio & Viel 2022).
+Spectroscopic confirmation is needed to verify these
+redshifts.
+However, some early attempts at spectro-
+scopic follow-up using facilities such as ALMA have
+yielded upper limits or tentative low-SNR detections
+(e.g., Bakx et al. 2022; Kaasinen et al. 2022; Yoon et al.
+2022; Fujimoto et al. 2022). JWST/NIRSpec has proven
+capable of spectroscopically detecting the rest-frame op-
+tical light from galaxies up to z ∼ 9 − 10 (Carnall et al.
+2022; Roberts-Borsani et al. 2022), but this might not
+be well-suited for rapidly validating redshifts in statis-
+tical samples.
+A complimentary approach, born from
+similar rest-frame optical colors of z > 10 Lyman Break
+Galaxies (LBGs) and dusty galaxies (Howell et al. 2010;
+Casey et al. 2014), is to use far-IR/sub-mm followup ob-
+servations of cold dust continuum (Zavala et al. 2022)
+and/or far-IR cooling lines (Fujimoto et al. 2022) to
+identify or rule out z < 6 IR-luminous galaxies lurking
+within z > 10 candidate catalogs. Dusty sources have
+posed a problem to the fidelity of high−z galaxy cata-
+logs since selection from Hubble Space Telescope (HST)
+extragalactic deep fields (e.g., Dunlop et al. 2007). HST
+samples at z ∼ 6 − 7 were contaminated by z ∼ 2 dusty,
+star-forming galaxies. Now, z ∼ 3−6 dusty galaxies may
+be contaminating z > 7 − 10 JWST samples. A con-
+tributing factor to this contamination is that both pop-
+ulations have similar, and uncertain, number densities:
+dusty star-forming galaxies at z ∼ 3 − 6 have volume
+number densities n ∼ 10−5 − 10−6 Mpc−3 (Micha�lowski
+et al. 2017; Koprowski et al. 2017; Rowan-Robinson et al.
+2018; Dudzeviˇci¯ut˙e et al. 2020; Gruppioni et al. 2020;
+Manning et al. 2022; Long et al. 2022), similar to early
+measurements of bright z > 10 LBGs (Finkelstein et al.
+2022a; Naidu et al. 2022a; Harikane et al. 2022; Bouwens
+et al. 2022).
+Disentangling these two populations is
+therefore also crucial for reducing uncertainties in their
+respective number densities, which are currently inflated
+by sample purity (e.g., Bouwens et al. 2022).
+Of particular concern within the rest-frame optical/near-
+IR spectra of IR-luminous galaxies is the relative con-
+tribution of strong narrow and broad emission lines
+to broadband filter fluxes, which could mask red con-
+tinuum slopes produced by dust attenuation.
+Strong
+nebular lines can change JWST/NIRCam colors (Za-
+ckrisson et al. 2008; Schaerer & de Barros 2009; Stark
+et al. 2013; Wilkins et al. 2013, 2020, 2022). This may
+be a promising tool for pseudo-spectroscopy of lower
+redshift galaxies using narrow-band filters, but in this
+particular instance is a source of potential population
+confusion for broadband high-redshift surveys. Indeed,
+some of the hottest and most luminous dusty galaxies at
+z > 1 exhibit very high rest-frame optical line equivalent
+widths (EWs) (Finnerty et al. 2020). These arise from
+a combination of low dust-attenuated continuum levels
+with bright lines emergent from less obscured regions,
+as well as ionized outflows driven by Active Galactic
+Nuclei (AGN). To what extent do these strong lines
+contribute to JWST photometry?
+In this letter, we take an empirically-grounded ap-
+proach and quantify the contamination from emission
+lines from hot dust-obscured galaxies to JWST/NIRCam
+photometry. In Section 2 we describe Keck II/NIRES
+observations of a sample of 4 luminous, IR galaxies
+(log LIR/L⊙∼ 12.5) and 17 hot dust, obscured galaxies
+(DOGs, log LIR/L⊙ > 13) between z ∼ 1 − 4, which
+we stack to derive an average optical spectrum (Section
+3). We quantify the contribution of strong and broad
+optical emission lines to NIRCam fluxes in Section 4,
+and discuss their impact on distinguishing between such
+sources at z < 6 and z > 10 LBGs.
+Section 5 sum-
+marizes our main findings.
+Throughout this work we
+adopt a ΛCDM cosmology with Ωm = 0.3, ΩΛ = 0.7
+and H0 = 70 km s−1 Mpc−1.
+2. SAMPLE AND DATA
+Hot DOGs were originally selected from WISE pho-
+tometry as W1W2-dropouts and include the most-
+luminous galaxies in the Universe (Eisenhardt et al.
+2012; Wu et al. 2012). This extreme population is expe-
+riencing a rapid phase of both supermassive black-hole
+and stellar mass assembly (Eisenhardt et al. 2012), and
+are mostly found at z ∼ 2−3 with log LIR/L⊙ ≥ 13 (Wu
+et al. 2012; Assef et al. 2015; Tsai et al. 2015). Most
+hot DOGs exhibit strong ionized outflows in optical
+spectroscopy (Wu et al. 2018; Finnerty et al. 2020; Jun
+et al. 2020), as implied by broad line components likely
+driven by radiative AGN feedback Wu et al. (2018).
+These sources are rare with only ∼ 1000 over the full
+WISE All-sky survey (Cutri & et al. 2012).
+
+3
+(A)
+(B)
+(C)
+ [arbitrary units]
+Fν
+Wavelength [ m]
+μ
+Figure 1. Stacked rest-frame optical spectrum of z ∼ 1 − 4 IR-luminous galaxies detected with Keck II/NIRES. (A) The
+mean-weighted stacked spectrum (black). Shaded grey errors correspond to 16th-84th percentiles on the flux density derived
+from the boostrapped stack distribution per wavelength. The upper panel gives the number of galaxies included in the stack as
+a function of wavelength. On average, 70% (> 14) of the sample is represented between λrest = 0.34 − 0.80 µm. We compare
+against the stacked (N= 20) continuum-normalized DSFG spectrum from Casey et al. (2017) (blue). Panels (B) and (C) show
+zoom-ins on the [O III], Hβ and [N II], Hα, [SII], [OI] features respectively. We measure broad and narrow components as
+expected from the individual spectra (Finnerty et al. 2020). Line fits are shown in purple with solid lines indicating the total
+line+continuum fit and dashed lines for individual line profiles. 16th and 84th percentiles derived from 1000 bootstrapped stacks
+are shown in blue.
+The Hot DOG spectra were previously described in
+Finnerty et al. (2020). In brief, we obtained simulta-
+neous JHK spectra at R ∼ 2700 with Keck II/NIRES
+(Wilson et al. 2004) and reduced the data using SPEX-
+TOOL (Cushing et al. 2004). Flux calibration was per-
+formed by comparing the integrated flux with K′ pho-
+tometry, see Finnerty et al. (2020) for details.
+Our
+stacked spectrum uses the 17 sources with detections
+of [OIII], Hβ and/or [NII], Hα.
+In addition to the hot DOGs, we include in our anal-
+ysis previously unpublished Keck II/NIRES spectra of
+four z ∼ 1 − 2 galaxies with log M∗/M⊙ ∼ 11 and
+log LIR/L⊙ ∼ 12.5:
+GS 3 (z = 0.544, RA/DEC =
+03:32:08.66, -27:47:34.4), GS 7 (z = 1.042, RA/DEC
+= 03:32:26.49, -27:40:35.7), GN 1 (z = 1.432, RA/DEC
+= 12:36:45.83, +62:07:54.0), and GN 40 (z = 1.609,
+RA/DEC = 12:36:49.65, +62:07:38.6).
+These targets
+were selected for existing Spitzer/IRS mid-infrared spec-
+troscopy and bright IRAC Ch. 1 photometry from a
+24 µm-selected parent sample (Kirkpatrick et al. 2012,
+2015). GS 3, GS 7, and GN 40 are mid-IR AGN (Kirk-
+patrick et al. 2015), and GN 1 is a composite source with
+a mid-IR AGN fraction of 50%. These sources have LIR
+on-average an order of magnitude lower than those of
+the hot DOGs. We reduce the data for this sub-sample
+following the exact same procedures as the hot DOGs
+described in Finnerty et al. 2020.
+[O II], [OIII], Hβ
+and [NII], Hα are individually detected with EWs on-
+average lower than those of the hot DOGs but within
+their range.
+3. ANALYSIS
+In this work we compute synthetic photometry in
+broadband JWST/NIRCam filters for the rest-frame op-
+tical spectrum of our stacked spectrum redshifted be-
+tween z = 1 − 9. While we do not claim our sample to
+be definitively representative of all dusty systems owing
+to the extreme nature of hot DOGs, the final stack is
+derived from empirical data with no modelling required.
+While a more detailed study exploring a range of con-
+tinuum templates with added nebular lines is warranted,
+such analysis is beyond the scope of this work given the
+current lack of constraint on rest-frame optical spectra
+of dusty galaxies beyond z > 4 − 5. To supplement our
+analysis of very luminous, massive hot DOGs, we also
+compute synthetic photometry for the average dusty,
+star-forming galaxy (DSFG) spectrum from Casey et al.
+2017. The Casey et al. 2017 stack is constructed from
+Keck/MOSFIRE spectra of 20 LIRGs and ULIRGs with
+⟨z⟩ = 2.1, a more typical IR-bright galaxy population
+
+4
+selected from ground-based single dish sub-mm surveys
+(Casey et al. 2013).
+3.1. Stacking
+Prior to stacking the data, we convert the observed
+wavelength range of each spectrum to the rest-frame
+with spectroscopic redshifts derived from optical lines
+with low errors (∆z ∼ 10−3, Finnerty et al. 2020).
+Next, we rebin the spectra to a common wavelength grid
+corresponding to the lowest rest-frame spectral resolu-
+tion (R ∼ 6400). Finally, we calculate the sigma-clipped
+mean continuum flux from line-free regions which we use
+to normalize each spectra in the stack.
+We tested multiple stacking procedures and found
+that a mean noise-weighted continuum stack produced
+the cleanest continuum and highest line SNRs. In the
+stack, the input spectrum is first normalized by its
+sigma-clipped mean flux and then weighted by the spec-
+tral uncertainty per channel. This ensures that the stack
+is not dominated by particularly noisy spectral regions
+and/or the brightest spectra. As the goal of this analy-
+sis is the relative contribution of strong emission lines to
+photometry, we are not concerned with absolute normal-
+ization of the spectrum. To quantify the uncertainty on
+the continuum and line profiles, we repeat the stacking
+analysis 1000 times, using in each iteration 21 random
+samples of the input spectra with replacement (“boot-
+strapping”).
+From the bootstrapped uncertainties we
+determine that our final stacked spectrum is reliable be-
+tween λrest = 0.34 − 0.8 µm.
+The final stacked spectrum is shown in Figure 1. We
+detect strong [OIII], Hβ, [NII], Hα emission lines, as well
+as [S II], [O II], [O I], [Ne III] and Hγ. The [O I]6300/Hα
+line ratio is 0.5±0.1, which is on the high end of the dis-
+tribution measured for Seyfert galaxies in the Swift-BAT
+AGN Spectroscopic Survey (Koss et al. 2017). EWs for
+the strong lines around the [OIII], Hβ and [NII], Hα
+complexes are listed in Table 1. Quoted uncertainties
+correspond to the standard deviation of EWs measured
+for 100 realizations of the spectrum perturbed by the
+spectral uncertainty per channel, and assuming a 10%
+error on the continuum (uncertainties increase by a fac-
+tor of 2.3 assuming a 20% error on the continuum).
+We also compare our stacked spectrum to the stack
+from Casey et al. 2017 derived from 20 MOSFIRE spec-
+tra of DSFGs.1 The stack of Casey et al. 2017 exhibits
+narrower emission lines than our spectrum, and does
+not contain the broad outflow signatures founds in hot
+1
+Available
+at
+http://www.as.utexas.edu/~cmcasey/
+downloads.html
+DOG rest-frame optical spectra (Finnerty et al. 2020;
+Wu et al. 2018).
+3.2. Synthetic Photometry
+To test the effect of emission lines on JWST/NIRCam
+photometry, we subtract strong spectral features from
+the stack to produce a line-free continuum spectrum.
+To do so, we subtract the gaussian model fits from the
+stack.
+These lines include all shown in Figure 1 (B,
+C) and include the range of velocity components re-
+quired to fit individual hot DOGs, namely: broad [O
+III] and Hα emission, narrow [O I], [O III], [S II], Hα,
+and Hβ (Finnerty et al. 2020).
+We do not mask out
+[OII], [NeIII] and Hγ in this exercise as their lower EWs
+correspond to significantly less increase in broadband
+fluxes relative to [OIII]+Hβ and [NII]+Hα.
+Follow-
+ing Finnerty et al. 2020, we assume narrow and broad
+profiles across different lines arise from the same kine-
+matic components. This amounts to fixing [N II] widths
+to that of the corresponding Hα component. We also
+fix the [N II]λ6548, λ6584 ˚A ratio to 0.338 and the [O
+III]λ4959, λ5007 ˚A ratio to 0.335.
+In addition to the
+line-free stacked spectrum, we also compute a broad-line
+only spectrum (continuum + broad line emission).
+We calculate synthetic JWST/NIRCam photometry
+using the filter response profiles provided by the JWST
+User Documentation. We convolve each filter with both
+the stacked spectrum and line-subtracted stack for a
+range in redshift between z = 1−9 in steps of ∆z = 0.05.
+We then take the ratio of filter flux between the line
+stack and line-subtracted (or broad line only) stack to
+infer the increase in flux attributed to emission lines as
+a function of redshift.
+3.2.1. Composite DSFG spectrum from Casey et al. (2017)
+As both a check against our stack and a test for
+systems at lower LIR than the hot DOGs, we repeat
+our synthetic photometry calculations for the compos-
+ite DSFG spectrum from Casey et al. 2017. We scale
+their continuum-subtracted Hα flux in their stack to
+the equivalent of 100 M⊙ yr−1 in star-formation rate us-
+ing the FHα calibration of Murphy et al. (2011).
+As
+the change in flux density due to nebular emission is
+a function of the relative strength between lines and
+continuum, we add the scaled DSFG spectrum to the
+empirically-derived rest-frame 0.1 − 1 µm mean DSFG
+SED from Casey et al. 2014. For the line-free calcula-
+tion we simply mask Hα, [O III] and Hβ from the stack
+prior to performing synthetic photometry, equivalent to
+computing fluxes for the continuum DSFG SED without
+adding the lines.
+4. RESULTS AND DISCUSSION
+
+5
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+1.5
+2.0
+2.5
+3.0
+This work, broad + narrow
+This work, broad only
+F115W
+F150W
+F200W
+F277W
+F356W
+F410M
+F444W
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+1
+2
+3
+4
+5
+6
+7
+8
+9
+redshift
+1.5
+2.0
+2.5
+3.0
+Casey+2017 DSFGs
+This work, narrow only
+F , lines + cont / F , cont
+MAG = MAGlines + cont
+MAGcont
+Figure 2. Increase in JWST/NIRCam flux by strong rest-frame emission lines for the average SED of hot, dust-obscured and
+IR-lumionous galaxies between λrest = 0.34 − 0.8 µm as a function of redshift. (Top) Solid lines account for broad and narrow
+velocity components, whereas dashed lines include only the broad component. Maximally, strong nebular emission lines can
+boost the broadband flux between ∼ 25 − 80% (|∆MAG| = 0.2 − 0.6) from z ∼ 1 − 8. Medium-band filters such as F410M can
+be boosted by up to a factor of 2.5 when they overlap with strong emission lines at z ∼ 5.5. The increase in flux attributed
+to velocity-broadened features is ∼25% on-average (|∆MAG| = 0.2). The double-peak effect for a given filter arises from the
+Hα complex first passing through, followed by [OIII]+Hβ. (Bottom) Increase in flux attributed to strong line emission for the
+average DSFG spectrum of Casey et al. 2017 scaled to an Hα star-formation rate of 100 M⊙ yr−1 (solid) and the narrow velocity
+component in our stack (dashed). The increase in flux by strong Hα+[NII] in the DSFG stack is consistent with the narrow line
+component for these lines the hot DOG stack. At z ∼ 5, all of the NIRCam LW filters are boosted by ∼ 20% − 100% for hot
+DOGs and DSFGs.
+The results of our synthetic photometry are shown
+in Figure 2, which gives the flux ratio between our fidu-
+cial and line-subtracted stacked spectrum and the DSFG
+stack from Casey et al. (2017). For the former, we also
+show the increase in flux seperated between the nar-
+row and broad velocity components. On average, strong
+narrow+broad rest-frame optical lines increase NIRCam
+fluxes by factors of ∼ 1.2 − 1.7, with corresponding
+change in apparent magnitude by 0.2−0.6. The maximal
+increase in flux occurs when any particular wide-band
+filter is centered on the strong [NII]+Hα complex. The
+[OIII] and Hβ lines collectively increase the wide-band
+flux maximally by ∼20%. Their broad components and
+those of [NII] and Hα increase synthetic flux densities
+by 1.2× on average, accounting for ∼ 66% of the boost
+for [OIII]+Hβ and 25% for [NII]+Hα.
+Medium-band
+filters are more affected by the presence of strong emis-
+sion lines and can be dominated by factors of ∼ 2.5 (1
+mag) by emission lines when redshifted to the line’s rest
+wavelength. For example, the F410M flux is increased
+by a factor of 2.5 at z = 5.5. In fact, z = 5 is a spe-
+cial regime where a boost in flux density is seen for all
+NIRCam LW filters. While we do not show the increase
+in flux attributed to the relatively weaker [OII] line on
+Fig. 2, this effect is maximally ∼ 10% if we mask the
+line following the methods outlined for [OIII]+Hβ and
+[NII]+Hα.
+The strong lines in the DSFG stack from Casey et al.
+2017 which we have scaled to an Hα SFR of 100 M⊙ yr−1
+(see section 3.2.1) increase broadband fluxes by up to a
+factor of ∼ 1.5. Such boosting occurs over similar ranges
+in redshift and to the same degree as found for the nar-
+row line components in the hot DOG stack. This demon-
+strates that significant line contamination can be present
+in the NIRCam photometry for IR-luminous galaxies
+more normal than the relatively extreme hot DOGs.
+Given extreme levels of attenuation in massive dust-
+obscured galaxies at high-redshift, their rest-frame op-
+
+6
+10
+11
+12
+13
+14
+15
+16
+17
+Best-fit LBG redshift
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+Best-fit LBG 
+3.5 < zstack < 4
+4 < zstack < 5
+5 < zstack < 6
+0.25
+0.50
+0.75
+Transmission
+F115W
+F150W
+F200W
+F277W
+F356W
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Observed wavelength [um]
+100
+101
+102
+F  [nJy]
+CEERS
+COSMOS-Web
+z = 15 LBG Template (
+= -2.4)
+Stack (this work) at z = 3.75
+Figure 3.
+(Left) Allowed LBG redshift and UV slope β solutions when fitting the synthetic NIRCam flux densities of our stack
+redshifted to z = 3.5−4 (blue), z = 4−5 (green), and z = 5−6 (pink). Posterior contours are drawn at the 16th, 50th, and 84th
+percentiles. At z > 4 [NII]+Hα increase the NIRCam flux density more so than [OIII]+Hβ which does not allow the strong lines
+to mask the red continuum and therefore precludes LBG solutions with β < −1.5. Between 3.5 < z < 4 [NII]+Hα falls within
+F277W while [OIII]+Hβ is missed by F200W, allowing degenerate solutions with blue β ≤ −2 LBGs at z ∼ 14 − 17. (Right)
+Illustration of the degeneracy between z ∼ 15 candidates and z < 4 dusty galaxies with strong rest-frame optical/emission lines.
+In this example, we redshift our stacked spectrum to z = 3.75 where strong line emission boosts the F277W filter flux by 60%.
+We then compute F200W, F277W, and F356W JWST/NIRCam photometry (black circles), assuming non-detections in F115W
+and F150W. We fit the synthetic photometry from the stack (circles) with an LBG template (red), deriving a photometric
+redshift of zphot = 15 and UV spectral index β = −2.4. Strong emission lines mask the red slope of the dusty template between
+F277W and F356W, and the SED is further confused with the Lyman break falling halfway between F200W. Such scenarios are
+possible given the relative filter depths of JWST Cycle 1 NIRCam extragalactic surveys in CEERS (blue, Bagley et al. 2022;
+Finkelstein et al. 2022b) and COSMOS-Web (pink, Casey et al. 2022) for example.
+tical spectra contain a combination of significantly red-
+dened continuum with ≲ 5% of the total un-obscured
+light escaping from the least obscured regions (Chap-
+man et al. 2005; Howell et al. 2010). With a combina-
+tion of strong lines emergent from less obscured regions
+on top of the very red continuum, ∼ 0.1 − 1 µm pho-
+tometry of dusty galaxies can mimic that of ultra-high
+redshift LBG candidates in large surveys (Zavala et al.
+2022; Fujimoto et al. 2022). To quantify the parame-
+ter space where this confusion is significant, we fit an
+LBG template to the synthetic NIRCam flux densities
+derived from our stacked spectrum. We first normalize
+the stack to a continuum flux on the order of ∼ 10 nJy
+over λobs = 2 − 3.5 µm, and assume it to be undetected
+in F115W and F150W. This represents a plausible sce-
+nario given the relative filter depths of JWST Cycle 1
+extragalactic deep fields (Bagley et al. 2022; Finkelstein
+et al. 2022b; Casey et al. 2022), and is similar to CEERS-
+93316 (Donnan et al. 2022) (CEERS2 2159, Finkelstein
+et al. 2022b) − a z = 16.4 LBG candidate selected from
+CEERS (Finkelstein et al. 2022a; Bagley et al. 2022).
+CEERS-93316 has a tentative 2.6σ SCUBA-2 detection
+(Zavala et al. 2022) and environmental evidence (Naidu
+et al. 2022b) both indicating a possible lower redshift
+solution at z ∼ 4.8.
+Figure 3 (Left) shows the 2D posterior distribution
+in redshift and UV slope β for LBG template fits to
+our stacked spectrum’s synthetic NIRCam flux densi-
+ties. We repeat the fitting analysis 1000 times after per-
+turbing the input spectrum by the spectral uncertainty,
+and in three redshift ranges for the stack: 3.5 < z < 4,
+4 < z < 5, and 5 < z < 6.
+The cumulative EW of
+Hα+[NII] is greater than EW([OIII]+Hβ) by a factor
+of ∼ 3 which precludes LBG fits with β < −1.5 when
+the stack is redshifted to z > 4. This is because both
+features fall within a broadband filter and so the strong
+lines do not mask the red continuum in the stack. At
+3.5 < z < 4 F277W picks up the strong Hα+[NII] emis-
+sion while [OIII]+Hβ is missed by F200W. This pro-
+duces degenerate photometry with z ∼ 16 LBGs. At
+z ∼ 16, the Lyman break falls halfway between F200W
+mimicking the red slope of the dusty galaxy stack while
+the very blue continuum mimics the F277W flux den-
+sity of the line-contaminated stack. In summary, the hot
+DOG stack can reproduce very blue UV slopes β ∼ −2.5
+for zstack ∼ 3.5 − 4 but not for zstack > 4. This supports
+
+7
+Table 1. Strong optical emission line characteristics
+in our stacked spectrum
+Line
+EW (˚A)
+FWHM (km s−1)
+Hβ
+45 ± 12
+1450
+Hαnarrow
+222 ± 27
+730
+Hαbroad
+540 ± 80
+4000
+[OIII]5007
+127 ± 19
+1000
+[OIII]4959
+43 ± 8
+1000
+[OIII]5007,broad
+144 ± 49
+7300
+[OIII]4959,broad
+48 ± 32
+7300
+[OI]6300
+109 ± 20
+1600
+[OI]6363
+38 ± 13
+1600
+[NII]6548
+35 ± 8
+730
+[NII]6583
+102 ± 17
+730
+[SII]6716
+103 ± 18
+1200
+[SII]6730
+43 ± 11
+1200
+AV (Hαnarrow /Hβ)
+6 ± 1
+AV (Hαtot /Hβ)
+10 ± 1
+the purity of the very blue NIRCam samples of Cullen
+et al. 2022 and Topping et al. 2022, which predominantly
+have β < −1.5 and 7 < z < 14.
+As further demonstration of the confusion between our
+stack at z ∼ 3.5 − 4 and z ∼ 16 LBGs, we show in Fig-
+ure 3 (Right) the LBG fit to our stack’s JWST/NIRCam
+flux densities. At zstack = 3.75 we find a best-fit LBG
+solution with z = 16 and UV spectral index β = −2.4.
+Lower redshift (z < 4) solutions with red continuum
+slopes and flux densities dominated by strong emis-
+sion lines should be considered when fitting the very
+blue (β < −2) spectral energy distributions (SEDs) of
+z ∼ 16 candidates. The lower redshift solutions could be
+ruled out with medium-band filters, longer wavelength
+sampling using NIRCam’s redder filters, MIRI observa-
+tions, and/or far-IR/sub-mm follow-up to detect cold
+dust continuum and fine-structure lines (Fujimoto et al.
+2022). Measurements that strongly rule out UV spectral
+indices β < −1.5 and only allow lower−z solutions at
+z > 4 should be particularly constraining against mas-
+sive, IR-luminous interlopers with strong optical lines
+provided they sample the SED with more than three
+filters.
+Based on the first analysis of JWST deep field obser-
+vations at 5σ point-source depths between ∼ 28 − 29
+MAG, the projected sky density of candidates at z > 10
+is approximately 350 ±120 deg−2 (Donnan et al. 2022;
+Finkelstein et al. 2022a; Naidu et al. 2022a; Harikane
+et al. 2022). Although preliminary, these source counts
+represent the population which could potentially be con-
+taminated by low−z dusty interlopers. In contrast, the
+sky density of luminous IR galaxies with log LIR/L⊙ >
+12 (12.5) and z ∼ 3 − 4 is 400 deg−2 (100 deg−2) (Casey
+et al. 2018; Zavala et al. 2021). If we assume the sam-
+ples of Finnerty et al. (2020) and Casey et al. (2017)
+include a range of physically possible rest-frame opti-
+cal properties for IR-bright galaxies (log LIR/L⊙ > 11),
+then their similar number counts to ultra high-redshift
+LBG candidates may be reason to be concerned about
+contamination.
+The fainter dusty galaxy population
+with log LIR/L⊙ < 11 are much more numerous based
+on the general shape of 1 mm number counts (Fujimoto
+et al. 2016; Gonz´alez-L´opez et al. 2020), and may also
+be an important source of contamination as galaxies
+fainter in the IR are less likely to be significantly ob-
+scured in the rest-frame optical. Further spectroscopic
+follow up is required to assess the purity of ultra-high
+redshift catalogs.
+In the meantime, F150W dropouts
+(z > 10) with β ∼ −2 and no/poor SED constraint
+above NIRCam/F356W should be checked against pos-
+sible intermediate-redshift dusty galaxy solutions.
+5. SUMMARY AND CONCLUSION
+In this Letter we test the response of JWST NIR-
+Cam filters over broad rest-frame optical emission lines
+in the average spectrum of hot, dust-obscured galaxies
+at z ∼ 1 − 4 and dusty, star-forming galaxies. As an
+empirical approach, we stack a sample of 21 IR lumi-
+nous galaxies with rest-frame optical spectra from Keck
+II/NIRES which we then compute synthetic photometry
+for between z = 1 − 9. Our main results are as follows:
+1. We measure broad rest-frame optical emission
+lines in the stack of z ∼ 1 − 4 hot, dust-obscured
+galaxies. In particular, we measure [OIII] and Hα
+EWs between 100 − 500 ˚A which are high relative
+to normal star-forming galaxies at high-redshift.
+2. After masking out strong emission features from
+the spectrum, we measure synthetic NIRCam pho-
+tometry with and without the lines. Narrow and
+broad components for [OIII] and Hβ increase the
+measured filter flux by 30%, and Hα+[NII] by 60%
+on average.
+Narrowband filters such as F410M
+can have their flux increased by a factor of 2 − 3
+(0.7 − 1.2 MAG).
+3. Rest-frame optical photometry of dusty galaxies
+with strong nebular lines at z ∼ 3.5 − 4 could
+
+8
+be important contaminants in F150W dropout
+(z > 10 candidate) catalogs as the strong lines
+can help mask red UV spectral indices. However,
+UV spectral indices β < −1.5 are difficult for our
+stacked spectrum to reproduce for interloper red-
+shifts z > 4.
+Distinguishing between different galaxy populations
+with JWST imaging is a key first step towards test-
+ing various aspects of galaxy formation.
+While this
+work has focused just on JWST’s NIRCam filters, the
+inclusion of deep MIRI photometry extending to longer
+wavelengths will add significant constraint on various
+redshift solutions to photometric fitting codes. In the
+absence of high SNR coverage in redder filters, far-
+IR/sub-mm followup can help identify dusty galaxies.
+On the near horizon, ToLTEC on the Large Millimeter
+Telescope (LMT) Alfonso Serrano will map multiple ex-
+tragalactic fields (COSMOS, UDS, GOODS-S) down to
+the LIRG limit at 1.1, 1.4, and 2 mm as part of “The
+TolTEC Ultra-Deep Galaxy Survey,” a public legacy
+program.
+These public data sets are well suited to
+quickly identify sub-mm bright DSFG counterparts to
+JWST sources.
+L.F. is a member of Student Researchers United
+(SRU-UAW). The data presented herein were obtained
+at the W. M. Keck Observatory, which is operated as
+a scientific partnership among the California Institute
+of Technology, the University of California, and the
+National Aeronautics and Space Administration. The
+Observatory was made possible by the generous finan-
+cial support of the W. M. Keck Foundation. We wish
+to acknowledge the critical importance of the current
+and recent Maunakea Observatories daycrew, techni-
+cians, telescope operators, computer support, and office
+staff employees, especially during the challenging times
+presented by the COVID-19 pandemic.
+Their exper-
+tise, ingenuity, and dedication is indispensable to the
+continued successful operation of these observatories.
+The authors wish to recognize and acknowledge the
+very significant cultural role and reverence that the
+summit of Maunakea has always had within the indige-
+nous Hawaiian community. We are most fortunate to
+have the opportunity to conduct observations from this
+mountain.
+
+9
+REFERENCES
+Assef, R. J., Eisenhardt, P. R. M., Stern, D., et al. 2015,
+ApJ, 804, 27, doi: 10.1088/0004-637X/804/1/27
+Bagley, M. B., Finkelstein, S. L., Koekemoer, A. M., et al.
+2022, arXiv e-prints, arXiv:2211.02495.
+https://arxiv.org/abs/2211.02495
+Bakx, T. J. L. C., Zavala, J. A., Mitsuhashi, I., et al. 2022,
+arXiv e-prints, arXiv:2208.13642.
+https://arxiv.org/abs/2208.13642
+Bouwens, R., Illingworth, G., Oesch, P., et al. 2022, arXiv
+e-prints, arXiv:2212.06683.
+https://arxiv.org/abs/2212.06683
+Boylan-Kolchin, M. 2022, arXiv e-prints, arXiv:2208.01611.
+https://arxiv.org/abs/2208.01611
+Bradley, L. D., Coe, D., Brammer, G., et al. 2022, arXiv
+e-prints, arXiv:2210.01777.
+https://arxiv.org/abs/2210.01777
+Carnall, A. C., Begley, R., McLeod, D. J., et al. 2022,
+MNRAS, doi: 10.1093/mnrasl/slac136
+Casey, C. M., Chen, C.-C., Cowie, L. L., et al. 2013,
+MNRAS, 436, 1919, doi: 10.1093/mnras/stt1673
+Casey, C. M., Scoville, N. Z., Sanders, D. B., et al. 2014,
+ApJ, 796, 95, doi: 10.1088/0004-637X/796/2/95
+Casey, C. M., Cooray, A., Killi, M., et al. 2017, ApJ, 840,
+101, doi: 10.3847/1538-4357/aa6cb1
+Casey, C. M., Zavala, J. A., Spilker, J., et al. 2018, ApJ,
+862, 77, doi: 10.3847/1538-4357/aac82d
+Casey, C. M., Kartaltepe, J. S., Drakos, N. E., et al. 2022,
+arXiv e-prints, arXiv:2211.07865.
+https://arxiv.org/abs/2211.07865
+Chapman, S. C., Blain, A. W., Smail, I., & Ivison, R. J.
+2005, ApJ, 622, 772, doi: 10.1086/428082
+Cullen, F., McLure, R. J., McLeod, D. J., et al. 2022, arXiv
+e-prints, arXiv:2208.04914.
+https://arxiv.org/abs/2208.04914
+Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004,
+PASP, 116, 362, doi: 10.1086/382907
+Cutri, R. M., & et al. 2012, VizieR Online Data Catalog,
+II/311
+Donnan, C. T., McLeod, D. J., Dunlop, J. S., et al. 2022,
+arXiv e-prints, arXiv:2207.12356.
+https://arxiv.org/abs/2207.12356
+Dudzeviˇci¯ut˙e, U., Smail, I., Swinbank, A. M., et al. 2020,
+MNRAS, 494, 3828, doi: 10.1093/mnras/staa769
+Dunlop, J. S., Cirasuolo, M., & McLure, R. J. 2007,
+MNRAS, 376, 1054,
+doi: 10.1111/j.1365-2966.2007.11453.x
+Eisenhardt, P. R. M., Wu, J., Tsai, C.-W., et al. 2012, ApJ,
+755, 173, doi: 10.1088/0004-637X/755/2/173
+Finkelstein, S. L., Bagley, M. B., Arrabal Haro, P., et al.
+2022a, arXiv e-prints, arXiv:2207.12474.
+https://arxiv.org/abs/2207.12474
+Finkelstein, S. L., Bagley, M. B., Ferguson, H. C., et al.
+2022b, arXiv e-prints, arXiv:2211.05792.
+https://arxiv.org/abs/2211.05792
+Finnerty, L., Larson, K., Soifer, B. T., et al. 2020, ApJ,
+905, 16, doi: 10.3847/1538-4357/abc3bf
+Fujimoto, S., Ouchi, M., Ono, Y., et al. 2016, ApJS, 222, 1,
+doi: 10.3847/0067-0049/222/1/1
+Fujimoto, S., Finkelstein, S. L., Burgarella, D., et al. 2022
+Gonz´alez-L´opez, J., Novak, M., Decarli, R., et al. 2020,
+ApJ, 897, 91, doi: 10.3847/1538-4357/ab765b
+Gruppioni, C., B´ethermin, M., Loiacono, F., et al. 2020,
+A&A, 643, A8, doi: 10.1051/0004-6361/202038487
+Harikane, Y., Ouchi, M., Oguri, M., et al. 2022, arXiv
+e-prints, arXiv:2208.01612.
+https://arxiv.org/abs/2208.01612
+Howell, J. H., Armus, L., Mazzarella, J. M., et al. 2010,
+ApJ, 715, 572, doi: 10.1088/0004-637X/715/1/572
+Jun, H. D., Assef, R. J., Bauer, F. E., et al. 2020, ApJ, 888,
+110, doi: 10.3847/1538-4357/ab5e7b
+Kaasinen, M., van Marrewijk, J., Popping, G., et al. 2022,
+arXiv e-prints, arXiv:2210.03754.
+https://arxiv.org/abs/2210.03754
+Kirkpatrick, A., Pope, A., Sajina, A., et al. 2015, ApJ, 814,
+9, doi: 10.1088/0004-637X/814/1/9
+Kirkpatrick, A., Pope, A., Alexander, D. M., et al. 2012,
+ApJ, 759, 139, doi: 10.1088/0004-637X/759/2/139
+Koprowski, M. P., Dunlop, J. S., Micha�lowski, M. J., et al.
+2017, MNRAS, 471, 4155, doi: 10.1093/mnras/stx1843
+Koss, M., Trakhtenbrot, B., Ricci, C., et al. 2017, ApJ, 850,
+74, doi: 10.3847/1538-4357/aa8ec9
+Labbe, I., van Dokkum, P., Nelson, E., et al. 2022, arXiv
+e-prints, arXiv:2207.12446.
+https://arxiv.org/abs/2207.12446
+Long, A. S., Casey, C. M., Lagos, C. d. P., et al. 2022,
+arXiv e-prints, arXiv:2211.02072.
+https://arxiv.org/abs/2211.02072
+Maio, U., & Viel, M. 2022
+Manning, S. M., Casey, C. M., Zavala, J. A., et al. 2022,
+ApJ, 925, 23, doi: 10.3847/1538-4357/ac366a
+Micha�lowski, M. J., Dunlop, J. S., Koprowski, M. P., et al.
+2017, MNRAS, 469, 492, doi: 10.1093/mnras/stx861
+Murphy, E. J., Chary, R. R., Dickinson, M., et al. 2011,
+ApJ, 732, 126, doi: 10.1088/0004-637X/732/2/126
+Naidu, R. P., Oesch, P. A., van Dokkum, P., et al. 2022a,
+arXiv e-prints, arXiv:2207.09434.
+https://arxiv.org/abs/2207.09434
+
+10
+Naidu, R. P., Oesch, P. A., Setton, D. J., et al. 2022b,
+arXiv e-prints, arXiv:2208.02794.
+https://arxiv.org/abs/2208.02794
+Roberts-Borsani, G., Treu, T., Chen, W., et al. 2022, arXiv
+e-prints, arXiv:2210.15639.
+https://arxiv.org/abs/2210.15639
+Rowan-Robinson, M., Wang, L., Farrah, D., et al. 2018,
+A&A, 619, A169, doi: 10.1051/0004-6361/201832671
+Schaerer, D., & de Barros, S. 2009, A&A, 502, 423,
+doi: 10.1051/0004-6361/200911781
+Stark, D. P., Schenker, M. A., Ellis, R., et al. 2013, ApJ,
+763, 129, doi: 10.1088/0004-637X/763/2/129
+Topping, M. W., Stark, D. P., Endsley, R., et al. 2022,
+arXiv e-prints, arXiv:2208.01610.
+https://arxiv.org/abs/2208.01610
+Tsai, C.-W., Eisenhardt, P. R. M., Wu, J., et al. 2015, ApJ,
+805, 90, doi: 10.1088/0004-637X/805/2/90
+Wilkins, S. M., Coulton, W., Caruana, J., et al. 2013,
+MNRAS, 435, 2885, doi: 10.1093/mnras/stt1471
+Wilkins, S. M., Lovell, C. C., Fairhurst, C., et al. 2020,
+MNRAS, 493, 6079, doi: 10.1093/mnras/staa649
+Wilkins, S. M., Vijayan, A. P., Lovell, C. C., et al. 2022,
+MNRAS, 517, 3227, doi: 10.1093/mnras/stac2548
+Wilson, J. C., Henderson, C. P., Herter, T. L., et al. 2004,
+in Society of Photo-Optical Instrumentation Engineers
+(SPIE) Conference Series, Vol. 5492, Ground-based
+Instrumentation for Astronomy, ed. A. F. M. Moorwood
+& M. Iye, 1295–1305
+Wu, J., Tsai, C.-W., Sayers, J., et al. 2012, ApJ, 756, 96,
+doi: 10.1088/0004-637X/756/1/96
+Wu, J., Jun, H. D., Assef, R. J., et al. 2018, ApJ, 852, 96,
+doi: 10.3847/1538-4357/aa9ff3
+Yoon, I., Carilli, C. L., Fujimoto, S., et al. 2022, arXiv
+e-prints, arXiv:2210.08413.
+https://arxiv.org/abs/2210.08413
+Yu-Yang Hsiao, T., Coe, D., Abdurrouf, et al. 2022, arXiv
+e-prints, arXiv:2210.14123.
+https://arxiv.org/abs/2210.14123
+Zackrisson, E., Bergvall, N., & Leitet, E. 2008, ApJL, 676,
+L9, doi: 10.1086/587030
+Zavala, J. A., Casey, C. M., Manning, S. M., et al. 2021,
+ApJ, 909, 165, doi: 10.3847/1538-4357/abdb27
+Zavala, J. A., Buat, V., Casey, C. M., et al. 2022, arXiv
+e-prints, arXiv:2208.01816.
+https://arxiv.org/abs/2208.01816
+
diff --git a/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/load_file.txt b/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5c08c24eb1138cc9b04f4793859035ce1d772b9f
--- /dev/null
+++ b/zNAyT4oBgHgl3EQfO_a9/content/tmp_files/load_file.txt
@@ -0,0 +1,978 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf,len=977
+page_content='Draft version January 3, 2023  Typeset using LATEX twocolumn style in AASTeX61  BROAD EMISSION LINES IN OPTICAL SPECTRA OF HOT DUST-OBSCURED GALAXIES CAN  CONTRIBUTE SIGNIFICANTLY TO JWST/NIRCAM PHOTOMETRY  Jed McKinney,1 Luke Finnerty,2 Caitlin M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Casey,1 Maximilien Franco,1 Arianna S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Long,1, ∗  Seiji Fujimoto,1, 3, 4, ∗ Jorge A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Zavala,5 Olivia Cooper,1 Hollis Akins,1 Alexandra Pope,6 Lee Armus,7  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Soifer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 Kirsten Larson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='9 Keith Matthews,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 Jason Melbourne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 and Michael Cushing10  1Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The University of Texas at Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2515 Speedway Blvd Stop C1400,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Austin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' TX 78712,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' USA  2Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 430 Portola Plaza,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Los Angeles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' CA 90095,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' USA  3Cosmic Dawn Center (DAWN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Jagtvej 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' DK2200 Copenhagen N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Denmark  4Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' University of Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Lyngbyvej 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' DK2100 Copenhagen Ø,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Denmark  5National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2-21-1 Osawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Japan  6Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' University of Massachusetts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Amherst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' MA 01003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  7IPAC, California Institute of Technology, 1200 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Pasadena, CA 91125, USA  8Division of Physics, Math, and Astronomy, California Institute of Technology, 1200 E California Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Pasadena, CA 91125, USA  9AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA  10Department of Physics and Astronomy, University of Toledo, 2801 West Bancroft St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Toledo, OH 43606, USA  ABSTRACT  Selecting the first galaxies at z > 7 − 10 from JWST surveys is complicated by z < 6 contaminants with degenerate  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' For example, strong optical nebular emission lines at z < 6 may mimic JWST/NIRCam photometry of  z > 7 − 10 Lyman Break Galaxies (LBGs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Dust-obscured 3 < z < 6 galaxies in particular are potentially important  contaminants, and their faint rest-optical spectra have been historically difficult to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A lack of optical emission  line and continuum measures for 3 < z < 6 dusty galaxies now makes it difficult to test their expected JWST/NIRCam  photometry for degenerate solutions with NIRCam dropouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Towards this end, we quantify the contribution by strong  emission lines to NIRCam photometry in a physically motivated manner by stacking 21 Keck II/NIRES spectra of  hot, dust-obscured, massive (log M∗/M⊙ ≳ 10 − 11) and infrared (IR) luminous galaxies at z ∼ 1 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We derive an  average spectrum and measure strong narrow (broad) [OIII]5007 and Hα features with equivalent widths of 130 ± 20 ˚A  (150±50 ˚A) and 220±30 ˚A (540±80 ˚A) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' These features can increase broadband NIRCam fluxes by factors  of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='7 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Due to significant dust-attenuation (AV ∼ 6), we find Hα+[NII] to be significantly brighter  than [OIII]+Hβ, and therefore find that emission-line dominated contaminants of high−z galaxy searches can only  reproduce moderately blue perceived UV continua of Sλ ∝ λβ with β > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 and z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' While there are some redshifts  (z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='75) where our stack is more degenerate with the photometry of z > 10 LBGs between λrest ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 µm,  redder filter coverage beyond λobs > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 µm and far-IR/sub-mm follow-up may be useful for breaking the degeneracy  and making a crucial separation between two fairly unconstrained populations, dust-obscured galaxies at z∼ 3 − 6 and  LBGs at z> 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  ∗ NASA Hubble Fellow  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='00017v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='GA]  30 Dec 2022  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' INTRODUCTION  A major objective baked into the design of JWST is  detecting the light from the first galaxies residing at  ultra-high redshifts (z > 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Delivering on its promise,  more than 30 galaxy candidates with photometric red-  shift solutions favoring z > 10 were identified within the  first months of publicly available data (Bradley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Harikane  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Yu-Yang Hsiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Assessing the fidelity of these samples is criti-  cal, particularly because the statistics assuming current  z > 10 − 15 candidates are real may or may not violate  ΛCDM predictions (Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Boylan-Kolchin  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Labbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Maio & Viel 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Spectroscopic confirmation is needed to verify these  redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  However, some early attempts at spectro-  scopic follow-up using facilities such as ALMA have  yielded upper limits or tentative low-SNR detections  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bakx et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Kaasinen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' JWST/NIRSpec has proven  capable of spectroscopically detecting the rest-frame op-  tical light from galaxies up to z ∼ 9 − 10 (Carnall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Roberts-Borsani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022), but this might not  be well-suited for rapidly validating redshifts in statis-  tical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  A complimentary approach, born from  similar rest-frame optical colors of z > 10 Lyman Break  Galaxies (LBGs) and dusty galaxies (Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2014), is to use far-IR/sub-mm followup ob-  servations of cold dust continuum (Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022)  and/or far-IR cooling lines (Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022) to  identify or rule out z < 6 IR-luminous galaxies lurking  within z > 10 candidate catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Dusty sources have  posed a problem to the fidelity of high−z galaxy cata-  logs since selection from Hubble Space Telescope (HST)  extragalactic deep fields (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Dunlop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' HST  samples at z ∼ 6 − 7 were contaminated by z ∼ 2 dusty,  star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Now, z ∼ 3−6 dusty galaxies may  be contaminating z > 7 − 10 JWST samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A con-  tributing factor to this contamination is that both pop-  ulations have similar, and uncertain, number densities:  dusty star-forming galaxies at z ∼ 3 − 6 have volume  number densities n ∼ 10−5 − 10−6 Mpc−3 (Micha�lowski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Koprowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Rowan-Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Dudzeviˇci¯ut˙e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Gruppioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Manning et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022), similar to early  measurements of bright z > 10 LBGs (Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Harikane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Bouwens  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Disentangling these two populations is  therefore also crucial for reducing uncertainties in their  respective number densities, which are currently inflated  by sample purity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bouwens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Of particular concern within the rest-frame optical/near-  IR spectra of IR-luminous galaxies is the relative con-  tribution of strong narrow and broad emission lines  to broadband filter fluxes, which could mask red con-  tinuum slopes produced by dust attenuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Strong  nebular lines can change JWST/NIRCam colors (Za-  ckrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Schaerer & de Barros 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Stark  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Wilkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013, 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This may  be a promising tool for pseudo-spectroscopy of lower  redshift galaxies using narrow-band filters, but in this  particular instance is a source of potential population  confusion for broadband high-redshift surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Indeed,  some of the hottest and most luminous dusty galaxies at  z > 1 exhibit very high rest-frame optical line equivalent  widths (EWs) (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' These arise from  a combination of low dust-attenuated continuum levels  with bright lines emergent from less obscured regions,  as well as ionized outflows driven by Active Galactic  Nuclei (AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' To what extent do these strong lines  contribute to JWST photometry?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  In this letter, we take an empirically-grounded ap-  proach and quantify the contamination from emission  lines from hot dust-obscured galaxies to JWST/NIRCam  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In Section 2 we describe Keck II/NIRES  observations of a sample of 4 luminous, IR galaxies  (log LIR/L⊙∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5) and 17 hot dust, obscured galaxies  (DOGs, log LIR/L⊙ > 13) between z ∼ 1 − 4, which  we stack to derive an average optical spectrum (Section  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We quantify the contribution of strong and broad  optical emission lines to NIRCam fluxes in Section 4,  and discuss their impact on distinguishing between such  sources at z < 6 and z > 10 LBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Section 5 sum-  marizes our main findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Throughout this work we  adopt a ΛCDM cosmology with Ωm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='7  and H0 = 70 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' SAMPLE AND DATA  Hot DOGs were originally selected from WISE pho-  tometry as W1W2-dropouts and include the most-  luminous galaxies in the Universe (Eisenhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This extreme population is expe-  riencing a rapid phase of both supermassive black-hole  and stellar mass assembly (Eisenhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012), and  are mostly found at z ∼ 2−3 with log LIR/L⊙ ≥ 13 (Wu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Tsai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Most  hot DOGs exhibit strong ionized outflows in optical  spectroscopy (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Jun  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020), as implied by broad line components likely  driven by radiative AGN feedback Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  These sources are rare with only ∼ 1000 over the full  WISE All-sky survey (Cutri & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3  (A)  (B)  (C)  [arbitrary units]  Fν  Wavelength [ m]  μ  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Stacked rest-frame optical spectrum of z ∼ 1 − 4 IR-luminous galaxies detected with Keck II/NIRES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (A) The  mean-weighted stacked spectrum (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Shaded grey errors correspond to 16th-84th percentiles on the flux density derived  from the boostrapped stack distribution per wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The upper panel gives the number of galaxies included in the stack as  a function of wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' On average, 70% (> 14) of the sample is represented between λrest = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='34 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='80 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We compare  against the stacked (N= 20) continuum-normalized DSFG spectrum from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2017) (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Panels (B) and (C) show  zoom-ins on the [O III], Hβ and [N II], Hα, [SII], [OI] features respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We measure broad and narrow components as  expected from the individual spectra (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Line fits are shown in purple with solid lines indicating the total  line+continuum fit and dashed lines for individual line profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 16th and 84th percentiles derived from 1000 bootstrapped stacks  are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The Hot DOG spectra were previously described in  Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In brief, we obtained simulta-  neous JHK spectra at R ∼ 2700 with Keck II/NIRES  (Wilson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2004) and reduced the data using SPEX-  TOOL (Cushing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Flux calibration was per-  formed by comparing the integrated flux with K′ pho-  tometry, see Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2020) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Our  stacked spectrum uses the 17 sources with detections  of [OIII], Hβ and/or [NII], Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  In addition to the hot DOGs, we include in our anal-  ysis previously unpublished Keck II/NIRES spectra of  four z ∼ 1 − 2 galaxies with log M∗/M⊙ ∼ 11 and  log LIR/L⊙ ∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5:  GS 3 (z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='544, RA/DEC =  03:32:08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='66, -27:47:34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4), GS 7 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='042, RA/DEC  = 03:32:26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='49, -27:40:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='7), GN 1 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='432, RA/DEC  = 12:36:45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='83, +62:07:54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0), and GN 40 (z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='609,  RA/DEC = 12:36:49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='65, +62:07:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  These targets  were selected for existing Spitzer/IRS mid-infrared spec-  troscopy and bright IRAC Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 1 photometry from a  24 µm-selected parent sample (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012,  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' GS 3, GS 7, and GN 40 are mid-IR AGN (Kirk-  patrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015), and GN 1 is a composite source with  a mid-IR AGN fraction of 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' These sources have LIR  on-average an order of magnitude lower than those of  the hot DOGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We reduce the data for this sub-sample  following the exact same procedures as the hot DOGs  described in Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  [O II], [OIII], Hβ  and [NII], Hα are individually detected with EWs on-  average lower than those of the hot DOGs but within  their range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' ANALYSIS  In this work we compute synthetic photometry in  broadband JWST/NIRCam filters for the rest-frame op-  tical spectrum of our stacked spectrum redshifted be-  tween z = 1 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' While we do not claim our sample to  be definitively representative of all dusty systems owing  to the extreme nature of hot DOGs, the final stack is  derived from empirical data with no modelling required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  While a more detailed study exploring a range of con-  tinuum templates with added nebular lines is warranted,  such analysis is beyond the scope of this work given the  current lack of constraint on rest-frame optical spectra  of dusty galaxies beyond z > 4 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' To supplement our  analysis of very luminous, massive hot DOGs, we also  compute synthetic photometry for the average dusty,  star-forming galaxy (DSFG) spectrum from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017 stack is constructed from  Keck/MOSFIRE spectra of 20 LIRGs and ULIRGs with  ⟨z⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1, a more typical IR-bright galaxy population  4  selected from ground-based single dish sub-mm surveys  (Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Stacking  Prior to stacking the data, we convert the observed  wavelength range of each spectrum to the rest-frame  with spectroscopic redshifts derived from optical lines  with low errors (∆z ∼ 10−3, Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Next, we rebin the spectra to a common wavelength grid  corresponding to the lowest rest-frame spectral resolu-  tion (R ∼ 6400).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Finally, we calculate the sigma-clipped  mean continuum flux from line-free regions which we use  to normalize each spectra in the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We tested multiple stacking procedures and found  that a mean noise-weighted continuum stack produced  the cleanest continuum and highest line SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In the  stack, the input spectrum is first normalized by its  sigma-clipped mean flux and then weighted by the spec-  tral uncertainty per channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This ensures that the stack  is not dominated by particularly noisy spectral regions  and/or the brightest spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' As the goal of this analy-  sis is the relative contribution of strong emission lines to  photometry, we are not concerned with absolute normal-  ization of the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' To quantify the uncertainty on  the continuum and line profiles, we repeat the stacking  analysis 1000 times, using in each iteration 21 random  samples of the input spectra with replacement (“boot-  strapping”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  From the bootstrapped uncertainties we  determine that our final stacked spectrum is reliable be-  tween λrest = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='34 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The final stacked spectrum is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We  detect strong [OIII], Hβ, [NII], Hα emission lines, as well  as [S II], [O II], [O I], [Ne III] and Hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The [O I]6300/Hα  line ratio is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1, which is on the high end of the dis-  tribution measured for Seyfert galaxies in the Swift-BAT  AGN Spectroscopic Survey (Koss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' EWs for  the strong lines around the [OIII], Hβ and [NII], Hα  complexes are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Quoted uncertainties  correspond to the standard deviation of EWs measured  for 100 realizations of the spectrum perturbed by the  spectral uncertainty per channel, and assuming a 10%  error on the continuum (uncertainties increase by a fac-  tor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3 assuming a 20% error on the continuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We also compare our stacked spectrum to the stack  from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017 derived from 20 MOSFIRE spec-  tra of DSFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1 The stack of Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017 exhibits  narrower emission lines than our spectrum, and does  not contain the broad outflow signatures founds in hot  1  Available  at  http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='edu/~cmcasey/  downloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='html  DOG rest-frame optical spectra (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Synthetic Photometry  To test the effect of emission lines on JWST/NIRCam  photometry, we subtract strong spectral features from  the stack to produce a line-free continuum spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  To do so, we subtract the gaussian model fits from the  stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  These lines include all shown in Figure 1 (B,  C) and include the range of velocity components re-  quired to fit individual hot DOGs, namely: broad [O  III] and Hα emission, narrow [O I], [O III], [S II], Hα,  and Hβ (Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We do not mask out  [OII], [NeIII] and Hγ in this exercise as their lower EWs  correspond to significantly less increase in broadband  fluxes relative to [OIII]+Hβ and [NII]+Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Follow-  ing Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020, we assume narrow and broad  profiles across different lines arise from the same kine-  matic components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This amounts to fixing [N II] widths  to that of the corresponding Hα component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We also  fix the [N II]λ6548, λ6584 ˚A ratio to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='338 and the [O  III]λ4959, λ5007 ˚A ratio to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='335.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  In addition to the  line-free stacked spectrum, we also compute a broad-line  only spectrum (continuum + broad line emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We calculate synthetic JWST/NIRCam photometry  using the filter response profiles provided by the JWST  User Documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We convolve each filter with both  the stacked spectrum and line-subtracted stack for a  range in redshift between z = 1−9 in steps of ∆z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We then take the ratio of filter flux between the line  stack and line-subtracted (or broad line only) stack to  infer the increase in flux attributed to emission lines as  a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Composite DSFG spectrum from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2017)  As both a check against our stack and a test for  systems at lower LIR than the hot DOGs, we repeat  our synthetic photometry calculations for the compos-  ite DSFG spectrum from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We scale  their continuum-subtracted Hα flux in their stack to  the equivalent of 100 M⊙ yr−1 in star-formation rate us-  ing the FHα calibration of Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  As  the change in flux density due to nebular emission is  a function of the relative strength between lines and  continuum, we add the scaled DSFG spectrum to the  empirically-derived rest-frame 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1 − 1 µm mean DSFG  SED from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' For the line-free calcula-  tion we simply mask Hα, [O III] and Hβ from the stack  prior to performing synthetic photometry, equivalent to  computing fluxes for the continuum DSFG SED without  adding the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' RESULTS AND DISCUSSION  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  This work, broad + narrow  This work, broad only  F115W  F150W  F200W  F277W  F356W  F410M  F444W  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2  1  2  3  4  5  6  7  8  9  redshift  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  Casey+2017 DSFGs  This work, narrow only  F , lines + cont / F , cont  MAG = MAGlines + cont  MAGcont  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Increase in JWST/NIRCam flux by strong rest-frame emission lines for the average SED of hot, dust-obscured and  IR-lumionous galaxies between λrest = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='34 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8 µm as a function of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (Top) Solid lines account for broad and narrow  velocity components, whereas dashed lines include only the broad component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Maximally, strong nebular emission lines can  boost the broadband flux between ∼ 25 − 80% (|∆MAG| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6) from z ∼ 1 − 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Medium-band filters such as F410M can  be boosted by up to a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 when they overlap with strong emission lines at z ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The increase in flux attributed  to velocity-broadened features is ∼25% on-average (|∆MAG| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The double-peak effect for a given filter arises from the  Hα complex first passing through, followed by [OIII]+Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (Bottom) Increase in flux attributed to strong line emission for the  average DSFG spectrum of Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017 scaled to an Hα star-formation rate of 100 M⊙ yr−1 (solid) and the narrow velocity  component in our stack (dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The increase in flux by strong Hα+[NII] in the DSFG stack is consistent with the narrow line  component for these lines the hot DOG stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' At z ∼ 5, all of the NIRCam LW filters are boosted by ∼ 20% − 100% for hot  DOGs and DSFGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The results of our synthetic photometry are shown  in Figure 2, which gives the flux ratio between our fidu-  cial and line-subtracted stacked spectrum and the DSFG  stack from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' For the former, we also  show the increase in flux seperated between the nar-  row and broad velocity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' On average, strong  narrow+broad rest-frame optical lines increase NIRCam  fluxes by factors of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='7, with corresponding  change in apparent magnitude by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The maximal  increase in flux occurs when any particular wide-band  filter is centered on the strong [NII]+Hα complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The  [OIII] and Hβ lines collectively increase the wide-band  flux maximally by ∼20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Their broad components and  those of [NII] and Hα increase synthetic flux densities  by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2× on average, accounting for ∼ 66% of the boost  for [OIII]+Hβ and 25% for [NII]+Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Medium-band  filters are more affected by the presence of strong emis-  sion lines and can be dominated by factors of ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 (1  mag) by emission lines when redshifted to the line’s rest  wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' For example, the F410M flux is increased  by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In fact, z = 5 is a spe-  cial regime where a boost in flux density is seen for all  NIRCam LW filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' While we do not show the increase  in flux attributed to the relatively weaker [OII] line on  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2, this effect is maximally ∼ 10% if we mask the  line following the methods outlined for [OIII]+Hβ and  [NII]+Hα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The strong lines in the DSFG stack from Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2017 which we have scaled to an Hα SFR of 100 M⊙ yr−1  (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1) increase broadband fluxes by up to a  factor of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Such boosting occurs over similar ranges  in redshift and to the same degree as found for the nar-  row line components in the hot DOG stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This demon-  strates that significant line contamination can be present  in the NIRCam photometry for IR-luminous galaxies  more normal than the relatively extreme hot DOGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Given extreme levels of attenuation in massive dust-  obscured galaxies at high-redshift, their rest-frame op-  6  10  11  12  13  14  15  16  17  Best-fit LBG redshift  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  Best-fit LBG  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 < zstack < 4  4 < zstack < 5  5 < zstack < 6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='75  Transmission  F115W  F150W  F200W  F277W  F356W  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='0  Observed wavelength [um]  100  101  102  F  [nJy]  CEERS  COSMOS-Web  z = 15 LBG Template (  = -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4)  Stack (this work) at z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='75  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  (Left) Allowed LBG redshift and UV slope β solutions when fitting the synthetic NIRCam flux densities of our stack  redshifted to z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5−4 (blue), z = 4−5 (green), and z = 5−6 (pink).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Posterior contours are drawn at the 16th, 50th, and 84th  percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' At z > 4 [NII]+Hα increase the NIRCam flux density more so than [OIII]+Hβ which does not allow the strong lines  to mask the red continuum and therefore precludes LBG solutions with β < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 < z < 4 [NII]+Hα falls within  F277W while [OIII]+Hβ is missed by F200W, allowing degenerate solutions with blue β ≤ −2 LBGs at z ∼ 14 − 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (Right)  Illustration of the degeneracy between z ∼ 15 candidates and z < 4 dusty galaxies with strong rest-frame optical/emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  In this example, we redshift our stacked spectrum to z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='75 where strong line emission boosts the F277W filter flux by 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  We then compute F200W, F277W, and F356W JWST/NIRCam photometry (black circles), assuming non-detections in F115W  and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We fit the synthetic photometry from the stack (circles) with an LBG template (red), deriving a photometric  redshift of zphot = 15 and UV spectral index β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Strong emission lines mask the red slope of the dusty template between  F277W and F356W, and the SED is further confused with the Lyman break falling halfway between F200W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Such scenarios are  possible given the relative filter depths of JWST Cycle 1 NIRCam extragalactic surveys in CEERS (blue, Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022b) and COSMOS-Web (pink, Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022) for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  tical spectra contain a combination of significantly red-  dened continuum with ≲ 5% of the total un-obscured  light escaping from the least obscured regions (Chap-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' With a combina-  tion of strong lines emergent from less obscured regions  on top of the very red continuum, ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1 − 1 µm pho-  tometry of dusty galaxies can mimic that of ultra-high  redshift LBG candidates in large surveys (Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' To quantify the parame-  ter space where this confusion is significant, we fit an  LBG template to the synthetic NIRCam flux densities  derived from our stacked spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We first normalize  the stack to a continuum flux on the order of ∼ 10 nJy  over λobs = 2 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 µm, and assume it to be undetected  in F115W and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This represents a plausible sce-  nario given the relative filter depths of JWST Cycle 1  extragalactic deep fields (Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Finkelstein  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022), and is similar to CEERS-  93316 (Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022) (CEERS2 2159, Finkelstein  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022b) − a z = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4 LBG candidate selected from  CEERS (Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  CEERS-93316 has a tentative 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='6σ SCUBA-2 detection  (Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022) and environmental evidence (Naidu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022b) both indicating a possible lower redshift  solution at z ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Figure 3 (Left) shows the 2D posterior distribution  in redshift and UV slope β for LBG template fits to  our stacked spectrum’s synthetic NIRCam flux densi-  ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We repeat the fitting analysis 1000 times after per-  turbing the input spectrum by the spectral uncertainty,  and in three redshift ranges for the stack: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 < z < 4,  4 < z < 5, and 5 < z < 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The cumulative EW of  Hα+[NII] is greater than EW([OIII]+Hβ) by a factor  of ∼ 3 which precludes LBG fits with β < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 when  the stack is redshifted to z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This is because both  features fall within a broadband filter and so the strong  lines do not mask the red continuum in the stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' At  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 < z < 4 F277W picks up the strong Hα+[NII] emis-  sion while [OIII]+Hβ is missed by F200W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This pro-  duces degenerate photometry with z ∼ 16 LBGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' At  z ∼ 16, the Lyman break falls halfway between F200W  mimicking the red slope of the dusty galaxy stack while  the very blue continuum mimics the F277W flux den-  sity of the line-contaminated stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In summary, the hot  DOG stack can reproduce very blue UV slopes β ∼ −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5  for zstack ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 − 4 but not for zstack > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' This supports  7  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Strong optical emission line characteristics  in our stacked spectrum  Line  EW (˚A)  FWHM (km s−1)  Hβ  45 ± 12  1450  Hαnarrow  222 ± 27  730  Hαbroad  540 ± 80  4000  [OIII]5007  127 ± 19  1000  [OIII]4959  43 ± 8  1000  [OIII]5007,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='broad  144 ± 49  7300  [OIII]4959,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='broad  48 ± 32  7300  [OI]6300  109 ± 20  1600  [OI]6363  38 ± 13  1600  [NII]6548  35 ± 8  730  [NII]6583  102 ± 17  730  [SII]6716  103 ± 18  1200  [SII]6730  43 ± 11  1200  AV (Hαnarrow /Hβ)  6 ± 1  AV (Hαtot /Hβ)  10 ± 1  the purity of the very blue NIRCam samples of Cullen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022 and Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, which predominantly  have β < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 and 7 < z < 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  As further demonstration of the confusion between our  stack at z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 − 4 and z ∼ 16 LBGs, we show in Fig-  ure 3 (Right) the LBG fit to our stack’s JWST/NIRCam  flux densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' At zstack = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='75 we find a best-fit LBG  solution with z = 16 and UV spectral index β = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Lower redshift (z < 4) solutions with red continuum  slopes and flux densities dominated by strong emis-  sion lines should be considered when fitting the very  blue (β < −2) spectral energy distributions (SEDs) of  z ∼ 16 candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The lower redshift solutions could be  ruled out with medium-band filters, longer wavelength  sampling using NIRCam’s redder filters, MIRI observa-  tions, and/or far-IR/sub-mm follow-up to detect cold  dust continuum and fine-structure lines (Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Measurements that strongly rule out UV spectral  indices β < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 and only allow lower−z solutions at  z > 4 should be particularly constraining against mas-  sive, IR-luminous interlopers with strong optical lines  provided they sample the SED with more than three  filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Based on the first analysis of JWST deep field obser-  vations at 5σ point-source depths between ∼ 28 − 29  MAG, the projected sky density of candidates at z > 10  is approximately 350 ±120 deg−2 (Donnan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Finkelstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Naidu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Harikane  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Although preliminary, these source counts  represent the population which could potentially be con-  taminated by low−z dusty interlopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In contrast, the  sky density of luminous IR galaxies with log LIR/L⊙ >  12 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5) and z ∼ 3 − 4 is 400 deg−2 (100 deg−2) (Casey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Zavala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' If we assume the sam-  ples of Finnerty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2020) and Casey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' (2017)  include a range of physically possible rest-frame opti-  cal properties for IR-bright galaxies (log LIR/L⊙ > 11),  then their similar number counts to ultra high-redshift  LBG candidates may be reason to be concerned about  contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The fainter dusty galaxy population  with log LIR/L⊙ < 11 are much more numerous based  on the general shape of 1 mm number counts (Fujimoto  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Gonz´alez-L´opez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020), and may also  be an important source of contamination as galaxies  fainter in the IR are less likely to be significantly ob-  scured in the rest-frame optical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Further spectroscopic  follow up is required to assess the purity of ultra-high  redshift catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  In the meantime, F150W dropouts  (z > 10) with β ∼ −2 and no/poor SED constraint  above NIRCam/F356W should be checked against pos-  sible intermediate-redshift dusty galaxy solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' SUMMARY AND CONCLUSION  In this Letter we test the response of JWST NIR-  Cam filters over broad rest-frame optical emission lines  in the average spectrum of hot, dust-obscured galaxies  at z ∼ 1 − 4 and dusty, star-forming galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' As an  empirical approach, we stack a sample of 21 IR lumi-  nous galaxies with rest-frame optical spectra from Keck  II/NIRES which we then compute synthetic photometry  for between z = 1 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Our main results are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We measure broad rest-frame optical emission  lines in the stack of z ∼ 1 − 4 hot, dust-obscured  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In particular, we measure [OIII] and Hα  EWs between 100 − 500 ˚A which are high relative  to normal star-forming galaxies at high-redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' After masking out strong emission features from  the spectrum, we measure synthetic NIRCam pho-  tometry with and without the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Narrow and  broad components for [OIII] and Hβ increase the  measured filter flux by 30%, and Hα+[NII] by 60%  on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Narrowband filters such as F410M  can have their flux increased by a factor of 2 − 3  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='7 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2 MAG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Rest-frame optical photometry of dusty galaxies  with strong nebular lines at z ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 − 4 could  8  be important contaminants in F150W dropout  (z > 10 candidate) catalogs as the strong lines  can help mask red UV spectral indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' However,  UV spectral indices β < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='5 are difficult for our  stacked spectrum to reproduce for interloper red-  shifts z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Distinguishing between different galaxy populations  with JWST imaging is a key first step towards test-  ing various aspects of galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  While this  work has focused just on JWST’s NIRCam filters, the  inclusion of deep MIRI photometry extending to longer  wavelengths will add significant constraint on various  redshift solutions to photometric fitting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' In the  absence of high SNR coverage in redder filters, far-  IR/sub-mm followup can help identify dusty galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  On the near horizon, ToLTEC on the Large Millimeter  Telescope (LMT) Alfonso Serrano will map multiple ex-  tragalactic fields (COSMOS, UDS, GOODS-S) down to  the LIRG limit at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='4, and 2 mm as part of “The  TolTEC Ultra-Deep Galaxy Survey,” a public legacy  program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  These public data sets are well suited to  quickly identify sub-mm bright DSFG counterparts to  JWST sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' is a member of Student Researchers United  (SRU-UAW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The data presented herein were obtained  at the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Keck Observatory, which is operated as  a scientific partnership among the California Institute  of Technology, the University of California, and the  National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' The  Observatory was made possible by the generous finan-  cial support of the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Keck Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We wish  to acknowledge the critical importance of the current  and recent Maunakea Observatories daycrew, techni-  cians, telescope operators, computer support, and office  staff employees, especially during the challenging times  presented by the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  Their exper-  tise, ingenuity, and dedication is indispensable to the  continued successful operation of these observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  The authors wish to recognize and acknowledge the  very significant cultural role and reverence that the  summit of Maunakea has always had within the indige-  nous Hawaiian community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' We are most fortunate to  have the opportunity to conduct observations from this  mountain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  9  REFERENCES  Assef, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Eisenhardt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Stern, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015,  ApJ, 804, 27, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/804/1/27  Bagley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Koekemoer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022, arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02495.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02495  Bakx, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Mitsuhashi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='13642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='13642  Bouwens, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Illingworth, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='06683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='06683  Boylan-Kolchin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01611  Bradley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Coe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Brammer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01777  Carnall, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Begley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', McLeod, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  MNRAS, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnrasl/slac136  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Cowie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013,  MNRAS, 436, 1919, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/stt1673  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Scoville, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Sanders, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2014,  ApJ, 796, 95, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/796/2/95  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Cooray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Killi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017, ApJ, 840,  101, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/aa6cb1  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Spilker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018, ApJ,  862, 77, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/aac82d  Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Kartaltepe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Drakos, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='07865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='07865  Chapman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Blain, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & Ivison, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2005, ApJ, 622, 772, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1086/428082  Cullen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', McLure, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', McLeod, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='04914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='04914  Cushing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Vacca, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & Rayner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2004,  PASP, 116, 362, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1086/382907  Cutri, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012, VizieR Online Data Catalog,  II/311  Donnan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', McLeod, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12356  Dudzeviˇci¯ut˙e, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Smail, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Swinbank, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020,  MNRAS, 494, 3828, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/staa769  Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Cirasuolo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & McLure, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2007,  MNRAS, 376, 1054,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='11453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='x  Eisenhardt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Tsai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012, ApJ,  755, 173, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/755/2/173  Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bagley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Arrabal Haro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022a, arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12474.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12474  Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bagley, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ferguson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2022b, arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='05792.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='05792  Finnerty, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Larson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Soifer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020, ApJ,  905, 16, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/abc3bf  Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ono, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2016, ApJS, 222, 1,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/0067-0049/222/1/1  Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Finkelstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Burgarella, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022  Gonz´alez-L´opez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Novak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Decarli, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020,  ApJ, 897, 91, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/ab765b  Gruppioni, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', B´ethermin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Loiacono, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020,  A&A, 643, A8, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1051/0004-6361/202038487  Harikane, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ouchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Oguri, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01612.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01612  Howell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Armus, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Mazzarella, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2010,  ApJ, 715, 572, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/715/1/572  Jun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Assef, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bauer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020, ApJ, 888,  110, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/ab5e7b  Kaasinen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', van Marrewijk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Popping, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='03754.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='03754  Kirkpatrick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Pope, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Sajina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015, ApJ, 814,  9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/814/1/9  Kirkpatrick, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Pope, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Alexander, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012,  ApJ, 759, 139, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/759/2/139  Koprowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Micha�lowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2017, MNRAS, 471, 4155, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/stx1843  Koss, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Trakhtenbrot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ricci, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2017, ApJ, 850,  74, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/aa8ec9  Labbe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', van Dokkum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Nelson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='12446  Long, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Lagos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02072  Maio, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & Viel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022  Manning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  ApJ, 925, 23, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/ac366a  Micha�lowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Dunlop, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Koprowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  2017, MNRAS, 469, 492, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/stx861  Murphy, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Chary, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Dickinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2011,  ApJ, 732, 126, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/732/2/126  Naidu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', van Dokkum, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022a,  arXiv e-prints, arXiv:2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='09434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='09434  10  Naidu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Oesch, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Setton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022b,  arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='02794  Roberts-Borsani, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Treu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='15639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='15639  Rowan-Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Farrah, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018,  A&A, 619, A169, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1051/0004-6361/201832671  Schaerer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & de Barros, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2009, A&A, 502, 423,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1051/0004-6361/200911781  Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Schenker, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Ellis, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013, ApJ,  763, 129, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/763/2/129  Topping, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Stark, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Endsley, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  arXiv e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01610  Tsai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Eisenhardt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2015, ApJ,  805, 90, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/805/2/90  Wilkins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Coulton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Caruana, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2013,  MNRAS, 435, 2885, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/stt1471  Wilkins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Lovell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Fairhurst, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2020,  MNRAS, 493, 6079, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/staa649  Wilkins, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Vijayan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Lovell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022,  MNRAS, 517, 3227, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1093/mnras/stac2548  Wilson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Henderson, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Herter, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2004,  in Society of Photo-Optical Instrumentation Engineers  (SPIE) Conference Series, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 5492, Ground-based  Instrumentation for Astronomy, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Moorwood  & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' Iye, 1295–1305  Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Tsai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Sayers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2012, ApJ, 756, 96,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1088/0004-637X/756/1/96  Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Jun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Assef, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2018, ApJ, 852, 96,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/aa9ff3  Yoon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Carilli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Fujimoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='08413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='08413  Yu-Yang Hsiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Coe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Abdurrouf, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='14123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='14123  Zackrisson, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Bergvall, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', & Leitet, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2008, ApJL, 676,  L9, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='1086/587030  Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Manning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2021,  ApJ, 909, 165, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='3847/1538-4357/abdb27  Zavala, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Buat, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', Casey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content=' 2022, arXiv  e-prints, arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='  https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='org/abs/2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
+page_content='01816' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNAyT4oBgHgl3EQfO_a9/content/2301.00017v1.pdf'}
diff --git a/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/2301.03500v1.pdf.txt b/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/2301.03500v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2f7aff0584fb14818621140c835c2d696472b176
--- /dev/null
+++ b/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/2301.03500v1.pdf.txt
@@ -0,0 +1,407 @@
+arXiv:2301.03500v1  [math.DG]  9 Jan 2023
+Generalized Ricci solitons and Einstein metrics on weak
+K-contact manifolds
+Vladimir Rovenski∗
+Abstract
+We study metric structures on a smooth manifold (introduced in our recent works [8,
+10] and called a weak contact metric structure and a weak K-structure) which generalize
+the metric contact and K-contact structures, and allow a new look at the classical theory.
+First, we characterize weak K-contact manifolds among all weak contact metric manifolds
+by the property well known for K-contact manifolds, and prove that a Riemannian manifold
+endowed with a unit Killing vector field forms a weak K-contact structure.
+Second, we
+find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a
+generalized Ricci soliton structure to be an Einstein manifold.
+Keywords: weak K-contact manifold, unit Killing vector field, Einstein metric, generalized
+Ricci soliton
+Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15
+1
+Introduction
+Contact geometry is of growing interest due to its important role in mechanics in explaining
+physical phenomena. In addition, many recent articles have been motivated by the question of
+how interesting self-similar solutions of the Ricci flow equation, i.e. Ricci solitons, can be for
+contact metric geometry. Some of them find conditions when a contact manifold equipped with
+a Ricci-type soliton structure carries a canonical (e.g., Einstein or constant curvature) metric,
+e.g. [2, 5, 7].
+K-contact manifolds (i.e. metric contact manifolds whose characteristic vector field generates
+a 1-parameter group of isometries) have been studied by several geometers, e.g., [1, 11], and it is
+seen that the K-contact structure is intermediate between the contact and Sasakian structures.
+The characteristic vector field ξ of the K-contact structure is a unit Killing vector field, and
+the influence of constant length Killing vector fields on the geometry of Riemannian manifolds
+has been studied by several authors from different points of view, e.g., [3, 4, 6]. An interesting
+result related to the above question is that a K-contact manifold equipped with generalised
+Ricci soliton structure has an Einstein metric, e.g., [5].
+In [10], we introduced the “weakened” metric structures on a smooth manifold (replacing the
+complex structure on the characteristic distribution with a nonsingular skew-symmetric tensor).
+They generalize the metric contact, K-contact, Sasakian and cosymplectic structures, and allow
+a new look at the classical theory. In [10], we build retraction of weak structures with positive
+partial Ricci curvature onto the subspace of classical structures. In [8], we proved that a weak
+Sasakian manifold is a weak K-contact manifold; and a weak almost contact metric manifold is
+weak Sasakian if and only if it is a Sasakian manifold. In this article we study weak K-contact
+manifolds using their Jacobi operator Rξ and Ricci curvature in the ξ-direction. Our goal is to
+show that the weak K-contact structure can be a useful tool for studying unit Killing vector
+fields on Riemannian manifolds, and that some results for K-contact manifolds can be extended
+∗Department of Mathematics, University of Haifa, Israel
+e-mail: vrovenski@univ.haifa.ac.il
+1
+
+to the case of weak K-contact manifolds. For example, we answer the question of when a weak
+K-contact manifold carries a generalized Ricci soliton structure or just an Einstein metric.
+The article is organized as follows. In Section 2, following the introductory Section 1, we
+recall basics of weak contact metric manifolds. In Section 3, we characterize (in Theorem 3.1)
+weak K-contact manifolds among all weakly contact metric manifolds by the property ϕ = −∇ξ
+(well known for K-contact manifolds), and prove (in Theorem 3.2) that a Riemannian manifold
+endowed with a unit Killing vector field forms a weak K-contact structure. In Section 4, for a
+weak K-contact manifold, we calculate (in Proposition 4.1) the Ricci curvature in the ξ-direction,
+then find (in Theorem 4.1) sufficient condition for such a manifold with parallel Ricci tensor to
+be an Einstein manifold. In Section 5, we find (in Theorem 5.1) sufficient conditions for a weak
+K-contact manifold admitting a generalized Ricci soliton structure to be an Einstein manifold.
+2
+Preliminaries
+Here, we recall (see [8, 9, 10]) basics of some metric structures that generalize the almost contact
+metric structure. A weak almost contact structure on a (2n + 1)-dimensional smooth manifold
+M is a set (ϕ, Q, ξ, η), where ϕ is a (1, 1)-tensor, Q is a nonsingular (1, 1)-tensor, ξ is the
+characteristic vector field and η is a dual 1-form, i.e. η(ξ) = 1, satisfying
+ϕ2 = −Q + η ⊗ ξ,
+Q ξ = ξ.
+(1)
+The form η determines a smooth 2n-dimensional contact distribution D := ker η, the collection
+of subspaces Dm = {X ∈ TmM : η(X) = 0} for m ∈ M. We assume that D is ϕ-invariant,
+ϕX ∈ D,
+X ∈ D,
+(2)
+as in the theory of almost contact structure [1, 11], where Q = id TM. By (1) and (2), the
+distribution D is invariant for Q: Q(D) = D. If there is a Riemannian metric g on M such that
+g(ϕX, ϕY ) = g(X, Q Y ) − η(X) η(Q Y ),
+X, Y ∈ XM,
+(3)
+then (ϕ, Q, ξ, η, g) is called a weak almost contact metric structure on M, and g is called a
+compatible metric.
+A weak almost contact manifold M(ϕ, Q, ξ, η) endowed with a compati-
+ble Riemannian metric g is called a weak almost contact metric manifold and is denoted by
+M(ϕ, Q, ξ, η, g).
+Putting Y = ξ in (3) and using Q ξ = ξ, we get, as in the classical theory, η(X) = g(X, ξ).
+In particular, ξ is g-orthogonal to D for any compatible metric g.
+For a weak almost contact structure on a smooth manifold M, the tensor ϕ has rank 2n and
+ϕ ξ = 0,
+η ◦ ϕ = 0,
+[Q, ϕ] = 0;
+moreover, for a weak almost contact metric structure, ϕ is skew-symmetric and Q is self-adjoint.
+Remark 2.1. According to [10], a weak almost contact structure admits a compatible metric if
+ϕ in (1)–(2) has a skew-symmetric representation, i.e. for any x ∈ M there exist a neighborhood
+Ux ⊂ M and a frame {ei} on Ux, for which ϕ has a skew-symmetric matrix. By (3), we get
+g(X, Q X) = g(ϕX, ϕX) > 0 for any nonzero vector X ∈ D, thus Q is positive definite.
+A weak contact metric structure is defined as a weak almost contact metric structure satis-
+fying Φ = dη, where Φ(X, Y ) = g(X, ϕY ) (X, Y ∈ XM) is the fundamental 2-form, and
+dη(X, Y ) = 1
+2 {X(η(Y )) − Y (η(X)) − η([X, Y ])},
+X, Y ∈ XM.
+(4)
+For weak contact metric structure, the distribution D is nowhere integrable, since g([X, ϕX], ξ) =
+2 dη(ϕX, X) = g(ϕX, ϕX) > 0 for any nonzero X ∈ D.
+2
+
+The Nijenhuis torsion [ϕ, ϕ] of ϕ is given by
+[ϕ, ϕ](X, Y ) = ϕ2[X, Y ] + [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ],
+X, Y ∈ XM.
+A weak almost contact structure (ϕ, Q, ξ, η) is called normal if the following tensor is zero:
+N (1)(X, Y ) = [ϕ, ϕ](X, Y ) + 2 dη(X, Y ) ξ,
+X, Y ∈ XM.
+(5)
+A weak K-contact manifold is defined as a weak contact metric manifold whose characteristic
+vector field ξ is Killing, i.e.
+(£ξ g)(X, Y ) := ξ(g(X, Y )) − g([ξ, X], Y ) − g(X, [ξ, Y ]) = g(∇Xξ, Y ) + g(∇Y ξ, X) = 0.
+(6)
+Here £ξ is the Lie derivative in the ξ-direction and ∇ is the Levi-Civita connection. A normal
+(weak) contact metric manifold is called a (weak) Sasakian manifold.
+The following tensors N (2), N (3) and N (4) are well known in the classical theory, see [1, 11]:
+N (2)(X, Y ) = (£ϕX η)(Y ) − (£ϕY η)(X)
+(4)
+= 2 dη(ϕX, Y ) − 2 dη(ϕY, X),
+N (3)(X) = (£ξ ϕ)X = [ξ, ϕX] − ϕ[ξ, X],
+N (4)(X) = (£ξ η)(X) = ξ(η(X)) − η([ξ, X])
+(4)
+= 2 dη(ξ, X).
+For a weak contact metric manifold, the tensors N (2) and N (4) vanish and the integral curves
+of ξ are geodesics; moreover, N (3) vanishes if and only if ξ is a Killing vector field, see [8].
+Definition 2.1 (see [8]). Two weak almost contact structures (ϕ, Q, ξ, η) and (ϕ′, Q′, ξ, η) on
+M are said to be homothetically equivalent if the following is valid for some real λ > 0:
+ϕ =
+√
+λ ϕ′,
+(7a)
+Q | D = λ Q′| D.
+(7b)
+Two weak contact metric structures (ϕ, Q, ξ, η, g) and (ϕ′, Q′, ξ, η, g′) on M are said to be ho-
+mothetically equivalent if they satisfy conditions (7a,b) and
+g| D = λ− 1
+2 g′| D,
+g(ξ, ·) = g′(ξ, ·).
+(7c)
+Lemma 2.1 (see [8]). Let (ϕ, Q, ξ, η) be a weak almost contact structure such that
+Q | D = λ idD,
+for some real λ > 0. Then the following is true:
+• (ϕ′, ξ, η) is an almost contact structure, where ϕ′ is given by (7a).
+• If (ϕ, Q, ξ, η, g) is a weak contact metric structure satisfying (7a,c), then (ϕ′, ξ, η, g′) is a
+contact metric structure.
+The following “small” (1,1)-tensor: ˜Q = Q − id is essential for the weak contact structure,
+and ˜Q = 0 means the classical contact geometry. Note that [ ˜Q, ϕ] = 0 and ˜Q ξ = 0.
+Lemma 2.2 ([8]). For a weak contact metric manifold, we have
+g((∇X ϕ)Y, Z) = 1
+2 g(N (1)(Y, Z), ϕX) + g(ϕX, ϕY ) η(Z) − g(ϕX, ϕZ) η(Y )
++ 1
+2 N (5)(X, Y, Z),
+(9)
+where the skew-symmetric with respect to Y and Z tensor N (5)(X, Y, Z) supplements the sequence
+of tensors N (i) (i = 1, 2, 3, 4) and is given by
+N (5)(X, Y, Z) = (ϕZ) (g(X, ˜QY )) − (ϕY ) (g(X, ˜QZ))
++ g([X, ϕZ], ˜QY ) − g([X, ϕY ], ˜QZ) + g([Y, ϕZ] − [Z, ϕY ] − ϕ[Y, Z], ˜QX).
+3
+
+In particular, (∇ξ ϕ)Y = 1
+2 N (5)(ξ, Y, ·) and
+N (5)(X, ξ, Z) = g(N (3)(Z), ˜QX),
+N (5)(ξ, Y, Z) = g([ξ, ϕZ], ˜QY ) − g([ξ, ϕY ], ˜QZ),
+N (5)(ξ, ξ, Z) = N (5)(ξ, Y, ξ) = 0.
+(10)
+3
+Killing vector fields of unit length
+Lemma 3.1. On a weak K-contact manifold, we have N (1)(ξ, ·) = 0 and
+N (5)(ξ, · , ·)
+=
+N (5)( · , ξ, ·) = 0,
+(11)
+£ξ ˜Q
+=
+∇ξ ˜Q = 0,
+(12)
+∇ξ ϕ
+=
+0.
+(13)
+Proof. By (5) and dη(ξ, ·) = 0 we have
+N (1)(ξ, X) = [ϕ, ϕ](X, ξ) = ϕ2[X, ξ] − ϕ[ϕX, ξ] = ϕN (3)(X) = 0.
+By [8, Lemma 3.1] with h = 0, we get N (5)(ξ, · , ·) = 0, £ξ ˜Q = 0 and
+g(Q ∇X ξ, Z) = g(ϕZ, QX) − 1
+2 N (5)(X, ξ, ϕZ).
+By (10)1 with N (3) = 0, we get N (5)( · , ξ, ·) = 0. We use [ϕ, ˜Q] = 0 to obtain ∇ξ ˜Q = 0:
+(£ξ ˜Q)X = [ξ, ˜QX] − ˜Q[ξ, X] = (∇ξ ˜Q)X + [ϕ, ˜Q]X = (∇ξ ˜Q)X.
+This completes the proof of (11) and (12). Next, from (9) with X = ξ we get (13).
+In the next theorem, we characterize weak K-contact manifolds among all weak contact
+metric manifolds by the following well known property of K-contact manifolds, see [1]:
+∇ ξ = −ϕ.
+(14)
+Theorem 3.1. A weak contact manifold is weak K-contact (that is ξ is a Killing vector field)
+if and only if (14) is valid.
+Proof. Let a weak contact manifold satisfy (14). By skew-symmetry of ϕ, we get (£ξ g)(X, Y ) =
+g(∇X ξ, Y ) + g(∇Y ξ, X) = −g(ϕX, Y ) − g(ϕY, X) = 0, thus, ξ is a Killing vector field.
+Conversely, let our manifold be weak K-contact. By (9) with Y = ξ, using N (1)(ξ, ·) = 0
+and N (5)(X, ξ, Z) = 0, we get
+g((∇X ϕ) ξ, Z) = 1
+2 g(N (1)(ξ, Z), ϕX) − g(ϕX, ϕZ) + 1
+2 N (5)(X, ξ, Z) = g(ϕ2X, Z).
+Hence, (∇X ϕ) ξ = ϕ2X. From this and 0 = ∇X (ϕ ξ) = (∇X ϕ) ξ + ϕ∇X ξ, we obtain ϕ∇X ξ =
+−ϕ2X. Since ϕ is nondegenerate on D, this completes the proof of (14).
+It is well known that a Riemannian manifold with a unit Killing vector field and the property
+RX,ξ ξ = X (X⊥ ξ) is a K-contact manifold, e.g., [11, Theorem 3.1] or [1, Proposition 7.4].
+We generalize this result in the following
+Theorem 3.2. A Riemannian manifold admitting a unit Killing vector field ξ is a weak K-
+contact manifold with certainly defined tensors: ϕ = −∇ ξ, see (14), and QX = RX,ξ ξ for
+X ∈ D.
+4
+
+Proof. Let η = g( · , ξ) and D = ker η. Put ϕX = −∇X ξ and QX = RX,ξ ξ for X ∈ D. Since ξ
+is a unit Killing vector field, we get ∇ξ ξ = 0 and ∇X∇Y ξ − ∇∇X Y ξ = Rξ,X ξ. Thus, ϕ ξ = 0
+and
+ϕ2X = ∇∇X ξ ξ = Rξ,X ξ = −QX
+(X ∈ D).
+Put Q ξ = ξ. Therefore, (1) is valid. Since ξ is Killing, we obtain
+dη(X, Y ) = 1
+2 (g(∇X ξ, Y ) − g(∇Y ξ, X)) = −g(∇Y ξ, X) = g(X, ϕY ) = Φ(X, Y ),
+that completes the proof.
+The sectional curvature of the plane containing ξ is called mixed, see [9].
+Corollary 3.1. Weak K-contact structure (ϕ, Q, ξ, η, g) with constant mixed sectional curvature
+K(ξ, X) = λ > 0 (X ∈ D) is homothetically equivalent to a K-contact structure (ϕ′, ξ, η, g′) after
+the transformation (7a,c).
+Proof. Note that K(ξ, X) = λ (X ∈ D) if and only if RX,ξ ξ = λ X (X ∈ D).
+By QX =
+RX,ξ ξ (X ∈ D), see Theorem 3.2, we get QX = λX (X ∈ D). By Lemma 2.1(ii), (ϕ′, ξ, η, g′)
+is a contact metric structure. Using (6), we get (£ξ g′)(X, Y ) = λ(£ξ g)(X, Y ) (X, Y ∈ D) and
+(£ξ g′)(ξ, ·) = 0. By £ξ g = 0, we get £ξ g′ = 0; thus (ϕ′, ξ, η, g′) is a K-contact structure.
+4
+The Jacobi operator and Ricci curvature in the ξ-direction
+Here, (M, ϕ, Q, ξ, η, g) is a (2n + 1)-dimensional weak K-contact manifold. The Jacobi operator
+Rξ in the ξ-direction is defined as Rξ(X) := RX, ξ ξ, e.g., [9]. The Ricci curvature in the ξ-
+direction is given by Ric(ξ, ξ) = � 2n
+i=1 g(Rei, ξ ξ, ei), where (ei) is any local orthonormal basis
+of D.
+Proposition 4.1. For a weak K-manifold, the following equalities are true:
+Rξ, X Y = (∇X ϕ)Y,
+(15)
+RX, ξ ξ = −ϕ2X = X − η(X) ξ + ˜Q X,
+(16)
+Ric(ξ, ξ) = 2 n + trace ˜Q.
+(17)
+Proof. Using (14), we derive
+RZ, X ξ = ∇Z(∇X ξ) − ∇X(∇Z ξ) − ∇[Z,X] ξ)
+= ∇X(ϕZ) − ∇Z(ϕX) + ϕ([Z, X]) = (∇X ϕ)Z − (∇Z ϕ)X.
+(18)
+Note that (∇XΦ)(Y, Z) = −g((∇X ϕ)Y, Z). Using condition dΦ = d2η = 0, we get
+(∇XΦ)(Y, Z) + (∇Y Φ)(Z, X) + (∇ZΦ)(X, Y ) = 0.
+(19)
+From (18), using (19) and skew-symmetry of Φ, we get (15):
+g(Rξ, X Y, Z) = g(RY, Z ξ, X)
+(18)
+= (∇Z Φ)(X, Y ) + (∇Y Φ)(Z, X)
+(19)
+= −(∇X Φ)(Y, Z) = g((∇X ϕ)Y, Z).
+By (15) with Y = ξ, using ϕ ξ = 0 and (14), we find
+Rξ,X ξ = (∇X ϕ) ξ = −ϕ∇X ξ = ϕ2X.
+This and (1)1 yield (16) for the Jacobi operator in the ξ-direction. By this, we get
+Ric(ξ, ξ)
+(16)
+= −
+� 2n
+i=1 g(ϕ2ei, ei)
+(1)
+=
+� 2n
+i=1 g(ei + ˜Qei, ei),
+where (ei) is any local orthonormal basis of D. Thus, (17) is true.
+5
+
+Corollary 4.1. For a weak K-contact manifold we have Ric(ξ, ξ) > 0.
+Proof. For any unit vector X ∈ D we get
+0 < g(ϕX, ϕX) = g(QX, X) = 1 + g( ˜QX, X),
+thus trace ˜Q > −2 n. Therefore, from (17) we get the statement.
+The following theorem generalizes a well known result, e.g., [11, Proposition 5.1].
+Theorem 4.1. A weak K-contact manifold with (∇ Ric)(ξ, ·) = 0 (in particular, the Ricci
+tensor is parallel) and trace ˜Q = const is an Einstein manifold of scalar curvature (2 n+1)(2 n+
+trace ˜Q).
+Proof. Differentiating (17) and using (14) and the conditions, we have
+0 = ∇Y (Ric(ξ, ξ)) = (∇Y Ric)(ξ, ξ) + 2 Ric(∇Y ξ, ξ)) = −2 Ric(ϕY, ξ)),
+hence Ric(Y, ξ) = (2 n + trace ˜Q) η(Y ). Differentiating this, then using
+X(η(Y )) = g(∇Xξ, Y ) = −g(ϕX, Y ) + g(∇XY, ξ)
+and assuming ∇XY = 0 at x ∈ M, gives
+(2 n + trace ˜Q) g(ϕY, X) = ∇X (Ric(Y, ξ)) = (∇X Ric)(Y, ξ) + 2 Ric(Y, ∇X ξ) = −2 Ric(Y, ϕX),
+and hence Ric(Y, ϕX) = (2 n + trace ˜Q) g(Y, ϕX). Therefore, we obtain
+Ric(X, Y ) = (2 n + trace ˜Q) g(X, Y )
+for any vector fields X and Y on M, which means that (M, g) is an Einstein manifold. Using
+the definition of scalar curvature, τ = trace Ric, we find τ = (2 n + 1)(2 n + trace ˜Q).
+5
+Generalized Ricci solitons
+The generalized Ricci soliton equation in a Riemannian manifold (M, g) is defined by [7],
+£X g = −2 c1X♭ ⊗ X♭ + 2 c2 Ric +2 λ g
+(20)
+for some smooth vector field X and real c1, c2 and λ. If X = ∇f in (20), then using the definition
+Hessf(X, Y ) = 1
+2 (£∇f g)(X,Y), we get the generalized gradient Ricci soliton equation
+Hessf = −c1df ⊗ df + c2 Ric +λ g .
+(21)
+For different values of c1, c2 and λ, equation (20) is a generalization of Killing equation (c1 =
+c2 = λ = 0), equation for homotheties (c1 = c2 = 0), Ricci soliton equation (c1 = 0, c2 = −1),
+vacuum near-horizon geometry equation (c1 = 1, c2 = 1/2), e.g., [5].
+First, we formulate some lemmas.
+Lemma 5.1. For a weak K-contact manifold the following holds:
+(£ξ(£X g))(Y, ξ) = g(X, Y ) + g(∇ξ∇ξ X, Y ) + Y g(∇ξ X, ξ)
+for all smooth vector fields X, Y with Y orthogonal to ξ.
+Proof. This uses the equalities ∇ξ ξ = 0 and (16), and is the same as for [5, Lemma 3.1].
+6
+
+Lemma 5.2 (see, for example, [5]). Let (M; g) be a Riemannian manifold and f be a smooth
+function on M. Then the following holds for every vector field Y :
+£ξ(df ⊗ df)(Y, ξ) = Y (ξ(f)) ξ(f) + Y (f) ξ(ξ(f)).
+Lemma 5.3. Let a weak K-contact manifold satisfies the generalized Ricci soliton equation.
+Then
+∇ξ∇f = (λ + 2 c2n + c2 trace ˜Q) ξ − c1ξ(f) ∇f.
+Proof. This uses (17) and (21), and is similar to the proof of [5, Lemma 3.3]. By (17) we get
+λ η(Y ) + c2 Ric(ξ, Y ) = (λ + 2 c2n + c2 trace ˜Q) η(Y ).
+(22)
+Using (21) and (22), we get
+Hessf(ξ, Y ) = −c1ξ(f) g(∇f, Y ) + (λ + 2 c2n + c2 trace ˜Q) η(Y ).
+(23)
+Thus, (23) and the definition of the Hessian (21) complete the proof.
+Recall that the Ricci curvature of any K-contact manifold satisfies the following condition:
+Ric(X, ξ) = 0
+(X ∈ D).
+(24)
+The following theorem generalizes [5, Theorem 3.1].
+Theorem 5.1. Let a weak K-contact manifold satisfies the generalized gradient Ricci soliton
+equation (21) with c1(λ+2 c2n+c2 trace ˜Q) ̸= −1. Suppose that conditions trace ˜Q = const and
+(24) are true. Then f = const. Furthermore, if c2 ̸= 0, then the manifold is an Einstein one.
+Proof. Let Y ∈ D. Then by Lemma 5.1 with X = ∇f, we obtain
+2 (£ξ(Hessf))(Y, ξ) = Y (f) + g(∇ξ∇ξ∇f, Y ) + Y g(∇ξ∇f, ξ).
+(25)
+Using Lemma 5.3 in (25) and the property ∇ξ ξ = 0 yields
+2 (£ξ(Hessf))(Y, ξ) = Y (f) − c1g(∇ξ(ξ(f)∇f), Y )
++ (λ + 2 cn2 + c2 trace ˜Q) Y − c1Y (ξ(f)2).
+(26)
+Using Lemma 5.3 with Y ∈ D, from (26) it follows that
+2 (£ξ(Hessf))(Y, ξ) = Y (f) − c1ξ(ξ(f)) Y (f) + c2
+1ξ(f)2Y (f) − 2 c1ξ(f)Y (ξ(f)).
+(27)
+Since ξ is a Killing vector field, thus £ξ g = 0, this implies £ξ Ric = 0. Using the above fact
+and applying the Lie derivative to equation (21), gives
+2 (£ξ(Hessf))(Y, ξ) = −2 c1(£ξ(df ⊗ df))(Y, ξ).
+(28)
+Using (27), (28) and Lemma 5.2, we obtain
+Y (f)
+�
+1 + c1ξ(ξ(f)) + c2
+1 ξ(f)2�
+= 0.
+(29)
+By Lemma 5.3, we get
+c1ξ(ξ(f)) = c1 ξ(g(ξ, ∇f)) = c1g(ξ, ∇ξ∇f) = c1(λ + 2 c2n + c2 trace ˜Q) − c2
+1 ξ(f)2.
+(30)
+Using (29) in (30), we get Y (f)(c1(λ + 2 c2n + c2 trace ˜Q) + 1) = 0. This implies Y (f) = 0
+provided by c1(λ + 2 c2n + c2 trace ˜Q) + 1 ̸= 0. Hence, ∇f is parallel to ξ. Taking the covariant
+derivative of ∇f = ξ(f) ξ and using (14), we obtain
+g(∇X ∇f, Y ) = X(ξ(f)) η(Y ) − ξ(f) g(ϕX, Y ),
+X, Y ∈ XM.
+From this, by symmetry of Hessf, i.e. g(∇X ∇f, Y ) = g(∇Y ∇f, X), we get ξ(f) g(ϕX, Y ) = 0.
+For Y = ϕX for some X ̸= 0, since g(ϕX, ϕX) > 0, we get ξ(f) = 0; so ∇f = 0, i.e. f = const.
+Thus, from (21) and c2 ̸= 0 it follows that the manifold is an Einstein manifold.
+7
+
+6
+Conclusion
+It is shown that the weak K-contact structure is a useful tool for studying unit Killing vector
+fields on Riemannian manifolds and that some results for K-contact manifolds can be extended
+to the case of weak K-contact manifolds. In conclusion, we ask about “weak” analogues of
+the following results mentioned in [5, Remark 3.2]: a compact K-contact Einstein manifold is
+a Sasakian manifold, thus, a compact K-contact manifold admitting generalized Ricci soliton
+sructure is a Sasakian manifold.
+References
+[1] D. Blair, Riemannian geometry of contact and symplectic manifolds, 2nd edition, Springer-
+Verlag, New York, 2010.
+[2]
+D.E. Blair,
+A Survey of Riemannian contact geometry,
+Complex Manifolds, 6 (2019),
+31–64. https://doi.org/10.1515/coma-2019-0002
+[3] V.N. Berestovskij, and Yu.G. Nikonorov, Killing vector fields of constant length on Rie-
+mannian manifolds, Sib. Math. J., 49:3 (2008), 395–407.
+[4] S. Deshmukh, and O. Belova, On Killing vector fields on Riemannian manifolds, Mathe-
+matics, 9, 259 (2021), 1–17. https://doi.org/10.3390/math9030259
+[5] G. Ghosh, and U.C. De, Generalized Ricci soliton on K-contact manifolds, Math. Sci.
+Appl. E-Notes, 8 (2020), 165–169.
+[6] Y.G. Nikonorov, Spectral properties of Killing vector fields of constant length and bounded
+Killing vector fields, in Operator Theory and Differential Equations. Trends in Mathematics
+(eds. A.G. Kusraev and Z.D. Totieva), Birkh¨auser, Cham, (2021), 143–154.
+[7]
+P. Nurowski, and M. Randall,
+Generalised Ricci solitons,
+J. Geom. Anal., 26 (2016),
+1280–1345.
+[8] V. Rovenski, and D.S. Patra, On the rigidity of the Sasakian structure and characterization
+of cosymplectic manifolds, preprint, arXiv:2203.04597 v2, 2022, 15 pp.
+[9] V. Rovenski, and P.G. Walczak, Extrinsic geometry of foliations, Progress in Mathematics,
+vol. 339, Birkh¨auser, Cham, 2021.
+[10] V. Rovenski, and R. Wolak, New metric structures on g-foliations, Indagationes Mathe-
+maticae, 33 (2022), 518–532.
+[11] K. Yano, and M. Kon, Structures on Manifolds, Vol. 3 of Series in Pure Math. World
+Scientific Publ. Co., Singapore, 1985.
+8
+
diff --git a/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/load_file.txt b/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..e25c0e670e63e9ee0ef40be39cd8304940f92b1e
--- /dev/null
+++ b/zdE1T4oBgHgl3EQf4QX8/content/tmp_files/load_file.txt
@@ -0,0 +1,308 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf,len=307
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='03500v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='DG]  9 Jan 2023  Generalized Ricci solitons and Einstein metrics on weak  K-contact manifolds  Vladimir Rovenski∗  Abstract  We study metric structures on a smooth manifold (introduced in our recent works [8,  10] and called a weak contact metric structure and a weak K-structure) which generalize  the metric contact and K-contact structures, and allow a new look at the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  First, we characterize weak K-contact manifolds among all weak contact metric manifolds  by the property well known for K-contact manifolds, and prove that a Riemannian manifold  endowed with a unit Killing vector field forms a weak K-contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Second, we  find sufficient conditions for a weak K-contact manifold with parallel Ricci tensor or with a  generalized Ricci soliton structure to be an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Keywords: weak K-contact manifold, unit Killing vector field, Einstein metric, generalized  Ricci soliton  Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15  1  Introduction  Contact geometry is of growing interest due to its important role in mechanics in explaining  physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In addition, many recent articles have been motivated by the question of  how interesting self-similar solutions of the Ricci flow equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Ricci solitons, can be for  contact metric geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Some of them find conditions when a contact manifold equipped with  a Ricci-type soliton structure carries a canonical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', Einstein or constant curvature) metric,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' [2, 5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  K-contact manifolds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' metric contact manifolds whose characteristic vector field generates  a 1-parameter group of isometries) have been studied by several geometers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [1, 11], and it is  seen that the K-contact structure is intermediate between the contact and Sasakian structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The characteristic vector field ξ of the K-contact structure is a unit Killing vector field, and  the influence of constant length Killing vector fields on the geometry of Riemannian manifolds  has been studied by several authors from different points of view, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [3, 4, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' An interesting  result related to the above question is that a K-contact manifold equipped with generalised  Ricci soliton structure has an Einstein metric, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  In [10], we introduced the “weakened” metric structures on a smooth manifold (replacing the  complex structure on the characteristic distribution with a nonsingular skew-symmetric tensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  They generalize the metric contact, K-contact, Sasakian and cosymplectic structures, and allow  a new look at the classical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In [10], we build retraction of weak structures with positive  partial Ricci curvature onto the subspace of classical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In [8], we proved that a weak  Sasakian manifold is a weak K-contact manifold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' and a weak almost contact metric manifold is  weak Sasakian if and only if it is a Sasakian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In this article we study weak K-contact  manifolds using their Jacobi operator Rξ and Ricci curvature in the ξ-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Our goal is to  show that the weak K-contact structure can be a useful tool for studying unit Killing vector  fields on Riemannian manifolds, and that some results for K-contact manifolds can be extended  ∗Department of Mathematics, University of Haifa, Israel  e-mail: vrovenski@univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='haifa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='il  1  to the case of weak K-contact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For example, we answer the question of when a weak  K-contact manifold carries a generalized Ricci soliton structure or just an Einstein metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In Section 2, following the introductory Section 1, we  recall basics of weak contact metric manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In Section 3, we characterize (in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1)  weak K-contact manifolds among all weakly contact metric manifolds by the property ϕ = −∇ξ  (well known for K-contact manifolds), and prove (in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2) that a Riemannian manifold  endowed with a unit Killing vector field forms a weak K-contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In Section 4, for a  weak K-contact manifold, we calculate (in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1) the Ricci curvature in the ξ-direction,  then find (in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1) sufficient condition for such a manifold with parallel Ricci tensor to  be an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In Section 5, we find (in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1) sufficient conditions for a weak  K-contact manifold admitting a generalized Ricci soliton structure to be an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  2  Preliminaries  Here, we recall (see [8, 9, 10]) basics of some metric structures that generalize the almost contact  metric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A weak almost contact structure on a (2n + 1)-dimensional smooth manifold  M is a set (ϕ, Q, ξ, η), where ϕ is a (1, 1)-tensor, Q is a nonsingular (1, 1)-tensor, ξ is the  characteristic vector field and η is a dual 1-form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' η(ξ) = 1, satisfying  ϕ2 = −Q + η ⊗ ξ,  Q ξ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (1)  The form η determines a smooth 2n-dimensional contact distribution D := ker η, the collection  of subspaces Dm = {X ∈ TmM : η(X) = 0} for m ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' We assume that D is ϕ-invariant,  ϕX ∈ D,  X ∈ D,  (2)  as in the theory of almost contact structure [1, 11], where Q = id TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By (1) and (2), the  distribution D is invariant for Q: Q(D) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' If there is a Riemannian metric g on M such that  g(ϕX, ϕY ) = g(X, Q Y ) − η(X) η(Q Y ),  X, Y ∈ XM,  (3)  then (ϕ, Q, ξ, η, g) is called a weak almost contact metric structure on M, and g is called a  compatible metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  A weak almost contact manifold M(ϕ, Q, ξ, η) endowed with a compati-  ble Riemannian metric g is called a weak almost contact metric manifold and is denoted by  M(ϕ, Q, ξ, η, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Putting Y = ξ in (3) and using Q ξ = ξ, we get, as in the classical theory, η(X) = g(X, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  In particular, ξ is g-orthogonal to D for any compatible metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  For a weak almost contact structure on a smooth manifold M, the tensor ϕ has rank 2n and  ϕ ξ = 0,  η ◦ ϕ = 0,  [Q, ϕ] = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  moreover, for a weak almost contact metric structure, ϕ is skew-symmetric and Q is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' According to [10], a weak almost contact structure admits a compatible metric if  ϕ in (1)–(2) has a skew-symmetric representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' for any x ∈ M there exist a neighborhood  Ux ⊂ M and a frame {ei} on Ux, for which ϕ has a skew-symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By (3), we get  g(X, Q X) = g(ϕX, ϕX) > 0 for any nonzero vector X ∈ D, thus Q is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  A weak contact metric structure is defined as a weak almost contact metric structure satis-  fying Φ = dη, where Φ(X, Y ) = g(X, ϕY ) (X, Y ∈ XM) is the fundamental 2-form, and  dη(X, Y ) = 1  2 {X(η(Y )) − Y (η(X)) − η([X, Y ])},  X, Y ∈ XM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (4)  For weak contact metric structure, the distribution D is nowhere integrable, since g([X, ϕX], ξ) =  2 dη(ϕX, X) = g(ϕX, ϕX) > 0 for any nonzero X ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  2  The Nijenhuis torsion [ϕ, ϕ] of ϕ is given by  [ϕ, ϕ](X, Y ) = ϕ2[X, Y ] + [ϕX, ϕY ] − ϕ[ϕX, Y ] − ϕ[X, ϕY ],  X, Y ∈ XM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  A weak almost contact structure (ϕ, Q, ξ, η) is called normal if the following tensor is zero:  N (1)(X, Y ) = [ϕ, ϕ](X, Y ) + 2 dη(X, Y ) ξ,  X, Y ∈ XM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (5)  A weak K-contact manifold is defined as a weak contact metric manifold whose characteristic  vector field ξ is Killing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (£ξ g)(X, Y ) := ξ(g(X, Y )) − g([ξ, X], Y ) − g(X, [ξ, Y ]) = g(∇Xξ, Y ) + g(∇Y ξ, X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (6)  Here £ξ is the Lie derivative in the ξ-direction and ∇ is the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A normal  (weak) contact metric manifold is called a (weak) Sasakian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The following tensors N (2), N (3) and N (4) are well known in the classical theory, see [1, 11]:  N (2)(X, Y ) = (£ϕX η)(Y ) − (£ϕY η)(X)  (4)  = 2 dη(ϕX, Y ) − 2 dη(ϕY, X),  N (3)(X) = (£ξ ϕ)X = [ξ, ϕX] − ϕ[ξ, X],  N (4)(X) = (£ξ η)(X) = ξ(η(X)) − η([ξ, X])  (4)  = 2 dη(ξ, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  For a weak contact metric manifold, the tensors N (2) and N (4) vanish and the integral curves  of ξ are geodesics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' moreover, N (3) vanishes if and only if ξ is a Killing vector field, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1 (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Two weak almost contact structures (ϕ, Q, ξ, η) and (ϕ′, Q′, ξ, η) on  M are said to be homothetically equivalent if the following is valid for some real λ > 0:  ϕ =  √  λ ϕ′,  (7a)  Q | D = λ Q′| D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (7b)  Two weak contact metric structures (ϕ, Q, ξ, η, g) and (ϕ′, Q′, ξ, η, g′) on M are said to be ho-  mothetically equivalent if they satisfy conditions (7a,b) and  g| D = λ− 1  2 g′| D,  g(ξ, ·) = g′(ξ, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (7c)  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1 (see [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let (ϕ, Q, ξ, η) be a weak almost contact structure such that  Q | D = λ idD,  for some real λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Then the following is true:  (ϕ′, ξ, η) is an almost contact structure, where ϕ′ is given by (7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  If (ϕ, Q, ξ, η, g) is a weak contact metric structure satisfying (7a,c), then (ϕ′, ξ, η, g′) is a  contact metric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The following “small” (1,1)-tensor: ˜Q = Q − id is essential for the weak contact structure,  and ˜Q = 0 means the classical contact geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Note that [ ˜Q, ϕ] = 0 and ˜Q ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2 ([8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For a weak contact metric manifold, we have  g((∇X ϕ)Y, Z) = 1  2 g(N (1)(Y, Z), ϕX) + g(ϕX, ϕY ) η(Z) − g(ϕX, ϕZ) η(Y )  + 1  2 N (5)(X, Y, Z),  (9)  where the skew-symmetric with respect to Y and Z tensor N (5)(X, Y, Z) supplements the sequence  of tensors N (i) (i = 1, 2, 3, 4) and is given by  N (5)(X, Y, Z) = (ϕZ) (g(X, ˜QY )) − (ϕY ) (g(X, ˜QZ))  + g([X, ϕZ], ˜QY ) − g([X, ϕY ], ˜QZ) + g([Y, ϕZ] − [Z, ϕY ] − ϕ[Y, Z], ˜QX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  3  In particular, (∇ξ ϕ)Y = 1  2 N (5)(ξ, Y, ·) and  N (5)(X, ξ, Z) = g(N (3)(Z), ˜QX),  N (5)(ξ, Y, Z) = g([ξ, ϕZ], ˜QY ) − g([ξ, ϕY ], ˜QZ),  N (5)(ξ, ξ, Z) = N (5)(ξ, Y, ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (10)  3  Killing vector fields of unit length  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' On a weak K-contact manifold, we have N (1)(ξ, ·) = 0 and  N (5)(ξ, · , ·)  =  N (5)( · , ξ, ·) = 0,  (11)  £ξ ˜Q  =  ∇ξ ˜Q = 0,  (12)  ∇ξ ϕ  =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By (5) and dη(ξ, ·) = 0 we have  N (1)(ξ, X) = [ϕ, ϕ](X, ξ) = ϕ2[X, ξ] − ϕ[ϕX, ξ] = ϕN (3)(X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  By [8, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1] with h = 0, we get N (5)(ξ, · , ·) = 0, £ξ ˜Q = 0 and  g(Q ∇X ξ, Z) = g(ϕZ, QX) − 1  2 N (5)(X, ξ, ϕZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  By (10)1 with N (3) = 0, we get N (5)( · , ξ, ·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' We use [ϕ, ˜Q] = 0 to obtain ∇ξ ˜Q = 0:  (£ξ ˜Q)X = [ξ, ˜QX] − ˜Q[ξ, X] = (∇ξ ˜Q)X + [ϕ, ˜Q]X = (∇ξ ˜Q)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  This completes the proof of (11) and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Next, from (9) with X = ξ we get (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  In the next theorem, we characterize weak K-contact manifolds among all weak contact  metric manifolds by the following well known property of K-contact manifolds, see [1]:  ∇ ξ = −ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (14)  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A weak contact manifold is weak K-contact (that is ξ is a Killing vector field)  if and only if (14) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let a weak contact manifold satisfy (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By skew-symmetry of ϕ, we get (£ξ g)(X, Y ) =  g(∇X ξ, Y ) + g(∇Y ξ, X) = −g(ϕX, Y ) − g(ϕY, X) = 0, thus, ξ is a Killing vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Conversely, let our manifold be weak K-contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By (9) with Y = ξ, using N (1)(ξ, ·) = 0  and N (5)(X, ξ, Z) = 0, we get  g((∇X ϕ) ξ, Z) = 1  2 g(N (1)(ξ, Z), ϕX) − g(ϕX, ϕZ) + 1  2 N (5)(X, ξ, Z) = g(ϕ2X, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Hence, (∇X ϕ) ξ = ϕ2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' From this and 0 = ∇X (ϕ ξ) = (∇X ϕ) ξ + ϕ∇X ξ, we obtain ϕ∇X ξ =  −ϕ2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Since ϕ is nondegenerate on D, this completes the proof of (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  It is well known that a Riemannian manifold with a unit Killing vector field and the property  RX,ξ ξ = X (X⊥ ξ) is a K-contact manifold, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [11, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1] or [1, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  We generalize this result in the following  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A Riemannian manifold admitting a unit Killing vector field ξ is a weak K-  contact manifold with certainly defined tensors: ϕ = −∇ ξ, see (14), and QX = RX,ξ ξ for  X ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let η = g( · , ξ) and D = ker η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Put ϕX = −∇X ξ and QX = RX,ξ ξ for X ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Since ξ  is a unit Killing vector field, we get ∇ξ ξ = 0 and ∇X∇Y ξ − ∇∇X Y ξ = Rξ,X ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Thus, ϕ ξ = 0  and  ϕ2X = ∇∇X ξ ξ = Rξ,X ξ = −QX  (X ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Put Q ξ = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Therefore, (1) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Since ξ is Killing, we obtain  dη(X, Y ) = 1  2 (g(∇X ξ, Y ) − g(∇Y ξ, X)) = −g(∇Y ξ, X) = g(X, ϕY ) = Φ(X, Y ),  that completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The sectional curvature of the plane containing ξ is called mixed, see [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Weak K-contact structure (ϕ, Q, ξ, η, g) with constant mixed sectional curvature  K(ξ, X) = λ > 0 (X ∈ D) is homothetically equivalent to a K-contact structure (ϕ′, ξ, η, g′) after  the transformation (7a,c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Note that K(ξ, X) = λ (X ∈ D) if and only if RX,ξ ξ = λ X (X ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  By QX =  RX,ξ ξ (X ∈ D), see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2, we get QX = λX (X ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1(ii), (ϕ′, ξ, η, g′)  is a contact metric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Using (6), we get (£ξ g′)(X, Y ) = λ(£ξ g)(X, Y ) (X, Y ∈ D) and  (£ξ g′)(ξ, ·) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By £ξ g = 0, we get £ξ g′ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' thus (ϕ′, ξ, η, g′) is a K-contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  4  The Jacobi operator and Ricci curvature in the ξ-direction  Here, (M, ϕ, Q, ξ, η, g) is a (2n + 1)-dimensional weak K-contact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' The Jacobi operator  Rξ in the ξ-direction is defined as Rξ(X) := RX, ξ ξ, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' The Ricci curvature in the ξ-  direction is given by Ric(ξ, ξ) = � 2n  i=1 g(Rei, ξ ξ, ei), where (ei) is any local orthonormal basis  of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For a weak K-manifold, the following equalities are true:  Rξ, X Y = (∇X ϕ)Y,  (15)  RX, ξ ξ = −ϕ2X = X − η(X) ξ + ˜Q X,  (16)  Ric(ξ, ξ) = 2 n + trace ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (17)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Using (14), we derive  RZ, X ξ = ∇Z(∇X ξ) − ∇X(∇Z ξ) − ∇[Z,X] ξ)  = ∇X(ϕZ) − ∇Z(ϕX) + ϕ([Z, X]) = (∇X ϕ)Z − (∇Z ϕ)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (18)  Note that (∇XΦ)(Y, Z) = −g((∇X ϕ)Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Using condition dΦ = d2η = 0, we get  (∇XΦ)(Y, Z) + (∇Y Φ)(Z, X) + (∇ZΦ)(X, Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (19)  From (18), using (19) and skew-symmetry of Φ, we get (15):  g(Rξ, X Y, Z) = g(RY, Z ξ, X)  (18)  = (∇Z Φ)(X, Y ) + (∇Y Φ)(Z, X)  (19)  = −(∇X Φ)(Y, Z) = g((∇X ϕ)Y, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  By (15) with Y = ξ, using ϕ ξ = 0 and (14), we find  Rξ,X ξ = (∇X ϕ) ξ = −ϕ∇X ξ = ϕ2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  This and (1)1 yield (16) for the Jacobi operator in the ξ-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By this, we get  Ric(ξ, ξ)  (16)  = −  � 2n  i=1 g(ϕ2ei, ei)  (1)  =  � 2n  i=1 g(ei + ˜Qei, ei),  where (ei) is any local orthonormal basis of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Thus, (17) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  5  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For a weak K-contact manifold we have Ric(ξ, ξ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For any unit vector X ∈ D we get  0 < g(ϕX, ϕX) = g(QX, X) = 1 + g( ˜QX, X),  thus trace ˜Q > −2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Therefore, from (17) we get the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  The following theorem generalizes a well known result, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [11, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A weak K-contact manifold with (∇ Ric)(ξ, ·) = 0 (in particular, the Ricci  tensor is parallel) and trace ˜Q = const is an Einstein manifold of scalar curvature (2 n+1)(2 n+  trace ˜Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Differentiating (17) and using (14) and the conditions, we have  0 = ∇Y (Ric(ξ, ξ)) = (∇Y Ric)(ξ, ξ) + 2 Ric(∇Y ξ, ξ)) = −2 Ric(ϕY, ξ)),  hence Ric(Y, ξ) = (2 n + trace ˜Q) η(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Differentiating this, then using  X(η(Y )) = g(∇Xξ, Y ) = −g(ϕX, Y ) + g(∇XY, ξ)  and assuming ∇XY = 0 at x ∈ M, gives  (2 n + trace ˜Q) g(ϕY, X) = ∇X (Ric(Y, ξ)) = (∇X Ric)(Y, ξ) + 2 Ric(Y, ∇X ξ) = −2 Ric(Y, ϕX),  and hence Ric(Y, ϕX) = (2 n + trace ˜Q) g(Y, ϕX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Therefore, we obtain  Ric(X, Y ) = (2 n + trace ˜Q) g(X, Y )  for any vector fields X and Y on M, which means that (M, g) is an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Using  the definition of scalar curvature, τ = trace Ric, we find τ = (2 n + 1)(2 n + trace ˜Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  5  Generalized Ricci solitons  The generalized Ricci soliton equation in a Riemannian manifold (M, g) is defined by [7],  £X g = −2 c1X♭ ⊗ X♭ + 2 c2 Ric +2 λ g  (20)  for some smooth vector field X and real c1, c2 and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' If X = ∇f in (20), then using the definition  Hessf(X, Y ) = 1  2 (£∇f g)(X,Y), we get the generalized gradient Ricci soliton equation  Hessf = −c1df ⊗ df + c2 Ric +λ g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (21)  For different values of c1, c2 and λ, equation (20) is a generalization of Killing equation (c1 =  c2 = λ = 0), equation for homotheties (c1 = c2 = 0), Ricci soliton equation (c1 = 0, c2 = −1),  vacuum near-horizon geometry equation (c1 = 1, c2 = 1/2), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  First, we formulate some lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' For a weak K-contact manifold the following holds:  (£ξ(£X g))(Y, ξ) = g(X, Y ) + g(∇ξ∇ξ X, Y ) + Y g(∇ξ X, ξ)  for all smooth vector fields X, Y with Y orthogonal to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' This uses the equalities ∇ξ ξ = 0 and (16), and is the same as for [5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  6  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2 (see, for example, [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' g) be a Riemannian manifold and f be a smooth  function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Then the following holds for every vector field Y :  £ξ(df ⊗ df)(Y, ξ) = Y (ξ(f)) ξ(f) + Y (f) ξ(ξ(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let a weak K-contact manifold satisfies the generalized Ricci soliton equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Then  ∇ξ∇f = (λ + 2 c2n + c2 trace ˜Q) ξ − c1ξ(f) ∇f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' This uses (17) and (21), and is similar to the proof of [5, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' By (17) we get  λ η(Y ) + c2 Ric(ξ, Y ) = (λ + 2 c2n + c2 trace ˜Q) η(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (22)  Using (21) and (22), we get  Hessf(ξ, Y ) = −c1ξ(f) g(∇f, Y ) + (λ + 2 c2n + c2 trace ˜Q) η(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (23)  Thus, (23) and the definition of the Hessian (21) complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Recall that the Ricci curvature of any K-contact manifold satisfies the following condition:  Ric(X, ξ) = 0  (X ∈ D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (24)  The following theorem generalizes [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let a weak K-contact manifold satisfies the generalized gradient Ricci soliton  equation (21) with c1(λ+2 c2n+c2 trace ˜Q) ̸= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Suppose that conditions trace ˜Q = const and  (24) are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Then f = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Furthermore, if c2 ̸= 0, then the manifold is an Einstein one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Let Y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1 with X = ∇f, we obtain  2 (£ξ(Hessf))(Y, ξ) = Y (f) + g(∇ξ∇ξ∇f, Y ) + Y g(∇ξ∇f, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (25)  Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3 in (25) and the property ∇ξ ξ = 0 yields  2 (£ξ(Hessf))(Y, ξ) = Y (f) − c1g(∇ξ(ξ(f)∇f), Y )  + (λ + 2 cn2 + c2 trace ˜Q) Y − c1Y (ξ(f)2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (26)  Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3 with Y ∈ D, from (26) it follows that  2 (£ξ(Hessf))(Y, ξ) = Y (f) − c1ξ(ξ(f)) Y (f) + c2  1ξ(f)2Y (f) − 2 c1ξ(f)Y (ξ(f)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (27)  Since ξ is a Killing vector field, thus £ξ g = 0, this implies £ξ Ric = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Using the above fact  and applying the Lie derivative to equation (21), gives  2 (£ξ(Hessf))(Y, ξ) = −2 c1(£ξ(df ⊗ df))(Y, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (28)  Using (27), (28) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2, we obtain  Y (f)  �  1 + c1ξ(ξ(f)) + c2  1 ξ(f)2�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (29)  By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3, we get  c1ξ(ξ(f)) = c1 ξ(g(ξ, ∇f)) = c1g(ξ, ∇ξ∇f) = c1(λ + 2 c2n + c2 trace ˜Q) − c2  1 ξ(f)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  (30)  Using (29) in (30), we get Y (f)(c1(λ + 2 c2n + c2 trace ˜Q) + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' This implies Y (f) = 0  provided by c1(λ + 2 c2n + c2 trace ˜Q) + 1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Hence, ∇f is parallel to ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Taking the covariant  derivative of ∇f = ξ(f) ξ and using (14), we obtain  g(∇X ∇f, Y ) = X(ξ(f)) η(Y ) − ξ(f) g(ϕX, Y ),  X, Y ∈ XM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  From this, by symmetry of Hessf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' g(∇X ∇f, Y ) = g(∇Y ∇f, X), we get ξ(f) g(ϕX, Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  For Y = ϕX for some X ̸= 0, since g(ϕX, ϕX) > 0, we get ξ(f) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' so ∇f = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' f = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Thus, from (21) and c2 ̸= 0 it follows that the manifold is an Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  7  6  Conclusion  It is shown that the weak K-contact structure is a useful tool for studying unit Killing vector  fields on Riemannian manifolds and that some results for K-contact manifolds can be extended  to the case of weak K-contact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' In conclusion, we ask about “weak” analogues of  the following results mentioned in [5, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='2]: a compact K-contact Einstein manifold is  a Sasakian manifold, thus, a compact K-contact manifold admitting generalized Ricci soliton  sructure is a Sasakian manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  References  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Blair, Riemannian geometry of contact and symplectic manifolds, 2nd edition, Springer-  Verlag, New York, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [2]  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Blair,  A Survey of Riemannian contact geometry,  Complex Manifolds, 6 (2019),  31–64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='1515/coma-2019-0002  [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Berestovskij, and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Nikonorov, Killing vector fields of constant length on Rie-  mannian manifolds, Sib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', 49:3 (2008), 395–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Deshmukh, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Belova, On Killing vector fields on Riemannian manifolds, Mathe-  matics, 9, 259 (2021), 1–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='3390/math9030259  [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Ghosh, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' De, Generalized Ricci soliton on K-contact manifolds, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' E-Notes, 8 (2020), 165–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Nikonorov, Spectral properties of Killing vector fields of constant length and bounded  Killing vector fields, in Operator Theory and Differential Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Trends in Mathematics  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Kusraev and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Totieva), Birkh¨auser, Cham, (2021), 143–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [7]  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Nurowski, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Randall,  Generalised Ricci solitons,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', 26 (2016),  1280–1345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [8] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Rovenski, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Patra, On the rigidity of the Sasakian structure and characterization  of cosymplectic manifolds, preprint, arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='04597 v2, 2022, 15 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [9] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Rovenski, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Walczak, Extrinsic geometry of foliations, Progress in Mathematics,  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' 339, Birkh¨auser, Cham, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [10] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Rovenski, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Wolak, New metric structures on g-foliations, Indagationes Mathe-  maticae, 33 (2022), 518–532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Yano, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Kon, Structures on Manifolds, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' 3 of Series in Pure Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' World  Scientific Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=' Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content=', Singapore, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
+page_content='  8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQf4QX8/content/2301.03500v1.pdf'}
diff --git a/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/2301.03274v1.pdf.txt b/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/2301.03274v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1ed53bb631b65968beb47869e3ad957ecf410ad3
--- /dev/null
+++ b/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/2301.03274v1.pdf.txt
@@ -0,0 +1,2393 @@
+Astronomy & Astrophysics manuscript no. bf-hsc
+©ESO 2023
+January 10, 2023
+Correction of the brighter-fatter effect on the CCDs of Hyper
+Suprime-Cam
+Pierre Astier1 and Nicolas Regnault1
+LPNHE, (CNRS/IN2P3, Sorbonne Université, Université Paris Cité), Laboratoire de Physique Nucléaire et de Hautes
+Énergies, F-75005, Paris, France
+Received Mont DD, YYYY; accepted Mont DD, YYYY
+ABSTRACT
+The brighter-fatter effect affects all CCD sensors to various degrees. Deep-depleted thick sensors are seriously affected
+and the measurement of galaxy shapes for cosmic shear measurements requires an accurate correction of the effect in
+science images. We describe the whole correction chain we have implemented for the CCDs of the Hyper Suprime-Cam
+imager on the Subaru Telescope. We derive non linearity corrections from a new sequence of flat field images, and
+measure their statistics, namely their two-pixel function. We constrain an electrostatic model from flat field statistics
+that we use to correct science images. We find evidence that some fraction of the observed variance and some covariances
+is not due to the combination of Poisson statistics and electrostatics – and the cause remains elusive. We then have
+to ignore some measurements when deriving the electrostatic model. Over a wide range of image qualities and in the
+5 bands of the imager, stars in corrected science images exhibit size variations with flux small enough to predict the
+point spread function for faint objects to an accuracy better than 10−3 for the trace of second moments – and even
+better for the ellipticity and the fourth radial moment. This performance is sufficient for upcoming large-scale cosmic
+shear surveys such as Rubin/LSST.
+1. Introduction
+The brighter-fatter (BF) effect refers to a dynamical image
+distortion that affects CCD sensors. The most spectacu-
+lar manifestation of the effect is that bright stars appear
+slightly bigger in size than faint ones, a manifestation that
+is reflected the very name of the effect. All studies of the
+effect have attributed it to distortions of the drift electric
+field sourced by the charges stored in the pixel potential
+wells during image integration. This modifies the apparent
+shape of bright objects and the two-point statistics of uni-
+form exposures, and thick CCDs are more vulnerable to
+the effect than thinner sensors. Evidence of the effect and
+the physical explanation can be found in Guyonnet et al.
+(2015, and references therein, G15 hereafter), together with
+the relation of the effect with non-trivial flat field statistics.
+Electrostatic calculations are shown to reproduce the data
+in Rasmussen et al. (2016) and in Lage et al. (2021) for a
+specific sensor.
+On deep-depleted thick CCDs, bright stars thus gener-
+ally appear bigger by a few percent than faint stars, com-
+promising the modeling of the image point spread function
+(PSF) at a level that is not tolerable for large-scale cosmic
+shear measurements (see e.g., Mandelbaum et al. 2018).
+The correction method proposed in G15 relies on flat field
+statistics to constrain the correction applied to science im-
+ages, which mostly consists of correcting the recorded image
+for dynamically displaced pixel boundaries. The method
+has been implemented for DECam in Gruen et al. (2015),
+and for Hyper Suprime-Cam (HSC) on the Subaru tele-
+scope in Coulton et al. (2018, C18 hereafter) with some
+minor differences with respect to G15.
+Send offprint requests to: pierre.astier@in2p3.fr
+The method proposed in G15 relies on first-order per-
+turbations both in the modeling of flat field correlations
+and when correcting science images. In Astier et al. (2019,
+A19 hereafter), the relation between pixel area alterations
+and flat field statistics is extended to higher orders, which
+removes significant biases from the analysis. In the same pa-
+per, correcting for non-linearity of the video chain is shown
+to play a potentially important role when constraining the
+BF effect from flat fields. One other evolution since the
+G15 proposal is that detailed electrostatic calculations have
+been shown to reproduce the measured flat-field statistics,
+when the mandatory manufacturing data is available (see,
+in particular, Lage et al. 2021). However, the CCD ven-
+dors do not necessarily release this data, or they do not
+even have it available to the required level of accuracy. As
+we show in this paper, there is also some detectable demo-
+graphic variability among the CCDs of the HSC camera,
+which are all of a unique type from a single vendor. Gruen
+et al. (2015) also detected some variability among DECam
+CCDs. Thus, constraining the image corrections from mea-
+surements of the actual sensors is still in order.
+In the present paper, we revisit the BF correction for
+the HSC camera described in C18 and applied to science
+images in Mandelbaum et al. (2018). We take advantage
+of a new flat-field sequence that allows us to re-determine
+both the non-linearity correction and the two-point correla-
+tion function of flat fields. We apply to these images some
+potentially important corrections (described in A19) and
+we fit a variance and covariance model that has been im-
+proved since C18. We also propose a different approach for
+transforming the information extracted from flat fields into
+the correction of science images. Finally, we test the correc-
+tion of the images separately over a broad range of image
+qualities and in the five bands of the camera.
+Article number, page 1 of 18
+arXiv:2301.03274v1  [astro-ph.IM]  9 Jan 2023
+
+A&A proofs: manuscript no. bf-hsc
+The flow of the paper is as follows. We first detail in
+§ 2 why it is necessary to correct non-linearities prior to
+measuring flat field statistics and the non-linearity mea-
+surement itself. In §3, we describe the measurements of flat
+field statistics and the fit of the measurements, as well as
+the variability observed among the sensors. The informa-
+tion extracted from flat field statistics is fundamentally in-
+sufficient to correct the science images, thus, we describe
+in § 4 the electrostatic model we use to derive the correc-
+tion from the flat-field results and the outcome of different
+fits we perform. Once we obtain models that allow us to
+correct science images, we apply those to real data, as de-
+scribed in § 5, and we compare the various outcomes. We
+face the evidence that the BF correction is inadequate for
+the y band and we compute a physically motivated reduced
+correction for this band, which we eventually apply to the
+science data. In § 6, we compute some PSF modeling quality
+indicators commonly used in the context of shear estima-
+tion and conclude that the quality of our correction exceeds
+the requirements for a large-scale cosmic shear survey such
+as Rubin/LSST. It also fulfills the less stringent require-
+ments of the PSF fidelity implied by photometry accuracy
+of high-redshift faint supernovae in order to estimate their
+distances, which is our initial motivation for this work.
+2. Non-linearity correction
+2.1. Importance of the non-linearity correction
+Following A19, we assume that the effective area A of a
+pixel is linearly altered by the charge content of the sensor:
+δA = A g
+�
+i,j
+aijQij,
+(1)
+where aij is a characteristic of the sensor and Qij denotes
+the charge content of the image, which evolves as light in-
+tegration goes on. The indices i and j refer to distances
+in pixel units along the serial and parallel directions, re-
+spectively. In the same coordinates, δA applies to the pixel
+located at (i = 0, j = 0). Conventionally, aij is expressed
+in el−1, Qij in ADU and g is the gain in el/ADU. We note
+that, equivalently, we may set g = 1 and express Q in elec-
+trons. From parity symmetry, we assume that:
+aij = a|i||j|,
+(2)
+so that we measure the aij on the i, j ⩾ 0 quadrant. If
+we consider a uniform image, with all Qij identical, then
+δA has to be zero, because of translation symmetry. This
+imposes the following “sum rule”:
+�
+−∞<i,j<+∞
+aij = 0,
+(3)
+where the sum extends to the four quadrants. This sum rule
+can also be regarded as a consequence of area conservation.
+Since aij is the fractional pixel area change for a unit source
+charge, we refer to these quantities as “area coefficients.”
+Since same-sign charges repel each other, a pixel shrinks as
+it fills up, and the self-interaction coefficient a00 is negative.
+For regular CCD operating conditions, all the other coeffi-
+cients turn out to be positive. The sum rule hence indicates
+that |a00| is much larger (in absolute value) than any other
+coefficient.
+Following Eq. 1, the expression of pixel covariances in
+uniform exposures reads (see the derivation in A19):
+(4)
+Cij(µ) = µ
+g
+�
+δi0δj0 + aijµg + 2
+3[a ⊗ a]ij(µg)2
++ 1
+3[a ⊗ a ⊗ a]ij(µg)3 + · · ·
+�
++ nij/g2,
+where µ is the average (in ADU) of the uniform exposure.
+nij refers to noises (expressed in el2), and n00 refers to
+the usual read noise. From this expression, the relation be-
+tween the variance of uniform exposures and their average
+(usually called the photon transfer curve or PTC) can be
+expressed as:
+C00 = n00/g2 + µ/g + a00µ2 + O(µ3),
+(5)
+where the first term is the read noise, the second is the
+Poisson term, and the quadratic term is the first and largest
+contribution of the BF effect: the slope of the PTC decays
+with the signal level (because a00 < 0). The fact that the
+PTC of CCDs grows less rapidly than the signal level was
+initially noted in Downing et al. (2006), who also noted that
+thanks to positive covariances, grouping data into bigger
+pixels tends to restore the Poisson behavior. These non-
+trivial flat-field statistics and the brighter-fatter effect per
+se were unified under the same physics in Antilogus et al.
+(2014).
+The analog readout chain can be affected by non-
+linearity and a common distortion is a quadratic contri-
+bution:
+µm = µt + kµ2
+t,
+(6)
+where µm and µt refer to measured and true values, and k
+is a small quantity describing the distortion. The last term
+can equivalently refer to either value of µ. For the variances,
+we have:
+Vm ≃ Vt
+�∂µm
+∂µt
+�2
+= Vt(1 + 2kµ)2
+≃ Vt(1 + 4kµ)
+= (n00/g2 + µt/g + a00µ2
+t)(1 + 4kµt)
+= n00/g2 + µm/g + (a00 + 3k/g)µ2
+m + O(µ3).
+(7)
+Thus, comparing Eqs. 7 and 5, we find that a quadratic non-
+linearity, if it is not corrected for, biases the measurement
+of a00, which is the largest coefficient describing the BF
+effect. The same calculation applied to covariances shows
+that there is no effect at the µ2 level, hence the measure-
+ment of the other aij coefficients is much less sensitive to a
+quadratic non-linearity than a00.
+2.2. Measurement of response non-linearity
+The Hyper Suprime-Cam instrument is installed at the
+prime focus of the 8.2-m Subaru telescope on the Mauna-
+Kea summit in Hawaii. A detailed description of the in-
+strument can be found in Miyazaki et al. (2018) and
+here we provide only the salient features for our work.
+The HSC camera contains 104 science sensors, which are
+deep-depleted Hamamatsu 2k×4k CCDs with four read-out
+Article number, page 2 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+0.24
+0.26
+0.28
+0.30
+0.32
+MJD - 58786 (days)
+0
+10000
+20000
+30000
+40000
+CCD signal  (ADU)
+ sensor 42 channel 1
+Fig. 1. CCD signal values as a function of time for a single
+channel during a sequence of uniform exposures illuminated by a
+lamp in the dome. The intensities are deliberately not ordered in
+a monotonic way, so that the drift of the illumination system can
+be separated from the non-linearity. The sequence was designed
+by N. Yasuda (IPMU).
+0.26
+0.27
+0.28
+0.29
+0.30
+0.31
+0.32
+0.33
+MJD - 58786 (days)
+0.9875
+0.9900
+0.9925
+0.9950
+0.9975
+1.0000
+1.0025
+1.0050
+Spline modeling of the illumination
+Illumination correction for sensor 42 channels
+channel 1
+channel 2
+channel 3
+channel 4
+Fig. 2. Spline correction from the minimization of Eq. 8 for
+four different channels, normalized to its average. Since this is
+presumably due to lamp variations, it should be similar for all
+channels. The maximum difference is in the 10−4 range.
+channels on each sensor. The sensors are 200 µm thick, the
+pixel size is 15 µm, which corresponds to 0.185′′, on aver-
+age, over the focal plane. The saturation of sensors occurs
+around 120,000 el and the video channels have a typical gain
+of 3.2 el/ADU. All channels may a priori exhibit different
+non-linearities.
+In order to measure the non-linearity of the camera
+video chains, we rely on a new sequence acquired on
+2019/10/30, of uniform exposures known as “dome flats”
+Fig. 3. Fit residuals as a function of date (top) and signal level
+(bottom) for all channels of all sensors. There are no obvious
+needs for a more sophisticated model as a function of date nor
+signal level. The rms residual is about 3 10−4µ.
+obtained at night by illuminating a screen attached to the
+dome, toward which the telescope is pointed. The signal
+level is controlled by varying the exposure time. The illu-
+mination system is not equipped with a photodiode or any
+similar device and we use the exposure time as a proxy for
+the amount of light received by the CCD. Obviously, this is
+an acceptable proxy only if the illumination system is stable
+over the whole sequence or if its variations can be modeled.
+In order to disentangle a drift of the lamp intensity from
+a genuine non-linearity, the various exposure times are not
+ordered time-wise in a monotonic way, as shown in Fig. 1.
+For a given video channel, we collect from every expo-
+sure (labeled i) an average signal, µi (in ADUs), an expo-
+sure time, Ti, and the (Julian) date at which it was ac-
+quired, di. We minimize the quantity:
+�
+i
+�
+(Ti + O)S(di) − µi − kµ2
+i
+�2 ,
+(8)
+where S represents a seven-knot spline meant to describe
+the lamp variation with time, O is a potential systematic
+Article number, page 3 of 18
+
+20
+15
+10
+-10
+0.26
+0.27
+0.28
+0.29
+0.30
+0.31
+0.32
+0.33
+MJD-58786 (days)
+20
+non-linearity fit residual (ADUs)
+15
+10
+-10
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+CCD signal μ (ADUs)A&A proofs: manuscript no. bf-hsc
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+4
+3
+2
+1
+0
+1
+k (ADUs
+1)
+1e
+7
+0
+20
+40
+60
+80
+100
+channel counts
+3
+2
+1
+0
+1
+1e
+7
+Fig. 4. Values of measured quadratic non linearity k (in Eq. 8)
+in the focal plane (top) and their distribution (bottom). Most
+of the channels cluster around 10−7, which corresponds to a
+correction of ∼ 0.4% around maximum CCD signal.
+shutter offset, k represents the quadratic non-linearity, and
+Ti is the required exposure time. The shutter offset is meant
+to account for a (small) difference between the required
+and realized exposure times. We do not find any compelling
+evidence for a global shutter offset, nor a spatial trend in
+the focal plane, that could arise from such a large blade
+shutter. Hence, we set O = 0. Since the illumination may
+evolve along the sequence in slightly different ways across
+the focal plane, we do not enforce the spline S to be the
+same for all channels. We eventually compare the outcome
+of different fits (on the same sensor) in Fig. 2. The fact that
+these corrections are almost indistinguishable indicates that
+the non-linearity and the lamp variations can be robustly
+separated from this data set.
+Figure 3 displays the fit residuals (from Eq. 8) as a func-
+tion of signal level or of date. Those are globally small. A
+more flexible model as a function of date would not reduce
+much these residuals, highly correlated over the channels
+at the same date. We can attribute this scatter to random
+fluctuations of the lamp or of the shutter. The fit resid-
+uals as a function of signal level do not strongly call for
+a more flexible or more general non-linearity model than
+a quadratic correction: the largest systematic residual is
+found around 10,000 ADUs and amounts to a few ADUs.
+However, for some yet unknown reason, we had to discard
+exposures before di − 58786 < 0.255 because they exhibit
+large residuals (tens of ADUs) consistently over channels.
+One might attribute this to some erratic behavior of the
+lamp that vanishes after about 0.5 h. While these first ex-
+posures are rejected from the non-linearity analysis, they
+are used to measure the flat field statistics in what follows.
+In Fig. 4, we display the values of quadratic non-
+linearity in the HSC focal plane in order to visualize the ab-
+sence of geometrical trend that could occur from a spatially
+variable evolution of the illuminating system. The distribu-
+tion peaks around k = 10−7 ADU−1, which corresponds to
+a correction of ∼ 0.4 % at the maximum CCD signal. We
+go on to show soon that a00 averages to −1.3 10−6 and the
+channel gains are around 3; if one ignores quadratic non-
+linearity (Eq. 7), the bias affecting a00 amounts to ∼8%. a00
+is the best measured area coefficient and essentially drives
+the scale of the BF correction. We might note that such
+a bias unavoidably causes a violation of the sum rule and,
+hence, translating such a set of area coefficients into a cor-
+rection is somehow arbitrary because sensible image cor-
+rections have to derive from areas changes that sum up to
+zero. We also note that that eliminating this trivial cause of
+sum rule violation allows us to detect more subtle problems
+in 4.3.
+3. Measurement and fit of flat-field statistics
+3.1. Measurements
+Measuring the variances and covariances of flat fields al-
+lows one to measure the dynamical alteration of pixel areas
+due to electrostatic distortions; namely, we measure vari-
+ances and covariances of flat field exposures (which we may
+call the two-point function of the exposure) at various sig-
+nal levels, in order to measure the aij coefficients (Eq. 1)
+by fitting Eq. 4 to the measurements. We closely follow
+the procedure described in A19: we first correct for non-
+linearity (see the previous section) and correct for deferred
+signals.
+A deferred signal is by definition collected after (in the
+serial time chain) the pixel it belongs to. In order to mea-
+sure those, we closely follow the procedure described in A19,
+and find that there are no significant deferred signals be-
+yond the nearest serial neighbor. We display a few examples
+of the nearest neighbor deferred signals in Fig. 5: those are
+small and almost linear, indicating that they mostly origi-
+nate from electronics. We tabulate one correction curve per
+channel, and for correcting the effect, we typically “place
+back” less than 0.1 % of a pixel into the preceding one.
+Failing to carry out this correction results in the covariance,
+C10, exhibiting a linear contribution (∝ µ), which cannot
+be absorbed by the model of Eq. 4, hence biasing a10. For
+our CCDs, at the maximum signal level, the contribution of
+uncorrected deferred signals to C10 would represent about
+25% of the electrostatic value.
+When computing image statistics, we mask any outlier
+pixels detected in the images and use the pixel mask ap-
+plied to science images at the epoch of the sequence. These
+pixel masks flag pixels that repeatedly depart from their
+neighbors in uniform exposures and pixels on the sides of
+the focal plane which are heavily vignetted (or fully blind).
+Article number, page 4 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+5
+0
+5
+10
+15
+20
+ 
+Chan 1
+Chan 2
+0
+10000
+20000
+30000
+40000
+ 
+5
+0
+5
+10
+15
+20
+Chan 3
+0
+10000
+20000
+30000
+40000
+Chan 4
+First overscan pixel (ADU)
+Last row real pixel (ADU)
+Fig. 5. Nearest serial neighbor deferred signals for the four chan-
+nels of sensor 42. The values are small and vary almost linearly
+with the signal. The model continuous lines are the cubic poly-
+nomial model that we use for correction.
+We account for the masked pixels in covariance computa-
+tions in a standard way, described in Appendix A of A19.
+The bias of variance and covariances due to masking out-
+lier pixels from images themselves is compensated for as
+described in §5.2 of A19.
+We measure each video channel independently and pro-
+duce variance and covariance curves using the 66 flat pairs
+of the sequence. We measure the covariances on the dif-
+ference of flat pairs acquired at the same signal level to
+eliminate contributions from illumination non-uniformity
+or sensor sensitivity variations. These 66 flat pairs were
+acquired in g band. Contrarily to C18, we carry out the
+measurements on all HSC science sensors.
+3.2. Fits of the covariance curves
+We then fit the measurements using Eq. 4, with weights de-
+rived from shot noise. This equation contains terms up to
+µ4, and including one extra order would change the predic-
+tions by a few 10−5, which is about 100 times less that the
+smallest shot noise. Fig. 6 displays the measurements and
+fits for a few (i, j) pairs and for the four channels of a sensor.
+One can readily note that the slopes for C10 (serial neigh-
+bors) and C01 (parallel neighbors) differ by a factor of more
+than 4, which is due to the very different mechanisms that
+confine the charges into the pixel wells for the serial and the
+parallel directions: while the boundaries between columns
+are static and implanted, the boundaries between rows re-
+sult from the potentials of the parallel clock stripes that
+are swung during image read-out in order to move charges
+towards the serial register and the output amplifier. The
+parallel pixel boundaries are then somehow weaker than
+the serial ones and this is reflected on the pixel covariances
+of uniform exposures. We perform the measurements in the
+0.28
+0.29
+0.30
+0.31
+0.32
+C00/ -0.002(channel-1) (ADUs2)
+channel 1
+channel 2
+channel 3
+channel 4
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+C01/ -0.002(channel-1) (ADUs2)
+0
+5000
+10000
+15000
+20000
+25000
+30000
+ (ADUs)
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+0.007
+C10/ -0.002(channel-1) (ADUs2)
+Fig. 6. Measurement and fits of the variance (C00) and the
+two nearest neighbor covariances (C10 and C01) for the four
+channels of sensor 42. The channels have been offset for visual
+convenience. The decay of C00/µ with µ, originally noted in
+Downing et al. (2006), is the most obvious effect of the BF effect.
+The spread of measurements is compatible with shot noise.
+range i, j < 10 and fit Eq. 4 to all the measurements of a
+given channel at once.
+In order to question whether the fitted model (Eq. 4)
+properly describes the data, we plot in Fig. 7 the fit residu-
+als of the variance and the nearest covariances. Since we are
+studying effects related to sensors, we turned the averages
+and covariances into gain-free quantities by converting them
+into electrons, using the gains obtained from the fits in the
+previous paragraph. An examination of Eq. 4 shows that
+the coefficients of the µ3 terms are entirely determined by
+combinations of the coefficients of the µ2 terms. The resid-
+uals in Fig. 7 exhibit some curvature, which is compatible
+with a mismatch between the coefficients of µ2 and µ3. For
+the residuals of C00 (top), the curvature of the residuals
+represents about 15% of the cubic term of Eq. 4 (thus il-
+Article number, page 5 of 18
+
+A&A proofs: manuscript no. bf-hsc
+Fig. 7. Residuals of fits of variance and covariance measure-
+ments, all expressed in electrons using the gains from the fits,
+so that the data from different sensors can be averaged. The red
+signs are binned averages of the residuals from individual fits.
+lustrating the importance of these higher order terms). For
+C01, the curvature represents about 30% of the cubic term,
+while they have similar values for the residuals to C10. We
+will propose later an explanation for the curvature of resid-
+uals of C00 and C01, but this explanation does not apply to
+C10.
+3.3. Demography of HSC sensors
+We may now investigate the demography of channels and
+sensors regarding the BF effect. In Fig. 8, we display scat-
+ter plots of the well-measured a coefficients. We can see the
+correlations, which are similar at the channel and sensor
+level, indicating that the variations are weaker within sen-
+sors. We can note that a00 does not correlate to the other
+coefficients, and that the apparent correlations are in a di-
+rection that does not preserve the “sum rule.” This may
+indicate that they are not due to genuine variations of the
+BF effect among the sensors. We may also observe that on
+average the sum of a is positive:
+�
+�
+−10<i,j<10
+aij
+�
+channels
+≃ 7 10−8.
+(9)
+Since all aij at larger distances are positive, the sum to in-
+finity is even farther from 0. Since < a00 >≃ −1.3 10−6, the
+violation of the sum rule is hence at least 5% of a00 and we
+eventually see that it reaches ∼10% once we have a model to
+evaluate the large-distance contributions. This means that
+the sum of all covariances rises with signal level faster than
+Poisson, at variance with expectations from statistics (see
+Eq. 8 in A19). Our measurements are thus affected by some
+noise, which increases with signal level and contributes as
+µ2.
+Since the variations of well-measured aij is at most 10
+% across channels, we decide to average the measurements
+in order to average the shot noise at large distance. Once
+equipped with this average, we fit an electrostatic model to
+it that, by construction, satisfies the sum rule. In the next
+section, we show that the mismatch between the model and
+the data is indeed localized.
+4. Electrostatic fit
+4.1. From the area coefficients to science image corrections
+We study the BF effect in order to suppress its impacts from
+the science images. The scheme proposed in G15 consists
+of evaluating from the science image itself the motions of
+pixel boundaries with respect to a perfect grid, as well as
+evaluating via interpolation how much charge was flowing
+over these pixel boundaries during integration, and then
+placing this amount of charge back where it belongs.
+We can readily note that the measurements we have per-
+formed so far constrain the change in the area of a pixel, but
+do not tell how its shape is altered. If we describe the shape
+change by different motions of the serial and parallel sides
+of a pixel, this simplistic shape description already requires
+two quantities per pixel, while we only have one. Defin-
+ing a shape change by distinct serial and parallel boundary
+shifts means in turn that we have to “split” the aij coef-
+ficients (into aN
+ij, aW
+ij , aS
+ij, aE
+ij) and rely on area-conserving
+symmetries such as aW
+10 ≡ −aE
+20. We can also regard aij as
+the discrete divergence of the pixel boundary displacement
+field and we are faced with the ill-posed problem of deter-
+mining a vector field from its divergence, in a a 2d discrete
+space.
+Various implementations of the correction method pro-
+posed in G15 differ in the way they perform this promotion
+of pixel area change into two directional components. In
+G15 and in Gruen et al. (2015), some ratios are imposed,
+which were estimated from the area changes themselves.
+C18 argue that the motions of pixel boundaries is a 2D vec-
+tor field that results from the (perturbation) electric field
+sourced by the charges that represent the image, and, hence,
+the 2d displacement field is (as is the 3d electric field) curl-
+free. The “scalar” aij field can then be transformed into a 2d
+vector curl-free field. While this curl-free assumption seems
+appealing, we show in § 4.4 that it turns out to be violated
+by a solution of the Poisson equation. We may guess (and
+Article number, page 6 of 18
+
+0.006
+ -model)/μ(e)
+0.004
+0.002
+0.000
+8
+0.002
+0.004
+-0.006
+0.004
+model)/μ(e)
+0.002
+0.000
+0.002
+Co1
+-0.004
+0.006
+0.004
+0.002
+0.000
+-0.002
+0.004
+0
+20000
+40000
+60000
+80000
+100000
+μ (e)Pierre Astier and Nicolas Regnault: BF on HSC
+1.3
+1.2
+a00 
+1e
+6
+3.0
+3.5
+4.0
+4.5
+5.0
+5.5
+6.0
+6.5
+a10
+1e
+8
+1.3
+1.2
+a00 
+1e
+6
+1.50
+1.55
+1.60
+1.65
+1.70
+1.75
+a01
+1e
+7
+4
+6
+a10 
+1e
+8
+1.50
+1.55
+1.60
+1.65
+1.70
+1.75
+a01
+1e
+7
+4
+6
+a10 
+1e
+8
+4.8
+5.0
+5.2
+5.4
+5.6
+5.8
+a11
+1e
+8
+1.5
+1.6
+1.7
+a01 
+1e
+7
+4.8
+5.0
+5.2
+5.4
+5.6
+5.8
+a11
+1e
+8
+1.5
+1.6
+1.7
+a01 
+1e
+7
+4
+2
+0
+2
+4
+a
+1e
+7
+Fig. 8. Distribution of area coefficients over the ∼400 hundred channels of HSC (blue) and the averages over each sensor (orange).
+All aij coefficients are expressed in el−1. We may note that the trends for channels and sensors are similar, which indicates some
+homogeneity within sensors (except for � a, where the scatter is dominated by shot noise). � a is the sum for −10 < i, j < 10.
+we show below) that the pixel boundaries motions are pro-
+portional to the integral over pixel boundaries of the per-
+turbating electric field. These integrals would themselves be
+curl-free (as is the electric field) if the integration paths were
+all identical for serial and parallel boundaries. The funda-
+mental anisotropy (e.g., from the ratio of a01 and a10) indi-
+cates that this is not true at small distances, and questions
+the curl-free hypothesis applied to the pixel boundaries dis-
+placement field. Even without this anisotropy, we might also
+question whether the curl-free property of the continuous
+3D electric field is precisely transferred to a curl-like com-
+bination of finite differences over the pixel lattice.
+Our practical approach is to promote the discrete aij
+scalar field into a discrete 2D vector field relying on electro-
+statics; namely, to propose to fit the geometrical parameters
+of a simple electrostatic model of the perturbating electric
+field to the data and to evaluate the 2D vector field we need
+from the model. We note that since the model reports ac-
+tual pixel areas, the sum rule is by construction satisfied by
+the outcome of a fit.
+4.2. The electrostatic model
+We are interested in pixel boundary shifts under the in-
+fluence of stored charge and here we sketch a first-order
+perturbation scheme that allows us to predict these shifts
+from a few geometrical quantities. The first ingredient is
+the electric field sourced by the stored charges inside the
+sensor, which we refer above as the “perturbating” electric
+field (because it adds to the drift field that defines pix-
+els). We assume that the field sourced by collected charges
+causes no rearrangement of charge within the bulk of the
+device. We model this perturbating field as sourced by a
+charge between two infinite grounded equipotentials figur-
+ing the light entrance window (on which the drift voltage is
+applied) and the parallel clock stripes. This is a text-book
+problem and using the image charge technique, we can write
+the corresponding potential as:
+φ(ρ, z) = Q
+4πϵ
+∞
+�
+n=−∞
+1/
+�
+ρ2 + (2nt + zq − z)2
+− 1/
+�
+ρ2 + (2nt − zq − z)2,
+(10)
+where ρ2 = x2 + y2, t is the sensor thickness, the source
+charge Q is located at (0, 0, zq), and the equipotential planes
+are at z = 0 (representing the parallel clock stripes) and
+z = t (the light entrance face of the sensor). We can check
+that φ(ρ, 0) = φ(ρ, t) = 0: in these cases, each positive term
+(first line) of the series has an exact opposite among the
+negative terms (second line), which is how the image tech-
+nique works. This expression for the potential converges
+poorly with n for ρ much larger than t and alternative ex-
+pressions should then be used (Pumplin 1969). For the fit,
+we do not need to evaluate the model at ρ > t.
+The field that results from the normal operation of the
+sensor drives the charges into the pixel wells. The drift
+lines on the pixel boundaries have a peculiar point at which
+the field is null and the potential has a saddle point. This
+point is commonly located a few microns away from the
+clock stripes (at z=0). These points are located at differ-
+ent heights above serial and parallel boundaries, and this
+difference contributes to the anisotropy of flat field small-
+distance covariances. We display in Fig. 9 a contour crafted
+to relate the pixel boundary shifts to the source charge us-
+ing Gauss’s theorem, applied to the total electric field, the
+drift field plus the perturbation introduced by the charge.
+This contour is made from four segments (numbers match-
+ing Fig. 9). Segment 1 corresponds to the perturbed drift
+line that separates two pixels and the integral of the trans-
+verse electric field is null. Segment 2 represents the short
+path between the unperturbed zero-field point and the per-
+turbed zero-field point and is very close to a drift line; the
+integral of the electric field is then a second order quantity,
+Article number, page 7 of 18
+
+A&A proofs: manuscript no. bf-hsc
+1
+2
+3
+4
+Q
+Fig. 9. Contour to which we are going to apply Gauss’s theorem
+to relate the boundary shift (the length of segment 4) to the
+electric field sourced by the charge Q.
+which we ignore. Along segment 3, the integral of the un-
+perturbed field vanishes because it is the unperturbed drift
+line; we are thus left with the integral of the perturbation
+field. Over segment 4, the integral of the field is the product
+of the drift field, Ed, times the boundary displacement, d,
+that we wish to estimate.
+So, Gauss’s theorem is finally expressed as:
+� z=z0
+z=t
+ET
+Q(xb, yb, z)dz + dEd =
+�
+C
+ρ/ϵ
+(11)
+where z0 is the zero-field point altitude, xb and yb are the
+coordinates of the unperturbed drift line, ET
+Q is the field
+transverse to the boundary sourced by the charge Q, Ed
+is the drift field (at the top), and the right-hand side is
+the bulk charge contained inside the contour, because the
+Silicon material can contain residual impurities sourcing a
+small bulk electric charge. We do not know this right-hand
+side, but we can assume that (to first order) it is propor-
+tional to d, our perturbation parameter. So, we eventually
+obtain:
+d ∝
+� z=z0
+z=t
+ET
+Q(xb, yb, z)dz.
+(12)
+In other words, the boundary displacement is proportional
+to the integral of the perturbating (transverse) field over
+the unperturbed trajectory. This is usually called the Born
+approximation. In a charge-free material, the normalization
+is expressed as 1/Ed. The expression assumes that charges
+are produced when light enters the sensor, at z = t. For the
+reddest bands, we should instead account for the conversion
+depth of photons in the sensor, as we will do in §5.3. This
+expression should be averaged over the pixel side in the
+direction perpendicular to the Fig. 9.
+In order to constrain the rhs of the expression, it is
+tempting to carry out the the measurements at different
+values of Ed by changing the drift voltage applied to the
+entrance side of the sensor. However, changing the drift ac-
+tually alters Ed but also z0 (for both flavors of boundaries),
+so measuring the sensor under different drift voltages would
+not help significantly at constraining the model. We may,
+in principle, predict the proportionality coefficient of Eq. 12
+because the (empty CCD) charge density, thickness, applied
+drift voltage, and drift field are related by basic electrostat-
+ics. We nonetheless stick to fitting the global scale because
+precisely estimating the drift electric field involves at least
+the clock stripe sizes and their potentials during image in-
+tegration. In this specific study, we have to fit the overall
+model scale because we were not able to find the drift and
+parallel clock potentials applied to the sensors.
+Q
+a
+b
+d
+(0,1)
+(1,1)
+(1,0)
+zs
+t
+y
+x
+z
+Fig. 10. Schematic of the geometry of the electrostatic fit.
+We add some flexibility to the model by allowing source
+charges to be extended and we stick to a very crude model: a
+uniformly charged rectangle. This refinement, as compared
+to point sources, only influences the very first neighbors.
+So the model eventually has six parameters (see Fig. 10): a
+global normalization factor, the height of the source charge
+zQ, the height of the zero-field points over parallel and serial
+pixel boundaries zp and zs, and the sides of the (uniform)
+rectangular source charge a and b.
+The calculations of the model also require the thickness
+of the sensor (200 µm) and the pixel side (15 µm). For
+the practical implementation of the integrals from Eq. 12,
+we settle for integrating analytically the terms of the series
+similar to Eq. 10 for the electric field and emulating the
+rectangular source by splitting the charge into nine equally
+spaced point charges (only for n = −1, 0, 1). We sum the
+series of the expressions in Eq. 10 up to n = ±11. This
+seems to be sufficient for up to 10 pixels because increas-
+ing to n = 12 only changes outputs at the sixth decimal
+place. This electrostatic modeling approach was initially
+presented in Le Breton (2017), which we adapted Figs. 9
+and 10 from.
+Article number, page 8 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+model
+10
+9
+10
+8
+10
+7
+10
+6
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+data (a00
+a00)
+10
+9
+10
+8
+10
+7
+10
+6
+0
+2
+4
+6
+8
+i
+0
+2
+4
+6
+8
+j
+data-model
+4
+3
+2
+1
+0
+1
+2
+3
+4
+1e
+9
+Fig. 11. Data (top), fitted electrostatic model (middle), and
+difference (bottom). The sign of a00 has been flipped in the top
+plots, but not in the difference. We can note the clear difference
+of the two nearest neighbors, an anisotropy that is already visible
+in Fig. 6. We expect a poor fit because the data violates the sum
+rule, while the fit cannot.
+4.3. Fit of the electrostatic model to the data
+We performed a least-squares fit to the average area coef-
+ficient data, using the spread over channels to weight the
+squares. We first fit the average data obtained at the previ-
+ous section. The data, fit, and difference are shown in Fig.
+11. We can see that the measurements reach a signal-to-
+noise ratio of about 1 at a distance of 8 to 9 pixels from the
+source. Beyond the small separations, the measurements
+exhibit a radial symmetry as expected from electrostatics
+and displayed as well by the model. The residuals do not
+average to zero and the data is larger than the model. This
+reflects that the data violates the sum rule but the fit can-
+not. The sum of the model area coefficients is −7 10−8 up
+to i, j < 10 (and tends to 0 with increasing bounds, so the
+contribution of unmeasured aij is 7 10−8). Since the sum of
+measured aij is 7 10−8, the data violates the sum rule by
+about 1.4 10−7, which is more than 10% of a00. The details
+of the electrostatic model cannot change the contribution
+of unmeasured aij significantly (at i > 9 or j > 9) and
+certainly cannot flip its sign. The excess of the data with
+respect to the model is concentrated on the 3 first serial
+pixels and the first parallel neighbor.
+Some excess of variance and covariance along the serial
+direction is expected if a video signal experiences rapid gain
+variation (or “gain noise”): rapid gain changes contribute a
+variance component that scales as the square of signal level
+and, hence, artificially increase the value of a00 (see Eq. 5).
+If gain fluctuations last longer than the time to read a pixel,
+they also bias covariances along the serial direction. Since
+the residuals seem to decay along the serial direction, possi-
+ble gain variations have also to decay rapidly. They cannot
+be invoked to explain the excess on the first parallel neigh-
+bor because while serial pixels are read out microseconds
+apart, milliseconds separate neighboring lines.
+We now attempt a fit that ignores the variance and the
+two next serial pixels, that is, a00, a10 and a20. The fit dis-
+played in Fig. 12 seems acceptable: the residuals are at most
+1% of the largest used coefficient a01 ≃ 1.6 10−7, and an
+even lower fraction of a00. The residuals exhibit some sort
+of low-level “chessboard pattern”, which is not entirely sys-
+tematic. Assuming it is real, we have no proposition for its
+source, and we could not invent a small periodic variation
+of the physical size of pixels that would produce this 2 × 2
+pixel pattern of the two-pixel correlation functions.
+We display the measured and fitted values, and their ra-
+tio in Fig. 13. The model reproduces the data well and gives
+some confidence that the model delivers sensible values for
+the data we decided not to use. We note that the decay of
+signal with distance is reproduced very well by the model
+over more than two orders of magnitude of signal level. In
+Pumplin (1969), it is shown that the large-distance decay
+of the perturbating electric field from a point charge in
+the sensor depends essentially exponentially on the inverse
+thickness of the sensor. The other geometrical parameters
+of the model cannot alter significantly the logarithmic slope
+of this decay at large distances. So, the evaluation of the
+model at large distances, needed to gauge the compliance
+of data to the sum rule, is a robust outcome of the model
+if the slopes of data and model match.
+In Table 1, we display the measured and fitted largest
+area coefficients of the data and the two fitted models. We
+can see in particular the size of the discrepancies along
+the serial direction. We can now extract from the fitted
+model the pixel boundary displacement caused by a unit
+charge. This allows us to compute for a given science image,
+the accumulated boundary displacements resulting from all
+present charges. We will test the quality of the BF effect
+correction derived from the models on science images in the
+following section. We now question the curl-free hypothesis,
+and then discuss the gain noise issue.
+Article number, page 9 of 18
+
+A&A proofs: manuscript no. bf-hsc
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+model
+10
+9
+10
+8
+10
+7
+10
+6
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+data (a00
+a00)
+10
+9
+10
+8
+10
+7
+10
+6
+0
+2
+4
+6
+8
+i
+0
+2
+4
+6
+8
+j
+data-model
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+1e
+9
+Fig. 12. Data (top), fitted electrostatic model (middle), and
+difference (bottom). This fit ignores the three first serial pixels,
+marked with crosses in the bottom plot. Compared to Fig. 11,
+we see that the data excess in a01 has disappeared. We note
+that the color scale of the difference is zoomed-in, as compared
+to Fig. 11
+4.4. Considering whether the pixel boundary displacement
+field is curl-free
+In C18, the authors assume that the boundary displacement
+field is curl-free. Our definition of discrete curl is expressed
+as:
+ci,j = (aN
+i,j+1 − aN
+i,j) − (aW
+i+1,j − aW
+i,j).
+(13)
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+i2 + j2
+0
+10
+9
+10
+8
+10
+7
+10
+6
+|aij|
+fit
+data
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+i2 + j2
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+(data-model)/|model|
+Fig. 13. Data and fit (left) as a function of distance, difference
+normalized to the model (right). The fit ignores the three first
+serial pixels, which are labeled with crosses in the rhs plot. The
+dashed line in the lhs plot separates the logarithmic and linear
+scale regions.
+Table 1. Values of the largest area coefficients for the data and
+two fitted models
+i = 0
+i = 1
+i = 2
+d
+2.60 10−8
+1.88 10−8
+9.95 10−9
+j = 2
+m1
+2.57 10−8
+1.78 10−8
+8.66 10−9
+m2
+2.77 10−8
+1.93 10−8
+9.57 10−9
+d
+1.61 10−7
+5.19 10−8
+1.85 10−8
+j = 1
+m1
+1.55 10−7
+5.23 10−8
+1.57 10−8
+m2
+1.60 10−7
+5.24 10−8
+1.68 10−8
+d
+−1.24 10−6
+4.79 10−8
+2.56 10−8
+j = 0
+m1
+−1.29 10−6
+3.90 10−8
+1.99 10−8
+m2
+−1.37 10−6
+3.43 10−8
+2.08 10−8
+Notes. The three rows for each value of j refer to the data,
+model 1, and model 2. Model 1 fits all the measured aij, while
+model 2 ignores the data from the bottom row.
+In Fig. 14, we display the discrete 2D curl of the field, the
+displacement of one boundary (arbitrarily chosen as the
+parallel one, the serial one is in fact smaller) and their ra-
+tio. We used the electrostatic solution displayed in Fig. 12,
+which (by definition) satisfies the Poisson equation and,
+hence, has a curl-free 3D electric field. We can readily see
+that the curl-free assumption may lead to sizable errors
+(several tens of percent) in the correction.
+One reason for the displacement field to be rotational is
+that drift paths along serial and parallel boundaries do not
+end at the same distance from the parallel clock stripes.
+The importance of this difference vanishes with distance
+because at large distances, the field varies less steeply with
+z than at short distances and, hence, the fractional con-
+tribution to the boundary displacement of the end of the
+drift path decays with distance. When it comes to science
+images, and especially when the image quality is good, the
+short-distance boundary displacements dominate the effec-
+tive correction for stars.
+Article number, page 10 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+curl value
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+top boundary shift
+0
+2
+4
+6
+8
+i
+0
+2
+4
+6
+8
+j
+ratio
+0.0
+0.5
+1.0
+1e
+7
+1
+2
+3
+4
+1e
+7
+0.2
+0.0
+0.2
+0.4
+Fig. 14. Values of the (discrete) curl, the top boundary dis-
+placement and their ratio. One can see that assuming a curl-free
+displacement field is acceptable only at distances larger than ∼
+3 pixels.
+4.5. Gain noise
+We measure a00 = −1.24 10−6, while the fit of Fig. 12
+indicates a00 = −1.37 10−6. This means that the model
+predicts a variance that is smaller than the measured one,
+and the leading difference is quadratic in signal level. This
+difference of a00 values also contributes to the apparent
+curvature of the residuals displayed in Fig. 3: it roughly
+explains the curvature for C00 and C01 – but not for C10.
+While questioning the area coefficients along the serial
+direction, it is tempting to relate the excess we observe
+to charge transfer issues; however, we provide a few argu-
+ments against this explanation. First, we have measured
+and corrected deferred signals and we find that the linear
+dependence displayed in Fig. 5, causes, if uncorrected, a
+linear contribution to C10 (∝ µ); however, the a10 excess
+we observe corresponds to a quadratic contribution to C10
+(∝ µ2). Second, any mechanism relying on imperfect charge
+transfer will unavoidably affect C00 and C10 in comparable
+and opposite amounts, but the data in Table 1 indicate that
+the excess of a00 is about 20 times larger that the one of
+a10, and they have the same sign. Third, serial charge trans-
+fer inefficiency is usually associated to localized and rare
+defects that is typically 1 or more frequently 0 along the
+serial register. The distributions of area coefficients (Fig. 8)
+do not seem quantized.
+So, assuming that this difference of a00 values is due
+to gain variations during the read out, the relative gain
+variations should be about 3.5 10−4 rms. Gain variations
+of the HSC electronics have been studied during the HSC
+camera fabrication process and are reported in Miyatake
+et al. (2012). They are evaluated as ∼ 2.4 10−5, however,
+this is only considering the video chain and not the CCD
+(and perhaps in a context that is different from what the
+instrument actually faces).
+The CCD readout chain of HSC relies on correlated dou-
+ble sampling, an approach that integrates the video signal
+during logical gates provided by the clocking system. The
+collected signal is hence vulnerable to fluctuations of the
+timing of the integration gates. We cross-correlated the sub-
+images from different channels of the same sensor in order
+to diagnose synchronous correlations, including variations
+of the gains (whatever the cause). We did not find any com-
+pelling signal for any sensor. We were hoping that cross-
+correlations would deliver the size of gain fluctuations, or
+any other common-mode fluctuation of channel response,
+thus allowing us to subtract this contributions from flat-
+field statistics.
+In summary, we are not able to provide a convincing
+cause of the sum rule violation. Because the electrostatic
+fit indicates an excess of fluctuations along the serial direc-
+tion, we ignore three area coefficients. This excess is obvi-
+ously puzzling, albeit compatible with rapid gain fluctua-
+tions (with some ad hoc time decay). Fortunately, we do
+not have to rely in any way on this hypothesis to use the
+electrostatic model.
+5. Correction of science images
+5.1. Data set and reduction
+We process the images of the Cosmos field from the ultra-
+deep part of the Subaru Strategic Program, acquired be-
+tween 2014 and 2019. The ultra-deep part of the survey is
+geared at obtaining extremely deep images by co-adding
+a large number of observations, as well as detecting high-
+redshift supernovae and measuring their light curves. The
+sample of images covers the five bands: g, r, i, z and y of
+the camera, and also covers a broad range of image qual-
+ities. We use data acquired with the new i and r filters,
+which are called r2 and i2 in HSC parlance and we stick to
+these names. This image sample constitutes a representa-
+tive playground for testing the quality of the brighter-fatter
+correction on HSC science images.
+Article number, page 11 of 18
+
+A&A proofs: manuscript no. bf-hsc
+The image reduction we perform is fairly standard for
+the brighter-fatter correction. We first average the overscan
+and subtract it from the actual image data. We then cor-
+rect each image segment for non-linearity. Then we have
+to express the image in electrons in order to perform the
+BF correction, so we multiply each image segment by its
+corresponding gain (as determined when fitting the covari-
+ance data), perform the correction, and divide back each
+segment by the gain. We note that the BF correction has
+to deliver an image that has not been corrected by chan-
+nel gains because our flats contain the gain differences and,
+hence, they should be applied to an unscaled image. We pre-
+fer to use flats that encode the relative gains because their
+evolution with time encodes a possible evolution of gains.
+Using approximate gains for the BF correction is clearly a
+second-order issue, while using approximate relative gains
+on a sensor can cause artificial steps in the sky background
+at the channel boundaries.
+The brighter-fatter correction consists in computing the
+boundary shifts by summing all the actions of all image
+charges on all boundaries. For parallel boundaries, the shifts
+are expressed as:
+δN
+ij = 1/2
+�
+kl
+aN
+k,lQi−k,j−l,
+(14)
+and similarly for serial boundaries. The factor of 1/2 ac-
+counts for the fact that source charges alter pixel shapes
+only once they have reached the pixel wells; thus, on aver-
+age, during half of the integration time. In Eq. 1, the charge
+on the rhs refers to the charge accumulated so far during the
+integration and, hence, the time-averaged boundary shift
+should be derived from the time-averaged charge content.
+The charge to displace from one pixel to its neighbor is com-
+puted from the pixel boundary shift and the charge flowing
+over the same pixel boundary. We compute the latter as
+the average between the two pixels that share the bound-
+ary. We tried a quadratic order interpolator (involving 4
+pixels), and did not find a decisive difference. We should
+note that the vast majority of our images are well sampled
+(the image quality is larger than 3 pixels) and sharper im-
+ages could react differently. Ultimately, the correction of
+the BF effect in the images themselves requires that an ac-
+curate estimation of the charge at pixel boundaries can be
+devised.
+Once the BF correction has been applied, we divide back
+each channel section by its gain and apply the flat field
+constructed from dome flats accumulated over about one
+month. For the y band, we have to subtract a fringe pattern
+constructed from a large set of science images.
+On each individual CCD from each exposure, we run
+SExtractor (Bertin & Arnouts 1996) to detect objects and
+measure the Gaussian second moments of these objects
+from an unweighted 2D Gaussian fit to the light distribu-
+tion. Because we do not integrate the Gaussian over pixels,
+we use the fast procedure described in Astier et al. (2013)
+to solve the normal equations. These second moments are
+similar if not identical to the “SDSS adaptive moments.” We
+identify stars in the image from the distribution of moments
+(see Astier et al. 2013). Because we have to accommodate
+significant variations of the PSF over the field of a single
+CCD, the cut that selects stars is broad enough to avoid re-
+jecting genuine stars because of the BF effect. We get three
+moments per star: MXX, MY Y and MXY , where X and Y
+refer to the serial and parallel directions on the sensor. As
+an indicator of PSF size, we use:
+IQ2 ≡ TP SF /2 = (MXX + MY Y )/2.
+(15)
+We then need color measurements of our stars, which
+we obtain by averaging fluxes over images and calibrating
+instrumental magnitudes over the common footprint of our
+star catalog and the field D2 from Betoule et al. (2013).
+This is not a critical step since only small color corrections
+are involved in what follows and we, hence, do not need
+colors measured to better than ∼0.1 mag.
+5.2. Processings and results
+0
+10000
+20000
+30000
+fmax
+0.02
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+MXX
+< MXX >
+0
+10000
+20000
+30000
+fmax
+MYY
+< MYY >
+0.00
+0.25
+0.50
+0.75
+1.00
+i2-z
+0.08
+0.06
+0.04
+0.02
+0.00
+0.02
+MXX
+< MXX >
+0.00
+0.25
+0.50
+0.75
+1.00
+i2-z
+MYY
+< MYY >
+Fig. 15. Differences between the star second moments and their
+expectations from the spatial smoothing, as a function of fmax
+(top) and color (bottom). We have selected stars with 3 < IQ2 <
+4 pix2, in the z band. The color range was chosen so that the
+color dependence is roughly linear. One can see that the slope
+of the top right plot is slightly steeper than the top left plot,
+reflecting the anisotropy of pixel covariances in flat fields.
+In order to assess the variation of star moments with
+their flux, we report the variation of star sizes with peak
+flux fmax, rather than with the flux itself. The variation
+with fmax is of practical interest, because fmax determines
+if a given star can enter into the PSF modeling. In order
+to isolate the variation of moments with peak flux we have
+to account for two other variations: the spatial variation in
+every exposure (mostly due to optics), and the color depen-
+dence of apparent star sizes. We remove the spatial depen-
+dence by fitting a second-order 2D polynomial to the star
+moments measured on a given CCD in a given exposure.
+We then interpolate this crude model at the star position
+and study the residual of the measurement to the model. We
+model MXX and MY Y independently. We apply a conserva-
+tive cut at fmax < 35000 ADU, in order to avoid any effect
+of sensor saturation. The dependence on fmax and color
+Article number, page 12 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+of these residuals are shown in Fig. 15, for a processing in
+which no BF correction was applied. We see that for stars
+selected at (MXX + MY Y )/2 ≃ 3.5 pix2, the total increase
+of moments with fmax is ∼ 0.13 pix2, which is about 3.7%
+for stars varying from 0 to saturation. This slope depends
+on the selection of image quality. Figure 15 also indicates
+that the variation of apparent size with color in the z band
+are not considerably smaller than the ones with flux. Since
+the fmax-color correlation coefficient is about −0.18 (in z
+band), we should account for color-induced size variations.
+In order to account for both the peak flux and color
+dependence of the moments, we regress the star moment
+residuals against both fmax and color:
+MAA − M expected
+AA
+= αfmax + βc + γ,
+(16)
+where A stands for X or Y . We then use α (i.e., the BF
+slope) as a color-independent indicator of the BF effect in-
+tensity (both before and after correction). The linear cor-
+rection in color can only work within a finite color range,
+and the selected color indicators and their range are pro-
+vided in Table 2. For this fit, we ignore the measurements
+with fmax<5000 because the measured moments could be
+affected by a biased background estimation. Out of about
+15 106 star measurements, the flux and color cuts retain
+2.4 106 measurements contributing to the slope measure-
+ments that are reported here.
+Table 2. Color indicators and color ranges used for the various
+HSC bands
+band
+color
+cmin
+cmax
+g
+g-r2
+0.3
+1.1
+r2
+r2-i2
+1.0
+2.0
+i2
+i2-z
+0.0
+1.0
+z
+i2-z
+0.0
+1.0
+y
+z-y
+0.0
+0.5
+Notes. For each band, the analysis selects stars with color be-
+tween cmin and cmax in order to perform the regression of Eq.
+16.
+In order to study the quality of the BF correction on
+real science conditions, we bin stars in moments bins and
+restrict the range to (MXX +MY Y )/2 < 10, which roughly
+corresponds to an image quality of 1.35′′ FWHM. We do
+not apply a lower cut because the best image qualities are
+precious for at least cosmic shear. The bin selection operates
+on the expected moments at the location of the star rather
+than the measured ones, so that the quantity used to select
+the bins contents is statistically independent of flux and
+color.
+The measurements of α (accounting for color correc-
+tions) for both CCD directions and in IQ2 bins are dis-
+played in Fig. 16. We can notice an increase of the slopes
+with IQ2, as well as the fact that the y band is notably
+less affected by the BF effect than the other bands. In the
+y band, the photons convert deeper in the sensor bulk and
+the charges experience a shorter drift path than for bluer
+bands, thus reducing the action of the perturbating electric
+field. For the y band, the starting point of the integral of
+Eq. 12 is noticeably smaller that the sensor thickness t.
+For some time, we ignore this subtlety and correct all
+bands using a same model for all bands, derived from co-
+2
+3
+4
+5
+6
+ (BF slope) in pix2/ADU
+1e
+6
+MXX
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pix2)
+2
+3
+4
+5
+6
+ (BF slope) in pix2/ADU
+1e
+6
+MYY
+Fig. 16. BF slopes (with color correction) in IQ2 bins for the
+five bands of HSC, without any BF correction, for the serial
+(top) and parallel (bottom) directions. We can note a roughly
+linear increase of the slopes with IQ2, and that for the best
+image qualities, the slope for MY Y is larger than for MXX, as
+expected from the anisotropy of the nearest neighbor aij (as seen
+in Fig. 11)
+variances measured in the g band. We first apply the cor-
+rection derived from “model 1” (displayed in Fig. 11) and
+the residual BF slopes are displayed in Fig. 17. Although
+the BF slopes are significantly reduced with respect to the
+raw data (Fig. 16), the residual slopes are large at low IQ2,
+indicating that the correction is underestimated at small
+distances. This correction leaves about 10% of the BF ef-
+fect at the best image qualities and about 3% at the upper
+end.
+We then apply the “model 2” correction (Fig.
+12),
+namely, the one that ignores the three first suspicious mea-
+surements along the serial direction. The corresponding BF
+slopes are displayed in Fig. 18. The quality of the correction,
+in particular at the lowest IQ is improved. Ignoring the y
+band, we can interpret the figure as a global small overcor-
+rection that leaves BF slopes about 30 times smaller than
+in the raw data.
+5.3. Brighter-fatter correction for the y-band
+When the energy of photons becomes comparable to the
+silicon band gap, the absorption cross-section tends to van-
+ish and silicon becomes transparent. The band gap is about
+E = 1.2 eV corresponding to λ = 1.1 µm. In the y band, the
+Article number, page 13 of 18
+
+A&A proofs: manuscript no. bf-hsc
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+ (BF slope) in pix2/ADU
+1e
+6
+MXX
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pix2)
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+ (BF slope) in pix2/ADU
+1e
+6
+MYY
+Fig. 17. BF slopes (with color correction) in IQ2 bins for the
+five bands of HSC, with the BF correction derived from “model
+1” (Fig. 11) which uses all the covariance measurements. One
+can note a residual BF slope essentially independent of image
+quality, for all bands, except y that deserves a specific treatment
+(detailed in 5.3).
+absorption length of photons becomes comparable to the
+sensor thickness and we can no longer assume that charges
+produced by converted photons drift all the way from the
+entrance window (z=t in the coordinates of §4.2). The flat-
+field data was acquired in g band, where the mean free path
+of photons is well below 1 µm, and the approximation of im-
+mediate conversion is adequate. This approximation works
+up to i band, may be questioned for z band and does not
+seem to apply to y band.
+Once we have our electrostatic model, the prediction for
+boundary motions in y band can be computed by altering
+the lower bound of the integral of Eq. 12 :
+d(zc) = k
+� z=z0
+z=zc
+ET
+Q(xb, yb, z)dz,
+(17)
+where zc is where the photon converts and k is the nor-
+malization determined when fitting the model to the area
+coefficient data (obtained in g band in our case). Since the
+integrand can vary rapidly with z, d(zc) ̸= d(zc), it is un-
+wise to evaluate these integrals at the average conversion
+depth of photons in the y band. We instead compute the
+average of d(zc) for a realistic distribution of zc:
+dy =
+� zc=szb
+zc=t
+d(zc)dN/dzc dzc,
+(18)
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ (BF slope) in pix2/ADU
+1e
+6
+MXX
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pix2)
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+ (BF slope) in pix2/ADU
+1e
+6
+MYY
+Fig. 18. BF slopes (with color correction) in IQ2 bins for the
+five bands of HSC, with the BF correction derived from “model
+2” (Fig. 12) which uses all the covariance measurements but the
+three first serial pixels. We face a small over correction, and the
+BF slopes have been reduced by a factor of about 30 as compared
+to the raw values of Fig. 16. The y band still requires a specific
+treatment.
+where dN/d zc is the distribution of conversion points (and
+should integrate to unity over the integration domain), and
+zb refers to zs or zp, depending on which boundary we are
+considering. The distribution of conversion depths depends
+on photon absorption length and some assumption for the
+object spectrum. The latter may be regarded as inconve-
+nient, but this is a small chromatic dependence of a small
+flux dependence and, hence, a second-order effect.
+We choose the spectrum of a color 0 (in AB magnitudes)
+object to compute the conversion depth distribution, noting
+that z−y = 0 lies inside the observed star color distribution,
+which has an average of < z−y >≃ 0.13. For the absorption
+length as a function of wavelength, we use the expression
+from Rajkanan et al. (1979) for Silicon at 173oK, which
+typically predicts 103 µm at λ = 950 nm. The conversion
+depth distributions for z and y bands is displayed in Fig.
+19, and we can note that in y band, the conversions are
+spread over the whole thickness and the depth distribution
+is stable when changing the color by about the width of
+the z − y distribution of stars. In Fig. 20, we compare the
+electrostatic model 2 integrated over the full thickness with
+the same model for the y band, using Eq. 17, with the
+distribution of Fig. 19. The largest relative changes happen
+Article number, page 14 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+0
+25
+50
+75
+100
+125
+150
+175
+200
+Conversion depth ( m)
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+Differential probablilty ( m
+1)
+Conversion depth distribution in HSC CCDs
+y (z-y=0)
+y (z-y=0.2)
+z
+Fig. 19. Distributions of conversion depth for a zero color ob-
+ject in the HSC CCDs. The y distribution is much flatter and
+leads to a globally reduced BF effect. Changing the color of the
+object by about the width of the our z − y distribution does not
+significantly alter the expected depth distribution in the y band.
+For the z band, most of the conversions occur at small depths,
+where the perturbating electric field is small. For the i2 band,
+the average conversion depth is about 10 µm.
+at large distances, where the perturbating electric field has
+a sizable contribution over most of the drift path.
+Next, we apply this modified BF correction for the y
+band to the actual science data. We can see in Fig. 21 that
+the y band thus behaves similarly to the other bands, which
+indicates that the reduced BF effect in y likely originates in
+shorter drift paths in this very red band. This also indicates
+that for these CCDs, flat field correlations measured in a
+blue band can be used to predict (via a model) the BF
+correction for a red band. Alternatively, we could certainly
+consider measuring flat-field statistics in y band. We have
+not applied the same treatment to the z band, where the
+correction is much smaller than for y, and the need for
+reducing the BF correction is considerably less obvious. We
+will however apply the BF model for z band in our final
+image processing.
+6. Discussion
+6.1. Quality of the BF correction
+The practical uses of PSF models crucially depend on the
+size of the model capable of faithfully representing the ac-
+tual PSF size for faint objects. Two applications may come
+to mind: first, the measurement of faint supernovae for cos-
+mological applications where fluxes have to be measured
+using PSF photometry and supernova fluxes are calibrated
+against the ones of bright stars; second, the measurement
+of galaxy shapes where one is interested in the intrinsic
+shape, namely, “before” it is smeared by the PSF. An in-
+accurate PSF size results in general in a biased shape. For
+both of these applications, the gauge is the difference of the
+PSF model size to the real size of a faint object, relative
+to the PSF size. The LSST Dark Energy Science Collabo-
+ration et al. (2018) set the maximum acceptable PSF size
+bias (δT/T) to 10−3 for the ten-year Rubin/LSST survey
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+model 2 (full drift)
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+j
+y band
+0
+2
+4
+6
+8
+i
+0
+2
+4
+6
+8
+j
+(m1-m2)/m1
+10
+9
+10
+8
+10
+7
+10
+6
+10
+9
+10
+8
+10
+7
+10
+6
+0.1
+0.0
+0.1
+0.2
+0.3
+Fig. 20. Area coefficients for “model 2” (displayed in Fig. 12)
+based on the same model with shorter drift paths meant to de-
+scribe the y band (center) and the relative difference at the bot-
+tom. The differences tend to increase with distance to the source.
+for all sources of PSF size bias, where T = MXX + MY Y
+is the trace of the second moment matrix. So, in what fol-
+lows, the PSF “size” is, in fact, the “area”. Regarding PSF
+photometry, the relative flux bias caused by PSF size bias
+reads δf/f = 1/2(δT/T). Shear measurements also require
+an accurate estimation of the PSF ellipticity. When sur-
+veys measure the same object with the same orientation of
+the sensors, sensor-induced ellipticities are transferred to
+shape measurements. For the Euclid mission the require-
+ment on ellipticities transferred to objects is 5 10−5 of the
+Article number, page 15 of 18
+
+A&A proofs: manuscript no. bf-hsc
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+ (BF slope) in pix2/ADU
+1e
+6
+MXX
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pix2)
+0.7
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.0
+ (BF slope) in pix2/ADU
+1e
+6
+MYY
+Fig. 21. BF slopes computed after correcting images with model
+2, with the modifications described above for y band. With re-
+spect to Fig. 18, only the y data has changed. We may note that
+all bands now behave in a similar way.
+rms (Table 1 of Cropper et al. 2013). In the context of the
+BF effect, we concentrate on the X/Y ellipticity, namely,
+(MXX − MY Y )/T. Cropper et al. (2013) also provide a re-
+quirement for the PSF size for Euclid, which reads 5 10−4
+rms, comparable to the Rubin 10−3 bound.
+In Fig. 22, we see that on corrected images, the relative
+PSF size difference is below 10−3 in all bands at all levels
+of image quality. In Fig. 23, we see that in the corrected
+images, the residual PSF ellipticity δ(MXX − MY Y )/TP SF
+is mostly below 10−4 across the range of image quality and
+meets the 5 10−5 r.m.s bound. We thus note that the need
+to omit three nearby serial pixels when fitting model 2 did
+not significantly degrade the quality of the BF correction in
+one direction. In Fig. 24, we display the slope of the radial
+fourth moment inaccuracies of the PSF, which also induces
+adverse effects on shear measurements, with a factor on
+the order of 1, as for the PSF size (see Zhang et al. 2021).
+The residuals are even smaller than for the PSF size. We
+note that our residual trends after correction all indicate an
+overcorrection of the BF effect and all this could thus be
+improved by adjusting the overall scale of the correction.
+We also note that these results were obtained in a blind
+way: we did not change the procedure after having seen
+them.
+Gruen et al. (2015) evaluated their image correction for
+DECam in ways similar to ours (their Fig. 12): their method
+overcorrects the size of objects and leaves a negative BF
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+0.006
+0.008
+0.010
+0.012
+0.014
+TPSF/TPSF
+raw
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+TPSF/TPSF
+1e
+3
+model 2 y
+Fig. 22. Relative change of the PSF size between an average
+PSF star and a faint object, as a function of TP SF /2. For the
+PSF size extimator, we use as well the trace of the second mo-
+ment matrix of the PSF. The top plot refers to the raw data,
+the bottom one to the data corrected by model 2 with the short-
+ened drift paths in y band. We assume that an average PSF star
+peaks at 1/3 of the saturation.
+slope, about -1/3 of the uncorrected one. This is certainly
+too large for a large-scale cosmic shear survey. The cor-
+rected ellipticities seem significantly better (and probably
+small enough for a DES-like survey) but they are weakly
+affected by the BF effect in the test sample (as compared
+with the top plot of our Fig. 23). Mandelbaum et al. (2018)
+assessed the correction derived in C18 for HSC images: the
+(linear) size of corrected stars varies with magnitude by
+about 0.2 % over three magnitudes (an eyeball estimation
+from their fig 6). Then, TP SF would evolve twice as much
+and this is small enough for the analysis of the first year
+of the HSC survey. Regarding the ellipticities, trends with
+regard to the flux are not provided. For both DECam and
+HSC, the methods overcorrect the BF trend and both as-
+sessments ignore chromatic contributions to star size varia-
+tions. If the color-flux correlation is similar to ours, correct-
+ing sizes for color decreases the apparent BF slope, hence
+degrading the performance of the BF correction in both
+instances.
+We have insisted on the importance of higher order
+terms in Eq. 4 when analyzing the covariance curves. Those
+terms result from the integration over time of Eq. 1. Re-
+garding the correction of the images, we did not carry out
+integration over time, and we just considered that half of
+Article number, page 16 of 18
+
+Pierre Astier and Nicolas Regnault: BF on HSC
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+0.0008
+0.0007
+0.0006
+0.0005
+0.0004
+0.0003
+0.0002
+0.0001
+0.0000
+ePSF
+raw
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+1
+0
+1
+2
+3
+ePSF
+1e
+4
+model 2 y
+Fig. 23. Change of the PSF ellipticity eP SF
+≡ (MXX −
+MY Y )/TP SF between an average PSF star and a faint object,
+as a function of TP SF /2. The top plot refers to the raw data,
+the bottom one to the data corrected by model 2 with the short-
+ened drift paths in y band. We assume that an average PSF star
+peaks at 1/3 of the saturation.
+the end-of-exposure image is a good approximation of the
+source of electrostatic distortions, hence, neglecting second
+order effects which are important for covariance curves. We
+anticipate that neglected second order effects in image cor-
+rection should manifest themselves as structured residuals
+to the moments versus fmax linear fits. We analyzed these
+residuals in band and image quality bins, summing differ-
+ent bands at the same image quality, and we could not find
+any hint of departure from linearity. This is fortunate since
+accounting for next to leading order effects in image cor-
+rection is more difficult than for the shape of covariance
+curves.
+6.2. Possible further developments
+While the obtained performance of the BF correction seems
+sufficient, we may consider potential avenues for improve-
+ments. First, in our analysis, the overall normalization of
+the BF correction model primarily depends on a01, which
+is the best measured area coefficient actually used in the
+electrostatic fit (the model 2 fit). This coefficient is slightly
+biased by an erroneous value for a00 through the µ3 terms
+in the covariance fit to a level compatible with the small
+over-correction we are facing. It may then seem legitimate
+to actually tune the overall normalization of the correction
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+0.007
+0.008
+M4PSF/M4PSF
+raw
+g
+r2
+i2
+z
+y
+2
+3
+4
+5
+6
+7
+8
+9
+TPSF/2 (pixel2)
+4.0
+3.5
+3.0
+2.5
+2.0
+1.5
+1.0
+0.5
+0.0
+M4PSF/M4PSF
+1e
+4
+model 2 y
+Fig. 24. Relative change of the radial fourth moment as a func-
+tion of image quality for all HSC bands, between a faint source
+and an average PSF star, before (top) and after (bottom) BF
+correction. We assumed that an average PSF star peaks at 1/3
+of the saturation.
+in order to bring the average BF slope to 0, at least for the
+best observing conditions.
+Second, the BF correction we implemented is the out-
+come of an analysis where three area coefficients of the BF
+effect (the three first serial measurements, including the
+largest coefficient) had to be ignored because they could
+not be accommodated by an electrostatic model. Those
+coefficients had to be derived from the measurements of
+other area coefficients through an electrostatic modeling.
+We might then anticipate that for a camera that does not
+suffer from this noise bias (which is first manifested by a
+violation of the sum rule), a better correction model could
+be constructed. In particular, an analysis of test data of
+the Vera Rubin Observatory camera indicates that the sum
+rule is satisfied on the integrated instrument.
+One appealing – and very different – approach could
+be to determine the electrostatic model from the science
+data by fitting the pixel distortion pattern that makes the
+PSF homothetic. The electrostatic modeling could proba-
+bly be computed sufficiently rapidly to be inserted into a
+PSF modeling fitting loop. The concerns about the preci-
+sion measurement of the two-point function of flat fields
+would become pointless.
+Article number, page 17 of 18
+
+A&A proofs: manuscript no. bf-hsc
+6.3. Computer codes
+Our code is split in three different parts: code to measure
+covariances and fit the covariance curves, code to perform
+the electrostatic fit, and code to process the images and in
+particular corrects those for the BF effect. Our public repos-
+itory1 contains the python code for the two first steps. The
+code that measures covariances did not evolve significantly
+since it was developed for A19. We publish here the elec-
+trostatic modeling code for the first time. Our image cor-
+rection code is fairly straightforward and would probably
+run much faster with the convolution in Eq. 14 computed
+in Fourier space. The parts of the analysis that are not in
+the repository can be made available upon request.
+Acknowledgements. We are indebt to the Subaru Telescope technical
+staff that very efficiently operates the observatory and its instruments.
+HSC images are made available on the SMOKA server2, that we have
+used extensively. We perform all our reductions and store our results
+at the Centre de Calcul de l’IN2P33, a computing facility of CNRS.
+This work benefited from useful discussions with N. Suzuky (LBL)
+and N. Yasuda (IPMU), and our colleagues from the Paris team. The
+manuscript eventually benefited from excellent suggestions from our
+referee.
+References
+Antilogus, P., Astier, P., Doherty, P., Guyonnet, A., & Regnault, N.
+2014, Journal of Instrumentation, 9, C3048
+Astier, P., Antilogus, P., Juramy, C., et al. 2019, A&A, 629, A36
+Astier, P., El Hage, P., Guy, J., et al. 2013, A&A, 557, A55
+Bertin, E. & Arnouts, S. 1996, A&AS, 117, 393
+Betoule, M., Marriner, J., Regnault, N., et al. 2013, A&A, 552, A124
+Coulton, W. R., Armstrong, R., Smith, K. M., Lupton, R. H., &
+Spergel, D. N. 2018, Astron. Journ., 155, 258
+Cropper, M., Hoekstra, H., Kitching, T., et al. 2013, MNRAS, 431,
+3103
+Downing, M.,
+Baade, B., Sinclaire, P., Deiries, S., & Christen, F.
+2006, Proc SPIE, 6276
+Gruen, D., Bernstein, G., Jarvis, M., et al. 2015, Journal of Instru-
+mentation, 10, C05032
+Guyonnet, A., Astier, P., Antilogus, P., Regnault, N., & Doherty, P.
+2015, A&A, 575, A41
+Lage, C., Bradshaw, A., Anthony Tyson, J., & LSST Dark Energy Sci-
+ence Collaboration. 2021, Journal of Applied Physics, 130, 164502
+Le
+Breton,
+R.
+2017,
+Theses,
+Université
+Pierre
+et
+Marie
+Curie
+-
+Paris
+VI,
+https://tel.archives-ouvertes.fr/tel-
+01720422/file/2017PA066298.pdf
+Mandelbaum, R., Miyatake, H., Hamana, T., et al. 2018, PASJ, 70,
+S25
+Miyatake, H., Aihara, H., Fujimori, H., et al. 2012, Physics Proce-
+dia, 37, 1413, proceedings of the 2nd International Conference on
+Technology and Instrumentation in Particle Physics (TIPP 2011)
+Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018, PASJ, 70,
+S1
+Pumplin, J. 1969, American Journal of Physics, 37, 737
+Rajkanan, K., Singh, R., & Shewchun, J. 1979, Solid-State Electronics,
+22, 793
+Rasmussen, A., Guyonnet, A., Lage, C., et al. 2016, in Proc. SPIE, Vol.
+9915, High Energy, Optical, and Infrared Detectors for Astronomy
+VII, 99151A
+The LSST Dark Energy Science Collaboration, Mandelbaum, R., Ei-
+fler, T., et al. 2018, arXiv, 1809.01669
+Zhang, T., Mandelbaum, R., & Collaboration, T. L. D. E. S. 2021,
+Monthly Notices of the Royal Astronomical Society, 510, 1978
+1 https://gitlab.in2p3.fr/astier/bfptc
+2 https://smoka.nao.ac.jp/index.jsp
+3 https://cc.in2p3.fr
+Article number, page 18 of 18
+
diff --git a/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/load_file.txt b/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..92424af697db0753a2e092605d2443d1ee64208a
--- /dev/null
+++ b/zdE1T4oBgHgl3EQfkgRW/content/tmp_files/load_file.txt
@@ -0,0 +1,1137 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf,len=1136
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  ©ESO 2023  January 10, 2023  Correction of the brighter-fatter effect on the CCDs of Hyper  Suprime-Cam  Pierre Astier1 and Nicolas Regnault1  LPNHE, (CNRS/IN2P3, Sorbonne Université, Université Paris Cité), Laboratoire de Physique Nucléaire et de Hautes  Énergies, F-75005, Paris, France  Received Mont DD, YYYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' accepted Mont DD, YYYY  ABSTRACT  The brighter-fatter effect affects all CCD sensors to various degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Deep-depleted thick sensors are seriously affected  and the measurement of galaxy shapes for cosmic shear measurements requires an accurate correction of the effect in  science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We describe the whole correction chain we have implemented for the CCDs of the Hyper Suprime-Cam  imager on the Subaru Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We derive non linearity corrections from a new sequence of flat field images, and  measure their statistics, namely their two-pixel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We constrain an electrostatic model from flat field statistics  that we use to correct science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We find evidence that some fraction of the observed variance and some covariances  is not due to the combination of Poisson statistics and electrostatics – and the cause remains elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We then have  to ignore some measurements when deriving the electrostatic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Over a wide range of image qualities and in the  5 bands of the imager, stars in corrected science images exhibit size variations with flux small enough to predict the  point spread function for faint objects to an accuracy better than 10−3 for the trace of second moments – and even  better for the ellipticity and the fourth radial moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This performance is sufficient for upcoming large-scale cosmic  shear surveys such as Rubin/LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Introduction  The brighter-fatter (BF) effect refers to a dynamical image  distortion that affects CCD sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The most spectacu-  lar manifestation of the effect is that bright stars appear  slightly bigger in size than faint ones, a manifestation that  is reflected the very name of the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' All studies of the  effect have attributed it to distortions of the drift electric  field sourced by the charges stored in the pixel potential  wells during image integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This modifies the apparent  shape of bright objects and the two-point statistics of uni-  form exposures, and thick CCDs are more vulnerable to  the effect than thinner sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Evidence of the effect and  the physical explanation can be found in Guyonnet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (2015, and references therein, G15 hereafter), together with  the relation of the effect with non-trivial flat field statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Electrostatic calculations are shown to reproduce the data  in Rasmussen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2016) and in Lage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2021) for a  specific sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  On deep-depleted thick CCDs, bright stars thus gener-  ally appear bigger by a few percent than faint stars, com-  promising the modeling of the image point spread function  (PSF) at a level that is not tolerable for large-scale cosmic  shear measurements (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The correction method proposed in G15 relies on flat field  statistics to constrain the correction applied to science im-  ages, which mostly consists of correcting the recorded image  for dynamically displaced pixel boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The method  has been implemented for DECam in Gruen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2015),  and for Hyper Suprime-Cam (HSC) on the Subaru tele-  scope in Coulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2018, C18 hereafter) with some  minor differences with respect to G15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Send offprint requests to: pierre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='astier@in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='fr  The method proposed in G15 relies on first-order per-  turbations both in the modeling of flat field correlations  and when correcting science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Astier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2019,  A19 hereafter), the relation between pixel area alterations  and flat field statistics is extended to higher orders, which  removes significant biases from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the same pa-  per, correcting for non-linearity of the video chain is shown  to play a potentially important role when constraining the  BF effect from flat fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' One other evolution since the  G15 proposal is that detailed electrostatic calculations have  been shown to reproduce the measured flat-field statistics,  when the mandatory manufacturing data is available (see,  in particular, Lage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' However, the CCD ven-  dors do not necessarily release this data, or they do not  even have it available to the required level of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' As  we show in this paper, there is also some detectable demo-  graphic variability among the CCDs of the HSC camera,  which are all of a unique type from a single vendor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Gruen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2015) also detected some variability among DECam  CCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Thus, constraining the image corrections from mea-  surements of the actual sensors is still in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In the present paper, we revisit the BF correction for  the HSC camera described in C18 and applied to science  images in Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We take advantage  of a new flat-field sequence that allows us to re-determine  both the non-linearity correction and the two-point correla-  tion function of flat fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We apply to these images some  potentially important corrections (described in A19) and  we fit a variance and covariance model that has been im-  proved since C18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We also propose a different approach for  transforming the information extracted from flat fields into  the correction of science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Finally, we test the correc-  tion of the images separately over a broad range of image  qualities and in the five bands of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 1 of 18  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='03274v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='IM]  9 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  The flow of the paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We first detail in  § 2 why it is necessary to correct non-linearities prior to  measuring flat field statistics and the non-linearity mea-  surement itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In §3, we describe the measurements of flat  field statistics and the fit of the measurements, as well as  the variability observed among the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The informa-  tion extracted from flat field statistics is fundamentally in-  sufficient to correct the science images, thus, we describe  in § 4 the electrostatic model we use to derive the correc-  tion from the flat-field results and the outcome of different  fits we perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Once we obtain models that allow us to  correct science images, we apply those to real data, as de-  scribed in § 5, and we compare the various outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  face the evidence that the BF correction is inadequate for  the y band and we compute a physically motivated reduced  correction for this band, which we eventually apply to the  science data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In § 6, we compute some PSF modeling quality  indicators commonly used in the context of shear estima-  tion and conclude that the quality of our correction exceeds  the requirements for a large-scale cosmic shear survey such  as Rubin/LSST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' It also fulfills the less stringent require-  ments of the PSF fidelity implied by photometry accuracy  of high-redshift faint supernovae in order to estimate their  distances, which is our initial motivation for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Non-linearity correction  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Importance of the non-linearity correction  Following A19, we assume that the effective area A of a  pixel is linearly altered by the charge content of the sensor:  δA = A g  �  i,j  aijQij,  (1)  where aij is a characteristic of the sensor and Qij denotes  the charge content of the image, which evolves as light in-  tegration goes on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The indices i and j refer to distances  in pixel units along the serial and parallel directions, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the same coordinates, δA applies to the pixel  located at (i = 0, j = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Conventionally, aij is expressed  in el−1, Qij in ADU and g is the gain in el/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We note  that, equivalently, we may set g = 1 and express Q in elec-  trons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' From parity symmetry, we assume that:  aij = a|i||j|,  (2)  so that we measure the aij on the i, j ⩾ 0 quadrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' If  we consider a uniform image, with all Qij identical, then  δA has to be zero, because of translation symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  imposes the following “sum rule”:  �  −∞<i,j<+∞  aij = 0,  (3)  where the sum extends to the four quadrants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This sum rule  can also be regarded as a consequence of area conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Since aij is the fractional pixel area change for a unit source  charge, we refer to these quantities as “area coefficients.”  Since same-sign charges repel each other, a pixel shrinks as  it fills up, and the self-interaction coefficient a00 is negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  For regular CCD operating conditions, all the other coeffi-  cients turn out to be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The sum rule hence indicates  that |a00| is much larger (in absolute value) than any other  coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1, the expression of pixel covariances in  uniform exposures reads (see the derivation in A19):  (4)  Cij(µ) = µ  g  �  δi0δj0 + aijµg + 2  3[a ⊗ a]ij(µg)2  + 1  3[a ⊗ a ⊗ a]ij(µg)3 + · · ·  �  + nij/g2,  where µ is the average (in ADU) of the uniform exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  nij refers to noises (expressed in el2), and n00 refers to  the usual read noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' From this expression, the relation be-  tween the variance of uniform exposures and their average  (usually called the photon transfer curve or PTC) can be  expressed as:  C00 = n00/g2 + µ/g + a00µ2 + O(µ3),  (5)  where the first term is the read noise, the second is the  Poisson term, and the quadratic term is the first and largest  contribution of the BF effect: the slope of the PTC decays  with the signal level (because a00 < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The fact that the  PTC of CCDs grows less rapidly than the signal level was  initially noted in Downing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2006), who also noted that  thanks to positive covariances, grouping data into bigger  pixels tends to restore the Poisson behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These non-  trivial flat-field statistics and the brighter-fatter effect per  se were unified under the same physics in Antilogus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The analog readout chain can be affected by non-  linearity and a common distortion is a quadratic contri-  bution:  µm = µt + kµ2  t,  (6)  where µm and µt refer to measured and true values, and k  is a small quantity describing the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The last term  can equivalently refer to either value of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the variances,  we have:  Vm ≃ Vt  �∂µm  ∂µt  �2  = Vt(1 + 2kµ)2  ≃ Vt(1 + 4kµ)  = (n00/g2 + µt/g + a00µ2  t)(1 + 4kµt)  = n00/g2 + µm/g + (a00 + 3k/g)µ2  m + O(µ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (7)  Thus, comparing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 7 and 5, we find that a quadratic non-  linearity, if it is not corrected for, biases the measurement  of a00, which is the largest coefficient describing the BF  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The same calculation applied to covariances shows  that there is no effect at the µ2 level, hence the measure-  ment of the other aij coefficients is much less sensitive to a  quadratic non-linearity than a00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Measurement of response non-linearity  The Hyper Suprime-Cam instrument is installed at the  prime focus of the 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2-m Subaru telescope on the Mauna-  Kea summit in Hawaii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' A detailed description of the in-  strument can be found in Miyazaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2018) and  here we provide only the salient features for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The HSC camera contains 104 science sensors, which are  deep-depleted Hamamatsu 2k×4k CCDs with four read-out  Article number, page 2 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='32  MJD - 58786 (days)  0  10000  20000  30000  40000  CCD signal  (ADU)  sensor 42 channel 1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' CCD signal values as a function of time for a single  channel during a sequence of uniform exposures illuminated by a  lamp in the dome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The intensities are deliberately not ordered in  a monotonic way, so that the drift of the illumination system can  be separated from the non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The sequence was designed  by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Yasuda (IPMU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='33  MJD - 58786 (days)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='9875  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='9900  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='9925  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='9950  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='9975  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0025  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0050  Spline modeling of the illumination  Illumination correction for sensor 42 channels  channel 1  channel 2  channel 3  channel 4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Spline correction from the minimization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8 for  four different channels, normalized to its average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since this is  presumably due to lamp variations, it should be similar for all  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The maximum difference is in the 10−4 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  channels on each sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The sensors are 200 µm thick, the  pixel size is 15 µm, which corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='185′′, on aver-  age, over the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The saturation of sensors occurs  around 120,000 el and the video channels have a typical gain  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2 el/ADU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' All channels may a priori exhibit different  non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to measure the non-linearity of the camera  video chains, we rely on a new sequence acquired on  2019/10/30, of uniform exposures known as “dome flats”  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Fit residuals as a function of date (top) and signal level  (bottom) for all channels of all sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' There are no obvious  needs for a more sophisticated model as a function of date nor  signal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The rms residual is about 3 10−4µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  obtained at night by illuminating a screen attached to the  dome, toward which the telescope is pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The signal  level is controlled by varying the exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The illu-  mination system is not equipped with a photodiode or any  similar device and we use the exposure time as a proxy for  the amount of light received by the CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Obviously, this is  an acceptable proxy only if the illumination system is stable  over the whole sequence or if its variations can be modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to disentangle a drift of the lamp intensity from  a genuine non-linearity, the various exposure times are not  ordered time-wise in a monotonic way, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  For a given video channel, we collect from every expo-  sure (labeled i) an average signal, µi (in ADUs), an expo-  sure time, Ti, and the (Julian) date at which it was ac-  quired, di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We minimize the quantity:  �  i  �  (Ti + O)S(di) − µi − kµ2  i  �2 ,  (8)  where S represents a seven-knot spline meant to describe  the lamp variation with time, O is a potential systematic  Article number, page 3 of 18  20  15  10  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='27  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='33  MJD-58786 (days)  20  non-linearity fit residual (ADUs)  15  10  10  0  5000  10000  15000  20000  25000  30000  35000  40000  CCD signal μ (ADUs)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  4  3  2  1  0  1  k (ADUs  1)  1e  7  0  20  40  60  80  100  channel counts  3  2  1  0  1  1e  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Values of measured quadratic non linearity k (in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8)  in the focal plane (top) and their distribution (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Most  of the channels cluster around 10−7, which corresponds to a  correction of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4% around maximum CCD signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  shutter offset, k represents the quadratic non-linearity, and  Ti is the required exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The shutter offset is meant  to account for a (small) difference between the required  and realized exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We do not find any compelling  evidence for a global shutter offset, nor a spatial trend in  the focal plane, that could arise from such a large blade  shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Hence, we set O = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since the illumination may  evolve along the sequence in slightly different ways across  the focal plane, we do not enforce the spline S to be the  same for all channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We eventually compare the outcome  of different fits (on the same sensor) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The fact that  these corrections are almost indistinguishable indicates that  the non-linearity and the lamp variations can be robustly  separated from this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Figure 3 displays the fit residuals (from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8) as a func-  tion of signal level or of date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Those are globally small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' A  more flexible model as a function of date would not reduce  much these residuals, highly correlated over the channels  at the same date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can attribute this scatter to random  fluctuations of the lamp or of the shutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The fit resid-  uals as a function of signal level do not strongly call for  a more flexible or more general non-linearity model than  a quadratic correction: the largest systematic residual is  found around 10,000 ADUs and amounts to a few ADUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  However, for some yet unknown reason, we had to discard  exposures before di − 58786 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='255 because they exhibit  large residuals (tens of ADUs) consistently over channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  One might attribute this to some erratic behavior of the  lamp that vanishes after about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' While these first ex-  posures are rejected from the non-linearity analysis, they  are used to measure the flat field statistics in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4, we display the values of quadratic non-  linearity in the HSC focal plane in order to visualize the ab-  sence of geometrical trend that could occur from a spatially  variable evolution of the illuminating system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The distribu-  tion peaks around k = 10−7 ADU−1, which corresponds to  a correction of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4 % at the maximum CCD signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  go on to show soon that a00 averages to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3 10−6 and the  channel gains are around 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' if one ignores quadratic non-  linearity (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 7), the bias affecting a00 amounts to ∼8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' a00  is the best measured area coefficient and essentially drives  the scale of the BF correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We might note that such  a bias unavoidably causes a violation of the sum rule and,  hence, translating such a set of area coefficients into a cor-  rection is somehow arbitrary because sensible image cor-  rections have to derive from areas changes that sum up to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We also note that that eliminating this trivial cause of  sum rule violation allows us to detect more subtle problems  in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Measurement and fit of flat-field statistics  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Measurements  Measuring the variances and covariances of flat fields al-  lows one to measure the dynamical alteration of pixel areas  due to electrostatic distortions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' namely, we measure vari-  ances and covariances of flat field exposures (which we may  call the two-point function of the exposure) at various sig-  nal levels, in order to measure the aij coefficients (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1)  by fitting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4 to the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We closely follow  the procedure described in A19: we first correct for non-  linearity (see the previous section) and correct for deferred  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  A deferred signal is by definition collected after (in the  serial time chain) the pixel it belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In order to mea-  sure those, we closely follow the procedure described in A19,  and find that there are no significant deferred signals be-  yond the nearest serial neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We display a few examples  of the nearest neighbor deferred signals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 5: those are  small and almost linear, indicating that they mostly origi-  nate from electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We tabulate one correction curve per  channel, and for correcting the effect, we typically “place  back” less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1 % of a pixel into the preceding one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Failing to carry out this correction results in the covariance,  C10, exhibiting a linear contribution (∝ µ), which cannot  be absorbed by the model of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4, hence biasing a10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For  our CCDs, at the maximum signal level, the contribution of  uncorrected deferred signals to C10 would represent about  25% of the electrostatic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  When computing image statistics, we mask any outlier  pixels detected in the images and use the pixel mask ap-  plied to science images at the epoch of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These  pixel masks flag pixels that repeatedly depart from their  neighbors in uniform exposures and pixels on the sides of  the focal plane which are heavily vignetted (or fully blind).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 4 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  5  0  5  10  15  20  Chan 1  Chan 2  0  10000  20000  30000  40000  5  0  5  10  15  20  Chan 3  0  10000  20000  30000  40000  Chan 4  First overscan pixel (ADU)  Last row real pixel (ADU)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Nearest serial neighbor deferred signals for the four chan-  nels of sensor 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The values are small and vary almost linearly  with the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The model continuous lines are the cubic poly-  nomial model that we use for correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We account for the masked pixels in covariance computa-  tions in a standard way, described in Appendix A of A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The bias of variance and covariances due to masking out-  lier pixels from images themselves is compensated for as  described in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2 of A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We measure each video channel independently and pro-  duce variance and covariance curves using the 66 flat pairs  of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We measure the covariances on the dif-  ference of flat pairs acquired at the same signal level to  eliminate contributions from illumination non-uniformity  or sensor sensitivity variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These 66 flat pairs were  acquired in g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Contrarily to C18, we carry out the  measurements on all HSC science sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Fits of the covariance curves  We then fit the measurements using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4, with weights de-  rived from shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This equation contains terms up to  µ4, and including one extra order would change the predic-  tions by a few 10−5, which is about 100 times less that the  smallest shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 6 displays the measurements and  fits for a few (i, j) pairs and for the four channels of a sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  One can readily note that the slopes for C10 (serial neigh-  bors) and C01 (parallel neighbors) differ by a factor of more  than 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' which is due to the very different mechanisms that  confine the charges into the pixel wells for the serial and the  parallel directions: while the boundaries between columns  are static and implanted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' the boundaries between rows re-  sult from the potentials of the parallel clock stripes that  are swung during image read-out in order to move charges  towards the serial register and the output amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  parallel pixel boundaries are then somehow weaker than  the serial ones and this is reflected on the pixel covariances  of uniform exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We perform the measurements in the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='28  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='31  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='32  C00/ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002(channel-1) (ADUs2)  channel 1  channel 2  channel 3  channel 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='010  C01/ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002(channel-1) (ADUs2)  0  5000  10000  15000  20000  25000  30000  (ADUs)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='007  C10/ -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002(channel-1) (ADUs2)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Measurement and fits of the variance (C00) and the  two nearest neighbor covariances (C10 and C01) for the four  channels of sensor 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The channels have been offset for visual  convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The decay of C00/µ with µ, originally noted in  Downing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2006), is the most obvious effect of the BF effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The spread of measurements is compatible with shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  range i, j < 10 and fit Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4 to all the measurements of a  given channel at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to question whether the fitted model (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4)  properly describes the data, we plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 7 the fit residu-  als of the variance and the nearest covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since we are  studying effects related to sensors, we turned the averages  and covariances into gain-free quantities by converting them  into electrons, using the gains obtained from the fits in the  previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' An examination of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4 shows that  the coefficients of the µ3 terms are entirely determined by  combinations of the coefficients of the µ2 terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The resid-  uals in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 7 exhibit some curvature, which is compatible  with a mismatch between the coefficients of µ2 and µ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For  the residuals of C00 (top), the curvature of the residuals  represents about 15% of the cubic term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4 (thus il-  Article number, page 5 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Residuals of fits of variance and covariance measure-  ments, all expressed in electrons using the gains from the fits,  so that the data from different sensors can be averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The red  signs are binned averages of the residuals from individual fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  lustrating the importance of these higher order terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For  C01, the curvature represents about 30% of the cubic term,  while they have similar values for the residuals to C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  will propose later an explanation for the curvature of resid-  uals of C00 and C01, but this explanation does not apply to  C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Demography of HSC sensors  We may now investigate the demography of channels and  sensors regarding the BF effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8, we display scat-  ter plots of the well-measured a coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can see the  correlations, which are similar at the channel and sensor  level, indicating that the variations are weaker within sen-  sors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can note that a00 does not correlate to the other  coefficients, and that the apparent correlations are in a di-  rection that does not preserve the “sum rule.” This may  indicate that they are not due to genuine variations of the  BF effect among the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We may also observe that on  average the sum of a is positive:  �  �  −10<i,j<10  aij  �  channels  ≃ 7 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (9)  Since all aij at larger distances are positive, the sum to in-  finity is even farther from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since < a00 >≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3 10−6, the  violation of the sum rule is hence at least 5% of a00 and we  eventually see that it reaches ∼10% once we have a model to  evaluate the large-distance contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This means that  the sum of all covariances rises with signal level faster than  Poisson, at variance with expectations from statistics (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8 in A19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Our measurements are thus affected by some  noise, which increases with signal level and contributes as  µ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Since the variations of well-measured aij is at most 10  % across channels, we decide to average the measurements  in order to average the shot noise at large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Once  equipped with this average, we fit an electrostatic model to  it that, by construction, satisfies the sum rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the next  section, we show that the mismatch between the model and  the data is indeed localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Electrostatic fit  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' From the area coefficients to science image corrections  We study the BF effect in order to suppress its impacts from  the science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The scheme proposed in G15 consists  of evaluating from the science image itself the motions of  pixel boundaries with respect to a perfect grid, as well as  evaluating via interpolation how much charge was flowing  over these pixel boundaries during integration, and then  placing this amount of charge back where it belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We can readily note that the measurements we have per-  formed so far constrain the change in the area of a pixel, but  do not tell how its shape is altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' If we describe the shape  change by different motions of the serial and parallel sides  of a pixel, this simplistic shape description already requires  two quantities per pixel, while we only have one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Defin-  ing a shape change by distinct serial and parallel boundary  shifts means in turn that we have to “split” the aij coef-  ficients (into aN  ij, aW  ij , aS  ij, aE  ij) and rely on area-conserving  symmetries such as aW  10 ≡ −aE  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can also regard aij as  the discrete divergence of the pixel boundary displacement  field and we are faced with the ill-posed problem of deter-  mining a vector field from its divergence, in a a 2d discrete  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Various implementations of the correction method pro-  posed in G15 differ in the way they perform this promotion  of pixel area change into two directional components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In  G15 and in Gruen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2015), some ratios are imposed,  which were estimated from the area changes themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  C18 argue that the motions of pixel boundaries is a 2D vec-  tor field that results from the (perturbation) electric field  sourced by the charges that represent the image, and, hence,  the 2d displacement field is (as is the 3d electric field) curl-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The “scalar” aij field can then be transformed into a 2d  vector curl-free field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' While this curl-free assumption seems  appealing, we show in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4 that it turns out to be violated  by a solution of the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We may guess (and  Article number, page 6 of 18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  model)/μ(e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  model)/μ(e)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  Co1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0  20000  40000  60000  80000  100000  μ (e)Pierre Astier and Nicolas Regnault: BF on HSC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  a00  1e  6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  a10  1e  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  a00  1e  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='75  a01  1e  7  4  6  a10  1e  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='70  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='75  a01  1e  7  4  6  a10  1e  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  a11  1e  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='7  a01  1e  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  a11  1e  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='7  a01  1e  7  4  2  0  2  4  a  1e  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Distribution of area coefficients over the ∼400 hundred channels of HSC (blue) and the averages over each sensor (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  All aij coefficients are expressed in el−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We may note that the trends for channels and sensors are similar, which indicates some  homogeneity within sensors (except for � a, where the scatter is dominated by shot noise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' � a is the sum for −10 < i, j < 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  we show below) that the pixel boundaries motions are pro-  portional to the integral over pixel boundaries of the per-  turbating electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These integrals would themselves be  curl-free (as is the electric field) if the integration paths were  all identical for serial and parallel boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The funda-  mental anisotropy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', from the ratio of a01 and a10) indi-  cates that this is not true at small distances, and questions  the curl-free hypothesis applied to the pixel boundaries dis-  placement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Even without this anisotropy, we might also  question whether the curl-free property of the continuous  3D electric field is precisely transferred to a curl-like com-  bination of finite differences over the pixel lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Our practical approach is to promote the discrete aij  scalar field into a discrete 2D vector field relying on electro-  statics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' namely, to propose to fit the geometrical parameters  of a simple electrostatic model of the perturbating electric  field to the data and to evaluate the 2D vector field we need  from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We note that since the model reports ac-  tual pixel areas, the sum rule is by construction satisfied by  the outcome of a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The electrostatic model  We are interested in pixel boundary shifts under the in-  fluence of stored charge and here we sketch a first-order  perturbation scheme that allows us to predict these shifts  from a few geometrical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The first ingredient is  the electric field sourced by the stored charges inside the  sensor, which we refer above as the “perturbating” electric  field (because it adds to the drift field that defines pix-  els).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We assume that the field sourced by collected charges  causes no rearrangement of charge within the bulk of the  device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We model this perturbating field as sourced by a  charge between two infinite grounded equipotentials figur-  ing the light entrance window (on which the drift voltage is  applied) and the parallel clock stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This is a text-book  problem and using the image charge technique, we can write  the corresponding potential as:  φ(ρ, z) = Q  4πϵ  ∞  �  n=−∞  1/  �  ρ2 + (2nt + zq − z)2  − 1/  �  ρ2 + (2nt − zq − z)2,  (10)  where ρ2 = x2 + y2, t is the sensor thickness, the source  charge Q is located at (0, 0, zq), and the equipotential planes  are at z = 0 (representing the parallel clock stripes) and  z = t (the light entrance face of the sensor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can check  that φ(ρ, 0) = φ(ρ, t) = 0: in these cases, each positive term  (first line) of the series has an exact opposite among the  negative terms (second line), which is how the image tech-  nique works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This expression for the potential converges  poorly with n for ρ much larger than t and alternative ex-  pressions should then be used (Pumplin 1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the fit,  we do not need to evaluate the model at ρ > t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The field that results from the normal operation of the  sensor drives the charges into the pixel wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The drift  lines on the pixel boundaries have a peculiar point at which  the field is null and the potential has a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  point is commonly located a few microns away from the  clock stripes (at z=0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These points are located at differ-  ent heights above serial and parallel boundaries, and this  difference contributes to the anisotropy of flat field small-  distance covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We display in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 9 a contour crafted  to relate the pixel boundary shifts to the source charge us-  ing Gauss’s theorem, applied to the total electric field, the  drift field plus the perturbation introduced by the charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  This contour is made from four segments (numbers match-  ing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Segment 1 corresponds to the perturbed drift  line that separates two pixels and the integral of the trans-  verse electric field is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Segment 2 represents the short  path between the unperturbed zero-field point and the per-  turbed zero-field point and is very close to a drift line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' the  integral of the electric field is then a second order quantity,  Article number, page 7 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  1  2  3  4  Q  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Contour to which we are going to apply Gauss’s theorem  to relate the boundary shift (the length of segment 4) to the  electric field sourced by the charge Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  which we ignore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Along segment 3, the integral of the un-  perturbed field vanishes because it is the unperturbed drift  line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' we are thus left with the integral of the perturbation  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Over segment 4, the integral of the field is the product  of the drift field, Ed, times the boundary displacement, d,  that we wish to estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  So, Gauss’s theorem is finally expressed as:  � z=z0  z=t  ET  Q(xb, yb, z)dz + dEd =  �  C  ρ/ϵ  (11)  where z0 is the zero-field point altitude, xb and yb are the  coordinates of the unperturbed drift line, ET  Q is the field  transverse to the boundary sourced by the charge Q, Ed  is the drift field (at the top), and the right-hand side is  the bulk charge contained inside the contour, because the  Silicon material can contain residual impurities sourcing a  small bulk electric charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We do not know this right-hand  side, but we can assume that (to first order) it is propor-  tional to d, our perturbation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' So, we eventually  obtain:  d ∝  � z=z0  z=t  ET  Q(xb, yb, z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (12)  In other words, the boundary displacement is proportional  to the integral of the perturbating (transverse) field over  the unperturbed trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This is usually called the Born  approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In a charge-free material, the normalization  is expressed as 1/Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The expression assumes that charges  are produced when light enters the sensor, at z = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the  reddest bands, we should instead account for the conversion  depth of photons in the sensor, as we will do in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  expression should be averaged over the pixel side in the  direction perpendicular to the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to constrain the rhs of the expression, it is  tempting to carry out the the measurements at different  values of Ed by changing the drift voltage applied to the  entrance side of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' However, changing the drift ac-  tually alters Ed but also z0 (for both flavors of boundaries),  so measuring the sensor under different drift voltages would  not help significantly at constraining the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We may,  in principle, predict the proportionality coefficient of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12  because the (empty CCD) charge density, thickness, applied  drift voltage, and drift field are related by basic electrostat-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We nonetheless stick to fitting the global scale because  precisely estimating the drift electric field involves at least  the clock stripe sizes and their potentials during image in-  tegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In this specific study, we have to fit the overall  model scale because we were not able to find the drift and  parallel clock potentials applied to the sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Q  a  b  d  (0,1)  (1,1)  (1,0)  zs  t  y  x  z  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Schematic of the geometry of the electrostatic fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We add some flexibility to the model by allowing source  charges to be extended and we stick to a very crude model: a  uniformly charged rectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This refinement, as compared  to point sources, only influences the very first neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  So the model eventually has six parameters (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 10): a  global normalization factor, the height of the source charge  zQ, the height of the zero-field points over parallel and serial  pixel boundaries zp and zs, and the sides of the (uniform)  rectangular source charge a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The calculations of the model also require the thickness  of the sensor (200 µm) and the pixel side (15 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For  the practical implementation of the integrals from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12,  we settle for integrating analytically the terms of the series  similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 10 for the electric field and emulating the  rectangular source by splitting the charge into nine equally  spaced point charges (only for n = −1, 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We sum the  series of the expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 10 up to n = ±11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  seems to be sufficient for up to 10 pixels because increas-  ing to n = 12 only changes outputs at the sixth decimal  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This electrostatic modeling approach was initially  presented in Le Breton (2017), which we adapted Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 9  and 10 from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 8 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  0  2  4  6  8  0  2  4  6  8  j  model  10  9  10  8  10  7  10  6  0  2  4  6  8  0  2  4  6  8  j  data (a00  a00)  10  9  10  8  10  7  10  6  0  2  4  6  8  i  0  2  4  6  8  j  data-model  4  3  2  1  0  1  2  3  4  1e  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Data (top), fitted electrostatic model (middle), and  difference (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The sign of a00 has been flipped in the top  plots, but not in the difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can note the clear difference  of the two nearest neighbors, an anisotropy that is already visible  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We expect a poor fit because the data violates the sum  rule, while the fit cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Fit of the electrostatic model to the data  We performed a least-squares fit to the average area coef-  ficient data, using the spread over channels to weight the  squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We first fit the average data obtained at the previ-  ous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The data, fit, and difference are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can see that the measurements reach a signal-to-  noise ratio of about 1 at a distance of 8 to 9 pixels from the  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Beyond the small separations, the measurements  exhibit a radial symmetry as expected from electrostatics  and displayed as well by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The residuals do not  average to zero and the data is larger than the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  reflects that the data violates the sum rule but the fit can-  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The sum of the model area coefficients is −7 10−8 up  to i, j < 10 (and tends to 0 with increasing bounds, so the  contribution of unmeasured aij is 7 10−8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since the sum of  measured aij is 7 10−8, the data violates the sum rule by  about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4 10−7, which is more than 10% of a00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The details  of the electrostatic model cannot change the contribution  of unmeasured aij significantly (at i > 9 or j > 9) and  certainly cannot flip its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The excess of the data with  respect to the model is concentrated on the 3 first serial  pixels and the first parallel neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Some excess of variance and covariance along the serial  direction is expected if a video signal experiences rapid gain  variation (or “gain noise”): rapid gain changes contribute a  variance component that scales as the square of signal level  and, hence, artificially increase the value of a00 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  If gain fluctuations last longer than the time to read a pixel,  they also bias covariances along the serial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since  the residuals seem to decay along the serial direction, possi-  ble gain variations have also to decay rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' They cannot  be invoked to explain the excess on the first parallel neigh-  bor because while serial pixels are read out microseconds  apart, milliseconds separate neighboring lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We now attempt a fit that ignores the variance and the  two next serial pixels, that is, a00, a10 and a20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The fit dis-  played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12 seems acceptable: the residuals are at most  1% of the largest used coefficient a01 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6 10−7, and an  even lower fraction of a00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The residuals exhibit some sort  of low-level “chessboard pattern”, which is not entirely sys-  tematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Assuming it is real, we have no proposition for its  source, and we could not invent a small periodic variation  of the physical size of pixels that would produce this 2 × 2  pixel pattern of the two-pixel correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We display the measured and fitted values, and their ra-  tio in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The model reproduces the data well and gives  some confidence that the model delivers sensible values for  the data we decided not to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We note that the decay of  signal with distance is reproduced very well by the model  over more than two orders of magnitude of signal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In  Pumplin (1969), it is shown that the large-distance decay  of the perturbating electric field from a point charge in  the sensor depends essentially exponentially on the inverse  thickness of the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The other geometrical parameters  of the model cannot alter significantly the logarithmic slope  of this decay at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' So, the evaluation of the  model at large distances, needed to gauge the compliance  of data to the sum rule, is a robust outcome of the model  if the slopes of data and model match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In Table 1, we display the measured and fitted largest  area coefficients of the data and the two fitted models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  can see in particular the size of the discrepancies along  the serial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can now extract from the fitted  model the pixel boundary displacement caused by a unit  charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This allows us to compute for a given science image,  the accumulated boundary displacements resulting from all  present charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We will test the quality of the BF effect  correction derived from the models on science images in the  following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We now question the curl-free hypothesis,  and then discuss the gain noise issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 9 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  0  2  4  6  8  0  2  4  6  8  j  model  10  9  10  8  10  7  10  6  0  2  4  6  8  0  2  4  6  8  j  data (a00  a00)  10  9  10  8  10  7  10  6  0  2  4  6  8  i  0  2  4  6  8  j  data-model  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1e  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Data (top), fitted electrostatic model (middle), and  difference (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This fit ignores the three first serial pixels,  marked with crosses in the bottom plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Compared to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11,  we see that the data excess in a01 has disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We note  that the color scale of the difference is zoomed-in, as compared  to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Considering whether the pixel boundary displacement  field is curl-free  In C18, the authors assume that the boundary displacement  field is curl-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Our definition of discrete curl is expressed  as:  ci,j = (aN  i,j+1 − aN  i,j) − (aW  i+1,j − aW  i,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (13)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  i2 + j2  0  10  9  10  8  10  7  10  6  |aij|  fit  data  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  i2 + j2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  (data-model)/|model|  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Data and fit (left) as a function of distance, difference  normalized to the model (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The fit ignores the three first  serial pixels, which are labeled with crosses in the rhs plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  dashed line in the lhs plot separates the logarithmic and linear  scale regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Values of the largest area coefficients for the data and  two fitted models  i = 0  i = 1  i = 2  d  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='60 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='88 10−8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='95 10−9  j = 2  m1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='57 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='78 10−8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='66 10−9  m2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='77 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='93 10−8  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='57 10−9  d  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='61 10−7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='19 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='85 10−8  j = 1  m1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='55 10−7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='23 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='57 10−8  m2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='60 10−7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='24 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='68 10−8  d  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='24 10−6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='79 10−8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='56 10−8  j = 0  m1  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='29 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='90 10−8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='99 10−8  m2  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='37 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='43 10−8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='08 10−8  Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The three rows for each value of j refer to the data,  model 1, and model 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Model 1 fits all the measured aij, while  model 2 ignores the data from the bottom row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 14, we display the discrete 2D curl of the field, the  displacement of one boundary (arbitrarily chosen as the  parallel one, the serial one is in fact smaller) and their ra-  tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We used the electrostatic solution displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12,  which (by definition) satisfies the Poisson equation and,  hence, has a curl-free 3D electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can readily see  that the curl-free assumption may lead to sizable errors  (several tens of percent) in the correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  One reason for the displacement field to be rotational is  that drift paths along serial and parallel boundaries do not  end at the same distance from the parallel clock stripes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The importance of this difference vanishes with distance  because at large distances, the field varies less steeply with  z than at short distances and, hence, the fractional con-  tribution to the boundary displacement of the end of the  drift path decays with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' When it comes to science  images, and especially when the image quality is good, the  short-distance boundary displacements dominate the effec-  tive correction for stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 10 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  0  2  4  6  8  0  2  4  6  8  j  curl value  0  2  4  6  8  0  2  4  6  8  j  top boundary shift  0  2  4  6  8  i  0  2  4  6  8  j  ratio  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1e  7  1  2  3  4  1e  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Values of the (discrete) curl, the top boundary dis-  placement and their ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' One can see that assuming a curl-free  displacement field is acceptable only at distances larger than ∼  3 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Gain noise  We measure a00 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='24 10−6, while the fit of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12  indicates a00 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='37 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This means that the model  predicts a variance that is smaller than the measured one,  and the leading difference is quadratic in signal level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This  difference of a00 values also contributes to the apparent  curvature of the residuals displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 3: it roughly  explains the curvature for C00 and C01 – but not for C10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  While questioning the area coefficients along the serial  direction, it is tempting to relate the excess we observe  to charge transfer issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' however, we provide a few argu-  ments against this explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' First, we have measured  and corrected deferred signals and we find that the linear  dependence displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 5, causes, if uncorrected, a  linear contribution to C10 (∝ µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' however, the a10 excess  we observe corresponds to a quadratic contribution to C10  (∝ µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Second, any mechanism relying on imperfect charge  transfer will unavoidably affect C00 and C10 in comparable  and opposite amounts, but the data in Table 1 indicate that  the excess of a00 is about 20 times larger that the one of  a10, and they have the same sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Third, serial charge trans-  fer inefficiency is usually associated to localized and rare  defects that is typically 1 or more frequently 0 along the  serial register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The distributions of area coefficients (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 8)  do not seem quantized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  So, assuming that this difference of a00 values is due  to gain variations during the read out, the relative gain  variations should be about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5 10−4 rms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Gain variations  of the HSC electronics have been studied during the HSC  camera fabrication process and are reported in Miyatake  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' They are evaluated as ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4 10−5, however,  this is only considering the video chain and not the CCD  (and perhaps in a context that is different from what the  instrument actually faces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The CCD readout chain of HSC relies on correlated dou-  ble sampling, an approach that integrates the video signal  during logical gates provided by the clocking system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  collected signal is hence vulnerable to fluctuations of the  timing of the integration gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We cross-correlated the sub-  images from different channels of the same sensor in order  to diagnose synchronous correlations, including variations  of the gains (whatever the cause).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We did not find any com-  pelling signal for any sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We were hoping that cross-  correlations would deliver the size of gain fluctuations, or  any other common-mode fluctuation of channel response,  thus allowing us to subtract this contributions from flat-  field statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In summary, we are not able to provide a convincing  cause of the sum rule violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Because the electrostatic  fit indicates an excess of fluctuations along the serial direc-  tion, we ignore three area coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This excess is obvi-  ously puzzling, albeit compatible with rapid gain fluctua-  tions (with some ad hoc time decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Fortunately, we do  not have to rely in any way on this hypothesis to use the  electrostatic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Correction of science images  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Data set and reduction  We process the images of the Cosmos field from the ultra-  deep part of the Subaru Strategic Program, acquired be-  tween 2014 and 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The ultra-deep part of the survey is  geared at obtaining extremely deep images by co-adding  a large number of observations, as well as detecting high-  redshift supernovae and measuring their light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  sample of images covers the five bands: g, r, i, z and y of  the camera, and also covers a broad range of image qual-  ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We use data acquired with the new i and r filters,  which are called r2 and i2 in HSC parlance and we stick to  these names.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This image sample constitutes a representa-  tive playground for testing the quality of the brighter-fatter  correction on HSC science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 11 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  The image reduction we perform is fairly standard for  the brighter-fatter correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We first average the overscan  and subtract it from the actual image data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We then cor-  rect each image segment for non-linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Then we have  to express the image in electrons in order to perform the  BF correction, so we multiply each image segment by its  corresponding gain (as determined when fitting the covari-  ance data), perform the correction, and divide back each  segment by the gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We note that the BF correction has  to deliver an image that has not been corrected by chan-  nel gains because our flats contain the gain differences and,  hence, they should be applied to an unscaled image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We pre-  fer to use flats that encode the relative gains because their  evolution with time encodes a possible evolution of gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Using approximate gains for the BF correction is clearly a  second-order issue, while using approximate relative gains  on a sensor can cause artificial steps in the sky background  at the channel boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The brighter-fatter correction consists in computing the  boundary shifts by summing all the actions of all image  charges on all boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For parallel boundaries, the shifts  are expressed as:  δN  ij = 1/2  �  kl  aN  k,lQi−k,j−l,  (14)  and similarly for serial boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The factor of 1/2 ac-  counts for the fact that source charges alter pixel shapes  only once they have reached the pixel wells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' thus, on aver-  age, during half of the integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1, the charge  on the rhs refers to the charge accumulated so far during the  integration and, hence, the time-averaged boundary shift  should be derived from the time-averaged charge content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The charge to displace from one pixel to its neighbor is com-  puted from the pixel boundary shift and the charge flowing  over the same pixel boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We compute the latter as  the average between the two pixels that share the bound-  ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We tried a quadratic order interpolator (involving 4  pixels), and did not find a decisive difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We should  note that the vast majority of our images are well sampled  (the image quality is larger than 3 pixels) and sharper im-  ages could react differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Ultimately, the correction of  the BF effect in the images themselves requires that an ac-  curate estimation of the charge at pixel boundaries can be  devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Once the BF correction has been applied, we divide back  each channel section by its gain and apply the flat field  constructed from dome flats accumulated over about one  month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the y band, we have to subtract a fringe pattern  constructed from a large set of science images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  On each individual CCD from each exposure, we run  SExtractor (Bertin & Arnouts 1996) to detect objects and  measure the Gaussian second moments of these objects  from an unweighted 2D Gaussian fit to the light distribu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Because we do not integrate the Gaussian over pixels,  we use the fast procedure described in Astier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2013)  to solve the normal equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' These second moments are  similar if not identical to the “SDSS adaptive moments.” We  identify stars in the image from the distribution of moments  (see Astier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Because we have to accommodate  significant variations of the PSF over the field of a single  CCD, the cut that selects stars is broad enough to avoid re-  jecting genuine stars because of the BF effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We get three  moments per star: MXX, MY Y and MXY , where X and Y  refer to the serial and parallel directions on the sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' As  an indicator of PSF size, we use:  IQ2 ≡ TP SF /2 = (MXX + MY Y )/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  (15)  We then need color measurements of our stars, which  we obtain by averaging fluxes over images and calibrating  instrumental magnitudes over the common footprint of our  star catalog and the field D2 from Betoule et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  This is not a critical step since only small color corrections  are involved in what follows and we, hence, do not need  colors measured to better than ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Processings and results  0  10000  20000  30000  fmax  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='10  MXX  < MXX >  0  10000  20000  30000  fmax  MYY  < MYY >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  i2-z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='02  MXX  < MXX >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='00  i2-z  MYY  < MYY >  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Differences between the star second moments and their  expectations from the spatial smoothing, as a function of fmax  (top) and color (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We have selected stars with 3 < IQ2 <  4 pix2, in the z band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The color range was chosen so that the  color dependence is roughly linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' One can see that the slope  of the top right plot is slightly steeper than the top left plot,  reflecting the anisotropy of pixel covariances in flat fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to assess the variation of star moments with  their flux, we report the variation of star sizes with peak  flux fmax, rather than with the flux itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The variation  with fmax is of practical interest, because fmax determines  if a given star can enter into the PSF modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In order  to isolate the variation of moments with peak flux we have  to account for two other variations: the spatial variation in  every exposure (mostly due to optics), and the color depen-  dence of apparent star sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We remove the spatial depen-  dence by fitting a second-order 2D polynomial to the star  moments measured on a given CCD in a given exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We then interpolate this crude model at the star position  and study the residual of the measurement to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  model MXX and MY Y independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We apply a conserva-  tive cut at fmax < 35000 ADU, in order to avoid any effect  of sensor saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The dependence on fmax and color  Article number, page 12 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  of these residuals are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 15, for a processing in  which no BF correction was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We see that for stars  selected at (MXX + MY Y )/2 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5 pix2, the total increase  of moments with fmax is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='13 pix2, which is about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='7%  for stars varying from 0 to saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This slope depends  on the selection of image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Figure 15 also indicates  that the variation of apparent size with color in the z band  are not considerably smaller than the ones with flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since  the fmax-color correlation coefficient is about −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='18 (in z  band), we should account for color-induced size variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to account for both the peak flux and color  dependence of the moments, we regress the star moment  residuals against both fmax and color:  MAA − M expected  AA  = αfmax + βc + γ,  (16)  where A stands for X or Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We then use α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', the BF  slope) as a color-independent indicator of the BF effect in-  tensity (both before and after correction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The linear cor-  rection in color can only work within a finite color range,  and the selected color indicators and their range are pro-  vided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For this fit, we ignore the measurements  with fmax<5000 because the measured moments could be  affected by a biased background estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Out of about  15 106 star measurements, the flux and color cuts retain  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4 106 measurements contributing to the slope measure-  ments that are reported here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Color indicators and color ranges used for the various  HSC bands  band  color  cmin  cmax  g  g-r2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1  r2  r2-i2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  i2  i2-z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  z  i2-z  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  y  z-y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For each band, the analysis selects stars with color be-  tween cmin and cmax in order to perform the regression of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In order to study the quality of the BF correction on  real science conditions, we bin stars in moments bins and  restrict the range to (MXX +MY Y )/2 < 10, which roughly  corresponds to an image quality of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='35′′ FWHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We do  not apply a lower cut because the best image qualities are  precious for at least cosmic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The bin selection operates  on the expected moments at the location of the star rather  than the measured ones, so that the quantity used to select  the bins contents is statistically independent of flux and  color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The measurements of α (accounting for color correc-  tions) for both CCD directions and in IQ2 bins are dis-  played in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can notice an increase of the slopes  with IQ2, as well as the fact that the y band is notably  less affected by the BF effect than the other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the  y band, the photons convert deeper in the sensor bulk and  the charges experience a shorter drift path than for bluer  bands, thus reducing the action of the perturbating electric  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the y band, the starting point of the integral of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12 is noticeably smaller that the sensor thickness t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  For some time, we ignore this subtlety and correct all  bands using a same model for all bands, derived from co-  2  3  4  5  6  (BF slope) in pix2/ADU  1e  6  MXX  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pix2)  2  3  4  5  6  (BF slope) in pix2/ADU  1e  6  MYY  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' BF slopes (with color correction) in IQ2 bins for the  five bands of HSC, without any BF correction, for the serial  (top) and parallel (bottom) directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can note a roughly  linear increase of the slopes with IQ2, and that for the best  image qualities, the slope for MY Y is larger than for MXX, as  expected from the anisotropy of the nearest neighbor aij (as seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11)  variances measured in the g band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We first apply the cor-  rection derived from “model 1” (displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11) and  the residual BF slopes are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Although  the BF slopes are significantly reduced with respect to the  raw data (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 16), the residual slopes are large at low IQ2,  indicating that the correction is underestimated at small  distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This correction leaves about 10% of the BF ef-  fect at the best image qualities and about 3% at the upper  end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We then apply the “model 2” correction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  12),  namely, the one that ignores the three first suspicious mea-  surements along the serial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The corresponding BF  slopes are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The quality of the correction,  in particular at the lowest IQ is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Ignoring the y  band, we can interpret the figure as a global small overcor-  rection that leaves BF slopes about 30 times smaller than  in the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Brighter-fatter correction for the y-band  When the energy of photons becomes comparable to the  silicon band gap, the absorption cross-section tends to van-  ish and silicon becomes transparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The band gap is about  E = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2 eV corresponding to λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the y band, the  Article number, page 13 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  (BF slope) in pix2/ADU  1e  6  MXX  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pix2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  (BF slope) in pix2/ADU  1e  6  MYY  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' BF slopes (with color correction) in IQ2 bins for the  five bands of HSC, with the BF correction derived from “model  1” (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 11) which uses all the covariance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' One  can note a residual BF slope essentially independent of image  quality, for all bands, except y that deserves a specific treatment  (detailed in 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  absorption length of photons becomes comparable to the  sensor thickness and we can no longer assume that charges  produced by converted photons drift all the way from the  entrance window (z=t in the coordinates of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The flat-  field data was acquired in g band, where the mean free path  of photons is well below 1 µm, and the approximation of im-  mediate conversion is adequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This approximation works  up to i band, may be questioned for z band and does not  seem to apply to y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Once we have our electrostatic model, the prediction for  boundary motions in y band can be computed by altering  the lower bound of the integral of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12 :  d(zc) = k  � z=z0  z=zc  ET  Q(xb, yb, z)dz,  (17)  where zc is where the photon converts and k is the nor-  malization determined when fitting the model to the area  coefficient data (obtained in g band in our case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Since the  integrand can vary rapidly with z, d(zc) ̸= d(zc), it is un-  wise to evaluate these integrals at the average conversion  depth of photons in the y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We instead compute the  average of d(zc) for a realistic distribution of zc:  dy =  � zc=szb  zc=t  d(zc)dN/dzc dzc,  (18)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  (BF slope) in pix2/ADU  1e  6  MXX  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pix2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  (BF slope) in pix2/ADU  1e  6  MYY  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' BF slopes (with color correction) in IQ2 bins for the  five bands of HSC, with the BF correction derived from “model  2” (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12) which uses all the covariance measurements but the  three first serial pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We face a small over correction, and the  BF slopes have been reduced by a factor of about 30 as compared  to the raw values of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The y band still requires a specific  treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  where dN/d zc is the distribution of conversion points (and  should integrate to unity over the integration domain), and  zb refers to zs or zp, depending on which boundary we are  considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The distribution of conversion depths depends  on photon absorption length and some assumption for the  object spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The latter may be regarded as inconve-  nient, but this is a small chromatic dependence of a small  flux dependence and, hence, a second-order effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We choose the spectrum of a color 0 (in AB magnitudes)  object to compute the conversion depth distribution, noting  that z−y = 0 lies inside the observed star color distribution,  which has an average of < z−y >≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the absorption  length as a function of wavelength, we use the expression  from Rajkanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (1979) for Silicon at 173oK, which  typically predicts 103 µm at λ = 950 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The conversion  depth distributions for z and y bands is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  19, and we can note that in y band, the conversions are  spread over the whole thickness and the depth distribution  is stable when changing the color by about the width of  the z − y distribution of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 20, we compare the  electrostatic model 2 integrated over the full thickness with  the same model for the y band, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 17, with the  distribution of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The largest relative changes happen  Article number, page 14 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  0  25  50  75  100  125  150  175  200  Conversion depth ( m)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='025  Differential probablilty ( m  1)  Conversion depth distribution in HSC CCDs  y (z-y=0)  y (z-y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2)  z  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Distributions of conversion depth for a zero color ob-  ject in the HSC CCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The y distribution is much flatter and  leads to a globally reduced BF effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Changing the color of the  object by about the width of the our z − y distribution does not  significantly alter the expected depth distribution in the y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  For the z band, most of the conversions occur at small depths,  where the perturbating electric field is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the i2 band,  the average conversion depth is about 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  at large distances, where the perturbating electric field has  a sizable contribution over most of the drift path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Next, we apply this modified BF correction for the y  band to the actual science data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We can see in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 21 that  the y band thus behaves similarly to the other bands, which  indicates that the reduced BF effect in y likely originates in  shorter drift paths in this very red band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This also indicates  that for these CCDs, flat field correlations measured in a  blue band can be used to predict (via a model) the BF  correction for a red band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Alternatively, we could certainly  consider measuring flat-field statistics in y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We have  not applied the same treatment to the z band, where the  correction is much smaller than for y, and the need for  reducing the BF correction is considerably less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  will however apply the BF model for z band in our final  image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Discussion  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Quality of the BF correction  The practical uses of PSF models crucially depend on the  size of the model capable of faithfully representing the ac-  tual PSF size for faint objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Two applications may come  to mind: first, the measurement of faint supernovae for cos-  mological applications where fluxes have to be measured  using PSF photometry and supernova fluxes are calibrated  against the ones of bright stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' second, the measurement  of galaxy shapes where one is interested in the intrinsic  shape, namely, “before” it is smeared by the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' An in-  accurate PSF size results in general in a biased shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For  both of these applications, the gauge is the difference of the  PSF model size to the real size of a faint object, relative  to the PSF size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The LSST Dark Energy Science Collabo-  ration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2018) set the maximum acceptable PSF size  bias (δT/T) to 10−3 for the ten-year Rubin/LSST survey  0  2  4  6  8  0  2  4  6  8  j  model 2 (full drift)  0  2  4  6  8  0  2  4  6  8  j  y band  0  2  4  6  8  i  0  2  4  6  8  j  (m1-m2)/m1  10  9  10  8  10  7  10  6  10  9  10  8  10  7  10  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Area coefficients for “model 2” (displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12)  based on the same model with shorter drift paths meant to de-  scribe the y band (center) and the relative difference at the bot-  tom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The differences tend to increase with distance to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  for all sources of PSF size bias, where T = MXX + MY Y  is the trace of the second moment matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' So, in what fol-  lows, the PSF “size” is, in fact, the “area”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Regarding PSF  photometry, the relative flux bias caused by PSF size bias  reads δf/f = 1/2(δT/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Shear measurements also require  an accurate estimation of the PSF ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' When sur-  veys measure the same object with the same orientation of  the sensors, sensor-induced ellipticities are transferred to  shape measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the Euclid mission the require-  ment on ellipticities transferred to objects is 5 10−5 of the  Article number, page 15 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  (BF slope) in pix2/ADU  1e  6  MXX  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pix2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  (BF slope) in pix2/ADU  1e  6  MYY  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' BF slopes computed after correcting images with model  2, with the modifications described above for y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' With re-  spect to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 18, only the y data has changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We may note that  all bands now behave in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  rms (Table 1 of Cropper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In the context of the  BF effect, we concentrate on the X/Y ellipticity, namely,  (MXX − MY Y )/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Cropper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2013) also provide a re-  quirement for the PSF size for Euclid, which reads 5 10−4  rms, comparable to the Rubin 10−3 bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 22, we see that on corrected images, the relative  PSF size difference is below 10−3 in all bands at all levels  of image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 23, we see that in the corrected  images, the residual PSF ellipticity δ(MXX − MY Y )/TP SF  is mostly below 10−4 across the range of image quality and  meets the 5 10−5 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='s bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We thus note that the need  to omit three nearby serial pixels when fitting model 2 did  not significantly degrade the quality of the BF correction in  one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 24, we display the slope of the radial  fourth moment inaccuracies of the PSF, which also induces  adverse effects on shear measurements, with a factor on  the order of 1, as for the PSF size (see Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  The residuals are even smaller than for the PSF size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  note that our residual trends after correction all indicate an  overcorrection of the BF effect and all this could thus be  improved by adjusting the overall scale of the correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We also note that these results were obtained in a blind  way: we did not change the procedure after having seen  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Gruen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2015) evaluated their image correction for  DECam in ways similar to ours (their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 12): their method  overcorrects the size of objects and leaves a negative BF  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='012  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='014  TPSF/TPSF  raw  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  TPSF/TPSF  1e  3  model 2 y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Relative change of the PSF size between an average  PSF star and a faint object, as a function of TP SF /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For the  PSF size extimator, we use as well the trace of the second mo-  ment matrix of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The top plot refers to the raw data,  the bottom one to the data corrected by model 2 with the short-  ened drift paths in y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We assume that an average PSF star  peaks at 1/3 of the saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  slope, about -1/3 of the uncorrected one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This is certainly  too large for a large-scale cosmic shear survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The cor-  rected ellipticities seem significantly better (and probably  small enough for a DES-like survey) but they are weakly  affected by the BF effect in the test sample (as compared  with the top plot of our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' (2018)  assessed the correction derived in C18 for HSC images: the  (linear) size of corrected stars varies with magnitude by  about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2 % over three magnitudes (an eyeball estimation  from their fig 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Then, TP SF would evolve twice as much  and this is small enough for the analysis of the first year  of the HSC survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Regarding the ellipticities, trends with  regard to the flux are not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' For both DECam and  HSC, the methods overcorrect the BF trend and both as-  sessments ignore chromatic contributions to star size varia-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' If the color-flux correlation is similar to ours, correct-  ing sizes for color decreases the apparent BF slope, hence  degrading the performance of the BF correction in both  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We have insisted on the importance of higher order  terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 4 when analyzing the covariance curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Those  terms result from the integration over time of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Re-  garding the correction of the images, we did not carry out  integration over time, and we just considered that half of  Article number, page 16 of 18  Pierre Astier and Nicolas Regnault: BF on HSC  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0000  ePSF  raw  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  1  0  1  2  3  ePSF  1e  4  model 2 y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Change of the PSF ellipticity eP SF  ≡ (MXX −  MY Y )/TP SF between an average PSF star and a faint object,  as a function of TP SF /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The top plot refers to the raw data,  the bottom one to the data corrected by model 2 with the short-  ened drift paths in y band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We assume that an average PSF star  peaks at 1/3 of the saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  the end-of-exposure image is a good approximation of the  source of electrostatic distortions, hence, neglecting second  order effects which are important for covariance curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We  anticipate that neglected second order effects in image cor-  rection should manifest themselves as structured residuals  to the moments versus fmax linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We analyzed these  residuals in band and image quality bins, summing differ-  ent bands at the same image quality, and we could not find  any hint of departure from linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This is fortunate since  accounting for next to leading order effects in image cor-  rection is more difficult than for the shape of covariance  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Possible further developments  While the obtained performance of the BF correction seems  sufficient, we may consider potential avenues for improve-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' First, in our analysis, the overall normalization of  the BF correction model primarily depends on a01, which  is the best measured area coefficient actually used in the  electrostatic fit (the model 2 fit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' This coefficient is slightly  biased by an erroneous value for a00 through the µ3 terms  in the covariance fit to a level compatible with the small  over-correction we are facing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' It may then seem legitimate  to actually tune the overall normalization of the correction  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='003  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='007  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='008  M4PSF/M4PSF  raw  g  r2  i2  z  y  2  3  4  5  6  7  8  9  TPSF/2 (pixel2)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='0  M4PSF/M4PSF  1e  4  model 2 y  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Relative change of the radial fourth moment as a func-  tion of image quality for all HSC bands, between a faint source  and an average PSF star, before (top) and after (bottom) BF  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We assumed that an average PSF star peaks at 1/3  of the saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  in order to bring the average BF slope to 0, at least for the  best observing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Second, the BF correction we implemented is the out-  come of an analysis where three area coefficients of the BF  effect (the three first serial measurements, including the  largest coefficient) had to be ignored because they could  not be accommodated by an electrostatic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Those  coefficients had to be derived from the measurements of  other area coefficients through an electrostatic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  We might then anticipate that for a camera that does not  suffer from this noise bias (which is first manifested by a  violation of the sum rule), a better correction model could  be constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' In particular, an analysis of test data of  the Vera Rubin Observatory camera indicates that the sum  rule is satisfied on the integrated instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  One appealing – and very different – approach could  be to determine the electrostatic model from the science  data by fitting the pixel distortion pattern that makes the  PSF homothetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The electrostatic modeling could proba-  bly be computed sufficiently rapidly to be inserted into a  PSF modeling fitting loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The concerns about the preci-  sion measurement of the two-point function of flat fields  would become pointless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Article number, page 17 of 18  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' bf-hsc  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Computer codes  Our code is split in three different parts: code to measure  covariances and fit the covariance curves, code to perform  the electrostatic fit, and code to process the images and in  particular corrects those for the BF effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Our public repos-  itory1 contains the python code for the two first steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  code that measures covariances did not evolve significantly  since it was developed for A19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We publish here the elec-  trostatic modeling code for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Our image cor-  rection code is fairly straightforward and would probably  run much faster with the convolution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 14 computed  in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The parts of the analysis that are not in  the repository can be made available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We are indebt to the Subaru Telescope technical  staff that very efficiently operates the observatory and its instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  HSC images are made available on the SMOKA server2, that we have  used extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' We perform all our reductions and store our results  at the Centre de Calcul de l’IN2P33, a computing facility of CNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  This work benefited from useful discussions with N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Suzuky (LBL)  and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Yasuda (IPMU), and our colleagues from the Paris team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' The  manuscript eventually benefited from excellent suggestions from our  referee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  References  Antilogus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Astier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Doherty, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Guyonnet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & Regnault, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2014, Journal of Instrumentation, 9, C3048  Astier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Antilogus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Juramy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2019, A&A, 629, A36  Astier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', El Hage, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Guy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2013, A&A, 557, A55  Bertin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' & Arnouts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1996, A&AS, 117, 393  Betoule, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Marriner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Regnault, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2013, A&A, 552, A124  Coulton, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Armstrong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Smith, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Lupton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', &  Spergel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2018, Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' Journ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', 155, 258  Cropper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Hoekstra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Kitching, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2013, MNRAS, 431,  3103  Downing, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=',  Baade, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Sinclaire, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Deiries, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & Christen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2006, Proc SPIE, 6276  Gruen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Bernstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Jarvis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2015, Journal of Instru-  mentation, 10, C05032  Guyonnet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Astier, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Antilogus, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Regnault, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & Doherty, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2015, A&A, 575, A41  Lage, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Bradshaw, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Anthony Tyson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & LSST Dark Energy Sci-  ence Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2021, Journal of Applied Physics, 130, 164502  Le  Breton,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  2017,  Theses,  Université  Pierre  et  Marie  Curie Paris  VI,  https://tel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='fr/tel-  01720422/file/2017PA066298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='pdf  Mandelbaum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Miyatake, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Hamana, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2018, PASJ, 70,  S25  Miyatake, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Aihara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Fujimori, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2012, Physics Proce-  dia, 37, 1413, proceedings of the 2nd International Conference on  Technology and Instrumentation in Particle Physics (TIPP 2011)  Miyazaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Komiyama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Kawanomoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2018, PASJ, 70,  S1  Pumplin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1969, American Journal of Physics, 37, 737  Rajkanan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Singh, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & Shewchun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 1979, Solid-State Electronics,  22, 793  Rasmussen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Guyonnet, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Lage, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2016, in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' SPIE, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='  9915, High Energy, Optical, and Infrared Detectors for Astronomy  VII, 99151A  The LSST Dark Energy Science Collaboration, Mandelbaum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Ei-  fler, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2018, arXiv, 1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='01669  Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', Mandelbaum, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=', & Collaboration, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content=' 2021,  Monthly Notices of the Royal Astronomical Society, 510, 1978  1 https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='fr/astier/bfptc  2 https://smoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='nao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='jp/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='jsp  3 https://cc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='in2p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}
+page_content='fr  Article number, page 18 of 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdE1T4oBgHgl3EQfkgRW/content/2301.03274v1.pdf'}